TPTP Problem File: ITP216^4.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP216^4 : TPTP v8.2.0. Released v8.0.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer problem Time_Reasoning 00074_002979
% Version : [Des22] axioms.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des22] Desharnais (2022), Email to Geoff Sutcliffe
% Source : [Des22]
% Names : 0029_Time_Reasoning_00074_002979 [Des22]
% Status : Theorem
% Rating : 0.67 v8.2.0, 0.33 v8.1.0
% Syntax : Number of formulae : 7612 (2659 unt; 913 typ; 0 def)
% Number of atoms : 19157 (7619 equ; 6 cnn)
% Maximal formula atoms : 48 ( 2 avg)
% Number of connectives : 155299 (1816 ~; 175 |;1143 &;144024 @)
% ( 0 <=>;8141 =>; 0 <=; 0 <~>)
% Maximal formula depth : 38 ( 8 avg)
% Number of types : 13 ( 12 usr)
% Number of type conns : 6488 (6488 >; 0 *; 0 +; 0 <<)
% Number of symbols : 905 ( 901 usr; 20 con; 0-10 aty)
% Number of variables : 25679 (2885 ^;21417 !; 384 ?;25679 :)
% ( 993 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% from the van Emde Boas Trees session in the Archive of Formal
% proofs -
% www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
% 2022-02-17 16:14:04.262
%------------------------------------------------------------------------------
% Could-be-implicit typings (26)
thf(ty_t_Heap__Time__Monad_OHeap,type,
heap_Time_Heap: $tType > $tType ).
thf(ty_t_Code__Numeral_Onatural,type,
code_natural: $tType ).
thf(ty_t_Code__Numeral_Ointeger,type,
code_integer: $tType ).
thf(ty_t_Code__Evaluation_Oterm,type,
code_term: $tType ).
thf(ty_t_Heap_Oheap_Oheap__ext,type,
heap_ext: $tType > $tType ).
thf(ty_t_Product__Type_Ounit,type,
product_unit: $tType ).
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_Old__Datatype_Onode,type,
old_node: $tType > $tType > $tType ).
thf(ty_t_Multiset_Omultiset,type,
multiset: $tType > $tType ).
thf(ty_t_Typerep_Otyperep,type,
typerep: $tType ).
thf(ty_t_Assertions_Oassn,type,
assn: $tType ).
thf(ty_t_Predicate_Opred,type,
pred: $tType > $tType ).
thf(ty_t_Sum__Type_Osum,type,
sum_sum: $tType > $tType > $tType ).
thf(ty_t_Predicate_Oseq,type,
seq: $tType > $tType ).
thf(ty_t_Option_Ooption,type,
option: $tType > $tType ).
thf(ty_t_String_Ochar,type,
char: $tType ).
thf(ty_t_Heap_Oarray,type,
array: $tType > $tType ).
thf(ty_t_List_Olist,type,
list: $tType > $tType ).
thf(ty_t_Heap_Oref,type,
ref: $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Rat_Orat,type,
rat: $tType ).
thf(ty_t_Num_Onum,type,
num: $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_t_Int_Oint,type,
int: $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
thf(ty_tf_a,type,
a: $tType ).
% Explicit typings (887)
thf(sy_cl_Lattices_Obounded__lattice__top,type,
bounded_lattice_top:
!>[A: $tType] : $o ).
thf(sy_cl_Typerep_Otyperep,type,
typerep2:
!>[A: $tType] : $o ).
thf(sy_cl_Enum_Oenum,type,
enum:
!>[A: $tType] : $o ).
thf(sy_cl_Code__Evaluation_Oterm__of,type,
code_term_of:
!>[A: $tType] : $o ).
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_HOL_Oequal,type,
cl_HOL_Oequal:
!>[A: $tType] : $o ).
thf(sy_cl_Heap_Oheap,type,
heap:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Odvd,type,
dvd:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oone,type,
one:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oring,type,
ring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oplus,type,
plus:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
thf(sy_cl_Num_Onumeral,type,
numeral:
!>[A: $tType] : $o ).
thf(sy_cl_Power_Opower,type,
power:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Ofield,type,
field:
!>[A: $tType] : $o ).
thf(sy_cl_GCD_Oring__gcd,type,
ring_gcd:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Oinf,type,
inf:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Osup,type,
sup:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oring__1,type,
ring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ouminus,type,
uminus:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Oinverse,type,
inverse:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring,type,
semiring:
!>[A: $tType] : $o ).
thf(sy_cl_Nat_Oring__char__0,type,
ring_char_0:
!>[A: $tType] : $o ).
thf(sy_cl_Num_Oneg__numeral,type,
neg_numeral:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__ring,type,
comm_ring:
!>[A: $tType] : $o ).
thf(sy_cl_GCD_Osemiring__Gcd,type,
semiring_Gcd:
!>[A: $tType] : $o ).
thf(sy_cl_GCD_Osemiring__gcd,type,
semiring_gcd:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ono__bot,type,
no_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ono__top,type,
no_top:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__0,type,
semiring_0:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__1,type,
semiring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Omonoid__add,type,
monoid_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__ring__1,type,
comm_ring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Omonoid__mult,type,
monoid_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
thf(sy_cl_Parity_Oring__parity,type,
ring_parity:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oidom__abs__sgn,type,
idom_abs_sgn:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__ring,type,
ordered_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ozero__neq__one,type,
zero_neq_one:
!>[A: $tType] : $o ).
thf(sy_cl_Countable_Ocountable,type,
countable:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Ofield__char__0,type,
field_char_0:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__group__add,type,
ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Nat_Osemiring__char__0,type,
semiring_char_0:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder__bot,type,
order_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder__top,type,
order_top:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__semiring,type,
comm_semiring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ozero__less__one,type,
zero_less_one:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Odivision__ring,type,
division_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Osemigroup__add,type,
semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Num_Osemiring__numeral,type,
semiring_numeral:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemidom__divide,type,
semidom_divide:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemidom__modulo,type,
semidom_modulo:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_OInf,type,
complete_Inf:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_OSup,type,
complete_Sup:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Osemigroup__mult,type,
semigroup_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__semiring__0,type,
comm_semiring_0:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__semiring__1,type,
comm_semiring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__ring,type,
linordered_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__modulo,type,
semiring_modulo:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__add,type,
comm_monoid_add:
!>[A: $tType] : $o ).
thf(sy_cl_Parity_Osemiring__parity,type,
semiring_parity:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Olinordered__field,type,
linordered_field:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__semigroup__add,type,
ab_semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__diff,type,
comm_monoid_diff:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__mult,type,
comm_monoid_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oalgebraic__semidom,type,
algebraic_semidom:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__semigroup__mult,type,
ab_semigroup_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Obounded__lattice,type,
bounded_lattice:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Osemilattice__inf,type,
semilattice_inf:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Osemilattice__sup,type,
semilattice_sup:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Odense__linorder,type,
dense_linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Quickcheck__Random_Orandom,type,
quickcheck_random:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Partial__Order_Occpo,type,
comple9053668089753744459l_ccpo:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semiring__1,type,
linord6961819062388156250ring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Onormalization__semidom,type,
normal8620421768224518004emidom:
!>[A: $tType] : $o ).
thf(sy_cl_Bit__Operations_Osemiring__bits,type,
bit_semiring_bits:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__semiring__1__cancel,type,
comm_s4317794764714335236cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__ring__strict,type,
linord4710134922213307826strict:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
cancel1802427076303600483id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oring__1__no__zero__divisors,type,
ring_15535105094025558882visors:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
cancel2418104881723323429up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Olinordered__ab__group__add,type,
linord5086331880401160121up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__comm__monoid__add,type,
ordere6911136660526730532id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Archimedean__Field_Ofloor__ceiling,type,
archim2362893244070406136eiling:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__group__add__abs,type,
ordere166539214618696060dd_abs:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
ordere6658533253407199908up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__no__zero__divisors,type,
semiri3467727345109120633visors:
!>[A: $tType] : $o ).
thf(sy_cl_Boolean__Algebras_Oboolean__algebra,type,
boolea8198339166811842893lgebra:
!>[A: $tType] : $o ).
thf(sy_cl_Quickcheck__Exhaustive_Oexhaustive,type,
quickc658316121487927005ustive:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemidom__divide__unit__factor,type,
semido2269285787275462019factor:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
linord181362715937106298miring:
!>[A: $tType] : $o ).
thf(sy_cl_Bit__Operations_Oring__bit__operations,type,
bit_ri3973907225187159222ations:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
comple6319245703460814977attice:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Olinordered__ab__semigroup__add,type,
linord4140545234300271783up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semiring__1__strict,type,
linord715952674999750819strict:
!>[A: $tType] : $o ).
thf(sy_cl_Archimedean__Field_Oarchimedean__field,type,
archim462609752435547400_field:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Oeuclidean__semiring,type,
euclid3725896446679973847miring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni5634975068530333245id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__cancel__comm__monoid__add,type,
ordere8940638589300402666id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ostrict__ordered__comm__monoid__add,type,
strict7427464778891057005id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Quickcheck__Exhaustive_Ofull__exhaustive,type,
quickc3360725361186068524ustive:
!>[A: $tType] : $o ).
thf(sy_cl_Bit__Operations_Osemiring__bit__operations,type,
bit_se359711467146920520ations:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
ordere2412721322843649153imp_le:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
ordere580206878836729694up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__cancel__comm__monoid__diff,type,
ordere1170586879665033532d_diff:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add,type,
strict9044650504122735259up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Oeuclidean__ring__cancel,type,
euclid8851590272496341667cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Divides_Ounique__euclidean__semiring__numeral,type,
unique1627219031080169319umeral:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Oeuclidean__semiring__cancel,type,
euclid4440199948858584721cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Ounique__euclidean__semiring,type,
euclid3128863361964157862miring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type,
ordere1937475149494474687imp_le:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Ounique__euclidean__ring__with__nat,type,
euclid8789492081693882211th_nat:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Ounique__euclidean__semiring__with__nat,type,
euclid5411537665997757685th_nat:
!>[A: $tType] : $o ).
thf(sy_cl_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,type,
semiri1453513574482234551roduct:
!>[A: $tType] : $o ).
thf(sy_cl_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations,type,
bit_un5681908812861735899ations:
!>[A: $tType] : $o ).
thf(sy_c_Archimedean__Field_Oceiling,type,
archimedean_ceiling:
!>[A: $tType] : ( A > int ) ).
thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor,type,
archim6421214686448440834_floor:
!>[A: $tType] : ( A > int ) ).
thf(sy_c_Archimedean__Field_Ofrac,type,
archimedean_frac:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Archimedean__Field_Oround,type,
archimedean_round:
!>[A: $tType] : ( A > int ) ).
thf(sy_c_Array__Time_Oalloc,type,
array_alloc:
!>[A: $tType] : ( ( list @ A ) > ( heap_ext @ product_unit ) > ( product_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) ) ) ).
thf(sy_c_Array__Time_Ofreeze,type,
array_freeze:
!>[A: $tType] : ( ( array @ A ) > ( heap_Time_Heap @ ( list @ A ) ) ) ).
thf(sy_c_Array__Time_Oget,type,
array_get:
!>[A: $tType] : ( ( heap_ext @ product_unit ) > ( array @ A ) > ( list @ A ) ) ).
thf(sy_c_Array__Time_Olen,type,
array_len:
!>[A: $tType] : ( ( array @ A ) > ( heap_Time_Heap @ nat ) ) ).
thf(sy_c_Array__Time_Olen_H,type,
array_len2:
!>[B: $tType,A: $tType] : ( ( array @ B ) > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_Array__Time_Olength,type,
array_length:
!>[A: $tType] : ( ( heap_ext @ product_unit ) > ( array @ A ) > nat ) ).
thf(sy_c_Array__Time_Omake,type,
array_make:
!>[A: $tType] : ( nat > ( nat > A ) > ( heap_Time_Heap @ ( array @ A ) ) ) ).
thf(sy_c_Array__Time_Omake_H,type,
array_make2:
!>[B: $tType,A: $tType] : ( code_integer > ( B > A ) > ( heap_Time_Heap @ ( array @ A ) ) ) ).
thf(sy_c_Array__Time_Omap__entry,type,
array_map_entry:
!>[A: $tType] : ( nat > ( A > A ) > ( array @ A ) > ( heap_Time_Heap @ ( array @ A ) ) ) ).
thf(sy_c_Array__Time_Onew,type,
array_new:
!>[A: $tType] : ( nat > A > ( heap_Time_Heap @ ( array @ A ) ) ) ).
thf(sy_c_Array__Time_Onew_H,type,
array_new2:
!>[A: $tType] : ( code_integer > A > ( heap_Time_Heap @ ( array @ A ) ) ) ).
thf(sy_c_Array__Time_Onoteq,type,
array_noteq:
!>[A: $tType,B: $tType] : ( ( array @ A ) > ( array @ B ) > $o ) ).
thf(sy_c_Array__Time_Onth,type,
array_nth:
!>[A: $tType] : ( ( array @ A ) > nat > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_Array__Time_Onth_H,type,
array_nth2:
!>[A: $tType] : ( ( array @ A ) > code_integer > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_Array__Time_Oof__list,type,
array_of_list:
!>[A: $tType] : ( ( list @ A ) > ( heap_Time_Heap @ ( array @ A ) ) ) ).
thf(sy_c_Array__Time_Opresent,type,
array_present:
!>[A: $tType] : ( ( heap_ext @ product_unit ) > ( array @ A ) > $o ) ).
thf(sy_c_Array__Time_Oset,type,
array_set:
!>[A: $tType] : ( ( array @ A ) > ( list @ A ) > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) ) ).
thf(sy_c_Array__Time_Oswap,type,
array_swap:
!>[A: $tType] : ( nat > A > ( array @ A ) > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_Array__Time_Oupd,type,
array_upd:
!>[A: $tType] : ( nat > A > ( array @ A ) > ( heap_Time_Heap @ ( array @ A ) ) ) ).
thf(sy_c_Array__Time_Oupd_H,type,
array_upd2:
!>[A: $tType] : ( ( array @ A ) > code_integer > A > ( heap_Time_Heap @ product_unit ) ) ).
thf(sy_c_Array__Time_Oupdate,type,
array_update:
!>[A: $tType] : ( ( array @ A ) > nat > A > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) ) ).
thf(sy_c_Assertions_Oassn_ORep__assn,type,
rep_assn: assn > ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ).
thf(sy_c_Assertions_Oex__assn,type,
ex_assn:
!>[A: $tType] : ( ( A > assn ) > assn ) ).
thf(sy_c_Assertions_Oin__range,type,
in_range: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ).
thf(sy_c_Assertions_Oin__range__rel,type,
in_range_rel: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ).
thf(sy_c_Assertions_Oone__assn__raw,type,
one_assn_raw: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ).
thf(sy_c_Assertions_Oone__assn__raw__rel,type,
one_assn_raw_rel: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ).
thf(sy_c_Assertions_Oproper,type,
proper: ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) > $o ).
thf(sy_c_Assertions_Opure__assn,type,
pure_assn: $o > assn ).
thf(sy_c_Assertions_Opure__assn__raw,type,
pure_assn_raw:
!>[A: $tType,B: $tType] : ( $o > ( product_prod @ A @ ( set @ B ) ) > $o ) ).
thf(sy_c_Assertions_Opure__assn__raw__rel,type,
pure_assn_raw_rel:
!>[A: $tType,B: $tType] : ( ( product_prod @ $o @ ( product_prod @ A @ ( set @ B ) ) ) > ( product_prod @ $o @ ( product_prod @ A @ ( set @ B ) ) ) > $o ) ).
thf(sy_c_Assertions_OrelH,type,
relH: ( set @ nat ) > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) > $o ).
thf(sy_c_Assertions_Osnga__assn,type,
snga_assn:
!>[A: $tType] : ( ( array @ A ) > ( list @ A ) > assn ) ).
thf(sy_c_Assertions_Osnga__assn__raw,type,
snga_assn_raw:
!>[A: $tType] : ( ( array @ A ) > ( list @ A ) > ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) ).
thf(sy_c_Assertions_Osnga__assn__raw__rel,type,
snga_assn_raw_rel:
!>[A: $tType] : ( ( product_prod @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) > ( product_prod @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) > $o ) ).
thf(sy_c_Assertions_Osngr__assn,type,
sngr_assn:
!>[A: $tType] : ( ( ref @ A ) > A > assn ) ).
thf(sy_c_Assertions_Osngr__assn__raw,type,
sngr_assn_raw:
!>[A: $tType] : ( ( ref @ A ) > A > ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) ).
thf(sy_c_Assertions_Osngr__assn__raw__rel,type,
sngr_assn_raw_rel:
!>[A: $tType] : ( ( product_prod @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) > ( product_prod @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) > $o ) ).
thf(sy_c_Assertions_Otimes__assn__raw,type,
times_assn_raw: ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) > ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) > ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ).
thf(sy_c_Assertions_Otimes__assn__raw__rel,type,
times_assn_raw_rel: ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) > ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) > $o ).
thf(sy_c_Assertions_Owand__assn,type,
wand_assn: assn > assn > assn ).
thf(sy_c_Assertions_Owand__raw,type,
wand_raw: ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) > ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) > ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ).
thf(sy_c_Assertions_Owand__raw__rel,type,
wand_raw_rel: ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) > ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) > $o ).
thf(sy_c_BNF__Cardinal__Arithmetic_OCsum,type,
bNF_Cardinal_Csum:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > ( set @ ( product_prod @ B @ B ) ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
thf(sy_c_BNF__Cardinal__Arithmetic_Ocexp,type,
bNF_Cardinal_cexp:
!>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) ) ).
thf(sy_c_BNF__Cardinal__Arithmetic_Ocfinite,type,
bNF_Cardinal_cfinite:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_BNF__Cardinal__Arithmetic_Ocinfinite,type,
bNF_Ca4139267488887388095finite:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_BNF__Cardinal__Arithmetic_Ocone,type,
bNF_Cardinal_cone: set @ ( product_prod @ product_unit @ product_unit ) ).
thf(sy_c_BNF__Cardinal__Arithmetic_Ocprod,type,
bNF_Cardinal_cprod:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
thf(sy_c_BNF__Cardinal__Arithmetic_Ocsum,type,
bNF_Cardinal_csum:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) ) ).
thf(sy_c_BNF__Cardinal__Arithmetic_Octwo,type,
bNF_Cardinal_ctwo: set @ ( product_prod @ $o @ $o ) ).
thf(sy_c_BNF__Cardinal__Arithmetic_Oczero,type,
bNF_Cardinal_czero:
!>[A: $tType] : ( set @ ( product_prod @ A @ A ) ) ).
thf(sy_c_BNF__Cardinal__Order__Relation_OcardSuc,type,
bNF_Ca8387033319878233205ardSuc:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ).
thf(sy_c_BNF__Cardinal__Order__Relation_Ocard__of,type,
bNF_Ca6860139660246222851ard_of:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_BNF__Cardinal__Order__Relation_Ocard__order__on,type,
bNF_Ca8970107618336181345der_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_BNF__Cardinal__Order__Relation_Ocofinal,type,
bNF_Ca7293521722713021262ofinal:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_BNF__Cardinal__Order__Relation_OisCardSuc,type,
bNF_Ca6246979054910435723ardSuc:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) > $o ) ).
thf(sy_c_BNF__Cardinal__Order__Relation_OnatLeq,type,
bNF_Ca8665028551170535155natLeq: set @ ( product_prod @ nat @ nat ) ).
thf(sy_c_BNF__Cardinal__Order__Relation_OregularCard,type,
bNF_Ca7133664381575040944arCard:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
bNF_Ca3754400796208372196lChain:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
thf(sy_c_BNF__Def_OGr,type,
bNF_Gr:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_BNF__Def_OGrp,type,
bNF_Grp:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > A > B > $o ) ).
thf(sy_c_BNF__Def_Ocollect,type,
bNF_collect:
!>[B: $tType,A: $tType] : ( ( set @ ( B > ( set @ A ) ) ) > B > ( set @ A ) ) ).
thf(sy_c_BNF__Def_Oconvol,type,
bNF_convol:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B ) > ( A > C ) > A > ( product_prod @ B @ C ) ) ).
thf(sy_c_BNF__Def_Oeq__onp,type,
bNF_eq_onp:
!>[A: $tType] : ( ( A > $o ) > A > A > $o ) ).
thf(sy_c_BNF__Def_OfstOp,type,
bNF_fstOp:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( B > C > $o ) > ( product_prod @ A @ C ) > ( product_prod @ A @ B ) ) ).
thf(sy_c_BNF__Def_Opick__middlep,type,
bNF_pick_middlep:
!>[B: $tType,A: $tType,C: $tType] : ( ( B > A > $o ) > ( A > C > $o ) > B > C > A ) ).
thf(sy_c_BNF__Def_Orel__fun,type,
bNF_rel_fun:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C > $o ) > ( B > D > $o ) > ( A > B ) > ( C > D ) > $o ) ).
thf(sy_c_BNF__Def_OsndOp,type,
bNF_sndOp:
!>[C: $tType,A: $tType,B: $tType] : ( ( C > A > $o ) > ( A > B > $o ) > ( product_prod @ C @ B ) > ( product_prod @ A @ B ) ) ).
thf(sy_c_BNF__Def_Ovimage2p,type,
bNF_vimage2p:
!>[A: $tType,D: $tType,B: $tType,E: $tType,C: $tType] : ( ( A > D ) > ( B > E ) > ( D > E > C ) > A > B > C ) ).
thf(sy_c_BNF__Greatest__Fixpoint_OfromCard,type,
bNF_Gr5436034075474128252omCard:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ B @ B ) ) > B > A ) ).
thf(sy_c_BNF__Greatest__Fixpoint_Oimage2,type,
bNF_Greatest_image2:
!>[C: $tType,A: $tType,B: $tType] : ( ( set @ C ) > ( C > A ) > ( C > B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_BNF__Greatest__Fixpoint_OrelImage,type,
bNF_Gr4221423524335903396lImage:
!>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( B > A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_BNF__Greatest__Fixpoint_OrelInvImage,type,
bNF_Gr7122648621184425601vImage:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_BNF__Greatest__Fixpoint_OtoCard,type,
bNF_Greatest_toCard:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ B @ B ) ) > A > B ) ).
thf(sy_c_BNF__Greatest__Fixpoint_OtoCard__pred,type,
bNF_Gr1419584066657907630d_pred:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > $o ) ).
thf(sy_c_BNF__Wellorder__Constructions_OFunc,type,
bNF_Wellorder_Func:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( A > B ) ) ) ).
thf(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 ) ) ) ) ).
thf(sy_c_BNF__Wellorder__Constructions_Ocurr,type,
bNF_Wellorder_curr:
!>[A: $tType,B: $tType,C: $tType] : ( ( set @ A ) > ( ( product_prod @ A @ B ) > C ) > A > B > C ) ).
thf(sy_c_BNF__Wellorder__Constructions_Odir__image,type,
bNF_We2720479622203943262_image:
!>[A: $tType,A2: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > A2 ) > ( set @ ( product_prod @ A2 @ A2 ) ) ) ).
thf(sy_c_BNF__Wellorder__Constructions_OofilterIncl,type,
bNF_We413866401316099525erIncl:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ).
thf(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 ) ) ) ) ).
thf(sy_c_BNF__Wellorder__Constructions_OordLeq,type,
bNF_Wellorder_ordLeq:
!>[A: $tType,A2: $tType] : ( set @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A2 @ A2 ) ) ) ) ).
thf(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 ) ) ) ) ).
thf(sy_c_BNF__Wellorder__Embedding_Ocompat,type,
bNF_Wellorder_compat:
!>[A: $tType,A2: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A2 @ A2 ) ) > ( A > A2 ) > $o ) ).
thf(sy_c_BNF__Wellorder__Embedding_Oembed,type,
bNF_Wellorder_embed:
!>[A: $tType,A2: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A2 @ A2 ) ) > ( A > A2 ) > $o ) ).
thf(sy_c_BNF__Wellorder__Embedding_OembedS,type,
bNF_Wellorder_embedS:
!>[A: $tType,A2: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A2 @ A2 ) ) > ( A > A2 ) > $o ) ).
thf(sy_c_BNF__Wellorder__Embedding_Oiso,type,
bNF_Wellorder_iso:
!>[A: $tType,A2: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A2 @ A2 ) ) > ( A > A2 ) > $o ) ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel,type,
bNF_Wellorder_wo_rel:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel_OisMinim,type,
bNF_We4791949203932849705sMinim:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > A > $o ) ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Omax2,type,
bNF_We1388413361240627857o_max2:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > A > A ) ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Ominim,type,
bNF_We6954850376910717587_minim:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > A ) ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Osuc,type,
bNF_Wellorder_wo_suc:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > A ) ).
thf(sy_c_Basic__BNF__LFPs_Oprod_Osize__prod,type,
basic_BNF_size_prod:
!>[A: $tType,B: $tType] : ( ( A > nat ) > ( B > nat ) > ( product_prod @ A @ B ) > nat ) ).
thf(sy_c_Basic__BNF__LFPs_Osum_Osize__sum,type,
basic_BNF_size_sum:
!>[A: $tType,B: $tType] : ( ( A > nat ) > ( B > nat ) > ( sum_sum @ A @ B ) > nat ) ).
thf(sy_c_Basic__BNFs_Ofsts,type,
basic_fsts:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( set @ A ) ) ).
thf(sy_c_Basic__BNFs_Opred__fun,type,
basic_pred_fun:
!>[A: $tType,B: $tType] : ( ( A > $o ) > ( B > $o ) > ( A > B ) > $o ) ).
thf(sy_c_Basic__BNFs_Opred__prod,type,
basic_pred_prod:
!>[A: $tType,B: $tType] : ( ( A > $o ) > ( B > $o ) > ( product_prod @ A @ B ) > $o ) ).
thf(sy_c_Basic__BNFs_Orel__prod,type,
basic_rel_prod:
!>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( A > B > $o ) > ( C > D > $o ) > ( product_prod @ A @ C ) > ( product_prod @ B @ D ) > $o ) ).
thf(sy_c_Basic__BNFs_Osnds,type,
basic_snds:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( set @ B ) ) ).
thf(sy_c_Binomial_Obinomial,type,
binomial: nat > nat > nat ).
thf(sy_c_Binomial_Ogbinomial,type,
gbinomial:
!>[A: $tType] : ( A > nat > A ) ).
thf(sy_c_Bit__Operations_Oand__int__rel,type,
bit_and_int_rel: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) > $o ).
thf(sy_c_Bit__Operations_Oand__not__num,type,
bit_and_not_num: num > num > ( option @ num ) ).
thf(sy_c_Bit__Operations_Oand__not__num__rel,type,
bit_and_not_num_rel: ( product_prod @ num @ num ) > ( product_prod @ num @ num ) > $o ).
thf(sy_c_Bit__Operations_Oconcat__bit,type,
bit_concat_bit: nat > int > int > int ).
thf(sy_c_Bit__Operations_Oor__not__num__neg,type,
bit_or_not_num_neg: num > num > num ).
thf(sy_c_Bit__Operations_Oor__not__num__neg__rel,type,
bit_or3848514188828904588eg_rel: ( product_prod @ num @ num ) > ( product_prod @ num @ num ) > $o ).
thf(sy_c_Bit__Operations_Oring__bit__operations__class_Onot,type,
bit_ri4277139882892585799ns_not:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit,type,
bit_ri4674362597316999326ke_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand,type,
bit_se5824344872417868541ns_and:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit,type,
bit_se4197421643247451524op_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit,type,
bit_se8732182000553998342ip_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask,type,
bit_se2239418461657761734s_mask:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor,type,
bit_se1065995026697491101ons_or:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit,type,
bit_se4730199178511100633sh_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit,type,
bit_se5668285175392031749et_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit,type,
bit_se2584673776208193580ke_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit,type,
bit_se2638667681897837118et_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor,type,
bit_se5824344971392196577ns_xor:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit,type,
bit_se5641148757651400278ts_bit:
!>[A: $tType] : ( A > nat > $o ) ).
thf(sy_c_Bit__Operations_Osemiring__bits__class_Opossible__bit,type,
bit_se6407376104438227557le_bit:
!>[A: $tType] : ( ( itself @ A ) > nat > $o ) ).
thf(sy_c_Bit__Operations_Otake__bit__num,type,
bit_take_bit_num: nat > num > ( option @ num ) ).
thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num,type,
bit_un7362597486090784418nd_num: num > num > ( option @ num ) ).
thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num__rel,type,
bit_un4731106466462545111um_rel: ( product_prod @ num @ num ) > ( product_prod @ num @ num ) > $o ).
thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oor__num,type,
bit_un6697907153464112080or_num: num > num > num ).
thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oor__num__rel,type,
bit_un4773296044027857193um_rel: ( product_prod @ num @ num ) > ( product_prod @ num @ num ) > $o ).
thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num,type,
bit_un2480387367778600638or_num: num > num > ( option @ num ) ).
thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num__rel,type,
bit_un2901131394128224187um_rel: ( product_prod @ num @ num ) > ( product_prod @ num @ num ) > $o ).
thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra,type,
boolea2506097494486148201lgebra:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > ( A > A ) > A > A > $o ) ).
thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra__sym__diff,type,
boolea3799213064322606851m_diff:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > ( A > A ) > A > A > ( A > A > A ) > $o ) ).
thf(sy_c_Cancellation_Oiterate__add,type,
iterate_add:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Code__Numeral_ONat,type,
code_Nat: code_integer > code_natural ).
thf(sy_c_Code__Numeral_ONeg,type,
code_Neg: num > code_integer ).
thf(sy_c_Code__Numeral_OPos,type,
code_Pos: num > code_integer ).
thf(sy_c_Code__Numeral_OSuc,type,
code_Suc: code_natural > code_natural ).
thf(sy_c_Code__Numeral_Obit__cut__integer,type,
code_bit_cut_integer: code_integer > ( product_prod @ code_integer @ $o ) ).
thf(sy_c_Code__Numeral_Ocr__integer,type,
code_cr_integer: int > code_integer > $o ).
thf(sy_c_Code__Numeral_Ocr__natural,type,
code_cr_natural: nat > code_natural > $o ).
thf(sy_c_Code__Numeral_Odivmod__abs,type,
code_divmod_abs: code_integer > code_integer > ( product_prod @ code_integer @ code_integer ) ).
thf(sy_c_Code__Numeral_Odivmod__integer,type,
code_divmod_integer: code_integer > code_integer > ( product_prod @ code_integer @ code_integer ) ).
thf(sy_c_Code__Numeral_Odup,type,
code_dup: code_integer > code_integer ).
thf(sy_c_Code__Numeral_Ointeger_Oint__of__integer,type,
code_int_of_integer: code_integer > int ).
thf(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
code_integer_of_int: int > code_integer ).
thf(sy_c_Code__Numeral_Ointeger__of__nat,type,
code_integer_of_nat: nat > code_integer ).
thf(sy_c_Code__Numeral_Ointeger__of__natural,type,
code_i5400310926305786745atural: code_natural > code_integer ).
thf(sy_c_Code__Numeral_Ointeger__of__num,type,
code_integer_of_num: num > code_integer ).
thf(sy_c_Code__Numeral_Onat__of__integer,type,
code_nat_of_integer: code_integer > nat ).
thf(sy_c_Code__Numeral_Onatural_Ocase__natural,type,
code_case_natural:
!>[T: $tType] : ( T > ( code_natural > T ) > code_natural > T ) ).
thf(sy_c_Code__Numeral_Onatural_Onat__of__natural,type,
code_nat_of_natural: code_natural > nat ).
thf(sy_c_Code__Numeral_Onatural_Onatural__of__nat,type,
code_natural_of_nat: nat > code_natural ).
thf(sy_c_Code__Numeral_Onatural_Orec__natural,type,
code_rec_natural:
!>[T: $tType] : ( T > ( code_natural > T > T ) > code_natural > T ) ).
thf(sy_c_Code__Numeral_Onatural_Orec__set__natural,type,
code_rec_set_natural:
!>[T: $tType] : ( T > ( code_natural > T > T ) > code_natural > T > $o ) ).
thf(sy_c_Code__Numeral_Onatural__of__integer,type,
code_n4118661773612635043nteger: code_integer > code_natural ).
thf(sy_c_Code__Numeral_Onegative,type,
code_negative: num > code_integer ).
thf(sy_c_Code__Numeral_Onum__of__integer,type,
code_num_of_integer: code_integer > num ).
thf(sy_c_Code__Numeral_Opcr__integer,type,
code_pcr_integer: int > code_integer > $o ).
thf(sy_c_Code__Numeral_Opcr__natural,type,
code_pcr_natural: nat > code_natural > $o ).
thf(sy_c_Code__Numeral_Opositive,type,
code_positive: num > code_integer ).
thf(sy_c_Code__Numeral_Osize__nat,type,
code_size_nat: code_natural > nat ).
thf(sy_c_Code__Numeral_Osub,type,
code_sub: num > num > code_integer ).
thf(sy_c_Complete__Lattices_OInf__class_OInf,type,
complete_Inf_Inf:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Complete__Lattices_OSup__class_OSup,type,
complete_Sup_Sup:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Complete__Partial__Order_Occpo_Oadmissible,type,
comple1908693960933563346ssible:
!>[A: $tType] : ( ( ( set @ A ) > A ) > ( A > A > $o ) > ( A > $o ) > $o ) ).
thf(sy_c_Complete__Partial__Order_Occpo_Ofixp,type,
comple187402453842119260l_fixp:
!>[A: $tType] : ( ( ( set @ A ) > A ) > ( A > A > $o ) > ( A > A ) > A ) ).
thf(sy_c_Complete__Partial__Order_Occpo__class_Ofixp,type,
comple115746919287870866o_fixp:
!>[A: $tType] : ( ( A > A ) > A ) ).
thf(sy_c_Complete__Partial__Order_Ochain,type,
comple1602240252501008431_chain:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Complete__Partial__Order_Omonotone,type,
comple7038119648293358887notone:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > B > $o ) > ( A > B ) > $o ) ).
thf(sy_c_Countable_Ofrom__nat,type,
from_nat:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Countable_Onth__item,type,
nth_item:
!>[A: $tType] : ( nat > ( set @ ( old_node @ A @ product_unit ) ) ) ).
thf(sy_c_Countable_Onth__item__rel,type,
nth_item_rel: nat > nat > $o ).
thf(sy_c_Countable_Oto__nat,type,
to_nat:
!>[A: $tType] : ( A > nat ) ).
thf(sy_c_Divides_Oadjust__div,type,
adjust_div: ( product_prod @ int @ int ) > int ).
thf(sy_c_Divides_Odivmod__nat,type,
divmod_nat: nat > nat > ( product_prod @ nat @ nat ) ).
thf(sy_c_Divides_Oeucl__rel__int,type,
eucl_rel_int: int > int > ( product_prod @ int @ int ) > $o ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux,type,
unique5940410009612947441es_aux:
!>[A: $tType] : ( ( product_prod @ A @ A ) > $o ) ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod,type,
unique8689654367752047608divmod:
!>[A: $tType] : ( num > num > ( product_prod @ A @ A ) ) ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step,type,
unique1321980374590559556d_step:
!>[A: $tType] : ( num > ( product_prod @ A @ A ) > ( product_prod @ A @ A ) ) ).
thf(sy_c_Equiv__Relations_Ocongruent,type,
equiv_congruent:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
thf(sy_c_Equiv__Relations_Ocongruent2,type,
equiv_congruent2:
!>[A: $tType,B: $tType,C: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ B @ B ) ) > ( A > B > C ) > $o ) ).
thf(sy_c_Equiv__Relations_Oequiv,type,
equiv_equiv:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Equiv__Relations_Oequivp,type,
equiv_equivp:
!>[A: $tType] : ( ( A > A > $o ) > $o ) ).
thf(sy_c_Equiv__Relations_Oproj,type,
equiv_proj:
!>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ A ) ) > B > ( set @ A ) ) ).
thf(sy_c_Equiv__Relations_Oquotient,type,
equiv_quotient:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ ( set @ A ) ) ) ).
thf(sy_c_Euclidean__Division_Oeuclidean__semiring__class_Oeuclidean__size,type,
euclid6346220572633701492n_size:
!>[A: $tType] : ( A > nat ) ).
thf(sy_c_Euclidean__Division_Ounique__euclidean__semiring__class_Odivision__segment,type,
euclid7384307370059645450egment:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer,type,
comm_s3205402744901411588hammer:
!>[A: $tType] : ( A > nat > A ) ).
thf(sy_c_Factorial_Osemiring__char__0__class_Ofact,type,
semiring_char_0_fact:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Fields_Oinverse__class_Oinverse,type,
inverse_inverse:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Finite__Set_Ocard,type,
finite_card:
!>[B: $tType] : ( ( set @ B ) > nat ) ).
thf(sy_c_Finite__Set_Ocomp__fun__commute,type,
finite6289374366891150609ommute:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ocomp__fun__idem__axioms,type,
finite1981172056906289455axioms:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ocomp__fun__idem__on__axioms,type,
finite4980608107308702382axioms:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ofinite,type,
finite_finite2:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Finite__Set_Ofold,type,
finite_fold:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( set @ A ) > B ) ).
thf(sy_c_Finite__Set_Ofolding,type,
finite_folding:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ofolding__idem,type,
finite_folding_idem:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ofolding__idem__axioms,type,
finite7837460588564673216axioms:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ofolding__idem__on,type,
finite1890593828518410140dem_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ofolding__idem__on__axioms,type,
finite6916993218817215295axioms:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > B ) > $o ) ).
thf(sy_c_Fun_Obij__betw,type,
bij_betw:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) > $o ) ).
thf(sy_c_Fun_Ocomp,type,
comp:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
thf(sy_c_Fun_Ofcomp,type,
fcomp:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B ) > ( B > C ) > A > C ) ).
thf(sy_c_Fun_Ofun__upd,type,
fun_upd:
!>[A: $tType,B: $tType] : ( ( A > B ) > A > B > A > B ) ).
thf(sy_c_Fun_Oid,type,
id:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Fun_Oinj__on,type,
inj_on:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
thf(sy_c_Fun_Omap__fun,type,
map_fun:
!>[C: $tType,A: $tType,B: $tType,D: $tType] : ( ( C > A ) > ( B > D ) > ( A > B ) > C > D ) ).
thf(sy_c_Fun_Omono__on,type,
mono_on:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
thf(sy_c_Fun_Ooverride__on,type,
override_on:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( A > B ) > ( set @ A ) > A > B ) ).
thf(sy_c_Fun_Ostrict__mono__on,type,
strict_mono_on:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
thf(sy_c_Fun_Othe__inv__into,type,
the_inv_into:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > B > A ) ).
thf(sy_c_Fun__Def_Omax__strict,type,
fun_max_strict: set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ).
thf(sy_c_Fun__Def_Omax__weak,type,
fun_max_weak: set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ).
thf(sy_c_Fun__Def_Omin__strict,type,
fun_min_strict: set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ).
thf(sy_c_Fun__Def_Omin__weak,type,
fun_min_weak: set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ).
thf(sy_c_Fun__Def_Opair__leq,type,
fun_pair_leq: set @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) ).
thf(sy_c_Fun__Def_Opair__less,type,
fun_pair_less: set @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) ).
thf(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 ) ).
thf(sy_c_Fun__Def_Orp__inv__image,type,
fun_rp_inv_image:
!>[A: $tType,B: $tType] : ( ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) > ( B > A ) > ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) ) ).
thf(sy_c_GCD_OGcd__class_OGcd,type,
gcd_Gcd:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_GCD_OGcd__class_OLcm,type,
gcd_Lcm:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_GCD_Obezw,type,
bezw: nat > nat > ( product_prod @ int @ int ) ).
thf(sy_c_GCD_Obezw__rel,type,
bezw_rel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
thf(sy_c_GCD_Obounded__quasi__semilattice,type,
bounde8507323023520639062attice:
!>[A: $tType] : ( ( A > A > A ) > A > A > ( A > A ) > $o ) ).
thf(sy_c_GCD_Obounded__quasi__semilattice__set,type,
bounde6485984586167503788ce_set:
!>[A: $tType] : ( ( A > A > A ) > A > A > ( A > A ) > $o ) ).
thf(sy_c_GCD_Obounded__quasi__semilattice__set_OF,type,
bounde2362111253966948842tice_F:
!>[A: $tType] : ( ( A > A > A ) > A > A > ( set @ A ) > A ) ).
thf(sy_c_GCD_Ogcd__class_Ogcd,type,
gcd_gcd:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_GCD_Ogcd__class_Olcm,type,
gcd_lcm:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_GCD_Ogcd__nat__rel,type,
gcd_nat_rel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
thf(sy_c_GCD_Osemiring__gcd__class_OGcd__fin,type,
semiring_gcd_Gcd_fin:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_GCD_Osemiring__gcd__class_OLcm__fin,type,
semiring_gcd_Lcm_fin:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Groups_Oabel__semigroup,type,
abel_semigroup:
!>[A: $tType] : ( ( A > A > A ) > $o ) ).
thf(sy_c_Groups_Oabel__semigroup__axioms,type,
abel_s757365448890700780axioms:
!>[A: $tType] : ( ( A > A > A ) > $o ) ).
thf(sy_c_Groups_Oabs__class_Oabs,type,
abs_abs:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Groups_Ocomm__monoid,type,
comm_monoid:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Groups_Ocomm__monoid__axioms,type,
comm_monoid_axioms:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Groups_Ogroup,type,
group:
!>[A: $tType] : ( ( A > A > A ) > A > ( A > A ) > $o ) ).
thf(sy_c_Groups_Ogroup__axioms,type,
group_axioms:
!>[A: $tType] : ( ( A > A > A ) > A > ( A > A ) > $o ) ).
thf(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Omonoid,type,
monoid:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Groups_Omonoid__axioms,type,
monoid_axioms:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Groups_Oone__class_Oone,type,
one_one:
!>[A: $tType] : A ).
thf(sy_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Osemigroup,type,
semigroup:
!>[A: $tType] : ( ( A > A > A ) > $o ) ).
thf(sy_c_Groups_Osgn__class_Osgn,type,
sgn_sgn:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Groups_Otimes__class_Otimes,type,
times_times:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Ouminus__class_Ouminus,type,
uminus_uminus:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_Groups__Big_Ocomm__monoid__add_Osum,type,
groups3894954378712506084id_sum:
!>[A: $tType,B: $tType] : ( ( A > A > A ) > A > ( B > A ) > ( set @ B ) > A ) ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum,type,
groups7311177749621191930dd_sum:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > A ) ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H,type,
groups1027152243600224163dd_sum:
!>[C: $tType,A: $tType] : ( ( C > A ) > ( set @ C ) > A ) ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod,type,
groups7121269368397514597t_prod:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > A ) ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_H,type,
groups1962203154675924110t_prod:
!>[C: $tType,A: $tType] : ( ( C > A ) > ( set @ C ) > A ) ).
thf(sy_c_Groups__Big_Ocomm__monoid__set,type,
groups778175481326437816id_set:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Groups__Big_Ocomm__monoid__set_OF,type,
groups_comm_monoid_F:
!>[A: $tType,B: $tType] : ( ( A > A > A ) > A > ( B > A ) > ( set @ B ) > A ) ).
thf(sy_c_Groups__Big_Ocomm__monoid__set_OG,type,
groups_comm_monoid_G:
!>[A: $tType,B: $tType] : ( ( A > A > A ) > A > ( B > A ) > ( set @ B ) > A ) ).
thf(sy_c_Groups__List_Ocomm__monoid__list,type,
groups1828464146339083142d_list:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Groups__List_Ocomm__monoid__list__set,type,
groups4802862169904069756st_set:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum,type,
groups4207007520872428315er_sum:
!>[B: $tType,A: $tType] : ( ( B > A ) > A > ( list @ B ) > A ) ).
thf(sy_c_Groups__List_Omonoid__add_Osum__list,type,
groups4543113879258116180m_list:
!>[A: $tType] : ( ( A > A > A ) > A > ( list @ A ) > A ) ).
thf(sy_c_Groups__List_Omonoid__add__class_Osum__list,type,
groups8242544230860333062m_list:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_Groups__List_Omonoid__list,type,
groups_monoid_list:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Groups__List_Omonoid__list_OF,type,
groups_monoid_F:
!>[A: $tType] : ( ( A > A > A ) > A > ( list @ A ) > A ) ).
thf(sy_c_Groups__List_Omonoid__mult__class_Oprod__list,type,
groups5270119922927024881d_list:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_HOL_ONO__MATCH,type,
nO_MATCH:
!>[A: $tType,B: $tType] : ( A > B > $o ) ).
thf(sy_c_HOL_OThe,type,
the:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_HOL_OUniq,type,
uniq:
!>[A: $tType] : ( ( A > $o ) > $o ) ).
thf(sy_c_HOL_Odefault__class_Odefault,type,
default_default:
!>[A: $tType] : A ).
thf(sy_c_HOL_Oundefined,type,
undefined:
!>[A: $tType] : A ).
thf(sy_c_Heap_Oaddr__of__array,type,
addr_of_array:
!>[A: $tType] : ( ( array @ A ) > nat ) ).
thf(sy_c_Heap_Oaddr__of__ref,type,
addr_of_ref:
!>[A: $tType] : ( ( ref @ A ) > nat ) ).
thf(sy_c_Heap_Oarray_OArray,type,
array2:
!>[A: $tType] : ( nat > ( array @ A ) ) ).
thf(sy_c_Heap_Oheap_Oarrays,type,
arrays:
!>[Z: $tType] : ( ( heap_ext @ Z ) > typerep > nat > ( list @ nat ) ) ).
thf(sy_c_Heap_Oheap_Oarrays__update,type,
arrays_update:
!>[Z: $tType] : ( ( ( typerep > nat > ( list @ nat ) ) > typerep > nat > ( list @ nat ) ) > ( heap_ext @ Z ) > ( heap_ext @ Z ) ) ).
thf(sy_c_Heap_Oheap_Olim,type,
lim:
!>[Z: $tType] : ( ( heap_ext @ Z ) > nat ) ).
thf(sy_c_Heap_Oheap_Olim__update,type,
lim_update:
!>[Z: $tType] : ( ( nat > nat ) > ( heap_ext @ Z ) > ( heap_ext @ Z ) ) ).
thf(sy_c_Heap_Oheap_Orefs,type,
refs:
!>[Z: $tType] : ( ( heap_ext @ Z ) > typerep > nat > nat ) ).
thf(sy_c_Heap_Oref_ORef,type,
ref2:
!>[A: $tType] : ( nat > ( ref @ A ) ) ).
thf(sy_c_Heap__Time__Monad_OHeap_OHeap,type,
heap_Time_Heap2:
!>[A: $tType] : ( ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_Heap__Time__Monad_OHeap_Ocase__Heap,type,
heap_Time_case_Heap:
!>[A: $tType,B: $tType] : ( ( ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > B ) > ( heap_Time_Heap @ A ) > B ) ).
thf(sy_c_Heap__Time__Monad_OHeap_Omap__Heap,type,
heap_Time_map_Heap:
!>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( heap_Time_Heap @ A ) > ( heap_Time_Heap @ Aa ) ) ).
thf(sy_c_Heap__Time__Monad_OHeap_Opred__Heap,type,
heap_Time_pred_Heap:
!>[A: $tType] : ( ( A > $o ) > ( heap_Time_Heap @ A ) > $o ) ).
thf(sy_c_Heap__Time__Monad_OHeap_Orec__Heap,type,
heap_Time_rec_Heap:
!>[A: $tType,C: $tType] : ( ( ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > C ) > ( heap_Time_Heap @ A ) > C ) ).
thf(sy_c_Heap__Time__Monad_OHeap_Orel__Heap,type,
heap_Time_rel_Heap:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( heap_Time_Heap @ A ) > ( heap_Time_Heap @ B ) > $o ) ).
thf(sy_c_Heap__Time__Monad_OHeap_Oset__Heap,type,
heap_Time_set_Heap:
!>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( set @ A ) ) ).
thf(sy_c_Heap__Time__Monad_OHeap_Osize__Heap,type,
heap_Time_size_Heap:
!>[A: $tType] : ( ( A > nat ) > ( heap_Time_Heap @ A ) > nat ) ).
thf(sy_c_Heap__Time__Monad_OHeap__lub,type,
heap_Time_Heap_lub:
!>[A: $tType] : ( ( set @ ( heap_Time_Heap @ A ) ) > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_Heap__Time__Monad_OHeap__ord,type,
heap_Time_Heap_ord:
!>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_Time_Heap @ A ) > $o ) ).
thf(sy_c_Heap__Time__Monad_Oassert,type,
heap_Time_assert:
!>[A: $tType] : ( ( A > $o ) > A > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_Heap__Time__Monad_Obind,type,
heap_Time_bind:
!>[A: $tType,B: $tType] : ( ( heap_Time_Heap @ A ) > ( A > ( heap_Time_Heap @ B ) ) > ( heap_Time_Heap @ B ) ) ).
thf(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 ) ).
thf(sy_c_Heap__Time__Monad_Oexecute,type,
heap_Time_execute:
!>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ).
thf(sy_c_Heap__Time__Monad_Oguard,type,
heap_Time_guard:
!>[A: $tType] : ( ( ( heap_ext @ product_unit ) > $o ) > ( ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_Heap__Time__Monad_Oheap,type,
heap_Time_heap:
!>[A: $tType] : ( ( ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_Heap__Time__Monad_Olift,type,
heap_Time_lift:
!>[A: $tType,B: $tType] : ( ( A > B ) > A > ( heap_Time_Heap @ B ) ) ).
thf(sy_c_Heap__Time__Monad_Oraise,type,
heap_Time_raise:
!>[A: $tType] : ( ( list @ char ) > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_Heap__Time__Monad_Oreturn,type,
heap_Time_return:
!>[A: $tType] : ( A > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_Heap__Time__Monad_Osuccess,type,
heap_Time_success:
!>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_ext @ product_unit ) > $o ) ).
thf(sy_c_Heap__Time__Monad_Otap,type,
heap_Time_tap:
!>[A: $tType] : ( ( ( heap_ext @ product_unit ) > A ) > ( heap_Time_Heap @ A ) ) ).
thf(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 ) ) ) ) ).
thf(sy_c_Heap__Time__Monad_OtimeFrame__rel,type,
heap_T5500966940807335491me_rel:
!>[A: $tType] : ( ( product_prod @ nat @ ( 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 ) ) ) ) > $o ) ).
thf(sy_c_Heap__Time__Monad_Oureturn,type,
heap_Time_ureturn:
!>[A: $tType] : ( A > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_Heap__Time__Monad_Owait,type,
heap_Time_wait: nat > ( heap_Time_Heap @ product_unit ) ).
thf(sy_c_Hilbert__Choice_Obijection,type,
hilbert_bijection:
!>[A: $tType] : ( ( A > A ) > $o ) ).
thf(sy_c_Hilbert__Choice_Oinv__into,type,
hilbert_inv_into:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > B > A ) ).
thf(sy_c_Hoare__Triple_Ohoare__triple,type,
hoare_hoare_triple:
!>[A: $tType] : ( assn > ( heap_Time_Heap @ A ) > ( A > assn ) > $o ) ).
thf(sy_c_Hoare__Triple_Onew__addrs,type,
hoare_new_addrs: ( heap_ext @ product_unit ) > ( set @ nat ) > ( heap_ext @ product_unit ) > ( set @ nat ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Inductive_Ocomplete__lattice__class_Olfp,type,
complete_lattice_lfp:
!>[A: $tType] : ( ( A > A ) > A ) ).
thf(sy_c_Int_OAbs__Integ,type,
abs_Integ: ( product_prod @ nat @ nat ) > int ).
thf(sy_c_Int_ONeg,type,
neg: num > int ).
thf(sy_c_Int_OPos,type,
pos: num > int ).
thf(sy_c_Int_ORep__Integ,type,
rep_Integ: int > ( product_prod @ nat @ nat ) ).
thf(sy_c_Int_Odup,type,
dup: int > int ).
thf(sy_c_Int_Ointrel,type,
intrel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
thf(sy_c_Int_Onat,type,
nat2: int > nat ).
thf(sy_c_Int_Opcr__int,type,
pcr_int: ( product_prod @ nat @ nat ) > int > $o ).
thf(sy_c_Int_Opower__int,type,
power_int:
!>[A: $tType] : ( A > int > A ) ).
thf(sy_c_Int_Oring__1__class_OInts,type,
ring_1_Ints:
!>[A: $tType] : ( set @ A ) ).
thf(sy_c_Int_Oring__1__class_Oof__int,type,
ring_1_of_int:
!>[A: $tType] : ( int > A ) ).
thf(sy_c_Int_Osub,type,
sub: num > num > int ).
thf(sy_c_Lattices_Oinf__class_Oinf,type,
inf_inf:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Lattices_Osemilattice__neutr,type,
semilattice_neutr:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Lattices_Osup__class_Osup,type,
sup_sup:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Lattices__Big_Olinorder__class_OMax,type,
lattic643756798349783984er_Max:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Olinorder__class_OMin,type,
lattic643756798350308766er_Min:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin,type,
lattic7752659483105999362nf_fin:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Osemilattice__neutr__set_OF,type,
lattic5214292709420241887eutr_F:
!>[A: $tType] : ( ( A > A > A ) > A > ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Osemilattice__set,type,
lattic149705377957585745ce_set:
!>[A: $tType] : ( ( A > A > A ) > $o ) ).
thf(sy_c_Lattices__Big_Osemilattice__set_OF,type,
lattic1715443433743089157tice_F:
!>[A: $tType] : ( ( A > A > A ) > ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin,type,
lattic5882676163264333800up_fin:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Lifting_OQuotient,type,
quotient:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > ( B > A ) > ( A > B > $o ) > $o ) ).
thf(sy_c_List_Oappend,type,
append:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oarg__min__list,type,
arg_min_list:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( list @ A ) > A ) ).
thf(sy_c_List_Oarg__min__list__rel,type,
arg_min_list_rel:
!>[A: $tType,B: $tType] : ( ( product_prod @ ( A > B ) @ ( list @ A ) ) > ( product_prod @ ( A > B ) @ ( list @ A ) ) > $o ) ).
thf(sy_c_List_Obutlast,type,
butlast:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oconcat,type,
concat:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ A ) ) ).
thf(sy_c_List_Ocount__list,type,
count_list:
!>[A: $tType] : ( ( list @ A ) > A > nat ) ).
thf(sy_c_List_Odistinct,type,
distinct:
!>[A: $tType] : ( ( list @ A ) > $o ) ).
thf(sy_c_List_Odrop,type,
drop:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_OdropWhile,type,
dropWhile:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oenumerate,type,
enumerate:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ ( product_prod @ nat @ A ) ) ) ).
thf(sy_c_List_Oextract,type,
extract:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) ) ).
thf(sy_c_List_Ofilter,type,
filter:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Ofind,type,
find:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( option @ A ) ) ).
thf(sy_c_List_Ofold,type,
fold:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > ( list @ A ) > B > B ) ).
thf(sy_c_List_Ofoldl,type,
foldl:
!>[B: $tType,A: $tType] : ( ( B > A > B ) > B > ( list @ A ) > B ) ).
thf(sy_c_List_Ofoldr,type,
foldr:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > ( list @ A ) > B > B ) ).
thf(sy_c_List_Ogen__length,type,
gen_length:
!>[A: $tType] : ( nat > ( list @ A ) > nat ) ).
thf(sy_c_List_Olast,type,
last:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_List_Olenlex,type,
lenlex:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olex,type,
lex:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olexn,type,
lexn:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > nat > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olexord,type,
lexord:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olexordp,type,
lexordp:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Olinorder__class_Osort__key,type,
linorder_sort_key:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( list @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Olinorder__class_Osorted__list__of__set,type,
linord4507533701916653071of_set:
!>[A: $tType] : ( ( set @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist_OCons,type,
cons:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist_ONil,type,
nil:
!>[A: $tType] : ( list @ A ) ).
thf(sy_c_List_Olist_Ocase__list,type,
case_list:
!>[B: $tType,A: $tType] : ( B > ( A > ( list @ A ) > B ) > ( list @ A ) > B ) ).
thf(sy_c_List_Olist_Ohd,type,
hd:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_List_Olist_Olist__all,type,
list_all:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Olist_Olist__all2,type,
list_all2:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( list @ A ) > ( list @ B ) > $o ) ).
thf(sy_c_List_Olist_Omap,type,
map:
!>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( list @ A ) > ( list @ Aa ) ) ).
thf(sy_c_List_Olist_Oset,type,
set2:
!>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
thf(sy_c_List_Olist_Osize__list,type,
size_list:
!>[A: $tType] : ( ( A > nat ) > ( list @ A ) > nat ) ).
thf(sy_c_List_Olist_Otl,type,
tl:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist__update,type,
list_update:
!>[A: $tType] : ( ( list @ A ) > nat > A > ( list @ A ) ) ).
thf(sy_c_List_Olistrel,type,
listrel:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) ) ).
thf(sy_c_List_Olistrel1,type,
listrel1:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olistrel1p,type,
listrel1p:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Olistrelp,type,
listrelp:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( list @ A ) > ( list @ B ) > $o ) ).
thf(sy_c_List_Olists,type,
lists:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( list @ A ) ) ) ).
thf(sy_c_List_Omap__filter,type,
map_filter:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( list @ A ) > ( list @ B ) ) ).
thf(sy_c_List_Omap__project,type,
map_project:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ A ) > ( set @ B ) ) ).
thf(sy_c_List_Omap__tailrec__rev,type,
map_tailrec_rev:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( list @ A ) > ( list @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Omap__tailrec__rev__rel,type,
map_tailrec_rev_rel:
!>[A: $tType,B: $tType] : ( ( product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) > ( product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) > $o ) ).
thf(sy_c_List_Omeasures,type,
measures:
!>[A: $tType] : ( ( list @ ( A > nat ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_List_Omin__list,type,
min_list:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_List_Omin__list__rel,type,
min_list_rel:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Onth,type,
nth:
!>[A: $tType] : ( ( list @ A ) > nat > A ) ).
thf(sy_c_List_Onths,type,
nths:
!>[A: $tType] : ( ( list @ A ) > ( set @ nat ) > ( list @ A ) ) ).
thf(sy_c_List_Oord__class_Olexordp,type,
ord_lexordp:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Opartition,type,
partition:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ).
thf(sy_c_List_Oproduct,type,
product:
!>[A: $tType,B: $tType] : ( ( list @ A ) > ( list @ B ) > ( list @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_List_Oproduct__lists,type,
product_lists:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Oremdups,type,
remdups:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oremdups__adj,type,
remdups_adj:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oremdups__adj__rel,type,
remdups_adj_rel:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Oremove1,type,
remove1:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oreplicate,type,
replicate:
!>[A: $tType] : ( nat > A > ( list @ A ) ) ).
thf(sy_c_List_Orev,type,
rev:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Orotate,type,
rotate:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Orotate1,type,
rotate1:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oshuffles,type,
shuffles:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( set @ ( list @ A ) ) ) ).
thf(sy_c_List_Oshuffles__rel,type,
shuffles_rel:
!>[A: $tType] : ( ( product_prod @ ( list @ A ) @ ( list @ A ) ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) > $o ) ).
thf(sy_c_List_Osorted__wrt,type,
sorted_wrt:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Osorted__wrt__rel,type,
sorted_wrt_rel:
!>[A: $tType] : ( ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) > ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) > $o ) ).
thf(sy_c_List_Osplice,type,
splice:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Osplice__rel,type,
splice_rel:
!>[A: $tType] : ( ( product_prod @ ( list @ A ) @ ( list @ A ) ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) > $o ) ).
thf(sy_c_List_Osuccessively,type,
successively:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Osuccessively__rel,type,
successively_rel:
!>[A: $tType] : ( ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) > ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) > $o ) ).
thf(sy_c_List_Otake,type,
take:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_OtakeWhile,type,
takeWhile:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Othose,type,
those:
!>[A: $tType] : ( ( list @ ( option @ A ) ) > ( option @ ( list @ A ) ) ) ).
thf(sy_c_List_Otranspose,type,
transpose:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Otranspose__rel,type,
transpose_rel:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) > $o ) ).
thf(sy_c_List_Oupt,type,
upt: nat > nat > ( list @ nat ) ).
thf(sy_c_List_Oupto,type,
upto: int > int > ( list @ int ) ).
thf(sy_c_List_Oupto__rel,type,
upto_rel: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) > $o ).
thf(sy_c_List_Ozip,type,
zip:
!>[A: $tType,B: $tType] : ( ( list @ A ) > ( list @ B ) > ( list @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Map_Odom,type,
dom:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ A ) ) ).
thf(sy_c_Map_Ograph,type,
graph:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Map_Omap__add,type,
map_add:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( A > ( option @ B ) ) > A > ( option @ B ) ) ).
thf(sy_c_Map_Omap__comp,type,
map_comp:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > ( option @ C ) ) > ( A > ( option @ B ) ) > A > ( option @ C ) ) ).
thf(sy_c_Map_Omap__le,type,
map_le:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( A > ( option @ B ) ) > $o ) ).
thf(sy_c_Map_Omap__of,type,
map_of:
!>[A: $tType,B: $tType] : ( ( list @ ( product_prod @ A @ B ) ) > A > ( option @ B ) ) ).
thf(sy_c_Map_Omap__upds,type,
map_upds:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( list @ A ) > ( list @ B ) > A > ( option @ B ) ) ).
thf(sy_c_Map_Oran,type,
ran:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ B ) ) ).
thf(sy_c_Map_Orestrict__map,type,
restrict_map:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ A ) > A > ( option @ B ) ) ).
thf(sy_c_Misc_OCODE__ABORT,type,
cODE_ABORT:
!>[A: $tType] : ( ( product_unit > A ) > A ) ).
thf(sy_c_Misc_OEps__Opt,type,
eps_Opt:
!>[A: $tType] : ( ( A > $o ) > ( option @ A ) ) ).
thf(sy_c_Misc_Obijective,type,
bijective:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > $o ) ).
thf(sy_c_Misc_Obrk__rel,type,
brk_rel:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B ) ) ) ) ).
thf(sy_c_Misc_Odflt__None__set,type,
dflt_None_set:
!>[A: $tType] : ( ( set @ A ) > ( option @ ( set @ A ) ) ) ).
thf(sy_c_Misc_Ofilter__rev,type,
filter_rev:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Misc_Ofilter__rev__aux,type,
filter_rev_aux:
!>[A: $tType] : ( ( list @ A ) > ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Misc_Ofun__of__rel,type,
fun_of_rel:
!>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ A ) ) > B > A ) ).
thf(sy_c_Misc_Oinv__on,type,
inv_on:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > B > A ) ).
thf(sy_c_Misc_Olist__all__zip,type,
list_all_zip:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( list @ A ) > ( list @ B ) > $o ) ).
thf(sy_c_Misc_Olist__all__zip__rel,type,
list_all_zip_rel:
!>[A: $tType,B: $tType] : ( ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) > ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) > $o ) ).
thf(sy_c_Misc_Olist__collect__set,type,
list_collect_set:
!>[B: $tType,A: $tType] : ( ( B > ( set @ A ) ) > ( list @ B ) > ( set @ A ) ) ).
thf(sy_c_Misc_Omap__mmupd,type,
map_mmupd:
!>[B: $tType,A: $tType] : ( ( B > ( option @ A ) ) > ( set @ B ) > A > B > ( option @ A ) ) ).
thf(sy_c_Misc_Omap__to__set,type,
map_to_set:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Misc_Omerge,type,
merge:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Misc_Omerge__list,type,
merge_list:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) > ( list @ A ) ) ).
thf(sy_c_Misc_Omerge__list__rel,type,
merge_list_rel:
!>[A: $tType] : ( ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) > ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) > $o ) ).
thf(sy_c_Misc_Omerge__rel,type,
merge_rel:
!>[A: $tType] : ( ( product_prod @ ( list @ A ) @ ( list @ A ) ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) > $o ) ).
thf(sy_c_Misc_Omergesort,type,
mergesort:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Misc_Omergesort__by__rel,type,
mergesort_by_rel:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Misc_Omergesort__by__rel__merge,type,
merges9089515139780605204_merge:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Misc_Omergesort__by__rel__merge__rel,type,
merges2244889521215249637ge_rel:
!>[A: $tType] : ( ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) > ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) > $o ) ).
thf(sy_c_Misc_Omergesort__by__rel__rel,type,
mergesort_by_rel_rel:
!>[A: $tType] : ( ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) > ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) > $o ) ).
thf(sy_c_Misc_Omergesort__by__rel__split,type,
merges295452479951948502_split:
!>[A: $tType] : ( ( product_prod @ ( list @ A ) @ ( list @ A ) ) > ( list @ A ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ).
thf(sy_c_Misc_Omergesort__by__rel__split__rel,type,
merges7066485432131860899it_rel:
!>[A: $tType] : ( ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) > ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) > $o ) ).
thf(sy_c_Misc_Omergesort__remdups,type,
mergesort_remdups:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Misc_Opairself,type,
pairself:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( product_prod @ A @ A ) > ( product_prod @ B @ B ) ) ).
thf(sy_c_Misc_Opairself__rel,type,
pairself_rel:
!>[A: $tType,B: $tType] : ( ( product_prod @ ( A > B ) @ ( product_prod @ A @ A ) ) > ( product_prod @ ( A > B ) @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Misc_Opartition__rev,type,
partition_rev:
!>[A: $tType] : ( ( A > $o ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) > ( list @ A ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ).
thf(sy_c_Misc_Opartition__rev__rel,type,
partition_rev_rel:
!>[A: $tType] : ( ( product_prod @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) ) > ( product_prod @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) ) > $o ) ).
thf(sy_c_Misc_Oquicksort__by__rel,type,
quicksort_by_rel:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Misc_Oquicksort__by__rel__rel,type,
quicksort_by_rel_rel:
!>[A: $tType] : ( ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) > ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) > $o ) ).
thf(sy_c_Misc_Orel__of,type,
rel_of:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( ( product_prod @ A @ B ) > $o ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Misc_Orel__restrict,type,
rel_restrict:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Misc_Oremove__rev,type,
remove_rev:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Misc_Orevg,type,
revg:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Misc_Orevg__rel,type,
revg_rel:
!>[A: $tType] : ( ( product_prod @ ( list @ A ) @ ( list @ A ) ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) > $o ) ).
thf(sy_c_Misc_Oset__to__map,type,
set_to_map:
!>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ A ) ) > B > ( option @ A ) ) ).
thf(sy_c_Misc_Oslice,type,
slice:
!>[A: $tType] : ( nat > nat > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Misc_Osu__rel__fun,type,
su_rel_fun:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( A > B ) > $o ) ).
thf(sy_c_Misc_Othe__default,type,
the_default:
!>[A: $tType] : ( A > ( option @ A ) > A ) ).
thf(sy_c_Misc_Ouncurry,type,
uncurry:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
thf(sy_c_Misc_Ozipf,type,
zipf:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( list @ A ) > ( list @ B ) > ( list @ C ) ) ).
thf(sy_c_Misc_Ozipf__rel,type,
zipf_rel:
!>[A: $tType,B: $tType,C: $tType] : ( ( product_prod @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) > ( product_prod @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) > $o ) ).
thf(sy_c_Multiset_Oadd__mset,type,
add_mset:
!>[A: $tType] : ( A > ( multiset @ A ) > ( multiset @ A ) ) ).
thf(sy_c_Multiset_Ocomm__monoid__add_Osum__mset,type,
comm_monoid_sum_mset:
!>[A: $tType] : ( ( A > A > A ) > A > ( multiset @ A ) > A ) ).
thf(sy_c_Multiset_Ocomm__monoid__add__class_Osum__mset,type,
comm_m7189776963980413722m_mset:
!>[A: $tType] : ( ( multiset @ A ) > A ) ).
thf(sy_c_Multiset_Ocomm__monoid__mset,type,
comm_monoid_mset:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Multiset_Ocomm__monoid__mset_OF,type,
comm_monoid_F:
!>[A: $tType] : ( ( A > A > A ) > A > ( multiset @ A ) > A ) ).
thf(sy_c_Multiset_Ocomm__monoid__mult__class_Oprod__mset,type,
comm_m9189036328036947845d_mset:
!>[A: $tType] : ( ( multiset @ A ) > A ) ).
thf(sy_c_Multiset_Ofilter__mset,type,
filter_mset:
!>[A: $tType] : ( ( A > $o ) > ( multiset @ A ) > ( multiset @ A ) ) ).
thf(sy_c_Multiset_Ofold__mset,type,
fold_mset:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( multiset @ A ) > B ) ).
thf(sy_c_Multiset_Oimage__mset,type,
image_mset:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( multiset @ A ) > ( multiset @ B ) ) ).
thf(sy_c_Multiset_Ointer__mset,type,
inter_mset:
!>[A: $tType] : ( ( multiset @ A ) > ( multiset @ A ) > ( multiset @ A ) ) ).
thf(sy_c_Multiset_Olinorder__class_Opart,type,
linorder_part:
!>[B: $tType,A: $tType] : ( ( B > A ) > A > ( list @ B ) > ( product_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) ) ) ).
thf(sy_c_Multiset_Oms__strict,type,
ms_strict: set @ ( product_prod @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ).
thf(sy_c_Multiset_Oms__weak,type,
ms_weak: set @ ( product_prod @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ).
thf(sy_c_Multiset_Omset,type,
mset:
!>[A: $tType] : ( ( list @ A ) > ( multiset @ A ) ) ).
thf(sy_c_Multiset_Omset__set,type,
mset_set:
!>[B: $tType] : ( ( set @ B ) > ( multiset @ B ) ) ).
thf(sy_c_Multiset_Omult,type,
mult:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) ) ) ).
thf(sy_c_Multiset_Omult1,type,
mult1:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) ) ) ).
thf(sy_c_Multiset_Omulteqp__code,type,
multeqp_code:
!>[A: $tType] : ( ( A > A > $o ) > ( multiset @ A ) > ( multiset @ A ) > $o ) ).
thf(sy_c_Multiset_Omultiset_OAbs__multiset,type,
abs_multiset:
!>[A: $tType] : ( ( A > nat ) > ( multiset @ A ) ) ).
thf(sy_c_Multiset_Omultiset_Ocount,type,
count:
!>[A: $tType] : ( ( multiset @ A ) > A > nat ) ).
thf(sy_c_Multiset_Omultp,type,
multp:
!>[A: $tType] : ( ( A > A > $o ) > ( multiset @ A ) > ( multiset @ A ) > $o ) ).
thf(sy_c_Multiset_Omultp__code,type,
multp_code:
!>[A: $tType] : ( ( A > A > $o ) > ( multiset @ A ) > ( multiset @ A ) > $o ) ).
thf(sy_c_Multiset_Opcr__multiset,type,
pcr_multiset:
!>[C: $tType,B: $tType] : ( ( C > B > $o ) > ( C > nat ) > ( multiset @ B ) > $o ) ).
thf(sy_c_Multiset_Opw__leq,type,
pw_leq: ( multiset @ ( product_prod @ nat @ nat ) ) > ( multiset @ ( product_prod @ nat @ nat ) ) > $o ).
thf(sy_c_Multiset_Orepeat__mset,type,
repeat_mset:
!>[A: $tType] : ( nat > ( multiset @ A ) > ( multiset @ A ) ) ).
thf(sy_c_Multiset_Oreplicate__mset,type,
replicate_mset:
!>[A: $tType] : ( nat > A > ( multiset @ A ) ) ).
thf(sy_c_Multiset_Oset__mset,type,
set_mset:
!>[A: $tType] : ( ( multiset @ A ) > ( set @ A ) ) ).
thf(sy_c_Multiset_Osize__multiset,type,
size_multiset:
!>[A: $tType] : ( ( A > nat ) > ( multiset @ A ) > nat ) ).
thf(sy_c_Multiset_Osubset__eq__mset__impl,type,
subset_eq_mset_impl:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( option @ $o ) ) ).
thf(sy_c_Multiset_Osubset__eq__mset__impl__rel,type,
subset751672762298770561pl_rel:
!>[A: $tType] : ( ( product_prod @ ( list @ A ) @ ( list @ A ) ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) > $o ) ).
thf(sy_c_Multiset_Osubset__mset,type,
subset_mset:
!>[A: $tType] : ( ( multiset @ A ) > ( multiset @ A ) > $o ) ).
thf(sy_c_Multiset_Osubseteq__mset,type,
subseteq_mset:
!>[A: $tType] : ( ( multiset @ A ) > ( multiset @ A ) > $o ) ).
thf(sy_c_Multiset_Ounion__mset,type,
union_mset:
!>[A: $tType] : ( ( multiset @ A ) > ( multiset @ A ) > ( multiset @ A ) ) ).
thf(sy_c_Multiset_Owcount,type,
wcount:
!>[A: $tType] : ( ( A > nat ) > ( multiset @ A ) > A > nat ) ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Ocompow,type,
compow:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Nat_Onat_Ocase__nat,type,
case_nat:
!>[A: $tType] : ( A > ( nat > A ) > nat > A ) ).
thf(sy_c_Nat_Oold_Onat_Orec__nat,type,
rec_nat:
!>[T: $tType] : ( T > ( nat > T > T ) > nat > T ) ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
semiring_1_of_nat:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux,type,
semiri8178284476397505188at_aux:
!>[A: $tType] : ( ( A > A ) > nat > A > A ) ).
thf(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : ( A > nat ) ).
thf(sy_c_Nat__Bijection_Olist__decode,type,
nat_list_decode: nat > ( list @ nat ) ).
thf(sy_c_Nat__Bijection_Olist__decode__rel,type,
nat_list_decode_rel: nat > nat > $o ).
thf(sy_c_Nat__Bijection_Olist__encode,type,
nat_list_encode: ( list @ nat ) > nat ).
thf(sy_c_Nat__Bijection_Olist__encode__rel,type,
nat_list_encode_rel: ( list @ nat ) > ( list @ nat ) > $o ).
thf(sy_c_Nat__Bijection_Oprod__decode,type,
nat_prod_decode: nat > ( product_prod @ nat @ nat ) ).
thf(sy_c_Nat__Bijection_Oprod__decode__aux,type,
nat_prod_decode_aux: nat > nat > ( product_prod @ nat @ nat ) ).
thf(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
nat_pr5047031295181774490ux_rel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
thf(sy_c_Nat__Bijection_Oprod__encode,type,
nat_prod_encode: ( product_prod @ nat @ nat ) > nat ).
thf(sy_c_Nat__Bijection_Oset__decode,type,
nat_set_decode: nat > ( set @ nat ) ).
thf(sy_c_Nat__Bijection_Oset__encode,type,
nat_set_encode: ( set @ nat ) > nat ).
thf(sy_c_Nat__Bijection_Osum__decode,type,
nat_sum_decode: nat > ( sum_sum @ nat @ nat ) ).
thf(sy_c_Nat__Bijection_Otriangle,type,
nat_triangle: nat > nat ).
thf(sy_c_Num_OBitM,type,
bitM: num > num ).
thf(sy_c_Num_Oinc,type,
inc: num > num ).
thf(sy_c_Num_Onat__of__num,type,
nat_of_num: num > nat ).
thf(sy_c_Num_Oneg__numeral__class_Odbl,type,
neg_numeral_dbl:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec,type,
neg_numeral_dbl_dec:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc,type,
neg_numeral_dbl_inc:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Num_Oneg__numeral__class_Ois__num,type,
neg_numeral_is_num:
!>[A: $tType] : ( A > $o ) ).
thf(sy_c_Num_Oneg__numeral__class_Osub,type,
neg_numeral_sub:
!>[A: $tType] : ( num > num > A ) ).
thf(sy_c_Num_Onum_OBit0,type,
bit0: num > num ).
thf(sy_c_Num_Onum_OBit1,type,
bit1: num > num ).
thf(sy_c_Num_Onum_OOne,type,
one2: num ).
thf(sy_c_Num_Onum_Ocase__num,type,
case_num:
!>[A: $tType] : ( A > ( num > A ) > ( num > A ) > num > A ) ).
thf(sy_c_Num_Onum_Osize__num,type,
size_num: num > nat ).
thf(sy_c_Num_Onum__of__nat,type,
num_of_nat: nat > num ).
thf(sy_c_Num_Onumeral__class_Onumeral,type,
numeral_numeral:
!>[A: $tType] : ( num > A ) ).
thf(sy_c_Num_Opred__numeral,type,
pred_numeral: num > nat ).
thf(sy_c_Num_Oring__1__class_Oiszero,type,
ring_1_iszero:
!>[A: $tType] : ( A > $o ) ).
thf(sy_c_Num_Osqr,type,
sqr: num > num ).
thf(sy_c_Old__Datatype_OAtom,type,
old_Atom:
!>[A: $tType,B: $tType] : ( ( sum_sum @ A @ nat ) > ( set @ ( old_node @ A @ B ) ) ) ).
thf(sy_c_Old__Datatype_OIn0,type,
old_In0:
!>[A: $tType,B: $tType] : ( ( set @ ( old_node @ A @ B ) ) > ( set @ ( old_node @ A @ B ) ) ) ).
thf(sy_c_Old__Datatype_OIn1,type,
old_In1:
!>[A: $tType,B: $tType] : ( ( set @ ( old_node @ A @ B ) ) > ( set @ ( old_node @ A @ B ) ) ) ).
thf(sy_c_Old__Datatype_OLeaf,type,
old_Leaf:
!>[A: $tType,B: $tType] : ( A > ( set @ ( old_node @ A @ B ) ) ) ).
thf(sy_c_Old__Datatype_ONode,type,
old_Node:
!>[B: $tType,A: $tType] : ( set @ ( product_prod @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat ) ) ) ).
thf(sy_c_Old__Datatype_ONumb,type,
old_Numb:
!>[A: $tType,B: $tType] : ( nat > ( set @ ( old_node @ A @ B ) ) ) ).
thf(sy_c_Old__Datatype_OPush__Node,type,
old_Push_Node:
!>[B: $tType,A: $tType] : ( ( sum_sum @ B @ nat ) > ( old_node @ A @ B ) > ( old_node @ A @ B ) ) ).
thf(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 ) ) ) ).
thf(sy_c_Old__Datatype_Odprod,type,
old_dprod:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) ) > ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) ) > ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) ) ) ).
thf(sy_c_Old__Datatype_Odsum,type,
old_dsum:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) ) > ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) ) > ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) ) ) ).
thf(sy_c_Old__Datatype_Ondepth,type,
old_ndepth:
!>[A: $tType,B: $tType] : ( ( old_node @ A @ B ) > nat ) ).
thf(sy_c_Old__Datatype_Onode_OAbs__Node,type,
old_Abs_Node:
!>[B: $tType,A: $tType] : ( ( product_prod @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat ) ) > ( old_node @ A @ B ) ) ).
thf(sy_c_Option_Obind,type,
bind:
!>[A: $tType,B: $tType] : ( ( option @ A ) > ( A > ( option @ B ) ) > ( option @ B ) ) ).
thf(sy_c_Option_Ocombine__options,type,
combine_options:
!>[A: $tType] : ( ( A > A > A ) > ( option @ A ) > ( option @ A ) > ( option @ A ) ) ).
thf(sy_c_Option_Ois__none,type,
is_none:
!>[A: $tType] : ( ( option @ A ) > $o ) ).
thf(sy_c_Option_Ooption_ONone,type,
none:
!>[A: $tType] : ( option @ A ) ).
thf(sy_c_Option_Ooption_OSome,type,
some:
!>[A: $tType] : ( A > ( option @ A ) ) ).
thf(sy_c_Option_Ooption_Ocase__option,type,
case_option:
!>[B: $tType,A: $tType] : ( B > ( A > B ) > ( option @ A ) > B ) ).
thf(sy_c_Option_Ooption_Omap__option,type,
map_option:
!>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( option @ A ) > ( option @ Aa ) ) ).
thf(sy_c_Option_Ooption_Opred__option,type,
pred_option:
!>[A: $tType] : ( ( A > $o ) > ( option @ A ) > $o ) ).
thf(sy_c_Option_Ooption_Orec__option,type,
rec_option:
!>[C: $tType,A: $tType] : ( C > ( A > C ) > ( option @ A ) > C ) ).
thf(sy_c_Option_Ooption_Orel__option,type,
rel_option:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( option @ A ) > ( option @ B ) > $o ) ).
thf(sy_c_Option_Ooption_Oset__option,type,
set_option:
!>[A: $tType] : ( ( option @ A ) > ( set @ A ) ) ).
thf(sy_c_Option_Ooption_Osize__option,type,
size_option:
!>[A: $tType] : ( ( A > nat ) > ( option @ A ) > nat ) ).
thf(sy_c_Option_Ooption_Othe,type,
the2:
!>[A: $tType] : ( ( option @ A ) > A ) ).
thf(sy_c_Option_Othese,type,
these:
!>[A: $tType] : ( ( set @ ( option @ A ) ) > ( set @ A ) ) ).
thf(sy_c_Order__Relation_OAbove,type,
order_Above:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Order__Relation_OAboveS,type,
order_AboveS:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Order__Relation_OUnder,type,
order_Under:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Order__Relation_OUnderS,type,
order_UnderS:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Order__Relation_Oabove,type,
order_above:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
thf(sy_c_Order__Relation_OaboveS,type,
order_aboveS:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
thf(sy_c_Order__Relation_Olinear__order__on,type,
order_679001287576687338der_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Order__Relation_Oofilter,type,
order_ofilter:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > $o ) ).
thf(sy_c_Order__Relation_Opartial__order__on,type,
order_7125193373082350890der_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Order__Relation_Opreorder__on,type,
order_preorder_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Order__Relation_Orelation__of,type,
order_relation_of:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Order__Relation_Ounder,type,
order_under:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
thf(sy_c_Order__Relation_OunderS,type,
order_underS:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
thf(sy_c_Order__Relation_Owell__order__on,type,
order_well_order_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
thf(sy_c_Orderings_Oord_Omax,type,
max:
!>[A: $tType] : ( ( A > A > $o ) > A > A > A ) ).
thf(sy_c_Orderings_Oord_Omin,type,
min:
!>[A: $tType] : ( ( A > A > $o ) > A > A > A ) ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Omax,type,
ord_max:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Orderings_Oord__class_Omin,type,
ord_min:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Orderings_Oorder__class_Omono,type,
order_mono:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Orderings_Oorder__class_Ostrict__mono,type,
order_strict_mono:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Orderings_Otop__class_Otop,type,
top_top:
!>[A: $tType] : A ).
thf(sy_c_Partial__Function_Oflat__lub,type,
partial_flat_lub:
!>[A: $tType] : ( A > ( set @ A ) > A ) ).
thf(sy_c_Partial__Function_Oflat__ord,type,
partial_flat_ord:
!>[A: $tType] : ( A > A > A > $o ) ).
thf(sy_c_Partial__Function_Ofun__lub,type,
partial_fun_lub:
!>[C: $tType,B: $tType,A: $tType] : ( ( ( set @ C ) > B ) > ( set @ ( A > C ) ) > A > B ) ).
thf(sy_c_Partial__Function_Ofun__ord,type,
partial_fun_ord:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( C > A ) > ( C > B ) > $o ) ).
thf(sy_c_Partial__Function_Oimg__lub,type,
partial_img_lub:
!>[A: $tType,C: $tType,D: $tType,B: $tType] : ( ( A > C ) > ( D > B ) > ( ( set @ C ) > D ) > ( set @ A ) > B ) ).
thf(sy_c_Partial__Function_Oimg__ord,type,
partial_img_ord:
!>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( C > C > B ) > A > A > B ) ).
thf(sy_c_Partial__Function_Opartial__function__definitions,type,
partia7178651479351089652itions:
!>[A: $tType] : ( ( A > A > $o ) > ( ( set @ A ) > A ) > $o ) ).
thf(sy_c_Power_Opower_Opower,type,
power2:
!>[A: $tType] : ( A > ( A > A > A ) > A > nat > A ) ).
thf(sy_c_Power_Opower__class_Opower,type,
power_power:
!>[A: $tType] : ( A > nat > A ) ).
thf(sy_c_Predicate_OSeq,type,
seq2:
!>[A: $tType] : ( ( product_unit > ( seq @ A ) ) > ( pred @ A ) ) ).
thf(sy_c_Predicate_Obind,type,
bind2:
!>[A: $tType,B: $tType] : ( ( pred @ A ) > ( A > ( pred @ B ) ) > ( pred @ B ) ) ).
thf(sy_c_Predicate_Oiterate__upto,type,
iterate_upto:
!>[A: $tType] : ( ( code_natural > A ) > code_natural > code_natural > ( pred @ A ) ) ).
thf(sy_c_Predicate_Oiterate__upto__rel,type,
iterate_upto_rel:
!>[A: $tType] : ( ( product_prod @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) ) > ( product_prod @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) ) > $o ) ).
thf(sy_c_Predicate_Oseq_OEmpty,type,
empty:
!>[A: $tType] : ( seq @ A ) ).
thf(sy_c_Predicate_Oseq_OInsert,type,
insert:
!>[A: $tType] : ( A > ( pred @ A ) > ( seq @ A ) ) ).
thf(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
thf(sy_c_Product__Type_OSigma,type,
product_Sigma:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Product__Type_OUnity,type,
product_Unity: product_unit ).
thf(sy_c_Product__Type_Oapfst,type,
product_apfst:
!>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( product_prod @ A @ B ) > ( product_prod @ C @ B ) ) ).
thf(sy_c_Product__Type_Oapsnd,type,
product_apsnd:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( product_prod @ A @ B ) > ( product_prod @ A @ C ) ) ).
thf(sy_c_Product__Type_Ocurry,type,
product_curry:
!>[A: $tType,B: $tType,C: $tType] : ( ( ( product_prod @ A @ B ) > C ) > A > B > C ) ).
thf(sy_c_Product__Type_Ointernal__case__prod,type,
produc5280177257484947105e_prod:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
thf(sy_c_Product__Type_Omap__prod,type,
product_map_prod:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C ) > ( B > D ) > ( product_prod @ A @ B ) > ( product_prod @ C @ D ) ) ).
thf(sy_c_Product__Type_Oold_Obool_Orec__bool,type,
product_rec_bool:
!>[T: $tType] : ( T > T > $o > T ) ).
thf(sy_c_Product__Type_Oold_Obool_Orec__set__bool,type,
product_rec_set_bool:
!>[T: $tType] : ( T > T > $o > T > $o ) ).
thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
product_rec_prod:
!>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
thf(sy_c_Product__Type_Oold_Oprod_Orec__set__prod,type,
product_rec_set_prod:
!>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T > $o ) ).
thf(sy_c_Product__Type_Oold_Ounit_Orec__set__unit,type,
product_rec_set_unit:
!>[T: $tType] : ( T > product_unit > T > $o ) ).
thf(sy_c_Product__Type_Oold_Ounit_Orec__unit,type,
product_rec_unit:
!>[T: $tType] : ( T > product_unit > T ) ).
thf(sy_c_Product__Type_Oprod_Ocase__prod,type,
product_case_prod:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
thf(sy_c_Product__Type_Oprod_Ofst,type,
product_fst:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
thf(sy_c_Product__Type_Oprod_Osnd,type,
product_snd:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
thf(sy_c_Product__Type_Oprod_Oswap,type,
product_swap:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
thf(sy_c_Product__Type_Oproduct,type,
product_product:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Product__Type_Oscomp,type,
product_scomp:
!>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( A > ( product_prod @ B @ C ) ) > ( B > C > D ) > A > D ) ).
thf(sy_c_Product__Type_Ounit_OAbs__unit,type,
product_Abs_unit: $o > product_unit ).
thf(sy_c_Product__Type_Ounit_ORep__unit,type,
product_Rep_unit: product_unit > $o ).
thf(sy_c_Product__Type_Ounit_Ocase__unit,type,
product_case_unit:
!>[A: $tType] : ( A > product_unit > A ) ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Quicksort_Olinorder__class_Oquicksort,type,
linorder_quicksort:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Quicksort_Olinorder__class_Oquicksort__rel,type,
linord6200660962353139674rt_rel:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_Quotient_OQuotient3,type,
quotient3:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > ( B > A ) > $o ) ).
thf(sy_c_Random_Oinc__shift,type,
inc_shift: code_natural > code_natural > code_natural ).
thf(sy_c_Random_Oiterate,type,
iterate:
!>[B: $tType,A: $tType] : ( code_natural > ( B > A > ( product_prod @ B @ A ) ) > B > A > ( product_prod @ B @ A ) ) ).
thf(sy_c_Random_Oiterate__rel,type,
iterate_rel:
!>[B: $tType,A: $tType] : ( ( product_prod @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B ) ) > ( product_prod @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B ) ) > $o ) ).
thf(sy_c_Random_Olog,type,
log: code_natural > code_natural > code_natural ).
thf(sy_c_Random_Olog__rel,type,
log_rel: ( product_prod @ code_natural @ code_natural ) > ( product_prod @ code_natural @ code_natural ) > $o ).
thf(sy_c_Random_Ominus__shift,type,
minus_shift: code_natural > code_natural > code_natural > code_natural ).
thf(sy_c_Random_Onext,type,
next: ( product_prod @ code_natural @ code_natural ) > ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) ) ).
thf(sy_c_Random_Opick,type,
pick:
!>[A: $tType] : ( ( list @ ( product_prod @ code_natural @ A ) ) > code_natural > A ) ).
thf(sy_c_Random_Orange,type,
range: code_natural > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) ) ).
thf(sy_c_Random_Oselect,type,
select:
!>[A: $tType] : ( ( list @ A ) > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ A @ ( product_prod @ code_natural @ code_natural ) ) ) ).
thf(sy_c_Random_Oselect__weight,type,
select_weight:
!>[A: $tType] : ( ( list @ ( product_prod @ code_natural @ A ) ) > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ A @ ( product_prod @ code_natural @ code_natural ) ) ) ).
thf(sy_c_Random_Osplit__seed,type,
split_seed: ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ code_natural @ code_natural ) ) ).
thf(sy_c_Random__Pred_Obind,type,
random_bind:
!>[A: $tType,B: $tType] : ( ( ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ) ) > ( A > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ B ) @ ( product_prod @ code_natural @ code_natural ) ) ) > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ B ) @ ( product_prod @ code_natural @ code_natural ) ) ) ).
thf(sy_c_Random__Pred_Oempty,type,
random_empty:
!>[A: $tType] : ( ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ) ) ).
thf(sy_c_Random__Pred_Oiterate__upto,type,
random_iterate_upto:
!>[A: $tType] : ( ( code_natural > A ) > code_natural > code_natural > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ) ) ).
thf(sy_c_Random__Pred_Ounion,type,
random_union:
!>[A: $tType] : ( ( ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ) ) > ( ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ) ) > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ) ) ).
thf(sy_c_Rat_OAbs__Rat,type,
abs_Rat: ( product_prod @ int @ int ) > rat ).
thf(sy_c_Rat_OFract,type,
fract: int > int > rat ).
thf(sy_c_Rat_OFrct,type,
frct: ( product_prod @ int @ int ) > rat ).
thf(sy_c_Rat_ORep__Rat,type,
rep_Rat: rat > ( product_prod @ int @ int ) ).
thf(sy_c_Rat_Ofield__char__0__class_Oof__rat,type,
field_char_0_of_rat:
!>[A: $tType] : ( rat > A ) ).
thf(sy_c_Rat_Onormalize,type,
normalize: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ).
thf(sy_c_Rat_Oof__int,type,
of_int: int > rat ).
thf(sy_c_Rat_Opcr__rat,type,
pcr_rat: ( product_prod @ int @ int ) > rat > $o ).
thf(sy_c_Rat_Opositive,type,
positive: rat > $o ).
thf(sy_c_Rat_Oquotient__of,type,
quotient_of: rat > ( product_prod @ int @ int ) ).
thf(sy_c_Rat_Oratrel,type,
ratrel: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) > $o ).
thf(sy_c_Ref__Time_Oalloc,type,
ref_alloc:
!>[A: $tType] : ( A > ( heap_ext @ product_unit ) > ( product_prod @ ( ref @ A ) @ ( heap_ext @ product_unit ) ) ) ).
thf(sy_c_Ref__Time_Ochange,type,
ref_change:
!>[A: $tType] : ( ( A > A ) > ( ref @ A ) > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_Ref__Time_Oget,type,
ref_get:
!>[A: $tType] : ( ( heap_ext @ product_unit ) > ( ref @ A ) > A ) ).
thf(sy_c_Ref__Time_Olookup,type,
ref_lookup:
!>[A: $tType] : ( ( ref @ A ) > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_Ref__Time_Opresent,type,
ref_present:
!>[A: $tType] : ( ( heap_ext @ product_unit ) > ( ref @ A ) > $o ) ).
thf(sy_c_Ref__Time_Oref,type,
ref_ref:
!>[A: $tType] : ( A > ( heap_Time_Heap @ ( ref @ A ) ) ) ).
thf(sy_c_Ref__Time_Oset,type,
ref_set:
!>[A: $tType] : ( ( ref @ A ) > A > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) ) ).
thf(sy_c_Ref__Time_Oupdate,type,
ref_update:
!>[A: $tType] : ( ( ref @ A ) > A > ( heap_Time_Heap @ product_unit ) ) ).
thf(sy_c_Refine__Imp__Hol_Oassert_H,type,
refine_Imp_assert: $o > ( heap_Time_Heap @ product_unit ) ).
thf(sy_c_Refine__Imp__Hol_Orefines,type,
refine_Imp_refines:
!>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_Time_Heap @ A ) > $o ) ).
thf(sy_c_Relation_ODomain,type,
domain:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ A ) ) ).
thf(sy_c_Relation_ODomainp,type,
domainp:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > A > $o ) ).
thf(sy_c_Relation_OField,type,
field2:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) ) ).
thf(sy_c_Relation_OId,type,
id2:
!>[A: $tType] : ( set @ ( product_prod @ A @ A ) ) ).
thf(sy_c_Relation_OId__on,type,
id_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Relation_OImage,type,
image:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ A ) > ( set @ B ) ) ).
thf(sy_c_Relation_ORange,type,
range2:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ B ) ) ).
thf(sy_c_Relation_ORangep,type,
rangep:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > B > $o ) ).
thf(sy_c_Relation_Oantisym,type,
antisym:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Relation_Oantisymp,type,
antisymp:
!>[A: $tType] : ( ( A > A > $o ) > $o ) ).
thf(sy_c_Relation_Oasym,type,
asym:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Relation_Oasymp,type,
asymp:
!>[A: $tType] : ( ( A > A > $o ) > $o ) ).
thf(sy_c_Relation_Oconverse,type,
converse:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ B @ A ) ) ) ).
thf(sy_c_Relation_Oconversep,type,
conversep:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > B > A > $o ) ).
thf(sy_c_Relation_Oinv__image,type,
inv_image:
!>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Relation_Oirrefl,type,
irrefl:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Relation_Oirreflp,type,
irreflp:
!>[A: $tType] : ( ( A > A > $o ) > $o ) ).
thf(sy_c_Relation_Orefl__on,type,
refl_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Relation_Oreflp,type,
reflp:
!>[A: $tType] : ( ( A > A > $o ) > $o ) ).
thf(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 ) ) ) ).
thf(sy_c_Relation_Orelcompp,type,
relcompp:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( B > C > $o ) > A > C > $o ) ).
thf(sy_c_Relation_Osingle__valued,type,
single_valued:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > $o ) ).
thf(sy_c_Relation_Osingle__valuedp,type,
single_valuedp:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
thf(sy_c_Relation_Ototal__on,type,
total_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Relation_Otrans,type,
trans:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Relation_Otransp,type,
transp:
!>[A: $tType] : ( ( A > A > $o ) > $o ) ).
thf(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime,type,
algebr8660921524188924756oprime:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Rings_Odivide__class_Odivide,type,
divide_divide:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Rings_Odvd__class_Odvd,type,
dvd_dvd:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Rings_Omodulo__class_Omodulo,type,
modulo_modulo:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Rings_Onormalization__semidom__class_Onormalize,type,
normal6383669964737779283malize:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Rings_Ounit__factor__class_Ounit__factor,type,
unit_f5069060285200089521factor:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool,type,
zero_neq_one_of_bool:
!>[A: $tType] : ( $o > A ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_OPow,type,
pow:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
thf(sy_c_Set_Obind,type,
bind3:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ B ) ) ).
thf(sy_c_Set_Odisjnt,type,
disjnt:
!>[A: $tType] : ( ( set @ A ) > ( set @ A ) > $o ) ).
thf(sy_c_Set_Oimage,type,
image2:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
thf(sy_c_Set_Oinsert,type,
insert2:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Set_Ois__singleton,type,
is_singleton:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Set_Ovimage,type,
vimage:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ B ) > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat,type,
set_fo6178422350223883121st_nat:
!>[A: $tType] : ( ( nat > A > A ) > nat > nat > A > A ) ).
thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat__rel,type,
set_fo1817059534552279752at_rel:
!>[A: $tType] : ( ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) > ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) > $o ) ).
thf(sy_c_Set__Interval_Oord__class_OatLeast,type,
set_ord_atLeast:
!>[A: $tType] : ( A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost,type,
set_or1337092689740270186AtMost:
!>[A: $tType] : ( A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan,type,
set_or7035219750837199246ssThan:
!>[A: $tType] : ( A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OatMost,type,
set_ord_atMost:
!>[A: $tType] : ( A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan,type,
set_ord_greaterThan:
!>[A: $tType] : ( A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost,type,
set_or3652927894154168847AtMost:
!>[A: $tType] : ( A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan,type,
set_or5935395276787703475ssThan:
!>[A: $tType] : ( A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OlessThan,type,
set_ord_lessThan:
!>[A: $tType] : ( A > ( set @ A ) ) ).
thf(sy_c_String_Ochar_OChar,type,
char2: $o > $o > $o > $o > $o > $o > $o > $o > char ).
thf(sy_c_String_Ointeger__of__char,type,
integer_of_char: char > code_integer ).
thf(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of,type,
unique5772411509450598832har_of:
!>[A: $tType] : ( A > char ) ).
thf(sy_c_Sum__Type_OInl,type,
sum_Inl:
!>[A: $tType,B: $tType] : ( A > ( sum_sum @ A @ B ) ) ).
thf(sy_c_Sum__Type_OInr,type,
sum_Inr:
!>[B: $tType,A: $tType] : ( B > ( sum_sum @ A @ B ) ) ).
thf(sy_c_Sum__Type_OPlus,type,
sum_Plus:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( sum_sum @ A @ B ) ) ) ).
thf(sy_c_Sum__Type_Omap__sum,type,
sum_map_sum:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C ) > ( B > D ) > ( sum_sum @ A @ B ) > ( sum_sum @ C @ D ) ) ).
thf(sy_c_Sum__Type_Osum_Ocase__sum,type,
sum_case_sum:
!>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( B > C ) > ( sum_sum @ A @ B ) > C ) ).
thf(sy_c_Syntax__Match_Osyntax__fo__nomatch,type,
syntax7388354845996824322omatch:
!>[A: $tType,B: $tType] : ( A > B > $o ) ).
thf(sy_c_Time__Reasoning_Ofails,type,
time_fails:
!>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_ext @ product_unit ) > $o ) ).
thf(sy_c_Time__Reasoning_Othe__heap,type,
time_the_heap:
!>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) ) ).
thf(sy_c_Time__Reasoning_Othe__res,type,
time_the_res:
!>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_ext @ product_unit ) > A ) ).
thf(sy_c_Time__Reasoning_Otime,type,
time_time:
!>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_ext @ product_unit ) > nat ) ).
thf(sy_c_Transitive__Closure_Oacyclic,type,
transitive_acyclic:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Transitive__Closure_Ontrancl,type,
transitive_ntrancl:
!>[A: $tType] : ( nat > ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Transitive__Closure_Ortrancl,type,
transitive_rtrancl:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Transitive__Closure_Ortranclp,type,
transitive_rtranclp:
!>[A: $tType] : ( ( A > A > $o ) > A > A > $o ) ).
thf(sy_c_Transitive__Closure_Otrancl,type,
transitive_trancl:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Transitive__Closure_Otranclp,type,
transitive_tranclp:
!>[A: $tType] : ( ( A > A > $o ) > A > A > $o ) ).
thf(sy_c_Typedef_Otype__definition,type,
type_definition:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( A > B ) > ( set @ A ) > $o ) ).
thf(sy_c_Typerep_Otyperep__class_Otyperep,type,
typerep_typerep:
!>[A: $tType] : ( ( itself @ A ) > typerep ) ).
thf(sy_c_Wellfounded_Oacc,type,
acc:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) ) ).
thf(sy_c_Wellfounded_Oaccp,type,
accp:
!>[A: $tType] : ( ( A > A > $o ) > A > $o ) ).
thf(sy_c_Wellfounded_Ofinite__psubset,type,
finite_psubset:
!>[A: $tType] : ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ).
thf(sy_c_Wellfounded_Oless__than,type,
less_than: set @ ( product_prod @ nat @ nat ) ).
thf(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 ) ) ) ) ).
thf(sy_c_Wellfounded_Omax__ext,type,
max_ext:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ).
thf(sy_c_Wellfounded_Omax__extp,type,
max_extp:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > ( set @ A ) > $o ) ).
thf(sy_c_Wellfounded_Omeasure,type,
measure:
!>[A: $tType] : ( ( A > nat ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Wellfounded_Omin__ext,type,
min_ext:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ).
thf(sy_c_Wellfounded_Omlex__prod,type,
mlex_prod:
!>[A: $tType] : ( ( A > nat ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Wellfounded_Opred__nat,type,
pred_nat: set @ ( product_prod @ nat @ nat ) ).
thf(sy_c_Wellfounded_Owf,type,
wf:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Wellfounded_OwfP,type,
wfP:
!>[A: $tType] : ( ( A > A > $o ) > $o ) ).
thf(sy_c_Wfrec_Oadm__wf,type,
adm_wf:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( ( A > B ) > A > B ) > $o ) ).
thf(sy_c_Wfrec_Ocut,type,
cut:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ ( product_prod @ A @ A ) ) > A > A > B ) ).
thf(sy_c_Wfrec_Osame__fst,type,
same_fst:
!>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > ( set @ ( product_prod @ B @ B ) ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
thf(sy_c_Zorn_OChains,type,
chains:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( set @ A ) ) ) ).
thf(sy_c_Zorn_Oinit__seg__of,type,
init_seg_of:
!>[A: $tType] : ( set @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
thf(sy_c_Zorn_Opred__on_Ochain,type,
pred_chain:
!>[A: $tType] : ( ( set @ A ) > ( A > A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_fChoice,type,
fChoice:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_h,type,
h: heap_ext @ product_unit ).
thf(sy_v_xs,type,
xs: list @ a ).
% Relevant facts (5993)
thf(fact_0_fails__def,axiom,
! [A: $tType] :
( ( time_fails @ A )
= ( ^ [M: heap_Time_Heap @ A,H: heap_ext @ product_unit] :
( ( heap_Time_execute @ A @ M @ H )
= ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ).
% fails_def
thf(fact_1_Heap__eqI,axiom,
! [A: $tType,F: heap_Time_Heap @ A,G: heap_Time_Heap @ A] :
( ! [H2: heap_ext @ product_unit] :
( ( heap_Time_execute @ A @ F @ H2 )
= ( heap_Time_execute @ A @ G @ H2 ) )
=> ( F = G ) ) ).
% Heap_eqI
thf(fact_2_execute__raise,axiom,
! [A: $tType,S: list @ char] :
( ( heap_Time_execute @ A @ ( heap_Time_raise @ A @ S ) )
= ( ^ [Uu: heap_ext @ product_unit] : ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ).
% execute_raise
thf(fact_3_execute__guard_I1_J,axiom,
! [A: $tType,P: ( heap_ext @ product_unit ) > $o,H3: heap_ext @ product_unit,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
( ~ ( P @ H3 )
=> ( ( heap_Time_execute @ A @ ( heap_Time_guard @ A @ P @ F ) @ H3 )
= ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ).
% execute_guard(1)
thf(fact_4_execute__assert_I2_J,axiom,
! [A: $tType,P: A > $o,X: A,H3: heap_ext @ product_unit] :
( ~ ( P @ X )
=> ( ( heap_Time_execute @ A @ ( heap_Time_assert @ A @ P @ X ) @ H3 )
= ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ).
% execute_assert(2)
thf(fact_5_successI,axiom,
! [A: $tType,F: heap_Time_Heap @ A,H3: heap_ext @ product_unit] :
( ( ( heap_Time_execute @ A @ F @ H3 )
!= ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
=> ( heap_Time_success @ A @ F @ H3 ) ) ).
% successI
thf(fact_6_success__def,axiom,
! [A: $tType] :
( ( heap_Time_success @ A )
= ( ^ [F2: heap_Time_Heap @ A,H: heap_ext @ product_unit] :
( ( heap_Time_execute @ A @ F2 @ H )
!= ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ).
% success_def
thf(fact_7_Heap__ordE,axiom,
! [A: $tType,X: heap_Time_Heap @ A,Y: heap_Time_Heap @ A,H3: heap_ext @ product_unit] :
( ( heap_Time_Heap_ord @ A @ X @ Y )
=> ( ( ( heap_Time_execute @ A @ X @ H3 )
!= ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
=> ( ( heap_Time_execute @ A @ X @ H3 )
= ( heap_Time_execute @ A @ Y @ H3 ) ) ) ) ).
% Heap_ordE
thf(fact_8_Heap__ordI,axiom,
! [A: $tType,X: heap_Time_Heap @ A,Y: heap_Time_Heap @ A] :
( ! [H2: heap_ext @ product_unit] :
( ( ( heap_Time_execute @ A @ X @ H2 )
= ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
| ( ( heap_Time_execute @ A @ X @ H2 )
= ( heap_Time_execute @ A @ Y @ H2 ) ) )
=> ( heap_Time_Heap_ord @ A @ X @ Y ) ) ).
% Heap_ordI
thf(fact_9_execute__bind_I2_J,axiom,
! [A: $tType,B: $tType,F: heap_Time_Heap @ A,H3: heap_ext @ product_unit,G: A > ( heap_Time_Heap @ B )] :
( ( ( heap_Time_execute @ A @ F @ H3 )
= ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
=> ( ( heap_Time_execute @ B @ ( heap_Time_bind @ A @ B @ F @ G ) @ H3 )
= ( none @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ).
% execute_bind(2)
thf(fact_10_the__default_Osimps_I2_J,axiom,
! [A: $tType,X: A] :
( ( the_default @ A @ X @ ( none @ A ) )
= X ) ).
% the_default.simps(2)
thf(fact_11_raise__bind,axiom,
! [B: $tType,A: $tType,E2: list @ char,F: B > ( heap_Time_Heap @ A )] :
( ( heap_Time_bind @ B @ A @ ( heap_Time_raise @ B @ E2 ) @ F )
= ( heap_Time_raise @ A @ E2 ) ) ).
% raise_bind
thf(fact_12_heap_Oleq__refl,axiom,
! [A: $tType,X: heap_Time_Heap @ A] : ( heap_Time_Heap_ord @ A @ X @ X ) ).
% heap.leq_refl
thf(fact_13_heap_Oleq__trans,axiom,
! [A: $tType,X: heap_Time_Heap @ A,Y: heap_Time_Heap @ A,Z2: heap_Time_Heap @ A] :
( ( heap_Time_Heap_ord @ A @ X @ Y )
=> ( ( heap_Time_Heap_ord @ A @ Y @ Z2 )
=> ( heap_Time_Heap_ord @ A @ X @ Z2 ) ) ) ).
% heap.leq_trans
thf(fact_14_heap_Oleq__antisym,axiom,
! [A: $tType,X: heap_Time_Heap @ A,Y: heap_Time_Heap @ A] :
( ( heap_Time_Heap_ord @ A @ X @ Y )
=> ( ( heap_Time_Heap_ord @ A @ Y @ X )
=> ( X = Y ) ) ) ).
% heap.leq_antisym
thf(fact_15_assert__cong,axiom,
! [B: $tType,A: $tType,P: A > $o,P2: A > $o,F: A > ( heap_Time_Heap @ B ),F3: A > ( heap_Time_Heap @ B ),X: A] :
( ( P = P2 )
=> ( ! [X2: A] :
( ( P2 @ X2 )
=> ( ( F @ X2 )
= ( F3 @ X2 ) ) )
=> ( ( heap_Time_bind @ A @ B @ ( heap_Time_assert @ A @ P @ X ) @ F )
= ( heap_Time_bind @ A @ B @ ( heap_Time_assert @ A @ P2 @ X ) @ F3 ) ) ) ) ).
% assert_cong
thf(fact_16_success__ifI,axiom,
! [A: $tType,C2: $o,T2: heap_Time_Heap @ A,H3: heap_ext @ product_unit,E2: heap_Time_Heap @ A] :
( ( C2
=> ( heap_Time_success @ A @ T2 @ H3 ) )
=> ( ( ~ C2
=> ( heap_Time_success @ A @ E2 @ H3 ) )
=> ( heap_Time_success @ A @ ( if @ ( heap_Time_Heap @ A ) @ C2 @ T2 @ E2 ) @ H3 ) ) ) ).
% success_ifI
thf(fact_17_success__guardI,axiom,
! [A: $tType,P: ( heap_ext @ product_unit ) > $o,H3: heap_ext @ product_unit,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
( ( P @ H3 )
=> ( heap_Time_success @ A @ ( heap_Time_guard @ A @ P @ F ) @ H3 ) ) ).
% success_guardI
thf(fact_18_success__assertI,axiom,
! [A: $tType,P: A > $o,X: A,H3: heap_ext @ product_unit] :
( ( P @ X )
=> ( heap_Time_success @ A @ ( heap_Time_assert @ A @ P @ X ) @ H3 ) ) ).
% success_assertI
thf(fact_19_fails__bind,axiom,
! [A: $tType,B: $tType,M2: heap_Time_Heap @ B,F: B > ( heap_Time_Heap @ A ),H3: heap_ext @ product_unit] :
( ( time_fails @ A @ ( heap_Time_bind @ B @ A @ M2 @ F ) @ H3 )
= ( ~ ( time_fails @ B @ M2 @ H3 )
=> ( time_fails @ A @ ( F @ ( time_the_res @ B @ M2 @ H3 ) ) @ ( time_the_heap @ B @ M2 @ H3 ) ) ) ) ).
% fails_bind
thf(fact_20_success__of__listI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [Xs: list @ A,H3: heap_ext @ product_unit] : ( heap_Time_success @ ( array @ A ) @ ( array_of_list @ A @ Xs ) @ H3 ) ) ).
% success_of_listI
thf(fact_21_success__tapI,axiom,
! [A: $tType,F: ( heap_ext @ product_unit ) > A,H3: heap_ext @ product_unit] : ( heap_Time_success @ A @ ( heap_Time_tap @ A @ F ) @ H3 ) ).
% success_tapI
thf(fact_22_success__heapI,axiom,
! [A: $tType,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ),H3: heap_ext @ product_unit] : ( heap_Time_success @ A @ ( heap_Time_heap @ A @ F ) @ H3 ) ).
% success_heapI
thf(fact_23_success__ureturnI,axiom,
! [A: $tType,X: A,H3: heap_ext @ product_unit] : ( heap_Time_success @ A @ ( heap_Time_ureturn @ A @ X ) @ H3 ) ).
% success_ureturnI
thf(fact_24_bind__ureturn,axiom,
! [A: $tType,F: heap_Time_Heap @ A] :
( ( heap_Time_bind @ A @ A @ F @ ( heap_Time_ureturn @ A ) )
= F ) ).
% bind_ureturn
thf(fact_25_ureturn__bind,axiom,
! [A: $tType,B: $tType,X: B,F: B > ( heap_Time_Heap @ A )] :
( ( heap_Time_bind @ B @ A @ ( heap_Time_ureturn @ B @ X ) @ F )
= ( F @ X ) ) ).
% ureturn_bind
thf(fact_26_option_Osimps_I6_J,axiom,
! [A: $tType,C: $tType,F1: C,F22: A > C] :
( ( rec_option @ C @ A @ F1 @ F22 @ ( none @ A ) )
= F1 ) ).
% option.simps(6)
thf(fact_27_success__bind__effectI,axiom,
! [A: $tType,B: $tType,F: heap_Time_Heap @ A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,X: A,N: nat,G: A > ( heap_Time_Heap @ B )] :
( ( heap_Time_effect @ A @ F @ H3 @ H4 @ X @ N )
=> ( ( heap_Time_success @ B @ ( G @ X ) @ H4 )
=> ( heap_Time_success @ B @ ( heap_Time_bind @ A @ B @ F @ G ) @ H3 ) ) ) ).
% success_bind_effectI
thf(fact_28_execute__guard_I2_J,axiom,
! [A: $tType,P: ( heap_ext @ product_unit ) > $o,H3: heap_ext @ product_unit,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
( ( P @ H3 )
=> ( ( heap_Time_execute @ A @ ( heap_Time_guard @ A @ P @ F ) @ H3 )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( F @ H3 ) ) ) ) ).
% execute_guard(2)
thf(fact_29_combine__options__simps_I1_J,axiom,
! [A: $tType,F: A > A > A,Y: option @ A] :
( ( combine_options @ A @ F @ ( none @ A ) @ Y )
= Y ) ).
% combine_options_simps(1)
thf(fact_30_combine__options__simps_I2_J,axiom,
! [A: $tType,F: A > A > A,X: option @ A] :
( ( combine_options @ A @ F @ X @ ( none @ A ) )
= X ) ).
% combine_options_simps(2)
thf(fact_31_option_Oinject,axiom,
! [A: $tType,X22: A,Y2: A] :
( ( ( some @ A @ X22 )
= ( some @ A @ Y2 ) )
= ( X22 = Y2 ) ) ).
% option.inject
thf(fact_32_not__None__eq,axiom,
! [A: $tType,X: option @ A] :
( ( X
!= ( none @ A ) )
= ( ? [Y3: A] :
( X
= ( some @ A @ Y3 ) ) ) ) ).
% not_None_eq
thf(fact_33_not__Some__eq,axiom,
! [A: $tType,X: option @ A] :
( ( ! [Y3: A] :
( X
!= ( some @ A @ Y3 ) ) )
= ( X
= ( none @ A ) ) ) ).
% not_Some_eq
thf(fact_34_combine__options__simps_I3_J,axiom,
! [A: $tType,F: A > A > A,A3: A,B2: A] :
( ( combine_options @ A @ F @ ( some @ A @ A3 ) @ ( some @ A @ B2 ) )
= ( some @ A @ ( F @ A3 @ B2 ) ) ) ).
% combine_options_simps(3)
thf(fact_35_option_Osimps_I7_J,axiom,
! [C: $tType,A: $tType,F1: C,F22: A > C,X22: A] :
( ( rec_option @ C @ A @ F1 @ F22 @ ( some @ A @ X22 ) )
= ( F22 @ X22 ) ) ).
% option.simps(7)
thf(fact_36_effect__deterministic_I3_J,axiom,
! [A: $tType,F: heap_Time_Heap @ A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,A3: A,N: nat,H5: heap_ext @ product_unit,B2: A,N2: nat] :
( ( heap_Time_effect @ A @ F @ H3 @ H4 @ A3 @ N )
=> ( ( heap_Time_effect @ A @ F @ H3 @ H5 @ B2 @ N2 )
=> ( N = N2 ) ) ) ).
% effect_deterministic(3)
thf(fact_37_effect__deterministic_I2_J,axiom,
! [A: $tType,F: heap_Time_Heap @ A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,A3: A,N: nat,H5: heap_ext @ product_unit,B2: A,N2: nat] :
( ( heap_Time_effect @ A @ F @ H3 @ H4 @ A3 @ N )
=> ( ( heap_Time_effect @ A @ F @ H3 @ H5 @ B2 @ N2 )
=> ( H4 = H5 ) ) ) ).
% effect_deterministic(2)
thf(fact_38_effect__deterministic_I1_J,axiom,
! [A: $tType,F: heap_Time_Heap @ A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,A3: A,N: nat,H5: heap_ext @ product_unit,B2: A,N2: nat] :
( ( heap_Time_effect @ A @ F @ H3 @ H4 @ A3 @ N )
=> ( ( heap_Time_effect @ A @ F @ H3 @ H5 @ B2 @ N2 )
=> ( A3 = B2 ) ) ) ).
% effect_deterministic(1)
thf(fact_39_effect__ifE,axiom,
! [A: $tType,C2: $o,T2: heap_Time_Heap @ A,E2: heap_Time_Heap @ A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A,N: nat] :
( ( heap_Time_effect @ A @ ( if @ ( heap_Time_Heap @ A ) @ C2 @ T2 @ E2 ) @ H3 @ H4 @ R @ N )
=> ( ( C2
=> ~ ( heap_Time_effect @ A @ T2 @ H3 @ H4 @ R @ N ) )
=> ~ ( ~ C2
=> ~ ( heap_Time_effect @ A @ E2 @ H3 @ H4 @ R @ N ) ) ) ) ).
% effect_ifE
thf(fact_40_effect__ifI,axiom,
! [A: $tType,C2: $o,T2: heap_Time_Heap @ A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A,N: nat,E2: heap_Time_Heap @ A] :
( ( C2
=> ( heap_Time_effect @ A @ T2 @ H3 @ H4 @ R @ N ) )
=> ( ( ~ C2
=> ( heap_Time_effect @ A @ E2 @ H3 @ H4 @ R @ N ) )
=> ( heap_Time_effect @ A @ ( if @ ( heap_Time_Heap @ A ) @ C2 @ T2 @ E2 ) @ H3 @ H4 @ R @ N ) ) ) ).
% effect_ifI
thf(fact_41_combine__options__assoc,axiom,
! [A: $tType,F: A > A > A,X: option @ A,Y: option @ A,Z2: option @ A] :
( ! [X2: A,Y4: A,Z3: A] :
( ( F @ ( F @ X2 @ Y4 ) @ Z3 )
= ( F @ X2 @ ( F @ Y4 @ Z3 ) ) )
=> ( ( combine_options @ A @ F @ ( combine_options @ A @ F @ X @ Y ) @ Z2 )
= ( combine_options @ A @ F @ X @ ( combine_options @ A @ F @ Y @ Z2 ) ) ) ) ).
% combine_options_assoc
thf(fact_42_combine__options__commute,axiom,
! [A: $tType,F: A > A > A,X: option @ A,Y: option @ A] :
( ! [X2: A,Y4: A] :
( ( F @ X2 @ Y4 )
= ( F @ Y4 @ X2 ) )
=> ( ( combine_options @ A @ F @ X @ Y )
= ( combine_options @ A @ F @ Y @ X ) ) ) ).
% combine_options_commute
thf(fact_43_combine__options__left__commute,axiom,
! [A: $tType,F: A > A > A,Y: option @ A,X: option @ A,Z2: option @ A] :
( ! [X2: A,Y4: A] :
( ( F @ X2 @ Y4 )
= ( F @ Y4 @ X2 ) )
=> ( ! [X2: A,Y4: A,Z3: A] :
( ( F @ ( F @ X2 @ Y4 ) @ Z3 )
= ( F @ X2 @ ( F @ Y4 @ Z3 ) ) )
=> ( ( combine_options @ A @ F @ Y @ ( combine_options @ A @ F @ X @ Z2 ) )
= ( combine_options @ A @ F @ X @ ( combine_options @ A @ F @ Y @ Z2 ) ) ) ) ) ).
% combine_options_left_commute
thf(fact_44_mem__Collect__eq,axiom,
! [A: $tType,A3: A,P: A > $o] :
( ( member @ A @ A3 @ ( collect @ A @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( collect @ A
@ ^ [X3: A] : ( member @ A @ X3 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_46_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X2: A] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_47_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X2: A] :
( ( F @ X2 )
= ( G @ X2 ) )
=> ( F = G ) ) ).
% ext
thf(fact_48_option_Odistinct_I1_J,axiom,
! [A: $tType,X22: A] :
( ( none @ A )
!= ( some @ A @ X22 ) ) ).
% option.distinct(1)
thf(fact_49_option_OdiscI,axiom,
! [A: $tType,Option: option @ A,X22: A] :
( ( Option
= ( some @ A @ X22 ) )
=> ( Option
!= ( none @ A ) ) ) ).
% option.discI
thf(fact_50_option_Oexhaust,axiom,
! [A: $tType,Y: option @ A] :
( ( Y
!= ( none @ A ) )
=> ~ ! [X23: A] :
( Y
!= ( some @ A @ X23 ) ) ) ).
% option.exhaust
thf(fact_51_split__option__ex,axiom,
! [A: $tType] :
( ( ^ [P3: ( option @ A ) > $o] :
? [X4: option @ A] : ( P3 @ X4 ) )
= ( ^ [P4: ( option @ A ) > $o] :
( ( P4 @ ( none @ A ) )
| ? [X3: A] : ( P4 @ ( some @ A @ X3 ) ) ) ) ) ).
% split_option_ex
thf(fact_52_split__option__all,axiom,
! [A: $tType] :
( ( ^ [P3: ( option @ A ) > $o] :
! [X4: option @ A] : ( P3 @ X4 ) )
= ( ^ [P4: ( option @ A ) > $o] :
( ( P4 @ ( none @ A ) )
& ! [X3: A] : ( P4 @ ( some @ A @ X3 ) ) ) ) ) ).
% split_option_all
thf(fact_53_combine__options__cases,axiom,
! [A: $tType,B: $tType,X: option @ A,P: ( option @ A ) > ( option @ B ) > $o,Y: option @ B] :
( ( ( X
= ( none @ A ) )
=> ( P @ X @ Y ) )
=> ( ( ( Y
= ( none @ B ) )
=> ( P @ X @ Y ) )
=> ( ! [A5: A,B3: B] :
( ( X
= ( some @ A @ A5 ) )
=> ( ( Y
= ( some @ B @ B3 ) )
=> ( P @ X @ Y ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_54_success__effectE,axiom,
! [A: $tType,C2: heap_Time_Heap @ A,H3: heap_ext @ product_unit] :
( ( heap_Time_success @ A @ C2 @ H3 )
=> ~ ! [R2: A,H6: heap_ext @ product_unit,N3: nat] :
~ ( heap_Time_effect @ A @ C2 @ H3 @ H6 @ R2 @ N3 ) ) ).
% success_effectE
thf(fact_55_effect__success,axiom,
! [A: $tType,C2: heap_Time_Heap @ A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A,N: nat] :
( ( heap_Time_effect @ A @ C2 @ H3 @ H4 @ R @ N )
=> ( heap_Time_success @ A @ C2 @ H3 ) ) ).
% effect_success
thf(fact_56_effect__raiseE,axiom,
! [A: $tType,X: list @ char,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A,N: nat] :
~ ( heap_Time_effect @ A @ ( heap_Time_raise @ A @ X ) @ H3 @ H4 @ R @ N ) ).
% effect_raiseE
thf(fact_57_the__default_Osimps_I1_J,axiom,
! [A: $tType,Uu2: A,X: A] :
( ( the_default @ A @ Uu2 @ ( some @ A @ X ) )
= X ) ).
% the_default.simps(1)
thf(fact_58_Heap__cases,axiom,
! [A: $tType,F: heap_Time_Heap @ A,H3: heap_ext @ product_unit] :
( ! [X2: A,H6: product_prod @ ( heap_ext @ product_unit ) @ nat] :
( ( heap_Time_execute @ A @ F @ H3 )
!= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X2 @ H6 ) ) )
=> ( ( heap_Time_execute @ A @ F @ H3 )
= ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ).
% Heap_cases
thf(fact_59_successE,axiom,
! [A: $tType,F: heap_Time_Heap @ A,H3: heap_ext @ product_unit] :
( ( heap_Time_success @ A @ F @ H3 )
=> ~ ! [R2: A,H6: product_prod @ ( heap_ext @ product_unit ) @ nat] :
( ( heap_Time_execute @ A @ F @ H3 )
!= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R2 @ H6 ) ) ) ) ).
% successE
thf(fact_60_execute__heap,axiom,
! [A: $tType,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
( ( heap_Time_execute @ A @ ( heap_Time_heap @ A @ F ) )
= ( 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
thf(fact_61_some__opt__sym__eq__trivial,axiom,
! [A: $tType,X: A] :
( ( eps_Opt @ A
@ ( ^ [Y5: A,Z4: A] : Y5 = Z4
@ X ) )
= ( some @ A @ X ) ) ).
% some_opt_sym_eq_trivial
thf(fact_62_success__newI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [N: nat,X: A,H3: heap_ext @ product_unit] : ( heap_Time_success @ ( array @ A ) @ ( array_new @ A @ N @ X ) @ H3 ) ) ).
% success_newI
thf(fact_63_success__makeI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [N: nat,F: nat > A,H3: heap_ext @ product_unit] : ( heap_Time_success @ ( array @ A ) @ ( array_make @ A @ N @ F ) @ H3 ) ) ).
% success_makeI
thf(fact_64_mmupd__in__upd,axiom,
! [A: $tType,B: $tType,K: A,K2: set @ A,M2: A > ( option @ B ),V: B] :
( ( member @ A @ K @ K2 )
=> ( ( map_mmupd @ A @ B @ M2 @ K2 @ V @ K )
= ( some @ B @ V ) ) ) ).
% mmupd_in_upd
thf(fact_65_option_Ocollapse,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
!= ( none @ A ) )
=> ( ( some @ A @ ( the2 @ A @ Option ) )
= Option ) ) ).
% option.collapse
thf(fact_66_not__Some__eq2,axiom,
! [B: $tType,A: $tType,V: option @ ( product_prod @ A @ B )] :
( ( ! [X3: A,Y3: B] :
( V
!= ( some @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) ) ) )
= ( V
= ( none @ ( product_prod @ A @ B ) ) ) ) ).
% not_Some_eq2
thf(fact_67_effect__tapE,axiom,
! [A: $tType,F: ( heap_ext @ product_unit ) > A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A,N: nat] :
( ( heap_Time_effect @ A @ ( heap_Time_tap @ A @ F ) @ H3 @ H4 @ R @ N )
=> ~ ( ( H4 = H3 )
=> ( ( R
= ( F @ H3 ) )
=> ( N
!= ( one_one @ nat ) ) ) ) ) ).
% effect_tapE
thf(fact_68_effect__tapI,axiom,
! [A: $tType,H4: heap_ext @ product_unit,H3: heap_ext @ product_unit,R: A,F: ( heap_ext @ product_unit ) > A] :
( ( H4 = H3 )
=> ( ( R
= ( F @ H3 ) )
=> ( heap_Time_effect @ A @ ( heap_Time_tap @ A @ F ) @ H3 @ H4 @ R @ ( one_one @ nat ) ) ) ) ).
% effect_tapI
thf(fact_69_effect__assertI,axiom,
! [A: $tType,P: A > $o,X: A,H4: heap_ext @ product_unit,H3: heap_ext @ product_unit,R: A,N: nat] :
( ( P @ X )
=> ( ( H4 = H3 )
=> ( ( R = X )
=> ( ( N
= ( one_one @ nat ) )
=> ( heap_Time_effect @ A @ ( heap_Time_assert @ A @ P @ X ) @ H3 @ H4 @ R @ N ) ) ) ) ) ).
% effect_assertI
thf(fact_70_mmupd__notin__upd,axiom,
! [B: $tType,A: $tType,K: A,K2: set @ A,M2: A > ( option @ B ),V: B] :
( ~ ( member @ A @ K @ K2 )
=> ( ( map_mmupd @ A @ B @ M2 @ K2 @ V @ K )
= ( M2 @ K ) ) ) ).
% mmupd_notin_upd
thf(fact_71_fun__comp__eq__conv,axiom,
! [B: $tType,C: $tType,A: $tType,F: C > B,G: A > C,Fg: A > B] :
( ( ( comp @ C @ B @ A @ F @ G )
= Fg )
= ( ! [X3: A] :
( ( F @ ( G @ X3 ) )
= ( Fg @ X3 ) ) ) ) ).
% fun_comp_eq_conv
thf(fact_72_comp__cong__right,axiom,
! [C: $tType,B: $tType,A: $tType,X: A > B,Y: A > B,F: B > C] :
( ( X = Y )
=> ( ( comp @ B @ C @ A @ F @ X )
= ( comp @ B @ C @ A @ F @ Y ) ) ) ).
% comp_cong_right
thf(fact_73_comp__cong__left,axiom,
! [B: $tType,A: $tType,C: $tType,X: A > B,Y: A > B,F: C > A] :
( ( X = Y )
=> ( ( comp @ A @ B @ C @ X @ F )
= ( comp @ A @ B @ C @ Y @ F ) ) ) ).
% comp_cong_left
thf(fact_74_bex2I,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,S2: set @ ( product_prod @ A @ B ),P: A > B > $o] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ S2 )
=> ( ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ S2 )
=> ( P @ A3 @ B2 ) )
=> ? [A5: A,B3: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B3 ) @ S2 )
& ( P @ A5 @ B3 ) ) ) ) ).
% bex2I
thf(fact_75_pairself_Ocases,axiom,
! [B: $tType,A: $tType,X: product_prod @ ( A > B ) @ ( product_prod @ A @ A )] :
~ ! [F4: A > B,A5: A,B3: A] :
( X
!= ( product_Pair @ ( A > B ) @ ( product_prod @ A @ A ) @ F4 @ ( product_Pair @ A @ A @ A5 @ B3 ) ) ) ).
% pairself.cases
thf(fact_76_fails__array__new,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [N: nat,X: A,H3: heap_ext @ product_unit] :
~ ( time_fails @ ( array @ A ) @ ( array_new @ A @ N @ X ) @ H3 ) ) ).
% fails_array_new
thf(fact_77_timeFrame_Ocases,axiom,
! [A: $tType,X: product_prod @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
( ! [N3: nat,R2: A,H2: heap_ext @ product_unit,N4: nat] :
( X
!= ( product_Pair @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ N3 @ ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H2 @ N4 ) ) ) ) )
=> ~ ! [N3: nat] :
( X
!= ( 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
thf(fact_78_execute__assert_I1_J,axiom,
! [A: $tType,P: A > $o,X: A,H3: heap_ext @ product_unit] :
( ( P @ X )
=> ( ( heap_Time_execute @ A @ ( heap_Time_assert @ A @ P @ X ) @ H3 )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ ( one_one @ nat ) ) ) ) ) ) ).
% execute_assert(1)
thf(fact_79_execute__tap,axiom,
! [A: $tType,F: ( heap_ext @ product_unit ) > A,H3: heap_ext @ product_unit] :
( ( heap_Time_execute @ A @ ( heap_Time_tap @ A @ F ) @ H3 )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H3 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ ( one_one @ nat ) ) ) ) ) ).
% execute_tap
thf(fact_80_option_Osel,axiom,
! [A: $tType,X22: A] :
( ( the2 @ A @ ( some @ A @ X22 ) )
= X22 ) ).
% option.sel
thf(fact_81_option_Oexpand,axiom,
! [A: $tType,Option: option @ A,Option2: option @ A] :
( ( ( Option
= ( none @ A ) )
= ( Option2
= ( none @ A ) ) )
=> ( ( ( Option
!= ( none @ A ) )
=> ( ( Option2
!= ( none @ A ) )
=> ( ( the2 @ A @ Option )
= ( the2 @ A @ Option2 ) ) ) )
=> ( Option = Option2 ) ) ) ).
% option.expand
thf(fact_82_effect__def,axiom,
! [A: $tType] :
( ( heap_Time_effect @ A )
= ( ^ [C3: heap_Time_Heap @ A,H: heap_ext @ product_unit,H7: heap_ext @ product_unit,R3: A,N5: nat] :
( ( heap_Time_execute @ A @ C3 @ H )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H7 @ N5 ) ) ) ) ) ) ).
% effect_def
thf(fact_83_effectI,axiom,
! [A: $tType,C2: heap_Time_Heap @ A,H3: heap_ext @ product_unit,R: A,H4: heap_ext @ product_unit,N: nat] :
( ( ( heap_Time_execute @ A @ C2 @ H3 )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H4 @ N ) ) ) )
=> ( heap_Time_effect @ A @ C2 @ H3 @ H4 @ R @ N ) ) ).
% effectI
thf(fact_84_map__mmupd__def,axiom,
! [A: $tType,B: $tType] :
( ( map_mmupd @ B @ A )
= ( ^ [M: B > ( option @ A ),K3: set @ B,V2: A,K4: B] : ( if @ ( option @ A ) @ ( member @ B @ K4 @ K3 ) @ ( some @ A @ V2 ) @ ( M @ K4 ) ) ) ) ).
% map_mmupd_def
thf(fact_85_map__mmupdE,axiom,
! [B: $tType,A: $tType,M2: B > ( option @ A ),K2: set @ B,V: A,K: B,X: A] :
( ( ( map_mmupd @ B @ A @ M2 @ K2 @ V @ K )
= ( some @ A @ X ) )
=> ( ( ~ ( member @ B @ K @ K2 )
=> ( ( M2 @ K )
!= ( some @ A @ X ) ) )
=> ~ ( ( member @ B @ K @ K2 )
=> ( X != V ) ) ) ) ).
% map_mmupdE
thf(fact_86_success__bind__executeI,axiom,
! [A: $tType,B: $tType,F: heap_Time_Heap @ A,H3: heap_ext @ product_unit,X: A,H4: heap_ext @ product_unit,N: nat,G: A > ( heap_Time_Heap @ B )] :
( ( ( heap_Time_execute @ A @ F @ H3 )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H4 @ N ) ) ) )
=> ( ( heap_Time_success @ B @ ( G @ X ) @ H4 )
=> ( heap_Time_success @ B @ ( heap_Time_bind @ A @ B @ F @ G ) @ H3 ) ) ) ).
% success_bind_executeI
thf(fact_87_Eps__Opt__eq__Some__implies,axiom,
! [A: $tType,P: A > $o,X: A] :
( ( ( eps_Opt @ A @ P )
= ( some @ A @ X ) )
=> ( P @ X ) ) ).
% Eps_Opt_eq_Some_implies
thf(fact_88_Eps__Opt__eq__Some,axiom,
! [A: $tType,P: A > $o,X: A] :
( ! [X5: A] :
( ( P @ X )
=> ( ( P @ X5 )
=> ( X5 = X ) ) )
=> ( ( ( eps_Opt @ A @ P )
= ( some @ A @ X ) )
= ( P @ X ) ) ) ).
% Eps_Opt_eq_Some
thf(fact_89_option_Oexhaust__sel,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
!= ( none @ A ) )
=> ( Option
= ( some @ A @ ( the2 @ A @ Option ) ) ) ) ).
% option.exhaust_sel
thf(fact_90_effect__assertE,axiom,
! [A: $tType,P: A > $o,X: A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A,N: nat] :
( ( heap_Time_effect @ A @ ( heap_Time_assert @ A @ P @ X ) @ H3 @ H4 @ R @ N )
=> ~ ( ( P @ X )
=> ( ( R = X )
=> ( ( H4 = H3 )
=> ( N
!= ( one_one @ nat ) ) ) ) ) ) ).
% effect_assertE
thf(fact_91_comp__apply,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comp @ B @ A @ C )
= ( ^ [F2: B > A,G2: C > B,X3: C] : ( F2 @ ( G2 @ X3 ) ) ) ) ).
% comp_apply
thf(fact_92_prod_Oinject,axiom,
! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y2: B] :
( ( ( product_Pair @ A @ B @ X1 @ X22 )
= ( product_Pair @ A @ B @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_93_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,A6: A,B4: B] :
( ( ( product_Pair @ A @ B @ A3 @ B2 )
= ( product_Pair @ A @ B @ A6 @ B4 ) )
= ( ( A3 = A6 )
& ( B2 = B4 ) ) ) ).
% old.prod.inject
thf(fact_94_execute__bind_I1_J,axiom,
! [A: $tType,B: $tType,F: heap_Time_Heap @ A,H3: heap_ext @ product_unit,X: A,H4: heap_ext @ product_unit,N: nat,G: A > ( heap_Time_Heap @ B )] :
( ( ( heap_Time_execute @ A @ F @ H3 )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H4 @ N ) ) ) )
=> ( ( heap_Time_execute @ B @ ( heap_Time_bind @ A @ B @ F @ G ) @ H3 )
= ( heap_Time_timeFrame @ B @ N @ ( heap_Time_execute @ B @ ( G @ X ) @ H4 ) ) ) ) ).
% execute_bind(1)
thf(fact_95_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A5: A,B3: B,C4: C] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B3 @ C4 ) ) ) ).
% prod_cases3
thf(fact_96_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
~ ! [A5: A,B3: B,C4: C,D2: D] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B3 @ ( product_Pair @ C @ D @ C4 @ D2 ) ) ) ) ).
% prod_cases4
thf(fact_97_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
~ ! [A5: A,B3: B,C4: C,D2: D,E3: E] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C4 @ ( product_Pair @ D @ E @ D2 @ E3 ) ) ) ) ) ).
% prod_cases5
thf(fact_98_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F5: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F5 ) ) ) )] :
~ ! [A5: A,B3: B,C4: C,D2: D,E3: E,F4: F5] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F5 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F5 ) ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F5 ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ F5 ) @ D2 @ ( product_Pair @ E @ F5 @ E3 @ F4 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_99_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F5: $tType,G3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) ) )] :
~ ! [A5: A,B3: B,C4: C,D2: D,E3: E,F4: F5,G4: G3] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F5 @ G3 ) @ E3 @ ( product_Pair @ F5 @ G3 @ F4 @ G4 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_100_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A5: A,B3: B,C4: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B3 @ C4 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_101_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
( ! [A5: A,B3: B,C4: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B3 @ ( product_Pair @ C @ D @ C4 @ D2 ) ) ) )
=> ( P @ X ) ) ).
% prod_induct4
thf(fact_102_timeFrame_Osimps_I2_J,axiom,
! [A: $tType,N: nat] :
( ( heap_Time_timeFrame @ A @ N @ ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
= ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ).
% timeFrame.simps(2)
thf(fact_103_Pair__inject,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,A6: A,B4: B] :
( ( ( product_Pair @ A @ B @ A3 @ B2 )
= ( product_Pair @ A @ B @ A6 @ B4 ) )
=> ~ ( ( A3 = A6 )
=> ( B2 != B4 ) ) ) ).
% Pair_inject
thf(fact_104_prod__cases,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P5: product_prod @ A @ B] :
( ! [A5: A,B3: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B3 ) )
=> ( P @ P5 ) ) ).
% prod_cases
thf(fact_105_surj__pair,axiom,
! [A: $tType,B: $tType,P5: product_prod @ A @ B] :
? [X2: A,Y4: B] :
( P5
= ( product_Pair @ A @ B @ X2 @ Y4 ) ) ).
% surj_pair
thf(fact_106_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
~ ! [A5: A,B3: B] :
( Y
!= ( product_Pair @ A @ B @ A5 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_107_one__reorient,axiom,
! [A: $tType] :
( ( one @ A )
=> ! [X: A] :
( ( ( one_one @ A )
= X )
= ( X
= ( one_one @ A ) ) ) ) ).
% one_reorient
thf(fact_108_comp__eq__dest__lhs,axiom,
! [C: $tType,B: $tType,A: $tType,A3: C > B,B2: A > C,C2: A > B,V: A] :
( ( ( comp @ C @ B @ A @ A3 @ B2 )
= C2 )
=> ( ( A3 @ ( B2 @ V ) )
= ( C2 @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_109_comp__eq__elim,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,A3: C > B,B2: A > C,C2: D > B,D3: A > D] :
( ( ( comp @ C @ B @ A @ A3 @ B2 )
= ( comp @ D @ B @ A @ C2 @ D3 ) )
=> ! [V3: A] :
( ( A3 @ ( B2 @ V3 ) )
= ( C2 @ ( D3 @ V3 ) ) ) ) ).
% comp_eq_elim
thf(fact_110_comp__eq__dest,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,A3: C > B,B2: A > C,C2: D > B,D3: A > D,V: A] :
( ( ( comp @ C @ B @ A @ A3 @ B2 )
= ( comp @ D @ B @ A @ C2 @ D3 ) )
=> ( ( A3 @ ( B2 @ V ) )
= ( C2 @ ( D3 @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_111_comp__assoc,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType,F: D > B,G: C > D,H3: A > C] :
( ( comp @ C @ B @ A @ ( comp @ D @ B @ C @ F @ G ) @ H3 )
= ( comp @ D @ B @ A @ F @ ( comp @ C @ D @ A @ G @ H3 ) ) ) ).
% comp_assoc
thf(fact_112_comp__def,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comp @ B @ C @ A )
= ( ^ [F2: B > C,G2: A > B,X3: A] : ( F2 @ ( G2 @ X3 ) ) ) ) ).
% comp_def
thf(fact_113_prod__induct7,axiom,
! [G3: $tType,F5: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) ) )] :
( ! [A5: A,B3: B,C4: C,D2: D,E3: E,F4: F5,G4: G3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F5 @ G3 ) @ E3 @ ( product_Pair @ F5 @ G3 @ F4 @ G4 ) ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct7
thf(fact_114_prod__induct6,axiom,
! [F5: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F5 ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F5 ) ) ) )] :
( ! [A5: A,B3: B,C4: C,D2: D,E3: E,F4: F5] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F5 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F5 ) ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F5 ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ F5 ) @ D2 @ ( product_Pair @ E @ F5 @ E3 @ F4 ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct6
thf(fact_115_prod__induct5,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
( ! [A5: A,B3: B,C4: C,D2: D,E3: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C4 @ ( product_Pair @ D @ E @ D2 @ E3 ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct5
thf(fact_116_old_Oprod_Orec,axiom,
! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A3: A,B2: B] :
( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A3 @ B2 ) )
= ( F1 @ A3 @ B2 ) ) ).
% old.prod.rec
thf(fact_117_timeFrame_Oelims,axiom,
! [A: $tType,X: nat,Xa: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ),Y: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
( ( ( heap_Time_timeFrame @ A @ X @ Xa )
= Y )
=> ( ! [R2: A,H2: heap_ext @ product_unit,N4: nat] :
( ( Xa
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H2 @ N4 ) ) ) )
=> ( Y
!= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H2 @ ( plus_plus @ nat @ X @ N4 ) ) ) ) ) )
=> ~ ( ( Xa
= ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
=> ( Y
!= ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ) ).
% timeFrame.elims
thf(fact_118_execute__bind__eq__SomeI,axiom,
! [A: $tType,B: $tType,F: heap_Time_Heap @ A,H3: heap_ext @ product_unit,X: A,H4: heap_ext @ product_unit,N: nat,G: A > ( heap_Time_Heap @ B ),Y: B,H5: heap_ext @ product_unit,N2: nat] :
( ( ( heap_Time_execute @ A @ F @ H3 )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H4 @ N ) ) ) )
=> ( ( ( heap_Time_execute @ B @ ( G @ X ) @ H4 )
= ( some @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ Y @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H5 @ N2 ) ) ) )
=> ( ( heap_Time_execute @ B @ ( heap_Time_bind @ A @ B @ F @ G ) @ H3 )
= ( some @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ Y @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H5 @ ( plus_plus @ nat @ N @ N2 ) ) ) ) ) ) ) ).
% execute_bind_eq_SomeI
thf(fact_119_uncurry__apply,axiom,
! [B: $tType,A: $tType,C: $tType,F: B > C > A,A3: B,B2: C] :
( ( uncurry @ B @ C @ A @ F @ ( product_Pair @ B @ C @ A3 @ B2 ) )
= ( F @ A3 @ B2 ) ) ).
% uncurry_apply
thf(fact_120_timeFrame_Osimps_I1_J,axiom,
! [A: $tType,N: nat,R: A,H3: heap_ext @ product_unit,N2: nat] :
( ( heap_Time_timeFrame @ A @ N @ ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ N2 ) ) ) )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ ( plus_plus @ nat @ N @ N2 ) ) ) ) ) ).
% timeFrame.simps(1)
thf(fact_121_heap__def,axiom,
! [A: $tType] :
( ( heap_Time_heap @ A )
= ( ^ [F2: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] : ( heap_Time_Heap2 @ A @ ( 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 ) ) ) @ F2 ) ) ) ) ).
% heap_def
thf(fact_122_internal__case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,C2: B > C > A,A3: B,B2: C] :
( ( produc5280177257484947105e_prod @ B @ C @ A @ C2 @ ( product_Pair @ B @ C @ A3 @ B2 ) )
= ( C2 @ A3 @ B2 ) ) ).
% internal_case_prod_conv
thf(fact_123_option_Osplit__sel,axiom,
! [B: $tType,A: $tType,P: B > $o,F1: B,F22: A > B,Option: option @ A] :
( ( P @ ( case_option @ B @ A @ F1 @ F22 @ Option ) )
= ( ( ( Option
= ( none @ A ) )
=> ( P @ F1 ) )
& ( ( Option
= ( some @ A @ ( the2 @ A @ Option ) ) )
=> ( P @ ( F22 @ ( the2 @ A @ Option ) ) ) ) ) ) ).
% option.split_sel
thf(fact_124_option_Osplit__sel__asm,axiom,
! [B: $tType,A: $tType,P: B > $o,F1: B,F22: A > B,Option: option @ A] :
( ( P @ ( case_option @ B @ A @ F1 @ F22 @ Option ) )
= ( ~ ( ( ( Option
= ( none @ A ) )
& ~ ( P @ F1 ) )
| ( ( Option
= ( some @ A @ ( the2 @ A @ Option ) ) )
& ~ ( P @ ( F22 @ ( the2 @ A @ Option ) ) ) ) ) ) ) ).
% option.split_sel_asm
thf(fact_125_fold__atLeastAtMost__nat_Ocases,axiom,
! [A: $tType,X: product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) )] :
~ ! [F4: nat > A > A,A5: nat,B3: nat,Acc: A] :
( X
!= ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ F4 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A5 @ ( product_Pair @ nat @ A @ B3 @ Acc ) ) ) ) ).
% fold_atLeastAtMost_nat.cases
thf(fact_126_one__natural_Orsp,axiom,
( ( one_one @ nat )
= ( one_one @ nat ) ) ).
% one_natural.rsp
thf(fact_127_timeFrame_Opelims,axiom,
! [A: $tType,X: nat,Xa: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ),Y: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
( ( ( heap_Time_timeFrame @ A @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ ( heap_T5500966940807335491me_rel @ A ) @ ( product_Pair @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ X @ Xa ) )
=> ( ! [R2: A,H2: heap_ext @ product_unit,N4: nat] :
( ( Xa
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H2 @ N4 ) ) ) )
=> ( ( Y
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H2 @ ( plus_plus @ nat @ X @ N4 ) ) ) ) )
=> ~ ( accp @ ( product_prod @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ ( heap_T5500966940807335491me_rel @ A ) @ ( product_Pair @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ X @ ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H2 @ N4 ) ) ) ) ) ) )
=> ~ ( ( Xa
= ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
=> ( ( Y
= ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
=> ~ ( accp @ ( product_prod @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ ( heap_T5500966940807335491me_rel @ A ) @ ( 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
thf(fact_128_add__right__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ( plus_plus @ A @ B2 @ A3 )
= ( plus_plus @ A @ C2 @ A3 ) )
= ( B2 = C2 ) ) ) ).
% add_right_cancel
thf(fact_129_add__left__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ( plus_plus @ A @ A3 @ B2 )
= ( plus_plus @ A @ A3 @ C2 ) )
= ( B2 = C2 ) ) ) ).
% add_left_cancel
thf(fact_130_Heap_Oinject,axiom,
! [A: $tType,X: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ),Ya: ( 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
thf(fact_131_Heap__execute,axiom,
! [A: $tType,F: heap_Time_Heap @ A] :
( ( heap_Time_Heap2 @ A @ ( heap_Time_execute @ A @ F ) )
= F ) ).
% Heap_execute
thf(fact_132_timeFrame__assoc,axiom,
! [A: $tType,N: nat,N2: nat,F: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
( ( heap_Time_timeFrame @ A @ N @ ( heap_Time_timeFrame @ A @ N2 @ F ) )
= ( heap_Time_timeFrame @ A @ ( plus_plus @ nat @ N @ N2 ) @ F ) ) ).
% timeFrame_assoc
thf(fact_133_add__right__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ( plus_plus @ A @ B2 @ A3 )
= ( plus_plus @ A @ C2 @ A3 ) )
=> ( B2 = C2 ) ) ) ).
% add_right_imp_eq
thf(fact_134_add__left__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ( plus_plus @ A @ A3 @ B2 )
= ( plus_plus @ A @ A3 @ C2 ) )
=> ( B2 = C2 ) ) ) ).
% add_left_imp_eq
thf(fact_135_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( plus_plus @ A @ B2 @ ( plus_plus @ A @ A3 @ C2 ) )
= ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% ab_semigroup_add_class.add.left_commute
thf(fact_136_ab__semigroup__add__class_Oadd_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ( ( plus_plus @ A )
= ( ^ [A7: A,B5: A] : ( plus_plus @ A @ B5 @ A7 ) ) ) ) ).
% ab_semigroup_add_class.add.commute
thf(fact_137_add_Oright__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ( plus_plus @ A @ B2 @ A3 )
= ( plus_plus @ A @ C2 @ A3 ) )
= ( B2 = C2 ) ) ) ).
% add.right_cancel
thf(fact_138_add_Oleft__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ( plus_plus @ A @ A3 @ B2 )
= ( plus_plus @ A @ A3 @ C2 ) )
= ( B2 = C2 ) ) ) ).
% add.left_cancel
thf(fact_139_add_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
= ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% add.assoc
thf(fact_140_group__cancel_Oadd2,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [B6: A,K: A,B2: A,A3: A] :
( ( B6
= ( plus_plus @ A @ K @ B2 ) )
=> ( ( plus_plus @ A @ A3 @ B6 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% group_cancel.add2
thf(fact_141_group__cancel_Oadd1,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: A,K: A,A3: A,B2: A] :
( ( A4
= ( plus_plus @ A @ K @ A3 ) )
=> ( ( plus_plus @ A @ A4 @ B2 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% group_cancel.add1
thf(fact_142_add__mono__thms__linordered__semiring_I4_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus @ A @ I @ K )
= ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_143_Heap_Oexhaust,axiom,
! [A: $tType,Y: heap_Time_Heap @ A] :
~ ! [X2: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
( Y
!= ( heap_Time_Heap2 @ A @ X2 ) ) ).
% Heap.exhaust
thf(fact_144_execute_Osimps,axiom,
! [A: $tType,F: ( 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
thf(fact_145_option_Osimps_I5_J,axiom,
! [B: $tType,A: $tType,F1: B,F22: A > B,X22: A] :
( ( case_option @ B @ A @ F1 @ F22 @ ( some @ A @ X22 ) )
= ( F22 @ X22 ) ) ).
% option.simps(5)
thf(fact_146_option_Osimps_I4_J,axiom,
! [A: $tType,B: $tType,F1: B,F22: A > B] :
( ( case_option @ B @ A @ F1 @ F22 @ ( none @ A ) )
= F1 ) ).
% option.simps(4)
thf(fact_147_effect__bindE,axiom,
! [B: $tType,A: $tType,F: heap_Time_Heap @ B,G: B > ( heap_Time_Heap @ A ),H3: heap_ext @ product_unit,H5: heap_ext @ product_unit,R4: A,N: nat] :
( ( heap_Time_effect @ A @ ( heap_Time_bind @ B @ A @ F @ G ) @ H3 @ H5 @ R4 @ N )
=> ~ ! [H6: heap_ext @ product_unit,R2: B,N1: nat] :
( ( heap_Time_effect @ B @ F @ H3 @ H6 @ R2 @ N1 )
=> ! [N22: nat] :
( ( heap_Time_effect @ A @ ( G @ R2 ) @ H6 @ H5 @ R4 @ N22 )
=> ( N
!= ( plus_plus @ nat @ N1 @ N22 ) ) ) ) ) ).
% effect_bindE
thf(fact_148_effect__bindI,axiom,
! [B: $tType,A: $tType,F: heap_Time_Heap @ A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A,N: nat,G: A > ( heap_Time_Heap @ B ),H5: heap_ext @ product_unit,R4: B,N2: nat] :
( ( heap_Time_effect @ A @ F @ H3 @ H4 @ R @ N )
=> ( ( heap_Time_effect @ B @ ( G @ R ) @ H4 @ H5 @ R4 @ N2 )
=> ( heap_Time_effect @ B @ ( heap_Time_bind @ A @ B @ F @ G ) @ H3 @ H5 @ R4 @ ( plus_plus @ nat @ N @ N2 ) ) ) ) ).
% effect_bindI
thf(fact_149_option_Ocase__eq__if,axiom,
! [A: $tType,B: $tType] :
( ( case_option @ B @ A )
= ( ^ [F12: B,F23: A > B,Option3: option @ A] :
( if @ B
@ ( Option3
= ( none @ A ) )
@ F12
@ ( F23 @ ( the2 @ A @ Option3 ) ) ) ) ) ).
% option.case_eq_if
thf(fact_150_uncurry__curry__id,axiom,
! [C: $tType,B: $tType,A: $tType,F: ( product_prod @ A @ B ) > C] :
( ( uncurry @ A @ B @ C @ ( product_curry @ A @ B @ C @ F ) )
= F ) ).
% uncurry_curry_id
thf(fact_151_curry__uncurry__id,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > B > C] :
( ( product_curry @ A @ B @ C @ ( uncurry @ A @ B @ C @ F ) )
= F ) ).
% curry_uncurry_id
thf(fact_152_accp_Ocases,axiom,
! [A: $tType,R: A > A > $o,A3: A] :
( ( accp @ A @ R @ A3 )
=> ! [Y6: A] :
( ( R @ Y6 @ A3 )
=> ( accp @ A @ R @ Y6 ) ) ) ).
% accp.cases
thf(fact_153_accp_Osimps,axiom,
! [A: $tType] :
( ( accp @ A )
= ( ^ [R3: A > A > $o,A7: A] :
? [X3: A] :
( ( A7 = X3 )
& ! [Y3: A] :
( ( R3 @ Y3 @ X3 )
=> ( accp @ A @ R3 @ Y3 ) ) ) ) ) ).
% accp.simps
thf(fact_154_accpI,axiom,
! [A: $tType,R: A > A > $o,X: A] :
( ! [Y4: A] :
( ( R @ Y4 @ X )
=> ( accp @ A @ R @ Y4 ) )
=> ( accp @ A @ R @ X ) ) ).
% accpI
thf(fact_155_accp__induct,axiom,
! [A: $tType,R: A > A > $o,A3: A,P: A > $o] :
( ( accp @ A @ R @ A3 )
=> ( ! [X2: A] :
( ( accp @ A @ R @ X2 )
=> ( ! [Y6: A] :
( ( R @ Y6 @ X2 )
=> ( P @ Y6 ) )
=> ( P @ X2 ) ) )
=> ( P @ A3 ) ) ) ).
% accp_induct
thf(fact_156_accp__downward,axiom,
! [A: $tType,R: A > A > $o,B2: A,A3: A] :
( ( accp @ A @ R @ B2 )
=> ( ( R @ A3 @ B2 )
=> ( accp @ A @ R @ A3 ) ) ) ).
% accp_downward
thf(fact_157_not__accp__down,axiom,
! [A: $tType,R5: A > A > $o,X: A] :
( ~ ( accp @ A @ R5 @ X )
=> ~ ! [Z3: A] :
( ( R5 @ Z3 @ X )
=> ( accp @ A @ R5 @ Z3 ) ) ) ).
% not_accp_down
thf(fact_158_accp__induct__rule,axiom,
! [A: $tType,R: A > A > $o,A3: A,P: A > $o] :
( ( accp @ A @ R @ A3 )
=> ( ! [X2: A] :
( ( accp @ A @ R @ X2 )
=> ( ! [Y6: A] :
( ( R @ Y6 @ X2 )
=> ( P @ Y6 ) )
=> ( P @ X2 ) ) )
=> ( P @ A3 ) ) ) ).
% accp_induct_rule
thf(fact_159_fun_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,D: $tType,G: B > C,F: A > B,V: D > A] :
( ( comp @ B @ C @ D @ G @ ( comp @ A @ B @ D @ F @ V ) )
= ( comp @ A @ C @ D @ ( comp @ B @ C @ A @ G @ F ) @ V ) ) ).
% fun.map_comp
thf(fact_160_comp__cong,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,E: $tType,F: B > A,G: C > B,X: C,F3: D > A,G5: E > D,X6: E] :
( ( ( F @ ( G @ X ) )
= ( F3 @ ( G5 @ X6 ) ) )
=> ( ( comp @ B @ A @ C @ F @ G @ X )
= ( comp @ D @ A @ E @ F3 @ G5 @ X6 ) ) ) ).
% comp_cong
thf(fact_161_curryI,axiom,
! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A3: A,B2: B] :
( ( F @ ( product_Pair @ A @ B @ A3 @ B2 ) )
=> ( product_curry @ A @ B @ $o @ F @ A3 @ B2 ) ) ).
% curryI
thf(fact_162_curry__conv,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( product_curry @ B @ C @ A )
= ( ^ [F2: ( product_prod @ B @ C ) > A,A7: B,B5: C] : ( F2 @ ( product_Pair @ B @ C @ A7 @ B5 ) ) ) ) ).
% curry_conv
thf(fact_163_curryE,axiom,
! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A3: A,B2: B] :
( ( product_curry @ A @ B @ $o @ F @ A3 @ B2 )
=> ( F @ ( product_Pair @ A @ B @ A3 @ B2 ) ) ) ).
% curryE
thf(fact_164_curryD,axiom,
! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A3: A,B2: B] :
( ( product_curry @ A @ B @ $o @ F @ A3 @ B2 )
=> ( F @ ( product_Pair @ A @ B @ A3 @ B2 ) ) ) ).
% curryD
thf(fact_165_fold__atLeastAtMost__nat_Opinduct,axiom,
! [A: $tType,A0: nat > A > A,A1: nat,A22: nat,A32: A,P: ( nat > A > A ) > nat > nat > A > $o] :
( ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ A0 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A1 @ ( product_Pair @ nat @ A @ A22 @ A32 ) ) ) )
=> ( ! [F4: nat > A > A,A5: nat,B3: nat,Acc: A] :
( ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ F4 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A5 @ ( product_Pair @ nat @ A @ B3 @ Acc ) ) ) )
=> ( ( ~ ( ord_less @ nat @ B3 @ A5 )
=> ( P @ F4 @ ( plus_plus @ nat @ A5 @ ( one_one @ nat ) ) @ B3 @ ( F4 @ A5 @ Acc ) ) )
=> ( P @ F4 @ A5 @ B3 @ Acc ) ) )
=> ( P @ A0 @ A1 @ A22 @ A32 ) ) ) ).
% fold_atLeastAtMost_nat.pinduct
thf(fact_166_in__lex__prod,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,A6: A,B4: B,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( product_Pair @ A @ B @ A6 @ B4 ) ) @ ( lex_prod @ A @ B @ R @ S ) )
= ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A6 ) @ R )
| ( ( A3 = A6 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B2 @ B4 ) @ S ) ) ) ) ).
% in_lex_prod
thf(fact_167_rewriteR__comp__comp2,axiom,
! [C: $tType,B: $tType,E: $tType,D: $tType,A: $tType,G: C > B,H3: A > C,R1: D > B,R22: A > D,F: B > E,L: D > E] :
( ( ( comp @ C @ B @ A @ G @ H3 )
= ( comp @ D @ B @ A @ R1 @ R22 ) )
=> ( ( ( comp @ B @ E @ D @ F @ R1 )
= L )
=> ( ( comp @ C @ E @ A @ ( comp @ B @ E @ C @ F @ G ) @ H3 )
= ( comp @ D @ E @ A @ L @ R22 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_168_rewriteL__comp__comp2,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,E: $tType,F: C > B,G: A > C,L1: D > B,L2: A > D,H3: E > A,R: E > D] :
( ( ( comp @ C @ B @ A @ F @ G )
= ( comp @ D @ B @ A @ L1 @ L2 ) )
=> ( ( ( comp @ A @ D @ E @ L2 @ H3 )
= R )
=> ( ( comp @ C @ B @ E @ F @ ( comp @ A @ C @ E @ G @ H3 ) )
= ( comp @ D @ B @ E @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_169_rewriteR__comp__comp,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,G: C > B,H3: A > C,R: A > B,F: B > D] :
( ( ( comp @ C @ B @ A @ G @ H3 )
= R )
=> ( ( comp @ C @ D @ A @ ( comp @ B @ D @ C @ F @ G ) @ H3 )
= ( comp @ B @ D @ A @ F @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_170_rewriteL__comp__comp,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType,F: C > B,G: A > C,L: A > B,H3: D > A] :
( ( ( comp @ C @ B @ A @ F @ G )
= L )
=> ( ( comp @ C @ B @ D @ F @ ( comp @ A @ C @ D @ G @ H3 ) )
= ( comp @ A @ B @ D @ L @ H3 ) ) ) ).
% rewriteL_comp_comp
thf(fact_171_type__copy__map__cong0,axiom,
! [B: $tType,D: $tType,E: $tType,A: $tType,C: $tType,M3: B > A,G: C > B,X: C,N6: D > A,H3: C > D,F: A > E] :
( ( ( M3 @ ( G @ X ) )
= ( N6 @ ( H3 @ X ) ) )
=> ( ( comp @ B @ E @ C @ ( comp @ A @ E @ B @ F @ M3 ) @ G @ X )
= ( comp @ D @ E @ C @ ( comp @ A @ E @ D @ F @ N6 ) @ H3 @ X ) ) ) ).
% type_copy_map_cong0
thf(fact_172_comp__apply__eq,axiom,
! [B: $tType,D: $tType,A: $tType,C: $tType,F: B > A,G: C > B,X: C,H3: D > A,K: C > D] :
( ( ( F @ ( G @ X ) )
= ( H3 @ ( K @ X ) ) )
=> ( ( comp @ B @ A @ C @ F @ G @ X )
= ( comp @ D @ A @ C @ H3 @ K @ X ) ) ) ).
% comp_apply_eq
thf(fact_173_effect__returnE,axiom,
! [A: $tType,X: A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A,N: nat] :
( ( heap_Time_effect @ A @ ( heap_Time_return @ A @ X ) @ H3 @ H4 @ R @ N )
=> ~ ( ( R = X )
=> ( ( H4 = H3 )
=> ( N
!= ( one_one @ nat ) ) ) ) ) ).
% effect_returnE
thf(fact_174_effect__returnI,axiom,
! [A: $tType,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,X: A] :
( ( H3 = H4 )
=> ( heap_Time_effect @ A @ ( heap_Time_return @ A @ X ) @ H3 @ H4 @ X @ ( one_one @ nat ) ) ) ).
% effect_returnI
thf(fact_175_add_Oright__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( plus_plus @ A @ A3 @ C2 ) @ B2 ) ) ) ).
% add.right_commute
thf(fact_176_add__less__cancel__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ).
% add_less_cancel_right
thf(fact_177_add__less__cancel__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ).
% add_less_cancel_left
thf(fact_178_fold__if__return,axiom,
! [A: $tType,B2: $o,C2: A,D3: A] :
( ( B2
=> ( ( heap_Time_return @ A @ C2 )
= ( heap_Time_return @ A @ ( if @ A @ B2 @ C2 @ D3 ) ) ) )
& ( ~ B2
=> ( ( heap_Time_return @ A @ D3 )
= ( heap_Time_return @ A @ ( if @ A @ B2 @ C2 @ D3 ) ) ) ) ) ).
% fold_if_return
thf(fact_179_add__less__imp__less__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
=> ( ord_less @ A @ A3 @ B2 ) ) ) ).
% add_less_imp_less_right
thf(fact_180_add__less__imp__less__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) )
=> ( ord_less @ A @ A3 @ B2 ) ) ) ).
% add_less_imp_less_left
thf(fact_181_add__strict__right__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% add_strict_right_mono
thf(fact_182_add__strict__left__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ord_less @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% add_strict_left_mono
thf(fact_183_add__strict__mono,axiom,
! [A: $tType] :
( ( strict9044650504122735259up_add @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ C2 @ D3 )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ D3 ) ) ) ) ) ).
% add_strict_mono
thf(fact_184_add__mono__thms__linordered__field_I1_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( K = L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_185_add__mono__thms__linordered__field_I2_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_186_add__mono__thms__linordered__field_I5_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_187_success__returnI,axiom,
! [A: $tType,X: A,H3: heap_ext @ product_unit] : ( heap_Time_success @ A @ ( heap_Time_return @ A @ X ) @ H3 ) ).
% success_returnI
thf(fact_188_fails__return,axiom,
! [A: $tType,X: A,H3: heap_ext @ product_unit] :
~ ( time_fails @ A @ ( heap_Time_return @ A @ X ) @ H3 ) ).
% fails_return
thf(fact_189_add_Oright__assoc,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
= ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% add.right_assoc
thf(fact_190_nat__add__left__cancel__less,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ K @ M2 ) @ ( plus_plus @ nat @ K @ N ) )
= ( ord_less @ nat @ M2 @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_191_less__option__Some,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ ( option @ A ) @ ( some @ A @ X ) @ ( some @ A @ Y ) )
= ( ord_less @ A @ X @ Y ) ) ) ).
% less_option_Some
thf(fact_192_fold__atLeastAtMost__nat_Opelims,axiom,
! [A: $tType,X: nat > A > A,Xa: nat,Xb: nat,Xc: A,Y: A] :
( ( ( set_fo6178422350223883121st_nat @ A @ X @ Xa @ Xb @ Xc )
= Y )
=> ( ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ X @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ Xa @ ( product_Pair @ nat @ A @ Xb @ Xc ) ) ) )
=> ~ ( ( ( ( ord_less @ nat @ Xb @ Xa )
=> ( Y = Xc ) )
& ( ~ ( ord_less @ nat @ Xb @ Xa )
=> ( Y
= ( set_fo6178422350223883121st_nat @ A @ X @ ( plus_plus @ nat @ Xa @ ( one_one @ nat ) ) @ Xb @ ( X @ Xa @ Xc ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ X @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ Xa @ ( product_Pair @ nat @ A @ Xb @ Xc ) ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.pelims
thf(fact_193_fold__atLeastAtMost__nat_Opsimps,axiom,
! [A: $tType,F: nat > A > A,A3: nat,B2: nat,Acc2: A] :
( ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ F @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A3 @ ( product_Pair @ nat @ A @ B2 @ Acc2 ) ) ) )
=> ( ( ( ord_less @ nat @ B2 @ A3 )
=> ( ( set_fo6178422350223883121st_nat @ A @ F @ A3 @ B2 @ Acc2 )
= Acc2 ) )
& ( ~ ( ord_less @ nat @ B2 @ A3 )
=> ( ( set_fo6178422350223883121st_nat @ A @ F @ A3 @ B2 @ Acc2 )
= ( set_fo6178422350223883121st_nat @ A @ F @ ( plus_plus @ nat @ A3 @ ( one_one @ nat ) ) @ B2 @ ( F @ A3 @ Acc2 ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.psimps
thf(fact_194_in__measure,axiom,
! [A: $tType,X: A,Y: A,F: A > nat] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measure @ A @ F ) )
= ( ord_less @ nat @ ( F @ X ) @ ( F @ Y ) ) ) ).
% in_measure
thf(fact_195_less__add__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A] : ( ord_less @ A @ A3 @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% less_add_one
thf(fact_196_add__mono1,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( plus_plus @ A @ B2 @ ( one_one @ A ) ) ) ) ) ).
% add_mono1
thf(fact_197_lift__collapse,axiom,
! [A: $tType,B: $tType] :
( ( heap_Time_lift @ B @ A )
= ( ^ [F2: B > A,X3: B] : ( heap_Time_return @ A @ ( F2 @ X3 ) ) ) ) ).
% lift_collapse
thf(fact_198_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ K )
=> ( ord_less @ nat @ I @ K ) ) ).
% add_lessD1
thf(fact_199_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( ord_less @ nat @ K @ L )
=> ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_200_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_201_less__option__None,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: option @ A] :
~ ( ord_less @ ( option @ A ) @ X @ ( none @ A ) ) ) ).
% less_option_None
thf(fact_202_less__option__None__Some__code,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] : ( ord_less @ ( option @ A ) @ ( none @ A ) @ ( some @ A @ X ) ) ) ).
% less_option_None_Some_code
thf(fact_203_less__option__None__Some,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] : ( ord_less @ ( option @ A ) @ ( none @ A ) @ ( some @ A @ X ) ) ) ).
% less_option_None_Some
thf(fact_204_less__option__None__is__Some,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: option @ A] :
( ( ord_less @ ( option @ A ) @ ( none @ A ) @ X )
=> ? [Z3: A] :
( X
= ( some @ A @ Z3 ) ) ) ) ).
% less_option_None_is_Some
thf(fact_205_lift__def,axiom,
! [A: $tType,B: $tType] :
( ( heap_Time_lift @ A @ B )
= ( comp @ B @ ( heap_Time_Heap @ B ) @ A @ ( heap_Time_return @ B ) ) ) ).
% lift_def
thf(fact_206_fold__atLeastAtMost__nat_Osimps,axiom,
! [A: $tType] :
( ( set_fo6178422350223883121st_nat @ A )
= ( ^ [F2: nat > A > A,A7: nat,B5: nat,Acc3: A] : ( if @ A @ ( ord_less @ nat @ B5 @ A7 ) @ Acc3 @ ( set_fo6178422350223883121st_nat @ A @ F2 @ ( plus_plus @ nat @ A7 @ ( one_one @ nat ) ) @ B5 @ ( F2 @ A7 @ Acc3 ) ) ) ) ) ).
% fold_atLeastAtMost_nat.simps
thf(fact_207_fold__atLeastAtMost__nat_Oelims,axiom,
! [A: $tType,X: nat > A > A,Xa: nat,Xb: nat,Xc: A,Y: A] :
( ( ( set_fo6178422350223883121st_nat @ A @ X @ Xa @ Xb @ Xc )
= Y )
=> ( ( ( ord_less @ nat @ Xb @ Xa )
=> ( Y = Xc ) )
& ( ~ ( ord_less @ nat @ Xb @ Xa )
=> ( Y
= ( set_fo6178422350223883121st_nat @ A @ X @ ( plus_plus @ nat @ Xa @ ( one_one @ nat ) ) @ Xb @ ( X @ Xa @ Xc ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.elims
thf(fact_208_less__add__eq__less,axiom,
! [K: nat,L: nat,M2: nat,N: nat] :
( ( ord_less @ nat @ K @ L )
=> ( ( ( plus_plus @ nat @ M2 @ L )
= ( plus_plus @ nat @ K @ N ) )
=> ( ord_less @ nat @ M2 @ N ) ) ) ).
% less_add_eq_less
thf(fact_209_trans__less__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ord_less @ nat @ I @ ( plus_plus @ nat @ M2 @ J ) ) ) ).
% trans_less_add2
thf(fact_210_trans__less__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ord_less @ nat @ I @ ( plus_plus @ nat @ J @ M2 ) ) ) ).
% trans_less_add1
thf(fact_211_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_212_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_213_less__numeral__extra_I4_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% less_numeral_extra(4)
thf(fact_214_mlex__iff,axiom,
! [A: $tType,X: A,Y: A,F: A > nat,R5: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( mlex_prod @ A @ F @ R5 ) )
= ( ( ord_less @ nat @ ( F @ X ) @ ( F @ Y ) )
| ( ( ( F @ X )
= ( F @ Y ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R5 ) ) ) ) ).
% mlex_iff
thf(fact_215_mlex__less,axiom,
! [A: $tType,F: A > nat,X: A,Y: A,R5: set @ ( product_prod @ A @ A )] :
( ( ord_less @ nat @ ( F @ X ) @ ( F @ Y ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( mlex_prod @ A @ F @ R5 ) ) ) ).
% mlex_less
thf(fact_216_pairself_Opelims,axiom,
! [B: $tType,A: $tType,X: A > B,Xa: product_prod @ A @ A,Y: product_prod @ B @ B] :
( ( ( pairself @ A @ B @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( A > B ) @ ( product_prod @ A @ A ) ) @ ( pairself_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( product_prod @ A @ A ) @ X @ Xa ) )
=> ~ ! [A5: A,B3: A] :
( ( Xa
= ( product_Pair @ A @ A @ A5 @ B3 ) )
=> ( ( Y
= ( product_Pair @ B @ B @ ( X @ A5 ) @ ( X @ B3 ) ) )
=> ~ ( accp @ ( product_prod @ ( A > B ) @ ( product_prod @ A @ A ) ) @ ( pairself_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( product_prod @ A @ A ) @ X @ ( product_Pair @ A @ A @ A5 @ B3 ) ) ) ) ) ) ) ).
% pairself.pelims
thf(fact_217_execute__return,axiom,
! [A: $tType,X: A] :
( ( heap_Time_execute @ A @ ( heap_Time_return @ A @ X ) )
= ( 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 ) ) )
@ ^ [H: heap_ext @ product_unit] : ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H @ ( one_one @ nat ) ) ) ) ) ).
% execute_return
thf(fact_218_same__fstI,axiom,
! [B: $tType,A: $tType,P: A > $o,X: A,Y7: B,Y: B,R5: A > ( set @ ( product_prod @ B @ B ) )] :
( ( P @ X )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y7 @ Y ) @ ( R5 @ X ) )
=> ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y7 ) @ ( product_Pair @ A @ B @ X @ Y ) ) @ ( same_fst @ A @ B @ P @ R5 ) ) ) ) ).
% same_fstI
thf(fact_219_zero__less__two,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ) ).
% zero_less_two
thf(fact_220_is__num__normalize_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
= ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% is_num_normalize(1)
thf(fact_221_add_Osafe__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [X: A,Y: A,A3: A,B2: A] :
( ( syntax7388354845996824322omatch @ A @ A @ ( plus_plus @ A @ X @ Y ) @ A3 )
=> ( ( plus_plus @ A @ A3 @ B2 )
= ( plus_plus @ A @ B2 @ A3 ) ) ) ) ).
% add.safe_commute
thf(fact_222_bijective__def,axiom,
! [B: $tType,A: $tType] :
( ( bijective @ A @ B )
= ( ^ [R6: set @ ( product_prod @ A @ B )] :
( ! [X3: A,Y3: B,Z5: B] :
( ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R6 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Z5 ) @ R6 ) )
=> ( Y3 = Z5 ) )
& ! [X3: A,Y3: A,Z5: B] :
( ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Z5 ) @ R6 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y3 @ Z5 ) @ R6 ) )
=> ( X3 = Y3 ) ) ) ) ) ).
% bijective_def
thf(fact_223_ssubst__Pair__rhs,axiom,
! [B: $tType,A: $tType,R: A,S: B,R5: set @ ( product_prod @ A @ B ),S3: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S ) @ R5 )
=> ( ( S3 = S )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S3 ) @ R5 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_224_add__is__0,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus @ nat @ M2 @ N )
= ( zero_zero @ nat ) )
= ( ( M2
= ( zero_zero @ nat ) )
& ( N
= ( zero_zero @ nat ) ) ) ) ).
% add_is_0
thf(fact_225_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus @ nat @ M2 @ ( zero_zero @ nat ) )
= M2 ) ).
% Nat.add_0_right
thf(fact_226_timeFrame__zero,axiom,
! [A: $tType,H3: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
( ( heap_Time_timeFrame @ A @ ( zero_zero @ nat ) @ H3 )
= H3 ) ).
% timeFrame_zero
thf(fact_227_Heap__Time__Monad_Obind__bind,axiom,
! [C: $tType,B: $tType,A: $tType,F: heap_Time_Heap @ A,G: A > ( heap_Time_Heap @ C ),K: C > ( heap_Time_Heap @ B )] :
( ( heap_Time_bind @ C @ B @ ( heap_Time_bind @ A @ C @ F @ G ) @ K )
= ( heap_Time_bind @ A @ B @ F
@ ^ [X3: A] : ( heap_Time_bind @ C @ B @ ( G @ X3 ) @ K ) ) ) ).
% Heap_Time_Monad.bind_bind
thf(fact_228_not__gr__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N: A] :
( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% not_gr_zero
thf(fact_229_add__0,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A3 )
= A3 ) ) ).
% add_0
thf(fact_230_zero__eq__add__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X: A,Y: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ X @ Y ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_231_add__eq__0__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X: A,Y: A] :
( ( ( plus_plus @ A @ X @ Y )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_232_add__cancel__right__right,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A3: A,B2: A] :
( ( A3
= ( plus_plus @ A @ A3 @ B2 ) )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_right
thf(fact_233_add__cancel__right__left,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A3: A,B2: A] :
( ( A3
= ( plus_plus @ A @ B2 @ A3 ) )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_left
thf(fact_234_add__cancel__left__right,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A3: A,B2: A] :
( ( ( plus_plus @ A @ A3 @ B2 )
= A3 )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_right
thf(fact_235_add__cancel__left__left,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [B2: A,A3: A] :
( ( ( plus_plus @ A @ B2 @ A3 )
= A3 )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_left
thf(fact_236_double__zero__sym,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ A3 @ A3 ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% double_zero_sym
thf(fact_237_add_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% add.right_neutral
thf(fact_238_add__gr__0,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ M2 @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
| ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% add_gr_0
thf(fact_239_less__one,axiom,
! [N: nat] :
( ( ord_less @ nat @ N @ ( one_one @ nat ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% less_one
thf(fact_240_some__opt__eq__trivial,axiom,
! [A: $tType,X: A] :
( ( eps_Opt @ A
@ ^ [Y3: A] : Y3 = X )
= ( some @ A @ X ) ) ).
% some_opt_eq_trivial
thf(fact_241_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ A3 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_242_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A3 @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_243_less__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( plus_plus @ A @ B2 @ A3 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% less_add_same_cancel2
thf(fact_244_less__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( plus_plus @ A @ A3 @ B2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% less_add_same_cancel1
thf(fact_245_add__less__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel2
thf(fact_246_add__less__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ B2 @ A3 ) @ B2 )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel1
thf(fact_247_some__opt__false__trivial,axiom,
! [A: $tType] :
( ( eps_Opt @ A
@ ^ [Uu: A] : $false )
= ( none @ A ) ) ).
% some_opt_false_trivial
thf(fact_248_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X: A] :
( ( ( zero_zero @ A )
= X )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_249_fun_Omap__ident,axiom,
! [A: $tType,D: $tType,T2: D > A] :
( ( comp @ A @ A @ D
@ ^ [X3: A] : X3
@ T2 )
= T2 ) ).
% fun.map_ident
thf(fact_250_option_Odisc__eq__case_I2_J,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
!= ( none @ A ) )
= ( case_option @ $o @ A @ $false
@ ^ [Uu: A] : $true
@ Option ) ) ).
% option.disc_eq_case(2)
thf(fact_251_option_Odisc__eq__case_I1_J,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
= ( none @ A ) )
= ( case_option @ $o @ A @ $true
@ ^ [Uu: A] : $false
@ Option ) ) ).
% option.disc_eq_case(1)
thf(fact_252_option_Ocase__distrib,axiom,
! [C: $tType,B: $tType,A: $tType,H3: B > C,F1: B,F22: A > B,Option: option @ A] :
( ( H3 @ ( case_option @ B @ A @ F1 @ F22 @ Option ) )
= ( case_option @ C @ A @ ( H3 @ F1 )
@ ^ [X3: A] : ( H3 @ ( F22 @ X3 ) )
@ Option ) ) ).
% option.case_distrib
thf(fact_253_effect__LetI,axiom,
! [B: $tType,A: $tType,X: A,T2: A,F: A > ( heap_Time_Heap @ B ),H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: B,N: nat] :
( ( X = T2 )
=> ( ( heap_Time_effect @ B @ ( F @ X ) @ H3 @ H4 @ R @ N )
=> ( heap_Time_effect @ B @ ( F @ T2 ) @ H3 @ H4 @ R @ N ) ) ) ).
% effect_LetI
thf(fact_254_distrib__if__bind,axiom,
! [A: $tType,B: $tType,B2: $o,C2: heap_Time_Heap @ B,D3: heap_Time_Heap @ B,F: B > ( heap_Time_Heap @ A )] :
( ( B2
=> ( ( heap_Time_bind @ B @ A @ ( if @ ( heap_Time_Heap @ B ) @ B2 @ C2 @ D3 ) @ F )
= ( heap_Time_bind @ B @ A @ C2 @ F ) ) )
& ( ~ B2
=> ( ( heap_Time_bind @ B @ A @ ( if @ ( heap_Time_Heap @ B ) @ B2 @ C2 @ D3 ) @ F )
= ( heap_Time_bind @ B @ A @ D3 @ F ) ) ) ) ).
% distrib_if_bind
thf(fact_255_success__LetI,axiom,
! [A: $tType,B: $tType,X: A,T2: A,F: A > ( heap_Time_Heap @ B ),H3: heap_ext @ product_unit] :
( ( X = T2 )
=> ( ( heap_Time_success @ B @ ( F @ X ) @ H3 )
=> ( heap_Time_success @ B @ ( F @ T2 ) @ H3 ) ) ) ).
% success_LetI
thf(fact_256_curry__K,axiom,
! [B: $tType,C: $tType,A: $tType,C2: C] :
( ( product_curry @ A @ B @ C
@ ^ [X3: product_prod @ A @ B] : C2 )
= ( ^ [X3: A,Y3: B] : C2 ) ) ).
% curry_K
thf(fact_257_case__optionE,axiom,
! [A: $tType,P: $o,Q: A > $o,X: option @ A] :
( ( case_option @ $o @ A @ P @ Q @ X )
=> ( ( ( X
= ( none @ A ) )
=> ~ P )
=> ~ ! [Y4: A] :
( ( X
= ( some @ A @ Y4 ) )
=> ~ ( Q @ Y4 ) ) ) ) ).
% case_optionE
thf(fact_258_curry__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_curry @ A @ B @ C )
= ( ^ [C3: ( product_prod @ A @ B ) > C,X3: A,Y3: B] : ( C3 @ ( product_Pair @ A @ B @ X3 @ Y3 ) ) ) ) ).
% curry_def
thf(fact_259_combine__options__def,axiom,
! [A: $tType] :
( ( combine_options @ A )
= ( ^ [F2: A > A > A,X3: option @ A,Y3: option @ A] :
( case_option @ ( option @ A ) @ A @ Y3
@ ^ [Z5: A] :
( case_option @ ( option @ A ) @ A @ ( some @ A @ Z5 )
@ ^ [Aa2: A] : ( some @ A @ ( F2 @ Z5 @ Aa2 ) )
@ Y3 )
@ X3 ) ) ) ).
% combine_options_def
thf(fact_260_less__option__def,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ ( option @ A ) )
= ( ^ [X3: option @ A] :
( case_option @ $o @ A @ $false
@ ^ [Y3: A] :
( case_option @ $o @ A @ $true
@ ^ [Z5: A] : ( ord_less @ A @ Z5 @ Y3 )
@ X3 ) ) ) ) ) ).
% less_option_def
thf(fact_261_zero__less__iff__neq__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ N )
= ( N
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_iff_neq_zero
thf(fact_262_gr__implies__not__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [M2: A,N: A] :
( ( ord_less @ A @ M2 @ N )
=> ( N
!= ( zero_zero @ A ) ) ) ) ).
% gr_implies_not_zero
thf(fact_263_not__less__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N: A] :
~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% not_less_zero
thf(fact_264_gr__zeroI,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N: A] :
( ( N
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% gr_zeroI
thf(fact_265_add_Ogroup__left__neutral,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A3 )
= A3 ) ) ).
% add.group_left_neutral
thf(fact_266_add_Ocomm__neutral,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% add.comm_neutral
thf(fact_267_comm__monoid__add__class_Oadd__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A3 )
= A3 ) ) ).
% comm_monoid_add_class.add_0
thf(fact_268_zero__neq__one,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_zero @ A )
!= ( one_one @ A ) ) ) ).
% zero_neq_one
thf(fact_269_array__make,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_new @ A )
= ( ^ [N5: nat,X3: A] :
( array_make @ A @ N5
@ ^ [Uu: nat] : X3 ) ) ) ) ).
% array_make
thf(fact_270_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus @ nat @ ( zero_zero @ nat ) @ N )
= N ) ).
% plus_nat.add_0
thf(fact_271_add__eq__self__zero,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus @ nat @ M2 @ N )
= M2 )
=> ( N
= ( zero_zero @ nat ) ) ) ).
% add_eq_self_zero
thf(fact_272_less__numeral__extra_I1_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% less_numeral_extra(1)
thf(fact_273_ureturn__def,axiom,
! [A: $tType] :
( ( heap_Time_ureturn @ A )
= ( ^ [X3: A] :
( heap_Time_heap @ A
@ ^ [H: heap_ext @ product_unit] : ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H @ ( zero_zero @ nat ) ) ) ) ) ) ).
% ureturn_def
thf(fact_274_raise__def,axiom,
! [A: $tType] :
( ( heap_Time_raise @ A )
= ( ^ [S4: list @ char] :
( heap_Time_Heap2 @ A
@ ^ [Uu: heap_ext @ product_unit] : ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ).
% raise_def
thf(fact_275_bind__lift,axiom,
! [A: $tType,B: $tType,F: heap_Time_Heap @ B,G: B > A] :
( ( heap_Time_bind @ B @ A @ F @ ( heap_Time_lift @ B @ A @ G ) )
= ( heap_Time_bind @ B @ A @ F
@ ^ [X3: B] : ( heap_Time_return @ A @ ( G @ X3 ) ) ) ) ).
% bind_lift
thf(fact_276_guard__def,axiom,
! [A: $tType] :
( ( heap_Time_guard @ A )
= ( ^ [P4: ( heap_ext @ product_unit ) > $o,F2: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
( heap_Time_Heap2 @ A
@ ^ [H: heap_ext @ product_unit] : ( if @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( P4 @ H ) @ ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( F2 @ H ) ) @ ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ) ).
% guard_def
thf(fact_277_pos__add__strict,axiom,
! [A: $tType] :
( ( strict7427464778891057005id_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ B2 @ ( plus_plus @ A @ A3 @ C2 ) ) ) ) ) ).
% pos_add_strict
thf(fact_278_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ~ ! [C4: A] :
( ( B2
= ( plus_plus @ A @ A3 @ C4 ) )
=> ( C4
= ( zero_zero @ A ) ) ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_279_add__pos__pos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% add_pos_pos
thf(fact_280_add__neg__neg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_neg_neg
thf(fact_281_add__less__zeroD,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( plus_plus @ A @ X @ Y ) @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ X @ ( zero_zero @ A ) )
| ( ord_less @ A @ Y @ ( zero_zero @ A ) ) ) ) ) ).
% add_less_zeroD
thf(fact_282_zero__less__one,axiom,
! [A: $tType] :
( ( zero_less_one @ A )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% zero_less_one
thf(fact_283_not__one__less__zero,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% not_one_less_zero
thf(fact_284_execute__ureturn,axiom,
! [A: $tType,X: A] :
( ( heap_Time_execute @ A @ ( heap_Time_ureturn @ A @ X ) )
= ( 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 ) ) )
@ ^ [H: heap_ext @ product_unit] : ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H @ ( zero_zero @ nat ) ) ) ) ) ).
% execute_ureturn
thf(fact_285_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less @ nat @ I @ J )
=> ? [K5: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K5 )
& ( ( plus_plus @ nat @ I @ K5 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_286_pairself_Osimps,axiom,
! [B: $tType,A: $tType,F: A > B,A3: A,B2: A] :
( ( pairself @ A @ B @ F @ ( product_Pair @ A @ A @ A3 @ B2 ) )
= ( product_Pair @ B @ B @ ( F @ A3 ) @ ( F @ B2 ) ) ) ).
% pairself.simps
thf(fact_287_pairself_Oelims,axiom,
! [B: $tType,A: $tType,X: A > B,Xa: product_prod @ A @ A,Y: product_prod @ B @ B] :
( ( ( pairself @ A @ B @ X @ Xa )
= Y )
=> ~ ! [A5: A,B3: A] :
( ( Xa
= ( product_Pair @ A @ A @ A5 @ B3 ) )
=> ( Y
!= ( product_Pair @ B @ B @ ( X @ A5 ) @ ( X @ B3 ) ) ) ) ) ).
% pairself.elims
thf(fact_288_return__def,axiom,
! [A: $tType] :
( ( heap_Time_return @ A )
= ( ^ [X3: A] :
( heap_Time_heap @ A
@ ^ [H: heap_ext @ product_unit] : ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H @ ( one_one @ nat ) ) ) ) ) ) ).
% return_def
thf(fact_289_effect__ureturnI,axiom,
! [A: $tType,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,X: A] :
( ( H3 = H4 )
=> ( heap_Time_effect @ A @ ( heap_Time_ureturn @ A @ X ) @ H3 @ H4 @ X @ ( zero_zero @ nat ) ) ) ).
% effect_ureturnI
thf(fact_290_effect__ureturnE,axiom,
! [A: $tType,X: A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A,N: nat] :
( ( heap_Time_effect @ A @ ( heap_Time_ureturn @ A @ X ) @ H3 @ H4 @ R @ N )
=> ~ ( ( R = X )
=> ( ( H4 = H3 )
=> ( N
!= ( zero_zero @ nat ) ) ) ) ) ).
% effect_ureturnE
thf(fact_291_tap__def,axiom,
! [A: $tType] :
( ( heap_Time_tap @ A )
= ( ^ [F2: ( heap_ext @ product_unit ) > A] :
( heap_Time_Heap2 @ A
@ ^ [H: heap_ext @ product_unit] : ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F2 @ H ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H @ ( one_one @ nat ) ) ) ) ) ) ) ).
% tap_def
thf(fact_292_double__eq__0__iff,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ( plus_plus @ A @ A3 @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% double_eq_0_iff
thf(fact_293_return__bind,axiom,
! [B: $tType,A: $tType,X: B,F: B > ( heap_Time_Heap @ A )] :
( ( heap_Time_bind @ B @ A @ ( heap_Time_return @ B @ X ) @ F )
= ( heap_Time_bind @ product_unit @ A @ ( heap_Time_wait @ ( one_one @ nat ) )
@ ^ [Uu: product_unit] : ( F @ X ) ) ) ).
% return_bind
thf(fact_294_bind__return,axiom,
! [A: $tType,F: heap_Time_Heap @ A] :
( ( heap_Time_bind @ A @ A @ F @ ( heap_Time_return @ A ) )
= ( heap_Time_bind @ product_unit @ A @ ( heap_Time_wait @ ( one_one @ nat ) )
@ ^ [Uu: product_unit] : F ) ) ).
% bind_return
thf(fact_295_add__0__iff,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [B2: A,A3: A] :
( ( B2
= ( plus_plus @ A @ B2 @ A3 ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% add_0_iff
thf(fact_296_verit__sum__simplify,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% verit_sum_simplify
thf(fact_297_dbl__inc__simps_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_inc @ A @ ( zero_zero @ A ) )
= ( one_one @ A ) ) ) ).
% dbl_inc_simps(2)
thf(fact_298_disjE__realizer2,axiom,
! [B: $tType,A: $tType,P: $o,Q: A > $o,X: option @ A,R5: B > $o,F: B,G: A > B] :
( ( case_option @ $o @ A @ P @ Q @ X )
=> ( ( P
=> ( R5 @ F ) )
=> ( ! [Q2: A] :
( ( Q @ Q2 )
=> ( R5 @ ( G @ Q2 ) ) )
=> ( R5 @ ( case_option @ B @ A @ F @ G @ X ) ) ) ) ) ).
% disjE_realizer2
thf(fact_299_set__to__map__def,axiom,
! [A: $tType,B: $tType] :
( ( set_to_map @ B @ A )
= ( ^ [S5: set @ ( product_prod @ B @ A ),K4: B] :
( eps_Opt @ A
@ ^ [V2: A] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ K4 @ V2 ) @ S5 ) ) ) ) ).
% set_to_map_def
thf(fact_300_K__record__comp,axiom,
! [C: $tType,B: $tType,A: $tType,C2: B,F: A > C] :
( ( comp @ C @ B @ A
@ ^ [X3: C] : C2
@ F )
= ( ^ [X3: A] : C2 ) ) ).
% K_record_comp
thf(fact_301_pred__equals__eq2,axiom,
! [B: $tType,A: $tType,R5: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
( ( ( ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R5 ) )
= ( ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ S2 ) ) )
= ( R5 = S2 ) ) ).
% pred_equals_eq2
thf(fact_302_dbl__inc__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_inc @ A )
= ( ^ [X3: A] : ( plus_plus @ A @ ( plus_plus @ A @ X3 @ X3 ) @ ( one_one @ A ) ) ) ) ) ).
% dbl_inc_def
thf(fact_303_zero__natural_Orsp,axiom,
( ( zero_zero @ nat )
= ( zero_zero @ nat ) ) ).
% zero_natural.rsp
thf(fact_304_fails__wait,axiom,
! [X: nat,H3: heap_ext @ product_unit] :
~ ( time_fails @ product_unit @ ( heap_Time_wait @ X ) @ H3 ) ).
% fails_wait
thf(fact_305_Euclid__induct,axiom,
! [P: nat > nat > $o,A3: nat,B2: nat] :
( ! [A5: nat,B3: nat] :
( ( P @ A5 @ B3 )
= ( P @ B3 @ A5 ) )
=> ( ! [A5: nat] : ( P @ A5 @ ( zero_zero @ nat ) )
=> ( ! [A5: nat,B3: nat] :
( ( P @ A5 @ B3 )
=> ( P @ A5 @ ( plus_plus @ nat @ A5 @ B3 ) ) )
=> ( P @ A3 @ B2 ) ) ) ) ).
% Euclid_induct
thf(fact_306_lookup__chain,axiom,
! [B: $tType,A: $tType] :
( ( heap @ B )
=> ! [R: ref @ B,F: heap_Time_Heap @ A] :
( ( heap_Time_bind @ B @ A @ ( ref_lookup @ B @ R )
@ ^ [Uu: B] : F )
= ( heap_Time_bind @ product_unit @ A @ ( heap_Time_wait @ ( one_one @ nat ) )
@ ^ [Uu: product_unit] : F ) ) ) ).
% lookup_chain
thf(fact_307_Ints__odd__less__0,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
=> ( ( ord_less @ A @ ( plus_plus @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ).
% Ints_odd_less_0
thf(fact_308_num__of__nat__plus__distrib,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( num_of_nat @ ( plus_plus @ nat @ M2 @ N ) )
= ( plus_plus @ num @ ( num_of_nat @ M2 ) @ ( num_of_nat @ N ) ) ) ) ) ).
% num_of_nat_plus_distrib
thf(fact_309_divides__aux__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [Q3: A,R: A] :
( ( unique5940410009612947441es_aux @ A @ ( product_Pair @ A @ A @ Q3 @ R ) )
= ( R
= ( zero_zero @ A ) ) ) ) ).
% divides_aux_eq
thf(fact_310_sgn__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( sgn_sgn @ A @ A3 )
= ( one_one @ A ) ) ) ) ).
% sgn_pos
thf(fact_311_abs__add__one__gt__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] : ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( abs_abs @ A @ X ) ) ) ) ).
% abs_add_one_gt_zero
thf(fact_312_map__option__case,axiom,
! [A: $tType,B: $tType] :
( ( map_option @ B @ A )
= ( ^ [F2: B > A] :
( case_option @ ( option @ A ) @ B @ ( none @ A )
@ ^ [X3: B] : ( some @ A @ ( F2 @ X3 ) ) ) ) ) ).
% map_option_case
thf(fact_313_add__neg__numeral__special_I7_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( plus_plus @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% add_neg_numeral_special(7)
thf(fact_314_add__neg__numeral__special_I8_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% add_neg_numeral_special(8)
thf(fact_315_dbl__dec__simps_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_dec @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% dbl_dec_simps(3)
thf(fact_316_neg__equal__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( ( uminus_uminus @ A @ A3 )
= ( uminus_uminus @ A @ B2 ) )
= ( A3 = B2 ) ) ) ).
% neg_equal_iff_equal
thf(fact_317_add_Oinverse__inverse,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A3 ) )
= A3 ) ) ).
% add.inverse_inverse
thf(fact_318_abs__idempotent,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( abs_abs @ A @ ( abs_abs @ A @ A3 ) )
= ( abs_abs @ A @ A3 ) ) ) ).
% abs_idempotent
thf(fact_319_neg__equal__zero,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ( uminus_uminus @ A @ A3 )
= A3 )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% neg_equal_zero
thf(fact_320_equal__neg__zero,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( A3
= ( uminus_uminus @ A @ A3 ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% equal_neg_zero
thf(fact_321_neg__equal__0__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( ( uminus_uminus @ A @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% neg_equal_0_iff_equal
thf(fact_322_neg__0__equal__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( ( zero_zero @ A )
= ( uminus_uminus @ A @ A3 ) )
= ( ( zero_zero @ A )
= A3 ) ) ) ).
% neg_0_equal_iff_equal
thf(fact_323_add_Oinverse__neutral,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( uminus_uminus @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% add.inverse_neutral
thf(fact_324_neg__less__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ).
% neg_less_iff_less
thf(fact_325_minus__add__distrib,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( uminus_uminus @ A @ ( plus_plus @ A @ A3 @ B2 ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ).
% minus_add_distrib
thf(fact_326_minus__add__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ ( plus_plus @ A @ A3 @ B2 ) )
= B2 ) ) ).
% minus_add_cancel
thf(fact_327_add__minus__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( plus_plus @ A @ A3 @ ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) )
= B2 ) ) ).
% add_minus_cancel
thf(fact_328_abs__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ( ( abs_abs @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% abs_zero
thf(fact_329_abs__eq__0,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ( abs_abs @ A @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% abs_eq_0
thf(fact_330_abs__0__eq,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ( zero_zero @ A )
= ( abs_abs @ A @ A3 ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% abs_0_eq
thf(fact_331_abs__add__abs,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] :
( ( abs_abs @ A @ ( plus_plus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) )
= ( plus_plus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% abs_add_abs
thf(fact_332_abs__1,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( abs_abs @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% abs_1
thf(fact_333_abs__minus__cancel,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( abs_abs @ A @ ( uminus_uminus @ A @ A3 ) )
= ( abs_abs @ A @ A3 ) ) ) ).
% abs_minus_cancel
thf(fact_334_sgn__1,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( sgn_sgn @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% sgn_1
thf(fact_335_map__option__eq__Some,axiom,
! [A: $tType,B: $tType,F: B > A,Xo: option @ B,Y: A] :
( ( ( map_option @ B @ A @ F @ Xo )
= ( some @ A @ Y ) )
= ( ? [Z5: B] :
( ( Xo
= ( some @ B @ Z5 ) )
& ( ( F @ Z5 )
= Y ) ) ) ) ).
% map_option_eq_Some
thf(fact_336_None__eq__map__option__iff,axiom,
! [A: $tType,B: $tType,F: B > A,X: option @ B] :
( ( ( none @ A )
= ( map_option @ B @ A @ F @ X ) )
= ( X
= ( none @ B ) ) ) ).
% None_eq_map_option_iff
thf(fact_337_map__option__is__None,axiom,
! [A: $tType,B: $tType,F: B > A,Opt: option @ B] :
( ( ( map_option @ B @ A @ F @ Opt )
= ( none @ A ) )
= ( Opt
= ( none @ B ) ) ) ).
% map_option_is_None
thf(fact_338_option_Omap__disc__iff,axiom,
! [B: $tType,A: $tType,F: A > B,A3: option @ A] :
( ( ( map_option @ A @ B @ F @ A3 )
= ( none @ B ) )
= ( A3
= ( none @ A ) ) ) ).
% option.map_disc_iff
thf(fact_339_neg__less__0__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% neg_less_0_iff_less
thf(fact_340_neg__0__less__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% neg_0_less_iff_less
thf(fact_341_neg__less__pos,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A3 ) @ A3 )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% neg_less_pos
thf(fact_342_less__neg__neg,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% less_neg_neg
thf(fact_343_ab__left__minus,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% ab_left_minus
thf(fact_344_add_Oright__inverse,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ A3 @ ( uminus_uminus @ A @ A3 ) )
= ( zero_zero @ A ) ) ) ).
% add.right_inverse
thf(fact_345_zero__less__abs__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( abs_abs @ A @ A3 ) )
= ( A3
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_abs_iff
thf(fact_346_abs__neg__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( abs_abs @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( one_one @ A ) ) ) ).
% abs_neg_one
thf(fact_347_case__map__option,axiom,
! [B: $tType,A: $tType,C: $tType,G: A,H3: B > A,F: C > B,X: option @ C] :
( ( case_option @ A @ B @ G @ H3 @ ( map_option @ C @ B @ F @ X ) )
= ( case_option @ A @ C @ G @ ( comp @ B @ A @ C @ H3 @ F ) @ X ) ) ).
% case_map_option
thf(fact_348_dbl__inc__simps_I4_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_inc @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% dbl_inc_simps(4)
thf(fact_349_abs__sgn__eq__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( sgn_sgn @ A @ A3 ) )
= ( one_one @ A ) ) ) ) ).
% abs_sgn_eq_1
thf(fact_350_dbl__dec__simps_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_dec @ A @ ( zero_zero @ A ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% dbl_dec_simps(2)
thf(fact_351_sgn__neg,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( sgn_sgn @ A @ A3 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% sgn_neg
thf(fact_352_minus__equation__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( ( uminus_uminus @ A @ A3 )
= B2 )
= ( ( uminus_uminus @ A @ B2 )
= A3 ) ) ) ).
% minus_equation_iff
thf(fact_353_equation__minus__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( A3
= ( uminus_uminus @ A @ B2 ) )
= ( B2
= ( uminus_uminus @ A @ A3 ) ) ) ) ).
% equation_minus_iff
thf(fact_354_same__sgn__abs__add,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B2: A,A3: A] :
( ( ( sgn_sgn @ A @ B2 )
= ( sgn_sgn @ A @ A3 ) )
=> ( ( abs_abs @ A @ ( plus_plus @ A @ A3 @ B2 ) )
= ( plus_plus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) ) ) ).
% same_sgn_abs_add
thf(fact_355_sgn__minus__1,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( sgn_sgn @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% sgn_minus_1
thf(fact_356_option_Omap__ident,axiom,
! [A: $tType,T2: option @ A] :
( ( map_option @ A @ A
@ ^ [X3: A] : X3
@ T2 )
= T2 ) ).
% option.map_ident
thf(fact_357_abs__of__neg,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ A3 )
= ( uminus_uminus @ A @ A3 ) ) ) ) ).
% abs_of_neg
thf(fact_358_abs__sgn__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ( A3
= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( sgn_sgn @ A @ A3 ) )
= ( zero_zero @ A ) ) )
& ( ( A3
!= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( sgn_sgn @ A @ A3 ) )
= ( one_one @ A ) ) ) ) ) ).
% abs_sgn_eq
thf(fact_359_abs__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( abs_abs @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% abs_one
thf(fact_360_map__option_Ocompositionality,axiom,
! [B: $tType,C: $tType,A: $tType,F: B > C,G: A > B,Option: option @ A] :
( ( map_option @ B @ C @ F @ ( map_option @ A @ B @ G @ Option ) )
= ( map_option @ A @ C @ ( comp @ B @ C @ A @ F @ G ) @ Option ) ) ).
% map_option.compositionality
thf(fact_361_option_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,G: B > C,F: A > B,V: option @ A] :
( ( map_option @ B @ C @ G @ ( map_option @ A @ B @ F @ V ) )
= ( map_option @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ V ) ) ).
% option.map_comp
thf(fact_362_map__option_Ocomp,axiom,
! [C: $tType,B: $tType,A: $tType,F: B > C,G: A > B] :
( ( comp @ ( option @ B ) @ ( option @ C ) @ ( option @ A ) @ ( map_option @ B @ C @ F ) @ ( map_option @ A @ B @ G ) )
= ( map_option @ A @ C @ ( comp @ B @ C @ A @ F @ G ) ) ) ).
% map_option.comp
thf(fact_363_minus__less__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
= ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ A3 ) ) ) ).
% minus_less_iff
thf(fact_364_less__minus__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
= ( ord_less @ A @ B2 @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% less_minus_iff
thf(fact_365_is__num__normalize_I8_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [A3: A,B2: A] :
( ( uminus_uminus @ A @ ( plus_plus @ A @ A3 @ B2 ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% is_num_normalize(8)
thf(fact_366_add_Oinverse__distrib__swap,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( uminus_uminus @ A @ ( plus_plus @ A @ A3 @ B2 ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% add.inverse_distrib_swap
thf(fact_367_group__cancel_Oneg1,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A4: A,K: A,A3: A] :
( ( A4
= ( plus_plus @ A @ K @ A3 ) )
=> ( ( uminus_uminus @ A @ A4 )
= ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ) ).
% group_cancel.neg1
thf(fact_368_one__neq__neg__one,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ( ( one_one @ A )
!= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% one_neq_neg_one
thf(fact_369_map__option__cong,axiom,
! [B: $tType,A: $tType,X: option @ A,Y: option @ A,F: A > B,G: A > B] :
( ( X = Y )
=> ( ! [A5: A] :
( ( Y
= ( some @ A @ A5 ) )
=> ( ( F @ A5 )
= ( G @ A5 ) ) )
=> ( ( map_option @ A @ B @ F @ X )
= ( map_option @ A @ B @ G @ Y ) ) ) ) ).
% map_option_cong
thf(fact_370_option_Osimps_I9_J,axiom,
! [B: $tType,A: $tType,F: A > B,X22: A] :
( ( map_option @ A @ B @ F @ ( some @ A @ X22 ) )
= ( some @ B @ ( F @ X22 ) ) ) ).
% option.simps(9)
thf(fact_371_option_Osimps_I8_J,axiom,
! [A: $tType,B: $tType,F: A > B] :
( ( map_option @ A @ B @ F @ ( none @ A ) )
= ( none @ B ) ) ).
% option.simps(8)
thf(fact_372_same__sgn__sgn__add,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B2: A,A3: A] :
( ( ( sgn_sgn @ A @ B2 )
= ( sgn_sgn @ A @ A3 ) )
=> ( ( sgn_sgn @ A @ ( plus_plus @ A @ A3 @ B2 ) )
= ( sgn_sgn @ A @ A3 ) ) ) ) ).
% same_sgn_sgn_add
thf(fact_373_Ints__add,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A3: A,B2: A] :
( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
=> ( ( member @ A @ B2 @ ( ring_1_Ints @ A ) )
=> ( member @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( ring_1_Ints @ A ) ) ) ) ) ).
% Ints_add
thf(fact_374_success__lookupI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R: ref @ A,H3: heap_ext @ product_unit] : ( heap_Time_success @ A @ ( ref_lookup @ A @ R ) @ H3 ) ) ).
% success_lookupI
thf(fact_375_Ints__1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( member @ A @ ( one_one @ A ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_1
thf(fact_376_Ints__nonzero__abs__less1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] :
( ( member @ A @ X @ ( ring_1_Ints @ A ) )
=> ( ( ord_less @ A @ ( abs_abs @ A @ X ) @ ( one_one @ A ) )
=> ( X
= ( zero_zero @ A ) ) ) ) ) ).
% Ints_nonzero_abs_less1
thf(fact_377_sgn__if,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( sgn_sgn @ A )
= ( ^ [X3: A] :
( if @ A
@ ( X3
= ( zero_zero @ A ) )
@ ( zero_zero @ A )
@ ( if @ A @ ( ord_less @ A @ ( zero_zero @ A ) @ X3 ) @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ) ).
% sgn_if
thf(fact_378_sgn__1__neg,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ( sgn_sgn @ A @ A3 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% sgn_1_neg
thf(fact_379_abs__not__less__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ ( abs_abs @ A @ A3 ) @ ( zero_zero @ A ) ) ) ).
% abs_not_less_zero
thf(fact_380_abs__of__pos,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( abs_abs @ A @ A3 )
= A3 ) ) ) ).
% abs_of_pos
thf(fact_381_option_Orec__o__map,axiom,
! [B: $tType,C: $tType,A: $tType,G: C,Ga: B > C,F: A > B] :
( ( comp @ ( option @ B ) @ C @ ( option @ A ) @ ( rec_option @ C @ B @ G @ Ga ) @ ( map_option @ A @ B @ F ) )
= ( rec_option @ C @ A @ G
@ ^ [X3: A] : ( Ga @ ( F @ X3 ) ) ) ) ).
% option.rec_o_map
thf(fact_382_neg__eq__iff__add__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( ( uminus_uminus @ A @ A3 )
= B2 )
= ( ( plus_plus @ A @ A3 @ B2 )
= ( zero_zero @ A ) ) ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_383_eq__neg__iff__add__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( A3
= ( uminus_uminus @ A @ B2 ) )
= ( ( plus_plus @ A @ A3 @ B2 )
= ( zero_zero @ A ) ) ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_384_add_Oinverse__unique,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( ( plus_plus @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ A3 )
= B2 ) ) ) ).
% add.inverse_unique
thf(fact_385_ab__group__add__class_Oab__left__minus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% ab_group_add_class.ab_left_minus
thf(fact_386_add__eq__0__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( ( plus_plus @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
= ( B2
= ( uminus_uminus @ A @ A3 ) ) ) ) ).
% add_eq_0_iff
thf(fact_387_zero__neq__neg__one,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ( ( zero_zero @ A )
!= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% zero_neq_neg_one
thf(fact_388_less__minus__one__simps_I4_J,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% less_minus_one_simps(4)
thf(fact_389_less__minus__one__simps_I2_J,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) ) ) ).
% less_minus_one_simps(2)
thf(fact_390_option_Omap__sel,axiom,
! [B: $tType,A: $tType,A3: option @ A,F: A > B] :
( ( A3
!= ( none @ A ) )
=> ( ( the2 @ B @ ( map_option @ A @ B @ F @ A3 ) )
= ( F @ ( the2 @ A @ A3 ) ) ) ) ).
% option.map_sel
thf(fact_391_Ints__double__eq__0__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [A3: A] :
( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
=> ( ( ( plus_plus @ A @ A3 @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ) ).
% Ints_double_eq_0_iff
thf(fact_392_less__minus__one__simps_I3_J,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% less_minus_one_simps(3)
thf(fact_393_less__minus__one__simps_I1_J,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ).
% less_minus_one_simps(1)
thf(fact_394_sgn__1__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ( sgn_sgn @ A @ A3 )
= ( one_one @ A ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% sgn_1_pos
thf(fact_395_Ints__odd__nonzero,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [A3: A] :
( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
=> ( ( plus_plus @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ A3 )
!= ( zero_zero @ A ) ) ) ) ).
% Ints_odd_nonzero
thf(fact_396_map__option__o__empty,axiom,
! [C: $tType,B: $tType,A: $tType,F: C > B] :
( ( comp @ ( option @ C ) @ ( option @ B ) @ A @ ( map_option @ C @ B @ F )
@ ^ [X3: A] : ( none @ C ) )
= ( ^ [X3: A] : ( none @ B ) ) ) ).
% map_option_o_empty
thf(fact_397_execute__lookup,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R: ref @ A,H3: heap_ext @ product_unit] :
( ( heap_Time_execute @ A @ ( ref_lookup @ A @ R ) @ H3 )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( ref_get @ A @ H3 @ R ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ ( one_one @ nat ) ) ) ) ) ) ).
% execute_lookup
thf(fact_398_change__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( ref_change @ A )
= ( ^ [F2: A > A,R3: ref @ A] :
( heap_Time_bind @ A @ A @ ( ref_lookup @ A @ R3 )
@ ^ [X3: A] :
( heap_Time_bind @ product_unit @ A @ ( ref_update @ A @ R3 @ ( F2 @ X3 ) )
@ ^ [Uu: product_unit] : ( heap_Time_return @ A @ ( F2 @ X3 ) ) ) ) ) ) ) ).
% change_def
thf(fact_399_fails__assert_H,axiom,
! [P: $o,H3: heap_ext @ product_unit] :
( ( time_fails @ product_unit @ ( refine_Imp_assert @ P ) @ H3 )
= ~ P ) ).
% fails_assert'
thf(fact_400_effect__lookupI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [H4: heap_ext @ product_unit,H3: heap_ext @ product_unit,X: A,R: ref @ A,N: nat] :
( ( H4 = H3 )
=> ( ( X
= ( ref_get @ A @ H3 @ R ) )
=> ( ( N
= ( one_one @ nat ) )
=> ( heap_Time_effect @ A @ ( ref_lookup @ A @ R ) @ H3 @ H4 @ X @ N ) ) ) ) ) ).
% effect_lookupI
thf(fact_401_effect__lookupE,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R: ref @ A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,X: A,N: nat] :
( ( heap_Time_effect @ A @ ( ref_lookup @ A @ R ) @ H3 @ H4 @ X @ N )
=> ~ ( ( H4 = H3 )
=> ( ( X
= ( ref_get @ A @ H3 @ R ) )
=> ( N
!= ( one_one @ nat ) ) ) ) ) ) ).
% effect_lookupE
thf(fact_402_add_Ogroup__axioms,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( group @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) @ ( uminus_uminus @ A ) ) ) ).
% add.group_axioms
thf(fact_403_num__of__nat__double,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( num_of_nat @ ( plus_plus @ nat @ N @ N ) )
= ( bit0 @ ( num_of_nat @ N ) ) ) ) ).
% num_of_nat_double
thf(fact_404_Ints__nonzero__abs__ge1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] :
( ( member @ A @ X @ ( ring_1_Ints @ A ) )
=> ( ( X
!= ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( abs_abs @ A @ X ) ) ) ) ) ).
% Ints_nonzero_abs_ge1
thf(fact_405_lookup__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( ref_lookup @ A )
= ( ^ [R3: ref @ A] :
( heap_Time_tap @ A
@ ^ [H: heap_ext @ product_unit] : ( ref_get @ A @ H @ R3 ) ) ) ) ) ).
% lookup_def
thf(fact_406_map__conv__bind__option,axiom,
! [A: $tType,B: $tType] :
( ( map_option @ B @ A )
= ( ^ [F2: B > A,X3: option @ B] : ( bind @ B @ A @ X3 @ ( comp @ A @ ( option @ A ) @ B @ ( some @ A ) @ F2 ) ) ) ) ).
% map_conv_bind_option
thf(fact_407_nat__add__left__cancel__le,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ K @ M2 ) @ ( plus_plus @ nat @ K @ N ) )
= ( ord_less_eq @ nat @ M2 @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_408_less__eq__option__None__code,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: option @ A] : ( ord_less_eq @ ( option @ A ) @ ( none @ A ) @ X ) ) ).
% less_eq_option_None_code
thf(fact_409_bind__assoc,axiom,
! [B: $tType,A: $tType,C: $tType,X: option @ C,F: C > ( option @ B ),G: B > ( option @ A )] :
( ( bind @ B @ A @ ( bind @ C @ B @ X @ F ) @ G )
= ( bind @ C @ A @ X
@ ^ [Y3: C] : ( bind @ B @ A @ ( F @ Y3 ) @ G ) ) ) ).
% bind_assoc
thf(fact_410_le__zero__eq,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N: A] :
( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% le_zero_eq
thf(fact_411_add__le__cancel__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) )
= ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% add_le_cancel_left
thf(fact_412_add__le__cancel__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
= ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% add_le_cancel_right
thf(fact_413_neg__le__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% neg_le_iff_le
thf(fact_414_less__eq__option__Some,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ ( option @ A ) @ ( some @ A @ X ) @ ( some @ A @ Y ) )
= ( ord_less_eq @ A @ X @ Y ) ) ) ).
% less_eq_option_Some
thf(fact_415_less__eq__option__Some__None,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] :
~ ( ord_less_eq @ ( option @ A ) @ ( some @ A @ X ) @ ( none @ A ) ) ) ).
% less_eq_option_Some_None
thf(fact_416_bind__runit,axiom,
! [A: $tType,X: option @ A] :
( ( bind @ A @ A @ X @ ( some @ A ) )
= X ) ).
% bind_runit
thf(fact_417_bind__rzero,axiom,
! [B: $tType,A: $tType,X: option @ B] :
( ( bind @ B @ A @ X
@ ^ [X3: B] : ( none @ A ) )
= ( none @ A ) ) ).
% bind_rzero
thf(fact_418_add__le__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ B2 @ A3 ) @ B2 )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% add_le_same_cancel1
thf(fact_419_add__le__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% add_le_same_cancel2
thf(fact_420_le__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( plus_plus @ A @ A3 @ B2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% le_add_same_cancel1
thf(fact_421_le__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( plus_plus @ A @ B2 @ A3 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% le_add_same_cancel2
thf(fact_422_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_423_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ A3 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_424_neg__0__le__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% neg_0_le_iff_le
thf(fact_425_neg__le__0__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% neg_le_0_iff_le
thf(fact_426_less__eq__neg__nonpos,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% less_eq_neg_nonpos
thf(fact_427_neg__less__eq__nonneg,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ A3 )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% neg_less_eq_nonneg
thf(fact_428_abs__of__nonneg,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( abs_abs @ A @ A3 )
= A3 ) ) ) ).
% abs_of_nonneg
thf(fact_429_abs__le__self__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ A3 )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% abs_le_self_iff
thf(fact_430_abs__le__zero__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% abs_le_zero_iff
thf(fact_431_abs__of__nonpos,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ A3 )
= ( uminus_uminus @ A @ A3 ) ) ) ) ).
% abs_of_nonpos
thf(fact_432_ord__eq__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( A3 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ( C2 = D3 )
=> ( ord_less_eq @ A @ A3 @ D3 ) ) ) ) ) ).
% ord_eq_le_eq_trans
thf(fact_433_bind__option__cong__code,axiom,
! [B: $tType,A: $tType,X: option @ A,Y: option @ A,F: A > ( option @ B )] :
( ( X = Y )
=> ( ( bind @ A @ B @ X @ F )
= ( bind @ A @ B @ Y @ F ) ) ) ).
% bind_option_cong_code
thf(fact_434_group_Oleft__cancel,axiom,
! [A: $tType,F: A > A > A,Z2: A,Inverse: A > A,A3: A,B2: A,C2: A] :
( ( group @ A @ F @ Z2 @ Inverse )
=> ( ( ( F @ A3 @ B2 )
= ( F @ A3 @ C2 ) )
= ( B2 = C2 ) ) ) ).
% group.left_cancel
thf(fact_435_group_Oleft__inverse,axiom,
! [A: $tType,F: A > A > A,Z2: A,Inverse: A > A,A3: A] :
( ( group @ A @ F @ Z2 @ Inverse )
=> ( ( F @ ( Inverse @ A3 ) @ A3 )
= Z2 ) ) ).
% group.left_inverse
thf(fact_436_group_Oright__cancel,axiom,
! [A: $tType,F: A > A > A,Z2: A,Inverse: A > A,B2: A,A3: A,C2: A] :
( ( group @ A @ F @ Z2 @ Inverse )
=> ( ( ( F @ B2 @ A3 )
= ( F @ C2 @ A3 ) )
= ( B2 = C2 ) ) ) ).
% group.right_cancel
thf(fact_437_group_Oright__inverse,axiom,
! [A: $tType,F: A > A > A,Z2: A,Inverse: A > A,A3: A] :
( ( group @ A @ F @ Z2 @ Inverse )
=> ( ( F @ A3 @ ( Inverse @ A3 ) )
= Z2 ) ) ).
% group.right_inverse
thf(fact_438_group_Oinverse__unique,axiom,
! [A: $tType,F: A > A > A,Z2: A,Inverse: A > A,A3: A,B2: A] :
( ( group @ A @ F @ Z2 @ Inverse )
=> ( ( ( F @ A3 @ B2 )
= Z2 )
=> ( ( Inverse @ A3 )
= B2 ) ) ) ).
% group.inverse_unique
thf(fact_439_group_Oinverse__inverse,axiom,
! [A: $tType,F: A > A > A,Z2: A,Inverse: A > A,A3: A] :
( ( group @ A @ F @ Z2 @ Inverse )
=> ( ( Inverse @ ( Inverse @ A3 ) )
= A3 ) ) ).
% group.inverse_inverse
thf(fact_440_group_Oinverse__neutral,axiom,
! [A: $tType,F: A > A > A,Z2: A,Inverse: A > A] :
( ( group @ A @ F @ Z2 @ Inverse )
=> ( ( Inverse @ Z2 )
= Z2 ) ) ).
% group.inverse_neutral
thf(fact_441_group_Ogroup__left__neutral,axiom,
! [A: $tType,F: A > A > A,Z2: A,Inverse: A > A,A3: A] :
( ( group @ A @ F @ Z2 @ Inverse )
=> ( ( F @ Z2 @ A3 )
= A3 ) ) ).
% group.group_left_neutral
thf(fact_442_group_Oinverse__distrib__swap,axiom,
! [A: $tType,F: A > A > A,Z2: A,Inverse: A > A,A3: A,B2: A] :
( ( group @ A @ F @ Z2 @ Inverse )
=> ( ( Inverse @ ( F @ A3 @ B2 ) )
= ( F @ ( Inverse @ B2 ) @ ( Inverse @ A3 ) ) ) ) ).
% group.inverse_distrib_swap
thf(fact_443_accp__subset,axiom,
! [A: $tType,R12: A > A > $o,R23: A > A > $o] :
( ( ord_less_eq @ ( A > A > $o ) @ R12 @ R23 )
=> ( ord_less_eq @ ( A > $o ) @ ( accp @ A @ R23 ) @ ( accp @ A @ R12 ) ) ) ).
% accp_subset
thf(fact_444_le__some__optE,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [M2: A,X: option @ A] :
( ( ord_less_eq @ ( option @ A ) @ ( some @ A @ M2 ) @ X )
=> ~ ! [M4: A] :
( ( X
= ( some @ A @ M4 ) )
=> ~ ( ord_less_eq @ A @ M2 @ M4 ) ) ) ) ).
% le_some_optE
thf(fact_445_pred__subset__eq2,axiom,
! [B: $tType,A: $tType,R5: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( A > B > $o )
@ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R5 )
@ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ S2 ) )
= ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R5 @ S2 ) ) ).
% pred_subset_eq2
thf(fact_446_less__eq__option__def,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less_eq @ ( option @ A ) )
= ( ^ [X3: option @ A,Y3: option @ A] :
( case_option @ $o @ A @ $true
@ ^ [Z5: A] : ( case_option @ $o @ A @ $false @ ( ord_less_eq @ A @ Z5 ) @ Y3 )
@ X3 ) ) ) ) ).
% less_eq_option_def
thf(fact_447_zero__le,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% zero_le
thf(fact_448_add__mono__thms__linordered__semiring_I3_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( K = L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_449_add__mono__thms__linordered__semiring_I2_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_450_add__mono__thms__linordered__semiring_I1_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_451_add__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D3 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ D3 ) ) ) ) ) ).
% add_mono
thf(fact_452_add__left__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% add_left_mono
thf(fact_453_less__eqE,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ~ ! [C4: A] :
( B2
!= ( plus_plus @ A @ A3 @ C4 ) ) ) ) ).
% less_eqE
thf(fact_454_add__right__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% add_right_mono
thf(fact_455_le__iff__add,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A7: A,B5: A] :
? [C3: A] :
( B5
= ( plus_plus @ A @ A7 @ C3 ) ) ) ) ) ).
% le_iff_add
thf(fact_456_add__le__imp__le__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% add_le_imp_le_left
thf(fact_457_add__le__imp__le__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% add_le_imp_le_right
thf(fact_458_accp__subset__induct,axiom,
! [A: $tType,D4: A > $o,R5: A > A > $o,X: A,P: A > $o] :
( ( ord_less_eq @ ( A > $o ) @ D4 @ ( accp @ A @ R5 ) )
=> ( ! [X2: A,Z3: A] :
( ( D4 @ X2 )
=> ( ( R5 @ Z3 @ X2 )
=> ( D4 @ Z3 ) ) )
=> ( ( D4 @ X )
=> ( ! [X2: A] :
( ( D4 @ X2 )
=> ( ! [Z6: A] :
( ( R5 @ Z6 @ X2 )
=> ( P @ Z6 ) )
=> ( P @ X2 ) ) )
=> ( P @ X ) ) ) ) ) ).
% accp_subset_induct
thf(fact_459_le__numeral__extra_I4_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% le_numeral_extra(4)
thf(fact_460_subrelI,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ! [X2: A,Y4: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y4 ) @ R )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y4 ) @ S ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S ) ) ).
% subrelI
thf(fact_461_le__minus__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
= ( ord_less_eq @ A @ B2 @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% le_minus_iff
thf(fact_462_minus__le__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ A3 ) ) ) ).
% minus_le_iff
thf(fact_463_le__imp__neg__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% le_imp_neg_le
thf(fact_464_exists__leI,axiom,
! [N: nat,P: nat > $o] :
( ( ! [N7: nat] :
( ( ord_less @ nat @ N7 @ N )
=> ~ ( P @ N7 ) )
=> ( P @ N ) )
=> ? [N4: nat] :
( ( ord_less_eq @ nat @ N4 @ N )
& ( P @ N4 ) ) ) ).
% exists_leI
thf(fact_465_nat__le__iff__add,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M: nat,N5: nat] :
? [K4: nat] :
( N5
= ( plus_plus @ nat @ M @ K4 ) ) ) ) ).
% nat_le_iff_add
thf(fact_466_trans__le__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ I @ ( plus_plus @ nat @ M2 @ J ) ) ) ).
% trans_le_add2
thf(fact_467_trans__le__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ I @ ( plus_plus @ nat @ J @ M2 ) ) ) ).
% trans_le_add1
thf(fact_468_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_469_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ nat @ K @ L )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_470_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq @ nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus @ nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_471_add__leD2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M2 @ K ) @ N )
=> ( ord_less_eq @ nat @ K @ N ) ) ).
% add_leD2
thf(fact_472_add__leD1,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M2 @ K ) @ N )
=> ( ord_less_eq @ nat @ M2 @ N ) ) ).
% add_leD1
thf(fact_473_le__add2,axiom,
! [N: nat,M2: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ M2 @ N ) ) ).
% le_add2
thf(fact_474_le__add1,axiom,
! [N: nat,M2: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ N @ M2 ) ) ).
% le_add1
thf(fact_475_add__leE,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M2 @ K ) @ N )
=> ~ ( ( ord_less_eq @ nat @ M2 @ N )
=> ~ ( ord_less_eq @ nat @ K @ N ) ) ) ).
% add_leE
thf(fact_476_abs__le__D1,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ B2 )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% abs_le_D1
thf(fact_477_abs__ge__self,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ ( abs_abs @ A @ A3 ) ) ) ).
% abs_ge_self
thf(fact_478_less__eq__option__None,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: option @ A] : ( ord_less_eq @ ( option @ A ) @ ( none @ A ) @ X ) ) ).
% less_eq_option_None
thf(fact_479_less__eq__option__None__is__None,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: option @ A] :
( ( ord_less_eq @ ( option @ A ) @ X @ ( none @ A ) )
=> ( X
= ( none @ A ) ) ) ) ).
% less_eq_option_None_is_None
thf(fact_480_bind_Obind__lunit,axiom,
! [B: $tType,A: $tType,X: A,F: A > ( option @ B )] :
( ( bind @ A @ B @ ( some @ A @ X ) @ F )
= ( F @ X ) ) ).
% bind.bind_lunit
thf(fact_481_Option_Obind__cong,axiom,
! [B: $tType,A: $tType,X: option @ A,Y: option @ A,F: A > ( option @ B ),G: A > ( option @ B )] :
( ( X = Y )
=> ( ! [A5: A] :
( ( Y
= ( some @ A @ A5 ) )
=> ( ( F @ A5 )
= ( G @ A5 ) ) )
=> ( ( bind @ A @ B @ X @ F )
= ( bind @ A @ B @ Y @ G ) ) ) ) ).
% Option.bind_cong
thf(fact_482_bind__eq__Some__conv,axiom,
! [A: $tType,B: $tType,F: option @ B,G: B > ( option @ A ),X: A] :
( ( ( bind @ B @ A @ F @ G )
= ( some @ A @ X ) )
= ( ? [Y3: B] :
( ( F
= ( some @ B @ Y3 ) )
& ( ( G @ Y3 )
= ( some @ A @ X ) ) ) ) ) ).
% bind_eq_Some_conv
thf(fact_483_bind_Obind__lzero,axiom,
! [A: $tType,B: $tType,F: A > ( option @ B )] :
( ( bind @ A @ B @ ( none @ A ) @ F )
= ( none @ B ) ) ).
% bind.bind_lzero
thf(fact_484_map__option__bind,axiom,
! [A: $tType,B: $tType,C: $tType,F: B > A,X: option @ C,G: C > ( option @ B )] :
( ( map_option @ B @ A @ F @ ( bind @ C @ B @ X @ G ) )
= ( bind @ C @ A @ X @ ( comp @ ( option @ B ) @ ( option @ A ) @ C @ ( map_option @ B @ A @ F ) @ G ) ) ) ).
% map_option_bind
thf(fact_485_add__decreasing,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C2 ) @ B2 ) ) ) ) ).
% add_decreasing
thf(fact_486_add__increasing,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A3 @ C2 ) ) ) ) ) ).
% add_increasing
thf(fact_487_add__decreasing2,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C2 ) @ B2 ) ) ) ) ).
% add_decreasing2
thf(fact_488_add__increasing2,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A3 @ C2 ) ) ) ) ) ).
% add_increasing2
thf(fact_489_add__nonneg__nonneg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_490_add__nonpos__nonpos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_nonpos_nonpos
thf(fact_491_add__nonneg__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ( plus_plus @ A @ X @ Y )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_492_add__nonpos__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) )
=> ( ( ( plus_plus @ A @ X @ Y )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_493_not__one__le__zero,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% not_one_le_zero
thf(fact_494_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_495_zero__less__one__class_Ozero__le__one,axiom,
! [A: $tType] :
( ( zero_less_one @ A )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% zero_less_one_class.zero_le_one
thf(fact_496_add__mono__thms__linordered__field_I4_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_497_add__mono__thms__linordered__field_I3_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_498_add__le__less__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ C2 @ D3 )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ D3 ) ) ) ) ) ).
% add_le_less_mono
thf(fact_499_add__less__le__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D3 )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ D3 ) ) ) ) ) ).
% add_less_le_mono
thf(fact_500_le__minus__one__simps_I2_J,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) ) ) ).
% le_minus_one_simps(2)
thf(fact_501_le__minus__one__simps_I4_J,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% le_minus_one_simps(4)
thf(fact_502_abs__ge__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( abs_abs @ A @ A3 ) ) ) ).
% abs_ge_zero
thf(fact_503_abs__triangle__ineq,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( plus_plus @ A @ A3 @ B2 ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% abs_triangle_ineq
thf(fact_504_abs__leI,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ B2 ) ) ) ) ).
% abs_leI
thf(fact_505_abs__le__D2,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ B2 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ).
% abs_le_D2
thf(fact_506_abs__le__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ B2 )
= ( ( ord_less_eq @ A @ A3 @ B2 )
& ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ) ).
% abs_le_iff
thf(fact_507_abs__ge__minus__self,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ ( abs_abs @ A @ A3 ) ) ) ).
% abs_ge_minus_self
thf(fact_508_mono__nat__linear__lb,axiom,
! [F: nat > nat,M2: nat,K: nat] :
( ! [M5: nat,N3: nat] :
( ( ord_less @ nat @ M5 @ N3 )
=> ( ord_less @ nat @ ( F @ M5 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( F @ M2 ) @ K ) @ ( F @ ( plus_plus @ nat @ M2 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_509_nat__geq__1__eq__neqz,axiom,
! [X: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ X )
= ( X
!= ( zero_zero @ nat ) ) ) ).
% nat_geq_1_eq_neqz
thf(fact_510_bind__split__asm,axiom,
! [A: $tType,B: $tType,P: ( option @ A ) > $o,M2: option @ B,F: B > ( option @ A )] :
( ( P @ ( bind @ B @ A @ M2 @ F ) )
= ( ~ ( ( ( M2
= ( none @ B ) )
& ~ ( P @ ( none @ A ) ) )
| ? [X3: B] :
( ( M2
= ( some @ B @ X3 ) )
& ~ ( P @ ( F @ X3 ) ) ) ) ) ) ).
% bind_split_asm
thf(fact_511_bind__split,axiom,
! [A: $tType,B: $tType,P: ( option @ A ) > $o,M2: option @ B,F: B > ( option @ A )] :
( ( P @ ( bind @ B @ A @ M2 @ F ) )
= ( ( ( M2
= ( none @ B ) )
=> ( P @ ( none @ A ) ) )
& ! [V2: B] :
( ( M2
= ( some @ B @ V2 ) )
=> ( P @ ( F @ V2 ) ) ) ) ) ).
% bind_split
thf(fact_512_mlex__leq,axiom,
! [A: $tType,F: A > nat,X: A,Y: A,R5: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ nat @ ( F @ X ) @ ( F @ Y ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R5 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( mlex_prod @ A @ F @ R5 ) ) ) ) ).
% mlex_leq
thf(fact_513_bind__eq__None__conv,axiom,
! [B: $tType,A: $tType,A3: option @ B,F: B > ( option @ A )] :
( ( ( bind @ B @ A @ A3 @ F )
= ( none @ A ) )
= ( ( A3
= ( none @ B ) )
| ( ( F @ ( the2 @ B @ A3 ) )
= ( none @ A ) ) ) ) ).
% bind_eq_None_conv
thf(fact_514_bind__map__option,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > B,X: option @ C,G: B > ( option @ A )] :
( ( bind @ B @ A @ ( map_option @ C @ B @ F @ X ) @ G )
= ( bind @ C @ A @ X @ ( comp @ B @ ( option @ A ) @ C @ G @ F ) ) ) ).
% bind_map_option
thf(fact_515_add__neg__nonpos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_neg_nonpos
thf(fact_516_add__nonneg__pos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% add_nonneg_pos
thf(fact_517_add__nonpos__neg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_nonpos_neg
thf(fact_518_add__pos__nonneg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% add_pos_nonneg
thf(fact_519_add__strict__increasing,axiom,
! [A: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less @ A @ B2 @ ( plus_plus @ A @ A3 @ C2 ) ) ) ) ) ).
% add_strict_increasing
thf(fact_520_add__strict__increasing2,axiom,
! [A: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ B2 @ ( plus_plus @ A @ A3 @ C2 ) ) ) ) ) ).
% add_strict_increasing2
thf(fact_521_discrete,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( ord_less @ A )
= ( ^ [A7: A] : ( ord_less_eq @ A @ ( plus_plus @ A @ A7 @ ( one_one @ A ) ) ) ) ) ) ).
% discrete
thf(fact_522_le__minus__one__simps_I1_J,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ).
% le_minus_one_simps(1)
thf(fact_523_le__minus__one__simps_I3_J,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% le_minus_one_simps(3)
thf(fact_524_dense__eq0__I,axiom,
! [A: $tType] :
( ( ( ordere166539214618696060dd_abs @ A )
& ( dense_linorder @ A ) )
=> ! [X: A] :
( ! [E3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E3 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ X ) @ E3 ) )
=> ( X
= ( zero_zero @ A ) ) ) ) ).
% dense_eq0_I
thf(fact_525_abs__minus__le__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( abs_abs @ A @ A3 ) ) @ ( zero_zero @ A ) ) ) ).
% abs_minus_le_zero
thf(fact_526_success__changeI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [F: A > A,R: ref @ A,H3: heap_ext @ product_unit] : ( heap_Time_success @ A @ ( ref_change @ A @ F @ R ) @ H3 ) ) ).
% success_changeI
thf(fact_527_semiring__norm_I6_J,axiom,
! [M2: num,N: num] :
( ( plus_plus @ num @ ( bit0 @ M2 ) @ ( bit0 @ N ) )
= ( bit0 @ ( plus_plus @ num @ M2 @ N ) ) ) ).
% semiring_norm(6)
thf(fact_528_field__le__epsilon,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ! [E3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E3 )
=> ( ord_less_eq @ A @ X @ ( plus_plus @ A @ Y @ E3 ) ) )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% field_le_epsilon
thf(fact_529_ex__has__greatest__nat__lemma,axiom,
! [A: $tType,P: A > $o,K: A,F: A > nat,N: nat] :
( ( P @ K )
=> ( ! [X2: A] :
( ( P @ X2 )
=> ? [Y6: A] :
( ( P @ Y6 )
& ~ ( ord_less_eq @ nat @ ( F @ Y6 ) @ ( F @ X2 ) ) ) )
=> ? [Y4: A] :
( ( P @ Y4 )
& ~ ( ord_less @ nat @ ( F @ Y4 ) @ ( plus_plus @ nat @ ( F @ K ) @ N ) ) ) ) ) ).
% ex_has_greatest_nat_lemma
thf(fact_530_map__comp__def,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( map_comp @ B @ C @ A )
= ( ^ [F2: B > ( option @ C ),G2: A > ( option @ B ),K4: A] : ( case_option @ ( option @ C ) @ B @ ( none @ C ) @ F2 @ ( G2 @ K4 ) ) ) ) ).
% map_comp_def
thf(fact_531_convex__bound__lt,axiom,
! [A: $tType] :
( ( linord715952674999750819strict @ A )
=> ! [X: A,A3: A,Y: A,U: A,V: A] :
( ( ord_less @ A @ X @ A3 )
=> ( ( ord_less @ A @ Y @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ U )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ V )
=> ( ( ( plus_plus @ A @ U @ V )
= ( one_one @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ U @ X ) @ ( times_times @ A @ V @ Y ) ) @ A3 ) ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_532_map__comp__empty_I1_J,axiom,
! [C: $tType,B: $tType,A: $tType,M2: C > ( option @ B )] :
( ( map_comp @ C @ B @ A @ M2
@ ^ [X3: A] : ( none @ C ) )
= ( ^ [X3: A] : ( none @ B ) ) ) ).
% map_comp_empty(1)
thf(fact_533_map__comp__empty_I2_J,axiom,
! [B: $tType,D: $tType,C: $tType,M2: C > ( option @ B )] :
( ( map_comp @ B @ D @ C
@ ^ [X3: B] : ( none @ D )
@ M2 )
= ( ^ [X3: C] : ( none @ D ) ) ) ).
% map_comp_empty(2)
thf(fact_534_map__comp__None__iff,axiom,
! [A: $tType,C: $tType,B: $tType,M1: B > ( option @ A ),M22: C > ( option @ B ),K: C] :
( ( ( map_comp @ B @ A @ C @ M1 @ M22 @ K )
= ( none @ A ) )
= ( ( ( M22 @ K )
= ( none @ B ) )
| ? [K6: B] :
( ( ( M22 @ K )
= ( some @ B @ K6 ) )
& ( ( M1 @ K6 )
= ( none @ A ) ) ) ) ) ).
% map_comp_None_iff
thf(fact_535_Ints__eq__abs__less1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( member @ A @ X @ ( ring_1_Ints @ A ) )
=> ( ( member @ A @ Y @ ( ring_1_Ints @ A ) )
=> ( ( X = Y )
= ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ Y ) ) @ ( one_one @ A ) ) ) ) ) ) ).
% Ints_eq_abs_less1
thf(fact_536_nat0__intermed__int__val,axiom,
! [N: nat,F: nat > int,K: int] :
( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ N )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F @ ( plus_plus @ nat @ I2 @ ( one_one @ nat ) ) ) @ ( F @ I2 ) ) ) @ ( one_one @ int ) ) )
=> ( ( ord_less_eq @ int @ ( F @ ( zero_zero @ nat ) ) @ K )
=> ( ( ord_less_eq @ int @ K @ ( F @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq @ nat @ I2 @ N )
& ( ( F @ I2 )
= K ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_537_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus @ nat @ ( minus_minus @ nat @ I @ J ) @ K )
= ( minus_minus @ nat @ I @ ( plus_plus @ nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_538_nat__1__eq__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( ( one_one @ nat )
= ( times_times @ nat @ M2 @ N ) )
= ( ( M2
= ( one_one @ nat ) )
& ( N
= ( one_one @ nat ) ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_539_nat__mult__eq__1__iff,axiom,
! [M2: nat,N: nat] :
( ( ( times_times @ nat @ M2 @ N )
= ( one_one @ nat ) )
= ( ( M2
= ( one_one @ nat ) )
& ( N
= ( one_one @ nat ) ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_540_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ A3 @ A3 )
= ( zero_zero @ A ) ) ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_541_diff__zero,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% diff_zero
thf(fact_542_zero__diff,axiom,
! [A: $tType] :
( ( comm_monoid_diff @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ ( zero_zero @ A ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% zero_diff
thf(fact_543_diff__0__right,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% diff_0_right
thf(fact_544_diff__self,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ A3 @ A3 )
= ( zero_zero @ A ) ) ) ).
% diff_self
thf(fact_545_mult_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A] :
( ( times_times @ A @ A3 @ ( one_one @ A ) )
= A3 ) ) ).
% mult.right_neutral
thf(fact_546_mult__1,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( one_one @ A ) @ A3 )
= A3 ) ) ).
% mult_1
thf(fact_547_add__diff__cancel__right_H,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A3: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
= A3 ) ) ).
% add_diff_cancel_right'
thf(fact_548_add__diff__cancel__right,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ A3 @ B2 ) ) ) ).
% add_diff_cancel_right
thf(fact_549_add__diff__cancel__left_H,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A3: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ A3 )
= B2 ) ) ).
% add_diff_cancel_left'
thf(fact_550_add__diff__cancel__left,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) )
= ( minus_minus @ A @ A3 @ B2 ) ) ) ).
% add_diff_cancel_left
thf(fact_551_diff__add__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( plus_plus @ A @ ( minus_minus @ A @ A3 @ B2 ) @ B2 )
= A3 ) ) ).
% diff_add_cancel
thf(fact_552_add__diff__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
= A3 ) ) ).
% add_diff_cancel
thf(fact_553_minus__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( uminus_uminus @ A @ ( minus_minus @ A @ A3 @ B2 ) )
= ( minus_minus @ A @ B2 @ A3 ) ) ) ).
% minus_diff_eq
thf(fact_554_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq @ nat @ K @ J )
=> ( ( plus_plus @ nat @ I @ ( minus_minus @ nat @ J @ K ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_555_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq @ nat @ K @ J )
=> ( ( plus_plus @ nat @ ( minus_minus @ nat @ J @ K ) @ I )
= ( minus_minus @ nat @ ( plus_plus @ nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_556_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq @ nat @ K @ J )
=> ( ( minus_minus @ nat @ I @ ( minus_minus @ nat @ J @ K ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_557_zle__add1__eq__le,axiom,
! [W: int,Z2: int] :
( ( ord_less @ int @ W @ ( plus_plus @ int @ Z2 @ ( one_one @ int ) ) )
= ( ord_less_eq @ int @ W @ Z2 ) ) ).
% zle_add1_eq_le
thf(fact_558_map__comp__simps_I2_J,axiom,
! [B: $tType,C: $tType,A: $tType,M22: B > ( option @ A ),K: B,K7: A,M1: A > ( option @ C )] :
( ( ( M22 @ K )
= ( some @ A @ K7 ) )
=> ( ( map_comp @ A @ C @ B @ M1 @ M22 @ K )
= ( M1 @ K7 ) ) ) ).
% map_comp_simps(2)
thf(fact_559_map__comp__simps_I1_J,axiom,
! [B: $tType,A: $tType,C: $tType,M22: B > ( option @ A ),K: B,M1: A > ( option @ C )] :
( ( ( M22 @ K )
= ( none @ A ) )
=> ( ( map_comp @ A @ C @ B @ M1 @ M22 @ K )
= ( none @ C ) ) ) ).
% map_comp_simps(1)
thf(fact_560_diff__ge__0__iff__ge,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A3 @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% diff_ge_0_iff_ge
thf(fact_561_zero__comp__diff__simps_I1_J,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A3 @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% zero_comp_diff_simps(1)
thf(fact_562_diff__gt__0__iff__gt,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A3 @ B2 ) )
= ( ord_less @ A @ B2 @ A3 ) ) ) ).
% diff_gt_0_iff_gt
thf(fact_563_zero__comp__diff__simps_I2_J,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A3 @ B2 ) )
= ( ord_less @ A @ B2 @ A3 ) ) ) ).
% zero_comp_diff_simps(2)
thf(fact_564_le__add__diff__inverse,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( plus_plus @ A @ B2 @ ( minus_minus @ A @ A3 @ B2 ) )
= A3 ) ) ) ).
% le_add_diff_inverse
thf(fact_565_le__add__diff__inverse2,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( plus_plus @ A @ ( minus_minus @ A @ A3 @ B2 ) @ B2 )
= A3 ) ) ) ).
% le_add_diff_inverse2
thf(fact_566_mult__cancel__left1,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [C2: A,B2: A] :
( ( C2
= ( times_times @ A @ C2 @ B2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( B2
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_left1
thf(fact_567_mult__cancel__left2,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [C2: A,A3: A] :
( ( ( times_times @ A @ C2 @ A3 )
= C2 )
= ( ( C2
= ( zero_zero @ A ) )
| ( A3
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_left2
thf(fact_568_mult__cancel__right1,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [C2: A,B2: A] :
( ( C2
= ( times_times @ A @ B2 @ C2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( B2
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_right1
thf(fact_569_mult__cancel__right2,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [A3: A,C2: A] :
( ( ( times_times @ A @ A3 @ C2 )
= C2 )
= ( ( C2
= ( zero_zero @ A ) )
| ( A3
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_right2
thf(fact_570_diff__add__zero,axiom,
! [A: $tType] :
( ( comm_monoid_diff @ A )
=> ! [A3: A,B2: A] :
( ( minus_minus @ A @ A3 @ ( plus_plus @ A @ A3 @ B2 ) )
= ( zero_zero @ A ) ) ) ).
% diff_add_zero
thf(fact_571_diff__numeral__special_I9_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( minus_minus @ A @ ( one_one @ A ) @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% diff_numeral_special(9)
thf(fact_572_diff__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ ( zero_zero @ A ) @ A3 )
= ( uminus_uminus @ A @ A3 ) ) ) ).
% diff_0
thf(fact_573_mult__minus1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z2: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ Z2 )
= ( uminus_uminus @ A @ Z2 ) ) ) ).
% mult_minus1
thf(fact_574_mult__minus1__right,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z2: A] :
( ( times_times @ A @ Z2 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ Z2 ) ) ) ).
% mult_minus1_right
thf(fact_575_uminus__add__conv__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
= ( minus_minus @ A @ B2 @ A3 ) ) ) ).
% uminus_add_conv_diff
thf(fact_576_diff__minus__eq__add,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( minus_minus @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
= ( plus_plus @ A @ A3 @ B2 ) ) ) ).
% diff_minus_eq_add
thf(fact_577_diff__numeral__special_I12_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% diff_numeral_special(12)
thf(fact_578_le__imp__0__less,axiom,
! [Z2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
=> ( ord_less @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( one_one @ int ) @ Z2 ) ) ) ).
% le_imp_0_less
thf(fact_579_q__pos__lemma,axiom,
! [B4: int,Q4: int,R4: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( times_times @ int @ B4 @ Q4 ) @ R4 ) )
=> ( ( ord_less @ int @ R4 @ B4 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B4 )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ Q4 ) ) ) ) ).
% q_pos_lemma
thf(fact_580_zdiv__mono2__lemma,axiom,
! [B2: int,Q3: int,R: int,B4: int,Q4: int,R4: int] :
( ( ( plus_plus @ int @ ( times_times @ int @ B2 @ Q3 ) @ R )
= ( plus_plus @ int @ ( times_times @ int @ B4 @ Q4 ) @ R4 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( times_times @ int @ B4 @ Q4 ) @ R4 ) )
=> ( ( ord_less @ int @ R4 @ B4 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B4 )
=> ( ( ord_less_eq @ int @ B4 @ B2 )
=> ( ord_less_eq @ int @ Q3 @ Q4 ) ) ) ) ) ) ) ).
% zdiv_mono2_lemma
thf(fact_581_zdiv__mono2__neg__lemma,axiom,
! [B2: int,Q3: int,R: int,B4: int,Q4: int,R4: int] :
( ( ( plus_plus @ int @ ( times_times @ int @ B2 @ Q3 ) @ R )
= ( plus_plus @ int @ ( times_times @ int @ B4 @ Q4 ) @ R4 ) )
=> ( ( ord_less @ int @ ( plus_plus @ int @ ( times_times @ int @ B4 @ Q4 ) @ R4 ) @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ R @ B2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R4 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B4 )
=> ( ( ord_less_eq @ int @ B4 @ B2 )
=> ( ord_less_eq @ int @ Q4 @ Q3 ) ) ) ) ) ) ) ).
% zdiv_mono2_neg_lemma
thf(fact_582_unique__quotient__lemma,axiom,
! [B2: int,Q4: int,R4: int,Q3: int,R: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R4 ) @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q3 ) @ R ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R4 )
=> ( ( ord_less @ int @ R4 @ B2 )
=> ( ( ord_less @ int @ R @ B2 )
=> ( ord_less_eq @ int @ Q4 @ Q3 ) ) ) ) ) ).
% unique_quotient_lemma
thf(fact_583_unique__quotient__lemma__neg,axiom,
! [B2: int,Q4: int,R4: int,Q3: int,R: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R4 ) @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q3 ) @ R ) )
=> ( ( ord_less_eq @ int @ R @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ B2 @ R )
=> ( ( ord_less @ int @ B2 @ R4 )
=> ( ord_less_eq @ int @ Q3 @ Q4 ) ) ) ) ) ).
% unique_quotient_lemma_neg
thf(fact_584_int__ge__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_eq @ int @ K @ I )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq @ int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus @ int @ I2 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_ge_induct
thf(fact_585_int__induct,axiom,
! [P: int > $o,K: int,I: int] :
( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq @ int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus @ int @ I2 @ ( one_one @ int ) ) ) ) )
=> ( ! [I2: int] :
( ( ord_less_eq @ int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus @ int @ I2 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_induct
thf(fact_586_zless__imp__add1__zle,axiom,
! [W: int,Z2: int] :
( ( ord_less @ int @ W @ Z2 )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ W @ ( one_one @ int ) ) @ Z2 ) ) ).
% zless_imp_add1_zle
thf(fact_587_add1__zle__eq,axiom,
! [W: int,Z2: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ W @ ( one_one @ int ) ) @ Z2 )
= ( ord_less @ int @ W @ Z2 ) ) ).
% add1_zle_eq
thf(fact_588_nat__eq__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M2 )
= ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M2 )
= N ) ) ) ).
% nat_eq_add_iff1
thf(fact_589_nat__eq__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M2 )
= ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( M2
= ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_590_nat__le__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M2 ) @ N ) ) ) ).
% nat_le_add_iff1
thf(fact_591_nat__le__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( ord_less_eq @ nat @ M2 @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_le_add_iff2
thf(fact_592_nat__diff__add__eq1,axiom,
! [J: nat,I: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M2 ) @ N ) ) ) ).
% nat_diff_add_eq1
thf(fact_593_nat__diff__add__eq2,axiom,
! [I: nat,J: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( minus_minus @ nat @ M2 @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_594_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M3: nat] :
( ( P @ X )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq @ nat @ X2 @ M3 ) )
=> ~ ! [M5: nat] :
( ( P @ M5 )
=> ~ ! [X7: nat] :
( ( P @ X7 )
=> ( ord_less_eq @ nat @ X7 @ M5 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_595_subset__Collect__conv,axiom,
! [A: $tType,S2: set @ A,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ S2 @ ( collect @ A @ P ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ S2 )
=> ( P @ X3 ) ) ) ) ).
% subset_Collect_conv
thf(fact_596_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A3 @ C2 ) @ B2 )
= ( minus_minus @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_597_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( times_times @ A @ B2 @ ( times_times @ A @ A3 @ C2 ) )
= ( times_times @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% ab_semigroup_mult_class.mult.left_commute
thf(fact_598_ab__semigroup__mult__class_Omult_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ( ( times_times @ A )
= ( ^ [A7: A,B5: A] : ( times_times @ A @ B5 @ A7 ) ) ) ) ).
% ab_semigroup_mult_class.mult.commute
thf(fact_599_mult_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_mult @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( times_times @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
= ( times_times @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% mult.assoc
thf(fact_600_diff__eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ( minus_minus @ A @ A3 @ B2 )
= ( minus_minus @ A @ C2 @ D3 ) )
=> ( ( A3 = B2 )
= ( C2 = D3 ) ) ) ) ).
% diff_eq_diff_eq
thf(fact_601_one__integer_Orsp,axiom,
( ( one_one @ int )
= ( one_one @ int ) ) ).
% one_integer.rsp
thf(fact_602_int__gr__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less @ int @ K @ I )
=> ( ( P @ ( plus_plus @ int @ K @ ( one_one @ int ) ) )
=> ( ! [I2: int] :
( ( ord_less @ int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus @ int @ I2 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_gr_induct
thf(fact_603_zless__add1__eq,axiom,
! [W: int,Z2: int] :
( ( ord_less @ int @ W @ ( plus_plus @ int @ Z2 @ ( one_one @ int ) ) )
= ( ( ord_less @ int @ W @ Z2 )
| ( W = Z2 ) ) ) ).
% zless_add1_eq
thf(fact_604_odd__nonzero,axiom,
! [Z2: int] :
( ( plus_plus @ int @ ( plus_plus @ int @ ( one_one @ int ) @ Z2 ) @ Z2 )
!= ( zero_zero @ int ) ) ).
% odd_nonzero
thf(fact_605_odd__less__0__iff,axiom,
! [Z2: int] :
( ( ord_less @ int @ ( plus_plus @ int @ ( plus_plus @ int @ ( one_one @ int ) @ Z2 ) @ Z2 ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ Z2 @ ( zero_zero @ int ) ) ) ).
% odd_less_0_iff
thf(fact_606_eq__add__iff1,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,E2: A,C2: A,B2: A,D3: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ A3 @ E2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ B2 @ E2 ) @ D3 ) )
= ( ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A3 @ B2 ) @ E2 ) @ C2 )
= D3 ) ) ) ).
% eq_add_iff1
thf(fact_607_eq__add__iff2,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,E2: A,C2: A,B2: A,D3: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ A3 @ E2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ B2 @ E2 ) @ D3 ) )
= ( C2
= ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A3 ) @ E2 ) @ D3 ) ) ) ) ).
% eq_add_iff2
thf(fact_608_square__diff__square__factored,axiom,
! [A: $tType] :
( ( comm_ring @ A )
=> ! [X: A,Y: A] :
( ( minus_minus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) )
= ( times_times @ A @ ( plus_plus @ A @ X @ Y ) @ ( minus_minus @ A @ X @ Y ) ) ) ) ).
% square_diff_square_factored
thf(fact_609_left__add__mult__distrib,axiom,
! [I: nat,U: nat,J: nat,K: nat] :
( ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ K ) )
= ( plus_plus @ nat @ ( times_times @ nat @ ( plus_plus @ nat @ I @ J ) @ U ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_610_nat__less__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M2 ) @ N ) ) ) ).
% nat_less_add_iff1
thf(fact_611_nat__less__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( ord_less @ nat @ M2 @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_less_add_iff2
thf(fact_612_le__add__iff1,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A3: A,E2: A,C2: A,B2: A,D3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ E2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E2 ) @ D3 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A3 @ B2 ) @ E2 ) @ C2 ) @ D3 ) ) ) ).
% le_add_iff1
thf(fact_613_le__add__iff2,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A3: A,E2: A,C2: A,B2: A,D3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ E2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E2 ) @ D3 ) )
= ( ord_less_eq @ A @ C2 @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A3 ) @ E2 ) @ D3 ) ) ) ) ).
% le_add_iff2
thf(fact_614_less__add__iff1,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A3: A,E2: A,C2: A,B2: A,D3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ E2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E2 ) @ D3 ) )
= ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A3 @ B2 ) @ E2 ) @ C2 ) @ D3 ) ) ) ).
% less_add_iff1
thf(fact_615_less__add__iff2,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A3: A,E2: A,C2: A,B2: A,D3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ E2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E2 ) @ D3 ) )
= ( ord_less @ A @ C2 @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A3 ) @ E2 ) @ D3 ) ) ) ) ).
% less_add_iff2
thf(fact_616_square__diff__one__factored,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: A] :
( ( minus_minus @ A @ ( times_times @ A @ X @ X ) @ ( one_one @ A ) )
= ( times_times @ A @ ( plus_plus @ A @ X @ ( one_one @ A ) ) @ ( minus_minus @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% square_diff_one_factored
thf(fact_617_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ( minus_minus @ A @ A3 @ B2 )
= ( minus_minus @ A @ C2 @ D3 ) )
=> ( ( ord_less_eq @ A @ A3 @ B2 )
= ( ord_less_eq @ A @ C2 @ D3 ) ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_618_diff__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C2 ) @ ( minus_minus @ A @ B2 @ C2 ) ) ) ) ).
% diff_right_mono
thf(fact_619_diff__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ C2 @ A3 ) @ ( minus_minus @ A @ C2 @ B2 ) ) ) ) ).
% diff_left_mono
thf(fact_620_diff__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,D3: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ D3 @ C2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C2 ) @ ( minus_minus @ A @ B2 @ D3 ) ) ) ) ) ).
% diff_mono
thf(fact_621_eq__iff__diff__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( ^ [Y5: A,Z4: A] : Y5 = Z4 )
= ( ^ [A7: A,B5: A] :
( ( minus_minus @ A @ A7 @ B5 )
= ( zero_zero @ A ) ) ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_622_diff__strict__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ord_less @ A @ ( minus_minus @ A @ A3 @ C2 ) @ ( minus_minus @ A @ B2 @ C2 ) ) ) ) ).
% diff_strict_right_mono
thf(fact_623_diff__strict__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ord_less @ A @ ( minus_minus @ A @ C2 @ A3 ) @ ( minus_minus @ A @ C2 @ B2 ) ) ) ) ).
% diff_strict_left_mono
thf(fact_624_diff__eq__diff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ( minus_minus @ A @ A3 @ B2 )
= ( minus_minus @ A @ C2 @ D3 ) )
=> ( ( ord_less @ A @ A3 @ B2 )
= ( ord_less @ A @ C2 @ D3 ) ) ) ) ).
% diff_eq_diff_less
thf(fact_625_diff__strict__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,D3: A,C2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ D3 @ C2 )
=> ( ord_less @ A @ ( minus_minus @ A @ A3 @ C2 ) @ ( minus_minus @ A @ B2 @ D3 ) ) ) ) ) ).
% diff_strict_mono
thf(fact_626_diff__diff__eq,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 )
= ( minus_minus @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% diff_diff_eq
thf(fact_627_add__implies__diff,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( ( plus_plus @ A @ C2 @ B2 )
= A3 )
=> ( C2
= ( minus_minus @ A @ A3 @ B2 ) ) ) ) ).
% add_implies_diff
thf(fact_628_diff__add__eq__diff__diff__swap,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( minus_minus @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( minus_minus @ A @ A3 @ C2 ) @ B2 ) ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_629_diff__add__eq,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 )
= ( minus_minus @ A @ ( plus_plus @ A @ A3 @ C2 ) @ B2 ) ) ) ).
% diff_add_eq
thf(fact_630_diff__diff__eq2,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( minus_minus @ A @ A3 @ ( minus_minus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A3 @ C2 ) @ B2 ) ) ) ).
% diff_diff_eq2
thf(fact_631_add__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( plus_plus @ A @ A3 @ ( minus_minus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% add_diff_eq
thf(fact_632_eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( A3
= ( minus_minus @ A @ C2 @ B2 ) )
= ( ( plus_plus @ A @ A3 @ B2 )
= C2 ) ) ) ).
% eq_diff_eq
thf(fact_633_diff__eq__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ( minus_minus @ A @ A3 @ B2 )
= C2 )
= ( A3
= ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% diff_eq_eq
thf(fact_634_group__cancel_Osub1,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A4: A,K: A,A3: A,B2: A] :
( ( A4
= ( plus_plus @ A @ K @ A3 ) )
=> ( ( minus_minus @ A @ A4 @ B2 )
= ( plus_plus @ A @ K @ ( minus_minus @ A @ A3 @ B2 ) ) ) ) ) ).
% group_cancel.sub1
thf(fact_635_minus__diff__commute,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [B2: A,A3: A] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ B2 ) @ A3 )
= ( minus_minus @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ).
% minus_diff_commute
thf(fact_636_ring__class_Oring__distribs_I2_J,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_637_ring__class_Oring__distribs_I1_J,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( times_times @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_638_comm__semiring__class_Odistrib,axiom,
! [A: $tType] :
( ( comm_semiring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% comm_semiring_class.distrib
thf(fact_639_distrib__left,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( times_times @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ).
% distrib_left
thf(fact_640_distrib__right,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% distrib_right
thf(fact_641_combine__common__factor,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A3: A,E2: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( times_times @ A @ A3 @ E2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E2 ) @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ E2 ) @ C2 ) ) ) ).
% combine_common_factor
thf(fact_642_crossproduct__eq,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [W: A,Y: A,X: A,Z2: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ W @ Y ) @ ( times_times @ A @ X @ Z2 ) )
= ( plus_plus @ A @ ( times_times @ A @ W @ Z2 ) @ ( times_times @ A @ X @ Y ) ) )
= ( ( W = X )
| ( Y = Z2 ) ) ) ) ).
% crossproduct_eq
thf(fact_643_crossproduct__noteq,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ( A3 != B2 )
& ( C2 != D3 ) )
= ( ( plus_plus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ D3 ) )
!= ( plus_plus @ A @ ( times_times @ A @ A3 @ D3 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% crossproduct_noteq
thf(fact_644_comm__monoid__mult__class_Omult__1,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( one_one @ A ) @ A3 )
= A3 ) ) ).
% comm_monoid_mult_class.mult_1
thf(fact_645_mult_Ocomm__neutral,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A] :
( ( times_times @ A @ A3 @ ( one_one @ A ) )
= A3 ) ) ).
% mult.comm_neutral
thf(fact_646_abs__minus__commute,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] :
( ( abs_abs @ A @ ( minus_minus @ A @ A3 @ B2 ) )
= ( abs_abs @ A @ ( minus_minus @ A @ B2 @ A3 ) ) ) ) ).
% abs_minus_commute
thf(fact_647_Nat_Odiff__cancel,axiom,
! [K: nat,M2: nat,N: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ K @ M2 ) @ ( plus_plus @ nat @ K @ N ) )
= ( minus_minus @ nat @ M2 @ N ) ) ).
% Nat.diff_cancel
thf(fact_648_diff__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ K ) @ ( plus_plus @ nat @ N @ K ) )
= ( minus_minus @ nat @ M2 @ N ) ) ).
% diff_cancel2
thf(fact_649_diff__add__inverse,axiom,
! [N: nat,M2: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ N @ M2 ) @ N )
= M2 ) ).
% diff_add_inverse
thf(fact_650_diff__add__inverse2,axiom,
! [M2: nat,N: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ N )
= M2 ) ).
% diff_add_inverse2
thf(fact_651_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq @ nat @ ( minus_minus @ nat @ J @ K ) @ I )
= ( ord_less_eq @ nat @ J @ ( plus_plus @ nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_652_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq @ nat @ K @ J )
=> ( ( ord_less_eq @ nat @ I @ ( minus_minus @ nat @ J @ K ) )
= ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_653_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq @ nat @ K @ J )
=> ( ( minus_minus @ nat @ ( plus_plus @ nat @ I @ J ) @ K )
= ( plus_plus @ nat @ I @ ( minus_minus @ nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_654_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq @ nat @ K @ J )
=> ( ( minus_minus @ nat @ ( plus_plus @ nat @ J @ I ) @ K )
= ( plus_plus @ nat @ ( minus_minus @ nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_655_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ( minus_minus @ nat @ J @ I )
= K )
= ( J
= ( plus_plus @ nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_656_add__mult__distrib,axiom,
! [M2: nat,N: nat,K: nat] :
( ( times_times @ nat @ ( plus_plus @ nat @ M2 @ N ) @ K )
= ( plus_plus @ nat @ ( times_times @ nat @ M2 @ K ) @ ( times_times @ nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_657_add__mult__distrib2,axiom,
! [K: nat,M2: nat,N: nat] :
( ( times_times @ nat @ K @ ( plus_plus @ nat @ M2 @ N ) )
= ( plus_plus @ nat @ ( times_times @ nat @ K @ M2 ) @ ( times_times @ nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_658_mlex__leI,axiom,
! [A3: nat,A6: nat,B2: nat,B4: nat,N6: nat] :
( ( ord_less_eq @ nat @ A3 @ A6 )
=> ( ( ord_less_eq @ nat @ B2 @ B4 )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A3 @ N6 ) @ B2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ A6 @ N6 ) @ B4 ) ) ) ) ).
% mlex_leI
thf(fact_659_nat__mult__1,axiom,
! [N: nat] :
( ( times_times @ nat @ ( one_one @ nat ) @ N )
= N ) ).
% nat_mult_1
thf(fact_660_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times @ nat @ N @ ( one_one @ nat ) )
= N ) ).
% nat_mult_1_right
thf(fact_661_mult__eq__if,axiom,
( ( times_times @ nat )
= ( ^ [M: nat,N5: nat] :
( if @ nat
@ ( M
= ( zero_zero @ nat ) )
@ ( zero_zero @ nat )
@ ( plus_plus @ nat @ N5 @ ( times_times @ nat @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) @ N5 ) ) ) ) ) ).
% mult_eq_if
thf(fact_662_lambda__one,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ( ( ^ [X3: A] : X3 )
= ( times_times @ A @ ( one_one @ A ) ) ) ) ).
% lambda_one
thf(fact_663_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 @ ( plus_plus @ A @ X @ Y ) ) ) ) ) ).
% is_num_normalize(6)
thf(fact_664_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 )
=> ( ( plus_plus @ A @ X @ Y )
= ( plus_plus @ A @ Y @ X ) ) ) ) ) ).
% is_num_add_commute
thf(fact_665_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 )
=> ( ( plus_plus @ A @ X @ ( plus_plus @ A @ Y @ Z2 ) )
= ( plus_plus @ A @ Y @ ( plus_plus @ A @ X @ Z2 ) ) ) ) ) ) ).
% is_num_add_left_commute
thf(fact_666_is__num__normalize_I4_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( neg_numeral_is_num @ A @ ( one_one @ A ) ) ) ).
% is_num_normalize(4)
thf(fact_667_le__iff__diff__le__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A7: A,B5: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ A7 @ B5 ) @ ( zero_zero @ A ) ) ) ) ) ).
% le_iff_diff_le_0
thf(fact_668_less__iff__diff__less__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ( ( ord_less @ A )
= ( ^ [A7: A,B5: A] : ( ord_less @ A @ ( minus_minus @ A @ A7 @ B5 ) @ ( zero_zero @ A ) ) ) ) ) ).
% less_iff_diff_less_0
thf(fact_669_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ( minus_minus @ A @ B2 @ A3 )
= C2 )
= ( B2
= ( plus_plus @ A @ C2 @ A3 ) ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_670_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( plus_plus @ A @ A3 @ ( minus_minus @ A @ B2 @ A3 ) )
= B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_671_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( minus_minus @ A @ C2 @ ( minus_minus @ A @ B2 @ A3 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ C2 @ A3 ) @ B2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_672_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A3 )
= ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A3 ) @ C2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_673_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A3 ) @ C2 )
= ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A3 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_674_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( minus_minus @ A @ ( plus_plus @ A @ C2 @ B2 ) @ A3 )
= ( plus_plus @ A @ C2 @ ( minus_minus @ A @ B2 @ A3 ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_675_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( plus_plus @ A @ C2 @ ( minus_minus @ A @ B2 @ A3 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ C2 @ B2 ) @ A3 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_676_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ ( minus_minus @ A @ B2 @ A3 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A3 ) @ B2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_677_le__add__diff,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ord_less_eq @ A @ C2 @ ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A3 ) ) ) ) ).
% le_add_diff
thf(fact_678_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A3 ) @ A3 )
= B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_679_le__diff__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( minus_minus @ A @ C2 @ B2 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% le_diff_eq
thf(fact_680_diff__le__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 )
= ( ord_less_eq @ A @ A3 @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% diff_le_eq
thf(fact_681_add__le__imp__le__diff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I: A,K: A,N: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ N )
=> ( ord_less_eq @ A @ I @ ( minus_minus @ A @ N @ K ) ) ) ) ).
% add_le_imp_le_diff
thf(fact_682_add__le__add__imp__diff__le,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I: A,K: A,N: A,J: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ N )
=> ( ( ord_less_eq @ A @ N @ ( plus_plus @ A @ J @ K ) )
=> ( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ N )
=> ( ( ord_less_eq @ A @ N @ ( plus_plus @ A @ J @ K ) )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ N @ K ) @ J ) ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_683_less__diff__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less @ A @ A3 @ ( minus_minus @ A @ C2 @ B2 ) )
= ( ord_less @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% less_diff_eq
thf(fact_684_diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 )
= ( ord_less @ A @ A3 @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% diff_less_eq
thf(fact_685_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,B2: A] :
( ~ ( ord_less @ A @ A3 @ B2 )
=> ( ( plus_plus @ A @ B2 @ ( minus_minus @ A @ A3 @ B2 ) )
= A3 ) ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_686_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ( ( minus_minus @ A )
= ( ^ [A7: A,B5: A] : ( plus_plus @ A @ A7 @ ( uminus_uminus @ A @ B5 ) ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_687_diff__conv__add__uminus,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( minus_minus @ A )
= ( ^ [A7: A,B5: A] : ( plus_plus @ A @ A7 @ ( uminus_uminus @ A @ B5 ) ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_688_group__cancel_Osub2,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [B6: A,K: A,B2: A,A3: A] :
( ( B6
= ( plus_plus @ A @ K @ B2 ) )
=> ( ( minus_minus @ A @ A3 @ B6 )
= ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( minus_minus @ A @ A3 @ B2 ) ) ) ) ) ).
% group_cancel.sub2
thf(fact_689_abs__triangle__ineq2__sym,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ B2 @ A3 ) ) ) ) ).
% abs_triangle_ineq2_sym
thf(fact_690_abs__triangle__ineq3,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ A3 @ B2 ) ) ) ) ).
% abs_triangle_ineq3
thf(fact_691_abs__triangle__ineq2,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ A3 @ B2 ) ) ) ) ).
% abs_triangle_ineq2
thf(fact_692_add__scale__eq__noteq,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [R: A,A3: A,B2: A,C2: A,D3: A] :
( ( R
!= ( zero_zero @ A ) )
=> ( ( ( A3 = B2 )
& ( C2 != D3 ) )
=> ( ( plus_plus @ A @ A3 @ ( times_times @ A @ R @ C2 ) )
!= ( plus_plus @ A @ B2 @ ( times_times @ A @ R @ D3 ) ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_693_less__1__mult,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [M2: A,N: A] :
( ( ord_less @ A @ ( one_one @ A ) @ M2 )
=> ( ( ord_less @ A @ ( one_one @ A ) @ N )
=> ( ord_less @ A @ ( one_one @ A ) @ ( times_times @ A @ M2 @ N ) ) ) ) ) ).
% less_1_mult
thf(fact_694_square__eq__1__iff,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [X: A] :
( ( ( times_times @ A @ X @ X )
= ( one_one @ A ) )
= ( ( X
= ( one_one @ A ) )
| ( X
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% square_eq_1_iff
thf(fact_695_diff__add__0,axiom,
! [N: nat,M2: nat] :
( ( minus_minus @ nat @ N @ ( plus_plus @ nat @ N @ M2 ) )
= ( zero_zero @ nat ) ) ).
% diff_add_0
thf(fact_696_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less @ nat @ I @ ( minus_minus @ nat @ J @ K ) )
= ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_697_add__diff__inverse__nat,axiom,
! [M2: nat,N: nat] :
( ~ ( ord_less @ nat @ M2 @ N )
=> ( ( plus_plus @ nat @ N @ ( minus_minus @ nat @ M2 @ N ) )
= M2 ) ) ).
% add_diff_inverse_nat
thf(fact_698_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq @ nat @ K @ J )
=> ( ( ord_less @ nat @ ( minus_minus @ nat @ J @ K ) @ I )
= ( ord_less @ nat @ J @ ( plus_plus @ nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_699_mlex__snd__decrI,axiom,
! [A3: nat,A6: nat,B2: nat,B4: nat,N6: nat] :
( ( A3 = A6 )
=> ( ( ord_less @ nat @ B2 @ B4 )
=> ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A3 @ N6 ) @ B2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ A6 @ N6 ) @ B4 ) ) ) ) ).
% mlex_snd_decrI
thf(fact_700_mlex__fst__decrI,axiom,
! [A3: nat,A6: nat,B2: nat,N6: nat,B4: nat] :
( ( ord_less @ nat @ A3 @ A6 )
=> ( ( ord_less @ nat @ B2 @ N6 )
=> ( ( ord_less @ nat @ B4 @ N6 )
=> ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A3 @ N6 ) @ B2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ A6 @ N6 ) @ B4 ) ) ) ) ) ).
% mlex_fst_decrI
thf(fact_701_mlex__bound,axiom,
! [A3: nat,A4: nat,B2: nat,N6: nat] :
( ( ord_less @ nat @ A3 @ A4 )
=> ( ( ord_less @ nat @ B2 @ N6 )
=> ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A3 @ N6 ) @ B2 ) @ ( times_times @ nat @ A4 @ N6 ) ) ) ) ).
% mlex_bound
thf(fact_702_mult__eq__self__implies__10,axiom,
! [M2: nat,N: nat] :
( ( M2
= ( times_times @ nat @ M2 @ N ) )
=> ( ( N
= ( one_one @ nat ) )
| ( M2
= ( zero_zero @ nat ) ) ) ) ).
% mult_eq_self_implies_10
thf(fact_703_map__comp__Some__iff,axiom,
! [A: $tType,C: $tType,B: $tType,M1: B > ( option @ A ),M22: C > ( option @ B ),K: C,V: A] :
( ( ( map_comp @ B @ A @ C @ M1 @ M22 @ K )
= ( some @ A @ V ) )
= ( ? [K6: B] :
( ( ( M22 @ K )
= ( some @ B @ K6 ) )
& ( ( M1 @ K6 )
= ( some @ A @ V ) ) ) ) ) ).
% map_comp_Some_iff
thf(fact_704_field__le__mult__one__interval,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ! [Z3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z3 )
=> ( ( ord_less @ A @ Z3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ Z3 @ X ) @ Y ) ) )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% field_le_mult_one_interval
thf(fact_705_abs__diff__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,A3: A,R: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ A3 ) ) @ R )
= ( ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ R ) @ X )
& ( ord_less_eq @ A @ X @ ( plus_plus @ A @ A3 @ R ) ) ) ) ) ).
% abs_diff_le_iff
thf(fact_706_abs__diff__triangle__ineq,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( plus_plus @ A @ C2 @ D3 ) ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ ( minus_minus @ A @ A3 @ C2 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ B2 @ D3 ) ) ) ) ) ).
% abs_diff_triangle_ineq
thf(fact_707_abs__triangle__ineq4,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ A3 @ B2 ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% abs_triangle_ineq4
thf(fact_708_abs__diff__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,A3: A,R: A] :
( ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ A3 ) ) @ R )
= ( ( ord_less @ A @ ( minus_minus @ A @ A3 @ R ) @ X )
& ( ord_less @ A @ X @ ( plus_plus @ A @ A3 @ R ) ) ) ) ) ).
% abs_diff_less_iff
thf(fact_709_sum__squares__ge__zero,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) ) ) ) ).
% sum_squares_ge_zero
thf(fact_710_mult__left__le,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [C2: A,A3: A] :
( ( ord_less_eq @ A @ C2 @ ( one_one @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ A3 ) ) ) ) ).
% mult_left_le
thf(fact_711_mult__le__one,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% mult_le_one
thf(fact_712_mult__right__le__one__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ Y @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ X @ Y ) @ X ) ) ) ) ) ).
% mult_right_le_one_le
thf(fact_713_mult__left__le__one__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ Y @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ Y @ X ) @ X ) ) ) ) ) ).
% mult_left_le_one_le
thf(fact_714_not__sum__squares__lt__zero,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [X: A,Y: A] :
~ ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) ) @ ( zero_zero @ A ) ) ) ).
% not_sum_squares_lt_zero
thf(fact_715_nat__diff__split,axiom,
! [P: nat > $o,A3: nat,B2: nat] :
( ( P @ ( minus_minus @ nat @ A3 @ B2 ) )
= ( ( ( ord_less @ nat @ A3 @ B2 )
=> ( P @ ( zero_zero @ nat ) ) )
& ! [D5: nat] :
( ( A3
= ( plus_plus @ nat @ B2 @ D5 ) )
=> ( P @ D5 ) ) ) ) ).
% nat_diff_split
thf(fact_716_nat__diff__split__asm,axiom,
! [P: nat > $o,A3: nat,B2: nat] :
( ( P @ ( minus_minus @ nat @ A3 @ B2 ) )
= ( ~ ( ( ( ord_less @ nat @ A3 @ B2 )
& ~ ( P @ ( zero_zero @ nat ) ) )
| ? [D5: nat] :
( ( A3
= ( plus_plus @ nat @ B2 @ D5 ) )
& ~ ( P @ D5 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_717_dbl__dec__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_dec @ A )
= ( ^ [X3: A] : ( minus_minus @ A @ ( plus_plus @ A @ X3 @ X3 ) @ ( one_one @ A ) ) ) ) ) ).
% dbl_dec_def
thf(fact_718_mult__le__cancel__left1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ ( times_times @ A @ C2 @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ B2 ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_719_mult__le__cancel__left2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,A3: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A3 ) @ C2 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A3 @ ( one_one @ A ) ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ A3 ) ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_720_mult__le__cancel__right1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ ( times_times @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ B2 ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_721_mult__le__cancel__right2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,C2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ C2 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A3 @ ( one_one @ A ) ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ A3 ) ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_722_mult__less__cancel__left1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,B2: A] :
( ( ord_less @ A @ C2 @ ( times_times @ A @ C2 @ B2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( one_one @ A ) @ B2 ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_723_mult__less__cancel__left2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,A3: A] :
( ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ C2 )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A3 @ ( one_one @ A ) ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( one_one @ A ) @ A3 ) ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_724_mult__less__cancel__right1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,B2: A] :
( ( ord_less @ A @ C2 @ ( times_times @ A @ B2 @ C2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( one_one @ A ) @ B2 ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_725_mult__less__cancel__right2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,C2: A] :
( ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ C2 )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A3 @ ( one_one @ A ) ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( one_one @ A ) @ A3 ) ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_726_convex__bound__le,axiom,
! [A: $tType] :
( ( linord6961819062388156250ring_1 @ A )
=> ! [X: A,A3: A,Y: A,U: A,V: A] :
( ( ord_less_eq @ A @ X @ A3 )
=> ( ( ord_less_eq @ A @ Y @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ U )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ V )
=> ( ( ( plus_plus @ A @ U @ V )
= ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ U @ X ) @ ( times_times @ A @ V @ Y ) ) @ A3 ) ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_727_is__num_Ocases,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [A3: A] :
( ( neg_numeral_is_num @ A @ A3 )
=> ( ( A3
!= ( one_one @ A ) )
=> ( ! [X2: A] :
( ( A3
= ( uminus_uminus @ A @ X2 ) )
=> ~ ( neg_numeral_is_num @ A @ X2 ) )
=> ~ ! [X2: A,Y4: A] :
( ( A3
= ( plus_plus @ A @ X2 @ Y4 ) )
=> ( ( neg_numeral_is_num @ A @ X2 )
=> ~ ( neg_numeral_is_num @ A @ Y4 ) ) ) ) ) ) ) ).
% is_num.cases
thf(fact_728_is__num_Osimps,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_is_num @ A )
= ( ^ [A7: A] :
( ( A7
= ( one_one @ A ) )
| ? [X3: A] :
( ( A7
= ( uminus_uminus @ A @ X3 ) )
& ( neg_numeral_is_num @ A @ X3 ) )
| ? [X3: A,Y3: A] :
( ( A7
= ( plus_plus @ A @ X3 @ Y3 ) )
& ( neg_numeral_is_num @ A @ X3 )
& ( neg_numeral_is_num @ A @ Y3 ) ) ) ) ) ) ).
% is_num.simps
thf(fact_729_sum__squares__gt__zero__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) ) )
= ( ( X
!= ( zero_zero @ A ) )
| ( Y
!= ( zero_zero @ A ) ) ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_730_sum__squares__le__zero__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) ) @ ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_731_incr__lemma,axiom,
! [D3: int,Z2: int,X: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ord_less @ int @ Z2 @ ( plus_plus @ int @ X @ ( times_times @ int @ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ X @ Z2 ) ) @ ( one_one @ int ) ) @ D3 ) ) ) ) ).
% incr_lemma
thf(fact_732_decr__lemma,axiom,
! [D3: int,X: int,Z2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ord_less @ int @ ( minus_minus @ int @ X @ ( times_times @ int @ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ X @ Z2 ) ) @ ( one_one @ int ) ) @ D3 ) ) @ Z2 ) ) ).
% decr_lemma
thf(fact_733_sum__squares__eq__zero__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [X: A,Y: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_734_frac__unique__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,A3: A] :
( ( ( archimedean_frac @ A @ X )
= A3 )
= ( ( member @ A @ ( minus_minus @ A @ X @ A3 ) @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ A3 @ ( one_one @ A ) ) ) ) ) ).
% frac_unique_iff
thf(fact_735_scaling__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [U: A,V: A,R: A,S: A] :
( ( ord_less_eq @ A @ U @ V )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ R )
=> ( ( ord_less_eq @ A @ R @ S )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ U @ ( divide_divide @ A @ ( times_times @ A @ R @ ( minus_minus @ A @ V @ U ) ) @ S ) ) @ V ) ) ) ) ) ).
% scaling_mono
thf(fact_736_divide__le__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
= ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% divide_le_eq_1_neg
thf(fact_737_divide__le__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
= ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ).
% divide_le_eq_1_pos
thf(fact_738_div__by__1,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ A3 @ ( one_one @ A ) )
= A3 ) ) ).
% div_by_1
thf(fact_739_div__self,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A3 @ A3 )
= ( one_one @ A ) ) ) ) ).
% div_self
thf(fact_740_zero__eq__1__divide__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ( zero_zero @ A )
= ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% zero_eq_1_divide_iff
thf(fact_741_one__divide__eq__0__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ( divide_divide @ A @ ( one_one @ A ) @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% one_divide_eq_0_iff
thf(fact_742_eq__divide__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A3: A] :
( ( ( one_one @ A )
= ( divide_divide @ A @ B2 @ A3 ) )
= ( ( A3
!= ( zero_zero @ A ) )
& ( A3 = B2 ) ) ) ) ).
% eq_divide_eq_1
thf(fact_743_divide__eq__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A3: A] :
( ( ( divide_divide @ A @ B2 @ A3 )
= ( one_one @ A ) )
= ( ( A3
!= ( zero_zero @ A ) )
& ( A3 = B2 ) ) ) ) ).
% divide_eq_eq_1
thf(fact_744_divide__self__if,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( ( A3
= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A3 @ A3 )
= ( zero_zero @ A ) ) )
& ( ( A3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A3 @ A3 )
= ( one_one @ A ) ) ) ) ) ).
% divide_self_if
thf(fact_745_divide__self,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A3 @ A3 )
= ( one_one @ A ) ) ) ) ).
% divide_self
thf(fact_746_one__eq__divide__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A] :
( ( ( one_one @ A )
= ( divide_divide @ A @ A3 @ B2 ) )
= ( ( B2
!= ( zero_zero @ A ) )
& ( A3 = B2 ) ) ) ) ).
% one_eq_divide_iff
thf(fact_747_divide__eq__1__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A] :
( ( ( divide_divide @ A @ A3 @ B2 )
= ( one_one @ A ) )
= ( ( B2
!= ( zero_zero @ A ) )
& ( A3 = B2 ) ) ) ) ).
% divide_eq_1_iff
thf(fact_748_divide__minus1,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A] :
( ( divide_divide @ A @ X @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ X ) ) ) ).
% divide_minus1
thf(fact_749_zero__le__divide__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% zero_le_divide_1_iff
thf(fact_750_divide__le__0__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% divide_le_0_1_iff
thf(fact_751_zero__less__divide__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% zero_less_divide_1_iff
thf(fact_752_less__divide__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% less_divide_eq_1_pos
thf(fact_753_less__divide__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
= ( ord_less @ A @ B2 @ A3 ) ) ) ) ).
% less_divide_eq_1_neg
thf(fact_754_divide__less__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
= ( ord_less @ A @ B2 @ A3 ) ) ) ) ).
% divide_less_eq_1_pos
thf(fact_755_divide__less__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% divide_less_eq_1_neg
thf(fact_756_divide__less__0__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% divide_less_0_1_iff
thf(fact_757_nonzero__divide__mult__cancel__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ B2 @ ( times_times @ A @ A3 @ B2 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ A3 ) ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_758_nonzero__divide__mult__cancel__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A3 @ ( times_times @ A @ A3 @ B2 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ B2 ) ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_759_le__divide__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
= ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% le_divide_eq_1_pos
thf(fact_760_le__divide__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
= ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ).
% le_divide_eq_1_neg
thf(fact_761_int__distrib_I2_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times @ int @ W @ ( plus_plus @ int @ Z1 @ Z22 ) )
= ( plus_plus @ int @ ( times_times @ int @ W @ Z1 ) @ ( times_times @ int @ W @ Z22 ) ) ) ).
% int_distrib(2)
thf(fact_762_int__distrib_I1_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times @ int @ ( plus_plus @ int @ Z1 @ Z22 ) @ W )
= ( plus_plus @ int @ ( times_times @ int @ Z1 @ W ) @ ( times_times @ int @ Z22 @ W ) ) ) ).
% int_distrib(1)
thf(fact_763_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus @ int @ ( zero_zero @ int ) @ L )
= L ) ).
% plus_int_code(2)
thf(fact_764_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus @ int @ K @ ( zero_zero @ int ) )
= K ) ).
% plus_int_code(1)
thf(fact_765_zero__integer_Orsp,axiom,
( ( zero_zero @ int )
= ( zero_zero @ int ) ) ).
% zero_integer.rsp
thf(fact_766_add__divide__distrib,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ).
% add_divide_distrib
thf(fact_767_int__div__pos__eq,axiom,
! [A3: int,B2: int,Q3: int,R: int] :
( ( A3
= ( plus_plus @ int @ ( times_times @ int @ B2 @ Q3 ) @ R ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R )
=> ( ( ord_less @ int @ R @ B2 )
=> ( ( divide_divide @ int @ A3 @ B2 )
= Q3 ) ) ) ) ).
% int_div_pos_eq
thf(fact_768_int__div__neg__eq,axiom,
! [A3: int,B2: int,Q3: int,R: int] :
( ( A3
= ( plus_plus @ int @ ( times_times @ int @ B2 @ Q3 ) @ R ) )
=> ( ( ord_less_eq @ int @ R @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ B2 @ R )
=> ( ( divide_divide @ int @ A3 @ B2 )
= Q3 ) ) ) ) ).
% int_div_neg_eq
thf(fact_769_split__zdiv,axiom,
! [P: int > $o,N: int,K: int] :
( ( P @ ( divide_divide @ int @ N @ K ) )
= ( ( ( K
= ( zero_zero @ int ) )
=> ( P @ ( zero_zero @ int ) ) )
& ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ! [I3: int,J2: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J2 )
& ( ord_less @ int @ J2 @ K )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I3 ) @ J2 ) ) )
=> ( P @ I3 ) ) )
& ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ! [I3: int,J2: int] :
( ( ( ord_less @ int @ K @ J2 )
& ( ord_less_eq @ int @ J2 @ ( zero_zero @ int ) )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I3 ) @ J2 ) ) )
=> ( P @ I3 ) ) ) ) ) ).
% split_zdiv
thf(fact_770_frac__lt__1,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less @ A @ ( archimedean_frac @ A @ X ) @ ( one_one @ A ) ) ) ).
% frac_lt_1
thf(fact_771_frac__1__eq,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( archimedean_frac @ A @ ( plus_plus @ A @ X @ ( one_one @ A ) ) )
= ( archimedean_frac @ A @ X ) ) ) ).
% frac_1_eq
thf(fact_772_right__inverse__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ A3 @ B2 )
= ( one_one @ A ) )
= ( A3 = B2 ) ) ) ) ).
% right_inverse_eq
thf(fact_773_less__divide__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ A3 @ B2 ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B2 @ A3 ) ) ) ) ) ).
% less_divide_eq_1
thf(fact_774_divide__less__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ B2 @ A3 ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ A3 @ B2 ) )
| ( A3
= ( zero_zero @ A ) ) ) ) ) ).
% divide_less_eq_1
thf(fact_775_divide__add__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,X: A,Y: A] :
( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X @ Z2 ) @ Y )
= ( divide_divide @ A @ ( plus_plus @ A @ X @ ( times_times @ A @ Y @ Z2 ) ) @ Z2 ) ) ) ) ).
% divide_add_eq_iff
thf(fact_776_add__divide__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,X: A,Y: A] :
( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ X @ ( divide_divide @ A @ Y @ Z2 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ Z2 ) @ Y ) @ Z2 ) ) ) ) ).
% add_divide_eq_iff
thf(fact_777_add__num__frac,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,Z2: A,X: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ Z2 @ ( divide_divide @ A @ X @ Y ) )
= ( divide_divide @ A @ ( plus_plus @ A @ X @ ( times_times @ A @ Z2 @ Y ) ) @ Y ) ) ) ) ).
% add_num_frac
thf(fact_778_add__frac__num,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,X: A,Z2: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X @ Y ) @ Z2 )
= ( divide_divide @ A @ ( plus_plus @ A @ X @ ( times_times @ A @ Z2 @ Y ) ) @ Y ) ) ) ) ).
% add_frac_num
thf(fact_779_add__frac__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,Z2: A,X: A,W: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X @ Y ) @ ( divide_divide @ A @ W @ Z2 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ Z2 ) @ ( times_times @ A @ W @ Y ) ) @ ( times_times @ A @ Y @ Z2 ) ) ) ) ) ) ).
% add_frac_eq
thf(fact_780_add__divide__eq__if__simps_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,A3: A,B2: A] :
( ( ( Z2
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ A3 @ ( divide_divide @ A @ B2 @ Z2 ) )
= A3 ) )
& ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ A3 @ ( divide_divide @ A @ B2 @ Z2 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ Z2 ) @ B2 ) @ Z2 ) ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_781_add__divide__eq__if__simps_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,A3: A,B2: A] :
( ( ( Z2
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ A3 @ Z2 ) @ B2 )
= B2 ) )
& ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ A3 @ Z2 ) @ B2 )
= ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ B2 @ Z2 ) ) @ Z2 ) ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_782_less__half__sum,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ord_less @ A @ A3 @ ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ) ) ) ).
% less_half_sum
thf(fact_783_gt__half__sum,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ord_less @ A @ ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) @ B2 ) ) ) ).
% gt_half_sum
thf(fact_784_divide__eq__minus__1__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A] :
( ( ( divide_divide @ A @ A3 @ B2 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( ( B2
!= ( zero_zero @ A ) )
& ( A3
= ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% divide_eq_minus_1_iff
thf(fact_785_div__pos__geq,axiom,
! [L: int,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ( ord_less_eq @ int @ L @ K )
=> ( ( divide_divide @ int @ K @ L )
= ( plus_plus @ int @ ( divide_divide @ int @ ( minus_minus @ int @ K @ L ) @ L ) @ ( one_one @ int ) ) ) ) ) ).
% div_pos_geq
thf(fact_786_frac__eq,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ( archimedean_frac @ A @ X )
= X )
= ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
& ( ord_less @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% frac_eq
thf(fact_787_frac__add,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] :
( ( ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) )
=> ( ( archimedean_frac @ A @ ( plus_plus @ A @ X @ Y ) )
= ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) ) )
& ( ~ ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) )
=> ( ( archimedean_frac @ A @ ( plus_plus @ A @ X @ Y ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) ) ) ) ) ) ).
% frac_add
thf(fact_788_incr__mult__lemma,axiom,
! [D3: int,P: int > $o,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ! [X2: int] :
( ( P @ X2 )
=> ( P @ ( plus_plus @ int @ X2 @ D3 ) ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ! [X7: int] :
( ( P @ X7 )
=> ( P @ ( plus_plus @ int @ X7 @ ( times_times @ int @ K @ D3 ) ) ) ) ) ) ) ).
% incr_mult_lemma
thf(fact_789_divide__le__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ B2 @ A3 ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ A3 @ B2 ) )
| ( A3
= ( zero_zero @ A ) ) ) ) ) ).
% divide_le_eq_1
thf(fact_790_le__divide__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ A3 @ B2 ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ) ).
% le_divide_eq_1
thf(fact_791_minus__divide__add__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,X: A,Y: A] :
( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ X @ Z2 ) ) @ Y )
= ( divide_divide @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ X ) @ ( times_times @ A @ Y @ Z2 ) ) @ Z2 ) ) ) ) ).
% minus_divide_add_eq_iff
thf(fact_792_add__divide__eq__if__simps_I3_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,A3: A,B2: A] :
( ( ( Z2
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ Z2 ) ) @ B2 )
= B2 ) )
& ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ Z2 ) ) @ B2 )
= ( divide_divide @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ ( times_times @ A @ B2 @ Z2 ) ) @ Z2 ) ) ) ) ) ).
% add_divide_eq_if_simps(3)
thf(fact_793_frac__neg,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ( member @ A @ X @ ( ring_1_Ints @ A ) )
=> ( ( archimedean_frac @ A @ ( uminus_uminus @ A @ X ) )
= ( zero_zero @ A ) ) )
& ( ~ ( member @ A @ X @ ( ring_1_Ints @ A ) )
=> ( ( archimedean_frac @ A @ ( uminus_uminus @ A @ X ) )
= ( minus_minus @ A @ ( one_one @ A ) @ ( archimedean_frac @ A @ X ) ) ) ) ) ) ).
% frac_neg
thf(fact_794_div__mult__self4,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ B2 @ C2 ) @ A3 ) @ B2 )
= ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% div_mult_self4
thf(fact_795_div__mult__self3,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ C2 @ B2 ) @ A3 ) @ B2 )
= ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% div_mult_self3
thf(fact_796_div__mult__self2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) ) @ B2 )
= ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% div_mult_self2
thf(fact_797_div__mult__self1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ C2 @ B2 ) ) @ B2 )
= ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% div_mult_self1
thf(fact_798_div__minus1__right,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ A3 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ A3 ) ) ) ).
% div_minus1_right
thf(fact_799_div__pos__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less_eq @ int @ ( plus_plus @ int @ K @ L ) @ ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ K @ L )
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ).
% div_pos_neg_trivial
thf(fact_800_bits__div__by__1,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ A3 @ ( one_one @ A ) )
= A3 ) ) ).
% bits_div_by_1
thf(fact_801_dividend__less__times__div,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ nat @ M2 @ ( plus_plus @ nat @ N @ ( times_times @ nat @ N @ ( divide_divide @ nat @ M2 @ N ) ) ) ) ) ).
% dividend_less_times_div
thf(fact_802_dividend__less__div__times,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ nat @ M2 @ ( plus_plus @ nat @ N @ ( times_times @ nat @ ( divide_divide @ nat @ M2 @ N ) @ N ) ) ) ) ).
% dividend_less_div_times
thf(fact_803_split__div,axiom,
! [P: nat > $o,M2: nat,N: nat] :
( ( P @ ( divide_divide @ nat @ M2 @ N ) )
= ( ( ( N
= ( zero_zero @ nat ) )
=> ( P @ ( zero_zero @ nat ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ! [I3: nat,J2: nat] :
( ( ord_less @ nat @ J2 @ N )
=> ( ( M2
= ( plus_plus @ nat @ ( times_times @ nat @ N @ I3 ) @ J2 ) )
=> ( P @ I3 ) ) ) ) ) ) ).
% split_div
thf(fact_804_div__less__dividend,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ord_less @ nat @ ( divide_divide @ nat @ M2 @ N ) @ M2 ) ) ) ).
% div_less_dividend
thf(fact_805_div__eq__dividend__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ( ( divide_divide @ nat @ M2 @ N )
= M2 )
= ( N
= ( one_one @ nat ) ) ) ) ).
% div_eq_dividend_iff
thf(fact_806_div__add__self1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ B2 @ A3 ) @ B2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% div_add_self1
thf(fact_807_div__add__self2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% div_add_self2
thf(fact_808_upto_Opinduct,axiom,
! [A0: int,A1: int,P: int > int > $o] :
( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ A0 @ A1 ) )
=> ( ! [I2: int,J3: int] :
( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ I2 @ J3 ) )
=> ( ( ( ord_less_eq @ int @ I2 @ J3 )
=> ( P @ ( plus_plus @ int @ I2 @ ( one_one @ int ) ) @ J3 ) )
=> ( P @ I2 @ J3 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% upto.pinduct
thf(fact_809_eucl__rel__int_Osimps,axiom,
( eucl_rel_int
= ( ^ [A12: int,A23: int,A33: product_prod @ int @ int] :
( ? [K4: int] :
( ( A12 = K4 )
& ( A23
= ( zero_zero @ int ) )
& ( A33
= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ K4 ) ) )
| ? [L3: int,K4: int,Q5: int] :
( ( A12 = K4 )
& ( A23 = L3 )
& ( A33
= ( product_Pair @ int @ int @ Q5 @ ( zero_zero @ int ) ) )
& ( L3
!= ( zero_zero @ int ) )
& ( K4
= ( times_times @ int @ Q5 @ L3 ) ) )
| ? [R3: int,L3: int,K4: int,Q5: int] :
( ( A12 = K4 )
& ( A23 = L3 )
& ( A33
= ( product_Pair @ int @ int @ Q5 @ R3 ) )
& ( ( sgn_sgn @ int @ R3 )
= ( sgn_sgn @ int @ L3 ) )
& ( ord_less @ int @ ( abs_abs @ int @ R3 ) @ ( abs_abs @ int @ L3 ) )
& ( K4
= ( plus_plus @ int @ ( times_times @ int @ Q5 @ L3 ) @ R3 ) ) ) ) ) ) ).
% eucl_rel_int.simps
thf(fact_810_eucl__rel__int_Ocases,axiom,
! [A1: int,A22: int,A32: product_prod @ int @ int] :
( ( eucl_rel_int @ A1 @ A22 @ A32 )
=> ( ( ( A22
= ( zero_zero @ int ) )
=> ( A32
!= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ A1 ) ) )
=> ( ! [Q2: int] :
( ( A32
= ( product_Pair @ int @ int @ Q2 @ ( zero_zero @ int ) ) )
=> ( ( A22
!= ( zero_zero @ int ) )
=> ( A1
!= ( times_times @ int @ Q2 @ A22 ) ) ) )
=> ~ ! [R2: int,Q2: int] :
( ( A32
= ( product_Pair @ int @ int @ Q2 @ R2 ) )
=> ( ( ( sgn_sgn @ int @ R2 )
= ( sgn_sgn @ int @ A22 ) )
=> ( ( ord_less @ int @ ( abs_abs @ int @ R2 ) @ ( abs_abs @ int @ A22 ) )
=> ( A1
!= ( plus_plus @ int @ ( times_times @ int @ Q2 @ A22 ) @ R2 ) ) ) ) ) ) ) ) ).
% eucl_rel_int.cases
thf(fact_811_eucl__rel__int__iff,axiom,
! [K: int,L: int,Q3: int,R: int] :
( ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q3 @ R ) )
= ( ( K
= ( plus_plus @ int @ ( times_times @ int @ L @ Q3 ) @ R ) )
& ( ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R )
& ( ord_less @ int @ R @ L ) ) )
& ( ~ ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ( ( ord_less @ int @ L @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ L @ R )
& ( ord_less_eq @ int @ R @ ( zero_zero @ int ) ) ) )
& ( ~ ( ord_less @ int @ L @ ( zero_zero @ int ) )
=> ( Q3
= ( zero_zero @ int ) ) ) ) ) ) ) ).
% eucl_rel_int_iff
thf(fact_812_power_Opower__eq__if,axiom,
! [A: $tType] :
( ( power2 @ A )
= ( ^ [One: A,Times: A > A > A,P6: A,M: nat] :
( if @ A
@ ( M
= ( zero_zero @ nat ) )
@ One
@ ( Times @ P6 @ ( power2 @ A @ One @ Times @ P6 @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) ) ) ) ) ).
% power.power_eq_if
thf(fact_813_split__neg__lemma,axiom,
! [K: int,P: int > int > $o,N: int] :
( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ( ( P @ ( divide_divide @ int @ N @ K ) @ ( modulo_modulo @ int @ N @ K ) )
= ( ! [I3: int,J2: int] :
( ( ( ord_less @ int @ K @ J2 )
& ( ord_less_eq @ int @ J2 @ ( zero_zero @ int ) )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I3 ) @ J2 ) ) )
=> ( P @ I3 @ J2 ) ) ) ) ) ).
% split_neg_lemma
thf(fact_814_split__pos__lemma,axiom,
! [K: int,P: int > int > $o,N: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ( P @ ( divide_divide @ int @ N @ K ) @ ( modulo_modulo @ int @ N @ K ) )
= ( ! [I3: int,J2: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J2 )
& ( ord_less @ int @ J2 @ K )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I3 ) @ J2 ) ) )
=> ( P @ I3 @ J2 ) ) ) ) ) ).
% split_pos_lemma
thf(fact_815_mod__add__self2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,B2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
= ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% mod_add_self2
thf(fact_816_mod__add__self1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A3: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ B2 @ A3 ) @ B2 )
= ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% mod_add_self1
thf(fact_817_mod__by__1,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A3: A] :
( ( modulo_modulo @ A @ A3 @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% mod_by_1
thf(fact_818_bits__mod__by__1,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A] :
( ( modulo_modulo @ A @ A3 @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% bits_mod_by_1
thf(fact_819_mod__mult__self1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ C2 @ B2 ) ) @ B2 )
= ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% mod_mult_self1
thf(fact_820_mod__mult__self2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) ) @ B2 )
= ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% mod_mult_self2
thf(fact_821_mod__mult__self3,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ ( times_times @ A @ C2 @ B2 ) @ A3 ) @ B2 )
= ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% mod_mult_self3
thf(fact_822_mod__mult__self4,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ ( times_times @ A @ B2 @ C2 ) @ A3 ) @ B2 )
= ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% mod_mult_self4
thf(fact_823_mod__minus1__right,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A] :
( ( modulo_modulo @ A @ A3 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% mod_minus1_right
thf(fact_824_mod__int__unique,axiom,
! [K: int,L: int,Q3: int,R: int] :
( ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q3 @ R ) )
=> ( ( modulo_modulo @ int @ K @ L )
= R ) ) ).
% mod_int_unique
thf(fact_825_unique__quotient,axiom,
! [A3: int,B2: int,Q3: int,R: int,Q4: int,R4: int] :
( ( eucl_rel_int @ A3 @ B2 @ ( product_Pair @ int @ int @ Q3 @ R ) )
=> ( ( eucl_rel_int @ A3 @ B2 @ ( product_Pair @ int @ int @ Q4 @ R4 ) )
=> ( Q3 = Q4 ) ) ) ).
% unique_quotient
thf(fact_826_unique__remainder,axiom,
! [A3: int,B2: int,Q3: int,R: int,Q4: int,R4: int] :
( ( eucl_rel_int @ A3 @ B2 @ ( product_Pair @ int @ int @ Q3 @ R ) )
=> ( ( eucl_rel_int @ A3 @ B2 @ ( product_Pair @ int @ int @ Q4 @ R4 ) )
=> ( R = R4 ) ) ) ).
% unique_remainder
thf(fact_827_full__exhaustive__int_H_Ocases,axiom,
! [X: product_prod @ ( ( product_prod @ int @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ int @ int )] :
~ ! [F4: ( product_prod @ int @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D2: int,I2: int] :
( X
!= ( product_Pair @ ( ( product_prod @ int @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ int @ int ) @ F4 @ ( product_Pair @ int @ int @ D2 @ I2 ) ) ) ).
% full_exhaustive_int'.cases
thf(fact_828_exhaustive__int_H_Ocases,axiom,
! [X: product_prod @ ( int > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ int @ int )] :
~ ! [F4: int > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D2: int,I2: int] :
( X
!= ( product_Pair @ ( int > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ int @ int ) @ F4 @ ( product_Pair @ int @ int @ D2 @ I2 ) ) ) ).
% exhaustive_int'.cases
thf(fact_829_small__lazy_H_Ocases,axiom,
! [X: product_prod @ int @ int] :
~ ! [D2: int,I2: int] :
( X
!= ( product_Pair @ int @ int @ D2 @ I2 ) ) ).
% small_lazy'.cases
thf(fact_830_eucl__rel__int,axiom,
! [K: int,L: int] : ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ ( divide_divide @ int @ K @ L ) @ ( modulo_modulo @ int @ K @ L ) ) ) ).
% eucl_rel_int
thf(fact_831_mod__add__right__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 )
= ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% mod_add_right_eq
thf(fact_832_mod__add__left__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ C2 ) @ B2 ) @ C2 )
= ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% mod_add_left_eq
thf(fact_833_mod__add__cong,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,C2: A,A6: A,B2: A,B4: A] :
( ( ( modulo_modulo @ A @ A3 @ C2 )
= ( modulo_modulo @ A @ A6 @ C2 ) )
=> ( ( ( modulo_modulo @ A @ B2 @ C2 )
= ( modulo_modulo @ A @ B4 @ C2 ) )
=> ( ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
= ( modulo_modulo @ A @ ( plus_plus @ A @ A6 @ B4 ) @ C2 ) ) ) ) ) ).
% mod_add_cong
thf(fact_834_mod__add__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ C2 ) @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 )
= ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% mod_add_eq
thf(fact_835_eucl__rel__int__by0,axiom,
! [K: int] : ( eucl_rel_int @ K @ ( zero_zero @ int ) @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ K ) ) ).
% eucl_rel_int_by0
thf(fact_836_div__int__unique,axiom,
! [K: int,L: int,Q3: int,R: int] :
( ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q3 @ R ) )
=> ( ( divide_divide @ int @ K @ L )
= Q3 ) ) ).
% div_int_unique
thf(fact_837_mod__eqE,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ( modulo_modulo @ A @ A3 @ C2 )
= ( modulo_modulo @ A @ B2 @ C2 ) )
=> ~ ! [D2: A] :
( B2
!= ( plus_plus @ A @ A3 @ ( times_times @ A @ C2 @ D2 ) ) ) ) ) ).
% mod_eqE
thf(fact_838_div__add1__eq,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( plus_plus @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) @ ( divide_divide @ A @ ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ C2 ) @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 ) ) ) ) ).
% div_add1_eq
thf(fact_839_mult__div__mod__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [B2: A,A3: A] :
( ( plus_plus @ A @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A3 @ B2 ) ) @ ( modulo_modulo @ A @ A3 @ B2 ) )
= A3 ) ) ).
% mult_div_mod_eq
thf(fact_840_mod__mult__div__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B2: A] :
( ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A3 @ B2 ) ) )
= A3 ) ) ).
% mod_mult_div_eq
thf(fact_841_mod__div__mult__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B2: A] :
( ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) )
= A3 ) ) ).
% mod_div_mult_eq
thf(fact_842_div__mult__mod__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B2: A] :
( ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) @ ( modulo_modulo @ A @ A3 @ B2 ) )
= A3 ) ) ).
% div_mult_mod_eq
thf(fact_843_mod__div__decomp,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B2: A] :
( A3
= ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) @ ( modulo_modulo @ A @ A3 @ B2 ) ) ) ) ).
% mod_div_decomp
thf(fact_844_cancel__div__mod__rules_I1_J,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) @ ( modulo_modulo @ A @ A3 @ B2 ) ) @ C2 )
= ( plus_plus @ A @ A3 @ C2 ) ) ) ).
% cancel_div_mod_rules(1)
thf(fact_845_cancel__div__mod__rules_I2_J,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A3 @ B2 ) ) @ ( modulo_modulo @ A @ A3 @ B2 ) ) @ C2 )
= ( plus_plus @ A @ A3 @ C2 ) ) ) ).
% cancel_div_mod_rules(2)
thf(fact_846_div__mult1__eq,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) ) @ ( divide_divide @ A @ ( times_times @ A @ A3 @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 ) ) ) ) ).
% div_mult1_eq
thf(fact_847_eucl__rel__int__dividesI,axiom,
! [L: int,K: int,Q3: int] :
( ( L
!= ( zero_zero @ int ) )
=> ( ( K
= ( times_times @ int @ Q3 @ L ) )
=> ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q3 @ ( zero_zero @ int ) ) ) ) ) ).
% eucl_rel_int_dividesI
thf(fact_848_div__mod__decomp__int,axiom,
! [A4: int,N: int] :
( A4
= ( plus_plus @ int @ ( times_times @ int @ ( divide_divide @ int @ A4 @ N ) @ N ) @ ( modulo_modulo @ int @ A4 @ N ) ) ) ).
% div_mod_decomp_int
thf(fact_849_mod__pos__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less_eq @ int @ ( plus_plus @ int @ K @ L ) @ ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ K @ L )
= ( plus_plus @ int @ K @ L ) ) ) ) ).
% mod_pos_neg_trivial
thf(fact_850_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( modulo_modulo @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ B2 @ ( modulo_modulo @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 ) ) @ ( modulo_modulo @ A @ A3 @ B2 ) ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_851_int__mod__pos__eq,axiom,
! [A3: int,B2: int,Q3: int,R: int] :
( ( A3
= ( plus_plus @ int @ ( times_times @ int @ B2 @ Q3 ) @ R ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R )
=> ( ( ord_less @ int @ R @ B2 )
=> ( ( modulo_modulo @ int @ A3 @ B2 )
= R ) ) ) ) ).
% int_mod_pos_eq
thf(fact_852_int__mod__neg__eq,axiom,
! [A3: int,B2: int,Q3: int,R: int] :
( ( A3
= ( plus_plus @ int @ ( times_times @ int @ B2 @ Q3 ) @ R ) )
=> ( ( ord_less_eq @ int @ R @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ B2 @ R )
=> ( ( modulo_modulo @ int @ A3 @ B2 )
= R ) ) ) ) ).
% int_mod_neg_eq
thf(fact_853_split__zmod,axiom,
! [P: int > $o,N: int,K: int] :
( ( P @ ( modulo_modulo @ int @ N @ K ) )
= ( ( ( K
= ( zero_zero @ int ) )
=> ( P @ N ) )
& ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ! [I3: int,J2: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J2 )
& ( ord_less @ int @ J2 @ K )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I3 ) @ J2 ) ) )
=> ( P @ J2 ) ) )
& ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ! [I3: int,J2: int] :
( ( ( ord_less @ int @ K @ J2 )
& ( ord_less_eq @ int @ J2 @ ( zero_zero @ int ) )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I3 ) @ J2 ) ) )
=> ( P @ J2 ) ) ) ) ) ).
% split_zmod
thf(fact_854_zmod__zmult2__eq,axiom,
! [C2: int,A3: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ C2 )
=> ( ( modulo_modulo @ int @ A3 @ ( times_times @ int @ B2 @ C2 ) )
= ( plus_plus @ int @ ( times_times @ int @ B2 @ ( modulo_modulo @ int @ ( divide_divide @ int @ A3 @ B2 ) @ C2 ) ) @ ( modulo_modulo @ int @ A3 @ B2 ) ) ) ) ).
% zmod_zmult2_eq
thf(fact_855_zminus1__lemma,axiom,
! [A3: int,B2: int,Q3: int,R: int] :
( ( eucl_rel_int @ A3 @ B2 @ ( product_Pair @ int @ int @ Q3 @ R ) )
=> ( ( B2
!= ( zero_zero @ int ) )
=> ( eucl_rel_int @ ( uminus_uminus @ int @ A3 ) @ B2
@ ( product_Pair @ int @ int
@ ( if @ int
@ ( R
= ( zero_zero @ int ) )
@ ( uminus_uminus @ int @ Q3 )
@ ( minus_minus @ int @ ( uminus_uminus @ int @ Q3 ) @ ( one_one @ int ) ) )
@ ( if @ int
@ ( R
= ( zero_zero @ int ) )
@ ( zero_zero @ int )
@ ( minus_minus @ int @ B2 @ R ) ) ) ) ) ) ).
% zminus1_lemma
thf(fact_856_eucl__rel__int__remainderI,axiom,
! [R: int,L: int,K: int,Q3: int] :
( ( ( sgn_sgn @ int @ R )
= ( sgn_sgn @ int @ L ) )
=> ( ( ord_less @ int @ ( abs_abs @ int @ R ) @ ( abs_abs @ int @ L ) )
=> ( ( K
= ( plus_plus @ int @ ( times_times @ int @ Q3 @ L ) @ R ) )
=> ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q3 @ R ) ) ) ) ) ).
% eucl_rel_int_remainderI
thf(fact_857_verit__le__mono__div__int,axiom,
! [A4: int,B6: int,N: int] :
( ( ord_less @ int @ A4 @ B6 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less_eq @ int
@ ( plus_plus @ int @ ( divide_divide @ int @ A4 @ N )
@ ( if @ int
@ ( ( modulo_modulo @ int @ B6 @ N )
= ( zero_zero @ int ) )
@ ( one_one @ int )
@ ( zero_zero @ int ) ) )
@ ( divide_divide @ int @ B6 @ N ) ) ) ) ).
% verit_le_mono_div_int
thf(fact_858_bezw__0,axiom,
! [X: nat] :
( ( bezw @ X @ ( zero_zero @ nat ) )
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) ).
% bezw_0
thf(fact_859_verit__le__mono__div,axiom,
! [A4: nat,B6: nat,N: nat] :
( ( ord_less @ nat @ A4 @ B6 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ nat
@ ( plus_plus @ nat @ ( divide_divide @ nat @ A4 @ N )
@ ( if @ nat
@ ( ( modulo_modulo @ nat @ B6 @ N )
= ( zero_zero @ nat ) )
@ ( one_one @ nat )
@ ( zero_zero @ nat ) ) )
@ ( divide_divide @ nat @ B6 @ N ) ) ) ) ).
% verit_le_mono_div
thf(fact_860_zero__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X ) ) ) ).
% zero_le_ceiling
thf(fact_861_ceiling__less__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( zero_zero @ int ) )
= ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_zero
thf(fact_862_floor__add,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] :
( ( ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) )
=> ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ Y ) )
= ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y ) ) ) )
& ( ~ ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) )
=> ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ Y ) )
= ( plus_plus @ int @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y ) ) @ ( one_one @ int ) ) ) ) ) ) ).
% floor_add
thf(fact_863_cpmi,axiom,
! [D4: int,P: int > $o,P2: int > $o,B6: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ? [Z6: int] :
! [X2: int] :
( ( ord_less @ int @ X2 @ Z6 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ! [X2: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B6 )
=> ( X2
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X2 )
=> ( P @ ( minus_minus @ int @ X2 @ D4 ) ) ) )
=> ( ! [X2: int,K5: int] :
( ( P2 @ X2 )
= ( P2 @ ( minus_minus @ int @ X2 @ ( times_times @ int @ K5 @ D4 ) ) ) )
=> ( ( ? [X8: int] : ( P @ X8 ) )
= ( ? [X3: int] :
( ( member @ int @ X3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
& ( P2 @ X3 ) )
| ? [X3: int] :
( ( member @ int @ X3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
& ? [Y3: int] :
( ( member @ int @ Y3 @ B6 )
& ( P @ ( plus_plus @ int @ Y3 @ X3 ) ) ) ) ) ) ) ) ) ) ).
% cpmi
thf(fact_864_cppi,axiom,
! [D4: int,P: int > $o,P2: int > $o,A4: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ? [Z6: int] :
! [X2: int] :
( ( ord_less @ int @ Z6 @ X2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ! [X2: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A4 )
=> ( X2
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X2 )
=> ( P @ ( plus_plus @ int @ X2 @ D4 ) ) ) )
=> ( ! [X2: int,K5: int] :
( ( P2 @ X2 )
= ( P2 @ ( minus_minus @ int @ X2 @ ( times_times @ int @ K5 @ D4 ) ) ) )
=> ( ( ? [X8: int] : ( P @ X8 ) )
= ( ? [X3: int] :
( ( member @ int @ X3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
& ( P2 @ X3 ) )
| ? [X3: int] :
( ( member @ int @ X3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
& ? [Y3: int] :
( ( member @ int @ Y3 @ A4 )
& ( P @ ( minus_minus @ int @ Y3 @ X3 ) ) ) ) ) ) ) ) ) ) ).
% cppi
thf(fact_865_bset_I6_J,axiom,
! [D4: int,B6: set @ int,T2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ! [X7: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B6 )
=> ( X7
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ int @ X7 @ T2 )
=> ( ord_less_eq @ int @ ( minus_minus @ int @ X7 @ D4 ) @ T2 ) ) ) ) ).
% bset(6)
thf(fact_866_bset_I8_J,axiom,
! [D4: int,T2: int,B6: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ( member @ int @ ( minus_minus @ int @ T2 @ ( one_one @ int ) ) @ B6 )
=> ! [X7: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B6 )
=> ( X7
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ int @ T2 @ X7 )
=> ( ord_less_eq @ int @ T2 @ ( minus_minus @ int @ X7 @ D4 ) ) ) ) ) ) ).
% bset(8)
thf(fact_867_aset_I6_J,axiom,
! [D4: int,T2: int,A4: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ( member @ int @ ( plus_plus @ int @ T2 @ ( one_one @ int ) ) @ A4 )
=> ! [X7: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A4 )
=> ( X7
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ int @ X7 @ T2 )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ X7 @ D4 ) @ T2 ) ) ) ) ) ).
% aset(6)
thf(fact_868_atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,L: A,U: A] :
( ( member @ A @ I @ ( set_or1337092689740270186AtMost @ A @ L @ U ) )
= ( ( ord_less_eq @ A @ L @ I )
& ( ord_less_eq @ A @ I @ U ) ) ) ) ).
% atLeastAtMost_iff
thf(fact_869_Icc__eq__Icc,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,H3: A,L4: A,H4: A] :
( ( ( set_or1337092689740270186AtMost @ A @ L @ H3 )
= ( set_or1337092689740270186AtMost @ A @ L4 @ H4 ) )
= ( ( ( L = L4 )
& ( H3 = H4 ) )
| ( ~ ( ord_less_eq @ A @ L @ H3 )
& ~ ( ord_less_eq @ A @ L4 @ H4 ) ) ) ) ) ).
% Icc_eq_Icc
thf(fact_870_atLeastatMost__subset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D3 ) )
= ( ~ ( ord_less_eq @ A @ A3 @ B2 )
| ( ( ord_less_eq @ A @ C2 @ A3 )
& ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_871_floor__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archim6421214686448440834_floor @ A @ ( one_one @ A ) )
= ( one_one @ int ) ) ) ).
% floor_one
thf(fact_872_ceiling__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_ceiling @ A @ ( one_one @ A ) )
= ( one_one @ int ) ) ) ).
% ceiling_one
thf(fact_873_floor__add2,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] :
( ( ( member @ A @ X @ ( ring_1_Ints @ A ) )
| ( member @ A @ Y @ ( ring_1_Ints @ A ) ) )
=> ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ Y ) )
= ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y ) ) ) ) ) ).
% floor_add2
thf(fact_874_zero__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( zero_zero @ int ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( one_one @ A ) @ X ) ) ) ).
% zero_less_floor
thf(fact_875_floor__le__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( zero_zero @ int ) )
= ( ord_less @ A @ X @ ( one_one @ A ) ) ) ) ).
% floor_le_zero
thf(fact_876_one__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( one_one @ int ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( one_one @ A ) @ X ) ) ) ).
% one_le_floor
thf(fact_877_floor__less__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( one_one @ int ) )
= ( ord_less @ A @ X @ ( one_one @ A ) ) ) ) ).
% floor_less_one
thf(fact_878_ceiling__le__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( one_one @ int ) )
= ( ord_less_eq @ A @ X @ ( one_one @ A ) ) ) ) ).
% ceiling_le_one
thf(fact_879_one__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( one_one @ int ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( one_one @ A ) @ X ) ) ) ).
% one_less_ceiling
thf(fact_880_ceiling__add__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X @ ( one_one @ A ) ) )
= ( plus_plus @ int @ ( archimedean_ceiling @ A @ X ) @ ( one_one @ int ) ) ) ) ).
% ceiling_add_one
thf(fact_881_floor__diff__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( archim6421214686448440834_floor @ A @ ( minus_minus @ A @ X @ ( one_one @ A ) ) )
= ( minus_minus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( one_one @ int ) ) ) ) ).
% floor_diff_one
thf(fact_882_ceiling__diff__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( archimedean_ceiling @ A @ ( minus_minus @ A @ X @ ( one_one @ A ) ) )
= ( minus_minus @ int @ ( archimedean_ceiling @ A @ X ) @ ( one_one @ int ) ) ) ) ).
% ceiling_diff_one
thf(fact_883_atLeastatMost__psubset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ord_less @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D3 ) )
= ( ( ~ ( ord_less_eq @ A @ A3 @ B2 )
| ( ( ord_less_eq @ A @ C2 @ A3 )
& ( ord_less_eq @ A @ B2 @ D3 )
& ( ( ord_less @ A @ C2 @ A3 )
| ( ord_less @ A @ B2 @ D3 ) ) ) )
& ( ord_less_eq @ A @ C2 @ D3 ) ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_884_nat__mod__eq__iff,axiom,
! [X: nat,N: nat,Y: nat] :
( ( ( modulo_modulo @ nat @ X @ N )
= ( modulo_modulo @ nat @ Y @ N ) )
= ( ? [Q1: nat,Q22: nat] :
( ( plus_plus @ nat @ X @ ( times_times @ nat @ N @ Q1 ) )
= ( plus_plus @ nat @ Y @ ( times_times @ nat @ N @ Q22 ) ) ) ) ) ).
% nat_mod_eq_iff
thf(fact_885_le__floor__add,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ int @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y ) ) @ ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ Y ) ) ) ) ).
% le_floor_add
thf(fact_886_bset_I1_J,axiom,
! [D4: int,B6: set @ int,P: int > $o,Q: int > $o] :
( ! [X2: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B6 )
=> ( X2
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X2 )
=> ( P @ ( minus_minus @ int @ X2 @ D4 ) ) ) )
=> ( ! [X2: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B6 )
=> ( X2
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q @ X2 )
=> ( Q @ ( minus_minus @ int @ X2 @ D4 ) ) ) )
=> ! [X7: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B6 )
=> ( X7
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P @ X7 )
& ( Q @ X7 ) )
=> ( ( P @ ( minus_minus @ int @ X7 @ D4 ) )
& ( Q @ ( minus_minus @ int @ X7 @ D4 ) ) ) ) ) ) ) ).
% bset(1)
thf(fact_887_bset_I2_J,axiom,
! [D4: int,B6: set @ int,P: int > $o,Q: int > $o] :
( ! [X2: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B6 )
=> ( X2
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X2 )
=> ( P @ ( minus_minus @ int @ X2 @ D4 ) ) ) )
=> ( ! [X2: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B6 )
=> ( X2
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q @ X2 )
=> ( Q @ ( minus_minus @ int @ X2 @ D4 ) ) ) )
=> ! [X7: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B6 )
=> ( X7
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P @ X7 )
| ( Q @ X7 ) )
=> ( ( P @ ( minus_minus @ int @ X7 @ D4 ) )
| ( Q @ ( minus_minus @ int @ X7 @ D4 ) ) ) ) ) ) ) ).
% bset(2)
thf(fact_888_aset_I1_J,axiom,
! [D4: int,A4: set @ int,P: int > $o,Q: int > $o] :
( ! [X2: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A4 )
=> ( X2
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X2 )
=> ( P @ ( plus_plus @ int @ X2 @ D4 ) ) ) )
=> ( ! [X2: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A4 )
=> ( X2
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q @ X2 )
=> ( Q @ ( plus_plus @ int @ X2 @ D4 ) ) ) )
=> ! [X7: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A4 )
=> ( X7
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P @ X7 )
& ( Q @ X7 ) )
=> ( ( P @ ( plus_plus @ int @ X7 @ D4 ) )
& ( Q @ ( plus_plus @ int @ X7 @ D4 ) ) ) ) ) ) ) ).
% aset(1)
thf(fact_889_aset_I2_J,axiom,
! [D4: int,A4: set @ int,P: int > $o,Q: int > $o] :
( ! [X2: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A4 )
=> ( X2
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X2 )
=> ( P @ ( plus_plus @ int @ X2 @ D4 ) ) ) )
=> ( ! [X2: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A4 )
=> ( X2
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q @ X2 )
=> ( Q @ ( plus_plus @ int @ X2 @ D4 ) ) ) )
=> ! [X7: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A4 )
=> ( X7
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P @ X7 )
| ( Q @ X7 ) )
=> ( ( P @ ( plus_plus @ int @ X7 @ D4 ) )
| ( Q @ ( plus_plus @ int @ X7 @ D4 ) ) ) ) ) ) ) ).
% aset(2)
thf(fact_890_nat__mod__eq__lemma,axiom,
! [X: nat,N: nat,Y: nat] :
( ( ( modulo_modulo @ nat @ X @ N )
= ( modulo_modulo @ nat @ Y @ N ) )
=> ( ( ord_less_eq @ nat @ Y @ X )
=> ? [Q2: nat] :
( X
= ( plus_plus @ nat @ Y @ ( times_times @ nat @ N @ Q2 ) ) ) ) ) ).
% nat_mod_eq_lemma
thf(fact_891_mod__eq__nat2E,axiom,
! [M2: nat,Q3: nat,N: nat] :
( ( ( modulo_modulo @ nat @ M2 @ Q3 )
= ( modulo_modulo @ nat @ N @ Q3 ) )
=> ( ( ord_less_eq @ nat @ M2 @ N )
=> ~ ! [S6: nat] :
( N
!= ( plus_plus @ nat @ M2 @ ( times_times @ nat @ Q3 @ S6 ) ) ) ) ) ).
% mod_eq_nat2E
thf(fact_892_mod__eq__nat1E,axiom,
! [M2: nat,Q3: nat,N: nat] :
( ( ( modulo_modulo @ nat @ M2 @ Q3 )
= ( modulo_modulo @ nat @ N @ Q3 ) )
=> ( ( ord_less_eq @ nat @ N @ M2 )
=> ~ ! [S6: nat] :
( M2
!= ( plus_plus @ nat @ N @ ( times_times @ nat @ Q3 @ S6 ) ) ) ) ) ).
% mod_eq_nat1E
thf(fact_893_ceiling__add__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ ( plus_plus @ A @ X @ Y ) ) @ ( plus_plus @ int @ ( archimedean_ceiling @ A @ X ) @ ( archimedean_ceiling @ A @ Y ) ) ) ) ).
% ceiling_add_le
thf(fact_894_mod__mult2__eq,axiom,
! [M2: nat,N: nat,Q3: nat] :
( ( modulo_modulo @ nat @ M2 @ ( times_times @ nat @ N @ Q3 ) )
= ( plus_plus @ nat @ ( times_times @ nat @ N @ ( modulo_modulo @ nat @ ( divide_divide @ nat @ M2 @ N ) @ Q3 ) ) @ ( modulo_modulo @ nat @ M2 @ N ) ) ) ).
% mod_mult2_eq
thf(fact_895_div__mod__decomp,axiom,
! [A4: nat,N: nat] :
( A4
= ( plus_plus @ nat @ ( times_times @ nat @ ( divide_divide @ nat @ A4 @ N ) @ N ) @ ( modulo_modulo @ nat @ A4 @ N ) ) ) ).
% div_mod_decomp
thf(fact_896_one__add__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( one_one @ int ) )
= ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% one_add_floor
thf(fact_897_split__mod,axiom,
! [P: nat > $o,M2: nat,N: nat] :
( ( P @ ( modulo_modulo @ nat @ M2 @ N ) )
= ( ( ( N
= ( zero_zero @ nat ) )
=> ( P @ M2 ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ! [I3: nat,J2: nat] :
( ( ord_less @ nat @ J2 @ N )
=> ( ( M2
= ( plus_plus @ nat @ ( times_times @ nat @ N @ I3 ) @ J2 ) )
=> ( P @ J2 ) ) ) ) ) ) ).
% split_mod
thf(fact_898_bset_I3_J,axiom,
! [D4: int,T2: int,B6: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ( member @ int @ ( minus_minus @ int @ T2 @ ( one_one @ int ) ) @ B6 )
=> ! [X7: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B6 )
=> ( X7
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X7 = T2 )
=> ( ( minus_minus @ int @ X7 @ D4 )
= T2 ) ) ) ) ) ).
% bset(3)
thf(fact_899_bset_I4_J,axiom,
! [D4: int,T2: int,B6: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ( member @ int @ T2 @ B6 )
=> ! [X7: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B6 )
=> ( X7
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X7 != T2 )
=> ( ( minus_minus @ int @ X7 @ D4 )
!= T2 ) ) ) ) ) ).
% bset(4)
thf(fact_900_bset_I5_J,axiom,
! [D4: int,B6: set @ int,T2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ! [X7: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B6 )
=> ( X7
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less @ int @ X7 @ T2 )
=> ( ord_less @ int @ ( minus_minus @ int @ X7 @ D4 ) @ T2 ) ) ) ) ).
% bset(5)
thf(fact_901_bset_I7_J,axiom,
! [D4: int,T2: int,B6: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ( member @ int @ T2 @ B6 )
=> ! [X7: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B6 )
=> ( X7
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less @ int @ T2 @ X7 )
=> ( ord_less @ int @ T2 @ ( minus_minus @ int @ X7 @ D4 ) ) ) ) ) ) ).
% bset(7)
thf(fact_902_aset_I3_J,axiom,
! [D4: int,T2: int,A4: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ( member @ int @ ( plus_plus @ int @ T2 @ ( one_one @ int ) ) @ A4 )
=> ! [X7: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A4 )
=> ( X7
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X7 = T2 )
=> ( ( plus_plus @ int @ X7 @ D4 )
= T2 ) ) ) ) ) ).
% aset(3)
thf(fact_903_aset_I4_J,axiom,
! [D4: int,T2: int,A4: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ( member @ int @ T2 @ A4 )
=> ! [X7: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A4 )
=> ( X7
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X7 != T2 )
=> ( ( plus_plus @ int @ X7 @ D4 )
!= T2 ) ) ) ) ) ).
% aset(4)
thf(fact_904_aset_I5_J,axiom,
! [D4: int,T2: int,A4: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ( member @ int @ T2 @ A4 )
=> ! [X7: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A4 )
=> ( X7
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less @ int @ X7 @ T2 )
=> ( ord_less @ int @ ( plus_plus @ int @ X7 @ D4 ) @ T2 ) ) ) ) ) ).
% aset(5)
thf(fact_905_aset_I7_J,axiom,
! [D4: int,A4: set @ int,T2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ! [X7: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A4 )
=> ( X7
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less @ int @ T2 @ X7 )
=> ( ord_less @ int @ T2 @ ( plus_plus @ int @ X7 @ D4 ) ) ) ) ) ).
% aset(7)
thf(fact_906_aset_I8_J,axiom,
! [D4: int,A4: set @ int,T2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ! [X7: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A4 )
=> ( X7
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ int @ T2 @ X7 )
=> ( ord_less_eq @ int @ T2 @ ( plus_plus @ int @ X7 @ D4 ) ) ) ) ) ).
% aset(8)
thf(fact_907_ceiling__divide__lower,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q3: A,P5: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q3 )
=> ( ord_less @ A @ ( times_times @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ ( divide_divide @ A @ P5 @ Q3 ) ) ) @ ( one_one @ A ) ) @ Q3 ) @ P5 ) ) ) ).
% ceiling_divide_lower
thf(fact_908_floor__divide__upper,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q3: A,P5: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q3 )
=> ( ord_less @ A @ P5 @ ( times_times @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ P5 @ Q3 ) ) ) @ ( one_one @ A ) ) @ Q3 ) ) ) ) ).
% floor_divide_upper
thf(fact_909_power__decreasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,M2: nat,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ A @ B2 @ ( one_one @ A ) )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ B2 @ M2 ) @ ( power_power @ A @ B2 @ N ) )
= ( ord_less_eq @ nat @ N @ M2 ) ) ) ) ) ).
% power_decreasing_iff
thf(fact_910_divmod__nat__def,axiom,
( divmod_nat
= ( ^ [M: nat,N5: nat] : ( product_Pair @ nat @ nat @ ( divide_divide @ nat @ M @ N5 ) @ ( modulo_modulo @ nat @ M @ N5 ) ) ) ) ).
% divmod_nat_def
thf(fact_911_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 @ ( divide_divide @ B @ X @ Y ) @ A3 )
=> ( ( times_times @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% distrib_left_NO_MATCH
thf(fact_912_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 @ ( divide_divide @ B @ X @ Y ) @ C2 )
=> ( ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% distrib_right_NO_MATCH
thf(fact_913_Suc__times__mod__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M2 )
=> ( ( modulo_modulo @ nat @ ( suc @ ( times_times @ nat @ M2 @ N ) ) @ M2 )
= ( one_one @ nat ) ) ) ).
% Suc_times_mod_eq
thf(fact_914_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 ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ B2 @ A3 ) @ B2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% div_add_self1_no_field
thf(fact_915_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 ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% div_add_self2_no_field
thf(fact_916_power__one,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [N: nat] :
( ( power_power @ A @ ( one_one @ A ) @ N )
= ( one_one @ A ) ) ) ).
% power_one
thf(fact_917_add__Suc__right,axiom,
! [M2: nat,N: nat] :
( ( plus_plus @ nat @ M2 @ ( suc @ N ) )
= ( suc @ ( plus_plus @ nat @ M2 @ N ) ) ) ).
% add_Suc_right
thf(fact_918_power__one__right,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A] :
( ( power_power @ A @ A3 @ ( one_one @ nat ) )
= A3 ) ) ).
% power_one_right
thf(fact_919_power__inject__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,M2: nat,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ( ( power_power @ A @ A3 @ M2 )
= ( power_power @ A @ A3 @ N ) )
= ( M2 = N ) ) ) ) ).
% power_inject_exp
thf(fact_920_mult__Suc__right,axiom,
! [M2: nat,N: nat] :
( ( times_times @ nat @ M2 @ ( suc @ N ) )
= ( plus_plus @ nat @ M2 @ ( times_times @ nat @ M2 @ N ) ) ) ).
% mult_Suc_right
thf(fact_921_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus @ nat @ ( suc @ N ) @ ( one_one @ nat ) )
= N ) ).
% diff_Suc_1
thf(fact_922_of__int__eq__1__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Z2: int] :
( ( ( ring_1_of_int @ A @ Z2 )
= ( one_one @ A ) )
= ( Z2
= ( one_one @ int ) ) ) ) ).
% of_int_eq_1_iff
thf(fact_923_of__int__1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_of_int @ A @ ( one_one @ int ) )
= ( one_one @ A ) ) ) ).
% of_int_1
thf(fact_924_of__int__add,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [W: int,Z2: int] :
( ( ring_1_of_int @ A @ ( plus_plus @ int @ W @ Z2 ) )
= ( plus_plus @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ).
% of_int_add
thf(fact_925_power__strict__increasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,X: nat,Y: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ B2 )
=> ( ( ord_less @ A @ ( power_power @ A @ B2 @ X ) @ ( power_power @ A @ B2 @ Y ) )
= ( ord_less @ nat @ X @ Y ) ) ) ) ).
% power_strict_increasing_iff
thf(fact_926_left__minus__one__mult__self,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat,A3: A] :
( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ A3 ) )
= A3 ) ) ).
% left_minus_one_mult_self
thf(fact_927_minus__one__mult__self,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) )
= ( one_one @ A ) ) ) ).
% minus_one_mult_self
thf(fact_928_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq @ nat @ K @ J )
=> ( ( minus_minus @ nat @ ( suc @ ( minus_minus @ nat @ J @ K ) ) @ I )
= ( minus_minus @ nat @ ( suc @ J ) @ ( plus_plus @ nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_929_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq @ nat @ K @ J )
=> ( ( minus_minus @ nat @ I @ ( suc @ ( minus_minus @ nat @ J @ K ) ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_930_Suc__diff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ M2 )
=> ( ( suc @ ( minus_minus @ nat @ N @ M2 ) )
= ( minus_minus @ nat @ N @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) ) ) ) ) ).
% Suc_diff
thf(fact_931_Suc__mod__mult__self4,axiom,
! [N: nat,K: nat,M2: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ ( times_times @ nat @ N @ K ) @ M2 ) ) @ N )
= ( modulo_modulo @ nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_mod_mult_self4
thf(fact_932_Suc__mod__mult__self3,axiom,
! [K: nat,N: nat,M2: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ ( times_times @ nat @ K @ N ) @ M2 ) ) @ N )
= ( modulo_modulo @ nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_mod_mult_self3
thf(fact_933_Suc__mod__mult__self2,axiom,
! [M2: nat,N: nat,K: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ M2 @ ( times_times @ nat @ N @ K ) ) ) @ N )
= ( modulo_modulo @ nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_mod_mult_self2
thf(fact_934_Suc__mod__mult__self1,axiom,
! [M2: nat,K: nat,N: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ M2 @ ( times_times @ nat @ K @ N ) ) ) @ N )
= ( modulo_modulo @ nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_mod_mult_self1
thf(fact_935_ceiling__add__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z2: int] :
( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X @ ( ring_1_of_int @ A @ Z2 ) ) )
= ( plus_plus @ int @ ( archimedean_ceiling @ A @ X ) @ Z2 ) ) ) ).
% ceiling_add_of_int
thf(fact_936_power__strict__decreasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,M2: nat,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ A @ B2 @ ( one_one @ A ) )
=> ( ( ord_less @ A @ ( power_power @ A @ B2 @ M2 ) @ ( power_power @ A @ B2 @ N ) )
= ( ord_less @ nat @ N @ M2 ) ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_937_power__increasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,X: nat,Y: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ B2 @ X ) @ ( power_power @ A @ B2 @ Y ) )
= ( ord_less_eq @ nat @ X @ Y ) ) ) ) ).
% power_increasing_iff
thf(fact_938_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( suc @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
= N ) ) ).
% Suc_diff_1
thf(fact_939_of__int__1__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: int] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ ( ring_1_of_int @ A @ Z2 ) )
= ( ord_less_eq @ int @ ( one_one @ int ) @ Z2 ) ) ) ).
% of_int_1_le_iff
thf(fact_940_of__int__le__1__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) )
= ( ord_less_eq @ int @ Z2 @ ( one_one @ int ) ) ) ) ).
% of_int_le_1_iff
thf(fact_941_of__int__less__1__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: int] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) )
= ( ord_less @ int @ Z2 @ ( one_one @ int ) ) ) ) ).
% of_int_less_1_iff
thf(fact_942_of__int__1__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: int] :
( ( ord_less @ A @ ( one_one @ A ) @ ( ring_1_of_int @ A @ Z2 ) )
= ( ord_less @ int @ ( one_one @ int ) @ Z2 ) ) ) ).
% of_int_1_less_iff
thf(fact_943_power__gt1,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ ( suc @ N ) ) ) ) ) ).
% power_gt1
thf(fact_944_add__Suc__shift,axiom,
! [M2: nat,N: nat] :
( ( plus_plus @ nat @ ( suc @ M2 ) @ N )
= ( plus_plus @ nat @ M2 @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_945_add__Suc,axiom,
! [M2: nat,N: nat] :
( ( plus_plus @ nat @ ( suc @ M2 ) @ N )
= ( suc @ ( plus_plus @ nat @ M2 @ N ) ) ) ).
% add_Suc
thf(fact_946_nat__arith_Osuc1,axiom,
! [A4: nat,K: nat,A3: nat] :
( ( A4
= ( plus_plus @ nat @ K @ A3 ) )
=> ( ( suc @ A4 )
= ( plus_plus @ nat @ K @ ( suc @ A3 ) ) ) ) ).
% nat_arith.suc1
thf(fact_947_power__Suc__le__self,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ ( suc @ N ) ) @ A3 ) ) ) ) ).
% power_Suc_le_self
thf(fact_948_power__Suc__less__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( power_power @ A @ A3 @ ( suc @ N ) ) @ ( one_one @ A ) ) ) ) ) ).
% power_Suc_less_one
thf(fact_949_power__minus_H,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: A,N: nat] :
( ( nO_MATCH @ A @ A @ ( one_one @ A ) @ X )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ X ) @ N )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( power_power @ A @ X @ N ) ) ) ) ) ).
% power_minus'
thf(fact_950_one__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% one_le_power
thf(fact_951_left__right__inverse__power,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [X: A,Y: A,N: nat] :
( ( ( times_times @ A @ X @ Y )
= ( one_one @ A ) )
=> ( ( times_times @ A @ ( power_power @ A @ X @ N ) @ ( power_power @ A @ Y @ N ) )
= ( one_one @ A ) ) ) ) ).
% left_right_inverse_power
thf(fact_952_power__0,axiom,
! [A: $tType] :
( ( power @ A )
=> ! [A3: A] :
( ( power_power @ A @ A3 @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% power_0
thf(fact_953_power__one__over,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,N: nat] :
( ( power_power @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) @ N )
= ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% power_one_over
thf(fact_954_power__add,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A,M2: nat,N: nat] :
( ( power_power @ A @ A3 @ ( plus_plus @ nat @ M2 @ N ) )
= ( times_times @ A @ ( power_power @ A @ A3 @ M2 ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% power_add
thf(fact_955_nat__compl__induct_H,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N3: nat] :
( ! [Nn: nat] :
( ( ord_less_eq @ nat @ Nn @ N3 )
=> ( P @ Nn ) )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N ) ) ) ).
% nat_compl_induct'
thf(fact_956_nat__compl__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N3: nat] :
( ! [Nn: nat] :
( ( ord_less_eq @ nat @ Nn @ N3 )
=> ( P @ Nn ) )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N ) ) ) ).
% nat_compl_induct
thf(fact_957_one__is__add,axiom,
! [M2: nat,N: nat] :
( ( ( suc @ ( zero_zero @ nat ) )
= ( plus_plus @ nat @ M2 @ N ) )
= ( ( ( M2
= ( suc @ ( zero_zero @ nat ) ) )
& ( N
= ( zero_zero @ nat ) ) )
| ( ( M2
= ( zero_zero @ nat ) )
& ( N
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ).
% one_is_add
thf(fact_958_add__is__1,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus @ nat @ M2 @ N )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ( ( M2
= ( suc @ ( zero_zero @ nat ) ) )
& ( N
= ( zero_zero @ nat ) ) )
| ( ( M2
= ( zero_zero @ nat ) )
& ( N
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ).
% add_is_1
thf(fact_959_nat__in__between__eq_I2_J,axiom,
! [A3: nat,B2: nat] :
( ( ( ord_less_eq @ nat @ A3 @ B2 )
& ( ord_less @ nat @ B2 @ ( suc @ A3 ) ) )
= ( B2 = A3 ) ) ).
% nat_in_between_eq(2)
thf(fact_960_nat__in__between__eq_I1_J,axiom,
! [A3: nat,B2: nat] :
( ( ( ord_less @ nat @ A3 @ B2 )
& ( ord_less_eq @ nat @ B2 @ ( suc @ A3 ) ) )
= ( B2
= ( suc @ A3 ) ) ) ).
% nat_in_between_eq(1)
thf(fact_961_less__imp__Suc__add,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ? [K5: nat] :
( N
= ( suc @ ( plus_plus @ nat @ M2 @ K5 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_962_less__iff__Suc__add,axiom,
( ( ord_less @ nat )
= ( ^ [M: nat,N5: nat] :
? [K4: nat] :
( N5
= ( suc @ ( plus_plus @ nat @ M @ K4 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_963_less__add__Suc2,axiom,
! [I: nat,M2: nat] : ( ord_less @ nat @ I @ ( suc @ ( plus_plus @ nat @ M2 @ I ) ) ) ).
% less_add_Suc2
thf(fact_964_less__add__Suc1,axiom,
! [I: nat,M2: nat] : ( ord_less @ nat @ I @ ( suc @ ( plus_plus @ nat @ I @ M2 ) ) ) ).
% less_add_Suc1
thf(fact_965_less__natE,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ~ ! [Q2: nat] :
( N
!= ( suc @ ( plus_plus @ nat @ M2 @ Q2 ) ) ) ) ).
% less_natE
thf(fact_966_Suc__to__right,axiom,
! [N: nat,M2: nat] :
( ( ( suc @ N )
= M2 )
=> ( N
= ( minus_minus @ nat @ M2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% Suc_to_right
thf(fact_967_One__nat__def,axiom,
( ( one_one @ nat )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% One_nat_def
thf(fact_968_mult__Suc,axiom,
! [M2: nat,N: nat] :
( ( times_times @ nat @ ( suc @ M2 ) @ N )
= ( plus_plus @ nat @ N @ ( times_times @ nat @ M2 @ N ) ) ) ).
% mult_Suc
thf(fact_969_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus @ nat @ ( one_one @ nat ) ) ) ).
% Suc_eq_plus1_left
thf(fact_970_plus__1__eq__Suc,axiom,
( ( plus_plus @ nat @ ( one_one @ nat ) )
= suc ) ).
% plus_1_eq_Suc
thf(fact_971_Suc__eq__plus1,axiom,
( suc
= ( ^ [N5: nat] : ( plus_plus @ nat @ N5 @ ( one_one @ nat ) ) ) ) ).
% Suc_eq_plus1
thf(fact_972_diff__Suc__eq__diff__pred,axiom,
! [M2: nat,N: nat] :
( ( minus_minus @ nat @ M2 @ ( suc @ N ) )
= ( minus_minus @ nat @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_973_power__le__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( one_one @ A ) ) ) ) ) ).
% power_le_one
thf(fact_974_power__less__power__Suc,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less @ A @ ( power_power @ A @ A3 @ N ) @ ( times_times @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% power_less_power_Suc
thf(fact_975_power__gt1__lemma,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less @ A @ ( one_one @ A ) @ ( times_times @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% power_gt1_lemma
thf(fact_976_power__0__left,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N: nat] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( power_power @ A @ ( zero_zero @ A ) @ N )
= ( one_one @ A ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( power_power @ A @ ( zero_zero @ A ) @ N )
= ( zero_zero @ A ) ) ) ) ) ).
% power_0_left
thf(fact_977_power__strict__increasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N6: nat,A3: A] :
( ( ord_less @ nat @ N @ N6 )
=> ( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ A3 @ N6 ) ) ) ) ) ).
% power_strict_increasing
thf(fact_978_power__less__imp__less__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,M2: nat,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ( ord_less @ A @ ( power_power @ A @ A3 @ M2 ) @ ( power_power @ A @ A3 @ N ) )
=> ( ord_less @ nat @ M2 @ N ) ) ) ) ).
% power_less_imp_less_exp
thf(fact_979_power__minus,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A3: A,N: nat] :
( ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% power_minus
thf(fact_980_power__increasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N6: nat,A3: A] :
( ( ord_less_eq @ nat @ N @ N6 )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ A3 @ N6 ) ) ) ) ) ).
% power_increasing
thf(fact_981_floor__add__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z2: int] :
( ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ Z2 )
= ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ) ).
% floor_add_int
thf(fact_982_int__add__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z2: int,X: A] :
( ( plus_plus @ int @ Z2 @ ( archim6421214686448440834_floor @ A @ X ) )
= ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z2 ) @ X ) ) ) ) ).
% int_add_floor
thf(fact_983_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( P @ ( one_one @ nat ) )
=> ( ! [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_984_of__int__leD,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: int,X: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ ( ring_1_of_int @ A @ N ) ) @ X )
=> ( ( N
= ( zero_zero @ int ) )
| ( ord_less_eq @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% of_int_leD
thf(fact_985_of__int__lessD,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: int,X: A] :
( ( ord_less @ A @ ( abs_abs @ A @ ( ring_1_of_int @ A @ N ) ) @ X )
=> ( ( N
= ( zero_zero @ int ) )
| ( ord_less @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% of_int_lessD
thf(fact_986_floor__exists,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [Z3: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z3 ) @ X )
& ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ Z3 @ ( one_one @ int ) ) ) ) ) ) ).
% floor_exists
thf(fact_987_floor__exists1,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [X2: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ X2 ) @ X )
& ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ X2 @ ( one_one @ int ) ) ) )
& ! [Y6: int] :
( ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Y6 ) @ X )
& ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ Y6 @ ( one_one @ int ) ) ) ) )
=> ( Y6 = X2 ) ) ) ) ).
% floor_exists1
thf(fact_988_power__Suc__less,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% power_Suc_less
thf(fact_989_power__strict__decreasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N6: nat,A3: A] :
( ( ord_less @ nat @ N @ N6 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( power_power @ A @ A3 @ N6 ) @ ( power_power @ A @ A3 @ N ) ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_990_power__decreasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N6: nat,A3: A] :
( ( ord_less_eq @ nat @ N @ N6 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N6 ) @ ( power_power @ A @ A3 @ N ) ) ) ) ) ) ).
% power_decreasing
thf(fact_991_power__le__imp__le__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,M2: nat,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ M2 ) @ ( power_power @ A @ A3 @ N ) )
=> ( ord_less_eq @ nat @ M2 @ N ) ) ) ) ).
% power_le_imp_le_exp
thf(fact_992_self__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ A3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% self_le_power
thf(fact_993_one__less__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% one_less_power
thf(fact_994_nz__le__conv__less,axiom,
! [K: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( ord_less_eq @ nat @ K @ M2 )
=> ( ord_less @ nat @ ( minus_minus @ nat @ K @ ( suc @ ( zero_zero @ nat ) ) ) @ M2 ) ) ) ).
% nz_le_conv_less
thf(fact_995_ceiling__altdef,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_ceiling @ A )
= ( ^ [X3: A] :
( if @ int
@ ( X3
= ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X3 ) ) )
@ ( archim6421214686448440834_floor @ A @ X3 )
@ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( one_one @ int ) ) ) ) ) ) ).
% ceiling_altdef
thf(fact_996_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( minus_minus @ nat @ ( suc @ M2 ) @ N )
= ( minus_minus @ nat @ M2 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_997_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( N
= ( suc @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% Suc_pred'
thf(fact_998_add__eq__if,axiom,
( ( plus_plus @ nat )
= ( ^ [M: nat,N5: nat] :
( if @ nat
@ ( M
= ( zero_zero @ nat ) )
@ N5
@ ( suc @ ( plus_plus @ nat @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) @ N5 ) ) ) ) ) ).
% add_eq_if
thf(fact_999_Suc__n__minus__m__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( ord_less @ nat @ ( one_one @ nat ) @ M2 )
=> ( ( suc @ ( minus_minus @ nat @ N @ M2 ) )
= ( minus_minus @ nat @ N @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) ) ) ) ) ).
% Suc_n_minus_m_eq
thf(fact_1000_floor__unique,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z2: int,X: A] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z2 ) @ X )
=> ( ( ord_less @ A @ X @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) )
=> ( ( archim6421214686448440834_floor @ A @ X )
= Z2 ) ) ) ) ).
% floor_unique
thf(fact_1001_floor__eq__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,A3: int] :
( ( ( archim6421214686448440834_floor @ A @ X )
= A3 )
= ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A3 ) @ X )
& ( ord_less @ A @ X @ ( plus_plus @ A @ ( ring_1_of_int @ A @ A3 ) @ ( one_one @ A ) ) ) ) ) ) ).
% floor_eq_iff
thf(fact_1002_floor__split,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [P: int > $o,T2: A] :
( ( P @ ( archim6421214686448440834_floor @ A @ T2 ) )
= ( ! [I3: int] :
( ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ I3 ) @ T2 )
& ( ord_less @ A @ T2 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ I3 ) @ ( one_one @ A ) ) ) )
=> ( P @ I3 ) ) ) ) ) ).
% floor_split
thf(fact_1003_ceiling__correct,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X ) ) @ ( one_one @ A ) ) @ X )
& ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X ) ) ) ) ) ).
% ceiling_correct
thf(fact_1004_ceiling__unique,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z2: int,X: A] :
( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) @ X )
=> ( ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ Z2 ) )
=> ( ( archimedean_ceiling @ A @ X )
= Z2 ) ) ) ) ).
% ceiling_unique
thf(fact_1005_ceiling__eq__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,A3: int] :
( ( ( archimedean_ceiling @ A @ X )
= A3 )
= ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ A3 ) @ ( one_one @ A ) ) @ X )
& ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ A3 ) ) ) ) ) ).
% ceiling_eq_iff
thf(fact_1006_ceiling__split,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [P: int > $o,T2: A] :
( ( P @ ( archimedean_ceiling @ A @ T2 ) )
= ( ! [I3: int] :
( ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ I3 ) @ ( one_one @ A ) ) @ T2 )
& ( ord_less_eq @ A @ T2 @ ( ring_1_of_int @ A @ I3 ) ) )
=> ( P @ I3 ) ) ) ) ) ).
% ceiling_split
thf(fact_1007_less__floor__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z2: int,X: A] :
( ( ord_less @ int @ Z2 @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) @ X ) ) ) ).
% less_floor_iff
thf(fact_1008_floor__le__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z2: int] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ Z2 )
= ( ord_less @ A @ X @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) ) ) ) ).
% floor_le_iff
thf(fact_1009_ceiling__less__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z2: int] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ Z2 )
= ( ord_less_eq @ A @ X @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_iff
thf(fact_1010_le__ceiling__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z2: int,X: A] :
( ( ord_less_eq @ int @ Z2 @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) @ X ) ) ) ).
% le_ceiling_iff
thf(fact_1011_floor__correct,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X ) ) @ X )
& ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( one_one @ int ) ) ) ) ) ) ).
% floor_correct
thf(fact_1012_power__diff__power__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,N: nat,M2: nat] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( ( ord_less_eq @ nat @ N @ M2 )
=> ( ( divide_divide @ A @ ( power_power @ A @ A3 @ M2 ) @ ( power_power @ A @ A3 @ N ) )
= ( power_power @ A @ A3 @ ( minus_minus @ nat @ M2 @ N ) ) ) )
& ( ~ ( ord_less_eq @ nat @ N @ M2 )
=> ( ( divide_divide @ A @ ( power_power @ A @ A3 @ M2 ) @ ( power_power @ A @ A3 @ N ) )
= ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ) ) ) ).
% power_diff_power_eq
thf(fact_1013_power__eq__if,axiom,
! [A: $tType] :
( ( power @ A )
=> ( ( power_power @ A )
= ( ^ [P6: A,M: nat] :
( if @ A
@ ( M
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( times_times @ A @ P6 @ ( power_power @ A @ P6 @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% power_eq_if
thf(fact_1014_power__minus__mult,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [N: nat,A3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( times_times @ A @ ( power_power @ A @ A3 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) @ A3 )
= ( power_power @ A @ A3 @ N ) ) ) ) ).
% power_minus_mult
thf(fact_1015_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( plus_plus @ nat @ N @ K ) )
= ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( minus_minus @ nat @ N @ K ) ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_1016_option_Osize__gen_I2_J,axiom,
! [A: $tType,X: A > nat,X22: A] :
( ( size_option @ A @ X @ ( some @ A @ X22 ) )
= ( plus_plus @ nat @ ( X @ X22 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% option.size_gen(2)
thf(fact_1017_Heap_Osize__gen,axiom,
! [A: $tType,Xa: A > nat,X: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
( ( heap_Time_size_Heap @ A @ Xa @ ( heap_Time_Heap2 @ A @ X ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% Heap.size_gen
thf(fact_1018_num_Osize__gen_I2_J,axiom,
! [X22: num] :
( ( size_num @ ( bit0 @ X22 ) )
= ( plus_plus @ nat @ ( size_num @ X22 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% num.size_gen(2)
thf(fact_1019_normalize__negative,axiom,
! [Q3: int,P5: int] :
( ( ord_less @ int @ Q3 @ ( zero_zero @ int ) )
=> ( ( normalize @ ( product_Pair @ int @ int @ P5 @ Q3 ) )
= ( normalize @ ( product_Pair @ int @ int @ ( uminus_uminus @ int @ P5 ) @ ( uminus_uminus @ int @ Q3 ) ) ) ) ) ).
% normalize_negative
thf(fact_1020_option_Osize__gen_I1_J,axiom,
! [A: $tType,X: A > nat] :
( ( size_option @ A @ X @ ( none @ A ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% option.size_gen(1)
thf(fact_1021_Heap_Osize_I2_J,axiom,
! [A: $tType,X: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
( ( size_size @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap2 @ A @ X ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% Heap.size(2)
thf(fact_1022_num_Osize_I5_J,axiom,
! [X22: num] :
( ( size_size @ num @ ( bit0 @ X22 ) )
= ( plus_plus @ nat @ ( size_size @ num @ X22 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% num.size(5)
thf(fact_1023_option_Osize_I3_J,axiom,
! [A: $tType] :
( ( size_size @ ( option @ A ) @ ( none @ A ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% option.size(3)
thf(fact_1024_normalize__denom__zero,axiom,
! [P5: int] :
( ( normalize @ ( product_Pair @ int @ int @ P5 @ ( zero_zero @ int ) ) )
= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ).
% normalize_denom_zero
thf(fact_1025_option_Osize__neq,axiom,
! [A: $tType,X: option @ A] :
( ( size_size @ ( option @ A ) @ X )
!= ( zero_zero @ nat ) ) ).
% option.size_neq
thf(fact_1026_Heap_Osize__neq,axiom,
! [A: $tType,X: heap_Time_Heap @ A] :
( ( size_size @ ( heap_Time_Heap @ A ) @ X )
!= ( zero_zero @ nat ) ) ).
% Heap.size_neq
thf(fact_1027_option_Osize__gen__o__map,axiom,
! [B: $tType,A: $tType,F: B > nat,G: A > B] :
( ( comp @ ( option @ B ) @ nat @ ( option @ A ) @ ( size_option @ B @ F ) @ ( map_option @ A @ B @ G ) )
= ( size_option @ A @ ( comp @ B @ nat @ A @ F @ G ) ) ) ).
% option.size_gen_o_map
thf(fact_1028_normalize__denom__pos,axiom,
! [R: product_prod @ int @ int,P5: int,Q3: int] :
( ( ( normalize @ R )
= ( product_Pair @ int @ int @ P5 @ Q3 ) )
=> ( ord_less @ int @ ( zero_zero @ int ) @ Q3 ) ) ).
% normalize_denom_pos
thf(fact_1029_normalize__crossproduct,axiom,
! [Q3: int,S: int,P5: int,R: int] :
( ( Q3
!= ( zero_zero @ int ) )
=> ( ( S
!= ( zero_zero @ int ) )
=> ( ( ( normalize @ ( product_Pair @ int @ int @ P5 @ Q3 ) )
= ( normalize @ ( product_Pair @ int @ int @ R @ S ) ) )
=> ( ( times_times @ int @ P5 @ S )
= ( times_times @ int @ R @ Q3 ) ) ) ) ) ).
% normalize_crossproduct
thf(fact_1030_option_Osize_I4_J,axiom,
! [A: $tType,X22: A] :
( ( size_size @ ( option @ A ) @ ( some @ A @ X22 ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% option.size(4)
thf(fact_1031_divmod__nat__if,axiom,
( divmod_nat
= ( ^ [M: nat,N5: nat] :
( if @ ( product_prod @ nat @ nat )
@ ( ( N5
= ( zero_zero @ nat ) )
| ( ord_less @ nat @ M @ N5 ) )
@ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ M )
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [Q5: nat] : ( product_Pair @ nat @ nat @ ( suc @ Q5 ) )
@ ( divmod_nat @ ( minus_minus @ nat @ M @ N5 ) @ N5 ) ) ) ) ) ).
% divmod_nat_if
thf(fact_1032_sum__gp__multiplied,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [M2: nat,N: nat,X: A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) )
= ( minus_minus @ A @ ( power_power @ A @ X @ M2 ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) ) ) ) ).
% sum_gp_multiplied
thf(fact_1033_pair__lessI2,axiom,
! [A3: nat,B2: nat,S: nat,T2: nat] :
( ( ord_less_eq @ nat @ A3 @ B2 )
=> ( ( ord_less @ nat @ S @ T2 )
=> ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A3 @ S ) @ ( product_Pair @ nat @ nat @ B2 @ T2 ) ) @ fun_pair_less ) ) ) ).
% pair_lessI2
thf(fact_1034_pair__less__iff1,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X @ Y ) @ ( product_Pair @ nat @ nat @ X @ Z2 ) ) @ fun_pair_less )
= ( ord_less @ nat @ Y @ Z2 ) ) ).
% pair_less_iff1
thf(fact_1035_Suc__times__numeral__mod__eq,axiom,
! [K: num,N: nat] :
( ( ( numeral_numeral @ nat @ K )
!= ( one_one @ nat ) )
=> ( ( modulo_modulo @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ K ) @ N ) ) @ ( numeral_numeral @ nat @ K ) )
= ( one_one @ nat ) ) ) ).
% Suc_times_numeral_mod_eq
thf(fact_1036_one__int_Otransfer,axiom,
pcr_int @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) @ ( one_one @ int ) ).
% one_int.transfer
thf(fact_1037_less__than__iff,axiom,
! [X: nat,Y: nat] :
( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X @ Y ) @ less_than )
= ( ord_less @ nat @ X @ Y ) ) ).
% less_than_iff
thf(fact_1038_one__int__def,axiom,
( ( one_one @ int )
= ( abs_Integ @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) ) ) ).
% one_int_def
thf(fact_1039_sum__telescope_H_H,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M2: nat,N: nat,F: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K4: nat] : ( minus_minus @ A @ ( F @ K4 ) @ ( F @ ( minus_minus @ nat @ K4 @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ N ) )
= ( minus_minus @ A @ ( F @ N ) @ ( F @ M2 ) ) ) ) ) ).
% sum_telescope''
thf(fact_1040_case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,F: B > C > A,A3: B,B2: C] :
( ( product_case_prod @ B @ C @ A @ F @ ( product_Pair @ B @ C @ A3 @ B2 ) )
= ( F @ A3 @ B2 ) ) ).
% case_prod_conv
thf(fact_1041_case__prod__curry,axiom,
! [C: $tType,B: $tType,A: $tType,F: ( product_prod @ A @ B ) > C] :
( ( product_case_prod @ A @ B @ C @ ( product_curry @ A @ B @ C @ F ) )
= F ) ).
% case_prod_curry
thf(fact_1042_curry__case__prod,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > B > C] :
( ( product_curry @ A @ B @ C @ ( product_case_prod @ A @ B @ C @ F ) )
= F ) ).
% curry_case_prod
thf(fact_1043_distrib__right__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( semiring @ A ) )
=> ! [A3: A,B2: A,V: num] :
( ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( numeral_numeral @ A @ V ) )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ V ) ) @ ( times_times @ A @ B2 @ ( numeral_numeral @ A @ V ) ) ) ) ) ).
% distrib_right_numeral
thf(fact_1044_distrib__left__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( semiring @ A ) )
=> ! [V: num,B2: A,C2: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ V ) @ ( plus_plus @ A @ B2 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ V ) @ B2 ) @ ( times_times @ A @ ( numeral_numeral @ A @ V ) @ C2 ) ) ) ) ).
% distrib_left_numeral
thf(fact_1045_add__neg__numeral__simps_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num,N: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( uminus_uminus @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N ) ) ) ) ) ).
% add_neg_numeral_simps(3)
thf(fact_1046_add__numeral__left,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [V: num,W: num,Z2: A] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ V ) @ ( plus_plus @ A @ ( numeral_numeral @ A @ W ) @ Z2 ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ V @ W ) ) @ Z2 ) ) ) ).
% add_numeral_left
thf(fact_1047_numeral__plus__numeral,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [M2: num,N: num] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N ) )
= ( numeral_numeral @ A @ ( plus_plus @ num @ M2 @ N ) ) ) ) ).
% numeral_plus_numeral
thf(fact_1048_semiring__norm_I168_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [V: num,W: num,Y: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ Y ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ V @ W ) ) ) @ Y ) ) ) ).
% semiring_norm(168)
thf(fact_1049_diff__numeral__simps_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num,N: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( numeral_numeral @ A @ N ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ M2 @ N ) ) ) ) ) ).
% diff_numeral_simps(3)
thf(fact_1050_diff__numeral__simps_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num,N: num] :
( ( minus_minus @ A @ ( numeral_numeral @ A @ M2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( numeral_numeral @ A @ ( plus_plus @ num @ M2 @ N ) ) ) ) ).
% diff_numeral_simps(2)
thf(fact_1051_ceiling__add__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V: num] :
( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X @ ( numeral_numeral @ A @ V ) ) )
= ( plus_plus @ int @ ( archimedean_ceiling @ A @ X ) @ ( numeral_numeral @ int @ V ) ) ) ) ).
% ceiling_add_numeral
thf(fact_1052_power__add__numeral,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A,M2: num,N: num] :
( ( times_times @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ M2 ) ) @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ N ) ) )
= ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( plus_plus @ num @ M2 @ N ) ) ) ) ) ).
% power_add_numeral
thf(fact_1053_power__add__numeral2,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A,M2: num,N: num,B2: A] :
( ( times_times @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ M2 ) ) @ ( times_times @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ N ) ) @ B2 ) )
= ( times_times @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( plus_plus @ num @ M2 @ N ) ) ) @ B2 ) ) ) ).
% power_add_numeral2
thf(fact_1054_numeral__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num,X: A] :
( ( ord_less @ int @ ( numeral_numeral @ int @ V ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ V ) @ ( one_one @ A ) ) @ X ) ) ) ).
% numeral_less_floor
thf(fact_1055_floor__le__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V: num] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( numeral_numeral @ int @ V ) )
= ( ord_less @ A @ X @ ( plus_plus @ A @ ( numeral_numeral @ A @ V ) @ ( one_one @ A ) ) ) ) ) ).
% floor_le_numeral
thf(fact_1056_sum_Ocl__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [N: nat,M2: nat,G: nat > A] :
( ( ( ord_less @ nat @ ( suc @ N ) @ M2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ ( suc @ N ) @ M2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ) ).
% sum.cl_ivl_Suc
thf(fact_1057_ceiling__less__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V: num] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( numeral_numeral @ int @ V ) )
= ( ord_less_eq @ A @ X @ ( minus_minus @ A @ ( numeral_numeral @ A @ V ) @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_numeral
thf(fact_1058_numeral__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num,X: A] :
( ( ord_less_eq @ int @ ( numeral_numeral @ int @ V ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( minus_minus @ A @ ( numeral_numeral @ A @ V ) @ ( one_one @ A ) ) @ X ) ) ) ).
% numeral_le_ceiling
thf(fact_1059_neg__numeral__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num,X: A] :
( ( ord_less @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( one_one @ A ) ) @ X ) ) ) ).
% neg_numeral_less_floor
thf(fact_1060_floor__le__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V: num] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) )
= ( ord_less @ A @ X @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( one_one @ A ) ) ) ) ) ).
% floor_le_neg_numeral
thf(fact_1061_ceiling__less__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V: num] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) )
= ( ord_less_eq @ A @ X @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_neg_numeral
thf(fact_1062_neg__numeral__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num,X: A] :
( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( one_one @ A ) ) @ X ) ) ) ).
% neg_numeral_le_ceiling
thf(fact_1063_nested__case__prod__simp,axiom,
! [A: $tType,D: $tType,C: $tType,B: $tType] :
( ( product_case_prod @ B @ C @ ( D > A ) )
= ( ^ [F2: B > C > D > A,X3: product_prod @ B @ C,Y3: D] :
( product_case_prod @ B @ C @ A
@ ^ [A7: B,B5: C] : ( F2 @ A7 @ B5 @ Y3 )
@ X3 ) ) ) ).
% nested_case_prod_simp
thf(fact_1064_prod_Ocase__distrib,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,H3: C > D,F: A > B > C,Prod: product_prod @ A @ B] :
( ( H3 @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
= ( product_case_prod @ A @ B @ D
@ ^ [X12: A,X24: B] : ( H3 @ ( F @ X12 @ X24 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_1065_sum_OatLeast__Suc__atMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% sum.atLeast_Suc_atMost_Suc_shift
thf(fact_1066_sum_OatLeastAtMost__shift__bounds,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M2: nat,K: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M2 @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ K ) ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% sum.atLeastAtMost_shift_bounds
thf(fact_1067_split__cong,axiom,
! [C: $tType,B: $tType,A: $tType,Q3: product_prod @ A @ B,F: A > B > C,G: A > B > C,P5: product_prod @ A @ B] :
( ! [X2: A,Y4: B] :
( ( ( product_Pair @ A @ B @ X2 @ Y4 )
= Q3 )
=> ( ( F @ X2 @ Y4 )
= ( G @ X2 @ Y4 ) ) )
=> ( ( P5 = Q3 )
=> ( ( product_case_prod @ A @ B @ C @ F @ P5 )
= ( product_case_prod @ A @ B @ C @ G @ Q3 ) ) ) ) ).
% split_cong
thf(fact_1068_old_Oprod_Ocase,axiom,
! [A: $tType,C: $tType,B: $tType,F: A > B > C,X1: A,X22: B] :
( ( product_case_prod @ A @ B @ C @ F @ ( product_Pair @ A @ B @ X1 @ X22 ) )
= ( F @ X1 @ X22 ) ) ).
% old.prod.case
thf(fact_1069_uminus__int_Oabs__eq,axiom,
! [X: product_prod @ nat @ nat] :
( ( uminus_uminus @ int @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [X3: nat,Y3: nat] : ( product_Pair @ nat @ nat @ Y3 @ X3 )
@ X ) ) ) ).
% uminus_int.abs_eq
thf(fact_1070_case__prod__Pair__iden,axiom,
! [B: $tType,A: $tType,P5: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) @ P5 )
= P5 ) ).
% case_prod_Pair_iden
thf(fact_1071_sum_Oshift__bounds__cl__Suc__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% sum.shift_bounds_cl_Suc_ivl
thf(fact_1072_sum_Oshift__bounds__cl__nat__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M2: nat,K: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M2 @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G @ ( plus_plus @ nat @ I3 @ K ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% sum.shift_bounds_cl_nat_ivl
thf(fact_1073_cond__case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > B > C,G: ( product_prod @ A @ B ) > C] :
( ! [X2: A,Y4: B] :
( ( F @ X2 @ Y4 )
= ( G @ ( product_Pair @ A @ B @ X2 @ Y4 ) ) )
=> ( ( product_case_prod @ A @ B @ C @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_1074_case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F: ( product_prod @ A @ B ) > C] :
( ( product_case_prod @ A @ B @ C
@ ^ [X3: A,Y3: B] : ( F @ ( product_Pair @ A @ B @ X3 @ Y3 ) ) )
= F ) ).
% case_prod_eta
thf(fact_1075_case__prodE2,axiom,
! [B: $tType,A: $tType,C: $tType,Q: A > $o,P: B > C > A,Z2: product_prod @ B @ C] :
( ( Q @ ( product_case_prod @ B @ C @ A @ P @ Z2 ) )
=> ~ ! [X2: B,Y4: C] :
( ( Z2
= ( product_Pair @ B @ C @ X2 @ Y4 ) )
=> ~ ( Q @ ( P @ X2 @ Y4 ) ) ) ) ).
% case_prodE2
thf(fact_1076_eq__Abs__Integ,axiom,
! [Z2: int] :
~ ! [X2: nat,Y4: nat] :
( Z2
!= ( abs_Integ @ ( product_Pair @ nat @ nat @ X2 @ Y4 ) ) ) ).
% eq_Abs_Integ
thf(fact_1077_one__le__numeral,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N: num] : ( ord_less_eq @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) ) ) ).
% one_le_numeral
thf(fact_1078_not__numeral__less__one,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N: num] :
~ ( ord_less @ A @ ( numeral_numeral @ A @ N ) @ ( one_one @ A ) ) ) ).
% not_numeral_less_one
thf(fact_1079_one__plus__numeral__commute,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [X: num] :
( ( plus_plus @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ X ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ X ) @ ( one_one @ A ) ) ) ) ).
% one_plus_numeral_commute
thf(fact_1080_numeral__Bit0,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N: num] :
( ( numeral_numeral @ A @ ( bit0 @ N ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ N ) ) ) ) ).
% numeral_Bit0
thf(fact_1081_one__neq__neg__numeral,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [N: num] :
( ( one_one @ A )
!= ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% one_neq_neg_numeral
thf(fact_1082_numeral__neq__neg__one,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [N: num] :
( ( numeral_numeral @ A @ N )
!= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% numeral_neq_neg_one
thf(fact_1083_uncurry__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( uncurry @ A @ B @ C )
= ( product_case_prod @ A @ B @ C ) ) ).
% uncurry_def
thf(fact_1084_internal__case__prod__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( produc5280177257484947105e_prod @ A @ B @ C )
= ( product_case_prod @ A @ B @ C ) ) ).
% internal_case_prod_def
thf(fact_1085_sum_OatLeastAtMost__rev,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ N @ M2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ N @ M2 ) ) ) ) ).
% sum.atLeastAtMost_rev
thf(fact_1086_numeral__code_I2_J,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N: num] :
( ( numeral_numeral @ A @ ( bit0 @ N ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ N ) ) ) ) ).
% numeral_code(2)
thf(fact_1087_sum__shift__lb__Suc0__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F: nat > A,K: nat] :
( ( ( F @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ) ).
% sum_shift_lb_Suc0_0
thf(fact_1088_sum_OatLeast0__atMost__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ).
% sum.atLeast0_atMost_Suc
thf(fact_1089_sum_OatLeast0__atMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% sum.atLeast0_atMost_Suc_shift
thf(fact_1090_sum_OatLeast__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M2: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( plus_plus @ A @ ( G @ M2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ N ) ) ) ) ) ) ).
% sum.atLeast_Suc_atMost
thf(fact_1091_sum_Onat__ivl__Suc_H,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M2: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M2 @ ( suc @ N ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G @ ( suc @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ) ) ).
% sum.nat_ivl_Suc'
thf(fact_1092_neg__numeral__le__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( one_one @ A ) ) ) ).
% neg_numeral_le_one
thf(fact_1093_neg__one__le__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ M2 ) ) ) ).
% neg_one_le_numeral
thf(fact_1094_neg__numeral__le__neg__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% neg_numeral_le_neg_one
thf(fact_1095_not__numeral__le__neg__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] :
~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ M2 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% not_numeral_le_neg_one
thf(fact_1096_not__one__le__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] :
~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) ) ) ).
% not_one_le_neg_numeral
thf(fact_1097_not__neg__one__less__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] :
~ ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) ) ) ).
% not_neg_one_less_neg_numeral
thf(fact_1098_not__one__less__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] :
~ ( ord_less @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) ) ) ).
% not_one_less_neg_numeral
thf(fact_1099_not__numeral__less__neg__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] :
~ ( ord_less @ A @ ( numeral_numeral @ A @ M2 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% not_numeral_less_neg_one
thf(fact_1100_neg__one__less__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ M2 ) ) ) ).
% neg_one_less_numeral
thf(fact_1101_neg__numeral__less__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( one_one @ A ) ) ) ).
% neg_numeral_less_one
thf(fact_1102_sum_OSuc__reindex__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M2: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G @ M2 )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ) ) ).
% sum.Suc_reindex_ivl
thf(fact_1103_sum__Suc__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M2: nat,N: nat,F: nat > A] :
( ( ord_less_eq @ nat @ M2 @ ( suc @ N ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( minus_minus @ A @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( minus_minus @ A @ ( F @ ( suc @ N ) ) @ ( F @ M2 ) ) ) ) ) ).
% sum_Suc_diff
thf(fact_1104_sum__power__add,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,M2: nat,I4: set @ nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( power_power @ A @ X @ ( plus_plus @ nat @ M2 @ I3 ) )
@ I4 )
= ( times_times @ A @ ( power_power @ A @ X @ M2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ I4 ) ) ) ) ).
% sum_power_add
thf(fact_1105_sum_OatLeast__atMost__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ( comp @ nat @ A @ nat @ G
@ ^ [N5: nat] : ( minus_minus @ nat @ N5 @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% sum.atLeast_atMost_pred_shift
thf(fact_1106_sum__atLeastAtMost__code,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F: nat > A,A3: nat,B2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or1337092689740270186AtMost @ nat @ A3 @ B2 ) )
= ( set_fo6178422350223883121st_nat @ A
@ ^ [A7: nat] : ( plus_plus @ A @ ( F @ A7 ) )
@ A3
@ B2
@ ( zero_zero @ A ) ) ) ) ).
% sum_atLeastAtMost_code
thf(fact_1107_sum_Oub__add__nat,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M2: nat,N: nat,G: nat > A,P5: nat] :
( ( ord_less_eq @ nat @ M2 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( plus_plus @ nat @ N @ P5 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N @ P5 ) ) ) ) ) ) ) ).
% sum.ub_add_nat
thf(fact_1108_sum_OatLeastAtMost__shift__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M2: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M2 ) ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ) ).
% sum.atLeastAtMost_shift_0
thf(fact_1109_zero__int__def,axiom,
( ( zero_zero @ int )
= ( abs_Integ @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) ) ) ).
% zero_int_def
thf(fact_1110_pair__lessI1,axiom,
! [A3: nat,B2: nat,S: nat,T2: nat] :
( ( ord_less @ nat @ A3 @ B2 )
=> ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A3 @ S ) @ ( product_Pair @ nat @ nat @ B2 @ T2 ) ) @ fun_pair_less ) ) ).
% pair_lessI1
thf(fact_1111_zero__int_Otransfer,axiom,
pcr_int @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) @ ( zero_zero @ int ) ).
% zero_int.transfer
thf(fact_1112_sum__natinterval__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M2: nat,N: nat,F: nat > A] :
( ( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K4: nat] : ( minus_minus @ A @ ( F @ K4 ) @ ( F @ ( plus_plus @ nat @ K4 @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( minus_minus @ A @ ( F @ M2 ) @ ( F @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) )
& ( ~ ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K4: nat] : ( minus_minus @ A @ ( F @ K4 ) @ ( F @ ( plus_plus @ nat @ K4 @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_natinterval_diff
thf(fact_1113_convex__sum__bound__le,axiom,
! [A: $tType,B: $tType] :
( ( linordered_idom @ B )
=> ! [I4: set @ A,X: A > B,A3: A > B,B2: B,Delta: B] :
( ! [I2: A] :
( ( member @ A @ I2 @ I4 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( X @ I2 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ A @ B @ X @ I4 )
= ( one_one @ B ) )
=> ( ! [I2: A] :
( ( member @ A @ I2 @ I4 )
=> ( ord_less_eq @ B @ ( abs_abs @ B @ ( minus_minus @ B @ ( A3 @ I2 ) @ B2 ) ) @ Delta ) )
=> ( ord_less_eq @ B
@ ( abs_abs @ B
@ ( minus_minus @ B
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I3: A] : ( times_times @ B @ ( A3 @ I3 ) @ ( X @ I3 ) )
@ I4 )
@ B2 ) )
@ Delta ) ) ) ) ) ).
% convex_sum_bound_le
thf(fact_1114_sum__comp__morphism,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( comm_monoid_add @ B )
& ( comm_monoid_add @ A ) )
=> ! [H3: B > A,G: C > B,A4: set @ C] :
( ( ( H3 @ ( zero_zero @ B ) )
= ( zero_zero @ A ) )
=> ( ! [X2: B,Y4: B] :
( ( H3 @ ( plus_plus @ B @ X2 @ Y4 ) )
= ( plus_plus @ A @ ( H3 @ X2 ) @ ( H3 @ Y4 ) ) )
=> ( ( groups7311177749621191930dd_sum @ C @ A @ ( comp @ B @ A @ C @ H3 @ G ) @ A4 )
= ( H3 @ ( groups7311177749621191930dd_sum @ C @ B @ G @ A4 ) ) ) ) ) ) ).
% sum_comp_morphism
thf(fact_1115_sum__gp,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [N: nat,M2: nat,X: A] :
( ( ( ord_less @ nat @ N @ M2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M2 )
=> ( ( ( X
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ M2 ) ) ) )
& ( ( X
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ X @ M2 ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ) ) ).
% sum_gp
thf(fact_1116_divmod__digit__1_I1_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) )
=> ( ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) @ ( one_one @ A ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ) ).
% divmod_digit_1(1)
thf(fact_1117_sum__gp__offset,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [X: A,M2: nat,N: nat] :
( ( ( X
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( plus_plus @ nat @ M2 @ N ) ) )
= ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) ) )
& ( ( X
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( plus_plus @ nat @ M2 @ N ) ) )
= ( divide_divide @ A @ ( times_times @ A @ ( power_power @ A @ X @ M2 ) @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ).
% sum_gp_offset
thf(fact_1118_Sum__Icc__nat,axiom,
! [M2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X3: nat] : X3
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( divide_divide @ nat @ ( minus_minus @ nat @ ( times_times @ nat @ N @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) @ ( times_times @ nat @ M2 @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% Sum_Icc_nat
thf(fact_1119_arith__series__nat,axiom,
! [A3: nat,D3: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I3: nat] : ( plus_plus @ nat @ A3 @ ( times_times @ nat @ I3 @ D3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( divide_divide @ nat @ ( times_times @ nat @ ( suc @ N ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A3 ) @ ( times_times @ nat @ N @ D3 ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% arith_series_nat
thf(fact_1120_mod__double__modulus,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M2: A,X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ( modulo_modulo @ A @ X @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) )
= ( modulo_modulo @ A @ X @ M2 ) )
| ( ( modulo_modulo @ A @ X @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) )
= ( plus_plus @ A @ ( modulo_modulo @ A @ X @ M2 ) @ M2 ) ) ) ) ) ) ).
% mod_double_modulus
thf(fact_1121_sum_Odistrib,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: B > A,H3: B > A,A4: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X3: B] : ( plus_plus @ A @ ( G @ X3 ) @ ( H3 @ X3 ) )
@ A4 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 ) @ ( groups7311177749621191930dd_sum @ B @ A @ H3 @ A4 ) ) ) ) ).
% sum.distrib
thf(fact_1122_case__prodI,axiom,
! [A: $tType,B: $tType,F: A > B > $o,A3: A,B2: B] :
( ( F @ A3 @ B2 )
=> ( product_case_prod @ A @ B @ $o @ F @ ( product_Pair @ A @ B @ A3 @ B2 ) ) ) ).
% case_prodI
thf(fact_1123_case__prodI2,axiom,
! [B: $tType,A: $tType,P5: product_prod @ A @ B,C2: A > B > $o] :
( ! [A5: A,B3: B] :
( ( P5
= ( product_Pair @ A @ B @ A5 @ B3 ) )
=> ( C2 @ A5 @ B3 ) )
=> ( product_case_prod @ A @ B @ $o @ C2 @ P5 ) ) ).
% case_prodI2
thf(fact_1124_mem__case__prodI,axiom,
! [A: $tType,B: $tType,C: $tType,Z2: A,C2: B > C > ( set @ A ),A3: B,B2: C] :
( ( member @ A @ Z2 @ ( C2 @ A3 @ B2 ) )
=> ( member @ A @ Z2 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ ( product_Pair @ B @ C @ A3 @ B2 ) ) ) ) ).
% mem_case_prodI
thf(fact_1125_mem__case__prodI2,axiom,
! [C: $tType,B: $tType,A: $tType,P5: product_prod @ A @ B,Z2: C,C2: A > B > ( set @ C )] :
( ! [A5: A,B3: B] :
( ( P5
= ( product_Pair @ A @ B @ A5 @ B3 ) )
=> ( member @ C @ Z2 @ ( C2 @ A5 @ B3 ) ) )
=> ( member @ C @ Z2 @ ( product_case_prod @ A @ B @ ( set @ C ) @ C2 @ P5 ) ) ) ).
% mem_case_prodI2
thf(fact_1126_case__prodI2_H,axiom,
! [A: $tType,B: $tType,C: $tType,P5: product_prod @ A @ B,C2: A > B > C > $o,X: C] :
( ! [A5: A,B3: B] :
( ( ( product_Pair @ A @ B @ A5 @ B3 )
= P5 )
=> ( C2 @ A5 @ B3 @ X ) )
=> ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P5 @ X ) ) ).
% case_prodI2'
thf(fact_1127_one__eq__numeral__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: num] :
( ( ( one_one @ A )
= ( numeral_numeral @ A @ N ) )
= ( one2 = N ) ) ) ).
% one_eq_numeral_iff
thf(fact_1128_numeral__eq__one__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: num] :
( ( ( numeral_numeral @ A @ N )
= ( one_one @ A ) )
= ( N = one2 ) ) ) ).
% numeral_eq_one_iff
thf(fact_1129_of__nat__add,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [M2: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ M2 @ N ) )
= ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% of_nat_add
thf(fact_1130_of__nat__1,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A @ ( one_one @ nat ) )
= ( one_one @ A ) ) ) ).
% of_nat_1
thf(fact_1131_of__nat__1__eq__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: nat] :
( ( ( one_one @ A )
= ( semiring_1_of_nat @ A @ N ) )
= ( N
= ( one_one @ nat ) ) ) ) ).
% of_nat_1_eq_iff
thf(fact_1132_of__nat__eq__1__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: nat] :
( ( ( semiring_1_of_nat @ A @ N )
= ( one_one @ A ) )
= ( N
= ( one_one @ nat ) ) ) ) ).
% of_nat_eq_1_iff
thf(fact_1133_semiring__norm_I2_J,axiom,
( ( plus_plus @ num @ one2 @ one2 )
= ( bit0 @ one2 ) ) ).
% semiring_norm(2)
thf(fact_1134_numeral__le__one__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N: num] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ N ) @ ( one_one @ A ) )
= ( ord_less_eq @ num @ N @ one2 ) ) ) ).
% numeral_le_one_iff
thf(fact_1135_one__less__numeral__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N: num] :
( ( ord_less @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) )
= ( ord_less @ num @ one2 @ N ) ) ) ).
% one_less_numeral_iff
thf(fact_1136_numeral__eq__neg__one__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [N: num] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( N = one2 ) ) ) ).
% numeral_eq_neg_one_iff
thf(fact_1137_neg__one__eq__numeral__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [N: num] :
( ( ( uminus_uminus @ A @ ( one_one @ A ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( N = one2 ) ) ) ).
% neg_one_eq_numeral_iff
thf(fact_1138_of__nat__Suc,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [M2: nat] :
( ( semiring_1_of_nat @ A @ ( suc @ M2 ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ M2 ) ) ) ) ).
% of_nat_Suc
thf(fact_1139_Suc__numeral,axiom,
! [N: num] :
( ( suc @ ( numeral_numeral @ nat @ N ) )
= ( numeral_numeral @ nat @ ( plus_plus @ num @ N @ one2 ) ) ) ).
% Suc_numeral
thf(fact_1140_not__neg__one__le__neg__numeral__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] :
( ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) ) )
= ( M2 != one2 ) ) ) ).
% not_neg_one_le_neg_numeral_iff
thf(fact_1141_one__add__one,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ( ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ).
% one_add_one
thf(fact_1142_neg__numeral__less__neg__one__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( M2 != one2 ) ) ) ).
% neg_numeral_less_neg_one_iff
thf(fact_1143_bits__one__mod__two__eq__one,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( modulo_modulo @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ).
% bits_one_mod_two_eq_one
thf(fact_1144_one__mod__two__eq__one,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ( ( modulo_modulo @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ).
% one_mod_two_eq_one
thf(fact_1145_add__2__eq__Suc,axiom,
! [N: nat] :
( ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
= ( suc @ ( suc @ N ) ) ) ).
% add_2_eq_Suc
thf(fact_1146_add__2__eq__Suc_H,axiom,
! [N: nat] :
( ( plus_plus @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( suc @ ( suc @ N ) ) ) ).
% add_2_eq_Suc'
thf(fact_1147_Suc__1,axiom,
( ( suc @ ( one_one @ nat ) )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% Suc_1
thf(fact_1148_numeral__plus__one,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N: num] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ N ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( plus_plus @ num @ N @ one2 ) ) ) ) ).
% numeral_plus_one
thf(fact_1149_one__plus__numeral,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N: num] :
( ( plus_plus @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) )
= ( numeral_numeral @ A @ ( plus_plus @ num @ one2 @ N ) ) ) ) ).
% one_plus_numeral
thf(fact_1150_add__self__div__2,axiom,
! [M2: nat] :
( ( divide_divide @ nat @ ( plus_plus @ nat @ M2 @ M2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= M2 ) ).
% add_self_div_2
thf(fact_1151_bits__1__div__2,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( zero_zero @ A ) ) ) ).
% bits_1_div_2
thf(fact_1152_one__div__two__eq__zero,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ( ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( zero_zero @ A ) ) ) ).
% one_div_two_eq_zero
thf(fact_1153_add__neg__numeral__special_I9_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% add_neg_numeral_special(9)
thf(fact_1154_diff__numeral__special_I11_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( minus_minus @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ).
% diff_numeral_special(11)
thf(fact_1155_diff__numeral__special_I10_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% diff_numeral_special(10)
thf(fact_1156_not__mod__2__eq__0__eq__1,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
!= ( zero_zero @ A ) )
= ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ) ).
% not_mod_2_eq_0_eq_1
thf(fact_1157_not__mod__2__eq__1__eq__0,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
!= ( one_one @ A ) )
= ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( zero_zero @ A ) ) ) ) ).
% not_mod_2_eq_1_eq_0
thf(fact_1158_minus__1__div__2__eq,axiom,
! [A: $tType] :
( ( euclid8789492081693882211th_nat @ A )
=> ( ( divide_divide @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% minus_1_div_2_eq
thf(fact_1159_bits__minus__1__mod__2__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( modulo_modulo @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ).
% bits_minus_1_mod_2_eq
thf(fact_1160_minus__1__mod__2__eq,axiom,
! [A: $tType] :
( ( euclid8789492081693882211th_nat @ A )
=> ( ( modulo_modulo @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ).
% minus_1_mod_2_eq
thf(fact_1161_diff__numeral__special_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N: num] :
( ( minus_minus @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( numeral_numeral @ A @ ( plus_plus @ num @ one2 @ N ) ) ) ) ).
% diff_numeral_special(3)
thf(fact_1162_diff__numeral__special_I4_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( one_one @ A ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ M2 @ one2 ) ) ) ) ) ).
% diff_numeral_special(4)
thf(fact_1163_add__self__mod__2,axiom,
! [M2: nat] :
( ( modulo_modulo @ nat @ ( plus_plus @ nat @ M2 @ M2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( zero_zero @ nat ) ) ).
% add_self_mod_2
thf(fact_1164_power__minus1__even,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat] :
( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
= ( one_one @ A ) ) ) ).
% power_minus1_even
thf(fact_1165_mod2__gr__0,axiom,
! [M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( modulo_modulo @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( ( modulo_modulo @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( one_one @ nat ) ) ) ).
% mod2_gr_0
thf(fact_1166_lex__prod__def,axiom,
! [B: $tType,A: $tType] :
( ( lex_prod @ A @ B )
= ( ^ [Ra: set @ ( product_prod @ A @ A ),Rb: set @ ( product_prod @ B @ B )] :
( collect @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) )
@ ( product_case_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ $o
@ ( product_case_prod @ A @ B @ ( ( product_prod @ A @ B ) > $o )
@ ^ [A7: A,B5: B] :
( product_case_prod @ A @ B @ $o
@ ^ [A8: A,B7: B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ A8 ) @ Ra )
| ( ( A7 = A8 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B5 @ B7 ) @ Rb ) ) ) ) ) ) ) ) ) ).
% lex_prod_def
thf(fact_1167_same__fst__def,axiom,
! [B: $tType,A: $tType] :
( ( same_fst @ A @ B )
= ( ^ [P4: A > $o,R6: A > ( set @ ( product_prod @ B @ B ) )] :
( collect @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) )
@ ( product_case_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ $o
@ ( product_case_prod @ A @ B @ ( ( product_prod @ A @ B ) > $o )
@ ^ [X9: A,Y8: B] :
( product_case_prod @ A @ B @ $o
@ ^ [X3: A,Y3: B] :
( ( X9 = X3 )
& ( P4 @ X3 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y8 @ Y3 ) @ ( R6 @ X3 ) ) ) ) ) ) ) ) ) ).
% same_fst_def
thf(fact_1168_mem__case__prodE,axiom,
! [B: $tType,A: $tType,C: $tType,Z2: A,C2: B > C > ( set @ A ),P5: product_prod @ B @ C] :
( ( member @ A @ Z2 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ P5 ) )
=> ~ ! [X2: B,Y4: C] :
( ( P5
= ( product_Pair @ B @ C @ X2 @ Y4 ) )
=> ~ ( member @ A @ Z2 @ ( C2 @ X2 @ Y4 ) ) ) ) ).
% mem_case_prodE
thf(fact_1169_case__prodD,axiom,
! [A: $tType,B: $tType,F: A > B > $o,A3: A,B2: B] :
( ( product_case_prod @ A @ B @ $o @ F @ ( product_Pair @ A @ B @ A3 @ B2 ) )
=> ( F @ A3 @ B2 ) ) ).
% case_prodD
thf(fact_1170_case__prodE,axiom,
! [A: $tType,B: $tType,C2: A > B > $o,P5: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ $o @ C2 @ P5 )
=> ~ ! [X2: A,Y4: B] :
( ( P5
= ( product_Pair @ A @ B @ X2 @ Y4 ) )
=> ~ ( C2 @ X2 @ Y4 ) ) ) ).
% case_prodE
thf(fact_1171_add__One__commute,axiom,
! [N: num] :
( ( plus_plus @ num @ one2 @ N )
= ( plus_plus @ num @ N @ one2 ) ) ).
% add_One_commute
thf(fact_1172_case__prodD_H,axiom,
! [B: $tType,A: $tType,C: $tType,R5: A > B > C > $o,A3: A,B2: B,C2: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ R5 @ ( product_Pair @ A @ B @ A3 @ B2 ) @ C2 )
=> ( R5 @ A3 @ B2 @ C2 ) ) ).
% case_prodD'
thf(fact_1173_case__prodE_H,axiom,
! [B: $tType,A: $tType,C: $tType,C2: A > B > C > $o,P5: product_prod @ A @ B,Z2: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P5 @ Z2 )
=> ~ ! [X2: A,Y4: B] :
( ( P5
= ( product_Pair @ A @ B @ X2 @ Y4 ) )
=> ~ ( C2 @ X2 @ Y4 @ Z2 ) ) ) ).
% case_prodE'
thf(fact_1174_sum_OatLeast__int__atMost__int__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: int > A,M2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ int @ A @ G @ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ int @ A @ nat @ G @ ( semiring_1_of_nat @ int ) ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% sum.atLeast_int_atMost_int_shift
thf(fact_1175_less__int_Oabs__eq,axiom,
! [Xa: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
( ( ord_less @ int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
= ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X3: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V2: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X3 @ V2 ) @ ( plus_plus @ nat @ U2 @ Y3 ) ) )
@ Xa
@ X ) ) ).
% less_int.abs_eq
thf(fact_1176_less__eq__int_Oabs__eq,axiom,
! [Xa: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
( ( ord_less_eq @ int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
= ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X3: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V2: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X3 @ V2 ) @ ( plus_plus @ nat @ U2 @ Y3 ) ) )
@ Xa
@ X ) ) ).
% less_eq_int.abs_eq
thf(fact_1177_Collect__case__prod__mono,axiom,
! [B: $tType,A: $tType,A4: A > B > $o,B6: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ A4 @ B6 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A4 ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ B6 ) ) ) ) ).
% Collect_case_prod_mono
thf(fact_1178_zadd__int__left,axiom,
! [M2: nat,N: nat,Z2: int] :
( ( plus_plus @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N ) @ Z2 ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ M2 @ N ) ) @ Z2 ) ) ).
% zadd_int_left
thf(fact_1179_int__plus,axiom,
! [N: nat,M2: nat] :
( ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ N @ M2 ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N ) @ ( semiring_1_of_nat @ int @ M2 ) ) ) ).
% int_plus
thf(fact_1180_int__ops_I5_J,axiom,
! [A3: nat,B2: nat] :
( ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ A3 @ B2 ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% int_ops(5)
thf(fact_1181_int__ops_I2_J,axiom,
( ( semiring_1_of_nat @ int @ ( one_one @ nat ) )
= ( one_one @ int ) ) ).
% int_ops(2)
thf(fact_1182_zle__iff__zadd,axiom,
( ( ord_less_eq @ int )
= ( ^ [W2: int,Z5: int] :
? [N5: nat] :
( Z5
= ( plus_plus @ int @ W2 @ ( semiring_1_of_nat @ int @ N5 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_1183_numeral__One,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ( ( numeral_numeral @ A @ one2 )
= ( one_one @ A ) ) ) ).
% numeral_One
thf(fact_1184_numerals_I1_J,axiom,
( ( numeral_numeral @ nat @ one2 )
= ( one_one @ nat ) ) ).
% numerals(1)
thf(fact_1185_execute__bind__case,axiom,
! [A: $tType,B: $tType,F: heap_Time_Heap @ B,G: B > ( heap_Time_Heap @ A ),H3: heap_ext @ product_unit] :
( ( heap_Time_execute @ A @ ( heap_Time_bind @ B @ A @ F @ G ) @ H3 )
= ( 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 ) ) )
@ ^ [X3: B] :
( product_case_prod @ ( heap_ext @ product_unit ) @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
@ ^ [H7: heap_ext @ product_unit,N5: nat] : ( heap_Time_timeFrame @ A @ N5 @ ( heap_Time_execute @ A @ ( G @ X3 ) @ H7 ) ) ) )
@ ( heap_Time_execute @ B @ F @ H3 ) ) ) ).
% execute_bind_case
thf(fact_1186_Sum__Icc__int,axiom,
! [M2: int,N: int] :
( ( ord_less_eq @ int @ M2 @ N )
=> ( ( groups7311177749621191930dd_sum @ int @ int
@ ^ [X3: int] : X3
@ ( set_or1337092689740270186AtMost @ int @ M2 @ N ) )
= ( divide_divide @ int @ ( minus_minus @ int @ ( times_times @ int @ N @ ( plus_plus @ int @ N @ ( one_one @ int ) ) ) @ ( times_times @ int @ M2 @ ( minus_minus @ int @ M2 @ ( one_one @ int ) ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ).
% Sum_Icc_int
thf(fact_1187_Heap__Time__Monad_Obind__def,axiom,
! [B: $tType,A: $tType] :
( ( heap_Time_bind @ A @ B )
= ( ^ [F2: heap_Time_Heap @ A,G2: A > ( heap_Time_Heap @ B )] :
( heap_Time_Heap2 @ B
@ ^ [H: heap_ext @ product_unit] :
( 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 ) ) )
@ ^ [R3: A] :
( product_case_prod @ ( heap_ext @ product_unit ) @ nat @ ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
@ ^ [H7: heap_ext @ product_unit,N5: nat] : ( heap_Time_timeFrame @ B @ N5 @ ( heap_Time_execute @ B @ ( G2 @ R3 ) @ H7 ) ) ) )
@ ( heap_Time_execute @ A @ F2 @ H ) ) ) ) ) ).
% Heap_Time_Monad.bind_def
thf(fact_1188_zless__iff__Suc__zadd,axiom,
( ( ord_less @ int )
= ( ^ [W2: int,Z5: int] :
? [N5: nat] :
( Z5
= ( plus_plus @ int @ W2 @ ( semiring_1_of_nat @ int @ ( suc @ N5 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_1189_int__ops_I4_J,axiom,
! [A3: nat] :
( ( semiring_1_of_nat @ int @ ( suc @ A3 ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( one_one @ int ) ) ) ).
% int_ops(4)
thf(fact_1190_int__Suc,axiom,
! [N: nat] :
( ( semiring_1_of_nat @ int @ ( suc @ N ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N ) @ ( one_one @ int ) ) ) ).
% int_Suc
thf(fact_1191_int__bit__induct,axiom,
! [P: int > $o,K: int] :
( ( P @ ( zero_zero @ int ) )
=> ( ( P @ ( uminus_uminus @ int @ ( one_one @ int ) ) )
=> ( ! [K5: int] :
( ( P @ K5 )
=> ( ( K5
!= ( zero_zero @ int ) )
=> ( P @ ( times_times @ int @ K5 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ! [K5: int] :
( ( P @ K5 )
=> ( ( K5
!= ( uminus_uminus @ int @ ( one_one @ int ) ) )
=> ( P @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ K5 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) )
=> ( P @ K ) ) ) ) ) ).
% int_bit_induct
thf(fact_1192_uminus__numeral__One,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ one2 ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% uminus_numeral_One
thf(fact_1193_int__def,axiom,
( ( semiring_1_of_nat @ int )
= ( ^ [N5: nat] : ( abs_Integ @ ( product_Pair @ nat @ nat @ N5 @ ( zero_zero @ nat ) ) ) ) ) ).
% int_def
thf(fact_1194_num__of__nat__One,axiom,
! [N: nat] :
( ( ord_less_eq @ nat @ N @ ( one_one @ nat ) )
=> ( ( num_of_nat @ N )
= one2 ) ) ).
% num_of_nat_One
thf(fact_1195_neg__zdiv__mult__2,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq @ int @ A3 @ ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) )
= ( divide_divide @ int @ ( plus_plus @ int @ B2 @ ( one_one @ int ) ) @ A3 ) ) ) ).
% neg_zdiv_mult_2
thf(fact_1196_pos__zdiv__mult__2,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
=> ( ( divide_divide @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) )
= ( divide_divide @ int @ B2 @ A3 ) ) ) ).
% pos_zdiv_mult_2
thf(fact_1197_pos__zmod__mult__2,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
=> ( ( modulo_modulo @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) )
= ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ B2 @ A3 ) ) ) ) ) ).
% pos_zmod_mult_2
thf(fact_1198_mod__mult2__eq_H,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A,M2: nat,N: nat] :
( ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N ) ) )
= ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( modulo_modulo @ A @ ( divide_divide @ A @ A3 @ ( semiring_1_of_nat @ A @ M2 ) ) @ ( semiring_1_of_nat @ A @ N ) ) ) @ ( modulo_modulo @ A @ A3 @ ( semiring_1_of_nat @ A @ M2 ) ) ) ) ) ).
% mod_mult2_eq'
thf(fact_1199_numeral__num__of__nat__unfold,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N: nat] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( numeral_numeral @ A @ ( num_of_nat @ N ) )
= ( one_one @ A ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( numeral_numeral @ A @ ( num_of_nat @ N ) )
= ( semiring_1_of_nat @ A @ N ) ) ) ) ) ).
% numeral_num_of_nat_unfold
thf(fact_1200_double__arith__series,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,D3: A,N: nat] :
( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( plus_plus @ A @ A3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I3 ) @ D3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) )
= ( times_times @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ D3 ) ) ) ) ) ).
% double_arith_series
thf(fact_1201_double__gauss__sum,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [N: nat] :
( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) ) ) ) ).
% double_gauss_sum
thf(fact_1202_neg__zmod__mult__2,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq @ int @ A3 @ ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) )
= ( minus_minus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ ( plus_plus @ int @ B2 @ ( one_one @ int ) ) @ A3 ) ) @ ( one_one @ int ) ) ) ) ).
% neg_zmod_mult_2
thf(fact_1203_mult__2,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [Z2: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Z2 )
= ( plus_plus @ A @ Z2 @ Z2 ) ) ) ).
% mult_2
thf(fact_1204_mult__2__right,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [Z2: A] :
( ( times_times @ A @ Z2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( plus_plus @ A @ Z2 @ Z2 ) ) ) ).
% mult_2_right
thf(fact_1205_left__add__twice,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [A3: A,B2: A] :
( ( plus_plus @ A @ A3 @ ( plus_plus @ A @ A3 @ B2 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) @ B2 ) ) ) ).
% left_add_twice
thf(fact_1206_pos__eucl__rel__int__mult__2,axiom,
! [B2: int,A3: int,Q3: int,R: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( eucl_rel_int @ A3 @ B2 @ ( product_Pair @ int @ int @ Q3 @ R ) )
=> ( eucl_rel_int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) @ ( product_Pair @ int @ int @ Q3 @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ R ) ) ) ) ) ) ).
% pos_eucl_rel_int_mult_2
thf(fact_1207_one__power2,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( power_power @ A @ ( one_one @ A ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ).
% one_power2
thf(fact_1208_nat__1__add__1,axiom,
( ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% nat_1_add_1
thf(fact_1209_Suc__nat__number__of__add,axiom,
! [V: num,N: nat] :
( ( suc @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ V ) @ N ) )
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( plus_plus @ num @ V @ one2 ) ) @ N ) ) ).
% Suc_nat_number_of_add
thf(fact_1210_double__gauss__sum__from__Suc__0,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [N: nat] :
( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) ) ) ) ).
% double_gauss_sum_from_Suc_0
thf(fact_1211_arith__series,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A,D3: A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( plus_plus @ A @ A3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I3 ) @ D3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( divide_divide @ A @ ( times_times @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ D3 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% arith_series
thf(fact_1212_gauss__sum,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( divide_divide @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% gauss_sum
thf(fact_1213_nat__approx__posE,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [E2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E2 )
=> ~ ! [N3: nat] :
~ ( ord_less @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ ( suc @ N3 ) ) ) @ E2 ) ) ) ).
% nat_approx_posE
thf(fact_1214_neg__eucl__rel__int__mult__2,axiom,
! [B2: int,A3: int,Q3: int,R: int] :
( ( ord_less_eq @ int @ B2 @ ( zero_zero @ int ) )
=> ( ( eucl_rel_int @ ( plus_plus @ int @ A3 @ ( one_one @ int ) ) @ B2 @ ( product_Pair @ int @ int @ Q3 @ R ) )
=> ( eucl_rel_int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) @ ( product_Pair @ int @ int @ Q3 @ ( minus_minus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ R ) @ ( one_one @ int ) ) ) ) ) ) ).
% neg_eucl_rel_int_mult_2
thf(fact_1215_gauss__sum__from__Suc__0,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) )
= ( divide_divide @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% gauss_sum_from_Suc_0
thf(fact_1216_exp__add__not__zero__imp__right,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M2: nat,N: nat] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M2 @ N ) )
!= ( zero_zero @ A ) )
=> ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
!= ( zero_zero @ A ) ) ) ) ).
% exp_add_not_zero_imp_right
thf(fact_1217_exp__add__not__zero__imp__left,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M2: nat,N: nat] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M2 @ N ) )
!= ( zero_zero @ A ) )
=> ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 )
!= ( zero_zero @ A ) ) ) ) ).
% exp_add_not_zero_imp_left
thf(fact_1218_sum__power2__eq__zero__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% sum_power2_eq_zero_iff
thf(fact_1219_div__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,M2: nat,N: nat] :
( ( divide_divide @ A @ ( divide_divide @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( divide_divide @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M2 @ N ) ) ) ) ) ).
% div_exp_eq
thf(fact_1220_power2__eq__1__iff,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [A3: A] :
( ( ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) )
= ( ( A3
= ( one_one @ A ) )
| ( A3
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% power2_eq_1_iff
thf(fact_1221_plus__int_Oabs__eq,axiom,
! [Xa: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
( ( plus_plus @ int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X3: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X3 @ U2 ) @ ( plus_plus @ nat @ Y3 @ V2 ) ) )
@ Xa
@ X ) ) ) ).
% plus_int.abs_eq
thf(fact_1222_abs__square__eq__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] :
( ( ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) )
= ( ( abs_abs @ A @ X )
= ( one_one @ A ) ) ) ) ).
% abs_square_eq_1
thf(fact_1223_minus__int_Oabs__eq,axiom,
! [Xa: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
( ( minus_minus @ int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X3: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X3 @ V2 ) @ ( plus_plus @ nat @ Y3 @ U2 ) ) )
@ Xa
@ X ) ) ) ).
% minus_int.abs_eq
thf(fact_1224_nat__induct2,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ( P @ ( one_one @ nat ) )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( plus_plus @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct2
thf(fact_1225_times__int_Oabs__eq,axiom,
! [Xa: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
( ( times_times @ int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X3: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X3 @ U2 ) @ ( times_times @ nat @ Y3 @ V2 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X3 @ V2 ) @ ( times_times @ nat @ Y3 @ U2 ) ) ) )
@ Xa
@ X ) ) ) ).
% times_int.abs_eq
thf(fact_1226_sum__power2__le__zero__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% sum_power2_le_zero_iff
thf(fact_1227_sum__power2__ge__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sum_power2_ge_zero
thf(fact_1228_bits__stable__imp__add__self,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A] :
( ( ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A3 )
=> ( ( plus_plus @ A @ A3 @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= ( zero_zero @ A ) ) ) ) ).
% bits_stable_imp_add_self
thf(fact_1229_not__sum__power2__lt__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
~ ( ord_less @ A @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( zero_zero @ A ) ) ) ).
% not_sum_power2_lt_zero
thf(fact_1230_sum__power2__gt__zero__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( ( X
!= ( zero_zero @ A ) )
| ( Y
!= ( zero_zero @ A ) ) ) ) ) ).
% sum_power2_gt_zero_iff
thf(fact_1231_power2__sum,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [X: A,Y: A] :
( ( power_power @ A @ ( plus_plus @ A @ X @ Y ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( plus_plus @ A @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) @ Y ) ) ) ) ).
% power2_sum
thf(fact_1232_square__le__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X )
=> ( ( ord_less_eq @ A @ X @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ) ).
% square_le_1
thf(fact_1233_abs__square__le__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] :
( ( ord_less_eq @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) )
= ( ord_less_eq @ A @ ( abs_abs @ A @ X ) @ ( one_one @ A ) ) ) ) ).
% abs_square_le_1
thf(fact_1234_abs__square__less__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) )
= ( ord_less @ A @ ( abs_abs @ A @ X ) @ ( one_one @ A ) ) ) ) ).
% abs_square_less_1
thf(fact_1235_div__exp__mod__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N: nat,M2: nat] :
( ( modulo_modulo @ A @ ( divide_divide @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) )
= ( divide_divide @ A @ ( modulo_modulo @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ N @ M2 ) ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% div_exp_mod_exp_eq
thf(fact_1236_ex__power__ivl2,axiom,
! [B2: nat,K: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
=> ? [N3: nat] :
( ( ord_less @ nat @ ( power_power @ nat @ B2 @ N3 ) @ K )
& ( ord_less_eq @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% ex_power_ivl2
thf(fact_1237_ex__power__ivl1,axiom,
! [B2: nat,K: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ K )
=> ? [N3: nat] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ B2 @ N3 ) @ K )
& ( ord_less @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% ex_power_ivl1
thf(fact_1238_power2__diff,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [X: A,Y: A] :
( ( power_power @ A @ ( minus_minus @ A @ X @ Y ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) @ Y ) ) ) ) ).
% power2_diff
thf(fact_1239_power__minus1__odd,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat] :
( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% power_minus1_odd
thf(fact_1240_mask__eq__sum__exp,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [N: nat] :
( ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ A ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
@ ( collect @ nat
@ ^ [Q5: nat] : ( ord_less @ nat @ Q5 @ N ) ) ) ) ) ).
% mask_eq_sum_exp
thf(fact_1241_sum_Oin__pairs,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( plus_plus @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% sum.in_pairs
thf(fact_1242_gauss__sum__nat,axiom,
! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X3: nat] : X3
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( divide_divide @ nat @ ( times_times @ nat @ N @ ( suc @ N ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% gauss_sum_nat
thf(fact_1243_divmod__step__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [L: num,R: A,Q3: A] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ L ) @ R )
=> ( ( unique1321980374590559556d_step @ A @ L @ ( product_Pair @ A @ A @ Q3 @ R ) )
= ( product_Pair @ A @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q3 ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ R @ ( numeral_numeral @ A @ L ) ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ L ) @ R )
=> ( ( unique1321980374590559556d_step @ A @ L @ ( product_Pair @ A @ A @ Q3 @ R ) )
= ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q3 ) @ R ) ) ) ) ) ).
% divmod_step_eq
thf(fact_1244_divmod__step__def,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique1321980374590559556d_step @ A )
= ( ^ [L3: num] :
( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q5: A,R3: A] : ( if @ ( product_prod @ A @ A ) @ ( ord_less_eq @ A @ ( numeral_numeral @ A @ L3 ) @ R3 ) @ ( product_Pair @ A @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q5 ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ R3 @ ( numeral_numeral @ A @ L3 ) ) ) @ ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q5 ) @ R3 ) ) ) ) ) ) ).
% divmod_step_def
thf(fact_1245_divmod__step__nat__def,axiom,
( ( unique1321980374590559556d_step @ nat )
= ( ^ [L3: num] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [Q5: nat,R3: nat] : ( if @ ( product_prod @ nat @ nat ) @ ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ L3 ) @ R3 ) @ ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Q5 ) @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ R3 @ ( numeral_numeral @ nat @ L3 ) ) ) @ ( product_Pair @ nat @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Q5 ) @ R3 ) ) ) ) ) ).
% divmod_step_nat_def
thf(fact_1246_divmod__step__int__def,axiom,
( ( unique1321980374590559556d_step @ int )
= ( ^ [L3: num] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [Q5: int,R3: int] : ( if @ ( product_prod @ int @ int ) @ ( ord_less_eq @ int @ ( numeral_numeral @ int @ L3 ) @ R3 ) @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Q5 ) @ ( one_one @ int ) ) @ ( minus_minus @ int @ R3 @ ( numeral_numeral @ int @ L3 ) ) ) @ ( product_Pair @ int @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Q5 ) @ R3 ) ) ) ) ) ).
% divmod_step_int_def
thf(fact_1247_set__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5668285175392031749et_bit @ A @ ( zero_zero @ nat ) @ A3 )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% set_bit_0
thf(fact_1248_mlex__eq,axiom,
! [A: $tType] :
( ( mlex_prod @ A )
= ( ^ [F2: A > nat,R6: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X3: A,Y3: A] :
( ( ord_less @ nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) )
| ( ( ord_less_eq @ nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R6 ) ) ) ) ) ) ) ).
% mlex_eq
thf(fact_1249_unset__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_se2638667681897837118et_bit @ A @ ( suc @ N ) @ A3 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2638667681897837118et_bit @ A @ N @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% unset_bit_Suc
thf(fact_1250_flip__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_se8732182000553998342ip_bit @ A @ ( suc @ N ) @ A3 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se8732182000553998342ip_bit @ A @ N @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% flip_bit_Suc
thf(fact_1251_set__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_se5668285175392031749et_bit @ A @ ( suc @ N ) @ A3 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5668285175392031749et_bit @ A @ N @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% set_bit_Suc
thf(fact_1252_split__part,axiom,
! [B: $tType,A: $tType,P: $o,Q: A > B > $o] :
( ( product_case_prod @ A @ B @ $o
@ ^ [A7: A,B5: B] :
( P
& ( Q @ A7 @ B5 ) ) )
= ( ^ [Ab: product_prod @ A @ B] :
( P
& ( product_case_prod @ A @ B @ $o @ Q @ Ab ) ) ) ) ).
% split_part
thf(fact_1253_prod_Odisc__eq__case,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
( product_case_prod @ A @ B @ $o
@ ^ [Uu: A,Uv: B] : $true
@ Prod ) ).
% prod.disc_eq_case
thf(fact_1254_divmod__step__integer__def,axiom,
( ( unique1321980374590559556d_step @ code_integer )
= ( ^ [L3: num] :
( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [Q5: code_integer,R3: code_integer] : ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less_eq @ code_integer @ ( numeral_numeral @ code_integer @ L3 ) @ R3 ) @ ( product_Pair @ code_integer @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ Q5 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ R3 @ ( numeral_numeral @ code_integer @ L3 ) ) ) @ ( product_Pair @ code_integer @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ Q5 ) @ R3 ) ) ) ) ) ).
% divmod_step_integer_def
thf(fact_1255_signed__take__bit__rec,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4674362597316999326ke_bit @ A )
= ( ^ [N5: nat,A7: A] :
( if @ A
@ ( N5
= ( zero_zero @ nat ) )
@ ( uminus_uminus @ A @ ( modulo_modulo @ A @ A7 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
@ ( plus_plus @ A @ ( modulo_modulo @ A @ A7 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4674362597316999326ke_bit @ A @ ( minus_minus @ nat @ N5 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A7 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% signed_take_bit_rec
thf(fact_1256_round__unique,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: int] :
( ( ord_less @ A @ ( minus_minus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ Y ) )
=> ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Y ) @ ( plus_plus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) )
=> ( ( archimedean_round @ A @ X )
= Y ) ) ) ) ).
% round_unique
thf(fact_1257_rel__of__def,axiom,
! [B: $tType,A: $tType] :
( ( rel_of @ A @ B )
= ( ^ [M: A > ( option @ B ),P4: ( product_prod @ A @ B ) > $o] :
( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [K4: A,V2: B] :
( ( ( M @ K4 )
= ( some @ B @ V2 ) )
& ( P4 @ ( product_Pair @ A @ B @ K4 @ V2 ) ) ) ) ) ) ) ).
% rel_of_def
thf(fact_1258_dbl__simps_I4_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% dbl_simps(4)
thf(fact_1259_round__altdef,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A )
= ( ^ [X3: A] : ( if @ int @ ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( archimedean_frac @ A @ X3 ) ) @ ( archimedean_ceiling @ A @ X3 ) @ ( archim6421214686448440834_floor @ A @ X3 ) ) ) ) ) ).
% round_altdef
thf(fact_1260_round__unique_H,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,N: int] :
( ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ ( ring_1_of_int @ A @ N ) ) ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
=> ( ( archimedean_round @ A @ X )
= N ) ) ) ).
% round_unique'
thf(fact_1261_of__int__round__abs__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X ) ) @ X ) ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% of_int_round_abs_le
thf(fact_1262_divmod__algorithm__code_I5_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M2: num,N: num] :
( ( unique8689654367752047608divmod @ A @ ( bit0 @ M2 ) @ ( bit0 @ N ) )
= ( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q5: A,R3: A] : ( product_Pair @ A @ A @ Q5 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ R3 ) )
@ ( unique8689654367752047608divmod @ A @ M2 @ N ) ) ) ) ).
% divmod_algorithm_code(5)
thf(fact_1263_signed__take__bit__of__minus__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat] :
( ( bit_ri4674362597316999326ke_bit @ A @ N @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% signed_take_bit_of_minus_1
thf(fact_1264_signed__take__bit__Suc__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat] :
( ( bit_ri4674362597316999326ke_bit @ A @ ( suc @ N ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% signed_take_bit_Suc_1
thf(fact_1265_signed__take__bit__numeral__of__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [K: num] :
( ( bit_ri4674362597316999326ke_bit @ A @ ( numeral_numeral @ nat @ K ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% signed_take_bit_numeral_of_1
thf(fact_1266_round__1,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A @ ( one_one @ A ) )
= ( one_one @ int ) ) ) ).
% round_1
thf(fact_1267_divmod__algorithm__code_I2_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M2: num] :
( ( unique8689654367752047608divmod @ A @ M2 @ one2 )
= ( product_Pair @ A @ A @ ( numeral_numeral @ A @ M2 ) @ ( zero_zero @ A ) ) ) ) ).
% divmod_algorithm_code(2)
thf(fact_1268_dbl__simps_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl @ A @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ).
% dbl_simps(3)
thf(fact_1269_divmod__algorithm__code_I3_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N: num] :
( ( unique8689654367752047608divmod @ A @ one2 @ ( bit0 @ N ) )
= ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ one2 ) ) ) ) ).
% divmod_algorithm_code(3)
thf(fact_1270_less__eq__integer__code_I1_J,axiom,
ord_less_eq @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) ).
% less_eq_integer_code(1)
thf(fact_1271_sgn__integer__code,axiom,
( ( sgn_sgn @ code_integer )
= ( ^ [K4: code_integer] :
( if @ code_integer
@ ( K4
= ( zero_zero @ code_integer ) )
@ ( zero_zero @ code_integer )
@ ( if @ code_integer @ ( ord_less @ code_integer @ K4 @ ( zero_zero @ code_integer ) ) @ ( uminus_uminus @ code_integer @ ( one_one @ code_integer ) ) @ ( one_one @ code_integer ) ) ) ) ) ).
% sgn_integer_code
thf(fact_1272_exhaustive__integer_H_Ocases,axiom,
! [X: product_prod @ ( code_integer > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_integer @ code_integer )] :
~ ! [F4: code_integer > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D2: code_integer,I2: code_integer] :
( X
!= ( product_Pair @ ( code_integer > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_integer @ code_integer ) @ F4 @ ( product_Pair @ code_integer @ code_integer @ D2 @ I2 ) ) ) ).
% exhaustive_integer'.cases
thf(fact_1273_full__exhaustive__integer_H_Ocases,axiom,
! [X: product_prod @ ( ( product_prod @ code_integer @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_integer @ code_integer )] :
~ ! [F4: ( product_prod @ code_integer @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D2: code_integer,I2: code_integer] :
( X
!= ( product_Pair @ ( ( product_prod @ code_integer @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_integer @ code_integer ) @ F4 @ ( product_Pair @ code_integer @ code_integer @ D2 @ I2 ) ) ) ).
% full_exhaustive_integer'.cases
thf(fact_1274_times__integer__code_I1_J,axiom,
! [K: code_integer] :
( ( times_times @ code_integer @ K @ ( zero_zero @ code_integer ) )
= ( zero_zero @ code_integer ) ) ).
% times_integer_code(1)
thf(fact_1275_times__integer__code_I2_J,axiom,
! [L: code_integer] :
( ( times_times @ code_integer @ ( zero_zero @ code_integer ) @ L )
= ( zero_zero @ code_integer ) ) ).
% times_integer_code(2)
thf(fact_1276_minus__integer__code_I1_J,axiom,
! [K: code_integer] :
( ( minus_minus @ code_integer @ K @ ( zero_zero @ code_integer ) )
= K ) ).
% minus_integer_code(1)
thf(fact_1277_plus__integer__code_I2_J,axiom,
! [L: code_integer] :
( ( plus_plus @ code_integer @ ( zero_zero @ code_integer ) @ L )
= L ) ).
% plus_integer_code(2)
thf(fact_1278_plus__integer__code_I1_J,axiom,
! [K: code_integer] :
( ( plus_plus @ code_integer @ K @ ( zero_zero @ code_integer ) )
= K ) ).
% plus_integer_code(1)
thf(fact_1279_minus__integer__code_I2_J,axiom,
! [L: code_integer] :
( ( minus_minus @ code_integer @ ( zero_zero @ code_integer ) @ L )
= ( uminus_uminus @ code_integer @ L ) ) ).
% minus_integer_code(2)
thf(fact_1280_divmod__integer_H__def,axiom,
( ( unique8689654367752047608divmod @ code_integer )
= ( ^ [M: num,N5: num] : ( product_Pair @ code_integer @ code_integer @ ( divide_divide @ code_integer @ ( numeral_numeral @ code_integer @ M ) @ ( numeral_numeral @ code_integer @ N5 ) ) @ ( modulo_modulo @ code_integer @ ( numeral_numeral @ code_integer @ M ) @ ( numeral_numeral @ code_integer @ N5 ) ) ) ) ) ).
% divmod_integer'_def
thf(fact_1281_signed__take__bit__add,axiom,
! [N: nat,K: int,L: int] :
( ( bit_ri4674362597316999326ke_bit @ int @ N @ ( plus_plus @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) @ ( bit_ri4674362597316999326ke_bit @ int @ N @ L ) ) )
= ( bit_ri4674362597316999326ke_bit @ int @ N @ ( plus_plus @ int @ K @ L ) ) ) ).
% signed_take_bit_add
thf(fact_1282_dbl__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl @ A )
= ( ^ [X3: A] : ( plus_plus @ A @ X3 @ X3 ) ) ) ) ).
% dbl_def
thf(fact_1283_divmod__int__def,axiom,
( ( unique8689654367752047608divmod @ int )
= ( ^ [M: num,N5: num] : ( product_Pair @ int @ int @ ( divide_divide @ int @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N5 ) ) @ ( modulo_modulo @ int @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N5 ) ) ) ) ) ).
% divmod_int_def
thf(fact_1284_divmod__def,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique8689654367752047608divmod @ A )
= ( ^ [M: num,N5: num] : ( product_Pair @ A @ A @ ( divide_divide @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N5 ) ) @ ( modulo_modulo @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N5 ) ) ) ) ) ) ).
% divmod_def
thf(fact_1285_divmod_H__nat__def,axiom,
( ( unique8689654367752047608divmod @ nat )
= ( ^ [M: num,N5: num] : ( product_Pair @ nat @ nat @ ( divide_divide @ nat @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N5 ) ) @ ( modulo_modulo @ nat @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N5 ) ) ) ) ) ).
% divmod'_nat_def
thf(fact_1286_signed__take__bit__int__greater__eq,axiom,
! [K: int,N: nat] :
( ( ord_less @ int @ K @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ N ) ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) ) ) ).
% signed_take_bit_int_greater_eq
thf(fact_1287_divmod__divmod__step,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique8689654367752047608divmod @ A )
= ( ^ [M: num,N5: num] : ( if @ ( product_prod @ A @ A ) @ ( ord_less @ num @ M @ N5 ) @ ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ M ) ) @ ( unique1321980374590559556d_step @ A @ N5 @ ( unique8689654367752047608divmod @ A @ M @ ( bit0 @ N5 ) ) ) ) ) ) ) ).
% divmod_divmod_step
thf(fact_1288_round__def,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A )
= ( ^ [X3: A] : ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X3 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% round_def
thf(fact_1289_signed__take__bit__Suc,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_ri4674362597316999326ke_bit @ A @ ( suc @ N ) @ A3 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4674362597316999326ke_bit @ A @ N @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% signed_take_bit_Suc
thf(fact_1290_of__int__round__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X ) ) @ ( plus_plus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% of_int_round_le
thf(fact_1291_of__int__round__ge,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X ) ) ) ) ).
% of_int_round_ge
thf(fact_1292_of__int__round__gt,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less @ A @ ( minus_minus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X ) ) ) ) ).
% of_int_round_gt
thf(fact_1293_integer__of__int__code,axiom,
( code_integer_of_int
= ( ^ [K4: int] :
( if @ code_integer @ ( ord_less @ int @ K4 @ ( zero_zero @ int ) ) @ ( uminus_uminus @ code_integer @ ( code_integer_of_int @ ( uminus_uminus @ int @ K4 ) ) )
@ ( if @ code_integer
@ ( K4
= ( zero_zero @ int ) )
@ ( zero_zero @ code_integer )
@ ( if @ code_integer
@ ( ( modulo_modulo @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( zero_zero @ int ) )
@ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ ( code_integer_of_int @ ( divide_divide @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) )
@ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ ( code_integer_of_int @ ( divide_divide @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ code_integer ) ) ) ) ) ) ) ).
% integer_of_int_code
thf(fact_1294_divmod__algorithm__code_I6_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M2: num,N: num] :
( ( unique8689654367752047608divmod @ A @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
= ( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q5: A,R3: A] : ( product_Pair @ A @ A @ Q5 @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ R3 ) @ ( one_one @ A ) ) )
@ ( unique8689654367752047608divmod @ A @ M2 @ N ) ) ) ) ).
% divmod_algorithm_code(6)
thf(fact_1295_of__nat__code,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A )
= ( ^ [N5: nat] :
( semiri8178284476397505188at_aux @ A
@ ^ [I3: A] : ( plus_plus @ A @ I3 @ ( one_one @ A ) )
@ N5
@ ( zero_zero @ A ) ) ) ) ) ).
% of_nat_code
thf(fact_1296_signed__take__bit__Suc__minus__bit1,axiom,
! [N: nat,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ ( minus_minus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) @ ( one_one @ int ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% signed_take_bit_Suc_minus_bit1
thf(fact_1297_concat__bit__Suc,axiom,
! [N: nat,K: int,L: int] :
( ( bit_concat_bit @ ( suc @ N ) @ K @ L )
= ( plus_plus @ int @ ( modulo_modulo @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_concat_bit @ N @ ( divide_divide @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ L ) ) ) ) ).
% concat_bit_Suc
thf(fact_1298_even__succ__mod__exp,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( modulo_modulo @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( modulo_modulo @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ) ) ).
% even_succ_mod_exp
thf(fact_1299_even__succ__div__exp,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( divide_divide @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ) ).
% even_succ_div_exp
thf(fact_1300_take__bit__rec,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2584673776208193580ke_bit @ A )
= ( ^ [N5: nat,A7: A] :
( if @ A
@ ( N5
= ( zero_zero @ nat ) )
@ ( zero_zero @ A )
@ ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( minus_minus @ nat @ N5 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A7 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ A @ A7 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% take_bit_rec
thf(fact_1301_nat__dvd__1__iff__1,axiom,
! [M2: nat] :
( ( dvd_dvd @ nat @ M2 @ ( one_one @ nat ) )
= ( M2
= ( one_one @ nat ) ) ) ).
% nat_dvd_1_iff_1
thf(fact_1302_integer__of__int__eq__of__int,axiom,
( code_integer_of_int
= ( ring_1_of_int @ code_integer ) ) ).
% integer_of_int_eq_of_int
thf(fact_1303_dvd__add__triv__right__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ B2 @ A3 ) )
= ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% dvd_add_triv_right_iff
thf(fact_1304_dvd__add__triv__left__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ A3 @ B2 ) )
= ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% dvd_add_triv_left_iff
thf(fact_1305_dvd__add__times__triv__right__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ B2 @ ( times_times @ A @ C2 @ A3 ) ) )
= ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_1306_dvd__add__times__triv__left__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ ( times_times @ A @ C2 @ A3 ) @ B2 ) )
= ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_1307_unit__prod,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% unit_prod
thf(fact_1308_div__add,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( dvd_dvd @ A @ C2 @ A3 )
=> ( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ) ).
% div_add
thf(fact_1309_unit__div__1__div__1,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ ( one_one @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
= A3 ) ) ) ).
% unit_div_1_div_1
thf(fact_1310_unit__div__1__unit,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) @ ( one_one @ A ) ) ) ) ).
% unit_div_1_unit
thf(fact_1311_unit__div,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% unit_div
thf(fact_1312_take__bit__Suc__1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% take_bit_Suc_1
thf(fact_1313_take__bit__numeral__1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [L: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% take_bit_numeral_1
thf(fact_1314_semiring__norm_I9_J,axiom,
! [M2: num,N: num] :
( ( plus_plus @ num @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
= ( bit1 @ ( plus_plus @ num @ M2 @ N ) ) ) ).
% semiring_norm(9)
thf(fact_1315_semiring__norm_I7_J,axiom,
! [M2: num,N: num] :
( ( plus_plus @ num @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
= ( bit1 @ ( plus_plus @ num @ M2 @ N ) ) ) ).
% semiring_norm(7)
thf(fact_1316_semiring__norm_I16_J,axiom,
! [M2: num,N: num] :
( ( times_times @ num @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
= ( bit1 @ ( plus_plus @ num @ ( plus_plus @ num @ M2 @ N ) @ ( bit0 @ ( times_times @ num @ M2 @ N ) ) ) ) ) ).
% semiring_norm(16)
thf(fact_1317_unit__mult__div__div,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( times_times @ A @ B2 @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
= ( divide_divide @ A @ B2 @ A3 ) ) ) ) ).
% unit_mult_div_div
thf(fact_1318_unit__div__mult__self,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( times_times @ A @ ( divide_divide @ A @ B2 @ A3 ) @ A3 )
= B2 ) ) ) ).
% unit_div_mult_self
thf(fact_1319_take__bit__of__1__eq__0__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat] :
( ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( one_one @ A ) )
= ( zero_zero @ A ) )
= ( N
= ( zero_zero @ nat ) ) ) ) ).
% take_bit_of_1_eq_0_iff
thf(fact_1320_semiring__norm_I10_J,axiom,
! [M2: num,N: num] :
( ( plus_plus @ num @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
= ( bit0 @ ( plus_plus @ num @ ( plus_plus @ num @ M2 @ N ) @ one2 ) ) ) ).
% semiring_norm(10)
thf(fact_1321_semiring__norm_I8_J,axiom,
! [M2: num] :
( ( plus_plus @ num @ ( bit1 @ M2 ) @ one2 )
= ( bit0 @ ( plus_plus @ num @ M2 @ one2 ) ) ) ).
% semiring_norm(8)
thf(fact_1322_semiring__norm_I5_J,axiom,
! [M2: num] :
( ( plus_plus @ num @ ( bit0 @ M2 ) @ one2 )
= ( bit1 @ M2 ) ) ).
% semiring_norm(5)
thf(fact_1323_semiring__norm_I4_J,axiom,
! [N: num] :
( ( plus_plus @ num @ one2 @ ( bit1 @ N ) )
= ( bit0 @ ( plus_plus @ num @ N @ one2 ) ) ) ).
% semiring_norm(4)
thf(fact_1324_semiring__norm_I3_J,axiom,
! [N: num] :
( ( plus_plus @ num @ one2 @ ( bit0 @ N ) )
= ( bit1 @ N ) ) ).
% semiring_norm(3)
thf(fact_1325_odd__add,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A,B2: A] :
( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A3 @ B2 ) ) )
= ( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) )
!= ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) ) ) ).
% odd_add
thf(fact_1326_even__add,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A3 @ B2 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
= ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) ) ).
% even_add
thf(fact_1327_dbl__inc__simps_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_inc @ A @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( bit1 @ one2 ) ) ) ) ).
% dbl_inc_simps(3)
thf(fact_1328_even__plus__one__iff,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) )
= ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ).
% even_plus_one_iff
thf(fact_1329_even__diff,axiom,
! [A: $tType] :
( ( ring_parity @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ A @ A3 @ B2 ) )
= ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ).
% even_diff
thf(fact_1330_neg__one__even__power,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N )
= ( one_one @ A ) ) ) ) ).
% neg_one_even_power
thf(fact_1331_neg__one__odd__power,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% neg_one_odd_power
thf(fact_1332_even__diff__nat,axiom,
! [M2: nat,N: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M2 @ N ) )
= ( ( ord_less @ nat @ M2 @ N )
| ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M2 @ N ) ) ) ) ).
% even_diff_nat
thf(fact_1333_divmod__algorithm__code_I4_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N: num] :
( ( unique8689654367752047608divmod @ A @ one2 @ ( bit1 @ N ) )
= ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ one2 ) ) ) ) ).
% divmod_algorithm_code(4)
thf(fact_1334_mod__Suc__eq__mod__add3,axiom,
! [M2: nat,N: nat] :
( ( modulo_modulo @ nat @ M2 @ ( suc @ ( suc @ ( suc @ N ) ) ) )
= ( modulo_modulo @ nat @ M2 @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ N ) ) ) ).
% mod_Suc_eq_mod_add3
thf(fact_1335_Suc__mod__eq__add3__mod__numeral,axiom,
! [M2: nat,V: num] :
( ( modulo_modulo @ nat @ ( suc @ ( suc @ ( suc @ M2 ) ) ) @ ( numeral_numeral @ nat @ V ) )
= ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M2 ) @ ( numeral_numeral @ nat @ V ) ) ) ).
% Suc_mod_eq_add3_mod_numeral
thf(fact_1336_div__Suc__eq__div__add3,axiom,
! [M2: nat,N: nat] :
( ( divide_divide @ nat @ M2 @ ( suc @ ( suc @ ( suc @ N ) ) ) )
= ( divide_divide @ nat @ M2 @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ N ) ) ) ).
% div_Suc_eq_div_add3
thf(fact_1337_Suc__div__eq__add3__div__numeral,axiom,
! [M2: nat,V: num] :
( ( divide_divide @ nat @ ( suc @ ( suc @ ( suc @ M2 ) ) ) @ ( numeral_numeral @ nat @ V ) )
= ( divide_divide @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M2 ) @ ( numeral_numeral @ nat @ V ) ) ) ).
% Suc_div_eq_add3_div_numeral
thf(fact_1338_dbl__dec__simps_I4_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_dec @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit1 @ one2 ) ) ) ) ) ).
% dbl_dec_simps(4)
thf(fact_1339_even__succ__div__2,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% even_succ_div_2
thf(fact_1340_odd__succ__div__two,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ) ).
% odd_succ_div_two
thf(fact_1341_even__succ__div__two,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% even_succ_div_two
thf(fact_1342_odd__two__times__div__two__nat,axiom,
! [N: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ).
% odd_two_times_div_two_nat
thf(fact_1343_odd__two__times__div__two__succ,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ A ) )
= A3 ) ) ) ).
% odd_two_times_div_two_succ
thf(fact_1344_semiring__parity__class_Oeven__mask__iff,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ A ) ) )
= ( N
= ( zero_zero @ nat ) ) ) ) ).
% semiring_parity_class.even_mask_iff
thf(fact_1345_zmod__numeral__Bit1,axiom,
! [V: num,W: num] :
( ( modulo_modulo @ int @ ( numeral_numeral @ int @ ( bit1 @ V ) ) @ ( numeral_numeral @ int @ ( bit0 @ W ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ ( numeral_numeral @ int @ V ) @ ( numeral_numeral @ int @ W ) ) ) @ ( one_one @ int ) ) ) ).
% zmod_numeral_Bit1
thf(fact_1346_divmod__algorithm__code_I8_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M2: num,N: num] :
( ( ( ord_less @ num @ M2 @ N )
=> ( ( unique8689654367752047608divmod @ A @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
= ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ ( bit1 @ M2 ) ) ) ) )
& ( ~ ( ord_less @ num @ M2 @ N )
=> ( ( unique8689654367752047608divmod @ A @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
= ( unique1321980374590559556d_step @ A @ ( bit1 @ N ) @ ( unique8689654367752047608divmod @ A @ ( bit1 @ M2 ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ) ).
% divmod_algorithm_code(8)
thf(fact_1347_divmod__algorithm__code_I7_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M2: num,N: num] :
( ( ( ord_less_eq @ num @ M2 @ N )
=> ( ( unique8689654367752047608divmod @ A @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
= ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ ( bit0 @ M2 ) ) ) ) )
& ( ~ ( ord_less_eq @ num @ M2 @ N )
=> ( ( unique8689654367752047608divmod @ A @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
= ( unique1321980374590559556d_step @ A @ ( bit1 @ N ) @ ( unique8689654367752047608divmod @ A @ ( bit0 @ M2 ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ) ).
% divmod_algorithm_code(7)
thf(fact_1348_signed__take__bit__Suc__bit1,axiom,
! [N: nat,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N ) @ ( numeral_numeral @ int @ ( bit1 @ K ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ ( numeral_numeral @ int @ K ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% signed_take_bit_Suc_bit1
thf(fact_1349_uminus__integer__code_I1_J,axiom,
( ( uminus_uminus @ code_integer @ ( zero_zero @ code_integer ) )
= ( zero_zero @ code_integer ) ) ).
% uminus_integer_code(1)
thf(fact_1350_take__bit__integer_Oabs__eq,axiom,
! [Xa: nat,X: int] :
( ( bit_se2584673776208193580ke_bit @ code_integer @ Xa @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( bit_se2584673776208193580ke_bit @ int @ Xa @ X ) ) ) ).
% take_bit_integer.abs_eq
thf(fact_1351_abs__integer__code,axiom,
( ( abs_abs @ code_integer )
= ( ^ [K4: code_integer] : ( if @ code_integer @ ( ord_less @ code_integer @ K4 @ ( zero_zero @ code_integer ) ) @ ( uminus_uminus @ code_integer @ K4 ) @ K4 ) ) ) ).
% abs_integer_code
thf(fact_1352_less__integer__code_I1_J,axiom,
~ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) ) ).
% less_integer_code(1)
thf(fact_1353_unset__bit__integer_Oabs__eq,axiom,
! [Xa: nat,X: int] :
( ( bit_se2638667681897837118et_bit @ code_integer @ Xa @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( bit_se2638667681897837118et_bit @ int @ Xa @ X ) ) ) ).
% unset_bit_integer.abs_eq
thf(fact_1354_flip__bit__integer_Oabs__eq,axiom,
! [Xa: nat,X: int] :
( ( bit_se8732182000553998342ip_bit @ code_integer @ Xa @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( bit_se8732182000553998342ip_bit @ int @ Xa @ X ) ) ) ).
% flip_bit_integer.abs_eq
thf(fact_1355_set__bit__integer_Oabs__eq,axiom,
! [Xa: nat,X: int] :
( ( bit_se5668285175392031749et_bit @ code_integer @ Xa @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( bit_se5668285175392031749et_bit @ int @ Xa @ X ) ) ) ).
% set_bit_integer.abs_eq
thf(fact_1356_take__bit__add,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,A3: A,B2: A] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( plus_plus @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) @ ( bit_se2584673776208193580ke_bit @ A @ N @ B2 ) ) )
= ( bit_se2584673776208193580ke_bit @ A @ N @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ).
% take_bit_add
thf(fact_1357_zero__integer__def,axiom,
( ( zero_zero @ code_integer )
= ( code_integer_of_int @ ( zero_zero @ int ) ) ) ).
% zero_integer_def
thf(fact_1358_less__integer_Oabs__eq,axiom,
! [Xa: int,X: int] :
( ( ord_less @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
= ( ord_less @ int @ Xa @ X ) ) ).
% less_integer.abs_eq
thf(fact_1359_uminus__integer_Oabs__eq,axiom,
! [X: int] :
( ( uminus_uminus @ code_integer @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( uminus_uminus @ int @ X ) ) ) ).
% uminus_integer.abs_eq
thf(fact_1360_abs__integer_Oabs__eq,axiom,
! [X: int] :
( ( abs_abs @ code_integer @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( abs_abs @ int @ X ) ) ) ).
% abs_integer.abs_eq
thf(fact_1361_dvd__add__right__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( dvd_dvd @ A @ A3 @ C2 ) ) ) ) ).
% dvd_add_right_iff
thf(fact_1362_dvd__add__left__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ C2 )
=> ( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( dvd_dvd @ A @ A3 @ B2 ) ) ) ) ).
% dvd_add_left_iff
thf(fact_1363_dvd__add,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( dvd_dvd @ A @ A3 @ C2 )
=> ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ) ).
% dvd_add
thf(fact_1364_divide__integer_Oabs__eq,axiom,
! [Xa: int,X: int] :
( ( divide_divide @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( divide_divide @ int @ Xa @ X ) ) ) ).
% divide_integer.abs_eq
thf(fact_1365_dvd__unit__imp__unit,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ A3 @ ( one_one @ A ) ) ) ) ) ).
% dvd_unit_imp_unit
thf(fact_1366_unit__imp__dvd,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ B2 @ A3 ) ) ) ).
% unit_imp_dvd
thf(fact_1367_one__dvd,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A] : ( dvd_dvd @ A @ ( one_one @ A ) @ A3 ) ) ).
% one_dvd
thf(fact_1368_modulo__integer_Oabs__eq,axiom,
! [Xa: int,X: int] :
( ( modulo_modulo @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( modulo_modulo @ int @ Xa @ X ) ) ) ).
% modulo_integer.abs_eq
thf(fact_1369_sgn__integer_Oabs__eq,axiom,
! [X: int] :
( ( sgn_sgn @ code_integer @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( sgn_sgn @ int @ X ) ) ) ).
% sgn_integer.abs_eq
thf(fact_1370_concat__bit__assoc,axiom,
! [N: nat,K: int,M2: nat,L: int,R: int] :
( ( bit_concat_bit @ N @ K @ ( bit_concat_bit @ M2 @ L @ R ) )
= ( bit_concat_bit @ ( plus_plus @ nat @ M2 @ N ) @ ( bit_concat_bit @ N @ K @ L ) @ R ) ) ).
% concat_bit_assoc
thf(fact_1371_not__is__unit__0,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ~ ( dvd_dvd @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% not_is_unit_0
thf(fact_1372_pinf_I9_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D3: B,S: B] :
? [Z3: B] :
! [X7: B] :
( ( ord_less @ B @ Z3 @ X7 )
=> ( ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X7 @ S ) )
= ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X7 @ S ) ) ) ) ) ).
% pinf(9)
thf(fact_1373_pinf_I10_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D3: B,S: B] :
? [Z3: B] :
! [X7: B] :
( ( ord_less @ B @ Z3 @ X7 )
=> ( ( ~ ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X7 @ S ) ) )
= ( ~ ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X7 @ S ) ) ) ) ) ) ).
% pinf(10)
thf(fact_1374_minf_I9_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D3: B,S: B] :
? [Z3: B] :
! [X7: B] :
( ( ord_less @ B @ X7 @ Z3 )
=> ( ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X7 @ S ) )
= ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X7 @ S ) ) ) ) ) ).
% minf(9)
thf(fact_1375_minf_I10_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D3: B,S: B] :
? [Z3: B] :
! [X7: B] :
( ( ord_less @ B @ X7 @ Z3 )
=> ( ( ~ ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X7 @ S ) ) )
= ( ~ ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X7 @ S ) ) ) ) ) ) ).
% minf(10)
thf(fact_1376_unit__mult__right__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( ( times_times @ A @ B2 @ A3 )
= ( times_times @ A @ C2 @ A3 ) )
= ( B2 = C2 ) ) ) ) ).
% unit_mult_right_cancel
thf(fact_1377_unit__mult__left__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( ( times_times @ A @ A3 @ B2 )
= ( times_times @ A @ A3 @ C2 ) )
= ( B2 = C2 ) ) ) ) ).
% unit_mult_left_cancel
thf(fact_1378_mult__unit__dvd__iff_H,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
= ( dvd_dvd @ A @ B2 @ C2 ) ) ) ) ).
% mult_unit_dvd_iff'
thf(fact_1379_dvd__mult__unit__iff_H,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) )
= ( dvd_dvd @ A @ A3 @ C2 ) ) ) ) ).
% dvd_mult_unit_iff'
thf(fact_1380_mult__unit__dvd__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
= ( dvd_dvd @ A @ A3 @ C2 ) ) ) ) ).
% mult_unit_dvd_iff
thf(fact_1381_dvd__mult__unit__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ A3 @ ( times_times @ A @ C2 @ B2 ) )
= ( dvd_dvd @ A @ A3 @ C2 ) ) ) ) ).
% dvd_mult_unit_iff
thf(fact_1382_is__unit__mult__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ ( one_one @ A ) )
= ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
& ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) ) ) ) ).
% is_unit_mult_iff
thf(fact_1383_div__plus__div__distrib__dvd__left,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( dvd_dvd @ A @ C2 @ A3 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% div_plus_div_distrib_dvd_left
thf(fact_1384_div__plus__div__distrib__dvd__right,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% div_plus_div_distrib_dvd_right
thf(fact_1385_unit__div__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ B2 @ A3 )
= ( divide_divide @ A @ C2 @ A3 ) )
= ( B2 = C2 ) ) ) ) ).
% unit_div_cancel
thf(fact_1386_div__unit__dvd__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 )
= ( dvd_dvd @ A @ A3 @ C2 ) ) ) ) ).
% div_unit_dvd_iff
thf(fact_1387_dvd__div__unit__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ A3 @ ( divide_divide @ A @ C2 @ B2 ) )
= ( dvd_dvd @ A @ A3 @ C2 ) ) ) ) ).
% dvd_div_unit_iff
thf(fact_1388_xor__num_Ocases,axiom,
! [X: product_prod @ num @ num] :
( ( X
!= ( product_Pair @ num @ num @ one2 @ one2 ) )
=> ( ! [N3: num] :
( X
!= ( product_Pair @ num @ num @ one2 @ ( bit0 @ N3 ) ) )
=> ( ! [N3: num] :
( X
!= ( product_Pair @ num @ num @ one2 @ ( bit1 @ N3 ) ) )
=> ( ! [M5: num] :
( X
!= ( product_Pair @ num @ num @ ( bit0 @ M5 ) @ one2 ) )
=> ( ! [M5: num,N3: num] :
( X
!= ( product_Pair @ num @ num @ ( bit0 @ M5 ) @ ( bit0 @ N3 ) ) )
=> ( ! [M5: num,N3: num] :
( X
!= ( product_Pair @ num @ num @ ( bit0 @ M5 ) @ ( bit1 @ N3 ) ) )
=> ( ! [M5: num] :
( X
!= ( product_Pair @ num @ num @ ( bit1 @ M5 ) @ one2 ) )
=> ( ! [M5: num,N3: num] :
( X
!= ( product_Pair @ num @ num @ ( bit1 @ M5 ) @ ( bit0 @ N3 ) ) )
=> ~ ! [M5: num,N3: num] :
( X
!= ( product_Pair @ num @ num @ ( bit1 @ M5 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ).
% xor_num.cases
thf(fact_1389_plus__integer_Oabs__eq,axiom,
! [Xa: int,X: int] :
( ( plus_plus @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( plus_plus @ int @ Xa @ X ) ) ) ).
% plus_integer.abs_eq
thf(fact_1390_bezout__lemma__nat,axiom,
! [D3: nat,A3: nat,B2: nat,X: nat,Y: nat] :
( ( dvd_dvd @ nat @ D3 @ A3 )
=> ( ( dvd_dvd @ nat @ D3 @ B2 )
=> ( ( ( ( times_times @ nat @ A3 @ X )
= ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y ) @ D3 ) )
| ( ( times_times @ nat @ B2 @ X )
= ( plus_plus @ nat @ ( times_times @ nat @ A3 @ Y ) @ D3 ) ) )
=> ? [X2: nat,Y4: nat] :
( ( dvd_dvd @ nat @ D3 @ A3 )
& ( dvd_dvd @ nat @ D3 @ ( plus_plus @ nat @ A3 @ B2 ) )
& ( ( ( times_times @ nat @ A3 @ X2 )
= ( plus_plus @ nat @ ( times_times @ nat @ ( plus_plus @ nat @ A3 @ B2 ) @ Y4 ) @ D3 ) )
| ( ( times_times @ nat @ ( plus_plus @ nat @ A3 @ B2 ) @ X2 )
= ( plus_plus @ nat @ ( times_times @ nat @ A3 @ Y4 ) @ D3 ) ) ) ) ) ) ) ).
% bezout_lemma_nat
thf(fact_1391_bezout__add__nat,axiom,
! [A3: nat,B2: nat] :
? [D2: nat,X2: nat,Y4: nat] :
( ( dvd_dvd @ nat @ D2 @ A3 )
& ( dvd_dvd @ nat @ D2 @ B2 )
& ( ( ( times_times @ nat @ A3 @ X2 )
= ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y4 ) @ D2 ) )
| ( ( times_times @ nat @ B2 @ X2 )
= ( plus_plus @ nat @ ( times_times @ nat @ A3 @ Y4 ) @ D2 ) ) ) ) ).
% bezout_add_nat
thf(fact_1392_one__integer__def,axiom,
( ( one_one @ code_integer )
= ( code_integer_of_int @ ( one_one @ int ) ) ) ).
% one_integer_def
thf(fact_1393_less__eq__integer_Oabs__eq,axiom,
! [Xa: int,X: int] :
( ( ord_less_eq @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
= ( ord_less_eq @ int @ Xa @ X ) ) ).
% less_eq_integer.abs_eq
thf(fact_1394_times__integer_Oabs__eq,axiom,
! [Xa: int,X: int] :
( ( times_times @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( times_times @ int @ Xa @ X ) ) ) ).
% times_integer.abs_eq
thf(fact_1395_zdvd__period,axiom,
! [A3: int,D3: int,X: int,T2: int,C2: int] :
( ( dvd_dvd @ int @ A3 @ D3 )
=> ( ( dvd_dvd @ int @ A3 @ ( plus_plus @ int @ X @ T2 ) )
= ( dvd_dvd @ int @ A3 @ ( plus_plus @ int @ ( plus_plus @ int @ X @ ( times_times @ int @ C2 @ D3 ) ) @ T2 ) ) ) ) ).
% zdvd_period
thf(fact_1396_zdvd__reduce,axiom,
! [K: int,N: int,M2: int] :
( ( dvd_dvd @ int @ K @ ( plus_plus @ int @ N @ ( times_times @ int @ K @ M2 ) ) )
= ( dvd_dvd @ int @ K @ N ) ) ).
% zdvd_reduce
thf(fact_1397_minus__integer_Oabs__eq,axiom,
! [Xa: int,X: int] :
( ( minus_minus @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( minus_minus @ int @ Xa @ X ) ) ) ).
% minus_integer.abs_eq
thf(fact_1398_unity__coeff__ex,axiom,
! [A: $tType] :
( ( ( dvd @ A )
& ( semiring_0 @ A ) )
=> ! [P: A > $o,L: A] :
( ( ? [X3: A] : ( P @ ( times_times @ A @ L @ X3 ) ) )
= ( ? [X3: A] :
( ( dvd_dvd @ A @ L @ ( plus_plus @ A @ X3 @ ( zero_zero @ A ) ) )
& ( P @ X3 ) ) ) ) ) ).
% unity_coeff_ex
thf(fact_1399_unit__dvdE,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ~ ( ( A3
!= ( zero_zero @ A ) )
=> ! [C4: A] :
( B2
!= ( times_times @ A @ A3 @ C4 ) ) ) ) ) ).
% unit_dvdE
thf(fact_1400_inf__period_I4_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [D3: A,D4: A,T2: A] :
( ( dvd_dvd @ A @ D3 @ D4 )
=> ! [X7: A,K8: A] :
( ( ~ ( dvd_dvd @ A @ D3 @ ( plus_plus @ A @ X7 @ T2 ) ) )
= ( ~ ( dvd_dvd @ A @ D3 @ ( plus_plus @ A @ ( minus_minus @ A @ X7 @ ( times_times @ A @ K8 @ D4 ) ) @ T2 ) ) ) ) ) ) ).
% inf_period(4)
thf(fact_1401_inf__period_I3_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [D3: A,D4: A,T2: A] :
( ( dvd_dvd @ A @ D3 @ D4 )
=> ! [X7: A,K8: A] :
( ( dvd_dvd @ A @ D3 @ ( plus_plus @ A @ X7 @ T2 ) )
= ( dvd_dvd @ A @ D3 @ ( plus_plus @ A @ ( minus_minus @ A @ X7 @ ( times_times @ A @ K8 @ D4 ) ) @ T2 ) ) ) ) ) ).
% inf_period(3)
thf(fact_1402_unit__div__eq__0__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ) ).
% unit_div_eq_0_iff
thf(fact_1403_numeral__Bit1,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N: num] :
( ( numeral_numeral @ A @ ( bit1 @ N ) )
= ( plus_plus @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ N ) ) @ ( one_one @ A ) ) ) ) ).
% numeral_Bit1
thf(fact_1404_unit__eq__div1,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ A3 @ B2 )
= C2 )
= ( A3
= ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% unit_eq_div1
thf(fact_1405_unit__eq__div2,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( A3
= ( divide_divide @ A @ C2 @ B2 ) )
= ( ( times_times @ A @ A3 @ B2 )
= C2 ) ) ) ) ).
% unit_eq_div2
thf(fact_1406_div__mult__unit2,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ A3 )
=> ( ( divide_divide @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 ) ) ) ) ) ).
% div_mult_unit2
thf(fact_1407_unit__div__commute,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 )
= ( divide_divide @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) ) ) ) ).
% unit_div_commute
thf(fact_1408_unit__div__mult__swap,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
=> ( ( times_times @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 ) ) ) ) ).
% unit_div_mult_swap
thf(fact_1409_is__unit__div__mult2__eq,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 ) ) ) ) ) ).
% is_unit_div_mult2_eq
thf(fact_1410_unit__imp__mod__eq__0,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B2: A,A3: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( modulo_modulo @ A @ A3 @ B2 )
= ( zero_zero @ A ) ) ) ) ).
% unit_imp_mod_eq_0
thf(fact_1411_is__unit__power__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,N: nat] :
( ( dvd_dvd @ A @ ( power_power @ A @ A3 @ N ) @ ( one_one @ A ) )
= ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% is_unit_power_iff
thf(fact_1412_bezout__add__strong__nat,axiom,
! [A3: nat,B2: nat] :
( ( A3
!= ( zero_zero @ nat ) )
=> ? [D2: nat,X2: nat,Y4: nat] :
( ( dvd_dvd @ nat @ D2 @ A3 )
& ( dvd_dvd @ nat @ D2 @ B2 )
& ( ( times_times @ nat @ A3 @ X2 )
= ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y4 ) @ D2 ) ) ) ) ).
% bezout_add_strong_nat
thf(fact_1413_take__bit__Suc__bit1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,K: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ ( numeral_numeral @ A @ ( bit1 @ K ) ) )
= ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ).
% take_bit_Suc_bit1
thf(fact_1414_numeral__code_I3_J,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N: num] :
( ( numeral_numeral @ A @ ( bit1 @ N ) )
= ( plus_plus @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ N ) ) @ ( one_one @ A ) ) ) ) ).
% numeral_code(3)
thf(fact_1415_stable__imp__take__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A,N: nat] :
( ( ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A3 )
=> ( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ A3 )
= ( zero_zero @ A ) ) )
& ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ A3 )
= ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ A ) ) ) ) ) ) ) ).
% stable_imp_take_bit_eq
thf(fact_1416_is__unitE,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ~ ( ( A3
!= ( zero_zero @ A ) )
=> ! [B3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ ( one_one @ A ) @ A3 )
= B3 )
=> ( ( ( divide_divide @ A @ ( one_one @ A ) @ B3 )
= A3 )
=> ( ( ( times_times @ A @ A3 @ B3 )
= ( one_one @ A ) )
=> ( ( divide_divide @ A @ C2 @ A3 )
!= ( times_times @ A @ C2 @ B3 ) ) ) ) ) ) ) ) ) ) ).
% is_unitE
thf(fact_1417_is__unit__div__mult__cancel__left,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ A3 @ ( times_times @ A @ A3 @ B2 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ B2 ) ) ) ) ) ).
% is_unit_div_mult_cancel_left
thf(fact_1418_is__unit__div__mult__cancel__right,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ A3 @ ( times_times @ A @ B2 @ A3 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ B2 ) ) ) ) ) ).
% is_unit_div_mult_cancel_right
thf(fact_1419_odd__even__add,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A,B2: A] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 )
=> ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% odd_even_add
thf(fact_1420_odd__one,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( one_one @ A ) ) ) ).
% odd_one
thf(fact_1421_dvd__power__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [X: A,M2: nat,N: nat] :
( ( X
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ ( power_power @ A @ X @ M2 ) @ ( power_power @ A @ X @ N ) )
= ( ( dvd_dvd @ A @ X @ ( one_one @ A ) )
| ( ord_less_eq @ nat @ M2 @ N ) ) ) ) ) ).
% dvd_power_iff
thf(fact_1422_dvd__power,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [N: nat,X: A] :
( ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
| ( X
= ( one_one @ A ) ) )
=> ( dvd_dvd @ A @ X @ ( power_power @ A @ X @ N ) ) ) ) ).
% dvd_power
thf(fact_1423_Suc3__eq__add__3,axiom,
! [N: nat] :
( ( suc @ ( suc @ ( suc @ N ) ) )
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ N ) ) ).
% Suc3_eq_add_3
thf(fact_1424_dvd__mult__cancel2,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ( dvd_dvd @ nat @ ( times_times @ nat @ N @ M2 ) @ M2 )
= ( N
= ( one_one @ nat ) ) ) ) ).
% dvd_mult_cancel2
thf(fact_1425_dvd__mult__cancel1,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ( dvd_dvd @ nat @ ( times_times @ nat @ M2 @ N ) @ M2 )
= ( N
= ( one_one @ nat ) ) ) ) ).
% dvd_mult_cancel1
thf(fact_1426_dvd__minus__add,axiom,
! [Q3: nat,N: nat,R: nat,M2: nat] :
( ( ord_less_eq @ nat @ Q3 @ N )
=> ( ( ord_less_eq @ nat @ Q3 @ ( times_times @ nat @ R @ M2 ) )
=> ( ( dvd_dvd @ nat @ M2 @ ( minus_minus @ nat @ N @ Q3 ) )
= ( dvd_dvd @ nat @ M2 @ ( plus_plus @ nat @ N @ ( minus_minus @ nat @ ( times_times @ nat @ R @ M2 ) @ Q3 ) ) ) ) ) ) ).
% dvd_minus_add
thf(fact_1427_power__dvd__imp__le,axiom,
! [I: nat,M2: nat,N: nat] :
( ( dvd_dvd @ nat @ ( power_power @ nat @ I @ M2 ) @ ( power_power @ nat @ I @ N ) )
=> ( ( ord_less @ nat @ ( one_one @ nat ) @ I )
=> ( ord_less_eq @ nat @ M2 @ N ) ) ) ).
% power_dvd_imp_le
thf(fact_1428_aset_I10_J,axiom,
! [D3: int,D4: int,A4: set @ int,T2: int] :
( ( dvd_dvd @ int @ D3 @ D4 )
=> ! [X7: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A4 )
=> ( X7
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ~ ( dvd_dvd @ int @ D3 @ ( plus_plus @ int @ X7 @ T2 ) )
=> ~ ( dvd_dvd @ int @ D3 @ ( plus_plus @ int @ ( plus_plus @ int @ X7 @ D4 ) @ T2 ) ) ) ) ) ).
% aset(10)
thf(fact_1429_aset_I9_J,axiom,
! [D3: int,D4: int,A4: set @ int,T2: int] :
( ( dvd_dvd @ int @ D3 @ D4 )
=> ! [X7: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A4 )
=> ( X7
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( dvd_dvd @ int @ D3 @ ( plus_plus @ int @ X7 @ T2 ) )
=> ( dvd_dvd @ int @ D3 @ ( plus_plus @ int @ ( plus_plus @ int @ X7 @ D4 ) @ T2 ) ) ) ) ) ).
% aset(9)
thf(fact_1430_bset_I10_J,axiom,
! [D3: int,D4: int,B6: set @ int,T2: int] :
( ( dvd_dvd @ int @ D3 @ D4 )
=> ! [X7: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B6 )
=> ( X7
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ~ ( dvd_dvd @ int @ D3 @ ( plus_plus @ int @ X7 @ T2 ) )
=> ~ ( dvd_dvd @ int @ D3 @ ( plus_plus @ int @ ( minus_minus @ int @ X7 @ D4 ) @ T2 ) ) ) ) ) ).
% bset(10)
thf(fact_1431_bset_I9_J,axiom,
! [D3: int,D4: int,B6: set @ int,T2: int] :
( ( dvd_dvd @ int @ D3 @ D4 )
=> ! [X7: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B6 )
=> ( X7
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( dvd_dvd @ int @ D3 @ ( plus_plus @ int @ X7 @ T2 ) )
=> ( dvd_dvd @ int @ D3 @ ( plus_plus @ int @ ( minus_minus @ int @ X7 @ D4 ) @ T2 ) ) ) ) ) ).
% bset(9)
thf(fact_1432_num_Osize__gen_I3_J,axiom,
! [X32: num] :
( ( size_num @ ( bit1 @ X32 ) )
= ( plus_plus @ nat @ ( size_num @ X32 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% num.size_gen(3)
thf(fact_1433_num_Osize_I6_J,axiom,
! [X32: num] :
( ( size_size @ num @ ( bit1 @ X32 ) )
= ( plus_plus @ nat @ ( size_size @ num @ X32 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% num.size(6)
thf(fact_1434_odd__iff__mod__2__eq__one,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) )
= ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ) ).
% odd_iff_mod_2_eq_one
thf(fact_1435_Suc__mod__eq__add3__mod,axiom,
! [M2: nat,N: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( suc @ ( suc @ M2 ) ) ) @ N )
= ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M2 ) @ N ) ) ).
% Suc_mod_eq_add3_mod
thf(fact_1436_Suc__div__eq__add3__div,axiom,
! [M2: nat,N: nat] :
( ( divide_divide @ nat @ ( suc @ ( suc @ ( suc @ M2 ) ) ) @ N )
= ( divide_divide @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M2 ) @ N ) ) ).
% Suc_div_eq_add3_div
thf(fact_1437_even__diff__iff,axiom,
! [K: int,L: int] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ int @ K @ L ) )
= ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ K @ L ) ) ) ).
% even_diff_iff
thf(fact_1438_even__abs__add__iff,axiom,
! [K: int,L: int] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ ( abs_abs @ int @ K ) @ L ) )
= ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ K @ L ) ) ) ).
% even_abs_add_iff
thf(fact_1439_even__add__abs__iff,axiom,
! [K: int,L: int] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ K @ ( abs_abs @ int @ L ) ) )
= ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ K @ L ) ) ) ).
% even_add_abs_iff
thf(fact_1440_signed__take__bit__eq__take__bit__shift,axiom,
( ( bit_ri4674362597316999326ke_bit @ int )
= ( ^ [N5: nat,K4: int] : ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N5 ) @ ( plus_plus @ int @ K4 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N5 ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N5 ) ) ) ) ).
% signed_take_bit_eq_take_bit_shift
thf(fact_1441_take__bit__incr__eq,axiom,
! [N: nat,K: int] :
( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
!= ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ int ) ) )
=> ( ( bit_se2584673776208193580ke_bit @ int @ N @ ( plus_plus @ int @ K @ ( one_one @ int ) ) )
= ( plus_plus @ int @ ( one_one @ int ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ) ) ).
% take_bit_incr_eq
thf(fact_1442_oddE,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ~ ! [B3: A] :
( A3
!= ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) @ ( one_one @ A ) ) ) ) ) ).
% oddE
thf(fact_1443_parity__cases,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
!= ( zero_zero @ A ) ) )
=> ~ ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
!= ( one_one @ A ) ) ) ) ) ).
% parity_cases
thf(fact_1444_mod2__eq__if,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( zero_zero @ A ) ) )
& ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ) ) ).
% mod2_eq_if
thf(fact_1445_minus__one__power__iff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat] :
( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N )
= ( one_one @ A ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% minus_one_power_iff
thf(fact_1446_take__bit__int__greater__eq,axiom,
! [K: int,N: nat] :
( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ) ).
% take_bit_int_greater_eq
thf(fact_1447_take__bit__Suc__minus__1__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( suc @ N ) ) @ ( one_one @ A ) ) ) ) ).
% take_bit_Suc_minus_1_eq
thf(fact_1448_take__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ A3 )
= ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% take_bit_Suc
thf(fact_1449_take__bit__numeral__minus__1__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [K: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ K ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ K ) ) @ ( one_one @ A ) ) ) ) ).
% take_bit_numeral_minus_1_eq
thf(fact_1450_even__mask__div__iff_H,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [M2: nat,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) @ ( one_one @ A ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) )
= ( ord_less_eq @ nat @ M2 @ N ) ) ) ).
% even_mask_div_iff'
thf(fact_1451_even__mask__div__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M2: nat,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) @ ( one_one @ A ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) )
= ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
= ( zero_zero @ A ) )
| ( ord_less_eq @ nat @ M2 @ N ) ) ) ) ).
% even_mask_div_iff
thf(fact_1452_Code__Numeral_Onegative__def,axiom,
( code_negative
= ( comp @ code_integer @ code_integer @ num @ ( uminus_uminus @ code_integer ) @ ( numeral_numeral @ code_integer ) ) ) ).
% Code_Numeral.negative_def
thf(fact_1453_take__bit__numeral__minus__numeral__int,axiom,
! [M2: num,N: num] :
( ( bit_se2584673776208193580ke_bit @ int @ ( numeral_numeral @ nat @ M2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( case_option @ int @ num @ ( zero_zero @ int )
@ ^ [Q5: num] : ( bit_se2584673776208193580ke_bit @ int @ ( numeral_numeral @ nat @ M2 ) @ ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ M2 ) ) @ ( numeral_numeral @ int @ Q5 ) ) )
@ ( bit_take_bit_num @ ( numeral_numeral @ nat @ M2 ) @ N ) ) ) ).
% take_bit_numeral_minus_numeral_int
thf(fact_1454_signed__take__bit__numeral__minus__bit1,axiom,
! [L: num,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( pred_numeral @ L ) @ ( minus_minus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) @ ( one_one @ int ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% signed_take_bit_numeral_minus_bit1
thf(fact_1455_execute__change,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [F: A > A,R: ref @ A,H3: heap_ext @ product_unit] :
( ( heap_Time_execute @ A @ ( ref_change @ A @ F @ R ) @ H3 )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ ( ref_get @ A @ H3 @ R ) ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ ( ref_set @ A @ R @ ( F @ ( ref_get @ A @ H3 @ R ) ) @ H3 ) @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ) ) ) ) ).
% execute_change
thf(fact_1456_signed__take__bit__numeral__bit1,axiom,
! [L: num,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ int @ ( bit1 @ K ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( pred_numeral @ L ) @ ( numeral_numeral @ int @ K ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% signed_take_bit_numeral_bit1
thf(fact_1457_take__bit__Suc__minus__bit1,axiom,
! [N: nat,K: num] :
( ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K ) ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% take_bit_Suc_minus_bit1
thf(fact_1458_divmod__BitM__2__eq,axiom,
! [M2: num] :
( ( unique8689654367752047608divmod @ int @ ( bitM @ M2 ) @ ( bit0 @ one2 ) )
= ( product_Pair @ int @ int @ ( minus_minus @ int @ ( numeral_numeral @ int @ M2 ) @ ( one_one @ int ) ) @ ( one_one @ int ) ) ) ).
% divmod_BitM_2_eq
thf(fact_1459_flip__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se8732182000553998342ip_bit @ A @ ( zero_zero @ nat ) @ A3 )
= ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% flip_bit_0
thf(fact_1460_of__bool__eq_I2_J,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_neq_one_of_bool @ A @ $true )
= ( one_one @ A ) ) ) ).
% of_bool_eq(2)
thf(fact_1461_of__bool__eq__1__iff,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P: $o] :
( ( ( zero_neq_one_of_bool @ A @ P )
= ( one_one @ A ) )
= P ) ) ).
% of_bool_eq_1_iff
thf(fact_1462_of__bool__less__one__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [P: $o] :
( ( ord_less @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( one_one @ A ) )
= ~ P ) ) ).
% of_bool_less_one_iff
thf(fact_1463_of__bool__not__iff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [P: $o] :
( ( zero_neq_one_of_bool @ A @ ~ P )
= ( minus_minus @ A @ ( one_one @ A ) @ ( zero_neq_one_of_bool @ A @ P ) ) ) ) ).
% of_bool_not_iff
thf(fact_1464_take__bit__num__simps_I3_J,axiom,
! [N: nat,M2: num] :
( ( bit_take_bit_num @ ( suc @ N ) @ ( bit0 @ M2 ) )
= ( case_option @ ( option @ num ) @ num @ ( none @ num )
@ ^ [Q5: num] : ( some @ num @ ( bit0 @ Q5 ) )
@ ( bit_take_bit_num @ N @ M2 ) ) ) ).
% take_bit_num_simps(3)
thf(fact_1465_take__bit__num__simps_I4_J,axiom,
! [N: nat,M2: num] :
( ( bit_take_bit_num @ ( suc @ N ) @ ( bit1 @ M2 ) )
= ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_take_bit_num @ N @ M2 ) ) ) ) ).
% take_bit_num_simps(4)
thf(fact_1466_take__bit__num__simps_I6_J,axiom,
! [R: num,M2: num] :
( ( bit_take_bit_num @ ( numeral_numeral @ nat @ R ) @ ( bit0 @ M2 ) )
= ( case_option @ ( option @ num ) @ num @ ( none @ num )
@ ^ [Q5: num] : ( some @ num @ ( bit0 @ Q5 ) )
@ ( bit_take_bit_num @ ( pred_numeral @ R ) @ M2 ) ) ) ).
% take_bit_num_simps(6)
thf(fact_1467_take__bit__of__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( one_one @ A ) )
= ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% take_bit_of_1
thf(fact_1468_add__neg__numeral__special_I6_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( inc @ M2 ) ) ) ) ) ).
% add_neg_numeral_special(6)
thf(fact_1469_add__neg__numeral__special_I5_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( inc @ N ) ) ) ) ) ).
% add_neg_numeral_special(5)
thf(fact_1470_diff__numeral__special_I5_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ N ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( inc @ N ) ) ) ) ) ).
% diff_numeral_special(5)
thf(fact_1471_diff__numeral__special_I6_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num] :
( ( minus_minus @ A @ ( numeral_numeral @ A @ M2 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( numeral_numeral @ A @ ( inc @ M2 ) ) ) ) ).
% diff_numeral_special(6)
thf(fact_1472_take__bit__num__simps_I7_J,axiom,
! [R: num,M2: num] :
( ( bit_take_bit_num @ ( numeral_numeral @ nat @ R ) @ ( bit1 @ M2 ) )
= ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_take_bit_num @ ( pred_numeral @ R ) @ M2 ) ) ) ) ).
% take_bit_num_simps(7)
thf(fact_1473_take__bit__numeral__numeral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M2: num,N: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ M2 ) @ ( numeral_numeral @ A @ N ) )
= ( case_option @ A @ num @ ( zero_zero @ A ) @ ( numeral_numeral @ A ) @ ( bit_take_bit_num @ ( numeral_numeral @ nat @ M2 ) @ N ) ) ) ) ).
% take_bit_numeral_numeral
thf(fact_1474_bits__1__div__exp,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [N: nat] :
( ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( zero_neq_one_of_bool @ A
@ ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% bits_1_div_exp
thf(fact_1475_one__div__2__pow__eq,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N: nat] :
( ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( zero_neq_one_of_bool @ A
@ ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% one_div_2_pow_eq
thf(fact_1476_one__mod__2__pow__eq,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N: nat] :
( ( modulo_modulo @ A @ ( one_one @ A ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% one_mod_2_pow_eq
thf(fact_1477_mult__inc,axiom,
! [X: num,Y: num] :
( ( times_times @ num @ X @ ( inc @ Y ) )
= ( plus_plus @ num @ ( times_times @ num @ X @ Y ) @ X ) ) ).
% mult_inc
thf(fact_1478_add__inc,axiom,
! [X: num,Y: num] :
( ( plus_plus @ num @ X @ ( inc @ Y ) )
= ( inc @ ( plus_plus @ num @ X @ Y ) ) ) ).
% add_inc
thf(fact_1479_of__bool__less__eq__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [P: $o] : ( ord_less_eq @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( one_one @ A ) ) ) ).
% of_bool_less_eq_one
thf(fact_1480_of__bool__def,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_neq_one_of_bool @ A )
= ( ^ [P6: $o] : ( if @ A @ P6 @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ) ) ).
% of_bool_def
thf(fact_1481_split__of__bool,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P: A > $o,P5: $o] :
( ( P @ ( zero_neq_one_of_bool @ A @ P5 ) )
= ( ( P5
=> ( P @ ( one_one @ A ) ) )
& ( ~ P5
=> ( P @ ( zero_zero @ A ) ) ) ) ) ) ).
% split_of_bool
thf(fact_1482_split__of__bool__asm,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P: A > $o,P5: $o] :
( ( P @ ( zero_neq_one_of_bool @ A @ P5 ) )
= ( ~ ( ( P5
& ~ ( P @ ( one_one @ A ) ) )
| ( ~ P5
& ~ ( P @ ( zero_zero @ A ) ) ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_1483_add__One,axiom,
! [X: num] :
( ( plus_plus @ num @ X @ one2 )
= ( inc @ X ) ) ).
% add_One
thf(fact_1484_effect__updateI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [H4: heap_ext @ product_unit,R: ref @ A,V: A,H3: heap_ext @ product_unit,N: nat,X: product_unit] :
( ( H4
= ( ref_set @ A @ R @ V @ H3 ) )
=> ( ( N
= ( one_one @ nat ) )
=> ( heap_Time_effect @ product_unit @ ( ref_update @ A @ R @ V ) @ H3 @ H4 @ X @ N ) ) ) ) ).
% effect_updateI
thf(fact_1485_effect__updateE,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R4: ref @ A,V: A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: product_unit,N: nat] :
( ( heap_Time_effect @ product_unit @ ( ref_update @ A @ R4 @ V ) @ H3 @ H4 @ R @ N )
=> ~ ( ( H4
= ( ref_set @ A @ R4 @ V @ H3 ) )
=> ( N
!= ( one_one @ nat ) ) ) ) ) ).
% effect_updateE
thf(fact_1486_pred__numeral__def,axiom,
( pred_numeral
= ( ^ [K4: num] : ( minus_minus @ nat @ ( numeral_numeral @ nat @ K4 ) @ ( one_one @ nat ) ) ) ) ).
% pred_numeral_def
thf(fact_1487_BitM__plus__one,axiom,
! [N: num] :
( ( plus_plus @ num @ ( bitM @ N ) @ one2 )
= ( bit0 @ N ) ) ).
% BitM_plus_one
thf(fact_1488_one__plus__BitM,axiom,
! [N: num] :
( ( plus_plus @ num @ one2 @ ( bitM @ N ) )
= ( bit0 @ N ) ) ).
% one_plus_BitM
thf(fact_1489_numeral__inc,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [X: num] :
( ( numeral_numeral @ A @ ( inc @ X ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ X ) @ ( one_one @ A ) ) ) ) ).
% numeral_inc
thf(fact_1490_numeral__BitM,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N: num] :
( ( numeral_numeral @ A @ ( bitM @ N ) )
= ( minus_minus @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) @ ( one_one @ A ) ) ) ) ).
% numeral_BitM
thf(fact_1491_take__bit__numeral__minus__bit1,axiom,
! [L: num,K: num] :
( ( bit_se2584673776208193580ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K ) ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% take_bit_numeral_minus_bit1
thf(fact_1492_bits__induct,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [P: A > $o,A3: A] :
( ! [A5: A] :
( ( ( divide_divide @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A5 )
=> ( P @ A5 ) )
=> ( ! [A5: A,B3: $o] :
( ( P @ A5 )
=> ( ( ( divide_divide @ A @ ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ B3 ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A5 ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A5 )
=> ( P @ ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ B3 ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A5 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% bits_induct
thf(fact_1493_effect__changeE,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [F: A > A,R4: ref @ A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A,N: nat] :
( ( heap_Time_effect @ A @ ( ref_change @ A @ F @ R4 ) @ H3 @ H4 @ R @ N )
=> ~ ( ( H4
= ( ref_set @ A @ R4 @ ( F @ ( ref_get @ A @ H3 @ R4 ) ) @ H3 ) )
=> ( ( R
= ( F @ ( ref_get @ A @ H3 @ R4 ) ) )
=> ( N
!= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ) ) ) ) ).
% effect_changeE
thf(fact_1494_effect__changeI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [H4: heap_ext @ product_unit,R: ref @ A,F: A > A,H3: heap_ext @ product_unit,X: A,N: nat] :
( ( H4
= ( ref_set @ A @ R @ ( F @ ( ref_get @ A @ H3 @ R ) ) @ H3 ) )
=> ( ( X
= ( F @ ( ref_get @ A @ H3 @ R ) ) )
=> ( ( N
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
=> ( heap_Time_effect @ A @ ( ref_change @ A @ F @ R ) @ H3 @ H4 @ X @ N ) ) ) ) ) ).
% effect_changeI
thf(fact_1495_divide__int__unfold,axiom,
! [L: int,K: int,N: nat,M2: nat] :
( ( ( ( ( sgn_sgn @ int @ L )
= ( zero_zero @ int ) )
| ( ( sgn_sgn @ int @ K )
= ( zero_zero @ int ) )
| ( N
= ( zero_zero @ nat ) ) )
=> ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M2 ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( zero_zero @ int ) ) )
& ( ~ ( ( ( sgn_sgn @ int @ L )
= ( zero_zero @ int ) )
| ( ( sgn_sgn @ int @ K )
= ( zero_zero @ int ) )
| ( N
= ( zero_zero @ nat ) ) )
=> ( ( ( ( sgn_sgn @ int @ K )
= ( sgn_sgn @ int @ L ) )
=> ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M2 ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ M2 @ N ) ) ) )
& ( ( ( sgn_sgn @ int @ K )
!= ( sgn_sgn @ int @ L ) )
=> ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M2 ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( uminus_uminus @ int
@ ( semiring_1_of_nat @ int
@ ( plus_plus @ nat @ ( divide_divide @ nat @ M2 @ N )
@ ( zero_neq_one_of_bool @ nat
@ ~ ( dvd_dvd @ nat @ N @ M2 ) ) ) ) ) ) ) ) ) ) ).
% divide_int_unfold
thf(fact_1496_take__bit__numeral__bit1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [L: num,K: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ A @ ( bit1 @ K ) ) )
= ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( pred_numeral @ L ) @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ).
% take_bit_numeral_bit1
thf(fact_1497_Divides_Oadjust__div__eq,axiom,
! [Q3: int,R: int] :
( ( adjust_div @ ( product_Pair @ int @ int @ Q3 @ R ) )
= ( plus_plus @ int @ Q3
@ ( zero_neq_one_of_bool @ int
@ ( R
!= ( zero_zero @ int ) ) ) ) ) ).
% Divides.adjust_div_eq
thf(fact_1498_mask__numeral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: num] :
( ( bit_se2239418461657761734s_mask @ A @ ( numeral_numeral @ nat @ N ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2239418461657761734s_mask @ A @ ( pred_numeral @ N ) ) ) ) ) ) ).
% mask_numeral
thf(fact_1499_and__int__unfold,axiom,
( ( bit_se5824344872417868541ns_and @ int )
= ( ^ [K4: int,L3: int] :
( if @ int
@ ( ( K4
= ( zero_zero @ int ) )
| ( L3
= ( zero_zero @ int ) ) )
@ ( zero_zero @ int )
@ ( if @ int
@ ( K4
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
@ L3
@ ( if @ int
@ ( L3
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
@ K4
@ ( plus_plus @ int @ ( times_times @ int @ ( modulo_modulo @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ).
% and_int_unfold
thf(fact_1500_the__res__def,axiom,
! [A: $tType] :
( ( time_the_res @ A )
= ( ^ [M: heap_Time_Heap @ A,H: heap_ext @ product_unit] :
( case_option @ A @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( undefined @ A )
@ ( product_case_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ A
@ ^ [R3: A] :
( product_case_prod @ ( heap_ext @ product_unit ) @ nat @ A
@ ^ [X3: heap_ext @ product_unit,Xa4: nat] : R3 ) )
@ ( heap_Time_execute @ A @ M @ H ) ) ) ) ).
% the_res_def
thf(fact_1501_take__bit__eq__mask__iff__exp__dvd,axiom,
! [N: nat,K: int] :
( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
= ( bit_se2239418461657761734s_mask @ int @ N ) )
= ( dvd_dvd @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ ( plus_plus @ int @ K @ ( one_one @ int ) ) ) ) ).
% take_bit_eq_mask_iff_exp_dvd
thf(fact_1502_Code__Numeral_Opositive__def,axiom,
( code_positive
= ( numeral_numeral @ code_integer ) ) ).
% Code_Numeral.positive_def
thf(fact_1503_divide__int__def,axiom,
( ( divide_divide @ int )
= ( ^ [K4: int,L3: int] :
( if @ int
@ ( L3
= ( zero_zero @ int ) )
@ ( zero_zero @ int )
@ ( if @ int
@ ( ( sgn_sgn @ int @ K4 )
= ( sgn_sgn @ int @ L3 ) )
@ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ ( nat2 @ ( abs_abs @ int @ K4 ) ) @ ( nat2 @ ( abs_abs @ int @ L3 ) ) ) )
@ ( uminus_uminus @ int
@ ( semiring_1_of_nat @ int
@ ( plus_plus @ nat @ ( divide_divide @ nat @ ( nat2 @ ( abs_abs @ int @ K4 ) ) @ ( nat2 @ ( abs_abs @ int @ L3 ) ) )
@ ( zero_neq_one_of_bool @ nat
@ ~ ( dvd_dvd @ int @ L3 @ K4 ) ) ) ) ) ) ) ) ) ).
% divide_int_def
thf(fact_1504_sum__gp0,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat] :
( ( ( X
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ N ) )
= ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) )
& ( ( X
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ).
% sum_gp0
thf(fact_1505_atMost__eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ( set_ord_atMost @ A @ X )
= ( set_ord_atMost @ A @ Y ) )
= ( X = Y ) ) ) ).
% atMost_eq_iff
thf(fact_1506_atMost__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,K: A] :
( ( member @ A @ I @ ( set_ord_atMost @ A @ K ) )
= ( ord_less_eq @ A @ I @ K ) ) ) ).
% atMost_iff
thf(fact_1507_atMost__subset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ X ) @ ( set_ord_atMost @ A @ Y ) )
= ( ord_less_eq @ A @ X @ Y ) ) ) ).
% atMost_subset_iff
thf(fact_1508_bit_Oconj__one__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se5824344872417868541ns_and @ A @ X @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= X ) ) ).
% bit.conj_one_right
thf(fact_1509_and_Oright__neutral,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A] :
( ( bit_se5824344872417868541ns_and @ A @ A3 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= A3 ) ) ).
% and.right_neutral
thf(fact_1510_and_Oleft__neutral,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ A3 )
= A3 ) ) ).
% and.left_neutral
thf(fact_1511_and__numerals_I2_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( one_one @ A ) ) ) ).
% and_numerals(2)
thf(fact_1512_and__numerals_I8_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% and_numerals(8)
thf(fact_1513_Icc__subset__Iic__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [L: A,H3: A,H4: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ H3 ) @ ( set_ord_atMost @ A @ H4 ) )
= ( ~ ( ord_less_eq @ A @ L @ H3 )
| ( ord_less_eq @ A @ H3 @ H4 ) ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_1514_mask__Suc__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2239418461657761734s_mask @ A @ ( suc @ ( zero_zero @ nat ) ) )
= ( one_one @ A ) ) ) ).
% mask_Suc_0
thf(fact_1515_sum_OatMost__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ).
% sum.atMost_Suc
thf(fact_1516_take__bit__minus__one__eq__mask,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ).
% take_bit_minus_one_eq_mask
thf(fact_1517_and__numerals_I1_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( zero_zero @ A ) ) ) ).
% and_numerals(1)
thf(fact_1518_and__numerals_I5_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% and_numerals(5)
thf(fact_1519_nat__numeral__diff__1,axiom,
! [V: num] :
( ( minus_minus @ nat @ ( numeral_numeral @ nat @ V ) @ ( one_one @ nat ) )
= ( nat2 @ ( minus_minus @ int @ ( numeral_numeral @ int @ V ) @ ( one_one @ int ) ) ) ) ).
% nat_numeral_diff_1
thf(fact_1520_and__numerals_I7_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% and_numerals(7)
thf(fact_1521_mask__integer_Oabs__eq,axiom,
( ( bit_se2239418461657761734s_mask @ code_integer )
= ( ^ [X3: nat] : ( code_integer_of_int @ ( bit_se2239418461657761734s_mask @ int @ X3 ) ) ) ) ).
% mask_integer.abs_eq
thf(fact_1522_not__Iic__eq__Icc,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [H4: A,L: A,H3: A] :
( ( set_ord_atMost @ A @ H4 )
!= ( set_or1337092689740270186AtMost @ A @ L @ H3 ) ) ) ).
% not_Iic_eq_Icc
thf(fact_1523_atMost__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_atMost @ A )
= ( ^ [U2: A] :
( collect @ A
@ ^ [X3: A] : ( ord_less_eq @ A @ X3 @ U2 ) ) ) ) ) ).
% atMost_def
thf(fact_1524_and__eq__minus__1__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A,B2: A] :
( ( ( bit_se5824344872417868541ns_and @ A @ A3 @ B2 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( ( A3
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
& ( B2
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% and_eq_minus_1_iff
thf(fact_1525_atMost__atLeast0,axiom,
( ( set_ord_atMost @ nat )
= ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) ) ) ).
% atMost_atLeast0
thf(fact_1526_nat__one__as__int,axiom,
( ( one_one @ nat )
= ( nat2 @ ( one_one @ int ) ) ) ).
% nat_one_as_int
thf(fact_1527_not__Iic__le__Icc,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [H3: A,L4: A,H4: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ H3 ) @ ( set_or1337092689740270186AtMost @ A @ L4 @ H4 ) ) ) ).
% not_Iic_le_Icc
thf(fact_1528_nat__int__add,axiom,
! [A3: nat,B2: nat] :
( ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) )
= ( plus_plus @ nat @ A3 @ B2 ) ) ).
% nat_int_add
thf(fact_1529_option_Othe__def,axiom,
! [A: $tType] :
( ( the2 @ A )
= ( case_option @ A @ A @ ( undefined @ A )
@ ^ [X24: A] : X24 ) ) ).
% option.the_def
thf(fact_1530_nat__plus__as__int,axiom,
( ( plus_plus @ nat )
= ( ^ [A7: nat,B5: nat] : ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A7 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ) ).
% nat_plus_as_int
thf(fact_1531_nat__add__distrib,axiom,
! [Z2: int,Z7: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z7 )
=> ( ( nat2 @ ( plus_plus @ int @ Z2 @ Z7 ) )
= ( plus_plus @ nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z7 ) ) ) ) ) ).
% nat_add_distrib
thf(fact_1532_Suc__as__int,axiom,
( suc
= ( ^ [A7: nat] : ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A7 ) @ ( one_one @ int ) ) ) ) ) ).
% Suc_as_int
thf(fact_1533_nat__abs__triangle__ineq,axiom,
! [K: int,L: int] : ( ord_less_eq @ nat @ ( nat2 @ ( abs_abs @ int @ ( plus_plus @ int @ K @ L ) ) ) @ ( plus_plus @ nat @ ( nat2 @ ( abs_abs @ int @ K ) ) @ ( nat2 @ ( abs_abs @ int @ L ) ) ) ) ).
% nat_abs_triangle_ineq
thf(fact_1534_sum_OatMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% sum.atMost_Suc_shift
thf(fact_1535_sum__telescope,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [F: nat > A,I: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( minus_minus @ A @ ( F @ I3 ) @ ( F @ ( suc @ I3 ) ) )
@ ( set_ord_atMost @ nat @ I ) )
= ( minus_minus @ A @ ( F @ ( zero_zero @ nat ) ) @ ( F @ ( suc @ I ) ) ) ) ) ).
% sum_telescope
thf(fact_1536_the__heap__def,axiom,
! [A: $tType] :
( ( time_the_heap @ A )
= ( ^ [M: heap_Time_Heap @ A,H: heap_ext @ product_unit] :
( case_option @ ( heap_ext @ product_unit ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( undefined @ ( heap_ext @ product_unit ) )
@ ( product_case_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( heap_ext @ product_unit )
@ ^ [X3: A] :
( product_case_prod @ ( heap_ext @ product_unit ) @ nat @ ( heap_ext @ product_unit )
@ ^ [H7: heap_ext @ product_unit,Xa4: nat] : H7 ) )
@ ( heap_Time_execute @ A @ M @ H ) ) ) ) ).
% the_heap_def
thf(fact_1537_and__one__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5824344872417868541ns_and @ A @ A3 @ ( one_one @ A ) )
= ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% and_one_eq
thf(fact_1538_one__and__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( one_one @ A ) @ A3 )
= ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% one_and_eq
thf(fact_1539_take__bit__eq__mask__iff,axiom,
! [N: nat,K: int] :
( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
= ( bit_se2239418461657761734s_mask @ int @ N ) )
= ( ( bit_se2584673776208193580ke_bit @ int @ N @ ( plus_plus @ int @ K @ ( one_one @ int ) ) )
= ( zero_zero @ int ) ) ) ).
% take_bit_eq_mask_iff
thf(fact_1540_Suc__nat__eq__nat__zadd1,axiom,
! [Z2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
=> ( ( suc @ ( nat2 @ Z2 ) )
= ( nat2 @ ( plus_plus @ int @ ( one_one @ int ) @ Z2 ) ) ) ) ).
% Suc_nat_eq_nat_zadd1
thf(fact_1541_Divides_Oadjust__div__def,axiom,
( adjust_div
= ( product_case_prod @ int @ int @ int
@ ^ [Q5: int,R3: int] :
( plus_plus @ int @ Q5
@ ( zero_neq_one_of_bool @ int
@ ( R3
!= ( zero_zero @ int ) ) ) ) ) ) ).
% Divides.adjust_div_def
thf(fact_1542_sum__gp__basic,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat] :
( ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ N ) ) )
= ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) ) ) ).
% sum_gp_basic
thf(fact_1543_sum__power__shift,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [M2: nat,N: nat,X: A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( times_times @ A @ ( power_power @ A @ X @ M2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ) ) ).
% sum_power_shift
thf(fact_1544_sum_Otriangle__reindex__eq,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I3: nat,J2: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I3 @ J2 ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K4: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G @ I3 @ ( minus_minus @ nat @ K4 @ I3 ) )
@ ( set_ord_atMost @ nat @ K4 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% sum.triangle_reindex_eq
thf(fact_1545_and__int__rec,axiom,
( ( bit_se5824344872417868541ns_and @ int )
= ( ^ [K4: int,L3: int] :
( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K4 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L3 ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% and_int_rec
thf(fact_1546_mask__half__int,axiom,
! [N: nat] :
( ( divide_divide @ int @ ( bit_se2239418461657761734s_mask @ int @ N ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( bit_se2239418461657761734s_mask @ int @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ).
% mask_half_int
thf(fact_1547_sum_Oin__pairs__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( plus_plus @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% sum.in_pairs_0
thf(fact_1548_mask__nat__def,axiom,
( ( bit_se2239418461657761734s_mask @ nat )
= ( ^ [N5: nat] : ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) @ ( one_one @ nat ) ) ) ) ).
% mask_nat_def
thf(fact_1549_mask__eq__exp__minus__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2239418461657761734s_mask @ A )
= ( ^ [N5: nat] : ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N5 ) @ ( one_one @ A ) ) ) ) ) ).
% mask_eq_exp_minus_1
thf(fact_1550_sum_Ozero__middle,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [P5: nat,K: nat,G: nat > A,H3: nat > A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P5 )
=> ( ( ord_less_eq @ nat @ K @ P5 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J2: nat] : ( if @ A @ ( ord_less @ nat @ J2 @ K ) @ ( G @ J2 ) @ ( if @ A @ ( J2 = K ) @ ( zero_zero @ A ) @ ( H3 @ ( minus_minus @ nat @ J2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( set_ord_atMost @ nat @ P5 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J2: nat] : ( if @ A @ ( ord_less @ nat @ J2 @ K ) @ ( G @ J2 ) @ ( H3 @ J2 ) )
@ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P5 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_1551_and__minus__numerals_I3_J,axiom,
! [M2: num,N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M2 @ ( bitM @ N ) ) ) ) ).
% and_minus_numerals(3)
thf(fact_1552_and__minus__numerals_I7_J,axiom,
! [N: num,M2: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) @ ( numeral_numeral @ int @ M2 ) )
= ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M2 @ ( bitM @ N ) ) ) ) ).
% and_minus_numerals(7)
thf(fact_1553_and__minus__numerals_I4_J,axiom,
! [M2: num,N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M2 @ ( bit0 @ N ) ) ) ) ).
% and_minus_numerals(4)
thf(fact_1554_and__minus__numerals_I8_J,axiom,
! [N: num,M2: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) @ ( numeral_numeral @ int @ M2 ) )
= ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M2 @ ( bit0 @ N ) ) ) ) ).
% and_minus_numerals(8)
thf(fact_1555_gbinomial__partial__row__sum,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K4: nat] : ( times_times @ A @ ( gbinomial @ A @ A3 @ K4 ) @ ( minus_minus @ A @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ A @ K4 ) ) )
@ ( set_ord_atMost @ nat @ M2 ) )
= ( times_times @ A @ ( divide_divide @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( gbinomial @ A @ A3 @ ( plus_plus @ nat @ M2 @ ( one_one @ nat ) ) ) ) ) ) ).
% gbinomial_partial_row_sum
thf(fact_1556_gbinomial__r__part__sum,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ M2 ) ) @ ( one_one @ A ) ) ) @ ( set_ord_atMost @ nat @ M2 ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) ) ) ) ).
% gbinomial_r_part_sum
thf(fact_1557_and__num_Oelims,axiom,
! [X: num,Xa: num,Y: option @ num] :
( ( ( bit_un7362597486090784418nd_num @ X @ Xa )
= Y )
=> ( ( ( X = one2 )
=> ( ( Xa = one2 )
=> ( Y
!= ( some @ num @ one2 ) ) ) )
=> ( ( ( X = one2 )
=> ( ? [N3: num] :
( Xa
= ( bit0 @ N3 ) )
=> ( Y
!= ( none @ num ) ) ) )
=> ( ( ( X = one2 )
=> ( ? [N3: num] :
( Xa
= ( bit1 @ N3 ) )
=> ( Y
!= ( some @ num @ one2 ) ) ) )
=> ( ( ? [M5: num] :
( X
= ( bit0 @ M5 ) )
=> ( ( Xa = one2 )
=> ( Y
!= ( none @ num ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ! [N3: num] :
( ( Xa
= ( bit0 @ N3 ) )
=> ( Y
!= ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N3 ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ! [N3: num] :
( ( Xa
= ( bit1 @ N3 ) )
=> ( Y
!= ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N3 ) ) ) ) )
=> ( ( ? [M5: num] :
( X
= ( bit1 @ M5 ) )
=> ( ( Xa = one2 )
=> ( Y
!= ( some @ num @ one2 ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ! [N3: num] :
( ( Xa
= ( bit0 @ N3 ) )
=> ( Y
!= ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N3 ) ) ) ) )
=> ~ ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ! [N3: num] :
( ( Xa
= ( bit1 @ N3 ) )
=> ( Y
!= ( case_option @ ( option @ num ) @ num @ ( some @ num @ one2 )
@ ^ [N8: num] : ( some @ num @ ( bit1 @ N8 ) )
@ ( bit_un7362597486090784418nd_num @ M5 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_num.elims
thf(fact_1558_take__bit__Suc__from__most,axiom,
! [N: nat,K: int] :
( ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N ) @ K )
= ( plus_plus @ int @ ( times_times @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ ( zero_neq_one_of_bool @ int @ ( bit_se5641148757651400278ts_bit @ int @ K @ N ) ) ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ) ).
% take_bit_Suc_from_most
thf(fact_1559_gbinomial__1,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ! [A3: A] :
( ( gbinomial @ A @ A3 @ ( one_one @ nat ) )
= A3 ) ) ).
% gbinomial_1
thf(fact_1560_gbinomial__0_I1_J,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ! [A3: A] :
( ( gbinomial @ A @ A3 @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% gbinomial_0(1)
thf(fact_1561_and__nat__numerals_I2_J,axiom,
! [Y: num] :
( ( bit_se5824344872417868541ns_and @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit1 @ Y ) ) )
= ( one_one @ nat ) ) ).
% and_nat_numerals(2)
thf(fact_1562_and__nat__numerals_I4_J,axiom,
! [X: num] :
( ( bit_se5824344872417868541ns_and @ nat @ ( numeral_numeral @ nat @ ( bit1 @ X ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( one_one @ nat ) ) ).
% and_nat_numerals(4)
thf(fact_1563_bit__disjunctive__add__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,B2: A,N: nat] :
( ! [N3: nat] :
( ~ ( bit_se5641148757651400278ts_bit @ A @ A3 @ N3 )
| ~ ( bit_se5641148757651400278ts_bit @ A @ B2 @ N3 ) )
=> ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ A3 @ B2 ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ A3 @ N )
| ( bit_se5641148757651400278ts_bit @ A @ B2 @ N ) ) ) ) ) ).
% bit_disjunctive_add_iff
thf(fact_1564_bit__1__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( one_one @ A ) @ N )
= ( N
= ( zero_zero @ nat ) ) ) ) ).
% bit_1_iff
thf(fact_1565_not__bit__1__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
~ ( bit_se5641148757651400278ts_bit @ A @ ( one_one @ A ) @ ( suc @ N ) ) ) ).
% not_bit_1_Suc
thf(fact_1566_bit__numeral__simps_I1_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: num] :
~ ( bit_se5641148757651400278ts_bit @ A @ ( one_one @ A ) @ ( numeral_numeral @ nat @ N ) ) ) ).
% bit_numeral_simps(1)
thf(fact_1567_gbinomial__Suc__Suc,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K ) )
= ( plus_plus @ A @ ( gbinomial @ A @ A3 @ K ) @ ( gbinomial @ A @ A3 @ ( suc @ K ) ) ) ) ) ).
% gbinomial_Suc_Suc
thf(fact_1568_and__integer_Oabs__eq,axiom,
! [Xa: int,X: int] :
( ( bit_se5824344872417868541ns_and @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( bit_se5824344872417868541ns_and @ int @ Xa @ X ) ) ) ).
% and_integer.abs_eq
thf(fact_1569_gbinomial__addition__formula,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( gbinomial @ A @ A3 @ ( suc @ K ) )
= ( plus_plus @ A @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ K ) ) ) ) ).
% gbinomial_addition_formula
thf(fact_1570_gbinomial__absorb__comp,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( times_times @ A @ ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ K ) ) @ ( gbinomial @ A @ A3 @ K ) )
= ( times_times @ A @ A3 @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ K ) ) ) ) ).
% gbinomial_absorb_comp
thf(fact_1571_gbinomial__mult__1_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( times_times @ A @ ( gbinomial @ A @ A3 @ K ) @ A3 )
= ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ K ) @ ( gbinomial @ A @ A3 @ K ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ A3 @ ( suc @ K ) ) ) ) ) ) ).
% gbinomial_mult_1'
thf(fact_1572_gbinomial__mult__1,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( times_times @ A @ A3 @ ( gbinomial @ A @ A3 @ K ) )
= ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ K ) @ ( gbinomial @ A @ A3 @ K ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ A3 @ ( suc @ K ) ) ) ) ) ) ).
% gbinomial_mult_1
thf(fact_1573_Suc__times__gbinomial,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A3: A] :
( ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K ) ) )
= ( times_times @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( gbinomial @ A @ A3 @ K ) ) ) ) ).
% Suc_times_gbinomial
thf(fact_1574_gbinomial__absorption,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A3: A] :
( ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ A3 @ ( suc @ K ) ) )
= ( times_times @ A @ A3 @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ K ) ) ) ) ).
% gbinomial_absorption
thf(fact_1575_gbinomial__parallel__sum,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K4: nat] : ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ K4 ) ) @ K4 )
@ ( set_ord_atMost @ nat @ N ) )
= ( gbinomial @ A @ ( plus_plus @ A @ ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ N ) ) @ ( one_one @ A ) ) @ N ) ) ) ).
% gbinomial_parallel_sum
thf(fact_1576_gbinomial__rec,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K ) )
= ( times_times @ A @ ( gbinomial @ A @ A3 @ K ) @ ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) ) ) ) ) ).
% gbinomial_rec
thf(fact_1577_gbinomial__factors,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K ) )
= ( times_times @ A @ ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) ) @ ( gbinomial @ A @ A3 @ K ) ) ) ) ).
% gbinomial_factors
thf(fact_1578_gbinomial__negated__upper,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A7: A,K4: nat] : ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K4 ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( minus_minus @ A @ ( semiring_1_of_nat @ A @ K4 ) @ A7 ) @ ( one_one @ A ) ) @ K4 ) ) ) ) ) ).
% gbinomial_negated_upper
thf(fact_1579_gbinomial__index__swap,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N: nat] :
( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N ) ) @ ( one_one @ A ) ) @ K ) )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ N ) ) ) ) ).
% gbinomial_index_swap
thf(fact_1580_int__bit__bound,axiom,
! [K: int] :
~ ! [N3: nat] :
( ! [M6: nat] :
( ( ord_less_eq @ nat @ N3 @ M6 )
=> ( ( bit_se5641148757651400278ts_bit @ int @ K @ M6 )
= ( bit_se5641148757651400278ts_bit @ int @ K @ N3 ) ) )
=> ~ ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( bit_se5641148757651400278ts_bit @ int @ K @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ int @ K @ N3 ) ) ) ) ) ).
% int_bit_bound
thf(fact_1581_gbinomial__minus,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( gbinomial @ A @ ( uminus_uminus @ A @ A3 ) @ K )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ K ) ) ) ) ).
% gbinomial_minus
thf(fact_1582_gbinomial__reduce__nat,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( gbinomial @ A @ A3 @ K )
= ( plus_plus @ A @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ K ) ) ) ) ) ).
% gbinomial_reduce_nat
thf(fact_1583_gbinomial__sum__lower__neg,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K4: nat] : ( times_times @ A @ ( gbinomial @ A @ A3 @ K4 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K4 ) )
@ ( set_ord_atMost @ nat @ M2 ) )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M2 ) @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ M2 ) ) ) ) ).
% gbinomial_sum_lower_neg
thf(fact_1584_numeral__and__num,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M2: num,N: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N ) )
= ( case_option @ A @ num @ ( zero_zero @ A ) @ ( numeral_numeral @ A ) @ ( bit_un7362597486090784418nd_num @ M2 @ N ) ) ) ) ).
% numeral_and_num
thf(fact_1585_gbinomial__sum__up__index,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J2: nat] : ( gbinomial @ A @ ( semiring_1_of_nat @ A @ J2 ) @ K )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) @ ( plus_plus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ).
% gbinomial_sum_up_index
thf(fact_1586_gbinomial__partial__sum__poly,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M2: nat,A3: A,X: A,Y: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K4: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M2 ) @ A3 ) @ K4 ) @ ( power_power @ A @ X @ K4 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ M2 @ K4 ) ) )
@ ( set_ord_atMost @ nat @ M2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K4: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( uminus_uminus @ A @ A3 ) @ K4 ) @ ( power_power @ A @ ( uminus_uminus @ A @ X ) @ K4 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X @ Y ) @ ( minus_minus @ nat @ M2 @ K4 ) ) )
@ ( set_ord_atMost @ nat @ M2 ) ) ) ) ).
% gbinomial_partial_sum_poly
thf(fact_1587_even__bit__succ__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ A3 @ N )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ) ).
% even_bit_succ_iff
thf(fact_1588_odd__bit__iff__bit__pred,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N: nat] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( bit_se5641148757651400278ts_bit @ A @ A3 @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ N )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ) ).
% odd_bit_iff_bit_pred
thf(fact_1589_and__num_Osimps_I9_J,axiom,
! [M2: num,N: num] :
( ( bit_un7362597486090784418nd_num @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
= ( case_option @ ( option @ num ) @ num @ ( some @ num @ one2 )
@ ^ [N8: num] : ( some @ num @ ( bit1 @ N8 ) )
@ ( bit_un7362597486090784418nd_num @ M2 @ N ) ) ) ).
% and_num.simps(9)
thf(fact_1590_gbinomial__absorption_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( gbinomial @ A @ A3 @ K )
= ( times_times @ A @ ( divide_divide @ A @ A3 @ ( semiring_1_of_nat @ A @ K ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ) ) ).
% gbinomial_absorption'
thf(fact_1591_gbinomial__partial__sum__poly__xpos,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M2: nat,A3: A,X: A,Y: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K4: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M2 ) @ A3 ) @ K4 ) @ ( power_power @ A @ X @ K4 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ M2 @ K4 ) ) )
@ ( set_ord_atMost @ nat @ M2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K4: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( minus_minus @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ K4 ) @ A3 ) @ ( one_one @ A ) ) @ K4 ) @ ( power_power @ A @ X @ K4 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X @ Y ) @ ( minus_minus @ nat @ M2 @ K4 ) ) )
@ ( set_ord_atMost @ nat @ M2 ) ) ) ) ).
% gbinomial_partial_sum_poly_xpos
thf(fact_1592_gbinomial__sum__nat__pow2,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K4: nat] : ( divide_divide @ A @ ( gbinomial @ A @ ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ M2 @ K4 ) ) @ K4 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ K4 ) )
@ ( set_ord_atMost @ nat @ M2 ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) ) ) ).
% gbinomial_sum_nat_pow2
thf(fact_1593_bit__sum__mult__2__cases,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A,B2: A,N: nat] :
( ! [J3: nat] :
~ ( bit_se5641148757651400278ts_bit @ A @ A3 @ ( suc @ J3 ) )
=> ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ N )
= ( ( ( N
= ( zero_zero @ nat ) )
=> ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( bit_se5641148757651400278ts_bit @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) @ N ) ) ) ) ) ) ).
% bit_sum_mult_2_cases
thf(fact_1594_and__not__num_Osimps_I8_J,axiom,
! [M2: num,N: num] :
( ( bit_and_not_num @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
= ( case_option @ ( option @ num ) @ num @ ( some @ num @ one2 )
@ ^ [N8: num] : ( some @ num @ ( bit1 @ N8 ) )
@ ( bit_and_not_num @ M2 @ N ) ) ) ).
% and_not_num.simps(8)
thf(fact_1595_bit__rec,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A7: A,N5: nat] :
( ( ( N5
= ( zero_zero @ nat ) )
=> ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A7 ) )
& ( ( N5
!= ( zero_zero @ nat ) )
=> ( bit_se5641148757651400278ts_bit @ A @ ( divide_divide @ A @ A7 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( minus_minus @ nat @ N5 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% bit_rec
thf(fact_1596_and__nat__unfold,axiom,
( ( bit_se5824344872417868541ns_and @ nat )
= ( ^ [M: nat,N5: nat] :
( if @ nat
@ ( ( M
= ( zero_zero @ nat ) )
| ( N5
= ( zero_zero @ nat ) ) )
@ ( zero_zero @ nat )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( modulo_modulo @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ nat @ ( divide_divide @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% and_nat_unfold
thf(fact_1597_and__nat__rec,axiom,
( ( bit_se5824344872417868541ns_and @ nat )
= ( ^ [M: nat,N5: nat] :
( plus_plus @ nat
@ ( zero_neq_one_of_bool @ nat
@ ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M )
& ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) ) )
@ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ nat @ ( divide_divide @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% and_nat_rec
thf(fact_1598_set__bit__eq,axiom,
( ( bit_se5668285175392031749et_bit @ int )
= ( ^ [N5: nat,K4: int] :
( plus_plus @ int @ K4
@ ( times_times @ int
@ ( zero_neq_one_of_bool @ int
@ ~ ( bit_se5641148757651400278ts_bit @ int @ K4 @ N5 ) )
@ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N5 ) ) ) ) ) ).
% set_bit_eq
thf(fact_1599_and__not__num_Oelims,axiom,
! [X: num,Xa: num,Y: option @ num] :
( ( ( bit_and_not_num @ X @ Xa )
= Y )
=> ( ( ( X = one2 )
=> ( ( Xa = one2 )
=> ( Y
!= ( none @ num ) ) ) )
=> ( ( ( X = one2 )
=> ( ? [N3: num] :
( Xa
= ( bit0 @ N3 ) )
=> ( Y
!= ( some @ num @ one2 ) ) ) )
=> ( ( ( X = one2 )
=> ( ? [N3: num] :
( Xa
= ( bit1 @ N3 ) )
=> ( Y
!= ( none @ num ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ( ( Xa = one2 )
=> ( Y
!= ( some @ num @ ( bit0 @ M5 ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ! [N3: num] :
( ( Xa
= ( bit0 @ N3 ) )
=> ( Y
!= ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M5 @ N3 ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ! [N3: num] :
( ( Xa
= ( bit1 @ N3 ) )
=> ( Y
!= ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M5 @ N3 ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ( ( Xa = one2 )
=> ( Y
!= ( some @ num @ ( bit0 @ M5 ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ! [N3: num] :
( ( Xa
= ( bit0 @ N3 ) )
=> ( Y
!= ( case_option @ ( option @ num ) @ num @ ( some @ num @ one2 )
@ ^ [N8: num] : ( some @ num @ ( bit1 @ N8 ) )
@ ( bit_and_not_num @ M5 @ N3 ) ) ) ) )
=> ~ ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ! [N3: num] :
( ( Xa
= ( bit1 @ N3 ) )
=> ( Y
!= ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M5 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_not_num.elims
thf(fact_1600_and__not__num_Opelims,axiom,
! [X: num,Xa: num,Y: option @ num] :
( ( ( bit_and_not_num @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ X @ Xa ) )
=> ( ( ( X = one2 )
=> ( ( Xa = one2 )
=> ( ( Y
= ( none @ num ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ one2 @ one2 ) ) ) ) )
=> ( ( ( X = one2 )
=> ! [N3: num] :
( ( Xa
= ( bit0 @ N3 ) )
=> ( ( Y
= ( some @ num @ one2 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ one2 @ ( bit0 @ N3 ) ) ) ) ) )
=> ( ( ( X = one2 )
=> ! [N3: num] :
( ( Xa
= ( bit1 @ N3 ) )
=> ( ( Y
= ( none @ num ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ one2 @ ( bit1 @ N3 ) ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ( ( Xa = one2 )
=> ( ( Y
= ( some @ num @ ( bit0 @ M5 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ ( bit0 @ M5 ) @ one2 ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ! [N3: num] :
( ( Xa
= ( bit0 @ N3 ) )
=> ( ( Y
= ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M5 @ N3 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ ( bit0 @ M5 ) @ ( bit0 @ N3 ) ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ! [N3: num] :
( ( Xa
= ( bit1 @ N3 ) )
=> ( ( Y
= ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M5 @ N3 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ ( bit0 @ M5 ) @ ( bit1 @ N3 ) ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ( ( Xa = one2 )
=> ( ( Y
= ( some @ num @ ( bit0 @ M5 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ ( bit1 @ M5 ) @ one2 ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ! [N3: num] :
( ( Xa
= ( bit0 @ N3 ) )
=> ( ( Y
= ( case_option @ ( option @ num ) @ num @ ( some @ num @ one2 )
@ ^ [N8: num] : ( some @ num @ ( bit1 @ N8 ) )
@ ( bit_and_not_num @ M5 @ N3 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ ( bit1 @ M5 ) @ ( bit0 @ N3 ) ) ) ) ) )
=> ~ ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ! [N3: num] :
( ( Xa
= ( bit1 @ N3 ) )
=> ( ( Y
= ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M5 @ N3 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ ( bit1 @ M5 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_not_num.pelims
thf(fact_1601_and__num_Opelims,axiom,
! [X: num,Xa: num,Y: option @ num] :
( ( ( bit_un7362597486090784418nd_num @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ X @ Xa ) )
=> ( ( ( X = one2 )
=> ( ( Xa = one2 )
=> ( ( Y
= ( some @ num @ one2 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ one2 @ one2 ) ) ) ) )
=> ( ( ( X = one2 )
=> ! [N3: num] :
( ( Xa
= ( bit0 @ N3 ) )
=> ( ( Y
= ( none @ num ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ one2 @ ( bit0 @ N3 ) ) ) ) ) )
=> ( ( ( X = one2 )
=> ! [N3: num] :
( ( Xa
= ( bit1 @ N3 ) )
=> ( ( Y
= ( some @ num @ one2 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ one2 @ ( bit1 @ N3 ) ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ( ( Xa = one2 )
=> ( ( Y
= ( none @ num ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M5 ) @ one2 ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ! [N3: num] :
( ( Xa
= ( bit0 @ N3 ) )
=> ( ( Y
= ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N3 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M5 ) @ ( bit0 @ N3 ) ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ! [N3: num] :
( ( Xa
= ( bit1 @ N3 ) )
=> ( ( Y
= ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N3 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M5 ) @ ( bit1 @ N3 ) ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ( ( Xa = one2 )
=> ( ( Y
= ( some @ num @ one2 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M5 ) @ one2 ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ! [N3: num] :
( ( Xa
= ( bit0 @ N3 ) )
=> ( ( Y
= ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N3 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M5 ) @ ( bit0 @ N3 ) ) ) ) ) )
=> ~ ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ! [N3: num] :
( ( Xa
= ( bit1 @ N3 ) )
=> ( ( Y
= ( case_option @ ( option @ num ) @ num @ ( some @ num @ one2 )
@ ^ [N8: num] : ( some @ num @ ( bit1 @ N8 ) )
@ ( bit_un7362597486090784418nd_num @ M5 @ N3 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M5 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_num.pelims
thf(fact_1602_gbinomial__code,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A7: A,K4: nat] :
( if @ A
@ ( K4
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( divide_divide @ A
@ ( set_fo6178422350223883121st_nat @ A
@ ^ [L3: nat] : ( times_times @ A @ ( minus_minus @ A @ A7 @ ( semiring_1_of_nat @ A @ L3 ) ) )
@ ( zero_zero @ nat )
@ ( minus_minus @ nat @ K4 @ ( one_one @ nat ) )
@ ( one_one @ A ) )
@ ( semiring_char_0_fact @ A @ K4 ) ) ) ) ) ) ).
% gbinomial_code
thf(fact_1603_xor__num_Oelims,axiom,
! [X: num,Xa: num,Y: option @ num] :
( ( ( bit_un2480387367778600638or_num @ X @ Xa )
= Y )
=> ( ( ( X = one2 )
=> ( ( Xa = one2 )
=> ( Y
!= ( none @ num ) ) ) )
=> ( ( ( X = one2 )
=> ! [N3: num] :
( ( Xa
= ( bit0 @ N3 ) )
=> ( Y
!= ( some @ num @ ( bit1 @ N3 ) ) ) ) )
=> ( ( ( X = one2 )
=> ! [N3: num] :
( ( Xa
= ( bit1 @ N3 ) )
=> ( Y
!= ( some @ num @ ( bit0 @ N3 ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ( ( Xa = one2 )
=> ( Y
!= ( some @ num @ ( bit1 @ M5 ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ! [N3: num] :
( ( Xa
= ( bit0 @ N3 ) )
=> ( Y
!= ( map_option @ num @ num @ bit0 @ ( bit_un2480387367778600638or_num @ M5 @ N3 ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ! [N3: num] :
( ( Xa
= ( bit1 @ N3 ) )
=> ( Y
!= ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_un2480387367778600638or_num @ M5 @ N3 ) ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ( ( Xa = one2 )
=> ( Y
!= ( some @ num @ ( bit0 @ M5 ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ! [N3: num] :
( ( Xa
= ( bit0 @ N3 ) )
=> ( Y
!= ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_un2480387367778600638or_num @ M5 @ N3 ) ) ) ) ) )
=> ~ ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ! [N3: num] :
( ( Xa
= ( bit1 @ N3 ) )
=> ( Y
!= ( map_option @ num @ num @ bit0 @ ( bit_un2480387367778600638or_num @ M5 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% xor_num.elims
thf(fact_1604_choose__alternating__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ I3 ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I3 ) ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) ) ).
% choose_alternating_sum
thf(fact_1605_choose__linear__sum,axiom,
! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I3: nat] : ( times_times @ nat @ I3 @ ( binomial @ N @ I3 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( times_times @ nat @ N @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% choose_linear_sum
thf(fact_1606_binomial__r__part__sum,axiom,
! [M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat @ ( binomial @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) @ ( one_one @ nat ) ) ) @ ( set_ord_atMost @ nat @ M2 ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) ) ) ).
% binomial_r_part_sum
thf(fact_1607_sum_Otriangle__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I3: nat,J2: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I3 @ J2 ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K4: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G @ I3 @ ( minus_minus @ nat @ K4 @ I3 ) )
@ ( set_ord_atMost @ nat @ K4 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum.triangle_reindex
thf(fact_1608_lessThan__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ( set_ord_lessThan @ A @ X )
= ( set_ord_lessThan @ A @ Y ) )
= ( X = Y ) ) ) ).
% lessThan_eq_iff
thf(fact_1609_lessThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,K: A] :
( ( member @ A @ I @ ( set_ord_lessThan @ A @ K ) )
= ( ord_less @ A @ I @ K ) ) ) ).
% lessThan_iff
thf(fact_1610_binomial__n__n,axiom,
! [N: nat] :
( ( binomial @ N @ N )
= ( one_one @ nat ) ) ).
% binomial_n_n
thf(fact_1611_lessThan__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_lessThan @ A @ X ) @ ( set_ord_lessThan @ A @ Y ) )
= ( ord_less_eq @ A @ X @ Y ) ) ) ).
% lessThan_subset_iff
thf(fact_1612_binomial__Suc__Suc,axiom,
! [N: nat,K: nat] :
( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
= ( plus_plus @ nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).
% binomial_Suc_Suc
thf(fact_1613_binomial__n__0,axiom,
! [N: nat] :
( ( binomial @ N @ ( zero_zero @ nat ) )
= ( one_one @ nat ) ) ).
% binomial_n_0
thf(fact_1614_sum_OlessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N ) ) @ ( G @ N ) ) ) ) ).
% sum.lessThan_Suc
thf(fact_1615_choose__one,axiom,
! [N: nat] :
( ( binomial @ N @ ( one_one @ nat ) )
= N ) ).
% choose_one
thf(fact_1616_lessThan__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_lessThan @ A )
= ( ^ [U2: A] :
( collect @ A
@ ^ [X3: A] : ( ord_less @ A @ X3 @ U2 ) ) ) ) ) ).
% lessThan_def
thf(fact_1617_choose__mult__lemma,axiom,
! [M2: nat,R: nat,K: nat] :
( ( times_times @ nat @ ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ M2 @ R ) @ K ) @ ( plus_plus @ nat @ M2 @ K ) ) @ ( binomial @ ( plus_plus @ nat @ M2 @ K ) @ K ) )
= ( times_times @ nat @ ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ M2 @ R ) @ K ) @ K ) @ ( binomial @ ( plus_plus @ nat @ M2 @ R ) @ M2 ) ) ) ).
% choose_mult_lemma
thf(fact_1618_lessThan__Suc__atMost,axiom,
! [K: nat] :
( ( set_ord_lessThan @ nat @ ( suc @ K ) )
= ( set_ord_atMost @ nat @ K ) ) ).
% lessThan_Suc_atMost
thf(fact_1619_bit__integer_Oabs__eq,axiom,
! [X: int] :
( ( bit_se5641148757651400278ts_bit @ code_integer @ ( code_integer_of_int @ X ) )
= ( bit_se5641148757651400278ts_bit @ int @ X ) ) ).
% bit_integer.abs_eq
thf(fact_1620_lessThan__strict__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M2: A,N: A] :
( ( ord_less @ ( set @ A ) @ ( set_ord_lessThan @ A @ M2 ) @ ( set_ord_lessThan @ A @ N ) )
= ( ord_less @ A @ M2 @ N ) ) ) ).
% lessThan_strict_subset_iff
thf(fact_1621_Suc__times__binomial__add,axiom,
! [A3: nat,B2: nat] :
( ( times_times @ nat @ ( suc @ A3 ) @ ( binomial @ ( suc @ ( plus_plus @ nat @ A3 @ B2 ) ) @ ( suc @ A3 ) ) )
= ( times_times @ nat @ ( suc @ B2 ) @ ( binomial @ ( suc @ ( plus_plus @ nat @ A3 @ B2 ) ) @ A3 ) ) ) ).
% Suc_times_binomial_add
thf(fact_1622_binomial__absorb__comp,axiom,
! [N: nat,K: nat] :
( ( times_times @ nat @ ( minus_minus @ nat @ N @ K ) @ ( binomial @ N @ K ) )
= ( times_times @ nat @ N @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ K ) ) ) ).
% binomial_absorb_comp
thf(fact_1623_fact__fact__dvd__fact,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: nat,N: nat] : ( dvd_dvd @ A @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_char_0_fact @ A @ N ) ) @ ( semiring_char_0_fact @ A @ ( plus_plus @ nat @ K @ N ) ) ) ) ).
% fact_fact_dvd_fact
thf(fact_1624_sum_Onat__diff__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ N @ ( suc @ I3 ) ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum.nat_diff_reindex
thf(fact_1625_Iic__subset__Iio__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ A3 ) @ ( set_ord_lessThan @ A @ B2 ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ).
% Iic_subset_Iio_iff
thf(fact_1626_binomial__absorption,axiom,
! [K: nat,N: nat] :
( ( times_times @ nat @ ( suc @ K ) @ ( binomial @ N @ ( suc @ K ) ) )
= ( times_times @ nat @ N @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ K ) ) ) ).
% binomial_absorption
thf(fact_1627_sum__diff__distrib,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Q: A > nat,P: A > nat,N: A] :
( ! [X2: A] : ( ord_less_eq @ nat @ ( Q @ X2 ) @ ( P @ X2 ) )
=> ( ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ P @ ( set_ord_lessThan @ A @ N ) ) @ ( groups7311177749621191930dd_sum @ A @ nat @ Q @ ( set_ord_lessThan @ A @ N ) ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X3: A] : ( minus_minus @ nat @ ( P @ X3 ) @ ( Q @ X3 ) )
@ ( set_ord_lessThan @ A @ N ) ) ) ) ) ).
% sum_diff_distrib
thf(fact_1628_sum_OlessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% sum.lessThan_Suc_shift
thf(fact_1629_sum__lessThan__telescope,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [F: nat > A,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N5: nat] : ( minus_minus @ A @ ( F @ ( suc @ N5 ) ) @ ( F @ N5 ) )
@ ( set_ord_lessThan @ nat @ M2 ) )
= ( minus_minus @ A @ ( F @ M2 ) @ ( F @ ( zero_zero @ nat ) ) ) ) ) ).
% sum_lessThan_telescope
thf(fact_1630_sum__lessThan__telescope_H,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [F: nat > A,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N5: nat] : ( minus_minus @ A @ ( F @ N5 ) @ ( F @ ( suc @ N5 ) ) )
@ ( set_ord_lessThan @ nat @ M2 ) )
= ( minus_minus @ A @ ( F @ ( zero_zero @ nat ) ) @ ( F @ M2 ) ) ) ) ).
% sum_lessThan_telescope'
thf(fact_1631_sum_OatLeast1__atMost__eq,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K4: nat] : ( G @ ( suc @ K4 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum.atLeast1_atMost_eq
thf(fact_1632_sum__bounds__lt__plus1,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F: nat > A,Mm: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K4: nat] : ( F @ ( suc @ K4 ) )
@ ( set_ord_lessThan @ nat @ Mm ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ Mm ) ) ) ) ).
% sum_bounds_lt_plus1
thf(fact_1633_sum_Onested__swap_H,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: nat > nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ ( A3 @ I3 ) @ ( set_ord_lessThan @ nat @ I3 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J2: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( A3 @ I3 @ J2 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J2 ) @ N ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum.nested_swap'
thf(fact_1634_sum__choose__lower,axiom,
! [R: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K4: nat] : ( binomial @ ( plus_plus @ nat @ R @ K4 ) @ K4 )
@ ( set_ord_atMost @ nat @ N ) )
= ( binomial @ ( suc @ ( plus_plus @ nat @ R @ N ) ) @ N ) ) ).
% sum_choose_lower
thf(fact_1635_choose__rising__sum_I2_J,axiom,
! [N: nat,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [J2: nat] : ( binomial @ ( plus_plus @ nat @ N @ J2 ) @ N )
@ ( set_ord_atMost @ nat @ M2 ) )
= ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ N @ M2 ) @ ( one_one @ nat ) ) @ M2 ) ) ).
% choose_rising_sum(2)
thf(fact_1636_choose__rising__sum_I1_J,axiom,
! [N: nat,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [J2: nat] : ( binomial @ ( plus_plus @ nat @ N @ J2 ) @ N )
@ ( set_ord_atMost @ nat @ M2 ) )
= ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ N @ M2 ) @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ).
% choose_rising_sum(1)
thf(fact_1637_binomial__code,axiom,
( binomial
= ( ^ [N5: nat,K4: nat] : ( if @ nat @ ( ord_less @ nat @ N5 @ K4 ) @ ( zero_zero @ nat ) @ ( if @ nat @ ( ord_less @ nat @ N5 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K4 ) ) @ ( binomial @ N5 @ ( minus_minus @ nat @ N5 @ K4 ) ) @ ( divide_divide @ nat @ ( set_fo6178422350223883121st_nat @ nat @ ( times_times @ nat ) @ ( plus_plus @ nat @ ( minus_minus @ nat @ N5 @ K4 ) @ ( one_one @ nat ) ) @ N5 @ ( one_one @ nat ) ) @ ( semiring_char_0_fact @ nat @ K4 ) ) ) ) ) ) ).
% binomial_code
thf(fact_1638_choose__reduce__nat,axiom,
! [N: nat,K: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( binomial @ N @ K )
= ( plus_plus @ nat @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ K ) ) ) ) ) ).
% choose_reduce_nat
thf(fact_1639_times__binomial__minus1__eq,axiom,
! [K: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( times_times @ nat @ K @ ( binomial @ N @ K ) )
= ( times_times @ nat @ N @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ) ).
% times_binomial_minus1_eq
thf(fact_1640_one__diff__power__eq,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat] :
( ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ N ) )
= ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% one_diff_power_eq
thf(fact_1641_power__diff__1__eq,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat] :
( ( minus_minus @ A @ ( power_power @ A @ X @ N ) @ ( one_one @ A ) )
= ( times_times @ A @ ( minus_minus @ A @ X @ ( one_one @ A ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% power_diff_1_eq
thf(fact_1642_geometric__sum,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,N: nat] :
( ( X
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ X @ N ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ X @ ( one_one @ A ) ) ) ) ) ) ).
% geometric_sum
thf(fact_1643_sum_OatMost__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ N ) )
= ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% sum.atMost_shift
thf(fact_1644_vandermonde,axiom,
! [M2: nat,N: nat,R: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K4: nat] : ( times_times @ nat @ ( binomial @ M2 @ K4 ) @ ( binomial @ N @ ( minus_minus @ nat @ R @ K4 ) ) )
@ ( set_ord_atMost @ nat @ R ) )
= ( binomial @ ( plus_plus @ nat @ M2 @ N ) @ R ) ) ).
% vandermonde
thf(fact_1645_binomial__addition__formula,axiom,
! [N: nat,K: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( binomial @ N @ ( suc @ K ) )
= ( plus_plus @ nat @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ K ) ) ) ) ).
% binomial_addition_formula
thf(fact_1646_power__diff__sumr2,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat,Y: A] :
( ( minus_minus @ A @ ( power_power @ A @ X @ N ) @ ( power_power @ A @ Y @ N ) )
= ( times_times @ A @ ( minus_minus @ A @ X @ Y )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( power_power @ A @ Y @ ( minus_minus @ nat @ N @ ( suc @ I3 ) ) ) @ ( power_power @ A @ X @ I3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% power_diff_sumr2
thf(fact_1647_diff__power__eq__sum,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat,Y: A] :
( ( minus_minus @ A @ ( power_power @ A @ X @ ( suc @ N ) ) @ ( power_power @ A @ Y @ ( suc @ N ) ) )
= ( times_times @ A @ ( minus_minus @ A @ X @ Y )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [P6: nat] : ( times_times @ A @ ( power_power @ A @ X @ P6 ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ N @ P6 ) ) )
@ ( set_ord_lessThan @ nat @ ( suc @ N ) ) ) ) ) ) ).
% diff_power_eq_sum
thf(fact_1648_sum__gp__strict,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat] :
( ( ( X
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N ) )
= ( semiring_1_of_nat @ A @ N ) ) )
& ( ( X
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ N ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ).
% sum_gp_strict
thf(fact_1649_binomial,axiom,
! [A3: nat,B2: nat,N: nat] :
( ( power_power @ nat @ ( plus_plus @ nat @ A3 @ B2 ) @ N )
= ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K4: nat] : ( times_times @ nat @ ( times_times @ nat @ ( semiring_1_of_nat @ nat @ ( binomial @ N @ K4 ) ) @ ( power_power @ nat @ A3 @ K4 ) ) @ ( power_power @ nat @ B2 @ ( minus_minus @ nat @ N @ K4 ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ).
% binomial
thf(fact_1650_choose__two,axiom,
! [N: nat] :
( ( binomial @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( divide_divide @ nat @ ( times_times @ nat @ N @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% choose_two
thf(fact_1651_one__diff__power__eq_H,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat] :
( ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ N ) )
= ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( power_power @ A @ X @ ( minus_minus @ nat @ N @ ( suc @ I3 ) ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% one_diff_power_eq'
thf(fact_1652_binomial__ring,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,B2: A,N: nat] :
( ( power_power @ A @ ( plus_plus @ A @ A3 @ B2 ) @ N )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K4: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K4 ) ) @ ( power_power @ A @ A3 @ K4 ) ) @ ( power_power @ A @ B2 @ ( minus_minus @ nat @ N @ K4 ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% binomial_ring
thf(fact_1653_xor__num_Osimps_I6_J,axiom,
! [M2: num,N: num] :
( ( bit_un2480387367778600638or_num @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
= ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_un2480387367778600638or_num @ M2 @ N ) ) ) ) ).
% xor_num.simps(6)
thf(fact_1654_xor__num_Osimps_I8_J,axiom,
! [M2: num,N: num] :
( ( bit_un2480387367778600638or_num @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
= ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_un2480387367778600638or_num @ M2 @ N ) ) ) ) ).
% xor_num.simps(8)
thf(fact_1655_choose__alternating__linear__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( N
!= ( one_one @ nat ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ I3 ) @ ( semiring_1_of_nat @ A @ I3 ) ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I3 ) ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) ) ).
% choose_alternating_linear_sum
thf(fact_1656_fact__Suc__0,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A @ ( suc @ ( zero_zero @ nat ) ) )
= ( one_one @ A ) ) ) ).
% fact_Suc_0
thf(fact_1657_fact__1,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A @ ( one_one @ nat ) )
= ( one_one @ A ) ) ) ).
% fact_1
thf(fact_1658_fact__0,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% fact_0
thf(fact_1659_fact__code,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N5: nat] : ( semiring_1_of_nat @ A @ ( set_fo6178422350223883121st_nat @ nat @ ( times_times @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 @ ( one_one @ nat ) ) ) ) ) ) ).
% fact_code
thf(fact_1660_fact__reduce,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( semiring_char_0_fact @ A @ N )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ) ).
% fact_reduce
thf(fact_1661_fact__num__eq__if,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [M: nat] :
( if @ A
@ ( M
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% fact_num_eq_if
thf(fact_1662_xor__num_Opelims,axiom,
! [X: num,Xa: num,Y: option @ num] :
( ( ( bit_un2480387367778600638or_num @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ X @ Xa ) )
=> ( ( ( X = one2 )
=> ( ( Xa = one2 )
=> ( ( Y
= ( none @ num ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ one2 @ one2 ) ) ) ) )
=> ( ( ( X = one2 )
=> ! [N3: num] :
( ( Xa
= ( bit0 @ N3 ) )
=> ( ( Y
= ( some @ num @ ( bit1 @ N3 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ one2 @ ( bit0 @ N3 ) ) ) ) ) )
=> ( ( ( X = one2 )
=> ! [N3: num] :
( ( Xa
= ( bit1 @ N3 ) )
=> ( ( Y
= ( some @ num @ ( bit0 @ N3 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ one2 @ ( bit1 @ N3 ) ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ( ( Xa = one2 )
=> ( ( Y
= ( some @ num @ ( bit1 @ M5 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M5 ) @ one2 ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ! [N3: num] :
( ( Xa
= ( bit0 @ N3 ) )
=> ( ( Y
= ( map_option @ num @ num @ bit0 @ ( bit_un2480387367778600638or_num @ M5 @ N3 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M5 ) @ ( bit0 @ N3 ) ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ! [N3: num] :
( ( Xa
= ( bit1 @ N3 ) )
=> ( ( Y
= ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_un2480387367778600638or_num @ M5 @ N3 ) ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M5 ) @ ( bit1 @ N3 ) ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ( ( Xa = one2 )
=> ( ( Y
= ( some @ num @ ( bit0 @ M5 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M5 ) @ one2 ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ! [N3: num] :
( ( Xa
= ( bit0 @ N3 ) )
=> ( ( Y
= ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_un2480387367778600638or_num @ M5 @ N3 ) ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M5 ) @ ( bit0 @ N3 ) ) ) ) ) )
=> ~ ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ! [N3: num] :
( ( Xa
= ( bit1 @ N3 ) )
=> ( ( Y
= ( map_option @ num @ num @ bit0 @ ( bit_un2480387367778600638or_num @ M5 @ N3 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M5 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% xor_num.pelims
thf(fact_1663_dvd__fact,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ M2 )
=> ( ( ord_less_eq @ nat @ M2 @ N )
=> ( dvd_dvd @ nat @ M2 @ ( semiring_char_0_fact @ nat @ N ) ) ) ) ).
% dvd_fact
thf(fact_1664_fact__ge__1,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat] : ( ord_less_eq @ A @ ( one_one @ A ) @ ( semiring_char_0_fact @ A @ N ) ) ) ).
% fact_ge_1
thf(fact_1665_pochhammer__double,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [Z2: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Z2 ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) @ ( comm_s3205402744901411588hammer @ A @ Z2 @ N ) ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z2 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ N ) ) ) ) ).
% pochhammer_double
thf(fact_1666_fact__double,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [N: nat] :
( ( semiring_char_0_fact @ A @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A @ ( times_times @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) @ ( comm_s3205402744901411588hammer @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N ) ) @ ( semiring_char_0_fact @ A @ N ) ) ) ) ).
% fact_double
thf(fact_1667_pochhammer__code,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( comm_s3205402744901411588hammer @ A )
= ( ^ [A7: A,N5: nat] :
( if @ A
@ ( N5
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( set_fo6178422350223883121st_nat @ A
@ ^ [O: nat] : ( times_times @ A @ ( plus_plus @ A @ A7 @ ( semiring_1_of_nat @ A @ O ) ) )
@ ( zero_zero @ nat )
@ ( minus_minus @ nat @ N5 @ ( one_one @ nat ) )
@ ( one_one @ A ) ) ) ) ) ) ).
% pochhammer_code
thf(fact_1668_xor__Suc__0__eq,axiom,
! [N: nat] :
( ( bit_se5824344971392196577ns_xor @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ N @ ( zero_neq_one_of_bool @ nat @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
@ ( zero_neq_one_of_bool @ nat
@ ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% xor_Suc_0_eq
thf(fact_1669_Suc__0__xor__eq,axiom,
! [N: nat] :
( ( bit_se5824344971392196577ns_xor @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
= ( minus_minus @ nat @ ( plus_plus @ nat @ N @ ( zero_neq_one_of_bool @ nat @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
@ ( zero_neq_one_of_bool @ nat
@ ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% Suc_0_xor_eq
thf(fact_1670_push__bit__numeral__minus__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: num] :
( ( bit_se4730199178511100633sh_bit @ A @ ( numeral_numeral @ nat @ N ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ N ) ) ) ) ) ).
% push_bit_numeral_minus_1
thf(fact_1671_num__of__integer__code,axiom,
( code_num_of_integer
= ( ^ [K4: code_integer] :
( if @ num @ ( ord_less_eq @ code_integer @ K4 @ ( one_one @ code_integer ) ) @ one2
@ ( product_case_prod @ code_integer @ code_integer @ num
@ ^ [L3: code_integer,J2: code_integer] :
( if @ num
@ ( J2
= ( zero_zero @ code_integer ) )
@ ( plus_plus @ num @ ( code_num_of_integer @ L3 ) @ ( code_num_of_integer @ L3 ) )
@ ( plus_plus @ num @ ( plus_plus @ num @ ( code_num_of_integer @ L3 ) @ ( code_num_of_integer @ L3 ) ) @ one2 ) )
@ ( code_divmod_integer @ K4 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% num_of_integer_code
thf(fact_1672_Sum__Ico__nat,axiom,
! [M2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X3: nat] : X3
@ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) )
= ( divide_divide @ nat @ ( minus_minus @ nat @ ( times_times @ nat @ N @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) @ ( times_times @ nat @ M2 @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% Sum_Ico_nat
thf(fact_1673_ivl__diff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [I: A,N: A,M2: A] :
( ( ord_less_eq @ A @ I @ N )
=> ( ( minus_minus @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ I @ M2 ) @ ( set_or7035219750837199246ssThan @ A @ I @ N ) )
= ( set_or7035219750837199246ssThan @ A @ N @ M2 ) ) ) ) ).
% ivl_diff
thf(fact_1674_push__bit__push__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M2: nat,N: nat,A3: A] :
( ( bit_se4730199178511100633sh_bit @ A @ M2 @ ( bit_se4730199178511100633sh_bit @ A @ N @ A3 ) )
= ( bit_se4730199178511100633sh_bit @ A @ ( plus_plus @ nat @ M2 @ N ) @ A3 ) ) ) ).
% push_bit_push_bit
thf(fact_1675_lessThan__minus__lessThan,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [N: A,M2: A] :
( ( minus_minus @ ( set @ A ) @ ( set_ord_lessThan @ A @ N ) @ ( set_ord_lessThan @ A @ M2 ) )
= ( set_or7035219750837199246ssThan @ A @ M2 @ N ) ) ) ).
% lessThan_minus_lessThan
thf(fact_1676_pochhammer__1,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( one_one @ nat ) )
= A3 ) ) ).
% pochhammer_1
thf(fact_1677_atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,L: A,U: A] :
( ( member @ A @ I @ ( set_or7035219750837199246ssThan @ A @ L @ U ) )
= ( ( ord_less_eq @ A @ L @ I )
& ( ord_less @ A @ I @ U ) ) ) ) ).
% atLeastLessThan_iff
thf(fact_1678_ivl__subset,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [I: A,J: A,M2: A,N: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ I @ J ) @ ( set_or7035219750837199246ssThan @ A @ M2 @ N ) )
= ( ( ord_less_eq @ A @ J @ I )
| ( ( ord_less_eq @ A @ M2 @ I )
& ( ord_less_eq @ A @ J @ N ) ) ) ) ) ).
% ivl_subset
thf(fact_1679_pochhammer__0,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% pochhammer_0
thf(fact_1680_xor__numerals_I1_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( numeral_numeral @ A @ ( bit1 @ Y ) ) ) ) ).
% xor_numerals(1)
thf(fact_1681_xor__numerals_I2_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( numeral_numeral @ A @ ( bit0 @ Y ) ) ) ) ).
% xor_numerals(2)
thf(fact_1682_xor__numerals_I5_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( bit1 @ X ) ) ) ) ).
% xor_numerals(5)
thf(fact_1683_xor__numerals_I8_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( bit0 @ X ) ) ) ) ).
% xor_numerals(8)
thf(fact_1684_push__bit__of__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( one_one @ A ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% push_bit_of_1
thf(fact_1685_sum_Oop__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [N: nat,M2: nat,G: nat > A] :
( ( ( ord_less @ nat @ N @ M2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ N ) ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) @ ( G @ N ) ) ) ) ) ) ).
% sum.op_ivl_Suc
thf(fact_1686_xor__numerals_I4_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% xor_numerals(4)
thf(fact_1687_xor__numerals_I6_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% xor_numerals(6)
thf(fact_1688_flip__bit__nat__def,axiom,
( ( bit_se8732182000553998342ip_bit @ nat )
= ( ^ [M: nat,N5: nat] : ( bit_se5824344971392196577ns_xor @ nat @ N5 @ ( bit_se4730199178511100633sh_bit @ nat @ M @ ( one_one @ nat ) ) ) ) ) ).
% flip_bit_nat_def
thf(fact_1689_push__bit__integer_Oabs__eq,axiom,
! [Xa: nat,X: int] :
( ( bit_se4730199178511100633sh_bit @ code_integer @ Xa @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( bit_se4730199178511100633sh_bit @ int @ Xa @ X ) ) ) ).
% push_bit_integer.abs_eq
thf(fact_1690_atLeastLessThan__inj_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ( set_or7035219750837199246ssThan @ A @ A3 @ B2 )
= ( set_or7035219750837199246ssThan @ A @ C2 @ D3 ) )
=> ( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ C2 @ D3 )
=> ( B2 = D3 ) ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_1691_atLeastLessThan__inj_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ( set_or7035219750837199246ssThan @ A @ A3 @ B2 )
= ( set_or7035219750837199246ssThan @ A @ C2 @ D3 ) )
=> ( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ C2 @ D3 )
=> ( A3 = C2 ) ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_1692_atLeastLessThan__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ C2 @ D3 )
=> ( ( ( set_or7035219750837199246ssThan @ A @ A3 @ B2 )
= ( set_or7035219750837199246ssThan @ A @ C2 @ D3 ) )
= ( ( A3 = C2 )
& ( B2 = D3 ) ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_1693_flip__bit__eq__xor,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se8732182000553998342ip_bit @ A )
= ( ^ [N5: nat,A7: A] : ( bit_se5824344971392196577ns_xor @ A @ A7 @ ( bit_se4730199178511100633sh_bit @ A @ N5 @ ( one_one @ A ) ) ) ) ) ) ).
% flip_bit_eq_xor
thf(fact_1694_push__bit__add,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A,B2: A] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( plus_plus @ A @ A3 @ B2 ) )
= ( plus_plus @ A @ ( bit_se4730199178511100633sh_bit @ A @ N @ A3 ) @ ( bit_se4730199178511100633sh_bit @ A @ N @ B2 ) ) ) ) ).
% push_bit_add
thf(fact_1695_atLeastLessThan__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ A3 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D3 ) )
=> ( ( ord_less_eq @ A @ B2 @ A3 )
| ( ( ord_less_eq @ A @ C2 @ A3 )
& ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_1696_atLeastLessThanSuc__atLeastAtMost,axiom,
! [L: nat,U: nat] :
( ( set_or7035219750837199246ssThan @ nat @ L @ ( suc @ U ) )
= ( set_or1337092689740270186AtMost @ nat @ L @ U ) ) ).
% atLeastLessThanSuc_atLeastAtMost
thf(fact_1697_lessThan__atLeast0,axiom,
( ( set_ord_lessThan @ nat )
= ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) ) ) ).
% lessThan_atLeast0
thf(fact_1698_push__bit__take__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M2: nat,N: nat,A3: A] :
( ( bit_se4730199178511100633sh_bit @ A @ M2 @ ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) )
= ( bit_se2584673776208193580ke_bit @ A @ ( plus_plus @ nat @ M2 @ N ) @ ( bit_se4730199178511100633sh_bit @ A @ M2 @ A3 ) ) ) ) ).
% push_bit_take_bit
thf(fact_1699_pochhammer__fact,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( comm_semiring_1 @ A ) )
=> ( ( semiring_char_0_fact @ A )
= ( comm_s3205402744901411588hammer @ A @ ( one_one @ A ) ) ) ) ).
% pochhammer_fact
thf(fact_1700_sum_Oshift__bounds__Suc__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% sum.shift_bounds_Suc_ivl
thf(fact_1701_sum_Oshift__bounds__nat__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M2: nat,K: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M2 @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G @ ( plus_plus @ nat @ I3 @ K ) )
@ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% sum.shift_bounds_nat_ivl
thf(fact_1702_sum_Oivl__cong,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( comm_monoid_add @ A ) )
=> ! [A3: B,C2: B,B2: B,D3: B,G: B > A,H3: B > A] :
( ( A3 = C2 )
=> ( ( B2 = D3 )
=> ( ! [X2: B] :
( ( ord_less_eq @ B @ C2 @ X2 )
=> ( ( ord_less @ B @ X2 @ D3 )
=> ( ( G @ X2 )
= ( H3 @ X2 ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( set_or7035219750837199246ssThan @ B @ A3 @ B2 ) )
= ( groups7311177749621191930dd_sum @ B @ A @ H3 @ ( set_or7035219750837199246ssThan @ B @ C2 @ D3 ) ) ) ) ) ) ) ).
% sum.ivl_cong
thf(fact_1703_sum_OatLeastLessThan__concat,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M2: nat,N: nat,P5: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( ord_less_eq @ nat @ N @ P5 )
=> ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ P5 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ P5 ) ) ) ) ) ) ).
% sum.atLeastLessThan_concat
thf(fact_1704_sum__diff__nat__ivl,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M2: nat,N: nat,P5: nat,F: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( ord_less_eq @ nat @ N @ P5 )
=> ( ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or7035219750837199246ssThan @ nat @ M2 @ P5 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or7035219750837199246ssThan @ nat @ N @ P5 ) ) ) ) ) ) ).
% sum_diff_nat_ivl
thf(fact_1705_pochhammer__0__left,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [N: nat] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( comm_s3205402744901411588hammer @ A @ ( zero_zero @ A ) @ N )
= ( one_one @ A ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( comm_s3205402744901411588hammer @ A @ ( zero_zero @ A ) @ N )
= ( zero_zero @ A ) ) ) ) ) ).
% pochhammer_0_left
thf(fact_1706_sum_OatLeast__Suc__lessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% sum.atLeast_Suc_lessThan_Suc_shift
thf(fact_1707_sum_OatLeastLessThan__shift__bounds,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M2: nat,K: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M2 @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ K ) ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% sum.atLeastLessThan_shift_bounds
thf(fact_1708_concat__bit__eq,axiom,
( bit_concat_bit
= ( ^ [N5: nat,K4: int,L3: int] : ( plus_plus @ int @ ( bit_se2584673776208193580ke_bit @ int @ N5 @ K4 ) @ ( bit_se4730199178511100633sh_bit @ int @ N5 @ L3 ) ) ) ) ).
% concat_bit_eq
thf(fact_1709_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D3 ) )
= ( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A3 )
& ( ord_less @ A @ B2 @ D3 ) ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_1710_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ A3 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D3 ) )
= ( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A3 )
& ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% atLeastLessThan_subseteq_atLeastAtMost_iff
thf(fact_1711_sum__shift__lb__Suc0__0__upt,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F: nat > A,K: nat] :
( ( ( F @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ) ).
% sum_shift_lb_Suc0_0_upt
thf(fact_1712_sum_OatLeast0__lessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G @ N ) ) ) ) ).
% sum.atLeast0_lessThan_Suc
thf(fact_1713_sum_OatLeast__Suc__lessThan,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M2: nat,N: nat,G: nat > A] :
( ( ord_less @ nat @ M2 @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) )
= ( plus_plus @ A @ ( G @ M2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ N ) ) ) ) ) ) ).
% sum.atLeast_Suc_lessThan
thf(fact_1714_sum_OatLeastLessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: nat,B2: nat,G: nat > A] :
( ( ord_less_eq @ nat @ A3 @ B2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A3 @ ( suc @ B2 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A3 @ B2 ) ) @ ( G @ B2 ) ) ) ) ) ).
% sum.atLeastLessThan_Suc
thf(fact_1715_sum_Olast__plus,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M2: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( plus_plus @ A @ ( G @ N ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ) ) ).
% sum.last_plus
thf(fact_1716_pochhammer__rec,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ N ) )
= ( times_times @ A @ A3 @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ N ) ) ) ) ).
% pochhammer_rec
thf(fact_1717_pochhammer__Suc,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ N ) )
= ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ A3 @ N ) @ ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ N ) ) ) ) ) ).
% pochhammer_Suc
thf(fact_1718_pochhammer__rec_H,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [Z2: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ Z2 @ ( suc @ N ) )
= ( times_times @ A @ ( plus_plus @ A @ Z2 @ ( semiring_1_of_nat @ A @ N ) ) @ ( comm_s3205402744901411588hammer @ A @ Z2 @ N ) ) ) ) ).
% pochhammer_rec'
thf(fact_1719_pochhammer__product_H,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [Z2: A,N: nat,M2: nat] :
( ( comm_s3205402744901411588hammer @ A @ Z2 @ ( plus_plus @ nat @ N @ M2 ) )
= ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z2 @ N ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z2 @ ( semiring_1_of_nat @ A @ N ) ) @ M2 ) ) ) ) ).
% pochhammer_product'
thf(fact_1720_bit__iff__and__push__bit__not__eq__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A7: A,N5: nat] :
( ( bit_se5824344872417868541ns_and @ A @ A7 @ ( bit_se4730199178511100633sh_bit @ A @ N5 @ ( one_one @ A ) ) )
!= ( zero_zero @ A ) ) ) ) ) ).
% bit_iff_and_push_bit_not_eq_0
thf(fact_1721_sum__Suc__diff_H,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M2: nat,N: nat,F: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( minus_minus @ A @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) )
= ( minus_minus @ A @ ( F @ N ) @ ( F @ M2 ) ) ) ) ) ).
% sum_Suc_diff'
thf(fact_1722_sum_OatLeastLessThan__rev,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ M2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ ( suc @ I3 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ N @ M2 ) ) ) ) ).
% sum.atLeastLessThan_rev
thf(fact_1723_sum_Onested__swap,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: nat > nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ ( A3 @ I3 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J2: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( A3 @ I3 @ J2 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J2 ) @ N ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% sum.nested_swap
thf(fact_1724_sum_Onat__group,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,K: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [M: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ M @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ M @ K ) @ K ) ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ N @ K ) ) ) ) ) ).
% sum.nat_group
thf(fact_1725_numeral__xor__num,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M2: num,N: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N ) )
= ( case_option @ A @ num @ ( zero_zero @ A ) @ ( numeral_numeral @ A ) @ ( bit_un2480387367778600638or_num @ M2 @ N ) ) ) ) ).
% numeral_xor_num
thf(fact_1726_sum_Ohead__if,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [N: nat,M2: nat,G: nat > A] :
( ( ( ord_less @ nat @ N @ M2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) @ ( G @ N ) ) ) ) ) ) ).
% sum.head_if
thf(fact_1727_sum_OatLeast0__lessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% sum.atLeast0_lessThan_Suc_shift
thf(fact_1728_sum_OatLeastLessThan__shift__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M2 ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ).
% sum.atLeastLessThan_shift_0
thf(fact_1729_pochhammer__product,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [M2: nat,N: nat,Z2: A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( comm_s3205402744901411588hammer @ A @ Z2 @ N )
= ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z2 @ M2 ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z2 @ ( semiring_1_of_nat @ A @ M2 ) ) @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ) ).
% pochhammer_product
thf(fact_1730_sum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ M2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M2 ) ) ) ) ).
% sum.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_1731_sum_OatLeast__lessThan__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ( comp @ nat @ A @ nat @ G
@ ^ [N5: nat] : ( minus_minus @ nat @ N5 @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% sum.atLeast_lessThan_pred_shift
thf(fact_1732_pochhammer__absorb__comp,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [R: A,K: nat] :
( ( times_times @ A @ ( minus_minus @ A @ R @ ( semiring_1_of_nat @ A @ K ) ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ R ) @ K ) )
= ( times_times @ A @ R @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ R ) @ ( one_one @ A ) ) @ K ) ) ) ) ).
% pochhammer_absorb_comp
thf(fact_1733_pochhammer__same,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( comm_ring_1 @ A )
& ( semiri3467727345109120633visors @ A ) )
=> ! [N: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N ) ) @ N )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( semiring_char_0_fact @ A @ N ) ) ) ) ).
% pochhammer_same
thf(fact_1734_pochhammer__minus,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [B2: A,K: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ B2 ) @ K )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ B2 @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ K ) ) ) ) ).
% pochhammer_minus
thf(fact_1735_pochhammer__minus_H,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [B2: A,K: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ B2 @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ K )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ B2 ) @ K ) ) ) ) ).
% pochhammer_minus'
thf(fact_1736_gbinomial__pochhammer_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A7: A,K4: nat] : ( divide_divide @ A @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ A7 @ ( semiring_1_of_nat @ A @ K4 ) ) @ ( one_one @ A ) ) @ K4 ) @ ( semiring_char_0_fact @ A @ K4 ) ) ) ) ) ).
% gbinomial_pochhammer'
thf(fact_1737_gbinomial__pochhammer,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A7: A,K4: nat] : ( divide_divide @ A @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K4 ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ A7 ) @ K4 ) ) @ ( semiring_char_0_fact @ A @ K4 ) ) ) ) ) ).
% gbinomial_pochhammer
thf(fact_1738_xor__nat__unfold,axiom,
( ( bit_se5824344971392196577ns_xor @ nat )
= ( ^ [M: nat,N5: nat] :
( if @ nat
@ ( M
= ( zero_zero @ nat ) )
@ N5
@ ( if @ nat
@ ( N5
= ( zero_zero @ nat ) )
@ M
@ ( plus_plus @ nat @ ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( modulo_modulo @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ nat @ ( divide_divide @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% xor_nat_unfold
thf(fact_1739_num__of__integer_Oabs__eq,axiom,
! [X: int] :
( ( code_num_of_integer @ ( code_integer_of_int @ X ) )
= ( num_of_nat @ ( nat2 @ X ) ) ) ).
% num_of_integer.abs_eq
thf(fact_1740_signed__take__bit__code,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4674362597316999326ke_bit @ A )
= ( ^ [N5: nat,A7: A] : ( if @ A @ ( bit_se5641148757651400278ts_bit @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N5 ) @ A7 ) @ N5 ) @ ( plus_plus @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N5 ) @ A7 ) @ ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N5 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N5 ) @ A7 ) ) ) ) ) ).
% signed_take_bit_code
thf(fact_1741_xor__nat__rec,axiom,
( ( bit_se5824344971392196577ns_xor @ nat )
= ( ^ [M: nat,N5: nat] :
( plus_plus @ nat
@ ( zero_neq_one_of_bool @ nat
@ ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
!= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) ) ) )
@ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ nat @ ( divide_divide @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% xor_nat_rec
thf(fact_1742_xor__one__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5824344971392196577ns_xor @ A @ A3 @ ( one_one @ A ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A3 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) )
@ ( zero_neq_one_of_bool @ A
@ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ) ).
% xor_one_eq
thf(fact_1743_one__xor__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5824344971392196577ns_xor @ A @ ( one_one @ A ) @ A3 )
= ( minus_minus @ A @ ( plus_plus @ A @ A3 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) )
@ ( zero_neq_one_of_bool @ A
@ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ) ).
% one_xor_eq
thf(fact_1744_pochhammer__binomial__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [A3: A,B2: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ A3 @ B2 ) @ N )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K4: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K4 ) ) @ ( comm_s3205402744901411588hammer @ A @ A3 @ K4 ) ) @ ( comm_s3205402744901411588hammer @ A @ B2 @ ( minus_minus @ nat @ N @ K4 ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% pochhammer_binomial_sum
thf(fact_1745_divmod__integer__def,axiom,
( code_divmod_integer
= ( ^ [K4: code_integer,L3: code_integer] : ( product_Pair @ code_integer @ code_integer @ ( divide_divide @ code_integer @ K4 @ L3 ) @ ( modulo_modulo @ code_integer @ K4 @ L3 ) ) ) ) ).
% divmod_integer_def
thf(fact_1746_sum__power2,axiom,
! [K: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) )
= ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K ) @ ( one_one @ nat ) ) ) ).
% sum_power2
thf(fact_1747_nat__of__integer__code,axiom,
( code_nat_of_integer
= ( ^ [K4: code_integer] :
( if @ nat @ ( ord_less_eq @ code_integer @ K4 @ ( zero_zero @ code_integer ) ) @ ( zero_zero @ nat )
@ ( product_case_prod @ code_integer @ code_integer @ nat
@ ^ [L3: code_integer,J2: code_integer] :
( if @ nat
@ ( J2
= ( zero_zero @ code_integer ) )
@ ( plus_plus @ nat @ ( code_nat_of_integer @ L3 ) @ ( code_nat_of_integer @ L3 ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( code_nat_of_integer @ L3 ) @ ( code_nat_of_integer @ L3 ) ) @ ( one_one @ nat ) ) )
@ ( code_divmod_integer @ K4 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% nat_of_integer_code
thf(fact_1748_pochhammer__times__pochhammer__half,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [Z2: A,N: nat] :
( ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z2 @ ( suc @ N ) ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z2 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K4: nat] : ( plus_plus @ A @ Z2 @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ K4 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ nat ) ) ) ) ) ) ).
% pochhammer_times_pochhammer_half
thf(fact_1749_int__of__integer__code,axiom,
( code_int_of_integer
= ( ^ [K4: code_integer] :
( if @ int @ ( ord_less @ code_integer @ K4 @ ( zero_zero @ code_integer ) ) @ ( uminus_uminus @ int @ ( code_int_of_integer @ ( uminus_uminus @ code_integer @ K4 ) ) )
@ ( if @ int
@ ( K4
= ( zero_zero @ code_integer ) )
@ ( zero_zero @ int )
@ ( product_case_prod @ code_integer @ code_integer @ int
@ ^ [L3: code_integer,J2: code_integer] :
( if @ int
@ ( J2
= ( zero_zero @ code_integer ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( code_int_of_integer @ L3 ) )
@ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( code_int_of_integer @ L3 ) ) @ ( one_one @ int ) ) )
@ ( code_divmod_integer @ K4 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% int_of_integer_code
thf(fact_1750_divmod__integer__eq__cases,axiom,
( code_divmod_integer
= ( ^ [K4: code_integer,L3: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( K4
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) )
@ ( if @ ( product_prod @ code_integer @ code_integer )
@ ( L3
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ K4 )
@ ( comp @ code_integer @ ( ( product_prod @ code_integer @ code_integer ) > ( product_prod @ code_integer @ code_integer ) ) @ code_integer @ ( comp @ ( code_integer > code_integer ) @ ( ( 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 ) @ L3
@ ( if @ ( product_prod @ code_integer @ code_integer )
@ ( ( sgn_sgn @ code_integer @ K4 )
= ( sgn_sgn @ code_integer @ L3 ) )
@ ( code_divmod_abs @ K4 @ L3 )
@ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [R3: code_integer,S4: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( S4
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R3 ) @ ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R3 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ ( abs_abs @ code_integer @ L3 ) @ S4 ) ) )
@ ( code_divmod_abs @ K4 @ L3 ) ) ) ) ) ) ) ) ).
% divmod_integer_eq_cases
thf(fact_1751_time__def,axiom,
! [A: $tType] :
( ( time_time @ A )
= ( ^ [C3: heap_Time_Heap @ A,H: heap_ext @ product_unit] :
( case_option @ nat @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( zero_zero @ nat )
@ ( product_case_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ nat
@ ^ [X3: A] :
( product_case_prod @ ( heap_ext @ product_unit ) @ nat @ nat
@ ^ [Xa4: heap_ext @ product_unit,T3: nat] : T3 ) )
@ ( heap_Time_execute @ A @ C3 @ H ) ) ) ) ).
% time_def
thf(fact_1752_bit__cut__integer__def,axiom,
( code_bit_cut_integer
= ( ^ [K4: code_integer] :
( product_Pair @ code_integer @ $o @ ( divide_divide @ code_integer @ K4 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) )
@ ~ ( dvd_dvd @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ K4 ) ) ) ) ).
% bit_cut_integer_def
thf(fact_1753_integer__of__num_I3_J,axiom,
! [N: num] :
( ( code_integer_of_num @ ( bit1 @ N ) )
= ( plus_plus @ code_integer @ ( plus_plus @ code_integer @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) @ ( one_one @ code_integer ) ) ) ).
% integer_of_num(3)
thf(fact_1754_less__eq__int_Orep__eq,axiom,
( ( ord_less_eq @ int )
= ( ^ [X3: int,Xa4: int] :
( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [Y3: nat,Z5: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V2: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ Y3 @ V2 ) @ ( plus_plus @ nat @ U2 @ Z5 ) ) )
@ ( rep_Integ @ X3 )
@ ( rep_Integ @ Xa4 ) ) ) ) ).
% less_eq_int.rep_eq
thf(fact_1755_nat__of__integer__of__nat,axiom,
! [N: nat] :
( ( code_nat_of_integer @ ( semiring_1_of_nat @ code_integer @ N ) )
= N ) ).
% nat_of_integer_of_nat
thf(fact_1756_int__of__integer__of__nat,axiom,
! [N: nat] :
( ( code_int_of_integer @ ( semiring_1_of_nat @ code_integer @ N ) )
= ( semiring_1_of_nat @ int @ N ) ) ).
% int_of_integer_of_nat
thf(fact_1757_integer__of__int__int__of__integer,axiom,
! [K: code_integer] :
( ( code_integer_of_int @ ( code_int_of_integer @ K ) )
= K ) ).
% integer_of_int_int_of_integer
thf(fact_1758_int__of__integer__integer__of__int,axiom,
! [K: int] :
( ( code_int_of_integer @ ( code_integer_of_int @ K ) )
= K ) ).
% int_of_integer_integer_of_int
thf(fact_1759_int__of__integer__inverse,axiom,
! [X: code_integer] :
( ( code_integer_of_int @ ( code_int_of_integer @ X ) )
= X ) ).
% int_of_integer_inverse
thf(fact_1760_apsnd__conv,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > B,X: A,Y: C] :
( ( product_apsnd @ C @ B @ A @ F @ ( product_Pair @ A @ C @ X @ Y ) )
= ( product_Pair @ A @ B @ X @ ( F @ Y ) ) ) ).
% apsnd_conv
thf(fact_1761_prod_Oneutral__const,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [Uu: B] : ( one_one @ A )
@ A4 )
= ( one_one @ A ) ) ) ).
% prod.neutral_const
thf(fact_1762_of__int__integer__of,axiom,
! [K: code_integer] :
( ( ring_1_of_int @ code_integer @ ( code_int_of_integer @ K ) )
= K ) ).
% of_int_integer_of
thf(fact_1763_int__of__integer__of__int,axiom,
! [K: int] :
( ( code_int_of_integer @ ( ring_1_of_int @ code_integer @ K ) )
= K ) ).
% int_of_integer_of_int
thf(fact_1764_zero__integer_Orep__eq,axiom,
( ( code_int_of_integer @ ( zero_zero @ code_integer ) )
= ( zero_zero @ int ) ) ).
% zero_integer.rep_eq
thf(fact_1765_int__of__integer__numeral,axiom,
! [K: num] :
( ( code_int_of_integer @ ( numeral_numeral @ code_integer @ K ) )
= ( numeral_numeral @ int @ K ) ) ).
% int_of_integer_numeral
thf(fact_1766_plus__integer_Orep__eq,axiom,
! [X: code_integer,Xa: code_integer] :
( ( code_int_of_integer @ ( plus_plus @ code_integer @ X @ Xa ) )
= ( plus_plus @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% plus_integer.rep_eq
thf(fact_1767_one__integer_Orep__eq,axiom,
( ( code_int_of_integer @ ( one_one @ code_integer ) )
= ( one_one @ int ) ) ).
% one_integer.rep_eq
thf(fact_1768_uminus__integer_Orep__eq,axiom,
! [X: code_integer] :
( ( code_int_of_integer @ ( uminus_uminus @ code_integer @ X ) )
= ( uminus_uminus @ int @ ( code_int_of_integer @ X ) ) ) ).
% uminus_integer.rep_eq
thf(fact_1769_times__integer_Orep__eq,axiom,
! [X: code_integer,Xa: code_integer] :
( ( code_int_of_integer @ ( times_times @ code_integer @ X @ Xa ) )
= ( times_times @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% times_integer.rep_eq
thf(fact_1770_minus__integer_Orep__eq,axiom,
! [X: code_integer,Xa: code_integer] :
( ( code_int_of_integer @ ( minus_minus @ code_integer @ X @ Xa ) )
= ( minus_minus @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% minus_integer.rep_eq
thf(fact_1771_abs__integer_Orep__eq,axiom,
! [X: code_integer] :
( ( code_int_of_integer @ ( abs_abs @ code_integer @ X ) )
= ( abs_abs @ int @ ( code_int_of_integer @ X ) ) ) ).
% abs_integer.rep_eq
thf(fact_1772_divide__integer_Orep__eq,axiom,
! [X: code_integer,Xa: code_integer] :
( ( code_int_of_integer @ ( divide_divide @ code_integer @ X @ Xa ) )
= ( divide_divide @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% divide_integer.rep_eq
thf(fact_1773_modulo__integer_Orep__eq,axiom,
! [X: code_integer,Xa: code_integer] :
( ( code_int_of_integer @ ( modulo_modulo @ code_integer @ X @ Xa ) )
= ( modulo_modulo @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% modulo_integer.rep_eq
thf(fact_1774_sgn__integer_Orep__eq,axiom,
! [X: code_integer] :
( ( code_int_of_integer @ ( sgn_sgn @ code_integer @ X ) )
= ( sgn_sgn @ int @ ( code_int_of_integer @ X ) ) ) ).
% sgn_integer.rep_eq
thf(fact_1775_prod_OatMost__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ).
% prod.atMost_Suc
thf(fact_1776_prod_OlessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N ) ) @ ( G @ N ) ) ) ) ).
% prod.lessThan_Suc
thf(fact_1777_nat__of__integer__non__positive,axiom,
! [K: code_integer] :
( ( ord_less_eq @ code_integer @ K @ ( zero_zero @ code_integer ) )
=> ( ( code_nat_of_integer @ K )
= ( zero_zero @ nat ) ) ) ).
% nat_of_integer_non_positive
thf(fact_1778_prod_Ocl__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [N: nat,M2: nat,G: nat > A] :
( ( ( ord_less @ nat @ ( suc @ N ) @ M2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) ) )
= ( one_one @ A ) ) )
& ( ~ ( ord_less @ nat @ ( suc @ N ) @ M2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ) ).
% prod.cl_ivl_Suc
thf(fact_1779_prod_Oop__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [N: nat,M2: nat,G: nat > A] :
( ( ( ord_less @ nat @ N @ M2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ N ) ) )
= ( one_one @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) @ ( G @ N ) ) ) ) ) ) ).
% prod.op_ivl_Suc
thf(fact_1780_xor__integer_Orep__eq,axiom,
! [X: code_integer,Xa: code_integer] :
( ( code_int_of_integer @ ( bit_se5824344971392196577ns_xor @ code_integer @ X @ Xa ) )
= ( bit_se5824344971392196577ns_xor @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% xor_integer.rep_eq
thf(fact_1781_push__bit__integer_Orep__eq,axiom,
! [X: nat,Xa: code_integer] :
( ( code_int_of_integer @ ( bit_se4730199178511100633sh_bit @ code_integer @ X @ Xa ) )
= ( bit_se4730199178511100633sh_bit @ int @ X @ ( code_int_of_integer @ Xa ) ) ) ).
% push_bit_integer.rep_eq
thf(fact_1782_xor__integer_Oabs__eq,axiom,
! [Xa: int,X: int] :
( ( bit_se5824344971392196577ns_xor @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( bit_se5824344971392196577ns_xor @ int @ Xa @ X ) ) ) ).
% xor_integer.abs_eq
thf(fact_1783_nat__of__integer_Orep__eq,axiom,
( code_nat_of_integer
= ( ^ [X3: code_integer] : ( nat2 @ ( code_int_of_integer @ X3 ) ) ) ) ).
% nat_of_integer.rep_eq
thf(fact_1784_integer__eq__iff,axiom,
( ( ^ [Y5: code_integer,Z4: code_integer] : Y5 = Z4 )
= ( ^ [K4: code_integer,L3: code_integer] :
( ( code_int_of_integer @ K4 )
= ( code_int_of_integer @ L3 ) ) ) ) ).
% integer_eq_iff
thf(fact_1785_integer__eqI,axiom,
! [K: code_integer,L: code_integer] :
( ( ( code_int_of_integer @ K )
= ( code_int_of_integer @ L ) )
=> ( K = L ) ) ).
% integer_eqI
thf(fact_1786_int__of__integer__inject,axiom,
! [X: code_integer,Y: code_integer] :
( ( ( code_int_of_integer @ X )
= ( code_int_of_integer @ Y ) )
= ( X = Y ) ) ).
% int_of_integer_inject
thf(fact_1787_prod_Onot__neutral__contains__not__neutral,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: B > A,A4: set @ B] :
( ( ( groups7121269368397514597t_prod @ B @ A @ G @ A4 )
!= ( one_one @ A ) )
=> ~ ! [A5: B] :
( ( member @ B @ A5 @ A4 )
=> ( ( G @ A5 )
= ( one_one @ A ) ) ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_1788_prod_Oneutral,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,G: B > A] :
( ! [X2: B] :
( ( member @ B @ X2 @ A4 )
=> ( ( G @ X2 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A4 )
= ( one_one @ A ) ) ) ) ).
% prod.neutral
thf(fact_1789_apsnd__compose,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,F: C > B,G: D > C,X: product_prod @ A @ D] :
( ( product_apsnd @ C @ B @ A @ F @ ( product_apsnd @ D @ C @ A @ G @ X ) )
= ( product_apsnd @ D @ B @ A @ ( comp @ C @ B @ D @ F @ G ) @ X ) ) ).
% apsnd_compose
thf(fact_1790_prod__ge__1,axiom,
! [A: $tType,B: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A4: set @ B,F: B > A] :
( ! [X2: B] :
( ( member @ B @ X2 @ A4 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( F @ X2 ) ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F @ A4 ) ) ) ) ).
% prod_ge_1
thf(fact_1791_int__of__integer__less__iff,axiom,
! [X: code_integer,Y: code_integer] :
( ( ord_less @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Y ) )
= ( ord_less @ code_integer @ X @ Y ) ) ).
% int_of_integer_less_iff
thf(fact_1792_integer__less__iff,axiom,
( ( ord_less @ code_integer )
= ( ^ [K4: code_integer,L3: code_integer] : ( ord_less @ int @ ( code_int_of_integer @ K4 ) @ ( code_int_of_integer @ L3 ) ) ) ) ).
% integer_less_iff
thf(fact_1793_less__integer_Orep__eq,axiom,
( ( ord_less @ code_integer )
= ( ^ [X3: code_integer,Xa4: code_integer] : ( ord_less @ int @ ( code_int_of_integer @ X3 ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).
% less_integer.rep_eq
thf(fact_1794_less__eq__integer_Orep__eq,axiom,
( ( ord_less_eq @ code_integer )
= ( ^ [X3: code_integer,Xa4: code_integer] : ( ord_less_eq @ int @ ( code_int_of_integer @ X3 ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).
% less_eq_integer.rep_eq
thf(fact_1795_integer__less__eq__iff,axiom,
( ( ord_less_eq @ code_integer )
= ( ^ [K4: code_integer,L3: code_integer] : ( ord_less_eq @ int @ ( code_int_of_integer @ K4 ) @ ( code_int_of_integer @ L3 ) ) ) ) ).
% integer_less_eq_iff
thf(fact_1796_prod_Oshift__bounds__cl__Suc__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M2: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% prod.shift_bounds_cl_Suc_ivl
thf(fact_1797_prod_Oshift__bounds__Suc__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M2: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% prod.shift_bounds_Suc_ivl
thf(fact_1798_prod_Oshift__bounds__cl__nat__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M2: nat,K: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M2 @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G @ ( plus_plus @ nat @ I3 @ K ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% prod.shift_bounds_cl_nat_ivl
thf(fact_1799_prod_Oshift__bounds__nat__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M2: nat,K: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M2 @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G @ ( plus_plus @ nat @ I3 @ K ) )
@ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% prod.shift_bounds_nat_ivl
thf(fact_1800_atLeastLessThanPlusOne__atLeastAtMost__integer,axiom,
! [L: code_integer,U: code_integer] :
( ( set_or7035219750837199246ssThan @ code_integer @ L @ ( plus_plus @ code_integer @ U @ ( one_one @ code_integer ) ) )
= ( set_or1337092689740270186AtMost @ code_integer @ L @ U ) ) ).
% atLeastLessThanPlusOne_atLeastAtMost_integer
thf(fact_1801_prod__le__1,axiom,
! [B: $tType,A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A4: set @ B,F: B > A] :
( ! [X2: B] :
( ( member @ B @ X2 @ A4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ X2 ) )
& ( ord_less_eq @ A @ ( F @ X2 ) @ ( one_one @ A ) ) ) )
=> ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A4 ) @ ( one_one @ A ) ) ) ) ).
% prod_le_1
thf(fact_1802_prod_Oivl__cong,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( comm_monoid_mult @ A ) )
=> ! [A3: B,C2: B,B2: B,D3: B,G: B > A,H3: B > A] :
( ( A3 = C2 )
=> ( ( B2 = D3 )
=> ( ! [X2: B] :
( ( ord_less_eq @ B @ C2 @ X2 )
=> ( ( ord_less @ B @ X2 @ D3 )
=> ( ( G @ X2 )
= ( H3 @ X2 ) ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( set_or7035219750837199246ssThan @ B @ A3 @ B2 ) )
= ( groups7121269368397514597t_prod @ B @ A @ H3 @ ( set_or7035219750837199246ssThan @ B @ C2 @ D3 ) ) ) ) ) ) ) ).
% prod.ivl_cong
thf(fact_1803_take__bit__integer_Orep__eq,axiom,
! [X: nat,Xa: code_integer] :
( ( code_int_of_integer @ ( bit_se2584673776208193580ke_bit @ code_integer @ X @ Xa ) )
= ( bit_se2584673776208193580ke_bit @ int @ X @ ( code_int_of_integer @ Xa ) ) ) ).
% take_bit_integer.rep_eq
thf(fact_1804_and__integer_Orep__eq,axiom,
! [X: code_integer,Xa: code_integer] :
( ( code_int_of_integer @ ( bit_se5824344872417868541ns_and @ code_integer @ X @ Xa ) )
= ( bit_se5824344872417868541ns_and @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% and_integer.rep_eq
thf(fact_1805_bit__integer_Orep__eq,axiom,
( ( bit_se5641148757651400278ts_bit @ code_integer )
= ( ^ [X3: code_integer] : ( bit_se5641148757651400278ts_bit @ int @ ( code_int_of_integer @ X3 ) ) ) ) ).
% bit_integer.rep_eq
thf(fact_1806_prod_OatLeastLessThan__concat,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M2: nat,N: nat,P5: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( ord_less_eq @ nat @ N @ P5 )
=> ( ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ P5 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ P5 ) ) ) ) ) ) ).
% prod.atLeastLessThan_concat
thf(fact_1807_mask__integer_Orep__eq,axiom,
! [X: nat] :
( ( code_int_of_integer @ ( bit_se2239418461657761734s_mask @ code_integer @ X ) )
= ( bit_se2239418461657761734s_mask @ int @ X ) ) ).
% mask_integer.rep_eq
thf(fact_1808_integer__of__num__def,axiom,
( code_integer_of_num
= ( numeral_numeral @ code_integer ) ) ).
% integer_of_num_def
thf(fact_1809_unset__bit__integer_Orep__eq,axiom,
! [X: nat,Xa: code_integer] :
( ( code_int_of_integer @ ( bit_se2638667681897837118et_bit @ code_integer @ X @ Xa ) )
= ( bit_se2638667681897837118et_bit @ int @ X @ ( code_int_of_integer @ Xa ) ) ) ).
% unset_bit_integer.rep_eq
thf(fact_1810_set__bit__integer_Orep__eq,axiom,
! [X: nat,Xa: code_integer] :
( ( code_int_of_integer @ ( bit_se5668285175392031749et_bit @ code_integer @ X @ Xa ) )
= ( bit_se5668285175392031749et_bit @ int @ X @ ( code_int_of_integer @ Xa ) ) ) ).
% set_bit_integer.rep_eq
thf(fact_1811_flip__bit__integer_Orep__eq,axiom,
! [X: nat,Xa: code_integer] :
( ( code_int_of_integer @ ( bit_se8732182000553998342ip_bit @ code_integer @ X @ Xa ) )
= ( bit_se8732182000553998342ip_bit @ int @ X @ ( code_int_of_integer @ Xa ) ) ) ).
% flip_bit_integer.rep_eq
thf(fact_1812_prod_OatLeast__Suc__atMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M2: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% prod.atLeast_Suc_atMost_Suc_shift
thf(fact_1813_prod_OatLeast__Suc__lessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M2: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% prod.atLeast_Suc_lessThan_Suc_shift
thf(fact_1814_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
! [L: int,U: int] :
( ( set_or7035219750837199246ssThan @ int @ L @ ( plus_plus @ int @ U @ ( one_one @ int ) ) )
= ( set_or1337092689740270186AtMost @ int @ L @ U ) ) ).
% atLeastLessThanPlusOne_atLeastAtMost_int
thf(fact_1815_prod_OatLeastAtMost__shift__bounds,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M2: nat,K: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M2 @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ K ) ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% prod.atLeastAtMost_shift_bounds
thf(fact_1816_prod_OatLeastLessThan__shift__bounds,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M2: nat,K: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M2 @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ K ) ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% prod.atLeastLessThan_shift_bounds
thf(fact_1817_prod_OatLeastAtMost__rev,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat,M2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ N @ M2 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ N @ M2 ) ) ) ) ).
% prod.atLeastAtMost_rev
thf(fact_1818_prod_Onat__diff__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ N @ ( suc @ I3 ) ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% prod.nat_diff_reindex
thf(fact_1819_nat__of__integer_Oabs__eq,axiom,
! [X: int] :
( ( code_nat_of_integer @ ( code_integer_of_int @ X ) )
= ( nat2 @ X ) ) ).
% nat_of_integer.abs_eq
thf(fact_1820_num__of__integer_Orep__eq,axiom,
( code_num_of_integer
= ( ^ [X3: code_integer] : ( num_of_nat @ ( nat2 @ ( code_int_of_integer @ X3 ) ) ) ) ) ).
% num_of_integer.rep_eq
thf(fact_1821_prod_OatLeast0__atMost__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ).
% prod.atLeast0_atMost_Suc
thf(fact_1822_prod_OatLeast0__lessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G @ N ) ) ) ) ).
% prod.atLeast0_lessThan_Suc
thf(fact_1823_prod_OatLeast__Suc__lessThan,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M2: nat,N: nat,G: nat > A] :
( ( ord_less @ nat @ M2 @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) )
= ( times_times @ A @ ( G @ M2 ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ N ) ) ) ) ) ) ).
% prod.atLeast_Suc_lessThan
thf(fact_1824_prod_Onat__ivl__Suc_H,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M2: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M2 @ ( suc @ N ) )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) ) )
= ( times_times @ A @ ( G @ ( suc @ N ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ) ) ).
% prod.nat_ivl_Suc'
thf(fact_1825_prod_OatLeast__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M2: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( times_times @ A @ ( G @ M2 ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ N ) ) ) ) ) ) ).
% prod.atLeast_Suc_atMost
thf(fact_1826_prod_OatLeastLessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: nat,B2: nat,G: nat > A] :
( ( ord_less_eq @ nat @ A3 @ B2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A3 @ ( suc @ B2 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A3 @ B2 ) ) @ ( G @ B2 ) ) ) ) ) ).
% prod.atLeastLessThan_Suc
thf(fact_1827_divmod__abs__code_I6_J,axiom,
! [J: code_integer] :
( ( code_divmod_abs @ ( zero_zero @ code_integer ) @ J )
= ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) ) ) ).
% divmod_abs_code(6)
thf(fact_1828_prod_Olast__plus,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M2: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( times_times @ A @ ( G @ N ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ) ) ).
% prod.last_plus
thf(fact_1829_prod_OSuc__reindex__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M2: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) )
= ( times_times @ A @ ( G @ M2 )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ) ) ).
% prod.Suc_reindex_ivl
thf(fact_1830_prod_OatMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
= ( times_times @ A @ ( G @ ( zero_zero @ nat ) )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% prod.atMost_Suc_shift
thf(fact_1831_prod_OlessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
= ( times_times @ A @ ( G @ ( zero_zero @ nat ) )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% prod.lessThan_Suc_shift
thf(fact_1832_prod_OatLeastLessThan__rev,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat,M2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ M2 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ ( suc @ I3 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ N @ M2 ) ) ) ) ).
% prod.atLeastLessThan_rev
thf(fact_1833_prod_Onested__swap,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: nat > nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( groups7121269368397514597t_prod @ nat @ A @ ( A3 @ I3 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J2: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( A3 @ I3 @ J2 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J2 ) @ N ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% prod.nested_swap
thf(fact_1834_prod_OatLeast1__atMost__eq,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K4: nat] : ( G @ ( suc @ K4 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% prod.atLeast1_atMost_eq
thf(fact_1835_prod_Onat__group,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,K: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [M: nat] : ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ M @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ M @ K ) @ K ) ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ N @ K ) ) ) ) ) ).
% prod.nat_group
thf(fact_1836_fact__prod,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N5: nat] :
( semiring_1_of_nat @ A
@ ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X3: nat] : X3
@ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N5 ) ) ) ) ) ) ).
% fact_prod
thf(fact_1837_prod_Onested__swap_H,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: nat > nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( groups7121269368397514597t_prod @ nat @ A @ ( A3 @ I3 ) @ ( set_ord_lessThan @ nat @ I3 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J2: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( A3 @ I3 @ J2 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J2 ) @ N ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% prod.nested_swap'
thf(fact_1838_prod__atLeastAtMost__code,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [F: nat > A,A3: nat,B2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ F @ ( set_or1337092689740270186AtMost @ nat @ A3 @ B2 ) )
= ( set_fo6178422350223883121st_nat @ A
@ ^ [A7: nat] : ( times_times @ A @ ( F @ A7 ) )
@ A3
@ B2
@ ( one_one @ A ) ) ) ) ).
% prod_atLeastAtMost_code
thf(fact_1839_prod_Ohead__if,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [N: nat,M2: nat,G: nat > A] :
( ( ( ord_less @ nat @ N @ M2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( one_one @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) @ ( G @ N ) ) ) ) ) ) ).
% prod.head_if
thf(fact_1840_prod_Oub__add__nat,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M2: nat,N: nat,G: nat > A,P5: nat] :
( ( ord_less_eq @ nat @ M2 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( plus_plus @ nat @ N @ P5 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N @ P5 ) ) ) ) ) ) ) ).
% prod.ub_add_nat
thf(fact_1841_prod_OatLeast0__atMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( times_times @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% prod.atLeast0_atMost_Suc_shift
thf(fact_1842_prod_OatLeast0__lessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( times_times @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% prod.atLeast0_lessThan_Suc_shift
thf(fact_1843_prod_OatLeastLessThan__shift__0,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M2: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M2 ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ).
% prod.atLeastLessThan_shift_0
thf(fact_1844_integer__of__num__triv_I1_J,axiom,
( ( code_integer_of_num @ one2 )
= ( one_one @ code_integer ) ) ).
% integer_of_num_triv(1)
thf(fact_1845_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat,M2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ M2 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M2 ) ) ) ) ).
% prod.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_1846_prod_OatMost__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ N ) )
= ( times_times @ A @ ( G @ ( zero_zero @ nat ) )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% prod.atMost_shift
thf(fact_1847_pochhammer__prod,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( comm_s3205402744901411588hammer @ A )
= ( ^ [A7: A,N5: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( plus_plus @ A @ A7 @ ( semiring_1_of_nat @ A @ I3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N5 ) ) ) ) ) ).
% pochhammer_prod
thf(fact_1848_prod_OatLeast__atMost__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M2: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ( comp @ nat @ A @ nat @ G
@ ^ [N5: nat] : ( minus_minus @ nat @ N5 @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% prod.atLeast_atMost_pred_shift
thf(fact_1849_prod_OatLeast__lessThan__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M2: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ( comp @ nat @ A @ nat @ G
@ ^ [N5: nat] : ( minus_minus @ nat @ N5 @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% prod.atLeast_lessThan_pred_shift
thf(fact_1850_integer__of__num_I2_J,axiom,
! [N: num] :
( ( code_integer_of_num @ ( bit0 @ N ) )
= ( plus_plus @ code_integer @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) ) ).
% integer_of_num(2)
thf(fact_1851_divmod__abs__code_I5_J,axiom,
! [J: code_integer] :
( ( code_divmod_abs @ J @ ( zero_zero @ code_integer ) )
= ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( abs_abs @ code_integer @ J ) ) ) ).
% divmod_abs_code(5)
thf(fact_1852_prod_OatLeastAtMost__shift__0,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M2: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M2 ) ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ) ).
% prod.atLeastAtMost_shift_0
thf(fact_1853_sum_OatLeast__int__lessThan__int__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: int > A,M2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ int @ A @ G @ ( set_or7035219750837199246ssThan @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ int @ A @ nat @ G @ ( semiring_1_of_nat @ int ) ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% sum.atLeast_int_lessThan_int_shift
thf(fact_1854_pochhammer__Suc__prod,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% pochhammer_Suc_prod
thf(fact_1855_pochhammer__prod__rev,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( comm_s3205402744901411588hammer @ A )
= ( ^ [A7: A,N5: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( plus_plus @ A @ A7 @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N5 @ I3 ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N5 ) ) ) ) ) ).
% pochhammer_prod_rev
thf(fact_1856_fact__div__fact,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq @ nat @ N @ M2 )
=> ( ( divide_divide @ nat @ ( semiring_char_0_fact @ nat @ M2 ) @ ( semiring_char_0_fact @ nat @ N ) )
= ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X3: nat] : X3
@ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ M2 ) ) ) ) ).
% fact_div_fact
thf(fact_1857_integer__of__num__triv_I2_J,axiom,
( ( code_integer_of_num @ ( bit0 @ one2 ) )
= ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ).
% integer_of_num_triv(2)
thf(fact_1858_prod_Oin__pairs,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M2: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% prod.in_pairs
thf(fact_1859_prod_Oin__pairs__0,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( times_times @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% prod.in_pairs_0
thf(fact_1860_nat__of__integer__less__iff,axiom,
! [X: code_integer,Y: code_integer] :
( ( ord_less_eq @ code_integer @ ( zero_zero @ code_integer ) @ X )
=> ( ( ord_less_eq @ code_integer @ ( zero_zero @ code_integer ) @ Y )
=> ( ( ord_less @ nat @ ( code_nat_of_integer @ X ) @ ( code_nat_of_integer @ Y ) )
= ( ord_less @ code_integer @ X @ Y ) ) ) ) ).
% nat_of_integer_less_iff
thf(fact_1861_pochhammer__Suc__prod__rev,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N @ I3 ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% pochhammer_Suc_prod_rev
thf(fact_1862_divmod__integer__code,axiom,
( code_divmod_integer
= ( ^ [K4: code_integer,L3: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( K4
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) )
@ ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ L3 )
@ ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ K4 ) @ ( code_divmod_abs @ K4 @ L3 )
@ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [R3: code_integer,S4: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( S4
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R3 ) @ ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R3 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ L3 @ S4 ) ) )
@ ( code_divmod_abs @ K4 @ L3 ) ) )
@ ( if @ ( product_prod @ code_integer @ code_integer )
@ ( L3
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ K4 )
@ ( product_apsnd @ code_integer @ code_integer @ code_integer @ ( uminus_uminus @ code_integer )
@ ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less @ code_integer @ K4 @ ( zero_zero @ code_integer ) ) @ ( code_divmod_abs @ K4 @ L3 )
@ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [R3: code_integer,S4: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( S4
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R3 ) @ ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R3 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ L3 ) @ S4 ) ) )
@ ( code_divmod_abs @ K4 @ L3 ) ) ) ) ) ) ) ) ) ).
% divmod_integer_code
thf(fact_1863_bit__cut__integer__code,axiom,
( code_bit_cut_integer
= ( ^ [K4: code_integer] :
( if @ ( product_prod @ code_integer @ $o )
@ ( K4
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ $o @ ( zero_zero @ code_integer ) @ $false )
@ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ $o )
@ ^ [R3: code_integer,S4: code_integer] :
( product_Pair @ code_integer @ $o @ ( if @ code_integer @ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ K4 ) @ R3 @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R3 ) @ S4 ) )
@ ( S4
= ( one_one @ code_integer ) ) )
@ ( code_divmod_abs @ K4 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% bit_cut_integer_code
thf(fact_1864_prod_Ozero__middle,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [P5: nat,K: nat,G: nat > A,H3: nat > A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P5 )
=> ( ( ord_less_eq @ nat @ K @ P5 )
=> ( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J2: nat] : ( if @ A @ ( ord_less @ nat @ J2 @ K ) @ ( G @ J2 ) @ ( if @ A @ ( J2 = K ) @ ( one_one @ A ) @ ( H3 @ ( minus_minus @ nat @ J2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( set_ord_atMost @ nat @ P5 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J2: nat] : ( if @ A @ ( ord_less @ nat @ J2 @ K ) @ ( G @ J2 ) @ ( H3 @ J2 ) )
@ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P5 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_1865_xor__int__rec,axiom,
( ( bit_se5824344971392196577ns_xor @ int )
= ( ^ [K4: int,L3: int] :
( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K4 ) )
!= ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L3 ) ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ int @ ( divide_divide @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% xor_int_rec
thf(fact_1866_divmod__abs__def,axiom,
( code_divmod_abs
= ( ^ [K4: code_integer,L3: code_integer] : ( product_Pair @ code_integer @ code_integer @ ( divide_divide @ code_integer @ ( abs_abs @ code_integer @ K4 ) @ ( abs_abs @ code_integer @ L3 ) ) @ ( modulo_modulo @ code_integer @ ( abs_abs @ code_integer @ K4 ) @ ( abs_abs @ code_integer @ L3 ) ) ) ) ) ).
% divmod_abs_def
thf(fact_1867_less__int_Orep__eq,axiom,
( ( ord_less @ int )
= ( ^ [X3: int,Xa4: int] :
( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [Y3: nat,Z5: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V2: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ Y3 @ V2 ) @ ( plus_plus @ nat @ U2 @ Z5 ) ) )
@ ( rep_Integ @ X3 )
@ ( rep_Integ @ Xa4 ) ) ) ) ).
% less_int.rep_eq
thf(fact_1868_make_H__def,axiom,
! [B: $tType,A: $tType] :
( ( ( heap @ A )
& ( semiring_1 @ B ) )
=> ( ( array_make2 @ B @ A )
= ( ^ [I3: code_integer,F2: B > A] : ( array_make @ A @ ( code_nat_of_integer @ I3 ) @ ( comp @ B @ A @ nat @ F2 @ ( semiring_1_of_nat @ B ) ) ) ) ) ) ).
% make'_def
thf(fact_1869_new_H__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_new2 @ A )
= ( comp @ nat @ ( A > ( heap_Time_Heap @ ( array @ A ) ) ) @ code_integer @ ( array_new @ A ) @ code_nat_of_integer ) ) ) ).
% new'_def
thf(fact_1870_xor__int__unfold,axiom,
( ( bit_se5824344971392196577ns_xor @ int )
= ( ^ [K4: int,L3: int] :
( if @ int
@ ( K4
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
@ ( bit_ri4277139882892585799ns_not @ int @ L3 )
@ ( if @ int
@ ( L3
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
@ ( bit_ri4277139882892585799ns_not @ int @ K4 )
@ ( if @ int
@ ( K4
= ( zero_zero @ int ) )
@ L3
@ ( if @ int
@ ( L3
= ( zero_zero @ int ) )
@ K4
@ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( modulo_modulo @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ int @ ( divide_divide @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% xor_int_unfold
thf(fact_1871_take__bit__num__code,axiom,
( bit_take_bit_num
= ( ^ [N5: nat,M: num] :
( product_case_prod @ nat @ num @ ( option @ num )
@ ^ [A7: nat,X3: num] :
( case_nat @ ( option @ num ) @ ( none @ num )
@ ^ [O: nat] :
( case_num @ ( option @ num ) @ ( some @ num @ one2 )
@ ^ [P6: num] :
( case_option @ ( option @ num ) @ num @ ( none @ num )
@ ^ [Q5: num] : ( some @ num @ ( bit0 @ Q5 ) )
@ ( bit_take_bit_num @ O @ P6 ) )
@ ^ [P6: num] : ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_take_bit_num @ O @ P6 ) ) )
@ X3 )
@ A7 )
@ ( product_Pair @ nat @ num @ N5 @ M ) ) ) ) ).
% take_bit_num_code
thf(fact_1872_prod_Otriangle__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I3: nat,J2: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I3 @ J2 ) @ N ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K4: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G @ I3 @ ( minus_minus @ nat @ K4 @ I3 ) )
@ ( set_ord_atMost @ nat @ K4 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% prod.triangle_reindex
thf(fact_1873_case__nat__add__eq__if,axiom,
! [A: $tType,A3: A,F: nat > A,V: num,N: nat] :
( ( case_nat @ A @ A3 @ F @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ V ) @ N ) )
= ( F @ ( plus_plus @ nat @ ( pred_numeral @ V ) @ N ) ) ) ).
% case_nat_add_eq_if
thf(fact_1874_rec__nat__add__eq__if,axiom,
! [A: $tType,A3: A,F: nat > A > A,V: num,N: nat] :
( ( rec_nat @ A @ A3 @ F @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ V ) @ N ) )
= ( F @ ( plus_plus @ nat @ ( pred_numeral @ V ) @ N ) @ ( rec_nat @ A @ A3 @ F @ ( plus_plus @ nat @ ( pred_numeral @ V ) @ N ) ) ) ) ).
% rec_nat_add_eq_if
thf(fact_1875_bit_Ocompl__zero,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4277139882892585799ns_not @ A @ ( zero_zero @ A ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.compl_zero
thf(fact_1876_bit_Ocompl__one,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4277139882892585799ns_not @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% bit.compl_one
thf(fact_1877_bit_Oxor__cancel__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se5824344971392196577ns_xor @ A @ X @ ( bit_ri4277139882892585799ns_not @ A @ X ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.xor_cancel_right
thf(fact_1878_bit_Oxor__cancel__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se5824344971392196577ns_xor @ A @ ( bit_ri4277139882892585799ns_not @ A @ X ) @ X )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.xor_cancel_left
thf(fact_1879_bit_Oxor__one__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se5824344971392196577ns_xor @ A @ X @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( bit_ri4277139882892585799ns_not @ A @ X ) ) ) ).
% bit.xor_one_right
thf(fact_1880_bit_Oxor__one__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se5824344971392196577ns_xor @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X )
= ( bit_ri4277139882892585799ns_not @ A @ X ) ) ) ).
% bit.xor_one_left
thf(fact_1881_push__bit__minus__one__eq__not__mask,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ) ).
% push_bit_minus_one_eq_not_mask
thf(fact_1882_not__one__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4277139882892585799ns_not @ A @ ( one_one @ A ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% not_one_eq
thf(fact_1883_not__add__distrib,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A,B2: A] :
( ( bit_ri4277139882892585799ns_not @ A @ ( plus_plus @ A @ A3 @ B2 ) )
= ( minus_minus @ A @ ( bit_ri4277139882892585799ns_not @ A @ A3 ) @ B2 ) ) ) ).
% not_add_distrib
thf(fact_1884_not__diff__distrib,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A,B2: A] :
( ( bit_ri4277139882892585799ns_not @ A @ ( minus_minus @ A @ A3 @ B2 ) )
= ( plus_plus @ A @ ( bit_ri4277139882892585799ns_not @ A @ A3 ) @ B2 ) ) ) ).
% not_diff_distrib
thf(fact_1885_minus__eq__not__plus__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( uminus_uminus @ A )
= ( ^ [A7: A] : ( plus_plus @ A @ ( bit_ri4277139882892585799ns_not @ A @ A7 ) @ ( one_one @ A ) ) ) ) ) ).
% minus_eq_not_plus_1
thf(fact_1886_minus__eq__not__minus__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( uminus_uminus @ A )
= ( ^ [A7: A] : ( bit_ri4277139882892585799ns_not @ A @ ( minus_minus @ A @ A7 @ ( one_one @ A ) ) ) ) ) ) ).
% minus_eq_not_minus_1
thf(fact_1887_not__eq__complement,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4277139882892585799ns_not @ A )
= ( ^ [A7: A] : ( minus_minus @ A @ ( uminus_uminus @ A @ A7 ) @ ( one_one @ A ) ) ) ) ) ).
% not_eq_complement
thf(fact_1888_prod__int__eq,axiom,
! [I: nat,J: nat] :
( ( groups7121269368397514597t_prod @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or1337092689740270186AtMost @ nat @ I @ J ) )
= ( groups7121269368397514597t_prod @ int @ int
@ ^ [X3: int] : X3
@ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ I ) @ ( semiring_1_of_nat @ int @ J ) ) ) ) ).
% prod_int_eq
thf(fact_1889_prod__int__plus__eq,axiom,
! [I: nat,J: nat] :
( ( groups7121269368397514597t_prod @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or1337092689740270186AtMost @ nat @ I @ ( plus_plus @ nat @ I @ J ) ) )
= ( groups7121269368397514597t_prod @ int @ int
@ ^ [X3: int] : X3
@ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ I ) @ ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ I @ J ) ) ) ) ) ).
% prod_int_plus_eq
thf(fact_1890_push__bit__mask__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M2: nat,N: nat] :
( ( bit_se4730199178511100633sh_bit @ A @ M2 @ ( bit_se2239418461657761734s_mask @ A @ N ) )
= ( bit_se5824344872417868541ns_and @ A @ ( bit_se2239418461657761734s_mask @ A @ ( plus_plus @ nat @ N @ M2 ) ) @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ M2 ) ) ) ) ) ).
% push_bit_mask_eq
thf(fact_1891_prod_OatLeast__int__lessThan__int__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: int > A,M2: nat,N: nat] :
( ( groups7121269368397514597t_prod @ int @ A @ G @ ( set_or7035219750837199246ssThan @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ int @ A @ nat @ G @ ( semiring_1_of_nat @ int ) ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% prod.atLeast_int_lessThan_int_shift
thf(fact_1892_unset__bit__eq__and__not,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_se2638667681897837118et_bit @ A )
= ( ^ [N5: nat,A7: A] : ( bit_se5824344872417868541ns_and @ A @ A7 @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se4730199178511100633sh_bit @ A @ N5 @ ( one_one @ A ) ) ) ) ) ) ) ).
% unset_bit_eq_and_not
thf(fact_1893_prod_OatLeast__int__atMost__int__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: int > A,M2: nat,N: nat] :
( ( groups7121269368397514597t_prod @ int @ A @ G @ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ int @ A @ nat @ G @ ( semiring_1_of_nat @ int ) ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% prod.atLeast_int_atMost_int_shift
thf(fact_1894_int__numeral__not__and__num,axiom,
! [M2: num,N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( numeral_numeral @ int @ N ) )
= ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ N @ M2 ) ) ) ).
% int_numeral_not_and_num
thf(fact_1895_int__numeral__and__not__num,axiom,
! [M2: num,N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M2 ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) )
= ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M2 @ N ) ) ) ).
% int_numeral_and_not_num
thf(fact_1896_prod_Otriangle__reindex__eq,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I3: nat,J2: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I3 @ J2 ) @ N ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K4: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G @ I3 @ ( minus_minus @ nat @ K4 @ I3 ) )
@ ( set_ord_atMost @ nat @ K4 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% prod.triangle_reindex_eq
thf(fact_1897_and__not__numerals_I8_J,axiom,
! [M2: num,N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit1 @ M2 ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M2 ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ) ).
% and_not_numerals(8)
thf(fact_1898_not__int__rec,axiom,
( ( bit_ri4277139882892585799ns_not @ int )
= ( ^ [K4: int] : ( plus_plus @ int @ ( zero_neq_one_of_bool @ int @ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K4 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( divide_divide @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% not_int_rec
thf(fact_1899_Nitpick_Ocase__nat__unfold,axiom,
! [A: $tType] :
( ( case_nat @ A )
= ( ^ [X3: A,F2: nat > A,N5: nat] :
( if @ A
@ ( N5
= ( zero_zero @ nat ) )
@ X3
@ ( F2 @ ( minus_minus @ nat @ N5 @ ( one_one @ nat ) ) ) ) ) ) ).
% Nitpick.case_nat_unfold
thf(fact_1900_or__int__rec,axiom,
( ( bit_se1065995026697491101ons_or @ int )
= ( ^ [K4: int,L3: int] :
( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K4 )
| ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L3 ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( divide_divide @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% or_int_rec
thf(fact_1901_or__numerals_I4_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% or_numerals(4)
thf(fact_1902_or__numerals_I6_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% or_numerals(6)
thf(fact_1903_or__numerals_I7_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% or_numerals(7)
thf(fact_1904_or__nat__rec,axiom,
( ( bit_se1065995026697491101ons_or @ nat )
= ( ^ [M: nat,N5: nat] :
( plus_plus @ nat
@ ( zero_neq_one_of_bool @ nat
@ ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M )
| ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) ) )
@ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ nat @ ( divide_divide @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% or_nat_rec
thf(fact_1905_or__not__numerals_I9_J,axiom,
! [M2: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit1 @ M2 ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M2 ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ) ).
% or_not_numerals(9)
thf(fact_1906_bit_Odisj__one__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se1065995026697491101ons_or @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.disj_one_left
thf(fact_1907_bit_Odisj__one__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se1065995026697491101ons_or @ A @ X @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.disj_one_right
thf(fact_1908_or__numerals_I8_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( bit1 @ X ) ) ) ) ).
% or_numerals(8)
thf(fact_1909_or__numerals_I2_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( numeral_numeral @ A @ ( bit1 @ Y ) ) ) ) ).
% or_numerals(2)
thf(fact_1910_bit_Odisj__cancel__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se1065995026697491101ons_or @ A @ ( bit_ri4277139882892585799ns_not @ A @ X ) @ X )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.disj_cancel_left
thf(fact_1911_bit_Odisj__cancel__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se1065995026697491101ons_or @ A @ X @ ( bit_ri4277139882892585799ns_not @ A @ X ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.disj_cancel_right
thf(fact_1912_or__numerals_I5_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( bit1 @ X ) ) ) ) ).
% or_numerals(5)
thf(fact_1913_or__numerals_I1_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( numeral_numeral @ A @ ( bit1 @ Y ) ) ) ) ).
% or_numerals(1)
thf(fact_1914_or__integer_Oabs__eq,axiom,
! [Xa: int,X: int] :
( ( bit_se1065995026697491101ons_or @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( bit_se1065995026697491101ons_or @ int @ Xa @ X ) ) ) ).
% or_integer.abs_eq
thf(fact_1915_or__integer_Orep__eq,axiom,
! [X: code_integer,Xa: code_integer] :
( ( code_int_of_integer @ ( bit_se1065995026697491101ons_or @ code_integer @ X @ Xa ) )
= ( bit_se1065995026697491101ons_or @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% or_integer.rep_eq
thf(fact_1916_disjunctive__add,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A,B2: A] :
( ! [N3: nat] :
( ~ ( bit_se5641148757651400278ts_bit @ A @ A3 @ N3 )
| ~ ( bit_se5641148757651400278ts_bit @ A @ B2 @ N3 ) )
=> ( ( plus_plus @ A @ A3 @ B2 )
= ( bit_se1065995026697491101ons_or @ A @ A3 @ B2 ) ) ) ) ).
% disjunctive_add
thf(fact_1917_plus__and__or,axiom,
! [X: int,Y: int] :
( ( plus_plus @ int @ ( bit_se5824344872417868541ns_and @ int @ X @ Y ) @ ( bit_se1065995026697491101ons_or @ int @ X @ Y ) )
= ( plus_plus @ int @ X @ Y ) ) ).
% plus_and_or
thf(fact_1918_not__integer_Orep__eq,axiom,
! [X: code_integer] :
( ( code_int_of_integer @ ( bit_ri4277139882892585799ns_not @ code_integer @ X ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( code_int_of_integer @ X ) ) ) ).
% not_integer.rep_eq
thf(fact_1919_set__bit__nat__def,axiom,
( ( bit_se5668285175392031749et_bit @ nat )
= ( ^ [M: nat,N5: nat] : ( bit_se1065995026697491101ons_or @ nat @ N5 @ ( bit_se4730199178511100633sh_bit @ nat @ M @ ( one_one @ nat ) ) ) ) ) ).
% set_bit_nat_def
thf(fact_1920_not__integer_Oabs__eq,axiom,
! [X: int] :
( ( bit_ri4277139882892585799ns_not @ code_integer @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( bit_ri4277139882892585799ns_not @ int @ X ) ) ) ).
% not_integer.abs_eq
thf(fact_1921_set__bit__eq__or,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5668285175392031749et_bit @ A )
= ( ^ [N5: nat,A7: A] : ( bit_se1065995026697491101ons_or @ A @ A7 @ ( bit_se4730199178511100633sh_bit @ A @ N5 @ ( one_one @ A ) ) ) ) ) ) ).
% set_bit_eq_or
thf(fact_1922_bit_Ocomplement__unique,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A,X: A,Y: A] :
( ( ( bit_se5824344872417868541ns_and @ A @ A3 @ X )
= ( zero_zero @ A ) )
=> ( ( ( bit_se1065995026697491101ons_or @ A @ A3 @ X )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
=> ( ( ( bit_se5824344872417868541ns_and @ A @ A3 @ Y )
= ( zero_zero @ A ) )
=> ( ( ( bit_se1065995026697491101ons_or @ A @ A3 @ Y )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
=> ( X = Y ) ) ) ) ) ) ).
% bit.complement_unique
thf(fact_1923_bit_Ocompl__unique,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A,Y: A] :
( ( ( bit_se5824344872417868541ns_and @ A @ X @ Y )
= ( zero_zero @ A ) )
=> ( ( ( bit_se1065995026697491101ons_or @ A @ X @ Y )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
=> ( ( bit_ri4277139882892585799ns_not @ A @ X )
= Y ) ) ) ) ).
% bit.compl_unique
thf(fact_1924_or__one__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se1065995026697491101ons_or @ A @ A3 @ ( one_one @ A ) )
= ( plus_plus @ A @ A3 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ) ).
% or_one_eq
thf(fact_1925_one__or__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ A3 )
= ( plus_plus @ A @ A3 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ) ).
% one_or_eq
thf(fact_1926_mask__Suc__double,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( bit_se2239418461657761734s_mask @ A @ ( suc @ N ) )
= ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ) ) ).
% mask_Suc_double
thf(fact_1927_or__not__numerals_I5_J,axiom,
! [M2: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit0 @ M2 ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M2 ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ) ).
% or_not_numerals(5)
thf(fact_1928_Suc__0__or__eq,axiom,
! [N: nat] :
( ( bit_se1065995026697491101ons_or @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
= ( plus_plus @ nat @ N @ ( zero_neq_one_of_bool @ nat @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% Suc_0_or_eq
thf(fact_1929_or__Suc__0__eq,axiom,
! [N: nat] :
( ( bit_se1065995026697491101ons_or @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
= ( plus_plus @ nat @ N @ ( zero_neq_one_of_bool @ nat @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% or_Suc_0_eq
thf(fact_1930_or__not__numerals_I8_J,axiom,
! [M2: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit1 @ M2 ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M2 ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ) ).
% or_not_numerals(8)
thf(fact_1931_or__int__unfold,axiom,
( ( bit_se1065995026697491101ons_or @ int )
= ( ^ [K4: int,L3: int] :
( if @ int
@ ( ( K4
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
| ( L3
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) )
@ ( uminus_uminus @ int @ ( one_one @ int ) )
@ ( if @ int
@ ( K4
= ( zero_zero @ int ) )
@ L3
@ ( if @ int
@ ( L3
= ( zero_zero @ int ) )
@ K4
@ ( plus_plus @ int @ ( ord_max @ int @ ( modulo_modulo @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( divide_divide @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ).
% or_int_unfold
thf(fact_1932_bit_Oabstract__boolean__algebra__sym__diff__axioms,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( boolea3799213064322606851m_diff @ A @ ( bit_se5824344872417868541ns_and @ A ) @ ( bit_se1065995026697491101ons_or @ A ) @ ( bit_ri4277139882892585799ns_not @ A ) @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( bit_se5824344971392196577ns_xor @ A ) ) ) ).
% bit.abstract_boolean_algebra_sym_diff_axioms
thf(fact_1933_or__nat__unfold,axiom,
( ( bit_se1065995026697491101ons_or @ nat )
= ( ^ [M: nat,N5: nat] :
( if @ nat
@ ( M
= ( zero_zero @ nat ) )
@ N5
@ ( if @ nat
@ ( N5
= ( zero_zero @ nat ) )
@ M
@ ( plus_plus @ nat @ ( ord_max @ nat @ ( modulo_modulo @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ nat @ ( divide_divide @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% or_nat_unfold
thf(fact_1934_drop__bit__rec,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se4197421643247451524op_bit @ A )
= ( ^ [N5: nat,A7: A] :
( if @ A
@ ( N5
= ( zero_zero @ nat ) )
@ A7
@ ( bit_se4197421643247451524op_bit @ A @ ( minus_minus @ nat @ N5 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A7 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% drop_bit_rec
thf(fact_1935_image__atLeastZeroLessThan__integer,axiom,
! [U: code_integer] :
( ( ord_less_eq @ code_integer @ ( zero_zero @ code_integer ) @ U )
=> ( ( set_or7035219750837199246ssThan @ code_integer @ ( zero_zero @ code_integer ) @ U )
= ( image2 @ nat @ code_integer @ ( semiring_1_of_nat @ code_integer ) @ ( set_ord_lessThan @ nat @ ( code_nat_of_integer @ U ) ) ) ) ) ).
% image_atLeastZeroLessThan_integer
thf(fact_1936_uminus__int__def,axiom,
( ( uminus_uminus @ 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 )
@ ^ [X3: nat,Y3: nat] : ( product_Pair @ nat @ nat @ Y3 @ X3 ) ) ) ) ).
% uminus_int_def
thf(fact_1937_bit_Oabstract__boolean__algebra__axioms,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( boolea2506097494486148201lgebra @ A @ ( bit_se5824344872417868541ns_and @ A ) @ ( bit_se1065995026697491101ons_or @ A ) @ ( bit_ri4277139882892585799ns_not @ A ) @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.abstract_boolean_algebra_axioms
thf(fact_1938_map__fun__apply,axiom,
! [D: $tType,A: $tType,C: $tType,B: $tType] :
( ( map_fun @ B @ C @ D @ A )
= ( ^ [F2: B > C,G2: D > A,H: C > D,X3: B] : ( G2 @ ( H @ ( F2 @ X3 ) ) ) ) ) ).
% map_fun_apply
thf(fact_1939_drop__bit__drop__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M2: nat,N: nat,A3: A] :
( ( bit_se4197421643247451524op_bit @ A @ M2 @ ( bit_se4197421643247451524op_bit @ A @ N @ A3 ) )
= ( bit_se4197421643247451524op_bit @ A @ ( plus_plus @ nat @ M2 @ N ) @ A3 ) ) ) ).
% drop_bit_drop_bit
thf(fact_1940_image__add__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: A,I: A,J: A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ K ) @ ( set_or1337092689740270186AtMost @ A @ I @ J ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ K ) ) ) ) ).
% image_add_atLeastAtMost
thf(fact_1941_image__diff__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [D3: A,A3: A,B2: A] :
( ( image2 @ A @ A @ ( minus_minus @ A @ D3 ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ D3 @ B2 ) @ ( minus_minus @ A @ D3 @ A3 ) ) ) ) ).
% image_diff_atLeastAtMost
thf(fact_1942_image__add__atLeastLessThan,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: A,I: A,J: A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ K ) @ ( set_or7035219750837199246ssThan @ A @ I @ J ) )
= ( set_or7035219750837199246ssThan @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ K ) ) ) ) ).
% image_add_atLeastLessThan
thf(fact_1943_max__0__1_I1_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ( ( ord_max @ A @ ( zero_zero @ A ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% max_0_1(1)
thf(fact_1944_max__0__1_I2_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ( ( ord_max @ A @ ( one_one @ A ) @ ( zero_zero @ A ) )
= ( one_one @ A ) ) ) ).
% max_0_1(2)
thf(fact_1945_image__uminus__atLeastAtMost,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A,Y: A] :
( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% image_uminus_atLeastAtMost
thf(fact_1946_max__0__1_I6_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: num] :
( ( ord_max @ A @ ( numeral_numeral @ A @ X ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ X ) ) ) ).
% max_0_1(6)
thf(fact_1947_max__0__1_I5_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: num] :
( ( ord_max @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ X ) )
= ( numeral_numeral @ A @ X ) ) ) ).
% max_0_1(5)
thf(fact_1948_image__add__atMost,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [C2: A,A3: A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ C2 ) @ ( set_ord_atMost @ A @ A3 ) )
= ( set_ord_atMost @ A @ ( plus_plus @ A @ C2 @ A3 ) ) ) ) ).
% image_add_atMost
thf(fact_1949_image__add__atLeastAtMost_H,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: A,I: A,J: A] :
( ( image2 @ A @ A
@ ^ [N5: A] : ( plus_plus @ A @ N5 @ K )
@ ( set_or1337092689740270186AtMost @ A @ I @ J ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ K ) ) ) ) ).
% image_add_atLeastAtMost'
thf(fact_1950_image__minus__const__atLeastAtMost_H,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [D3: A,A3: A,B2: A] :
( ( image2 @ A @ A
@ ^ [T3: A] : ( minus_minus @ A @ T3 @ D3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ A3 @ D3 ) @ ( minus_minus @ A @ B2 @ D3 ) ) ) ) ).
% image_minus_const_atLeastAtMost'
thf(fact_1951_image__add__atLeastLessThan_H,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: A,I: A,J: A] :
( ( image2 @ A @ A
@ ^ [N5: A] : ( plus_plus @ A @ N5 @ K )
@ ( set_or7035219750837199246ssThan @ A @ I @ J ) )
= ( set_or7035219750837199246ssThan @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ K ) ) ) ) ).
% image_add_atLeastLessThan'
thf(fact_1952_drop__bit__of__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( bit_se4197421643247451524op_bit @ A @ N @ ( one_one @ A ) )
= ( zero_neq_one_of_bool @ A
@ ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% drop_bit_of_1
thf(fact_1953_image__mult__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [D3: A,A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ D3 )
=> ( ( image2 @ A @ A @ ( times_times @ A @ D3 ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ D3 @ A3 ) @ ( times_times @ A @ D3 @ B2 ) ) ) ) ) ).
% image_mult_atLeastAtMost
thf(fact_1954_image__divide__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [D3: A,A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ D3 )
=> ( ( image2 @ A @ A
@ ^ [C3: A] : ( divide_divide @ A @ C3 @ D3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( divide_divide @ A @ A3 @ D3 ) @ ( divide_divide @ A @ B2 @ D3 ) ) ) ) ) ).
% image_divide_atLeastAtMost
thf(fact_1955_drop__bit__integer_Orep__eq,axiom,
! [X: nat,Xa: code_integer] :
( ( code_int_of_integer @ ( bit_se4197421643247451524op_bit @ code_integer @ X @ Xa ) )
= ( bit_se4197421643247451524op_bit @ int @ X @ ( code_int_of_integer @ Xa ) ) ) ).
% drop_bit_integer.rep_eq
thf(fact_1956_drop__bit__integer_Oabs__eq,axiom,
! [Xa: nat,X: int] :
( ( bit_se4197421643247451524op_bit @ code_integer @ Xa @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( bit_se4197421643247451524op_bit @ int @ Xa @ X ) ) ) ).
% drop_bit_integer.abs_eq
thf(fact_1957_max__add__distrib__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( plus_plus @ A @ ( ord_max @ A @ X @ Y ) @ Z2 )
= ( ord_max @ A @ ( plus_plus @ A @ X @ Z2 ) @ ( plus_plus @ A @ Y @ Z2 ) ) ) ) ).
% max_add_distrib_left
thf(fact_1958_max__add__distrib__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( plus_plus @ A @ X @ ( ord_max @ A @ Y @ Z2 ) )
= ( ord_max @ A @ ( plus_plus @ A @ X @ Y ) @ ( plus_plus @ A @ X @ Z2 ) ) ) ) ).
% max_add_distrib_right
thf(fact_1959_max__diff__distrib__left,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( minus_minus @ A @ ( ord_max @ A @ X @ Y ) @ Z2 )
= ( ord_max @ A @ ( minus_minus @ A @ X @ Z2 ) @ ( minus_minus @ A @ Y @ Z2 ) ) ) ) ).
% max_diff_distrib_left
thf(fact_1960_translation__Compl,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,T2: set @ A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ ( uminus_uminus @ ( set @ A ) @ T2 ) )
= ( uminus_uminus @ ( set @ A ) @ ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ T2 ) ) ) ) ).
% translation_Compl
thf(fact_1961_nat__add__max__right,axiom,
! [M2: nat,N: nat,Q3: nat] :
( ( plus_plus @ nat @ M2 @ ( ord_max @ nat @ N @ Q3 ) )
= ( ord_max @ nat @ ( plus_plus @ nat @ M2 @ N ) @ ( plus_plus @ nat @ M2 @ Q3 ) ) ) ).
% nat_add_max_right
thf(fact_1962_nat__add__max__left,axiom,
! [M2: nat,N: nat,Q3: nat] :
( ( plus_plus @ nat @ ( ord_max @ nat @ M2 @ N ) @ Q3 )
= ( ord_max @ nat @ ( plus_plus @ nat @ M2 @ Q3 ) @ ( plus_plus @ nat @ N @ Q3 ) ) ) ).
% nat_add_max_left
thf(fact_1963_image__eq__imp__comp,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F: B > A,A4: set @ B,G: C > A,B6: set @ C,H3: A > D] :
( ( ( image2 @ B @ A @ F @ A4 )
= ( image2 @ C @ A @ G @ B6 ) )
=> ( ( image2 @ B @ D @ ( comp @ A @ D @ B @ H3 @ F ) @ A4 )
= ( image2 @ C @ D @ ( comp @ A @ D @ C @ H3 @ G ) @ B6 ) ) ) ).
% image_eq_imp_comp
thf(fact_1964_image__comp,axiom,
! [B: $tType,A: $tType,C: $tType,F: B > A,G: C > B,R: set @ C] :
( ( image2 @ B @ A @ F @ ( image2 @ C @ B @ G @ R ) )
= ( image2 @ C @ A @ ( comp @ B @ A @ C @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_1965_map__fun__def,axiom,
! [B: $tType,D: $tType,A: $tType,C: $tType] :
( ( map_fun @ C @ A @ B @ D )
= ( ^ [F2: C > A,G2: B > D,H: A > B] : ( comp @ A @ D @ C @ ( comp @ B @ D @ A @ G2 @ H ) @ F2 ) ) ) ).
% map_fun_def
thf(fact_1966_map__fun_Ocompositionality,axiom,
! [D: $tType,F5: $tType,C: $tType,E: $tType,B: $tType,A: $tType,F: E > C,G: D > F5,H3: C > A,I: B > D,Fun: A > B] :
( ( map_fun @ E @ C @ D @ F5 @ F @ G @ ( map_fun @ C @ A @ B @ D @ H3 @ I @ Fun ) )
= ( map_fun @ E @ A @ B @ F5 @ ( comp @ C @ A @ E @ H3 @ F ) @ ( comp @ D @ F5 @ B @ G @ I ) @ Fun ) ) ).
% map_fun.compositionality
thf(fact_1967_map__fun_Ocomp,axiom,
! [E: $tType,C: $tType,A: $tType,F5: $tType,D: $tType,B: $tType,F: E > C,G: D > F5,H3: C > A,I: B > D] :
( ( comp @ ( C > D ) @ ( E > F5 ) @ ( A > B ) @ ( map_fun @ E @ C @ D @ F5 @ F @ G ) @ ( map_fun @ C @ A @ B @ D @ H3 @ I ) )
= ( map_fun @ E @ A @ B @ F5 @ ( comp @ C @ A @ E @ H3 @ F ) @ ( comp @ D @ F5 @ B @ G @ I ) ) ) ).
% map_fun.comp
thf(fact_1968_translation__subtract__Compl,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,T2: set @ A] :
( ( image2 @ A @ A
@ ^ [X3: A] : ( minus_minus @ A @ X3 @ A3 )
@ ( uminus_uminus @ ( set @ A ) @ T2 ) )
= ( uminus_uminus @ ( set @ A )
@ ( image2 @ A @ A
@ ^ [X3: A] : ( minus_minus @ A @ X3 @ A3 )
@ T2 ) ) ) ) ).
% translation_subtract_Compl
thf(fact_1969_nat__minus__add__max,axiom,
! [N: nat,M2: nat] :
( ( plus_plus @ nat @ ( minus_minus @ nat @ N @ M2 ) @ M2 )
= ( ord_max @ nat @ N @ M2 ) ) ).
% nat_minus_add_max
thf(fact_1970_translation__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,S: set @ A,T2: set @ A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ ( minus_minus @ ( set @ A ) @ S @ T2 ) )
= ( minus_minus @ ( set @ A ) @ ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ S ) @ ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ T2 ) ) ) ) ).
% translation_diff
thf(fact_1971_translation__subtract__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,S: set @ A,T2: set @ A] :
( ( image2 @ A @ A
@ ^ [X3: A] : ( minus_minus @ A @ X3 @ A3 )
@ ( minus_minus @ ( set @ A ) @ S @ T2 ) )
= ( minus_minus @ ( set @ A )
@ ( image2 @ A @ A
@ ^ [X3: A] : ( minus_minus @ A @ X3 @ A3 )
@ S )
@ ( image2 @ A @ A
@ ^ [X3: A] : ( minus_minus @ A @ X3 @ A3 )
@ T2 ) ) ) ) ).
% translation_subtract_diff
thf(fact_1972_take__bit__drop__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M2: nat,N: nat,A3: A] :
( ( bit_se2584673776208193580ke_bit @ A @ M2 @ ( bit_se4197421643247451524op_bit @ A @ N @ A3 ) )
= ( bit_se4197421643247451524op_bit @ A @ N @ ( bit_se2584673776208193580ke_bit @ A @ ( plus_plus @ nat @ M2 @ N ) @ A3 ) ) ) ) ).
% take_bit_drop_bit
thf(fact_1973_bit__drop__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se4197421643247451524op_bit @ A @ N @ A3 ) )
= ( comp @ nat @ $o @ nat @ ( bit_se5641148757651400278ts_bit @ A @ A3 ) @ ( plus_plus @ nat @ N ) ) ) ) ).
% bit_drop_bit_eq
thf(fact_1974_div__push__bit__of__1__eq__drop__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A,N: nat] :
( ( divide_divide @ A @ A3 @ ( bit_se4730199178511100633sh_bit @ A @ N @ ( one_one @ A ) ) )
= ( bit_se4197421643247451524op_bit @ A @ N @ A3 ) ) ) ).
% div_push_bit_of_1_eq_drop_bit
thf(fact_1975_bit__iff__and__drop__bit__eq__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A7: A,N5: nat] :
( ( bit_se5824344872417868541ns_and @ A @ ( bit_se4197421643247451524op_bit @ A @ N5 @ A7 ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ) ) ).
% bit_iff_and_drop_bit_eq_1
thf(fact_1976_bits__ident,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( plus_plus @ A @ ( bit_se4730199178511100633sh_bit @ A @ N @ ( bit_se4197421643247451524op_bit @ A @ N @ A3 ) ) @ ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) )
= A3 ) ) ).
% bits_ident
thf(fact_1977_slice__eq__mask,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat,M2: nat,A3: A] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( bit_se2584673776208193580ke_bit @ A @ M2 @ ( bit_se4197421643247451524op_bit @ A @ N @ A3 ) ) )
= ( bit_se5824344872417868541ns_and @ A @ A3 @ ( bit_se5824344872417868541ns_and @ A @ ( bit_se2239418461657761734s_mask @ A @ ( plus_plus @ nat @ M2 @ N ) ) @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ) ) ) ).
% slice_eq_mask
thf(fact_1978_image__add__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [S2: set @ A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ ( zero_zero @ A ) ) @ S2 )
= S2 ) ) ).
% image_add_0
thf(fact_1979_times__int__def,axiom,
( ( times_times @ int )
= ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( 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 @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X3: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X3 @ U2 ) @ ( times_times @ nat @ Y3 @ V2 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X3 @ V2 ) @ ( times_times @ nat @ Y3 @ U2 ) ) ) ) ) ) ) ).
% times_int_def
thf(fact_1980_minus__int__def,axiom,
( ( minus_minus @ int )
= ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( 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 @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X3: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X3 @ V2 ) @ ( plus_plus @ nat @ Y3 @ U2 ) ) ) ) ) ) ).
% minus_int_def
thf(fact_1981_plus__int__def,axiom,
( ( plus_plus @ int )
= ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( 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 @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X3: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X3 @ U2 ) @ ( plus_plus @ nat @ Y3 @ V2 ) ) ) ) ) ) ).
% plus_int_def
thf(fact_1982_mlex__prod__def,axiom,
! [A: $tType] :
( ( mlex_prod @ A )
= ( ^ [F2: A > nat,R6: set @ ( product_prod @ A @ A )] :
( inv_image @ ( product_prod @ nat @ A ) @ A @ ( lex_prod @ nat @ A @ less_than @ R6 )
@ ^ [X3: A] : ( product_Pair @ nat @ A @ ( F2 @ X3 ) @ X3 ) ) ) ) ).
% mlex_prod_def
thf(fact_1983_rat__minus__code,axiom,
! [P5: rat,Q3: rat] :
( ( quotient_of @ ( minus_minus @ rat @ P5 @ Q3 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A7: int,C3: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B5: int,D5: int] : ( normalize @ ( product_Pair @ int @ int @ ( minus_minus @ int @ ( times_times @ int @ A7 @ D5 ) @ ( times_times @ int @ B5 @ C3 ) ) @ ( times_times @ int @ C3 @ D5 ) ) )
@ ( quotient_of @ Q3 ) )
@ ( quotient_of @ P5 ) ) ) ).
% rat_minus_code
thf(fact_1984_rat__plus__code,axiom,
! [P5: rat,Q3: rat] :
( ( quotient_of @ ( plus_plus @ rat @ P5 @ Q3 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A7: int,C3: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B5: int,D5: int] : ( normalize @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ A7 @ D5 ) @ ( times_times @ int @ B5 @ C3 ) ) @ ( times_times @ int @ C3 @ D5 ) ) )
@ ( quotient_of @ Q3 ) )
@ ( quotient_of @ P5 ) ) ) ).
% rat_plus_code
thf(fact_1985_image__Suc__atLeastAtMost,axiom,
! [I: nat,J: nat] :
( ( image2 @ nat @ nat @ suc @ ( set_or1337092689740270186AtMost @ nat @ I @ J ) )
= ( set_or1337092689740270186AtMost @ nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
% image_Suc_atLeastAtMost
thf(fact_1986_image__Suc__atLeastLessThan,axiom,
! [I: nat,J: nat] :
( ( image2 @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ I @ J ) )
= ( set_or7035219750837199246ssThan @ nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
% image_Suc_atLeastLessThan
thf(fact_1987_in__inv__image,axiom,
! [A: $tType,B: $tType,X: A,Y: A,R: set @ ( product_prod @ B @ B ),F: A > B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( inv_image @ B @ A @ R @ F ) )
= ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F @ X ) @ ( F @ Y ) ) @ R ) ) ).
% in_inv_image
thf(fact_1988_pair__imageI,axiom,
! [C: $tType,B: $tType,A: $tType,A3: A,B2: B,A4: set @ ( product_prod @ A @ B ),F: A > B > C] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ A4 )
=> ( member @ C @ ( F @ A3 @ B2 ) @ ( image2 @ ( product_prod @ A @ B ) @ C @ ( product_case_prod @ A @ B @ C @ F ) @ A4 ) ) ) ).
% pair_imageI
thf(fact_1989_int__of__integer__max,axiom,
! [K: code_integer,L: code_integer] :
( ( code_int_of_integer @ ( ord_max @ code_integer @ K @ L ) )
= ( ord_max @ int @ ( code_int_of_integer @ K ) @ ( code_int_of_integer @ L ) ) ) ).
% int_of_integer_max
thf(fact_1990_rat__one__code,axiom,
( ( quotient_of @ ( one_one @ rat ) )
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) ) ).
% rat_one_code
thf(fact_1991_of__nat__of__integer,axiom,
! [K: code_integer] :
( ( semiring_1_of_nat @ code_integer @ ( code_nat_of_integer @ K ) )
= ( ord_max @ code_integer @ ( zero_zero @ code_integer ) @ K ) ) ).
% of_nat_of_integer
thf(fact_1992_rat__zero__code,axiom,
( ( quotient_of @ ( zero_zero @ rat ) )
= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ).
% rat_zero_code
thf(fact_1993_quotient__of__number_I3_J,axiom,
! [K: num] :
( ( quotient_of @ ( numeral_numeral @ rat @ K ) )
= ( product_Pair @ int @ int @ ( numeral_numeral @ int @ K ) @ ( one_one @ int ) ) ) ).
% quotient_of_number(3)
thf(fact_1994_quotient__of__number_I4_J,axiom,
( ( quotient_of @ ( uminus_uminus @ rat @ ( one_one @ rat ) ) )
= ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( one_one @ int ) ) ) ).
% quotient_of_number(4)
thf(fact_1995_quotient__of__number_I5_J,axiom,
! [K: num] :
( ( quotient_of @ ( uminus_uminus @ rat @ ( numeral_numeral @ rat @ K ) ) )
= ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) @ ( one_one @ int ) ) ) ).
% quotient_of_number(5)
thf(fact_1996_diff__rat__def,axiom,
( ( minus_minus @ rat )
= ( ^ [Q5: rat,R3: rat] : ( plus_plus @ rat @ Q5 @ ( uminus_uminus @ rat @ R3 ) ) ) ) ).
% diff_rat_def
thf(fact_1997_quotient__of__div,axiom,
! [R: rat,N: int,D3: int] :
( ( ( quotient_of @ R )
= ( product_Pair @ int @ int @ N @ D3 ) )
=> ( R
= ( divide_divide @ rat @ ( ring_1_of_int @ rat @ N ) @ ( ring_1_of_int @ rat @ D3 ) ) ) ) ).
% quotient_of_div
thf(fact_1998_zero__notin__Suc__image,axiom,
! [A4: set @ nat] :
~ ( member @ nat @ ( zero_zero @ nat ) @ ( image2 @ nat @ nat @ suc @ A4 ) ) ).
% zero_notin_Suc_image
thf(fact_1999_None__notin__image__Some,axiom,
! [A: $tType,A4: set @ A] :
~ ( member @ ( option @ A ) @ ( none @ A ) @ ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ A4 ) ) ).
% None_notin_image_Some
thf(fact_2000_plus__integer__def,axiom,
( ( plus_plus @ code_integer )
= ( map_fun @ code_integer @ int @ ( int > int ) @ ( 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
thf(fact_2001_times__integer__def,axiom,
( ( times_times @ code_integer )
= ( map_fun @ code_integer @ int @ ( int > int ) @ ( 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
thf(fact_2002_minus__integer__def,axiom,
( ( minus_minus @ code_integer )
= ( map_fun @ code_integer @ int @ ( int > int ) @ ( 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
thf(fact_2003_divide__integer__def,axiom,
( ( divide_divide @ code_integer )
= ( map_fun @ code_integer @ int @ ( int > int ) @ ( 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
thf(fact_2004_modulo__integer__def,axiom,
( ( modulo_modulo @ code_integer )
= ( map_fun @ code_integer @ int @ ( int > int ) @ ( 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
thf(fact_2005_and__integer__def,axiom,
( ( bit_se5824344872417868541ns_and @ code_integer )
= ( map_fun @ code_integer @ int @ ( int > int ) @ ( 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
thf(fact_2006_or__integer__def,axiom,
( ( bit_se1065995026697491101ons_or @ code_integer )
= ( map_fun @ code_integer @ int @ ( int > int ) @ ( 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
thf(fact_2007_xor__integer__def,axiom,
( ( bit_se5824344971392196577ns_xor @ code_integer )
= ( map_fun @ code_integer @ int @ ( int > int ) @ ( 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
thf(fact_2008_image__int__atLeastAtMost,axiom,
! [A3: nat,B2: nat] :
( ( image2 @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or1337092689740270186AtMost @ nat @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% image_int_atLeastAtMost
thf(fact_2009_image__int__atLeastLessThan,axiom,
! [A3: nat,B2: nat] :
( ( image2 @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or7035219750837199246ssThan @ nat @ A3 @ B2 ) )
= ( set_or7035219750837199246ssThan @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% image_int_atLeastLessThan
thf(fact_2010_quotient__of__denom__pos,axiom,
! [R: rat,P5: int,Q3: int] :
( ( ( quotient_of @ R )
= ( product_Pair @ int @ int @ P5 @ Q3 ) )
=> ( ord_less @ int @ ( zero_zero @ int ) @ Q3 ) ) ).
% quotient_of_denom_pos
thf(fact_2011_image__Suc__atMost,axiom,
! [N: nat] :
( ( image2 @ nat @ nat @ suc @ ( set_ord_atMost @ nat @ N ) )
= ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ ( suc @ N ) ) ) ).
% image_Suc_atMost
thf(fact_2012_image__Suc__lessThan,axiom,
! [N: nat] :
( ( image2 @ nat @ nat @ suc @ ( set_ord_lessThan @ nat @ N ) )
= ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N ) ) ).
% image_Suc_lessThan
thf(fact_2013_inv__image__def,axiom,
! [A: $tType,B: $tType] :
( ( inv_image @ B @ A )
= ( ^ [R3: set @ ( product_prod @ B @ B ),F2: A > B] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) @ R3 ) ) ) ) ) ).
% inv_image_def
thf(fact_2014_rat__uminus__code,axiom,
! [P5: rat] :
( ( quotient_of @ ( uminus_uminus @ rat @ P5 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A7: int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ A7 ) )
@ ( quotient_of @ P5 ) ) ) ).
% rat_uminus_code
thf(fact_2015_rat__abs__code,axiom,
! [P5: rat] :
( ( quotient_of @ ( abs_abs @ rat @ P5 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A7: int] : ( product_Pair @ int @ int @ ( abs_abs @ int @ A7 ) )
@ ( quotient_of @ P5 ) ) ) ).
% rat_abs_code
thf(fact_2016_uminus__integer__def,axiom,
( ( uminus_uminus @ code_integer )
= ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int @ ( uminus_uminus @ int ) ) ) ).
% uminus_integer_def
thf(fact_2017_abs__integer__def,axiom,
( ( abs_abs @ code_integer )
= ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int @ ( abs_abs @ int ) ) ) ).
% abs_integer_def
thf(fact_2018_image__add__int__atLeastLessThan,axiom,
! [L: int,U: int] :
( ( image2 @ int @ int
@ ^ [X3: int] : ( plus_plus @ int @ X3 @ L )
@ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ ( minus_minus @ int @ U @ L ) ) )
= ( set_or7035219750837199246ssThan @ int @ L @ U ) ) ).
% image_add_int_atLeastLessThan
thf(fact_2019_sgn__integer__def,axiom,
( ( sgn_sgn @ code_integer )
= ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int @ ( sgn_sgn @ int ) ) ) ).
% sgn_integer_def
thf(fact_2020_not__integer__def,axiom,
( ( bit_ri4277139882892585799ns_not @ 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
thf(fact_2021_image__add__integer__atLeastLessThan,axiom,
! [L: code_integer,U: code_integer] :
( ( image2 @ code_integer @ code_integer
@ ^ [X3: code_integer] : ( plus_plus @ code_integer @ X3 @ L )
@ ( set_or7035219750837199246ssThan @ code_integer @ ( zero_zero @ code_integer ) @ ( minus_minus @ code_integer @ U @ L ) ) )
= ( set_or7035219750837199246ssThan @ code_integer @ L @ U ) ) ).
% image_add_integer_atLeastLessThan
thf(fact_2022_rat__times__code,axiom,
! [P5: rat,Q3: rat] :
( ( quotient_of @ ( times_times @ rat @ P5 @ Q3 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A7: int,C3: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B5: int,D5: int] : ( normalize @ ( product_Pair @ int @ int @ ( times_times @ int @ A7 @ B5 ) @ ( times_times @ int @ C3 @ D5 ) ) )
@ ( quotient_of @ Q3 ) )
@ ( quotient_of @ P5 ) ) ) ).
% rat_times_code
thf(fact_2023_rat__divide__code,axiom,
! [P5: rat,Q3: rat] :
( ( quotient_of @ ( divide_divide @ rat @ P5 @ Q3 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A7: int,C3: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B5: int,D5: int] : ( normalize @ ( product_Pair @ int @ int @ ( times_times @ int @ A7 @ D5 ) @ ( times_times @ int @ C3 @ B5 ) ) )
@ ( quotient_of @ Q3 ) )
@ ( quotient_of @ P5 ) ) ) ).
% rat_divide_code
thf(fact_2024_image__atLeastZeroLessThan__int,axiom,
! [U: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ U )
=> ( ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ U )
= ( image2 @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_ord_lessThan @ nat @ ( nat2 @ U ) ) ) ) ) ).
% image_atLeastZeroLessThan_int
thf(fact_2025_quotient__of__int,axiom,
! [A3: int] :
( ( quotient_of @ ( of_int @ A3 ) )
= ( product_Pair @ int @ int @ A3 @ ( one_one @ int ) ) ) ).
% quotient_of_int
thf(fact_2026_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 ) ) @ ( ( B > A ) > ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) )
@ ^ [R6: set @ ( product_prod @ A @ A ),S5: set @ ( product_prod @ A @ A ),F2: B > A] : ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( inv_image @ A @ B @ R6 @ F2 ) @ ( inv_image @ A @ B @ S5 @ F2 ) ) ) ) ).
% rp_inv_image_def
thf(fact_2027_rat__inverse__code,axiom,
! [P5: rat] :
( ( quotient_of @ ( inverse_inverse @ rat @ P5 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A7: int,B5: int] :
( if @ ( product_prod @ int @ int )
@ ( A7
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( times_times @ int @ ( sgn_sgn @ int @ A7 ) @ B5 ) @ ( abs_abs @ int @ A7 ) ) )
@ ( quotient_of @ P5 ) ) ) ).
% rat_inverse_code
thf(fact_2028_diff__numeral__special_I8_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( neg_numeral_sub @ A @ one2 @ M2 ) ) ) ).
% diff_numeral_special(8)
thf(fact_2029_diff__numeral__special_I7_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( neg_numeral_sub @ A @ N @ one2 ) ) ) ).
% diff_numeral_special(7)
thf(fact_2030_minus__sub__one__diff__one,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [M2: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( neg_numeral_sub @ A @ M2 @ one2 ) ) @ ( one_one @ A ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) ) ) ).
% minus_sub_one_diff_one
thf(fact_2031_add__neg__numeral__special_I4_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ N ) )
= ( neg_numeral_sub @ A @ N @ one2 ) ) ) ).
% add_neg_numeral_special(4)
thf(fact_2032_int__of__integer__sub,axiom,
! [K: num,L: num] :
( ( code_int_of_integer @ ( neg_numeral_sub @ code_integer @ K @ L ) )
= ( neg_numeral_sub @ int @ K @ L ) ) ).
% int_of_integer_sub
thf(fact_2033_inverse__1,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( inverse_inverse @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% inverse_1
thf(fact_2034_inverse__eq__1__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A] :
( ( ( inverse_inverse @ A @ X )
= ( one_one @ A ) )
= ( X
= ( one_one @ A ) ) ) ) ).
% inverse_eq_1_iff
thf(fact_2035_right__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ ( inverse_inverse @ A @ A3 ) )
= ( one_one @ A ) ) ) ) ).
% right_inverse
thf(fact_2036_left__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ A3 )
= ( one_one @ A ) ) ) ) ).
% left_inverse
thf(fact_2037_inverse__eq__divide__numeral,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W: num] :
( ( inverse_inverse @ A @ ( numeral_numeral @ A @ W ) )
= ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ W ) ) ) ) ).
% inverse_eq_divide_numeral
thf(fact_2038_add__neg__numeral__simps_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num,N: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( numeral_numeral @ A @ N ) )
= ( neg_numeral_sub @ A @ N @ M2 ) ) ) ).
% add_neg_numeral_simps(2)
thf(fact_2039_add__neg__numeral__simps_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num,N: num] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ M2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( neg_numeral_sub @ A @ M2 @ N ) ) ) ).
% add_neg_numeral_simps(1)
thf(fact_2040_semiring__norm_I166_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [V: num,W: num,Y: A] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ V ) @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ Y ) )
= ( plus_plus @ A @ ( neg_numeral_sub @ A @ V @ W ) @ Y ) ) ) ).
% semiring_norm(166)
thf(fact_2041_semiring__norm_I167_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [V: num,W: num,Y: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( plus_plus @ A @ ( numeral_numeral @ A @ W ) @ Y ) )
= ( plus_plus @ A @ ( neg_numeral_sub @ A @ W @ V ) @ Y ) ) ) ).
% semiring_norm(167)
thf(fact_2042_inverse__eq__divide__neg__numeral,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W: num] :
( ( inverse_inverse @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
= ( divide_divide @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ) ).
% inverse_eq_divide_neg_numeral
thf(fact_2043_diff__numeral__special_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num] :
( ( minus_minus @ A @ ( numeral_numeral @ A @ M2 ) @ ( one_one @ A ) )
= ( neg_numeral_sub @ A @ M2 @ one2 ) ) ) ).
% diff_numeral_special(2)
thf(fact_2044_diff__numeral__special_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N: num] :
( ( minus_minus @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) )
= ( neg_numeral_sub @ A @ one2 @ N ) ) ) ).
% diff_numeral_special(1)
thf(fact_2045_add__neg__numeral__special_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num] :
( ( plus_plus @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) )
= ( neg_numeral_sub @ A @ one2 @ M2 ) ) ) ).
% add_neg_numeral_special(1)
thf(fact_2046_add__neg__numeral__special_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( one_one @ A ) )
= ( neg_numeral_sub @ A @ one2 @ M2 ) ) ) ).
% add_neg_numeral_special(2)
thf(fact_2047_add__neg__numeral__special_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ M2 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( neg_numeral_sub @ A @ M2 @ one2 ) ) ) ).
% add_neg_numeral_special(3)
thf(fact_2048_inverse__unique,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A] :
( ( ( times_times @ A @ A3 @ B2 )
= ( one_one @ A ) )
=> ( ( inverse_inverse @ A @ A3 )
= B2 ) ) ) ).
% inverse_unique
thf(fact_2049_inverse__eq__divide,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( inverse_inverse @ A )
= ( divide_divide @ A @ ( one_one @ A ) ) ) ) ).
% inverse_eq_divide
thf(fact_2050_prod__inversef,axiom,
! [A: $tType,B: $tType] :
( ( field @ A )
=> ! [F: B > A,A4: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A @ ( comp @ A @ A @ B @ ( inverse_inverse @ A ) @ F ) @ A4 )
= ( inverse_inverse @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A4 ) ) ) ) ).
% prod_inversef
thf(fact_2051_inverse__le__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ X ) @ ( one_one @ A ) )
= ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
| ( ord_less_eq @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% inverse_le_1_iff
thf(fact_2052_one__less__inverse__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ X ) )
= ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
& ( ord_less @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% one_less_inverse_iff
thf(fact_2053_one__less__inverse,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% one_less_inverse
thf(fact_2054_division__ring__inverse__add,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
= ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ ( plus_plus @ A @ A3 @ B2 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ).
% division_ring_inverse_add
thf(fact_2055_inverse__add,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
= ( times_times @ A @ ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( inverse_inverse @ A @ A3 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ).
% inverse_add
thf(fact_2056_field__class_Ofield__inverse,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ A3 )
= ( one_one @ A ) ) ) ) ).
% field_class.field_inverse
thf(fact_2057_nonzero__inverse__eq__divide,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ A3 )
= ( divide_divide @ A @ ( one_one @ A ) @ A3 ) ) ) ) ).
% nonzero_inverse_eq_divide
thf(fact_2058_one__le__inverse__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ X ) )
= ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
& ( ord_less_eq @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% one_le_inverse_iff
thf(fact_2059_inverse__less__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ X ) @ ( one_one @ A ) )
= ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
| ( ord_less @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% inverse_less_1_iff
thf(fact_2060_one__le__inverse,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% one_le_inverse
thf(fact_2061_Frct__code__post_I5_J,axiom,
! [K: num] :
( ( frct @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( numeral_numeral @ int @ K ) ) )
= ( divide_divide @ rat @ ( one_one @ rat ) @ ( numeral_numeral @ rat @ K ) ) ) ).
% Frct_code_post(5)
thf(fact_2062_Code__Numeral_Odup__def,axiom,
( code_dup
= ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int
@ ^ [K4: int] : ( plus_plus @ int @ K4 @ K4 ) ) ) ).
% Code_Numeral.dup_def
thf(fact_2063_Frct__code__post_I6_J,axiom,
! [K: num,L: num] :
( ( frct @ ( product_Pair @ int @ int @ ( numeral_numeral @ int @ K ) @ ( numeral_numeral @ int @ L ) ) )
= ( divide_divide @ rat @ ( numeral_numeral @ rat @ K ) @ ( numeral_numeral @ rat @ L ) ) ) ).
% Frct_code_post(6)
thf(fact_2064_Frct__code__post_I4_J,axiom,
! [K: num] :
( ( frct @ ( product_Pair @ int @ int @ ( numeral_numeral @ int @ K ) @ ( one_one @ int ) ) )
= ( numeral_numeral @ rat @ K ) ) ).
% Frct_code_post(4)
thf(fact_2065_eq__numeral__iff__iszero_I8_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Y: num] :
( ( ( one_one @ A )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ Y ) ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ one2 @ Y ) ) ) ) ) ).
% eq_numeral_iff_iszero(8)
thf(fact_2066_eq__numeral__iff__iszero_I7_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: num] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) )
= ( one_one @ A ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ X @ one2 ) ) ) ) ) ).
% eq_numeral_iff_iszero(7)
thf(fact_2067_Inf_OINF__image,axiom,
! [B: $tType,A: $tType,C: $tType,Inf: ( set @ A ) > A,G: B > A,F: C > B,A4: set @ C] :
( ( Inf @ ( image2 @ B @ A @ G @ ( image2 @ C @ B @ F @ A4 ) ) )
= ( Inf @ ( image2 @ C @ A @ ( comp @ B @ A @ C @ G @ F ) @ A4 ) ) ) ).
% Inf.INF_image
thf(fact_2068_not__iszero__1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ~ ( ring_1_iszero @ A @ ( one_one @ A ) ) ) ).
% not_iszero_1
thf(fact_2069_Code__Numeral_Odup__code_I1_J,axiom,
( ( code_dup @ ( zero_zero @ code_integer ) )
= ( zero_zero @ code_integer ) ) ).
% Code_Numeral.dup_code(1)
thf(fact_2070_not__iszero__neg__1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ~ ( ring_1_iszero @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% not_iszero_neg_1
thf(fact_2071_dup_Orep__eq,axiom,
! [X: code_integer] :
( ( code_int_of_integer @ ( code_dup @ X ) )
= ( plus_plus @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ X ) ) ) ).
% dup.rep_eq
thf(fact_2072_dup_Oabs__eq,axiom,
! [X: int] :
( ( code_dup @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( plus_plus @ int @ X @ X ) ) ) ).
% dup.abs_eq
thf(fact_2073_eq__numeral__iff__iszero_I2_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: num,Y: num] :
( ( ( numeral_numeral @ A @ X )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ Y ) ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ X @ Y ) ) ) ) ) ).
% eq_numeral_iff_iszero(2)
thf(fact_2074_eq__numeral__iff__iszero_I3_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: num,Y: num] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) )
= ( numeral_numeral @ A @ Y ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ X @ Y ) ) ) ) ) ).
% eq_numeral_iff_iszero(3)
thf(fact_2075_Frct__code__post_I2_J,axiom,
! [A3: int] :
( ( frct @ ( product_Pair @ int @ int @ A3 @ ( zero_zero @ int ) ) )
= ( zero_zero @ rat ) ) ).
% Frct_code_post(2)
thf(fact_2076_Frct__code__post_I1_J,axiom,
! [A3: int] :
( ( frct @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ A3 ) )
= ( zero_zero @ rat ) ) ).
% Frct_code_post(1)
thf(fact_2077_Frct__code__post_I3_J,axiom,
( ( frct @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) )
= ( one_one @ rat ) ) ).
% Frct_code_post(3)
thf(fact_2078_eq__numeral__iff__iszero_I6_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Y: num] :
( ( ( one_one @ A )
= ( numeral_numeral @ A @ Y ) )
= ( ring_1_iszero @ A @ ( neg_numeral_sub @ A @ one2 @ Y ) ) ) ) ).
% eq_numeral_iff_iszero(6)
thf(fact_2079_eq__numeral__iff__iszero_I5_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: num] :
( ( ( numeral_numeral @ A @ X )
= ( one_one @ A ) )
= ( ring_1_iszero @ A @ ( neg_numeral_sub @ A @ X @ one2 ) ) ) ) ).
% eq_numeral_iff_iszero(5)
thf(fact_2080_Frct__code__post_I7_J,axiom,
! [A3: int,B2: int] :
( ( frct @ ( product_Pair @ int @ int @ ( uminus_uminus @ int @ A3 ) @ B2 ) )
= ( uminus_uminus @ rat @ ( frct @ ( product_Pair @ int @ int @ A3 @ B2 ) ) ) ) ).
% Frct_code_post(7)
thf(fact_2081_Frct__code__post_I8_J,axiom,
! [A3: int,B2: int] :
( ( frct @ ( product_Pair @ int @ int @ A3 @ ( uminus_uminus @ int @ B2 ) ) )
= ( uminus_uminus @ rat @ ( frct @ ( product_Pair @ int @ int @ A3 @ B2 ) ) ) ) ).
% Frct_code_post(8)
thf(fact_2082_Sup_OSUP__image,axiom,
! [B: $tType,A: $tType,C: $tType,Sup: ( set @ A ) > A,G: B > A,F: C > B,A4: set @ C] :
( ( Sup @ ( image2 @ B @ A @ G @ ( image2 @ C @ B @ F @ A4 ) ) )
= ( Sup @ ( image2 @ C @ A @ ( comp @ B @ A @ C @ G @ F ) @ A4 ) ) ) ).
% Sup.SUP_image
thf(fact_2083_Code__Numeral_Osub__code_I9_J,axiom,
! [M2: num,N: num] :
( ( code_sub @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
= ( minus_minus @ code_integer @ ( code_dup @ ( code_sub @ M2 @ N ) ) @ ( one_one @ code_integer ) ) ) ).
% Code_Numeral.sub_code(9)
thf(fact_2084_Code__Numeral_Osub__code_I8_J,axiom,
! [M2: num,N: num] :
( ( code_sub @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
= ( plus_plus @ code_integer @ ( code_dup @ ( code_sub @ M2 @ N ) ) @ ( one_one @ code_integer ) ) ) ).
% Code_Numeral.sub_code(8)
thf(fact_2085_less__eq__int__def,axiom,
( ( ord_less_eq @ int )
= ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > $o ) @ ( int > $o ) @ rep_Integ @ ( map_fun @ int @ ( product_prod @ nat @ nat ) @ $o @ $o @ rep_Integ @ ( id @ $o ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X3: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V2: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X3 @ V2 ) @ ( plus_plus @ nat @ U2 @ Y3 ) ) ) ) ) ) ).
% less_eq_int_def
thf(fact_2086_less__int__def,axiom,
( ( ord_less @ int )
= ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > $o ) @ ( int > $o ) @ rep_Integ @ ( map_fun @ int @ ( product_prod @ nat @ nat ) @ $o @ $o @ rep_Integ @ ( id @ $o ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X3: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V2: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X3 @ V2 ) @ ( plus_plus @ nat @ U2 @ Y3 ) ) ) ) ) ) ).
% less_int_def
thf(fact_2087_image__affinity__atLeastAtMost__div__diff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,M2: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image2 @ A @ A
@ ^ [X3: A] : ( minus_minus @ A @ ( divide_divide @ A @ X3 @ M2 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( image2 @ A @ A
@ ^ [X3: A] : ( minus_minus @ A @ ( divide_divide @ A @ X3 @ M2 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( divide_divide @ A @ A3 @ M2 ) @ C2 ) @ ( minus_minus @ A @ ( divide_divide @ A @ B2 @ M2 ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( image2 @ A @ A
@ ^ [X3: A] : ( minus_minus @ A @ ( divide_divide @ A @ X3 @ M2 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( divide_divide @ A @ B2 @ M2 ) @ C2 ) @ ( minus_minus @ A @ ( divide_divide @ A @ A3 @ M2 ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_div_diff
thf(fact_2088_image__affinity__atLeastAtMost__div,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,M2: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image2 @ A @ A
@ ^ [X3: A] : ( plus_plus @ A @ ( divide_divide @ A @ X3 @ M2 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( image2 @ A @ A
@ ^ [X3: A] : ( plus_plus @ A @ ( divide_divide @ A @ X3 @ M2 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( divide_divide @ A @ A3 @ M2 ) @ C2 ) @ ( plus_plus @ A @ ( divide_divide @ A @ B2 @ M2 ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( image2 @ A @ A
@ ^ [X3: A] : ( plus_plus @ A @ ( divide_divide @ A @ X3 @ M2 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( divide_divide @ A @ B2 @ M2 ) @ C2 ) @ ( plus_plus @ A @ ( divide_divide @ A @ A3 @ M2 ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_div
thf(fact_2089_image__affinity__atLeastAtMost__diff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,M2: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image2 @ A @ A
@ ^ [X3: A] : ( minus_minus @ A @ ( times_times @ A @ M2 @ X3 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( image2 @ A @ A
@ ^ [X3: A] : ( minus_minus @ A @ ( times_times @ A @ M2 @ X3 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( times_times @ A @ M2 @ A3 ) @ C2 ) @ ( minus_minus @ A @ ( times_times @ A @ M2 @ B2 ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( image2 @ A @ A
@ ^ [X3: A] : ( minus_minus @ A @ ( times_times @ A @ M2 @ X3 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( times_times @ A @ M2 @ B2 ) @ C2 ) @ ( minus_minus @ A @ ( times_times @ A @ M2 @ A3 ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_diff
thf(fact_2090_id__apply,axiom,
! [A: $tType] :
( ( id @ A )
= ( ^ [X3: A] : X3 ) ) ).
% id_apply
thf(fact_2091_bijective__Empty,axiom,
! [B: $tType,A: $tType] : ( bijective @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% bijective_Empty
thf(fact_2092_image__id,axiom,
! [A: $tType] :
( ( image2 @ A @ A @ ( id @ A ) )
= ( id @ ( set @ A ) ) ) ).
% image_id
thf(fact_2093_fun_Omap__id,axiom,
! [A: $tType,D: $tType,T2: D > A] :
( ( comp @ A @ A @ D @ ( id @ A ) @ T2 )
= T2 ) ).
% fun.map_id
thf(fact_2094_id__comp,axiom,
! [B: $tType,A: $tType,G: A > B] :
( ( comp @ B @ B @ A @ ( id @ B ) @ G )
= G ) ).
% id_comp
thf(fact_2095_comp__id,axiom,
! [B: $tType,A: $tType,F: A > B] :
( ( comp @ A @ B @ A @ F @ ( id @ A ) )
= F ) ).
% comp_id
thf(fact_2096_lessThan__0,axiom,
( ( set_ord_lessThan @ nat @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% lessThan_0
thf(fact_2097_apsnd__id,axiom,
! [B: $tType,A: $tType] :
( ( product_apsnd @ B @ B @ A @ ( id @ B ) )
= ( id @ ( product_prod @ A @ B ) ) ) ).
% apsnd_id
thf(fact_2098_map__mmupd__empty,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B ),V: B] :
( ( map_mmupd @ A @ B @ M2 @ ( bot_bot @ ( set @ A ) ) @ V )
= M2 ) ).
% map_mmupd_empty
thf(fact_2099_set__to__map__empty,axiom,
! [B: $tType,A: $tType] :
( ( set_to_map @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( ^ [X3: A] : ( none @ B ) ) ) ).
% set_to_map_empty
thf(fact_2100_atLeastatMost__empty__iff2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( ~ ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_2101_atLeastatMost__empty__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A] :
( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_2102_atLeastatMost__empty,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% atLeastatMost_empty
thf(fact_2103_atLeastLessThan__empty,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( set_or7035219750837199246ssThan @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% atLeastLessThan_empty
thf(fact_2104_atLeastLessThan__empty__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A] :
( ( ( set_or7035219750837199246ssThan @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% atLeastLessThan_empty_iff
thf(fact_2105_atLeastLessThan__empty__iff2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or7035219750837199246ssThan @ A @ A3 @ B2 ) )
= ( ~ ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% atLeastLessThan_empty_iff2
thf(fact_2106_prod_Oempty,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: B > A] :
( ( groups7121269368397514597t_prod @ B @ A @ G @ ( bot_bot @ ( set @ B ) ) )
= ( one_one @ A ) ) ) ).
% prod.empty
thf(fact_2107_group__add__class_Ominus__comp__minus,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( comp @ A @ A @ A @ ( uminus_uminus @ A ) @ ( uminus_uminus @ A ) )
= ( id @ A ) ) ) ).
% group_add_class.minus_comp_minus
thf(fact_2108_boolean__algebra__class_Ominus__comp__minus,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ( ( comp @ A @ A @ A @ ( uminus_uminus @ A ) @ ( uminus_uminus @ A ) )
= ( id @ A ) ) ) ).
% boolean_algebra_class.minus_comp_minus
thf(fact_2109_rel__of__empty,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o] :
( ( rel_of @ A @ B
@ ^ [X3: A] : ( none @ B )
@ P )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% rel_of_empty
thf(fact_2110_comp__the__Some,axiom,
! [A: $tType] :
( ( comp @ ( option @ A ) @ A @ A @ ( the2 @ A ) @ ( some @ A ) )
= ( id @ A ) ) ).
% comp_the_Some
thf(fact_2111_fun_Omap__id0,axiom,
! [A: $tType,D: $tType] :
( ( comp @ A @ A @ D @ ( id @ A ) )
= ( id @ ( D > A ) ) ) ).
% fun.map_id0
thf(fact_2112_Iio__eq__empty__iff,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( order_bot @ A ) )
=> ! [N: A] :
( ( ( set_ord_lessThan @ A @ N )
= ( bot_bot @ ( set @ A ) ) )
= ( N
= ( bot_bot @ A ) ) ) ) ).
% Iio_eq_empty_iff
thf(fact_2113_option_Omap__id0,axiom,
! [A: $tType] :
( ( map_option @ A @ A @ ( id @ A ) )
= ( id @ ( option @ A ) ) ) ).
% option.map_id0
thf(fact_2114_set__notEmptyE,axiom,
! [A: $tType,S2: set @ A] :
( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [X2: A] :
~ ( member @ A @ X2 @ S2 ) ) ).
% set_notEmptyE
thf(fact_2115_memb__imp__not__empty,axiom,
! [A: $tType,X: A,S2: set @ A] :
( ( member @ A @ X @ S2 )
=> ( S2
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% memb_imp_not_empty
thf(fact_2116_id__def,axiom,
! [A: $tType] :
( ( id @ A )
= ( ^ [X3: A] : X3 ) ) ).
% id_def
thf(fact_2117_eq__id__iff,axiom,
! [A: $tType,F: A > A] :
( ( ! [X3: A] :
( ( F @ X3 )
= X3 ) )
= ( F
= ( id @ A ) ) ) ).
% eq_id_iff
thf(fact_2118_comp__eq__id__dest,axiom,
! [C: $tType,B: $tType,A: $tType,A3: C > B,B2: A > C,C2: A > B,V: A] :
( ( ( comp @ C @ B @ A @ A3 @ B2 )
= ( comp @ B @ B @ A @ ( id @ B ) @ C2 ) )
=> ( ( A3 @ ( B2 @ V ) )
= ( C2 @ V ) ) ) ).
% comp_eq_id_dest
thf(fact_2119_pointfree__idE,axiom,
! [B: $tType,A: $tType,F: B > A,G: A > B,X: A] :
( ( ( comp @ B @ A @ A @ F @ G )
= ( id @ A ) )
=> ( ( F @ ( G @ X ) )
= X ) ) ).
% pointfree_idE
thf(fact_2120_option_Omap__id,axiom,
! [A: $tType,T2: option @ A] :
( ( map_option @ A @ A @ ( id @ A ) @ T2 )
= T2 ) ).
% option.map_id
thf(fact_2121_not__empty__eq__Iic__eq__empty,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [H3: A] :
( ( bot_bot @ ( set @ A ) )
!= ( set_ord_atMost @ A @ H3 ) ) ) ).
% not_empty_eq_Iic_eq_empty
thf(fact_2122_lessThan__non__empty,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [X: A] :
( ( set_ord_lessThan @ A @ X )
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% lessThan_non_empty
thf(fact_2123_subset__minus__empty,axiom,
! [A: $tType,A4: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B6 )
=> ( ( minus_minus @ ( set @ A ) @ A4 @ B6 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_minus_empty
thf(fact_2124_atLeastLessThan0,axiom,
! [M2: nat] :
( ( set_or7035219750837199246ssThan @ nat @ M2 @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% atLeastLessThan0
thf(fact_2125_lessThan__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan @ nat @ N )
= ( bot_bot @ ( set @ nat ) ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% lessThan_empty_iff
thf(fact_2126_set__to__map__empty__iff_I1_J,axiom,
! [B: $tType,A: $tType,S2: set @ ( product_prod @ A @ B )] :
( ( ( set_to_map @ A @ B @ S2 )
= ( ^ [X3: A] : ( none @ B ) ) )
= ( S2
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ).
% set_to_map_empty_iff(1)
thf(fact_2127_Code__Numeral_Osub__code_I1_J,axiom,
( ( code_sub @ one2 @ one2 )
= ( zero_zero @ code_integer ) ) ).
% Code_Numeral.sub_code(1)
thf(fact_2128_set__to__map__empty__iff_I2_J,axiom,
! [B: $tType,A: $tType,S2: set @ ( product_prod @ A @ B )] :
( ( ( ^ [X3: A] : ( none @ B ) )
= ( set_to_map @ A @ B @ S2 ) )
= ( S2
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ).
% set_to_map_empty_iff(2)
thf(fact_2129_less__integer__def,axiom,
( ( ord_less @ code_integer )
= ( map_fun @ code_integer @ int @ ( int > $o ) @ ( code_integer > $o ) @ code_int_of_integer @ ( map_fun @ code_integer @ int @ $o @ $o @ code_int_of_integer @ ( id @ $o ) ) @ ( ord_less @ int ) ) ) ).
% less_integer_def
thf(fact_2130_less__eq__integer__def,axiom,
( ( ord_less_eq @ code_integer )
= ( map_fun @ code_integer @ int @ ( int > $o ) @ ( code_integer > $o ) @ code_int_of_integer @ ( map_fun @ code_integer @ int @ $o @ $o @ code_int_of_integer @ ( id @ $o ) ) @ ( ord_less_eq @ int ) ) ) ).
% less_eq_integer_def
thf(fact_2131_Code__Numeral_Osub__code_I6_J,axiom,
! [M2: num,N: num] :
( ( code_sub @ ( bit0 @ M2 ) @ ( bit0 @ N ) )
= ( code_dup @ ( code_sub @ M2 @ N ) ) ) ).
% Code_Numeral.sub_code(6)
thf(fact_2132_Code__Numeral_Osub__code_I7_J,axiom,
! [M2: num,N: num] :
( ( code_sub @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
= ( code_dup @ ( code_sub @ M2 @ N ) ) ) ).
% Code_Numeral.sub_code(7)
thf(fact_2133_sub_Orep__eq,axiom,
! [X: num,Xa: num] :
( ( code_int_of_integer @ ( code_sub @ X @ Xa ) )
= ( minus_minus @ int @ ( numeral_numeral @ int @ X ) @ ( numeral_numeral @ int @ Xa ) ) ) ).
% sub.rep_eq
thf(fact_2134_sub_Oabs__eq,axiom,
( code_sub
= ( ^ [Xa4: num,X3: num] : ( code_integer_of_int @ ( minus_minus @ int @ ( numeral_numeral @ int @ Xa4 ) @ ( numeral_numeral @ int @ X3 ) ) ) ) ) ).
% sub.abs_eq
thf(fact_2135_image__mult__atLeastAtMost__if,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,X: A,Y: A] :
( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( image2 @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ C2 @ X ) @ ( times_times @ A @ C2 @ Y ) ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less_eq @ A @ X @ Y )
=> ( ( image2 @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ C2 @ Y ) @ ( times_times @ A @ C2 @ X ) ) ) )
& ( ~ ( ord_less_eq @ A @ X @ Y )
=> ( ( image2 @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% image_mult_atLeastAtMost_if
thf(fact_2136_image__mult__atLeastAtMost__if_H,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A,C2: A] :
( ( ( ord_less_eq @ A @ X @ Y )
=> ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( image2 @ A @ A
@ ^ [X3: A] : ( times_times @ A @ X3 @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ X @ C2 ) @ ( times_times @ A @ Y @ C2 ) ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( image2 @ A @ A
@ ^ [X3: A] : ( times_times @ A @ X3 @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ Y @ C2 ) @ ( times_times @ A @ X @ C2 ) ) ) ) ) )
& ( ~ ( ord_less_eq @ A @ X @ Y )
=> ( ( image2 @ A @ A
@ ^ [X3: A] : ( times_times @ A @ X3 @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% image_mult_atLeastAtMost_if'
thf(fact_2137_image__affinity__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,M2: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image2 @ A @ A
@ ^ [X3: A] : ( plus_plus @ A @ ( times_times @ A @ M2 @ X3 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( image2 @ A @ A
@ ^ [X3: A] : ( plus_plus @ A @ ( times_times @ A @ M2 @ X3 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( times_times @ A @ M2 @ A3 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ M2 @ B2 ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( image2 @ A @ A
@ ^ [X3: A] : ( plus_plus @ A @ ( times_times @ A @ M2 @ X3 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( times_times @ A @ M2 @ B2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ M2 @ A3 ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost
thf(fact_2138_and__int_Osimps,axiom,
( ( bit_se5824344872417868541ns_and @ int )
= ( ^ [K4: int,L3: int] :
( if @ int
@ ( ( member @ int @ K4 @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L3 @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
@ ( uminus_uminus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K4 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L3 ) ) ) )
@ ( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K4 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L3 ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% and_int.simps
thf(fact_2139_and__int_Oelims,axiom,
! [X: int,Xa: int,Y: int] :
( ( ( bit_se5824344872417868541ns_and @ int @ X @ Xa )
= Y )
=> ( ( ( ( member @ int @ X @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ Xa @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( Y
= ( uminus_uminus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa ) ) ) ) ) )
& ( ~ ( ( member @ int @ X @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ Xa @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( Y
= ( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ X @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ Xa @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% and_int.elims
thf(fact_2140_and__int_Opelims,axiom,
! [X: int,Xa: int,Y: int] :
( ( ( bit_se5824344872417868541ns_and @ int @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ X @ Xa ) )
=> ~ ( ( ( ( ( member @ int @ X @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ Xa @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( Y
= ( uminus_uminus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa ) ) ) ) ) )
& ( ~ ( ( member @ int @ X @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ Xa @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( Y
= ( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ X @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ Xa @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ X @ Xa ) ) ) ) ) ).
% and_int.pelims
thf(fact_2141_and__int_Opsimps,axiom,
! [K: int,L: int] :
( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ K @ L ) )
=> ( ( ( ( member @ int @ K @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( ( bit_se5824344872417868541ns_and @ int @ K @ L )
= ( uminus_uminus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L ) ) ) ) ) )
& ( ~ ( ( member @ int @ K @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( ( bit_se5824344872417868541ns_and @ int @ K @ L )
= ( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% and_int.psimps
thf(fact_2142_and__int_Opinduct,axiom,
! [A0: int,A1: int,P: int > int > $o] :
( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ A0 @ A1 ) )
=> ( ! [K5: int,L5: int] :
( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ K5 @ L5 ) )
=> ( ( ~ ( ( member @ int @ K5 @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L5 @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( P @ ( divide_divide @ int @ K5 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L5 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) )
=> ( P @ K5 @ L5 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% and_int.pinduct
thf(fact_2143_mask__mod__exp,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N: nat,M2: nat] :
( ( modulo_modulo @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ A ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) )
= ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( ord_min @ nat @ M2 @ N ) ) @ ( one_one @ A ) ) ) ) ).
% mask_mod_exp
thf(fact_2144_pair__leqI2,axiom,
! [A3: nat,B2: nat,S: nat,T2: nat] :
( ( ord_less_eq @ nat @ A3 @ B2 )
=> ( ( ord_less_eq @ nat @ S @ T2 )
=> ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A3 @ S ) @ ( product_Pair @ nat @ nat @ B2 @ T2 ) ) @ fun_pair_leq ) ) ) ).
% pair_leqI2
thf(fact_2145_min__eq__arg_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M2: A,N: A] :
( ( ( ord_min @ A @ M2 @ N )
= M2 )
= ( ord_less_eq @ A @ M2 @ N ) ) ) ).
% min_eq_arg(1)
thf(fact_2146_min__eq__arg_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M2: A,N: A] :
( ( ( ord_min @ A @ M2 @ N )
= N )
= ( ord_less_eq @ A @ N @ M2 ) ) ) ).
% min_eq_arg(2)
thf(fact_2147_min__arg__le_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [N: A,M2: A] :
( ( ord_less_eq @ A @ N @ ( ord_min @ A @ M2 @ N ) )
= ( ( ord_min @ A @ M2 @ N )
= N ) ) ) ).
% min_arg_le(1)
thf(fact_2148_min__arg__le_I2_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [M2: A,N: A] :
( ( ord_less_eq @ A @ M2 @ ( ord_min @ A @ M2 @ N ) )
= ( ( ord_min @ A @ M2 @ N )
= M2 ) ) ) ).
% min_arg_le(2)
thf(fact_2149_min__arg__not__ge_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M2: A,N: A] :
( ( ~ ( ord_less @ A @ ( ord_min @ A @ M2 @ N ) @ M2 ) )
= ( ( ord_min @ A @ M2 @ N )
= M2 ) ) ) ).
% min_arg_not_ge(1)
thf(fact_2150_min__arg__not__ge_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M2: A,N: A] :
( ( ~ ( ord_less @ A @ ( ord_min @ A @ M2 @ N ) @ N ) )
= ( ( ord_min @ A @ M2 @ N )
= N ) ) ) ).
% min_arg_not_ge(2)
thf(fact_2151_min__less__self__conv_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( ord_min @ A @ A3 @ B2 ) @ A3 )
= ( ord_less @ A @ B2 @ A3 ) ) ) ).
% min_less_self_conv(1)
thf(fact_2152_min__less__self__conv_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( ord_min @ A @ A3 @ B2 ) @ B2 )
= ( ord_less @ A @ A3 @ B2 ) ) ) ).
% min_less_self_conv(2)
thf(fact_2153_min__simps_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_min @ A @ A3 @ B2 )
= A3 ) ) ) ).
% min_simps(1)
thf(fact_2154_min__simps_I2_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( ord_min @ A @ A3 @ B2 )
= B2 ) ) ) ).
% min_simps(2)
thf(fact_2155_min__0__1_I2_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ( ( ord_min @ A @ ( one_one @ A ) @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% min_0_1(2)
thf(fact_2156_min__0__1_I1_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ( ( ord_min @ A @ ( zero_zero @ A ) @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% min_0_1(1)
thf(fact_2157_min__0__1_I5_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: num] :
( ( ord_min @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ X ) )
= ( one_one @ A ) ) ) ).
% min_0_1(5)
thf(fact_2158_min__0__1_I6_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: num] :
( ( ord_min @ A @ ( numeral_numeral @ A @ X ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% min_0_1(6)
thf(fact_2159_atLeastAtMost__singleton__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
= ( insert2 @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A3 = B2 )
& ( B2 = C2 ) ) ) ) ).
% atLeastAtMost_singleton_iff
thf(fact_2160_atLeastAtMost__singleton,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A] :
( ( set_or1337092689740270186AtMost @ A @ A3 @ A3 )
= ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% atLeastAtMost_singleton
thf(fact_2161_min__Suc__gt_I2_J,axiom,
! [A3: nat,B2: nat] :
( ( ord_less @ nat @ A3 @ B2 )
=> ( ( ord_min @ nat @ B2 @ ( suc @ A3 ) )
= ( suc @ A3 ) ) ) ).
% min_Suc_gt(2)
thf(fact_2162_min__Suc__gt_I1_J,axiom,
! [A3: nat,B2: nat] :
( ( ord_less @ nat @ A3 @ B2 )
=> ( ( ord_min @ nat @ ( suc @ A3 ) @ B2 )
= ( suc @ A3 ) ) ) ).
% min_Suc_gt(1)
thf(fact_2163_case__prod__Pair,axiom,
! [B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) )
= ( id @ ( product_prod @ A @ B ) ) ) ).
% case_prod_Pair
thf(fact_2164_single__Diff__lessThan,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [K: A] :
( ( minus_minus @ ( set @ A ) @ ( insert2 @ A @ K @ ( bot_bot @ ( set @ A ) ) ) @ ( set_ord_lessThan @ A @ K ) )
= ( insert2 @ A @ K @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% single_Diff_lessThan
thf(fact_2165_min__add__distrib__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( plus_plus @ A @ X @ ( ord_min @ A @ Y @ Z2 ) )
= ( ord_min @ A @ ( plus_plus @ A @ X @ Y ) @ ( plus_plus @ A @ X @ Z2 ) ) ) ) ).
% min_add_distrib_right
thf(fact_2166_min__add__distrib__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( plus_plus @ A @ ( ord_min @ A @ X @ Y ) @ Z2 )
= ( ord_min @ A @ ( plus_plus @ A @ X @ Z2 ) @ ( plus_plus @ A @ Y @ Z2 ) ) ) ) ).
% min_add_distrib_left
thf(fact_2167_min__diff__distrib__left,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( minus_minus @ A @ ( ord_min @ A @ X @ Y ) @ Z2 )
= ( ord_min @ A @ ( minus_minus @ A @ X @ Z2 ) @ ( minus_minus @ A @ Y @ Z2 ) ) ) ) ).
% min_diff_distrib_left
thf(fact_2168_bot__option__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( bot_bot @ ( option @ A ) )
= ( none @ A ) ) ) ).
% bot_option_def
thf(fact_2169_bot__empty__eq2,axiom,
! [B: $tType,A: $tType] :
( ( bot_bot @ ( A > B > $o ) )
= ( ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% bot_empty_eq2
thf(fact_2170_map__fun_Oidentity,axiom,
! [B: $tType,A: $tType] :
( ( map_fun @ A @ A @ B @ B
@ ^ [X3: A] : X3
@ ^ [X3: B] : X3 )
= ( id @ ( A > B ) ) ) ).
% map_fun.identity
thf(fact_2171_map__option_Oidentity,axiom,
! [A: $tType] :
( ( map_option @ A @ A
@ ^ [X3: A] : X3 )
= ( id @ ( option @ A ) ) ) ).
% map_option.identity
thf(fact_2172_minus__max__eq__min,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [X: A,Y: A] :
( ( uminus_uminus @ A @ ( ord_max @ A @ X @ Y ) )
= ( ord_min @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% minus_max_eq_min
thf(fact_2173_minus__min__eq__max,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [X: A,Y: A] :
( ( uminus_uminus @ A @ ( ord_min @ A @ X @ Y ) )
= ( ord_max @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% minus_min_eq_max
thf(fact_2174_insert__minus__eq,axiom,
! [A: $tType,X: A,Y: A,A4: set @ A] :
( ( X != Y )
=> ( ( minus_minus @ ( set @ A ) @ ( insert2 @ A @ X @ A4 ) @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert2 @ A @ X @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% insert_minus_eq
thf(fact_2175_set__minus__singleton__eq,axiom,
! [A: $tType,X: A,X10: set @ A] :
( ~ ( member @ A @ X @ X10 )
=> ( ( minus_minus @ ( set @ A ) @ X10 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X10 ) ) ).
% set_minus_singleton_eq
thf(fact_2176_atLeastAtMost__singleton_H,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A] :
( ( A3 = B2 )
=> ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
= ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% atLeastAtMost_singleton'
thf(fact_2177_remove__subset,axiom,
! [A: $tType,X: A,S2: set @ A] :
( ( member @ A @ X @ S2 )
=> ( ord_less @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ S2 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ S2 ) ) ).
% remove_subset
thf(fact_2178_nat__of__integer__def,axiom,
( code_nat_of_integer
= ( map_fun @ code_integer @ int @ nat @ nat @ code_int_of_integer @ ( id @ nat ) @ nat2 ) ) ).
% nat_of_integer_def
thf(fact_2179_bit__integer__def,axiom,
( ( bit_se5641148757651400278ts_bit @ code_integer )
= ( map_fun @ code_integer @ int @ ( nat > $o ) @ ( nat > $o ) @ code_int_of_integer @ ( id @ ( nat > $o ) ) @ ( bit_se5641148757651400278ts_bit @ int ) ) ) ).
% bit_integer_def
thf(fact_2180_mask__integer__def,axiom,
( ( bit_se2239418461657761734s_mask @ code_integer )
= ( map_fun @ nat @ nat @ int @ code_integer @ ( id @ nat ) @ code_integer_of_int @ ( bit_se2239418461657761734s_mask @ int ) ) ) ).
% mask_integer_def
thf(fact_2181_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( set_or7035219750837199246ssThan @ A )
= ( ^ [A7: A,B5: A] : ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A7 @ B5 ) @ ( insert2 @ A @ B5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% atLeastLessThan_eq_atLeastAtMost_diff
thf(fact_2182_atLeastAtMostPlus1__int__conv,axiom,
! [M2: int,N: int] :
( ( ord_less_eq @ int @ M2 @ ( plus_plus @ int @ ( one_one @ int ) @ N ) )
=> ( ( set_or1337092689740270186AtMost @ int @ M2 @ ( plus_plus @ int @ ( one_one @ int ) @ N ) )
= ( insert2 @ int @ ( plus_plus @ int @ ( one_one @ int ) @ N ) @ ( set_or1337092689740270186AtMost @ int @ M2 @ N ) ) ) ) ).
% atLeastAtMostPlus1_int_conv
thf(fact_2183_simp__from__to,axiom,
( ( set_or1337092689740270186AtMost @ int )
= ( ^ [I3: int,J2: int] : ( if @ ( set @ int ) @ ( ord_less @ int @ J2 @ I3 ) @ ( bot_bot @ ( set @ int ) ) @ ( insert2 @ int @ I3 @ ( set_or1337092689740270186AtMost @ int @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) @ J2 ) ) ) ) ) ).
% simp_from_to
thf(fact_2184_Code__Numeral_Osub__def,axiom,
( code_sub
= ( map_fun @ num @ num @ ( num > int ) @ ( num > code_integer ) @ ( id @ num ) @ ( map_fun @ num @ num @ int @ code_integer @ ( id @ num ) @ code_integer_of_int )
@ ^ [M: num,N5: num] : ( minus_minus @ int @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N5 ) ) ) ) ).
% Code_Numeral.sub_def
thf(fact_2185_num__of__integer__def,axiom,
( code_num_of_integer
= ( map_fun @ code_integer @ int @ num @ num @ code_int_of_integer @ ( id @ num ) @ ( comp @ nat @ num @ int @ num_of_nat @ nat2 ) ) ) ).
% num_of_integer_def
thf(fact_2186_take__bit__integer__def,axiom,
( ( bit_se2584673776208193580ke_bit @ code_integer )
= ( map_fun @ nat @ nat @ ( int > int ) @ ( 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
thf(fact_2187_drop__bit__integer__def,axiom,
( ( bit_se4197421643247451524op_bit @ code_integer )
= ( map_fun @ nat @ nat @ ( int > int ) @ ( 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
thf(fact_2188_push__bit__integer__def,axiom,
( ( bit_se4730199178511100633sh_bit @ code_integer )
= ( map_fun @ nat @ nat @ ( int > int ) @ ( 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
thf(fact_2189_unset__bit__integer__def,axiom,
( ( bit_se2638667681897837118et_bit @ code_integer )
= ( map_fun @ nat @ nat @ ( int > int ) @ ( 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
thf(fact_2190_set__bit__integer__def,axiom,
( ( bit_se5668285175392031749et_bit @ code_integer )
= ( map_fun @ nat @ nat @ ( int > int ) @ ( 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
thf(fact_2191_flip__bit__integer__def,axiom,
( ( bit_se8732182000553998342ip_bit @ code_integer )
= ( map_fun @ nat @ nat @ ( int > int ) @ ( 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
thf(fact_2192_pair__leqI1,axiom,
! [A3: nat,B2: nat,S: nat,T2: nat] :
( ( ord_less @ nat @ A3 @ B2 )
=> ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A3 @ S ) @ ( product_Pair @ nat @ nat @ B2 @ T2 ) ) @ fun_pair_leq ) ) ).
% pair_leqI1
thf(fact_2193_image__minus__const__atLeastLessThan__nat,axiom,
! [C2: nat,Y: nat,X: nat] :
( ( ( ord_less @ nat @ C2 @ Y )
=> ( ( image2 @ nat @ nat
@ ^ [I3: nat] : ( minus_minus @ nat @ I3 @ C2 )
@ ( set_or7035219750837199246ssThan @ nat @ X @ Y ) )
= ( set_or7035219750837199246ssThan @ nat @ ( minus_minus @ nat @ X @ C2 ) @ ( minus_minus @ nat @ Y @ C2 ) ) ) )
& ( ~ ( ord_less @ nat @ C2 @ Y )
=> ( ( ( ord_less @ nat @ X @ Y )
=> ( ( image2 @ nat @ nat
@ ^ [I3: nat] : ( minus_minus @ nat @ I3 @ C2 )
@ ( set_or7035219750837199246ssThan @ nat @ X @ Y ) )
= ( insert2 @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) )
& ( ~ ( ord_less @ nat @ X @ Y )
=> ( ( image2 @ nat @ nat
@ ^ [I3: nat] : ( minus_minus @ nat @ I3 @ C2 )
@ ( set_or7035219750837199246ssThan @ nat @ X @ Y ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% image_minus_const_atLeastLessThan_nat
thf(fact_2194_card__atLeastAtMost__int,axiom,
! [L: int,U: int] :
( ( finite_card @ int @ ( set_or1337092689740270186AtMost @ int @ L @ U ) )
= ( nat2 @ ( plus_plus @ int @ ( minus_minus @ int @ U @ L ) @ ( one_one @ int ) ) ) ) ).
% card_atLeastAtMost_int
thf(fact_2195_range__mod,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( image2 @ nat @ nat
@ ^ [M: nat] : ( modulo_modulo @ nat @ M @ N )
@ ( top_top @ ( set @ nat ) ) )
= ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% range_mod
thf(fact_2196_Some__image__these__eq,axiom,
! [A: $tType,A4: set @ ( option @ A )] :
( ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ ( these @ A @ A4 ) )
= ( collect @ ( option @ A )
@ ^ [X3: option @ A] :
( ( member @ ( option @ A ) @ X3 @ A4 )
& ( X3
!= ( none @ A ) ) ) ) ) ).
% Some_image_these_eq
thf(fact_2197_prod__mono2,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [B6: set @ A,A4: set @ A,F: A > B] :
( ( finite_finite2 @ A @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B6 )
=> ( ! [B3: A] :
( ( member @ A @ B3 @ ( minus_minus @ ( set @ A ) @ B6 @ A4 ) )
=> ( ord_less_eq @ B @ ( one_one @ B ) @ ( F @ B3 ) ) )
=> ( ! [A5: A] :
( ( member @ A @ A5 @ A4 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F @ A5 ) ) )
=> ( ord_less_eq @ B @ ( groups7121269368397514597t_prod @ A @ B @ F @ A4 ) @ ( groups7121269368397514597t_prod @ A @ B @ F @ B6 ) ) ) ) ) ) ) ).
% prod_mono2
thf(fact_2198_the__dflt__None__nonempty,axiom,
! [A: $tType,S2: set @ A] :
( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( dflt_None_set @ A @ S2 )
= ( some @ ( set @ A ) @ S2 ) ) ) ).
% the_dflt_None_nonempty
thf(fact_2199_the__dflt__None__empty,axiom,
! [A: $tType] :
( ( dflt_None_set @ A @ ( bot_bot @ ( set @ A ) ) )
= ( none @ ( set @ A ) ) ) ).
% the_dflt_None_empty
thf(fact_2200_atMost__UNIV__triv,axiom,
! [A: $tType] :
( ( set_ord_atMost @ ( set @ A ) @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ ( set @ A ) ) ) ) ).
% atMost_UNIV_triv
thf(fact_2201_finite__atLeastAtMost,axiom,
! [L: nat,U: nat] : ( finite_finite2 @ nat @ ( set_or1337092689740270186AtMost @ nat @ L @ U ) ) ).
% finite_atLeastAtMost
thf(fact_2202_finite__atLeastLessThan,axiom,
! [L: nat,U: nat] : ( finite_finite2 @ nat @ ( set_or7035219750837199246ssThan @ nat @ L @ U ) ) ).
% finite_atLeastLessThan
thf(fact_2203_int__of__integer__min,axiom,
! [K: code_integer,L: code_integer] :
( ( code_int_of_integer @ ( ord_min @ code_integer @ K @ L ) )
= ( ord_min @ int @ ( code_int_of_integer @ K ) @ ( code_int_of_integer @ L ) ) ) ).
% int_of_integer_min
thf(fact_2204_finite__atMost,axiom,
! [K: nat] : ( finite_finite2 @ nat @ ( set_ord_atMost @ nat @ K ) ) ).
% finite_atMost
thf(fact_2205_finite__lessThan,axiom,
! [K: nat] : ( finite_finite2 @ nat @ ( set_ord_lessThan @ nat @ K ) ) ).
% finite_lessThan
thf(fact_2206_finite__atLeastAtMost__int,axiom,
! [L: int,U: int] : ( finite_finite2 @ int @ ( set_or1337092689740270186AtMost @ int @ L @ U ) ) ).
% finite_atLeastAtMost_int
thf(fact_2207_finite__atLeastLessThan__int,axiom,
! [L: int,U: int] : ( finite_finite2 @ int @ ( set_or7035219750837199246ssThan @ int @ L @ U ) ) ).
% finite_atLeastLessThan_int
thf(fact_2208_finite__atLeastLessThan__integer,axiom,
! [L: code_integer,U: code_integer] : ( finite_finite2 @ code_integer @ ( set_or7035219750837199246ssThan @ code_integer @ L @ U ) ) ).
% finite_atLeastLessThan_integer
thf(fact_2209_finite__atLeastAtMost__integer,axiom,
! [L: code_integer,U: code_integer] : ( finite_finite2 @ code_integer @ ( set_or1337092689740270186AtMost @ code_integer @ L @ U ) ) ).
% finite_atLeastAtMost_integer
thf(fact_2210_finite__option__UNIV,axiom,
! [A: $tType] :
( ( finite_finite2 @ ( option @ A ) @ ( top_top @ ( set @ ( option @ A ) ) ) )
= ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% finite_option_UNIV
thf(fact_2211_card__eq__UNIV,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [S2: set @ A] :
( ( ( finite_card @ A @ S2 )
= ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) )
= ( S2
= ( top_top @ ( set @ A ) ) ) ) ) ).
% card_eq_UNIV
thf(fact_2212_card__eq__UNIV2,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [S2: set @ A] :
( ( ( finite_card @ A @ ( top_top @ ( set @ A ) ) )
= ( finite_card @ A @ S2 ) )
= ( S2
= ( top_top @ ( set @ A ) ) ) ) ) ).
% card_eq_UNIV2
thf(fact_2213_these__empty,axiom,
! [A: $tType] :
( ( these @ A @ ( bot_bot @ ( set @ ( option @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% these_empty
thf(fact_2214_these__insert__None,axiom,
! [A: $tType,A4: set @ ( option @ A )] :
( ( these @ A @ ( insert2 @ ( option @ A ) @ ( none @ A ) @ A4 ) )
= ( these @ A @ A4 ) ) ).
% these_insert_None
thf(fact_2215_the__dflt__None__set,axiom,
! [A: $tType,X: set @ A] :
( ( the_default @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( dflt_None_set @ A @ X ) )
= X ) ).
% the_dflt_None_set
thf(fact_2216_surj__plus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surj_plus
thf(fact_2217_infinite__Icc__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A] :
( ( ~ ( finite_finite2 @ A @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ).
% infinite_Icc_iff
thf(fact_2218_infinite__Ico__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A] :
( ( ~ ( finite_finite2 @ A @ ( set_or7035219750837199246ssThan @ A @ A3 @ B2 ) ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ).
% infinite_Ico_iff
thf(fact_2219_prod_Oinfinite,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,G: B > A] :
( ~ ( finite_finite2 @ B @ A4 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A4 )
= ( one_one @ A ) ) ) ) ).
% prod.infinite
thf(fact_2220_card__ge__UNIV,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [S2: set @ A] :
( ( ord_less_eq @ nat @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) @ ( finite_card @ A @ S2 ) )
= ( S2
= ( top_top @ ( set @ A ) ) ) ) ) ).
% card_ge_UNIV
thf(fact_2221_surj__diff__right,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A] :
( ( image2 @ A @ A
@ ^ [X3: A] : ( minus_minus @ A @ X3 @ A3 )
@ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surj_diff_right
thf(fact_2222_prod__eq__1__iff,axiom,
! [A: $tType,A4: set @ A,F: A > nat] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ( groups7121269368397514597t_prod @ A @ nat @ F @ A4 )
= ( one_one @ nat ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( ( F @ X3 )
= ( one_one @ nat ) ) ) ) ) ) ).
% prod_eq_1_iff
thf(fact_2223_prod_Odelta,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S2: set @ B,A3: B,B2: B > A] :
( ( finite_finite2 @ B @ S2 )
=> ( ( ( member @ B @ A3 @ S2 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K4: B] : ( if @ A @ ( K4 = A3 ) @ ( B2 @ K4 ) @ ( one_one @ A ) )
@ S2 )
= ( B2 @ A3 ) ) )
& ( ~ ( member @ B @ A3 @ S2 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K4: B] : ( if @ A @ ( K4 = A3 ) @ ( B2 @ K4 ) @ ( one_one @ A ) )
@ S2 )
= ( one_one @ A ) ) ) ) ) ) ).
% prod.delta
thf(fact_2224_prod_Odelta_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S2: set @ B,A3: B,B2: B > A] :
( ( finite_finite2 @ B @ S2 )
=> ( ( ( member @ B @ A3 @ S2 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K4: B] : ( if @ A @ ( A3 = K4 ) @ ( B2 @ K4 ) @ ( one_one @ A ) )
@ S2 )
= ( B2 @ A3 ) ) )
& ( ~ ( member @ B @ A3 @ S2 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K4: B] : ( if @ A @ ( A3 = K4 ) @ ( B2 @ K4 ) @ ( one_one @ A ) )
@ S2 )
= ( one_one @ A ) ) ) ) ) ) ).
% prod.delta'
thf(fact_2225_these__insert__Some,axiom,
! [A: $tType,X: A,A4: set @ ( option @ A )] :
( ( these @ A @ ( insert2 @ ( option @ A ) @ ( some @ A @ X ) @ A4 ) )
= ( insert2 @ A @ X @ ( these @ A @ A4 ) ) ) ).
% these_insert_Some
thf(fact_2226_these__image__Some__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( these @ A @ ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ A4 ) )
= A4 ) ).
% these_image_Some_eq
thf(fact_2227_sum_Oinsert,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,X: B,G: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ~ ( member @ B @ X @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert2 @ B @ X @ A4 ) )
= ( plus_plus @ A @ ( G @ X ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 ) ) ) ) ) ) ).
% sum.insert
thf(fact_2228_atLeastLessThan__singleton,axiom,
! [M2: nat] :
( ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ M2 ) )
= ( insert2 @ nat @ M2 @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% atLeastLessThan_singleton
thf(fact_2229_atMost__0,axiom,
( ( set_ord_atMost @ nat @ ( zero_zero @ nat ) )
= ( insert2 @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% atMost_0
thf(fact_2230_card__atLeastLessThan__int,axiom,
! [L: int,U: int] :
( ( finite_card @ int @ ( set_or7035219750837199246ssThan @ int @ L @ U ) )
= ( nat2 @ ( minus_minus @ int @ U @ L ) ) ) ).
% card_atLeastLessThan_int
thf(fact_2231_card__doubleton__eq__2__iff,axiom,
! [A: $tType,A3: A,B2: A] :
( ( ( finite_card @ A @ ( insert2 @ A @ A3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( A3 != B2 ) ) ).
% card_doubleton_eq_2_iff
thf(fact_2232_atMost__eq__UNIV__iff,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X: A] :
( ( ( set_ord_atMost @ A @ X )
= ( top_top @ ( set @ A ) ) )
= ( X
= ( top_top @ A ) ) ) ) ).
% atMost_eq_UNIV_iff
thf(fact_2233_card__le__if__inj__on__rel,axiom,
! [B: $tType,A: $tType,B6: set @ A,A4: set @ B,R: B > A > $o] :
( ( finite_finite2 @ A @ B6 )
=> ( ! [A5: B] :
( ( member @ B @ A5 @ A4 )
=> ? [B8: A] :
( ( member @ A @ B8 @ B6 )
& ( R @ A5 @ B8 ) ) )
=> ( ! [A13: B,A24: B,B3: A] :
( ( member @ B @ A13 @ A4 )
=> ( ( member @ B @ A24 @ A4 )
=> ( ( member @ A @ B3 @ B6 )
=> ( ( R @ A13 @ B3 )
=> ( ( R @ A24 @ B3 )
=> ( A13 = A24 ) ) ) ) ) )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ A4 ) @ ( finite_card @ A @ B6 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_2234_top__option__def,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ( ( top_top @ ( option @ A ) )
= ( some @ A @ ( top_top @ A ) ) ) ) ).
% top_option_def
thf(fact_2235_finite__if__eq__beyond__finite,axiom,
! [A: $tType,S2: set @ A,S3: set @ A] :
( ( finite_finite2 @ A @ S2 )
=> ( finite_finite2 @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [S4: set @ A] :
( ( minus_minus @ ( set @ A ) @ S4 @ S2 )
= ( minus_minus @ ( set @ A ) @ S3 @ S2 ) ) ) ) ) ).
% finite_if_eq_beyond_finite
thf(fact_2236_finite__range__Some,axiom,
! [A: $tType] :
( ( finite_finite2 @ ( option @ A ) @ ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ ( top_top @ ( set @ A ) ) ) )
= ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% finite_range_Some
thf(fact_2237_obtain__subset__with__card__n,axiom,
! [A: $tType,N: nat,S2: set @ A] :
( ( ord_less_eq @ nat @ N @ ( finite_card @ A @ S2 ) )
=> ~ ! [T4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ T4 @ S2 )
=> ( ( ( finite_card @ A @ T4 )
= N )
=> ~ ( finite_finite2 @ A @ T4 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_2238_atLeastAtMost__eq__UNIV__iff,axiom,
! [A: $tType] :
( ( bounded_lattice @ A )
=> ! [X: A,Y: A] :
( ( ( set_or1337092689740270186AtMost @ A @ X @ Y )
= ( top_top @ ( set @ A ) ) )
= ( ( X
= ( bot_bot @ A ) )
& ( Y
= ( top_top @ A ) ) ) ) ) ).
% atLeastAtMost_eq_UNIV_iff
thf(fact_2239_UNIV__option__conv,axiom,
! [A: $tType] :
( ( top_top @ ( set @ ( option @ A ) ) )
= ( insert2 @ ( option @ A ) @ ( none @ A ) @ ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ).
% UNIV_option_conv
thf(fact_2240_bounded__nat__set__is__finite,axiom,
! [N6: set @ nat,N: nat] :
( ! [X2: nat] :
( ( member @ nat @ X2 @ N6 )
=> ( ord_less @ nat @ X2 @ N ) )
=> ( finite_finite2 @ nat @ N6 ) ) ).
% bounded_nat_set_is_finite
thf(fact_2241_finite__nat__set__iff__bounded,axiom,
( ( finite_finite2 @ nat )
= ( ^ [N9: set @ nat] :
? [M: nat] :
! [X3: nat] :
( ( member @ nat @ X3 @ N9 )
=> ( ord_less @ nat @ X3 @ M ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_2242_finite__nat__set__iff__bounded__le,axiom,
( ( finite_finite2 @ nat )
= ( ^ [N9: set @ nat] :
? [M: nat] :
! [X3: nat] :
( ( member @ nat @ X3 @ N9 )
=> ( ord_less_eq @ nat @ X3 @ M ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_2243_surjD,axiom,
! [A: $tType,B: $tType,F: B > A,Y: A] :
( ( ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ? [X2: B] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_2244_surjE,axiom,
! [A: $tType,B: $tType,F: B > A,Y: A] :
( ( ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ~ ! [X2: B] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_2245_surjI,axiom,
! [B: $tType,A: $tType,G: B > A,F: A > B] :
( ! [X2: A] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image2 @ B @ A @ G @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surjI
thf(fact_2246_surj__def,axiom,
! [B: $tType,A: $tType,F: B > A] :
( ( ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
= ( ! [Y3: A] :
? [X3: B] :
( Y3
= ( F @ X3 ) ) ) ) ).
% surj_def
thf(fact_2247_these__empty__eq,axiom,
! [A: $tType,B6: set @ ( option @ A )] :
( ( ( these @ A @ B6 )
= ( bot_bot @ ( set @ A ) ) )
= ( ( B6
= ( bot_bot @ ( set @ ( option @ A ) ) ) )
| ( B6
= ( insert2 @ ( option @ A ) @ ( none @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) ) ) ) ) ).
% these_empty_eq
thf(fact_2248_these__not__empty__eq,axiom,
! [A: $tType,B6: set @ ( option @ A )] :
( ( ( these @ A @ B6 )
!= ( bot_bot @ ( set @ A ) ) )
= ( ( B6
!= ( bot_bot @ ( set @ ( option @ A ) ) ) )
& ( B6
!= ( insert2 @ ( option @ A ) @ ( none @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) ) ) ) ) ).
% these_not_empty_eq
thf(fact_2249_not__UNIV__eq__Icc,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [L4: A,H4: A] :
( ( top_top @ ( set @ A ) )
!= ( set_or1337092689740270186AtMost @ A @ L4 @ H4 ) ) ) ).
% not_UNIV_eq_Icc
thf(fact_2250_infinite__Iic,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_bot @ A ) )
=> ! [A3: A] :
~ ( finite_finite2 @ A @ ( set_ord_atMost @ A @ A3 ) ) ) ).
% infinite_Iic
thf(fact_2251_infinite__Iio,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_bot @ A ) )
=> ! [A3: A] :
~ ( finite_finite2 @ A @ ( set_ord_lessThan @ A @ A3 ) ) ) ).
% infinite_Iio
thf(fact_2252_not__UNIV__eq__Iic,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [H4: A] :
( ( top_top @ ( set @ A ) )
!= ( set_ord_atMost @ A @ H4 ) ) ) ).
% not_UNIV_eq_Iic
thf(fact_2253_card__1__singletonI,axiom,
! [A: $tType,S2: set @ A,X: A] :
( ( finite_finite2 @ A @ S2 )
=> ( ( ( finite_card @ A @ S2 )
= ( one_one @ nat ) )
=> ( ( member @ A @ X @ S2 )
=> ( S2
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% card_1_singletonI
thf(fact_2254_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [K4: nat] :
( ( P @ K4 )
& ( ord_less @ nat @ K4 @ I ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_2255_finite__less__ub,axiom,
! [F: nat > nat,U: nat] :
( ! [N3: nat] : ( ord_less_eq @ nat @ N3 @ ( F @ N3 ) )
=> ( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N5: nat] : ( ord_less_eq @ nat @ ( F @ N5 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_2256_card__insert__disjoint_H,axiom,
! [A: $tType,A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ~ ( member @ A @ X @ A4 )
=> ( ( minus_minus @ nat @ ( finite_card @ A @ ( insert2 @ A @ X @ A4 ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( finite_card @ A @ A4 ) ) ) ) ).
% card_insert_disjoint'
thf(fact_2257_in__these__eq,axiom,
! [A: $tType,X: A,A4: set @ ( option @ A )] :
( ( member @ A @ X @ ( these @ A @ A4 ) )
= ( member @ ( option @ A ) @ ( some @ A @ X ) @ A4 ) ) ).
% in_these_eq
thf(fact_2258_infinite__Icc,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ~ ( finite_finite2 @ A @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) ) ) ) ).
% infinite_Icc
thf(fact_2259_infinite__Ico,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ~ ( finite_finite2 @ A @ ( set_or7035219750837199246ssThan @ A @ A3 @ B2 ) ) ) ) ).
% infinite_Ico
thf(fact_2260_subset__eq__atLeast0__atMost__finite,axiom,
! [N6: set @ nat,N: nat] :
( ( ord_less_eq @ ( set @ nat ) @ N6 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( finite_finite2 @ nat @ N6 ) ) ).
% subset_eq_atLeast0_atMost_finite
thf(fact_2261_subset__eq__atLeast0__lessThan__finite,axiom,
! [N6: set @ nat,N: nat] :
( ( ord_less_eq @ ( set @ nat ) @ N6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( finite_finite2 @ nat @ N6 ) ) ).
% subset_eq_atLeast0_lessThan_finite
thf(fact_2262_atMost__Suc,axiom,
! [K: nat] :
( ( set_ord_atMost @ nat @ ( suc @ K ) )
= ( insert2 @ nat @ ( suc @ K ) @ ( set_ord_atMost @ nat @ K ) ) ) ).
% atMost_Suc
thf(fact_2263_surj__fun__eq,axiom,
! [B: $tType,C: $tType,A: $tType,F: B > A,X10: set @ B,G1: A > C,G22: A > C] :
( ( ( image2 @ B @ A @ F @ X10 )
= ( top_top @ ( set @ A ) ) )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ X10 )
=> ( ( comp @ A @ C @ B @ G1 @ F @ X2 )
= ( comp @ A @ C @ B @ G22 @ F @ X2 ) ) )
=> ( G1 = G22 ) ) ) ).
% surj_fun_eq
thf(fact_2264_comp__surj,axiom,
! [B: $tType,A: $tType,C: $tType,F: B > A,G: A > C] :
( ( ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( ( image2 @ A @ C @ G @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ C ) ) )
=> ( ( image2 @ B @ C @ ( comp @ A @ C @ B @ G @ F ) @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ C ) ) ) ) ) ).
% comp_surj
thf(fact_2265_fun_Oset__map,axiom,
! [B: $tType,A: $tType,D: $tType,F: A > B,V: D > A] :
( ( image2 @ D @ B @ ( comp @ A @ B @ D @ F @ V ) @ ( top_top @ ( set @ D ) ) )
= ( image2 @ A @ B @ F @ ( image2 @ D @ A @ V @ ( top_top @ ( set @ D ) ) ) ) ) ).
% fun.set_map
thf(fact_2266_fun_Omap__cong,axiom,
! [B: $tType,A: $tType,D: $tType,X: D > A,Ya: D > A,F: A > B,G: A > B] :
( ( X = Ya )
=> ( ! [Z3: A] :
( ( member @ A @ Z3 @ ( image2 @ D @ A @ Ya @ ( top_top @ ( set @ D ) ) ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( comp @ A @ B @ D @ F @ X )
= ( comp @ A @ B @ D @ G @ Ya ) ) ) ) ).
% fun.map_cong
thf(fact_2267_fun_Omap__cong0,axiom,
! [B: $tType,A: $tType,D: $tType,X: D > A,F: A > B,G: A > B] :
( ! [Z3: A] :
( ( member @ A @ Z3 @ ( image2 @ D @ A @ X @ ( top_top @ ( set @ D ) ) ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( comp @ A @ B @ D @ F @ X )
= ( comp @ A @ B @ D @ G @ X ) ) ) ).
% fun.map_cong0
thf(fact_2268_fun_Oinj__map__strong,axiom,
! [B: $tType,A: $tType,D: $tType,X: D > A,Xa: D > A,F: A > B,Fa: A > B] :
( ! [Z3: A,Za: A] :
( ( member @ A @ Z3 @ ( image2 @ D @ A @ X @ ( top_top @ ( set @ D ) ) ) )
=> ( ( member @ A @ Za @ ( image2 @ D @ A @ Xa @ ( top_top @ ( set @ D ) ) ) )
=> ( ( ( F @ Z3 )
= ( Fa @ Za ) )
=> ( Z3 = Za ) ) ) )
=> ( ( ( comp @ A @ B @ D @ F @ X )
= ( comp @ A @ B @ D @ Fa @ Xa ) )
=> ( X = Xa ) ) ) ).
% fun.inj_map_strong
thf(fact_2269_lessThan__Suc,axiom,
! [K: nat] :
( ( set_ord_lessThan @ nat @ ( suc @ K ) )
= ( insert2 @ nat @ K @ ( set_ord_lessThan @ nat @ K ) ) ) ).
% lessThan_Suc
thf(fact_2270_not__UNIV__le__Icc,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [L: A,H3: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ ( set_or1337092689740270186AtMost @ A @ L @ H3 ) ) ) ).
% not_UNIV_le_Icc
thf(fact_2271_surj__id,axiom,
! [A: $tType] :
( ( image2 @ A @ A @ ( id @ A ) @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% surj_id
thf(fact_2272_not__UNIV__le__Iic,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [H3: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ ( set_ord_atMost @ A @ H3 ) ) ) ).
% not_UNIV_le_Iic
thf(fact_2273_sum_Ofinite__Collect__op,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I4: set @ B,X: B > A,Y: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I3: B] :
( ( member @ B @ I3 @ I4 )
& ( ( X @ I3 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I3: B] :
( ( member @ B @ I3 @ I4 )
& ( ( Y @ I3 )
!= ( zero_zero @ A ) ) ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I3: B] :
( ( member @ B @ I3 @ I4 )
& ( ( plus_plus @ A @ ( X @ I3 ) @ ( Y @ I3 ) )
!= ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_2274_prod_Ofinite__Collect__op,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I4: set @ B,X: B > A,Y: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I3: B] :
( ( member @ B @ I3 @ I4 )
& ( ( X @ I3 )
!= ( one_one @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I3: B] :
( ( member @ B @ I3 @ I4 )
& ( ( Y @ I3 )
!= ( one_one @ A ) ) ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I3: B] :
( ( member @ B @ I3 @ I4 )
& ( ( times_times @ A @ ( X @ I3 ) @ ( Y @ I3 ) )
!= ( one_one @ A ) ) ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_2275_finite__atLeastZeroLessThan__int,axiom,
! [U: int] : ( finite_finite2 @ int @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ U ) ) ).
% finite_atLeastZeroLessThan_int
thf(fact_2276_card__eq__sum,axiom,
! [A: $tType] :
( ( finite_card @ A )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X3: A] : ( one_one @ nat ) ) ) ).
% card_eq_sum
thf(fact_2277_prod_Ointer__filter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,G: B > A,P: B > $o] :
( ( finite_finite2 @ B @ A4 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G
@ ( collect @ B
@ ^ [X3: B] :
( ( member @ B @ X3 @ A4 )
& ( P @ X3 ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X3: B] : ( if @ A @ ( P @ X3 ) @ ( G @ X3 ) @ ( one_one @ A ) )
@ A4 ) ) ) ) ).
% prod.inter_filter
thf(fact_2278_prod__gen__delta,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S2: set @ B,A3: B,B2: B > A,C2: A] :
( ( finite_finite2 @ B @ S2 )
=> ( ( ( member @ B @ A3 @ S2 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K4: B] : ( if @ A @ ( K4 = A3 ) @ ( B2 @ K4 ) @ C2 )
@ S2 )
= ( times_times @ A @ ( B2 @ A3 ) @ ( power_power @ A @ C2 @ ( minus_minus @ nat @ ( finite_card @ B @ S2 ) @ ( one_one @ nat ) ) ) ) ) )
& ( ~ ( member @ B @ A3 @ S2 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K4: B] : ( if @ A @ ( K4 = A3 ) @ ( B2 @ K4 ) @ C2 )
@ S2 )
= ( power_power @ A @ C2 @ ( finite_card @ B @ S2 ) ) ) ) ) ) ) ).
% prod_gen_delta
thf(fact_2279_finite__atLeastZeroLessThan__integer,axiom,
! [U: code_integer] : ( finite_finite2 @ code_integer @ ( set_or7035219750837199246ssThan @ code_integer @ ( zero_zero @ code_integer ) @ U ) ) ).
% finite_atLeastZeroLessThan_integer
thf(fact_2280_card__2__iff_H,axiom,
! [A: $tType,S2: set @ A] :
( ( ( finite_card @ A @ S2 )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( ? [X3: A] :
( ( member @ A @ X3 @ S2 )
& ? [Y3: A] :
( ( member @ A @ Y3 @ S2 )
& ( X3 != Y3 )
& ! [Z5: A] :
( ( member @ A @ Z5 @ S2 )
=> ( ( Z5 = X3 )
| ( Z5 = Y3 ) ) ) ) ) ) ) ).
% card_2_iff'
thf(fact_2281_sum_Orelated,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [R5: A > A > $o,S2: set @ B,H3: B > A,G: B > A] :
( ( R5 @ ( zero_zero @ A ) @ ( zero_zero @ A ) )
=> ( ! [X13: A,Y12: A,X23: A,Y22: A] :
( ( ( R5 @ X13 @ X23 )
& ( R5 @ Y12 @ Y22 ) )
=> ( R5 @ ( plus_plus @ A @ X13 @ Y12 ) @ ( plus_plus @ A @ X23 @ Y22 ) ) )
=> ( ( finite_finite2 @ B @ S2 )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ S2 )
=> ( R5 @ ( H3 @ X2 ) @ ( G @ X2 ) ) )
=> ( R5 @ ( groups7311177749621191930dd_sum @ B @ A @ H3 @ S2 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ S2 ) ) ) ) ) ) ) ).
% sum.related
thf(fact_2282_prod_Orelated,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [R5: A > A > $o,S2: set @ B,H3: B > A,G: B > A] :
( ( R5 @ ( one_one @ A ) @ ( one_one @ A ) )
=> ( ! [X13: A,Y12: A,X23: A,Y22: A] :
( ( ( R5 @ X13 @ X23 )
& ( R5 @ Y12 @ Y22 ) )
=> ( R5 @ ( times_times @ A @ X13 @ Y12 ) @ ( times_times @ A @ X23 @ Y22 ) ) )
=> ( ( finite_finite2 @ B @ S2 )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ S2 )
=> ( R5 @ ( H3 @ X2 ) @ ( G @ X2 ) ) )
=> ( R5 @ ( groups7121269368397514597t_prod @ B @ A @ H3 @ S2 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ S2 ) ) ) ) ) ) ) ).
% prod.related
thf(fact_2283_sum_Oinsert__if,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,X: B,G: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( ( member @ B @ X @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert2 @ B @ X @ A4 ) )
= ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 ) ) )
& ( ~ ( member @ B @ X @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert2 @ B @ X @ A4 ) )
= ( plus_plus @ A @ ( G @ X ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 ) ) ) ) ) ) ) ).
% sum.insert_if
thf(fact_2284_atLeast0__atMost__Suc,axiom,
! [N: nat] :
( ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) )
= ( insert2 @ nat @ ( suc @ N ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% atLeast0_atMost_Suc
thf(fact_2285_atLeast0__lessThan__Suc,axiom,
! [N: nat] :
( ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) )
= ( insert2 @ nat @ N @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% atLeast0_lessThan_Suc
thf(fact_2286_Icc__eq__insert__lb__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( set_or1337092689740270186AtMost @ nat @ M2 @ N )
= ( insert2 @ nat @ M2 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ N ) ) ) ) ).
% Icc_eq_insert_lb_nat
thf(fact_2287_atLeastAtMostSuc__conv,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ ( suc @ N ) )
=> ( ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) )
= ( insert2 @ nat @ ( suc @ N ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% atLeastAtMostSuc_conv
thf(fact_2288_atLeastAtMost__insertL,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( insert2 @ nat @ M2 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ N ) )
= ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ).
% atLeastAtMost_insertL
thf(fact_2289_prod_Oreindex__bij__witness__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S7: set @ B,T5: set @ C,S2: set @ B,I: C > B,J: B > C,T6: set @ C,G: B > A,H3: C > A] :
( ( finite_finite2 @ B @ S7 )
=> ( ( finite_finite2 @ C @ T5 )
=> ( ! [A5: B] :
( ( member @ B @ A5 @ ( minus_minus @ ( set @ B ) @ S2 @ S7 ) )
=> ( ( I @ ( J @ A5 ) )
= A5 ) )
=> ( ! [A5: B] :
( ( member @ B @ A5 @ ( minus_minus @ ( set @ B ) @ S2 @ S7 ) )
=> ( member @ C @ ( J @ A5 ) @ ( minus_minus @ ( set @ C ) @ T6 @ T5 ) ) )
=> ( ! [B3: C] :
( ( member @ C @ B3 @ ( minus_minus @ ( set @ C ) @ T6 @ T5 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: C] :
( ( member @ C @ B3 @ ( minus_minus @ ( set @ C ) @ T6 @ T5 ) )
=> ( member @ B @ ( I @ B3 ) @ ( minus_minus @ ( set @ B ) @ S2 @ S7 ) ) )
=> ( ! [A5: B] :
( ( member @ B @ A5 @ S7 )
=> ( ( G @ A5 )
= ( one_one @ A ) ) )
=> ( ! [B3: C] :
( ( member @ C @ B3 @ T5 )
=> ( ( H3 @ B3 )
= ( one_one @ A ) ) )
=> ( ! [A5: B] :
( ( member @ B @ A5 @ S2 )
=> ( ( H3 @ ( J @ A5 ) )
= ( G @ A5 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ S2 )
= ( groups7121269368397514597t_prod @ C @ A @ H3 @ T6 ) ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_2290_dflt__None__set__def,axiom,
! [A: $tType] :
( ( dflt_None_set @ A )
= ( ^ [S5: set @ A] :
( if @ ( option @ ( set @ A ) )
@ ( S5
= ( bot_bot @ ( set @ A ) ) )
@ ( none @ ( set @ A ) )
@ ( some @ ( set @ A ) @ S5 ) ) ) ) ).
% dflt_None_set_def
thf(fact_2291_sum__eq__1__iff,axiom,
! [A: $tType,A4: set @ A,F: A > nat] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ( groups7311177749621191930dd_sum @ A @ nat @ F @ A4 )
= ( one_one @ nat ) )
= ( ? [X3: A] :
( ( member @ A @ X3 @ A4 )
& ( ( F @ X3 )
= ( one_one @ nat ) )
& ! [Y3: A] :
( ( member @ A @ Y3 @ A4 )
=> ( ( X3 != Y3 )
=> ( ( F @ Y3 )
= ( zero_zero @ nat ) ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_2292_surj__Compl__image__subset,axiom,
! [A: $tType,B: $tType,F: B > A,A4: set @ B] :
( ( ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ ( image2 @ B @ A @ F @ A4 ) ) @ ( image2 @ B @ A @ F @ ( uminus_uminus @ ( set @ B ) @ A4 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_2293_card__atLeastZeroLessThan__int,axiom,
! [U: int] :
( ( finite_card @ int @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ U ) )
= ( nat2 @ U ) ) ).
% card_atLeastZeroLessThan_int
thf(fact_2294_atMost__nat__numeral,axiom,
! [K: num] :
( ( set_ord_atMost @ nat @ ( numeral_numeral @ nat @ K ) )
= ( insert2 @ nat @ ( numeral_numeral @ nat @ K ) @ ( set_ord_atMost @ nat @ ( pred_numeral @ K ) ) ) ) ).
% atMost_nat_numeral
thf(fact_2295_lessThan__nat__numeral,axiom,
! [K: num] :
( ( set_ord_lessThan @ nat @ ( numeral_numeral @ nat @ K ) )
= ( insert2 @ nat @ ( pred_numeral @ K ) @ ( set_ord_lessThan @ nat @ ( pred_numeral @ K ) ) ) ) ).
% lessThan_nat_numeral
thf(fact_2296_sum__Suc,axiom,
! [A: $tType,F: A > nat,A4: set @ A] :
( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X3: A] : ( suc @ ( F @ X3 ) )
@ A4 )
= ( plus_plus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F @ A4 ) @ ( finite_card @ A @ A4 ) ) ) ).
% sum_Suc
thf(fact_2297_prod_Osetdiff__irrelevant,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G
@ ( minus_minus @ ( set @ B ) @ A4
@ ( collect @ B
@ ^ [X3: B] :
( ( G @ X3 )
= ( one_one @ A ) ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A @ G @ A4 ) ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_2298_notin__range__Some,axiom,
! [A: $tType,X: option @ A] :
( ( ~ ( member @ ( option @ A ) @ X @ ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ ( top_top @ ( set @ A ) ) ) ) )
= ( X
= ( none @ A ) ) ) ).
% notin_range_Some
thf(fact_2299_less__1__prod2,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [I4: set @ A,I: A,F: A > B] :
( ( finite_finite2 @ A @ I4 )
=> ( ( member @ A @ I @ I4 )
=> ( ( ord_less @ B @ ( one_one @ B ) @ ( F @ I ) )
=> ( ! [I2: A] :
( ( member @ A @ I2 @ I4 )
=> ( ord_less_eq @ B @ ( one_one @ B ) @ ( F @ I2 ) ) )
=> ( ord_less @ B @ ( one_one @ B ) @ ( groups7121269368397514597t_prod @ A @ B @ F @ I4 ) ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_2300_less__1__prod,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [I4: set @ A,F: A > B] :
( ( finite_finite2 @ A @ I4 )
=> ( ( I4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [I2: A] :
( ( member @ A @ I2 @ I4 )
=> ( ord_less @ B @ ( one_one @ B ) @ ( F @ I2 ) ) )
=> ( ord_less @ B @ ( one_one @ B ) @ ( groups7121269368397514597t_prod @ A @ B @ F @ I4 ) ) ) ) ) ) ).
% less_1_prod
thf(fact_2301_sum_Oreindex__nontrivial,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,H3: B > C,G: C > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ! [X2: B,Y4: B] :
( ( member @ B @ X2 @ A4 )
=> ( ( member @ B @ Y4 @ A4 )
=> ( ( X2 != Y4 )
=> ( ( ( H3 @ X2 )
= ( H3 @ Y4 ) )
=> ( ( G @ ( H3 @ X2 ) )
= ( zero_zero @ A ) ) ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ C @ A @ G @ ( image2 @ B @ C @ H3 @ A4 ) )
= ( groups7311177749621191930dd_sum @ B @ A @ ( comp @ C @ A @ B @ G @ H3 ) @ A4 ) ) ) ) ) ).
% sum.reindex_nontrivial
thf(fact_2302_prod_Oreindex__nontrivial,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,H3: B > C,G: C > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ! [X2: B,Y4: B] :
( ( member @ B @ X2 @ A4 )
=> ( ( member @ B @ Y4 @ A4 )
=> ( ( X2 != Y4 )
=> ( ( ( H3 @ X2 )
= ( H3 @ Y4 ) )
=> ( ( G @ ( H3 @ X2 ) )
= ( one_one @ A ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ C @ A @ G @ ( image2 @ B @ C @ H3 @ A4 ) )
= ( groups7121269368397514597t_prod @ B @ A @ ( comp @ C @ A @ B @ G @ H3 ) @ A4 ) ) ) ) ) ).
% prod.reindex_nontrivial
thf(fact_2303_sum_Osubset__diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [B6: set @ B,A4: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ B6 @ A4 )
=> ( ( finite_finite2 @ B @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A4 @ B6 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ B6 ) ) ) ) ) ) ).
% sum.subset_diff
thf(fact_2304_prod_Osame__carrier,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C5: set @ B,A4: set @ B,B6: set @ B,G: B > A,H3: B > A] :
( ( finite_finite2 @ B @ C5 )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ C5 )
=> ( ( ord_less_eq @ ( set @ B ) @ B6 @ C5 )
=> ( ! [A5: B] :
( ( member @ B @ A5 @ ( minus_minus @ ( set @ B ) @ C5 @ A4 ) )
=> ( ( G @ A5 )
= ( one_one @ A ) ) )
=> ( ! [B3: B] :
( ( member @ B @ B3 @ ( minus_minus @ ( set @ B ) @ C5 @ B6 ) )
=> ( ( H3 @ B3 )
= ( one_one @ A ) ) )
=> ( ( ( groups7121269368397514597t_prod @ B @ A @ G @ A4 )
= ( groups7121269368397514597t_prod @ B @ A @ H3 @ B6 ) )
= ( ( groups7121269368397514597t_prod @ B @ A @ G @ C5 )
= ( groups7121269368397514597t_prod @ B @ A @ H3 @ C5 ) ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_2305_prod_Osame__carrierI,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C5: set @ B,A4: set @ B,B6: set @ B,G: B > A,H3: B > A] :
( ( finite_finite2 @ B @ C5 )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ C5 )
=> ( ( ord_less_eq @ ( set @ B ) @ B6 @ C5 )
=> ( ! [A5: B] :
( ( member @ B @ A5 @ ( minus_minus @ ( set @ B ) @ C5 @ A4 ) )
=> ( ( G @ A5 )
= ( one_one @ A ) ) )
=> ( ! [B3: B] :
( ( member @ B @ B3 @ ( minus_minus @ ( set @ B ) @ C5 @ B6 ) )
=> ( ( H3 @ B3 )
= ( one_one @ A ) ) )
=> ( ( ( groups7121269368397514597t_prod @ B @ A @ G @ C5 )
= ( groups7121269368397514597t_prod @ B @ A @ H3 @ C5 ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A4 )
= ( groups7121269368397514597t_prod @ B @ A @ H3 @ B6 ) ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_2306_prod_Omono__neutral__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T6: set @ B,S2: set @ B,G: B > A] :
( ( finite_finite2 @ B @ T6 )
=> ( ( ord_less_eq @ ( set @ B ) @ S2 @ T6 )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ ( minus_minus @ ( set @ B ) @ T6 @ S2 ) )
=> ( ( G @ X2 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ S2 )
= ( groups7121269368397514597t_prod @ B @ A @ G @ T6 ) ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_2307_prod_Omono__neutral__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T6: set @ B,S2: set @ B,G: B > A] :
( ( finite_finite2 @ B @ T6 )
=> ( ( ord_less_eq @ ( set @ B ) @ S2 @ T6 )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ ( minus_minus @ ( set @ B ) @ T6 @ S2 ) )
=> ( ( G @ X2 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ T6 )
= ( groups7121269368397514597t_prod @ B @ A @ G @ S2 ) ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_2308_prod_Omono__neutral__cong__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T6: set @ B,S2: set @ B,H3: B > A,G: B > A] :
( ( finite_finite2 @ B @ T6 )
=> ( ( ord_less_eq @ ( set @ B ) @ S2 @ T6 )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ ( minus_minus @ ( set @ B ) @ T6 @ S2 ) )
=> ( ( H3 @ X2 )
= ( one_one @ A ) ) )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ S2 )
=> ( ( G @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ S2 )
= ( groups7121269368397514597t_prod @ B @ A @ H3 @ T6 ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_2309_prod_Omono__neutral__cong__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T6: set @ B,S2: set @ B,G: B > A,H3: B > A] :
( ( finite_finite2 @ B @ T6 )
=> ( ( ord_less_eq @ ( set @ B ) @ S2 @ T6 )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ ( minus_minus @ ( set @ B ) @ T6 @ S2 ) )
=> ( ( G @ X2 )
= ( one_one @ A ) ) )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ S2 )
=> ( ( G @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ T6 )
= ( groups7121269368397514597t_prod @ B @ A @ H3 @ S2 ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_2310_atLeastLessThanSuc,axiom,
! [M2: nat,N: nat] :
( ( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ N ) )
= ( insert2 @ nat @ N @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) )
& ( ~ ( ord_less_eq @ nat @ M2 @ N )
=> ( ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ N ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeastLessThanSuc
thf(fact_2311_Option_Othese__def,axiom,
! [A: $tType] :
( ( these @ A )
= ( ^ [A9: set @ ( option @ A )] :
( image2 @ ( option @ A ) @ A @ ( the2 @ A )
@ ( collect @ ( option @ A )
@ ^ [X3: option @ A] :
( ( member @ ( option @ A ) @ X3 @ A9 )
& ( X3
!= ( none @ A ) ) ) ) ) ) ) ).
% Option.these_def
thf(fact_2312_atLeast0__atMost__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) )
= ( insert2 @ nat @ ( zero_zero @ nat ) @ ( image2 @ nat @ nat @ suc @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% atLeast0_atMost_Suc_eq_insert_0
thf(fact_2313_atLeast0__lessThan__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) )
= ( insert2 @ nat @ ( zero_zero @ nat ) @ ( image2 @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% atLeast0_lessThan_Suc_eq_insert_0
thf(fact_2314_atMost__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_ord_atMost @ nat @ ( suc @ N ) )
= ( insert2 @ nat @ ( zero_zero @ nat ) @ ( image2 @ nat @ nat @ suc @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% atMost_Suc_eq_insert_0
thf(fact_2315_lessThan__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_ord_lessThan @ nat @ ( suc @ N ) )
= ( insert2 @ nat @ ( zero_zero @ nat ) @ ( image2 @ nat @ nat @ suc @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% lessThan_Suc_eq_insert_0
thf(fact_2316_card__2__iff,axiom,
! [A: $tType,S2: set @ A] :
( ( ( finite_card @ A @ S2 )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( ? [X3: A,Y3: A] :
( ( S2
= ( insert2 @ A @ X3 @ ( insert2 @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( X3 != Y3 ) ) ) ) ).
% card_2_iff
thf(fact_2317_card__3__iff,axiom,
! [A: $tType,S2: set @ A] :
( ( ( finite_card @ A @ S2 )
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
= ( ? [X3: A,Y3: A,Z5: A] :
( ( S2
= ( insert2 @ A @ X3 @ ( insert2 @ A @ Y3 @ ( insert2 @ A @ Z5 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
& ( X3 != Y3 )
& ( Y3 != Z5 )
& ( X3 != Z5 ) ) ) ) ).
% card_3_iff
thf(fact_2318_sum__image__le,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ordere6911136660526730532id_add @ B )
=> ! [I4: set @ C,G: A > B,F: C > A] :
( ( finite_finite2 @ C @ I4 )
=> ( ! [I2: C] :
( ( member @ C @ I2 @ I4 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( G @ ( F @ I2 ) ) ) )
=> ( ord_less_eq @ B @ ( groups7311177749621191930dd_sum @ A @ B @ G @ ( image2 @ C @ A @ F @ I4 ) ) @ ( groups7311177749621191930dd_sum @ C @ B @ ( comp @ A @ B @ C @ G @ F ) @ I4 ) ) ) ) ) ).
% sum_image_le
thf(fact_2319_sum_Oremove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,X: B,G: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( member @ B @ X @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 )
= ( plus_plus @ A @ ( G @ X ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A4 @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_2320_sum_Oinsert__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,G: B > A,X: B] :
( ( finite_finite2 @ B @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert2 @ B @ X @ A4 ) )
= ( plus_plus @ A @ ( G @ X ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A4 @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_2321_atLeast1__atMost__eq__remove0,axiom,
! [N: nat] :
( ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
= ( minus_minus @ ( set @ nat ) @ ( set_ord_atMost @ nat @ N ) @ ( insert2 @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeast1_atMost_eq_remove0
thf(fact_2322_atLeast1__lessThan__eq__remove0,axiom,
! [N: nat] :
( ( set_or7035219750837199246ssThan @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
= ( minus_minus @ ( set @ nat ) @ ( set_ord_lessThan @ nat @ N ) @ ( insert2 @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeast1_lessThan_eq_remove0
thf(fact_2323_atLeastLessThan__nat__numeral,axiom,
! [M2: nat,K: num] :
( ( ( ord_less_eq @ nat @ M2 @ ( pred_numeral @ K ) )
=> ( ( set_or7035219750837199246ssThan @ nat @ M2 @ ( numeral_numeral @ nat @ K ) )
= ( insert2 @ nat @ ( pred_numeral @ K ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ ( pred_numeral @ K ) ) ) ) )
& ( ~ ( ord_less_eq @ nat @ M2 @ ( pred_numeral @ K ) )
=> ( ( set_or7035219750837199246ssThan @ nat @ M2 @ ( numeral_numeral @ nat @ K ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeastLessThan_nat_numeral
thf(fact_2324_sum_Odelta__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S2: set @ B,A3: B,B2: B > A,C2: B > A] :
( ( finite_finite2 @ B @ S2 )
=> ( ( ( member @ B @ A3 @ S2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K4: B] : ( if @ A @ ( K4 = A3 ) @ ( B2 @ K4 ) @ ( C2 @ K4 ) )
@ S2 )
= ( plus_plus @ A @ ( B2 @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S2 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member @ B @ A3 @ S2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K4: B] : ( if @ A @ ( K4 = A3 ) @ ( B2 @ K4 ) @ ( C2 @ K4 ) )
@ S2 )
= ( groups7311177749621191930dd_sum @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S2 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% sum.delta_remove
thf(fact_2325_prod__le__power,axiom,
! [B: $tType,A: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: set @ B,F: B > A,N: A,K: nat] :
( ! [I2: B] :
( ( member @ B @ I2 @ A4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ I2 ) )
& ( ord_less_eq @ A @ ( F @ I2 ) @ N ) ) )
=> ( ( ord_less_eq @ nat @ ( finite_card @ B @ A4 ) @ K )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ N )
=> ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A4 ) @ ( power_power @ A @ N @ K ) ) ) ) ) ) ).
% prod_le_power
thf(fact_2326_card__Diff__insert,axiom,
! [A: $tType,A3: A,A4: set @ A,B6: set @ A] :
( ( member @ A @ A3 @ A4 )
=> ( ~ ( member @ A @ A3 @ B6 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ B6 ) ) )
= ( minus_minus @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ B6 ) ) @ ( one_one @ nat ) ) ) ) ) ).
% card_Diff_insert
thf(fact_2327_card__insert__le__m1,axiom,
! [A: $tType,N: nat,Y: set @ A,X: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ Y ) @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ ( insert2 @ A @ X @ Y ) ) @ N ) ) ) ).
% card_insert_le_m1
thf(fact_2328_card__Diff__singleton__if,axiom,
! [A: $tType,X: A,A4: set @ A] :
( ( ( member @ A @ X @ A4 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A4 ) @ ( one_one @ nat ) ) ) )
& ( ~ ( member @ A @ X @ A4 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( finite_card @ A @ A4 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_2329_card__Diff__singleton,axiom,
! [A: $tType,X: A,A4: set @ A] :
( ( member @ A @ X @ A4 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A4 ) @ ( one_one @ nat ) ) ) ) ).
% card_Diff_singleton
thf(fact_2330_in__finite__psubset,axiom,
! [A: $tType,A4: set @ A,B6: set @ A] :
( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A4 @ B6 ) @ ( finite_psubset @ A ) )
= ( ( ord_less @ ( set @ A ) @ A4 @ B6 )
& ( finite_finite2 @ A @ B6 ) ) ) ).
% in_finite_psubset
thf(fact_2331_card__1__singletonE,axiom,
! [A: $tType,A4: set @ A] :
( ( ( finite_card @ A @ A4 )
= ( one_one @ nat ) )
=> ~ ! [X2: A] :
( A4
!= ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% card_1_singletonE
thf(fact_2332_integer__of__nat__def,axiom,
( code_integer_of_nat
= ( map_fun @ nat @ nat @ int @ code_integer @ ( id @ nat ) @ code_integer_of_int @ ( semiring_1_of_nat @ int ) ) ) ).
% integer_of_nat_def
thf(fact_2333_card__UNIV__unit,axiom,
( ( finite_card @ product_unit @ ( top_top @ ( set @ product_unit ) ) )
= ( one_one @ nat ) ) ).
% card_UNIV_unit
thf(fact_2334_card__lessThan,axiom,
! [U: nat] :
( ( finite_card @ nat @ ( set_ord_lessThan @ nat @ U ) )
= U ) ).
% card_lessThan
thf(fact_2335_card__atLeastLessThan,axiom,
! [L: nat,U: nat] :
( ( finite_card @ nat @ ( set_or7035219750837199246ssThan @ nat @ L @ U ) )
= ( minus_minus @ nat @ U @ L ) ) ).
% card_atLeastLessThan
thf(fact_2336_card__atMost,axiom,
! [U: nat] :
( ( finite_card @ nat @ ( set_ord_atMost @ nat @ U ) )
= ( suc @ U ) ) ).
% card_atMost
thf(fact_2337_Collect__const__case__prod,axiom,
! [B: $tType,A: $tType,P: $o] :
( ( P
=> ( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A7: A,B5: B] : P ) )
= ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) )
& ( ~ P
=> ( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A7: A,B5: B] : P ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% Collect_const_case_prod
thf(fact_2338_nat__of__integer__integer__of__nat,axiom,
! [N: nat] :
( ( code_nat_of_integer @ ( code_integer_of_nat @ N ) )
= N ) ).
% nat_of_integer_integer_of_nat
thf(fact_2339_card__atLeastAtMost,axiom,
! [L: nat,U: nat] :
( ( finite_card @ nat @ ( set_or1337092689740270186AtMost @ nat @ L @ U ) )
= ( minus_minus @ nat @ ( suc @ U ) @ L ) ) ).
% card_atLeastAtMost
thf(fact_2340_integer__of__nat_Orep__eq,axiom,
! [X: nat] :
( ( code_int_of_integer @ ( code_integer_of_nat @ X ) )
= ( semiring_1_of_nat @ int @ X ) ) ).
% integer_of_nat.rep_eq
thf(fact_2341_int__of__integer__integer__of__nat,axiom,
! [N: nat] :
( ( code_int_of_integer @ ( code_integer_of_nat @ N ) )
= ( semiring_1_of_nat @ int @ N ) ) ).
% int_of_integer_integer_of_nat
thf(fact_2342_top__empty__eq2,axiom,
! [B: $tType,A: $tType] :
( ( top_top @ ( A > B > $o ) )
= ( ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% top_empty_eq2
thf(fact_2343_int__of__integer__induct,axiom,
! [Y: int,P: int > $o] :
( ( member @ int @ Y @ ( top_top @ ( set @ int ) ) )
=> ( ! [X2: code_integer] : ( P @ ( code_int_of_integer @ X2 ) )
=> ( P @ Y ) ) ) ).
% int_of_integer_induct
thf(fact_2344_int__of__integer__cases,axiom,
! [Y: int] :
( ( member @ int @ Y @ ( top_top @ ( set @ int ) ) )
=> ~ ! [X2: code_integer] :
( Y
!= ( code_int_of_integer @ X2 ) ) ) ).
% int_of_integer_cases
thf(fact_2345_int__of__integer,axiom,
! [X: code_integer] : ( member @ int @ ( code_int_of_integer @ X ) @ ( top_top @ ( set @ int ) ) ) ).
% int_of_integer
thf(fact_2346_integer__of__int__inject,axiom,
! [X: int,Y: int] :
( ( member @ int @ X @ ( top_top @ ( set @ int ) ) )
=> ( ( member @ int @ Y @ ( top_top @ ( set @ int ) ) )
=> ( ( ( code_integer_of_int @ X )
= ( code_integer_of_int @ Y ) )
= ( X = Y ) ) ) ) ).
% integer_of_int_inject
thf(fact_2347_integer__of__int__induct,axiom,
! [P: code_integer > $o,X: code_integer] :
( ! [Y4: int] :
( ( member @ int @ Y4 @ ( top_top @ ( set @ int ) ) )
=> ( P @ ( code_integer_of_int @ Y4 ) ) )
=> ( P @ X ) ) ).
% integer_of_int_induct
thf(fact_2348_integer__of__int__cases,axiom,
! [X: code_integer] :
~ ! [Y4: int] :
( ( X
= ( code_integer_of_int @ Y4 ) )
=> ~ ( member @ int @ Y4 @ ( top_top @ ( set @ int ) ) ) ) ).
% integer_of_int_cases
thf(fact_2349_subset__card__intvl__is__intvl,axiom,
! [A4: set @ nat,K: nat] :
( ( ord_less_eq @ ( set @ nat ) @ A4 @ ( set_or7035219750837199246ssThan @ nat @ K @ ( plus_plus @ nat @ K @ ( finite_card @ nat @ A4 ) ) ) )
=> ( A4
= ( set_or7035219750837199246ssThan @ nat @ K @ ( plus_plus @ nat @ K @ ( finite_card @ nat @ A4 ) ) ) ) ) ).
% subset_card_intvl_is_intvl
thf(fact_2350_integer__of__int__inverse,axiom,
! [Y: int] :
( ( member @ int @ Y @ ( top_top @ ( set @ int ) ) )
=> ( ( code_int_of_integer @ ( code_integer_of_int @ Y ) )
= Y ) ) ).
% integer_of_int_inverse
thf(fact_2351_subset__eq__atLeast0__lessThan__card,axiom,
! [N6: set @ nat,N: nat] :
( ( ord_less_eq @ ( set @ nat ) @ N6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( ord_less_eq @ nat @ ( finite_card @ nat @ N6 ) @ N ) ) ).
% subset_eq_atLeast0_lessThan_card
thf(fact_2352_integer__of__nat__eq__of__nat,axiom,
( code_integer_of_nat
= ( semiring_1_of_nat @ code_integer ) ) ).
% integer_of_nat_eq_of_nat
thf(fact_2353_card__less__Suc2,axiom,
! [M3: set @ nat,I: nat] :
( ~ ( member @ nat @ ( zero_zero @ nat ) @ M3 )
=> ( ( finite_card @ nat
@ ( collect @ nat
@ ^ [K4: nat] :
( ( member @ nat @ ( suc @ K4 ) @ M3 )
& ( ord_less @ nat @ K4 @ I ) ) ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [K4: nat] :
( ( member @ nat @ K4 @ M3 )
& ( ord_less @ nat @ K4 @ ( suc @ I ) ) ) ) ) ) ) ).
% card_less_Suc2
thf(fact_2354_card__less__Suc,axiom,
! [M3: set @ nat,I: nat] :
( ( member @ nat @ ( zero_zero @ nat ) @ M3 )
=> ( ( suc
@ ( finite_card @ nat
@ ( collect @ nat
@ ^ [K4: nat] :
( ( member @ nat @ ( suc @ K4 ) @ M3 )
& ( ord_less @ nat @ K4 @ I ) ) ) ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [K4: nat] :
( ( member @ nat @ K4 @ M3 )
& ( ord_less @ nat @ K4 @ ( suc @ I ) ) ) ) ) ) ) ).
% card_less_Suc
thf(fact_2355_card__less,axiom,
! [M3: set @ nat,I: nat] :
( ( member @ nat @ ( zero_zero @ nat ) @ M3 )
=> ( ( finite_card @ nat
@ ( collect @ nat
@ ^ [K4: nat] :
( ( member @ nat @ K4 @ M3 )
& ( ord_less @ nat @ K4 @ ( suc @ I ) ) ) ) )
!= ( zero_zero @ nat ) ) ) ).
% card_less
thf(fact_2356_integer__of__nat__0,axiom,
( ( code_integer_of_nat @ ( zero_zero @ nat ) )
= ( zero_zero @ code_integer ) ) ).
% integer_of_nat_0
thf(fact_2357_integer__of__nat_Oabs__eq,axiom,
( code_integer_of_nat
= ( ^ [X3: nat] : ( code_integer_of_int @ ( semiring_1_of_nat @ int @ X3 ) ) ) ) ).
% integer_of_nat.abs_eq
thf(fact_2358_integer__of__nat__numeral,axiom,
! [N: num] :
( ( code_integer_of_nat @ ( numeral_numeral @ nat @ N ) )
= ( numeral_numeral @ code_integer @ N ) ) ).
% integer_of_nat_numeral
thf(fact_2359_integer__of__nat__1,axiom,
( ( code_integer_of_nat @ ( one_one @ nat ) )
= ( one_one @ code_integer ) ) ).
% integer_of_nat_1
thf(fact_2360_card__sum__le__nat__sum,axiom,
! [S2: set @ nat] :
( ord_less_eq @ nat
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X3: nat] : X3
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( finite_card @ nat @ S2 ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X3: nat] : X3
@ S2 ) ) ).
% card_sum_le_nat_sum
thf(fact_2361_set__encode__insert,axiom,
! [A4: set @ nat,N: nat] :
( ( finite_finite2 @ nat @ A4 )
=> ( ~ ( member @ nat @ N @ A4 )
=> ( ( nat_set_encode @ ( insert2 @ nat @ N @ A4 ) )
= ( plus_plus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( nat_set_encode @ A4 ) ) ) ) ) ).
% set_encode_insert
thf(fact_2362_card__greaterThanLessThan__int,axiom,
! [L: int,U: int] :
( ( finite_card @ int @ ( set_or5935395276787703475ssThan @ int @ L @ U ) )
= ( nat2 @ ( minus_minus @ int @ U @ ( plus_plus @ int @ L @ ( one_one @ int ) ) ) ) ) ).
% card_greaterThanLessThan_int
thf(fact_2363_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 )] :
( ( member @ ( product_prod @ nat @ nat ) @ Y @ YS )
=> ( ( member @ ( 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 )
=> ( ( member @ ( 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 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( insert2 @ ( product_prod @ nat @ nat ) @ X @ XS ) @ YS ) @ fun_max_weak ) ) ) ) ).
% wmax_insertI
thf(fact_2364_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 )] :
( ( member @ ( product_prod @ nat @ nat ) @ X @ XS )
=> ( ( member @ ( 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 )
=> ( ( member @ ( 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 )
=> ( member @ ( 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 @ ( insert2 @ ( product_prod @ nat @ nat ) @ Y @ YS ) ) @ fun_min_weak ) ) ) ) ).
% wmin_insertI
thf(fact_2365_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 )] :
( ( member @ ( product_prod @ nat @ nat ) @ X @ XS )
=> ( ( member @ ( 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 )
=> ( ( member @ ( 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 )
=> ( member @ ( 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 @ ( insert2 @ ( product_prod @ nat @ nat ) @ Y @ YS ) ) @ fun_min_strict ) ) ) ) ).
% smin_insertI
thf(fact_2366_prod_Oinsert_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I4: set @ B,P5: B > A,I: B] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X3: B] :
( ( member @ B @ X3 @ I4 )
& ( ( P5 @ X3 )
!= ( one_one @ A ) ) ) ) )
=> ( ( ( member @ B @ I @ I4 )
=> ( ( groups1962203154675924110t_prod @ B @ A @ P5 @ ( insert2 @ B @ I @ I4 ) )
= ( groups1962203154675924110t_prod @ B @ A @ P5 @ I4 ) ) )
& ( ~ ( member @ B @ I @ I4 )
=> ( ( groups1962203154675924110t_prod @ B @ A @ P5 @ ( insert2 @ B @ I @ I4 ) )
= ( times_times @ A @ ( P5 @ I ) @ ( groups1962203154675924110t_prod @ B @ A @ P5 @ I4 ) ) ) ) ) ) ) ).
% prod.insert'
thf(fact_2367_greaterThanLessThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,L: A,U: A] :
( ( member @ A @ I @ ( set_or5935395276787703475ssThan @ A @ L @ U ) )
= ( ( ord_less @ A @ L @ I )
& ( ord_less @ A @ I @ U ) ) ) ) ).
% greaterThanLessThan_iff
thf(fact_2368_finite__greaterThanLessThan__int,axiom,
! [L: int,U: int] : ( finite_finite2 @ int @ ( set_or5935395276787703475ssThan @ int @ L @ U ) ) ).
% finite_greaterThanLessThan_int
thf(fact_2369_greaterThanLessThan__empty__iff2,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% greaterThanLessThan_empty_iff2
thf(fact_2370_greaterThanLessThan__empty__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A] :
( ( ( set_or5935395276787703475ssThan @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% greaterThanLessThan_empty_iff
thf(fact_2371_greaterThanLessThan__empty,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,K: A] :
( ( ord_less_eq @ A @ L @ K )
=> ( ( set_or5935395276787703475ssThan @ A @ K @ L )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% greaterThanLessThan_empty
thf(fact_2372_infinite__Ioo__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A] :
( ( ~ ( finite_finite2 @ A @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ).
% infinite_Ioo_iff
thf(fact_2373_image__uminus__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A,Y: A] :
( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_or5935395276787703475ssThan @ A @ X @ Y ) )
= ( set_or5935395276787703475ssThan @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% image_uminus_greaterThanLessThan
thf(fact_2374_prod_Oempty_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [P5: B > A] :
( ( groups1962203154675924110t_prod @ B @ A @ P5 @ ( bot_bot @ ( set @ B ) ) )
= ( one_one @ A ) ) ) ).
% prod.empty'
thf(fact_2375_smin__emptyI,axiom,
! [X10: set @ ( product_prod @ nat @ nat )] :
( ( X10
!= ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ X10 @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ fun_min_strict ) ) ).
% smin_emptyI
thf(fact_2376_wmax__emptyI,axiom,
! [X10: set @ ( product_prod @ nat @ nat )] :
( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ X10 )
=> ( member @ ( 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 ) ) ) @ X10 ) @ fun_max_weak ) ) ).
% wmax_emptyI
thf(fact_2377_wmin__emptyI,axiom,
! [X10: set @ ( product_prod @ nat @ nat )] : ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ X10 @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ fun_min_weak ) ).
% wmin_emptyI
thf(fact_2378_prod_Onon__neutral_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: B > A,I4: set @ B] :
( ( groups1962203154675924110t_prod @ B @ A @ G
@ ( collect @ B
@ ^ [X3: B] :
( ( member @ B @ X3 @ I4 )
& ( ( G @ X3 )
!= ( one_one @ A ) ) ) ) )
= ( groups1962203154675924110t_prod @ B @ A @ G @ I4 ) ) ) ).
% prod.non_neutral'
thf(fact_2379_infinite__Ioo,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ~ ( finite_finite2 @ A @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) ) ) ) ).
% infinite_Ioo
thf(fact_2380_greaterThanLessThan__subseteq__greaterThanLessThan,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) @ ( set_or5935395276787703475ssThan @ A @ C2 @ D3 ) )
= ( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A3 )
& ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_greaterThanLessThan
thf(fact_2381_prod__decode__aux_Ocases,axiom,
! [X: product_prod @ nat @ nat] :
~ ! [K5: nat,M5: nat] :
( X
!= ( product_Pair @ nat @ nat @ K5 @ M5 ) ) ).
% prod_decode_aux.cases
thf(fact_2382_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
! [L: int,U: int] :
( ( set_or7035219750837199246ssThan @ int @ ( plus_plus @ int @ L @ ( one_one @ int ) ) @ U )
= ( set_or5935395276787703475ssThan @ int @ L @ U ) ) ).
% atLeastPlusOneLessThan_greaterThanLessThan_int
thf(fact_2383_prod_Omono__neutral__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S2: set @ B,T6: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S2 @ T6 )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ ( minus_minus @ ( set @ B ) @ T6 @ S2 ) )
=> ( ( G @ X2 )
= ( one_one @ A ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G @ S2 )
= ( groups1962203154675924110t_prod @ B @ A @ G @ T6 ) ) ) ) ) ).
% prod.mono_neutral_left'
thf(fact_2384_prod_Omono__neutral__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S2: set @ B,T6: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S2 @ T6 )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ ( minus_minus @ ( set @ B ) @ T6 @ S2 ) )
=> ( ( G @ X2 )
= ( one_one @ A ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G @ T6 )
= ( groups1962203154675924110t_prod @ B @ A @ G @ S2 ) ) ) ) ) ).
% prod.mono_neutral_right'
thf(fact_2385_prod_Omono__neutral__cong__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S2: set @ B,T6: set @ B,H3: B > A,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S2 @ T6 )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ ( minus_minus @ ( set @ B ) @ T6 @ S2 ) )
=> ( ( H3 @ I2 )
= ( one_one @ A ) ) )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ S2 )
=> ( ( G @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G @ S2 )
= ( groups1962203154675924110t_prod @ B @ A @ H3 @ T6 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left'
thf(fact_2386_prod_Omono__neutral__cong__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S2: set @ B,T6: set @ B,G: B > A,H3: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S2 @ T6 )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ ( minus_minus @ ( set @ B ) @ T6 @ S2 ) )
=> ( ( G @ X2 )
= ( one_one @ A ) ) )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ S2 )
=> ( ( G @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G @ T6 )
= ( groups1962203154675924110t_prod @ B @ A @ H3 @ S2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right'
thf(fact_2387_prod_Odistrib_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I4: set @ B,G: B > A,H3: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X3: B] :
( ( member @ B @ X3 @ I4 )
& ( ( G @ X3 )
!= ( one_one @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X3: B] :
( ( member @ B @ X3 @ I4 )
& ( ( H3 @ X3 )
!= ( one_one @ A ) ) ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A
@ ^ [I3: B] : ( times_times @ A @ ( G @ I3 ) @ ( H3 @ I3 ) )
@ I4 )
= ( times_times @ A @ ( groups1962203154675924110t_prod @ B @ A @ G @ I4 ) @ ( groups1962203154675924110t_prod @ B @ A @ H3 @ I4 ) ) ) ) ) ) ).
% prod.distrib'
thf(fact_2388_prod_OG__def,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ( ( groups1962203154675924110t_prod @ B @ A )
= ( ^ [P6: B > A,I5: set @ B] :
( if @ A
@ ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X3: B] :
( ( member @ B @ X3 @ I5 )
& ( ( P6 @ X3 )
!= ( one_one @ A ) ) ) ) )
@ ( groups7121269368397514597t_prod @ B @ A @ P6
@ ( collect @ B
@ ^ [X3: B] :
( ( member @ B @ X3 @ I5 )
& ( ( P6 @ X3 )
!= ( one_one @ A ) ) ) ) )
@ ( one_one @ A ) ) ) ) ) ).
% prod.G_def
thf(fact_2389_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D3 ) )
= ( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A3 )
& ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_atLeastAtMost_iff
thf(fact_2390_greaterThanLessThan__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D3 ) )
= ( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A3 )
& ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_atLeastLessThan_iff
thf(fact_2391_atLeastAtMost__diff__ends,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ ( insert2 @ A @ A3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) ) ) ).
% atLeastAtMost_diff_ends
thf(fact_2392_smax__insertI,axiom,
! [Y: product_prod @ nat @ nat,Y9: set @ ( product_prod @ nat @ nat ),X: product_prod @ nat @ nat,X10: set @ ( product_prod @ nat @ nat )] :
( ( member @ ( product_prod @ nat @ nat ) @ Y @ Y9 )
=> ( ( member @ ( 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 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ X10 @ Y9 ) @ fun_max_strict )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( insert2 @ ( product_prod @ nat @ nat ) @ X @ X10 ) @ Y9 ) @ fun_max_strict ) ) ) ) ).
% smax_insertI
thf(fact_2393_set__decode__plus__power__2,axiom,
! [N: nat,Z2: nat] :
( ~ ( member @ nat @ N @ ( nat_set_decode @ Z2 ) )
=> ( ( nat_set_decode @ ( plus_plus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ Z2 ) )
= ( insert2 @ nat @ N @ ( nat_set_decode @ Z2 ) ) ) ) ).
% set_decode_plus_power_2
thf(fact_2394_prod__decode__aux_Osimps,axiom,
( nat_prod_decode_aux
= ( ^ [K4: nat,M: nat] : ( if @ ( product_prod @ nat @ nat ) @ ( ord_less_eq @ nat @ M @ K4 ) @ ( product_Pair @ nat @ nat @ M @ ( minus_minus @ nat @ K4 @ M ) ) @ ( nat_prod_decode_aux @ ( suc @ K4 ) @ ( minus_minus @ nat @ M @ ( suc @ K4 ) ) ) ) ) ) ).
% prod_decode_aux.simps
thf(fact_2395_prod__decode__aux_Oelims,axiom,
! [X: nat,Xa: nat,Y: product_prod @ nat @ nat] :
( ( ( nat_prod_decode_aux @ X @ Xa )
= Y )
=> ( ( ( ord_less_eq @ nat @ Xa @ X )
=> ( Y
= ( product_Pair @ nat @ nat @ Xa @ ( minus_minus @ nat @ X @ Xa ) ) ) )
& ( ~ ( ord_less_eq @ nat @ Xa @ X )
=> ( Y
= ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus @ nat @ Xa @ ( suc @ X ) ) ) ) ) ) ) ).
% prod_decode_aux.elims
thf(fact_2396_min__rpair__set,axiom,
fun_reduction_pair @ ( 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
thf(fact_2397_triangle__Suc,axiom,
! [N: nat] :
( ( nat_triangle @ ( suc @ N ) )
= ( plus_plus @ nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).
% triangle_Suc
thf(fact_2398_finite__greaterThanLessThan,axiom,
! [L: nat,U: nat] : ( finite_finite2 @ nat @ ( set_or5935395276787703475ssThan @ nat @ L @ U ) ) ).
% finite_greaterThanLessThan
thf(fact_2399_finite__greaterThanLessThan__integer,axiom,
! [L: code_integer,U: code_integer] : ( finite_finite2 @ code_integer @ ( set_or5935395276787703475ssThan @ code_integer @ L @ U ) ) ).
% finite_greaterThanLessThan_integer
thf(fact_2400_card__greaterThanLessThan,axiom,
! [L: nat,U: nat] :
( ( finite_card @ nat @ ( set_or5935395276787703475ssThan @ nat @ L @ U ) )
= ( minus_minus @ nat @ U @ ( suc @ L ) ) ) ).
% card_greaterThanLessThan
thf(fact_2401_max__rpair__set,axiom,
fun_reduction_pair @ ( 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
thf(fact_2402_atLeastSucLessThan__greaterThanLessThan,axiom,
! [L: nat,U: nat] :
( ( set_or7035219750837199246ssThan @ nat @ ( suc @ L ) @ U )
= ( set_or5935395276787703475ssThan @ nat @ L @ U ) ) ).
% atLeastSucLessThan_greaterThanLessThan
thf(fact_2403_smax__emptyI,axiom,
! [Y9: set @ ( product_prod @ nat @ nat )] :
( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ Y9 )
=> ( ( Y9
!= ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) )
=> ( member @ ( 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 ) ) ) @ Y9 ) @ fun_max_strict ) ) ) ).
% smax_emptyI
thf(fact_2404_atLeastPlusOneLessThan__greaterThanLessThan__integer,axiom,
! [L: code_integer,U: code_integer] :
( ( set_or7035219750837199246ssThan @ code_integer @ ( plus_plus @ code_integer @ L @ ( one_one @ code_integer ) ) @ U )
= ( set_or5935395276787703475ssThan @ code_integer @ L @ U ) ) ).
% atLeastPlusOneLessThan_greaterThanLessThan_integer
thf(fact_2405_prod__decode__aux_Opelims,axiom,
! [X: nat,Xa: nat,Y: product_prod @ nat @ nat] :
( ( ( nat_prod_decode_aux @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ nat @ nat ) @ nat_pr5047031295181774490ux_rel @ ( product_Pair @ nat @ nat @ X @ Xa ) )
=> ~ ( ( ( ( ord_less_eq @ nat @ Xa @ X )
=> ( Y
= ( product_Pair @ nat @ nat @ Xa @ ( minus_minus @ nat @ X @ Xa ) ) ) )
& ( ~ ( ord_less_eq @ nat @ Xa @ X )
=> ( Y
= ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus @ nat @ Xa @ ( suc @ X ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ nat @ nat ) @ nat_pr5047031295181774490ux_rel @ ( product_Pair @ nat @ nat @ X @ Xa ) ) ) ) ) ).
% prod_decode_aux.pelims
thf(fact_2406_total__on__singleton,axiom,
! [A: $tType,X: A] : ( total_on @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% total_on_singleton
thf(fact_2407_prod__encode__prod__decode__aux,axiom,
! [K: nat,M2: nat] :
( ( nat_prod_encode @ ( nat_prod_decode_aux @ K @ M2 ) )
= ( plus_plus @ nat @ ( nat_triangle @ K ) @ M2 ) ) ).
% prod_encode_prod_decode_aux
thf(fact_2408_sum_Oinsert_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I4: set @ B,P5: B > A,I: B] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X3: B] :
( ( member @ B @ X3 @ I4 )
& ( ( P5 @ X3 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( ( member @ B @ I @ I4 )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ P5 @ ( insert2 @ B @ I @ I4 ) )
= ( groups1027152243600224163dd_sum @ B @ A @ P5 @ I4 ) ) )
& ( ~ ( member @ B @ I @ I4 )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ P5 @ ( insert2 @ B @ I @ I4 ) )
= ( plus_plus @ A @ ( P5 @ I ) @ ( groups1027152243600224163dd_sum @ B @ A @ P5 @ I4 ) ) ) ) ) ) ) ).
% sum.insert'
thf(fact_2409_max__ext_Omax__extI,axiom,
! [A: $tType,X10: set @ A,Y9: set @ A,R5: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ A @ X10 )
=> ( ( finite_finite2 @ A @ Y9 )
=> ( ( Y9
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ X10 )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ Y9 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Xa2 ) @ R5 ) ) )
=> ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X10 @ Y9 ) @ ( max_ext @ A @ R5 ) ) ) ) ) ) ).
% max_ext.max_extI
thf(fact_2410_total__on__def,axiom,
! [A: $tType] :
( ( total_on @ A )
= ( ^ [A9: set @ A,R3: set @ ( product_prod @ A @ A )] :
! [X3: A] :
( ( member @ A @ X3 @ A9 )
=> ! [Y3: A] :
( ( member @ A @ Y3 @ A9 )
=> ( ( X3 != Y3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R3 ) ) ) ) ) ) ) ).
% total_on_def
thf(fact_2411_total__onI,axiom,
! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A )] :
( ! [X2: A,Y4: A] :
( ( member @ A @ X2 @ A4 )
=> ( ( member @ A @ Y4 @ A4 )
=> ( ( X2 != Y4 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y4 ) @ R )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X2 ) @ R ) ) ) ) )
=> ( total_on @ A @ A4 @ R ) ) ).
% total_onI
thf(fact_2412_sum_Odistrib__triv_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I4: set @ B,G: B > A,H3: B > A] :
( ( finite_finite2 @ B @ I4 )
=> ( ( groups1027152243600224163dd_sum @ B @ A
@ ^ [I3: B] : ( plus_plus @ A @ ( G @ I3 ) @ ( H3 @ I3 ) )
@ I4 )
= ( plus_plus @ A @ ( groups1027152243600224163dd_sum @ B @ A @ G @ I4 ) @ ( groups1027152243600224163dd_sum @ B @ A @ H3 @ I4 ) ) ) ) ) ).
% sum.distrib_triv'
thf(fact_2413_le__prod__encode__1,axiom,
! [A3: nat,B2: nat] : ( ord_less_eq @ nat @ A3 @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ A3 @ B2 ) ) ) ).
% le_prod_encode_1
thf(fact_2414_le__prod__encode__2,axiom,
! [B2: nat,A3: nat] : ( ord_less_eq @ nat @ B2 @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ A3 @ B2 ) ) ) ).
% le_prod_encode_2
thf(fact_2415_sum_Odistrib_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I4: set @ B,G: B > A,H3: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X3: B] :
( ( member @ B @ X3 @ I4 )
& ( ( G @ X3 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X3: B] :
( ( member @ B @ X3 @ I4 )
& ( ( H3 @ X3 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A
@ ^ [I3: B] : ( plus_plus @ A @ ( G @ I3 ) @ ( H3 @ I3 ) )
@ I4 )
= ( plus_plus @ A @ ( groups1027152243600224163dd_sum @ B @ A @ G @ I4 ) @ ( groups1027152243600224163dd_sum @ B @ A @ H3 @ I4 ) ) ) ) ) ) ).
% sum.distrib'
thf(fact_2416_prod__encode__def,axiom,
( nat_prod_encode
= ( product_case_prod @ nat @ nat @ nat
@ ^ [M: nat,N5: nat] : ( plus_plus @ nat @ ( nat_triangle @ ( plus_plus @ nat @ M @ N5 ) ) @ M ) ) ) ).
% prod_encode_def
thf(fact_2417_max__ext_Ocases,axiom,
! [A: $tType,A1: set @ A,A22: set @ A,R5: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A1 @ A22 ) @ ( max_ext @ A @ R5 ) )
=> ~ ( ( finite_finite2 @ A @ A1 )
=> ( ( finite_finite2 @ A @ A22 )
=> ( ( A22
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [X7: A] :
( ( member @ A @ X7 @ A1 )
=> ? [Xa3: A] :
( ( member @ A @ Xa3 @ A22 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X7 @ Xa3 ) @ R5 ) ) ) ) ) ) ) ).
% max_ext.cases
thf(fact_2418_max__ext_Osimps,axiom,
! [A: $tType,A1: set @ A,A22: set @ A,R5: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A1 @ A22 ) @ ( max_ext @ A @ R5 ) )
= ( ( finite_finite2 @ A @ A1 )
& ( finite_finite2 @ A @ A22 )
& ( A22
!= ( bot_bot @ ( set @ A ) ) )
& ! [X3: A] :
( ( member @ A @ X3 @ A1 )
=> ? [Y3: A] :
( ( member @ A @ Y3 @ A22 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R5 ) ) ) ) ) ).
% max_ext.simps
thf(fact_2419_max__extp__eq,axiom,
! [A: $tType] :
( ( max_extp @ A )
= ( ^ [R3: A > A > $o,X3: set @ A,Y3: set @ A] : ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X3 @ Y3 ) @ ( max_ext @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R3 ) ) ) ) ) ) ).
% max_extp_eq
thf(fact_2420_max__extp__max__ext__eq,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
( ( max_extp @ A
@ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R5 ) )
= ( ^ [X3: set @ A,Y3: set @ A] : ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X3 @ Y3 ) @ ( max_ext @ A @ R5 ) ) ) ) ).
% max_extp_max_ext_eq
thf(fact_2421_one__div__numeral,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N: num] :
( ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) )
= ( product_fst @ A @ A @ ( unique8689654367752047608divmod @ A @ one2 @ N ) ) ) ) ).
% one_div_numeral
thf(fact_2422_sqr_Osimps_I3_J,axiom,
! [N: num] :
( ( sqr @ ( bit1 @ N ) )
= ( bit1 @ ( bit0 @ ( plus_plus @ num @ ( sqr @ N ) @ N ) ) ) ) ).
% sqr.simps(3)
thf(fact_2423_one__mod__numeral,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N: num] :
( ( modulo_modulo @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) )
= ( product_snd @ A @ A @ ( unique8689654367752047608divmod @ A @ one2 @ N ) ) ) ) ).
% one_mod_numeral
thf(fact_2424_card__UNION,axiom,
! [A: $tType,A4: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ A4 )
=> ( ! [X2: set @ A] :
( ( member @ ( set @ A ) @ X2 @ A4 )
=> ( finite_finite2 @ A @ X2 ) )
=> ( ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ A4 ) )
= ( nat2
@ ( groups7311177749621191930dd_sum @ ( set @ ( set @ A ) ) @ int
@ ^ [I5: set @ ( set @ A )] : ( times_times @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( plus_plus @ nat @ ( finite_card @ ( set @ A ) @ I5 ) @ ( one_one @ nat ) ) ) @ ( semiring_1_of_nat @ int @ ( finite_card @ A @ ( complete_Inf_Inf @ ( set @ A ) @ I5 ) ) ) )
@ ( collect @ ( set @ ( set @ A ) )
@ ^ [I5: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ I5 @ A4 )
& ( I5
!= ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% card_UNION
thf(fact_2425_Sup__atMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [Y: A] :
( ( complete_Sup_Sup @ A @ ( set_ord_atMost @ A @ Y ) )
= Y ) ) ).
% Sup_atMost
thf(fact_2426_Sup__lessThan,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [Y: A] :
( ( complete_Sup_Sup @ A @ ( set_ord_lessThan @ A @ Y ) )
= Y ) ) ).
% Sup_lessThan
thf(fact_2427_fst__apsnd,axiom,
! [B: $tType,C: $tType,A: $tType,F: C > B,X: product_prod @ A @ C] :
( ( product_fst @ A @ B @ ( product_apsnd @ C @ B @ A @ F @ X ) )
= ( product_fst @ A @ C @ X ) ) ).
% fst_apsnd
thf(fact_2428_fst__comp__apsnd,axiom,
! [C: $tType,B: $tType,A: $tType,F: B > C] :
( ( comp @ ( product_prod @ A @ C ) @ A @ ( product_prod @ A @ B ) @ ( product_fst @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F ) )
= ( product_fst @ A @ B ) ) ).
% fst_comp_apsnd
thf(fact_2429_apsnd__eq__conv,axiom,
! [B: $tType,C: $tType,A: $tType,F: C > B,X: product_prod @ A @ C,G: C > B] :
( ( ( product_apsnd @ C @ B @ A @ F @ X )
= ( product_apsnd @ C @ B @ A @ G @ X ) )
= ( ( F @ ( product_snd @ A @ C @ X ) )
= ( G @ ( product_snd @ A @ C @ X ) ) ) ) ).
% apsnd_eq_conv
thf(fact_2430_snd__apsnd,axiom,
! [A: $tType,C: $tType,B: $tType,F: C > A,X: product_prod @ B @ C] :
( ( product_snd @ B @ A @ ( product_apsnd @ C @ A @ B @ F @ X ) )
= ( F @ ( product_snd @ B @ C @ X ) ) ) ).
% snd_apsnd
thf(fact_2431_snd__comp__apsnd,axiom,
! [C: $tType,B: $tType,A: $tType,F: B > C] :
( ( comp @ ( product_prod @ A @ C ) @ C @ ( product_prod @ A @ B ) @ ( product_snd @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F ) )
= ( comp @ B @ C @ ( product_prod @ A @ B ) @ F @ ( product_snd @ A @ B ) ) ) ).
% snd_comp_apsnd
thf(fact_2432_fst__divmod__integer,axiom,
! [K: code_integer,L: code_integer] :
( ( product_fst @ code_integer @ code_integer @ ( code_divmod_integer @ K @ L ) )
= ( divide_divide @ code_integer @ K @ L ) ) ).
% fst_divmod_integer
thf(fact_2433_snd__divmod__integer,axiom,
! [K: code_integer,L: code_integer] :
( ( product_snd @ code_integer @ code_integer @ ( code_divmod_integer @ K @ L ) )
= ( modulo_modulo @ code_integer @ K @ L ) ) ).
% snd_divmod_integer
thf(fact_2434_Sup__atLeastAtMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( complete_Sup_Sup @ A @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= Y ) ) ) ).
% Sup_atLeastAtMost
thf(fact_2435_Inf__atLeastAtMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( complete_Inf_Inf @ A @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= X ) ) ) ).
% Inf_atLeastAtMost
thf(fact_2436_Sup__atLeastLessThan,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( complete_Sup_Sup @ A @ ( set_or7035219750837199246ssThan @ A @ X @ Y ) )
= Y ) ) ) ).
% Sup_atLeastLessThan
thf(fact_2437_Inf__atLeastLessThan,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( complete_Inf_Inf @ A @ ( set_or7035219750837199246ssThan @ A @ X @ Y ) )
= X ) ) ) ).
% Inf_atLeastLessThan
thf(fact_2438_prod_Ocollapse,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
( ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_2439_Inf__atMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A] :
( ( complete_Inf_Inf @ A @ ( set_ord_atMost @ A @ X ) )
= ( bot_bot @ A ) ) ) ).
% Inf_atMost
thf(fact_2440_Sup__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( complete_Sup_Sup @ A @ ( set_or5935395276787703475ssThan @ A @ X @ Y ) )
= Y ) ) ) ).
% Sup_greaterThanLessThan
thf(fact_2441_img__fst,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,S2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ S2 )
=> ( member @ A @ A3 @ ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ S2 ) ) ) ).
% img_fst
thf(fact_2442_Inf__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( complete_Inf_Inf @ A @ ( set_or5935395276787703475ssThan @ A @ X @ Y ) )
= X ) ) ) ).
% Inf_greaterThanLessThan
thf(fact_2443_img__snd,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,S2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ S2 )
=> ( member @ B @ B2 @ ( image2 @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ S2 ) ) ) ).
% img_snd
thf(fact_2444_fst__divmod__abs,axiom,
! [K: code_integer,L: code_integer] :
( ( product_fst @ code_integer @ code_integer @ ( code_divmod_abs @ K @ L ) )
= ( divide_divide @ code_integer @ ( abs_abs @ code_integer @ K ) @ ( abs_abs @ code_integer @ L ) ) ) ).
% fst_divmod_abs
thf(fact_2445_snd__divmod__abs,axiom,
! [K: code_integer,L: code_integer] :
( ( product_snd @ code_integer @ code_integer @ ( code_divmod_abs @ K @ L ) )
= ( modulo_modulo @ code_integer @ ( abs_abs @ code_integer @ K ) @ ( abs_abs @ code_integer @ L ) ) ) ).
% snd_divmod_abs
thf(fact_2446_Inf__atMostLessThan,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( top_top @ A ) @ X )
=> ( ( complete_Inf_Inf @ A @ ( set_ord_lessThan @ A @ X ) )
= ( bot_bot @ A ) ) ) ) ).
% Inf_atMostLessThan
thf(fact_2447_split__comp__eq,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,F: A > B > C,G: D > A] :
( ( ^ [U2: product_prod @ D @ B] : ( F @ ( G @ ( product_fst @ D @ B @ U2 ) ) @ ( product_snd @ D @ B @ U2 ) ) )
= ( product_case_prod @ D @ B @ C
@ ^ [X3: D] : ( F @ ( G @ X3 ) ) ) ) ).
% split_comp_eq
thf(fact_2448_case__prod__beta_H,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ C )
= ( ^ [F2: A > B > C,X3: product_prod @ A @ B] : ( F2 @ ( product_fst @ A @ B @ X3 ) @ ( product_snd @ A @ B @ X3 ) ) ) ) ).
% case_prod_beta'
thf(fact_2449_case__prod__unfold,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ C )
= ( ^ [C3: A > B > C,P6: product_prod @ A @ B] : ( C3 @ ( product_fst @ A @ B @ P6 ) @ ( product_snd @ A @ B @ P6 ) ) ) ) ).
% case_prod_unfold
thf(fact_2450_Product__Type_OCollect__case__prodD,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B,A4: A > B > $o] :
( ( member @ ( product_prod @ A @ B ) @ X @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A4 ) ) )
=> ( A4 @ ( product_fst @ A @ B @ X ) @ ( product_snd @ A @ B @ X ) ) ) ).
% Product_Type.Collect_case_prodD
thf(fact_2451_case__prod__beta,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( product_case_prod @ B @ C @ A )
= ( ^ [F2: B > C > A,P6: product_prod @ B @ C] : ( F2 @ ( product_fst @ B @ C @ P6 ) @ ( product_snd @ B @ C @ P6 ) ) ) ) ).
% case_prod_beta
thf(fact_2452_split__beta,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ C )
= ( ^ [F2: A > B > C,Prod2: product_prod @ A @ B] : ( F2 @ ( product_fst @ A @ B @ Prod2 ) @ ( product_snd @ A @ B @ Prod2 ) ) ) ) ).
% split_beta
thf(fact_2453_effect__heapI,axiom,
! [A: $tType,N: nat,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ),H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A] :
( ( N
= ( product_snd @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H3 ) ) ) )
=> ( ( H4
= ( product_fst @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H3 ) ) ) )
=> ( ( R
= ( product_fst @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H3 ) ) )
=> ( heap_Time_effect @ A @ ( heap_Time_heap @ A @ F ) @ H3 @ H4 @ R @ N ) ) ) ) ).
% effect_heapI
thf(fact_2454_effect__heapE,axiom,
! [A: $tType,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ),H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A,N: nat] :
( ( heap_Time_effect @ A @ ( heap_Time_heap @ A @ F ) @ H3 @ H4 @ R @ N )
=> ~ ( ( H4
= ( product_fst @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H3 ) ) ) )
=> ( ( N
= ( product_snd @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H3 ) ) ) )
=> ( R
!= ( product_fst @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H3 ) ) ) ) ) ) ).
% effect_heapE
thf(fact_2455_effect__guardI,axiom,
! [A: $tType,P: ( heap_ext @ product_unit ) > $o,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ),N: nat,R: A] :
( ( P @ H3 )
=> ( ( H4
= ( product_fst @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H3 ) ) ) )
=> ( ( N
= ( product_snd @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H3 ) ) ) )
=> ( ( R
= ( product_fst @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H3 ) ) )
=> ( heap_Time_effect @ A @ ( heap_Time_guard @ A @ P @ F ) @ H3 @ H4 @ R @ N ) ) ) ) ) ).
% effect_guardI
thf(fact_2456_effect__guardE,axiom,
! [A: $tType,P: ( heap_ext @ product_unit ) > $o,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ),H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A,N: nat] :
( ( heap_Time_effect @ A @ ( heap_Time_guard @ A @ P @ F ) @ H3 @ H4 @ R @ N )
=> ~ ( ( H4
= ( product_fst @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H3 ) ) ) )
=> ( ( N
= ( product_snd @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H3 ) ) ) )
=> ( ( R
= ( product_fst @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H3 ) ) )
=> ~ ( P @ H3 ) ) ) ) ) ).
% effect_guardE
thf(fact_2457_effectE,axiom,
! [A: $tType,C2: heap_Time_Heap @ A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A,N: nat] :
( ( heap_Time_effect @ A @ C2 @ H3 @ H4 @ R @ N )
=> ~ ( ( R
= ( product_fst @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( the2 @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( heap_Time_execute @ A @ C2 @ H3 ) ) ) )
=> ( ( H4
= ( product_fst @ ( 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 ) ) @ ( heap_Time_execute @ A @ C2 @ H3 ) ) ) ) )
=> ( ( N
= ( product_snd @ ( 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 ) ) @ ( heap_Time_execute @ A @ C2 @ H3 ) ) ) ) )
=> ~ ( heap_Time_success @ A @ C2 @ H3 ) ) ) ) ) ).
% effectE
thf(fact_2458_prod__eq__iff,axiom,
! [B: $tType,A: $tType] :
( ( ^ [Y5: product_prod @ A @ B,Z4: product_prod @ A @ B] : Y5 = Z4 )
= ( ^ [S4: product_prod @ A @ B,T3: product_prod @ A @ B] :
( ( ( product_fst @ A @ B @ S4 )
= ( product_fst @ A @ B @ T3 ) )
& ( ( product_snd @ A @ B @ S4 )
= ( product_snd @ A @ B @ T3 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_2459_prod__eqI,axiom,
! [B: $tType,A: $tType,P5: product_prod @ A @ B,Q3: product_prod @ A @ B] :
( ( ( product_fst @ A @ B @ P5 )
= ( product_fst @ A @ B @ Q3 ) )
=> ( ( ( product_snd @ A @ B @ P5 )
= ( product_snd @ A @ B @ Q3 ) )
=> ( P5 = Q3 ) ) ) ).
% prod_eqI
thf(fact_2460_prod_Oexpand,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B,Prod3: product_prod @ A @ B] :
( ( ( ( product_fst @ A @ B @ Prod )
= ( product_fst @ A @ B @ Prod3 ) )
& ( ( product_snd @ A @ B @ Prod )
= ( product_snd @ A @ B @ Prod3 ) ) )
=> ( Prod = Prod3 ) ) ).
% prod.expand
thf(fact_2461_All__prod__contract,axiom,
! [B: $tType,A: $tType,P: A > B > $o] :
( ( ! [A7: A,X8: B] : ( P @ A7 @ X8 ) )
= ( ! [Z5: product_prod @ A @ B] : ( P @ ( product_fst @ A @ B @ Z5 ) @ ( product_snd @ A @ B @ Z5 ) ) ) ) ).
% All_prod_contract
thf(fact_2462_Ex__prod__contract,axiom,
! [B: $tType,A: $tType,P: A > B > $o] :
( ( ? [A7: A,X8: B] : ( P @ A7 @ X8 ) )
= ( ? [Z5: product_prod @ A @ B] : ( P @ ( product_fst @ A @ B @ Z5 ) @ ( product_snd @ A @ B @ Z5 ) ) ) ) ).
% Ex_prod_contract
thf(fact_2463_prod_Osplit__sel,axiom,
! [C: $tType,B: $tType,A: $tType,P: C > $o,F: A > B > C,Prod: product_prod @ A @ B] :
( ( P @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
= ( ( Prod
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
=> ( P @ ( F @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ).
% prod.split_sel
thf(fact_2464_prod_Osplit__sel__asm,axiom,
! [C: $tType,B: $tType,A: $tType,P: C > $o,F: A > B > C,Prod: product_prod @ A @ B] :
( ( P @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
= ( ~ ( ( Prod
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
& ~ ( P @ ( F @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ) ).
% prod.split_sel_asm
thf(fact_2465_surjective__pairing,axiom,
! [B: $tType,A: $tType,T2: product_prod @ A @ B] :
( T2
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T2 ) @ ( product_snd @ A @ B @ T2 ) ) ) ).
% surjective_pairing
thf(fact_2466_prod_Oexhaust__sel,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
( Prod
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_2467_exI__realizer,axiom,
! [B: $tType,A: $tType,P: A > B > $o,Y: A,X: B] :
( ( P @ Y @ X )
=> ( P @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_2468_conjI__realizer,axiom,
! [A: $tType,B: $tType,P: A > $o,P5: A,Q: B > $o,Q3: B] :
( ( P @ P5 )
=> ( ( Q @ Q3 )
=> ( ( P @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P5 @ Q3 ) ) )
& ( Q @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P5 @ Q3 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_2469_execute__bind__success,axiom,
! [B: $tType,A: $tType,F: heap_Time_Heap @ A,H3: heap_ext @ product_unit,G: A > ( heap_Time_Heap @ B )] :
( ( heap_Time_success @ A @ F @ H3 )
=> ( ( heap_Time_execute @ B @ ( heap_Time_bind @ A @ B @ F @ G ) @ H3 )
= ( heap_Time_timeFrame @ B @ ( product_snd @ ( 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 ) ) @ ( heap_Time_execute @ A @ F @ H3 ) ) ) ) @ ( heap_Time_execute @ B @ ( G @ ( product_fst @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( the2 @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( heap_Time_execute @ A @ F @ H3 ) ) ) ) @ ( product_fst @ ( 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 ) ) @ ( heap_Time_execute @ A @ F @ H3 ) ) ) ) ) ) ) ) ).
% execute_bind_success
thf(fact_2470_sndI,axiom,
! [A: $tType,B: $tType,X: product_prod @ A @ B,Y: A,Z2: B] :
( ( X
= ( product_Pair @ A @ B @ Y @ Z2 ) )
=> ( ( product_snd @ A @ B @ X )
= Z2 ) ) ).
% sndI
thf(fact_2471_sndE,axiom,
! [A: $tType,B: $tType,X: product_prod @ A @ B,A3: A,B2: B,P: B > $o] :
( ( X
= ( product_Pair @ A @ B @ A3 @ B2 ) )
=> ( ( P @ ( product_snd @ A @ B @ X ) )
=> ( P @ B2 ) ) ) ).
% sndE
thf(fact_2472_snd__eqD,axiom,
! [B: $tType,A: $tType,X: B,Y: A,A3: A] :
( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
= A3 )
=> ( Y = A3 ) ) ).
% snd_eqD
thf(fact_2473_snd__conv,axiom,
! [Aa: $tType,A: $tType,X1: Aa,X22: A] :
( ( product_snd @ Aa @ A @ ( product_Pair @ Aa @ A @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_2474_Some__Inf,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A] :
( ( some @ A @ ( complete_Inf_Inf @ A @ A4 ) )
= ( complete_Inf_Inf @ ( option @ A ) @ ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ A4 ) ) ) ) ).
% Some_Inf
thf(fact_2475_fstI,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B,Y: A,Z2: B] :
( ( X
= ( product_Pair @ A @ B @ Y @ Z2 ) )
=> ( ( product_fst @ A @ B @ X )
= Y ) ) ).
% fstI
thf(fact_2476_fstE,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B,A3: A,B2: B,P: A > $o] :
( ( X
= ( product_Pair @ A @ B @ A3 @ B2 ) )
=> ( ( P @ ( product_fst @ A @ B @ X ) )
=> ( P @ A3 ) ) ) ).
% fstE
thf(fact_2477_fst__eqD,axiom,
! [B: $tType,A: $tType,X: A,Y: B,A3: A] :
( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X @ Y ) )
= A3 )
=> ( X = A3 ) ) ).
% fst_eqD
thf(fact_2478_fst__conv,axiom,
! [B: $tType,A: $tType,X1: A,X22: B] :
( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_2479_snd__fst__flip,axiom,
! [A: $tType,B: $tType] :
( ( product_snd @ B @ A )
= ( comp @ ( product_prod @ A @ B ) @ A @ ( product_prod @ B @ A ) @ ( product_fst @ A @ B )
@ ( product_case_prod @ B @ A @ ( product_prod @ A @ B )
@ ^ [X3: B,Y3: A] : ( product_Pair @ A @ B @ Y3 @ X3 ) ) ) ) ).
% snd_fst_flip
thf(fact_2480_fst__snd__flip,axiom,
! [B: $tType,A: $tType] :
( ( product_fst @ A @ B )
= ( comp @ ( product_prod @ B @ A ) @ A @ ( product_prod @ A @ B ) @ ( product_snd @ B @ A )
@ ( product_case_prod @ A @ B @ ( product_prod @ B @ A )
@ ^ [X3: A,Y3: B] : ( product_Pair @ B @ A @ Y3 @ X3 ) ) ) ) ).
% fst_snd_flip
thf(fact_2481_finite__UNION__then__finite,axiom,
! [A: $tType,B: $tType,B6: B > ( set @ A ),A4: set @ B,A3: B] :
( ( finite_finite2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B6 @ A4 ) ) )
=> ( ( member @ B @ A3 @ A4 )
=> ( finite_finite2 @ A @ ( B6 @ A3 ) ) ) ) ).
% finite_UNION_then_finite
thf(fact_2482_case__prod__comp,axiom,
! [D: $tType,A: $tType,C: $tType,B: $tType,F: D > C > A,G: B > D,X: product_prod @ B @ C] :
( ( product_case_prod @ B @ C @ A @ ( comp @ D @ ( C > A ) @ B @ F @ G ) @ X )
= ( F @ ( G @ ( product_fst @ B @ C @ X ) ) @ ( product_snd @ B @ C @ X ) ) ) ).
% case_prod_comp
thf(fact_2483_fn__snd__conv,axiom,
! [B: $tType,C: $tType,A: $tType,F: B > C] :
( ( ^ [X3: product_prod @ A @ B] : ( F @ ( product_snd @ A @ B @ X3 ) ) )
= ( product_case_prod @ A @ B @ C
@ ^ [Uu: A] : F ) ) ).
% fn_snd_conv
thf(fact_2484_snd__def,axiom,
! [B: $tType,A: $tType] :
( ( product_snd @ A @ B )
= ( product_case_prod @ A @ B @ B
@ ^ [X12: A,X24: B] : X24 ) ) ).
% snd_def
thf(fact_2485_fn__fst__conv,axiom,
! [B: $tType,C: $tType,A: $tType,F: A > C] :
( ( ^ [X3: product_prod @ A @ B] : ( F @ ( product_fst @ A @ B @ X3 ) ) )
= ( product_case_prod @ A @ B @ C
@ ^ [A7: A,Uu: B] : ( F @ A7 ) ) ) ).
% fn_fst_conv
thf(fact_2486_fst__def,axiom,
! [B: $tType,A: $tType] :
( ( product_fst @ A @ B )
= ( product_case_prod @ A @ B @ A
@ ^ [X12: A,X24: B] : X12 ) ) ).
% fst_def
thf(fact_2487_Collect__split__mono__strong,axiom,
! [B: $tType,A: $tType,X10: set @ A,A4: set @ ( product_prod @ A @ B ),Y9: set @ B,P: A > B > $o,Q: A > B > $o] :
( ( X10
= ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A4 ) )
=> ( ( Y9
= ( image2 @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A4 ) )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ X10 )
=> ! [Xa3: B] :
( ( member @ B @ Xa3 @ Y9 )
=> ( ( P @ X2 @ Xa3 )
=> ( Q @ X2 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A4 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A4 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ Q ) ) ) ) ) ) ) ).
% Collect_split_mono_strong
thf(fact_2488_Some__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ A] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( some @ A @ ( complete_Sup_Sup @ A @ A4 ) )
= ( complete_Sup_Sup @ ( option @ A ) @ ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ A4 ) ) ) ) ) ).
% Some_Sup
thf(fact_2489_Inf__option__def,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Inf_Inf @ ( option @ A ) )
= ( ^ [A9: set @ ( option @ A )] : ( if @ ( option @ A ) @ ( member @ ( option @ A ) @ ( none @ A ) @ A9 ) @ ( none @ A ) @ ( some @ A @ ( complete_Inf_Inf @ A @ ( these @ A @ A9 ) ) ) ) ) ) ) ).
% Inf_option_def
thf(fact_2490_Some__INF,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: B > A,A4: set @ B] :
( ( some @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A4 ) ) )
= ( complete_Inf_Inf @ ( option @ A )
@ ( image2 @ B @ ( option @ A )
@ ^ [X3: B] : ( some @ A @ ( F @ X3 ) )
@ A4 ) ) ) ) ).
% Some_INF
thf(fact_2491_INF__image,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( complete_Inf @ A )
=> ! [G: B > A,F: C > B,A4: set @ C] :
( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G @ ( image2 @ C @ B @ F @ A4 ) ) )
= ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ ( comp @ B @ A @ C @ G @ F ) @ A4 ) ) ) ) ).
% INF_image
thf(fact_2492_SUP__image,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( complete_Sup @ A )
=> ! [G: B > A,F: C > B,A4: set @ C] :
( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G @ ( image2 @ C @ B @ F @ A4 ) ) )
= ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ ( comp @ B @ A @ C @ G @ F ) @ A4 ) ) ) ) ).
% SUP_image
thf(fact_2493_in__snd__imageE,axiom,
! [A: $tType,B: $tType,Y: A,S2: set @ ( product_prod @ B @ A )] :
( ( member @ A @ Y @ ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ S2 ) )
=> ~ ! [X2: B] :
~ ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X2 @ Y ) @ S2 ) ) ).
% in_snd_imageE
thf(fact_2494_in__fst__imageE,axiom,
! [B: $tType,A: $tType,X: A,S2: set @ ( product_prod @ A @ B )] :
( ( member @ A @ X @ ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ S2 ) )
=> ~ ! [Y4: B] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ S2 ) ) ).
% in_fst_imageE
thf(fact_2495_Some__SUP,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ B )
=> ! [A4: set @ A,F: A > B] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( some @ B @ ( complete_Sup_Sup @ B @ ( image2 @ A @ B @ F @ A4 ) ) )
= ( complete_Sup_Sup @ ( option @ B )
@ ( image2 @ A @ ( option @ B )
@ ^ [X3: A] : ( some @ B @ ( F @ X3 ) )
@ A4 ) ) ) ) ) ).
% Some_SUP
thf(fact_2496_finite__range__prod,axiom,
! [A: $tType,C: $tType,B: $tType,F: B > ( product_prod @ A @ C )] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ ( comp @ ( product_prod @ A @ C ) @ A @ B @ ( product_fst @ A @ C ) @ F ) @ ( top_top @ ( set @ B ) ) ) )
=> ( ( finite_finite2 @ C @ ( image2 @ B @ C @ ( comp @ ( product_prod @ A @ C ) @ C @ B @ ( product_snd @ A @ C ) @ F ) @ ( top_top @ ( set @ B ) ) ) )
=> ( finite_finite2 @ ( product_prod @ A @ C ) @ ( image2 @ B @ ( product_prod @ A @ C ) @ F @ ( top_top @ ( set @ B ) ) ) ) ) ) ).
% finite_range_prod
thf(fact_2497_UN__atMost__UNIV,axiom,
( ( complete_Sup_Sup @ ( set @ nat ) @ ( image2 @ nat @ ( set @ nat ) @ ( set_ord_atMost @ nat ) @ ( top_top @ ( set @ nat ) ) ) )
= ( top_top @ ( set @ nat ) ) ) ).
% UN_atMost_UNIV
thf(fact_2498_UN__lessThan__UNIV,axiom,
( ( complete_Sup_Sup @ ( set @ nat ) @ ( image2 @ nat @ ( set @ nat ) @ ( set_ord_lessThan @ nat ) @ ( top_top @ ( set @ nat ) ) ) )
= ( top_top @ ( set @ nat ) ) ) ).
% UN_lessThan_UNIV
thf(fact_2499_Sup__option__def,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Sup_Sup @ ( option @ A ) )
= ( ^ [A9: set @ ( option @ A )] :
( if @ ( option @ A )
@ ( ( A9
= ( bot_bot @ ( set @ ( option @ A ) ) ) )
| ( A9
= ( insert2 @ ( option @ A ) @ ( none @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) ) ) )
@ ( none @ A )
@ ( some @ A @ ( complete_Sup_Sup @ A @ ( these @ A @ A9 ) ) ) ) ) ) ) ).
% Sup_option_def
thf(fact_2500_snd__image__mp,axiom,
! [B: $tType,A: $tType,A4: set @ ( product_prod @ B @ A ),B6: set @ A,X: B,Y: A] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ A4 ) @ B6 )
=> ( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y ) @ A4 )
=> ( member @ A @ Y @ B6 ) ) ) ).
% snd_image_mp
thf(fact_2501_fst__image__mp,axiom,
! [B: $tType,A: $tType,A4: set @ ( product_prod @ A @ B ),B6: set @ A,X: A,Y: B] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A4 ) @ B6 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ A4 )
=> ( member @ A @ X @ B6 ) ) ) ).
% fst_image_mp
thf(fact_2502_prod_OUnion__comp,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [B6: set @ ( set @ B ),G: B > A] :
( ! [X2: set @ B] :
( ( member @ ( set @ B ) @ X2 @ B6 )
=> ( finite_finite2 @ B @ X2 ) )
=> ( ! [A14: set @ B] :
( ( member @ ( set @ B ) @ A14 @ B6 )
=> ! [A25: set @ B] :
( ( member @ ( set @ B ) @ A25 @ B6 )
=> ( ( A14 != A25 )
=> ! [X2: B] :
( ( member @ B @ X2 @ A14 )
=> ( ( member @ B @ X2 @ A25 )
=> ( ( G @ X2 )
= ( one_one @ A ) ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( complete_Sup_Sup @ ( set @ B ) @ B6 ) )
= ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7121269368397514597t_prod @ ( set @ B ) @ A ) @ ( groups7121269368397514597t_prod @ B @ A ) @ G @ B6 ) ) ) ) ) ).
% prod.Union_comp
thf(fact_2503_UN__UN__finite__eq,axiom,
! [A: $tType,A4: nat > ( set @ A )] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ nat @ ( set @ A )
@ ^ [N5: nat] : ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A4 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N5 ) ) )
@ ( top_top @ ( set @ nat ) ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A4 @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% UN_UN_finite_eq
thf(fact_2504_UN__le__add__shift,axiom,
! [A: $tType,M3: nat > ( set @ A ),K: nat,N: nat] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ nat @ ( set @ A )
@ ^ [I3: nat] : ( M3 @ ( plus_plus @ nat @ I3 @ K ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ M3 @ ( set_or1337092689740270186AtMost @ nat @ K @ ( plus_plus @ nat @ N @ K ) ) ) ) ) ).
% UN_le_add_shift
thf(fact_2505_UN__le__add__shift__strict,axiom,
! [A: $tType,M3: nat > ( set @ A ),K: nat,N: nat] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ nat @ ( set @ A )
@ ^ [I3: nat] : ( M3 @ ( plus_plus @ nat @ I3 @ K ) )
@ ( set_ord_lessThan @ nat @ N ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ M3 @ ( set_or7035219750837199246ssThan @ nat @ K @ ( plus_plus @ nat @ N @ K ) ) ) ) ) ).
% UN_le_add_shift_strict
thf(fact_2506_rat__sgn__code,axiom,
! [P5: rat] :
( ( quotient_of @ ( sgn_sgn @ rat @ P5 ) )
= ( product_Pair @ int @ int @ ( sgn_sgn @ int @ ( product_fst @ int @ int @ ( quotient_of @ P5 ) ) ) @ ( one_one @ int ) ) ) ).
% rat_sgn_code
thf(fact_2507_bezw__non__0,axiom,
! [Y: nat,X: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ Y )
=> ( ( bezw @ X @ Y )
= ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Y @ ( modulo_modulo @ nat @ X @ Y ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Y @ ( modulo_modulo @ nat @ X @ Y ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Y @ ( modulo_modulo @ nat @ X @ Y ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X @ Y ) ) ) ) ) ) ) ).
% bezw_non_0
thf(fact_2508_UN__finite__subset,axiom,
! [A: $tType,A4: nat > ( set @ A ),C5: set @ A] :
( ! [N3: nat] : ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A4 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) @ C5 )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A4 @ ( top_top @ ( set @ nat ) ) ) ) @ C5 ) ) ).
% UN_finite_subset
thf(fact_2509_UN__finite2__eq,axiom,
! [A: $tType,A4: nat > ( set @ A ),B6: nat > ( set @ A ),K: nat] :
( ! [N3: nat] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A4 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ B6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ N3 @ K ) ) ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A4 @ ( top_top @ ( set @ nat ) ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ B6 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% UN_finite2_eq
thf(fact_2510_bezw_Oelims,axiom,
! [X: nat,Xa: nat,Y: product_prod @ int @ int] :
( ( ( bezw @ X @ Xa )
= Y )
=> ( ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( Y
= ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X @ Xa ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X @ Xa ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X @ Xa ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X @ Xa ) ) ) ) ) ) ) ) ) ).
% bezw.elims
thf(fact_2511_bezw_Osimps,axiom,
( bezw
= ( ^ [X3: nat,Y3: nat] :
( if @ ( product_prod @ int @ int )
@ ( Y3
= ( zero_zero @ nat ) )
@ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Y3 @ ( modulo_modulo @ nat @ X3 @ Y3 ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Y3 @ ( modulo_modulo @ nat @ X3 @ Y3 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Y3 @ ( modulo_modulo @ nat @ X3 @ Y3 ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X3 @ Y3 ) ) ) ) ) ) ) ) ).
% bezw.simps
thf(fact_2512_max__ext__def,axiom,
! [A: $tType] :
( ( max_ext @ A )
= ( ^ [R6: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
@ ( max_extp @ A
@ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R6 ) ) ) ) ) ) ).
% max_ext_def
thf(fact_2513_UN__finite2__subset,axiom,
! [A: $tType,A4: nat > ( set @ A ),B6: nat > ( set @ A ),K: nat] :
( ! [N3: nat] : ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A4 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ B6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ N3 @ K ) ) ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A4 @ ( top_top @ ( set @ nat ) ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ B6 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% UN_finite2_subset
thf(fact_2514_bezw_Opelims,axiom,
! [X: nat,Xa: nat,Y: product_prod @ int @ int] :
( ( ( bezw @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ nat @ nat ) @ bezw_rel @ ( product_Pair @ nat @ nat @ X @ Xa ) )
=> ~ ( ( ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( Y
= ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X @ Xa ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X @ Xa ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X @ Xa ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X @ Xa ) ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ nat @ nat ) @ bezw_rel @ ( product_Pair @ nat @ nat @ X @ Xa ) ) ) ) ) ).
% bezw.pelims
thf(fact_2515_snd__diag__id,axiom,
! [A: $tType,Z2: A] :
( ( comp @ ( product_prod @ A @ A ) @ A @ A @ ( product_snd @ A @ A )
@ ^ [X3: A] : ( product_Pair @ A @ A @ X3 @ X3 )
@ Z2 )
= ( id @ A @ Z2 ) ) ).
% snd_diag_id
thf(fact_2516_fst__diag__id,axiom,
! [A: $tType,Z2: A] :
( ( comp @ ( product_prod @ A @ A ) @ A @ A @ ( product_fst @ A @ A )
@ ^ [X3: A] : ( product_Pair @ A @ A @ X3 @ X3 )
@ Z2 )
= ( id @ A @ Z2 ) ) ).
% fst_diag_id
thf(fact_2517_size__prod__simp,axiom,
! [B: $tType,A: $tType] :
( ( basic_BNF_size_prod @ A @ B )
= ( ^ [F2: A > nat,G2: B > nat,P6: product_prod @ A @ B] : ( plus_plus @ nat @ ( plus_plus @ nat @ ( F2 @ ( product_fst @ A @ B @ P6 ) ) @ ( G2 @ ( product_snd @ A @ B @ P6 ) ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% size_prod_simp
thf(fact_2518_fst__diag__snd,axiom,
! [B: $tType,A: $tType] :
( ( comp @ ( product_prod @ B @ B ) @ B @ ( product_prod @ A @ B ) @ ( product_fst @ B @ B )
@ ( comp @ B @ ( product_prod @ B @ B ) @ ( product_prod @ A @ B )
@ ^ [X3: B] : ( product_Pair @ B @ B @ X3 @ X3 )
@ ( product_snd @ A @ B ) ) )
= ( product_snd @ A @ B ) ) ).
% fst_diag_snd
thf(fact_2519_snd__diag__fst,axiom,
! [B: $tType,A: $tType] :
( ( comp @ ( product_prod @ A @ A ) @ A @ ( product_prod @ A @ B ) @ ( product_snd @ A @ A )
@ ( comp @ A @ ( product_prod @ A @ A ) @ ( product_prod @ A @ B )
@ ^ [X3: A] : ( product_Pair @ A @ A @ X3 @ X3 )
@ ( product_fst @ A @ B ) ) )
= ( product_fst @ A @ B ) ) ).
% snd_diag_fst
thf(fact_2520_None__in__Inf,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A4: set @ ( option @ A )] :
( ( member @ ( option @ A ) @ ( none @ A ) @ A4 )
=> ( ( complete_Inf_Inf @ ( option @ A ) @ A4 )
= ( none @ A ) ) ) ) ).
% None_in_Inf
thf(fact_2521_empty__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Sup_Sup @ ( option @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) )
= ( none @ A ) ) ) ).
% empty_Sup
thf(fact_2522_singleton__None__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Sup_Sup @ ( option @ A ) @ ( insert2 @ ( option @ A ) @ ( none @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) ) )
= ( none @ A ) ) ) ).
% singleton_None_Sup
thf(fact_2523_INF__INT__eq2,axiom,
! [B: $tType,C: $tType,A: $tType,R: C > ( set @ ( product_prod @ A @ B ) ),S2: set @ C] :
( ( complete_Inf_Inf @ ( A > B > $o )
@ ( image2 @ C @ ( A > B > $o )
@ ^ [I3: C,X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( R @ I3 ) )
@ S2 ) )
= ( ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ C @ ( set @ ( product_prod @ A @ B ) ) @ R @ S2 ) ) ) ) ) ).
% INF_INT_eq2
thf(fact_2524_INF__Int__eq2,axiom,
! [B: $tType,A: $tType,S2: set @ ( set @ ( product_prod @ A @ B ) )] :
( ( complete_Inf_Inf @ ( A > B > $o )
@ ( image2 @ ( set @ ( product_prod @ A @ B ) ) @ ( A > B > $o )
@ ^ [I3: set @ ( product_prod @ A @ B ),X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ I3 )
@ S2 ) )
= ( ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) ) @ S2 ) ) ) ) ).
% INF_Int_eq2
thf(fact_2525_SUP__UN__eq2,axiom,
! [B: $tType,C: $tType,A: $tType,R: C > ( set @ ( product_prod @ A @ B ) ),S2: set @ C] :
( ( complete_Sup_Sup @ ( A > B > $o )
@ ( image2 @ C @ ( A > B > $o )
@ ^ [I3: C,X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( R @ I3 ) )
@ S2 ) )
= ( ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ C @ ( set @ ( product_prod @ A @ B ) ) @ R @ S2 ) ) ) ) ) ).
% SUP_UN_eq2
thf(fact_2526_SUP__Sup__eq2,axiom,
! [B: $tType,A: $tType,S2: set @ ( set @ ( product_prod @ A @ B ) )] :
( ( complete_Sup_Sup @ ( A > B > $o )
@ ( image2 @ ( set @ ( product_prod @ A @ B ) ) @ ( A > B > $o )
@ ^ [I3: set @ ( product_prod @ A @ B ),X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ I3 )
@ S2 ) )
= ( ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ S2 ) ) ) ) ).
% SUP_Sup_eq2
thf(fact_2527_Inf__INT__eq2,axiom,
! [B: $tType,A: $tType] :
( ( complete_Inf_Inf @ ( A > B > $o ) )
= ( ^ [S5: set @ ( A > B > $o ),X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ ( ( product_prod @ A @ B ) > $o ) @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) ) @ ( image2 @ ( A > B > $o ) @ ( ( product_prod @ A @ B ) > $o ) @ ( product_case_prod @ A @ B @ $o ) @ S5 ) ) ) ) ) ) ).
% Inf_INT_eq2
thf(fact_2528_Sup__SUP__eq2,axiom,
! [B: $tType,A: $tType] :
( ( complete_Sup_Sup @ ( A > B > $o ) )
= ( ^ [S5: set @ ( A > B > $o ),X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ ( ( product_prod @ A @ B ) > $o ) @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) ) @ ( image2 @ ( A > B > $o ) @ ( ( product_prod @ A @ B ) > $o ) @ ( product_case_prod @ A @ B @ $o ) @ S5 ) ) ) ) ) ) ).
% Sup_SUP_eq2
thf(fact_2529_fst__diag__fst,axiom,
! [B: $tType,A: $tType] :
( ( comp @ ( product_prod @ A @ A ) @ A @ ( product_prod @ A @ B ) @ ( product_fst @ A @ A )
@ ( comp @ A @ ( product_prod @ A @ A ) @ ( product_prod @ A @ B )
@ ^ [X3: A] : ( product_Pair @ A @ A @ X3 @ X3 )
@ ( product_fst @ A @ B ) ) )
= ( product_fst @ A @ B ) ) ).
% fst_diag_fst
thf(fact_2530_snd__diag__snd,axiom,
! [B: $tType,A: $tType] :
( ( comp @ ( product_prod @ B @ B ) @ B @ ( product_prod @ A @ B ) @ ( product_snd @ B @ B )
@ ( comp @ B @ ( product_prod @ B @ B ) @ ( product_prod @ A @ B )
@ ^ [X3: B] : ( product_Pair @ B @ B @ X3 @ X3 )
@ ( product_snd @ A @ B ) ) )
= ( product_snd @ A @ B ) ) ).
% snd_diag_snd
thf(fact_2531_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [B: $tType,A: $tType,P: A > B > $o,X: A,Y: B,A3: product_prod @ A @ B] :
( ( P @ X @ Y )
=> ( ( A3
= ( product_Pair @ A @ B @ X @ Y ) )
=> ( P @ ( product_fst @ A @ B @ A3 ) @ ( product_snd @ A @ B @ A3 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_2532_normalize__def,axiom,
( normalize
= ( ^ [P6: product_prod @ int @ int] :
( if @ ( product_prod @ int @ int ) @ ( ord_less @ int @ ( zero_zero @ int ) @ ( product_snd @ int @ int @ P6 ) ) @ ( product_Pair @ int @ int @ ( divide_divide @ int @ ( product_fst @ int @ int @ P6 ) @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P6 ) @ ( product_snd @ int @ int @ P6 ) ) ) @ ( divide_divide @ int @ ( product_snd @ int @ int @ P6 ) @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P6 ) @ ( product_snd @ int @ int @ P6 ) ) ) )
@ ( if @ ( product_prod @ int @ int )
@ ( ( product_snd @ int @ int @ P6 )
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( divide_divide @ int @ ( product_fst @ int @ int @ P6 ) @ ( uminus_uminus @ int @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P6 ) @ ( product_snd @ int @ int @ P6 ) ) ) ) @ ( divide_divide @ int @ ( product_snd @ int @ int @ P6 ) @ ( uminus_uminus @ int @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P6 ) @ ( product_snd @ int @ int @ P6 ) ) ) ) ) ) ) ) ) ).
% normalize_def
thf(fact_2533_range__prod,axiom,
! [C: $tType,B: $tType,A: $tType,F: C > ( product_prod @ A @ B )] :
( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ C @ ( product_prod @ A @ B ) @ F @ ( top_top @ ( set @ C ) ) )
@ ( product_Sigma @ A @ B @ ( image2 @ C @ A @ ( comp @ ( product_prod @ A @ B ) @ A @ C @ ( product_fst @ A @ B ) @ F ) @ ( top_top @ ( set @ C ) ) )
@ ^ [Uu: A] : ( image2 @ C @ B @ ( comp @ ( product_prod @ A @ B ) @ B @ C @ ( product_snd @ A @ B ) @ F ) @ ( top_top @ ( set @ C ) ) ) ) ) ).
% range_prod
thf(fact_2534_fun_Oin__rel,axiom,
! [B: $tType,A: $tType,D: $tType,R5: A > B > $o,A3: D > A,B2: D > B] :
( ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y5: D,Z4: D] : Y5 = Z4
@ R5
@ A3
@ B2 )
= ( ? [Z5: D > ( product_prod @ A @ B )] :
( ( member @ ( D > ( product_prod @ A @ B ) ) @ Z5
@ ( collect @ ( D > ( product_prod @ A @ B ) )
@ ^ [X3: D > ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ D @ ( product_prod @ A @ B ) @ X3 @ ( top_top @ ( set @ D ) ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R5 ) ) ) ) )
& ( ( comp @ ( product_prod @ A @ B ) @ A @ D @ ( product_fst @ A @ B ) @ Z5 )
= A3 )
& ( ( comp @ ( product_prod @ A @ B ) @ B @ D @ ( product_snd @ A @ B ) @ Z5 )
= B2 ) ) ) ) ).
% fun.in_rel
thf(fact_2535_apfst__apsnd,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,F: C > A,G: D > B,X: product_prod @ C @ D] :
( ( product_apfst @ C @ A @ B @ F @ ( product_apsnd @ D @ B @ C @ G @ X ) )
= ( product_Pair @ A @ B @ ( F @ ( product_fst @ C @ D @ X ) ) @ ( G @ ( product_snd @ C @ D @ X ) ) ) ) ).
% apfst_apsnd
thf(fact_2536_apsnd__apfst,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,F: C > B,G: D > A,X: product_prod @ D @ C] :
( ( product_apsnd @ C @ B @ A @ F @ ( product_apfst @ D @ A @ C @ G @ X ) )
= ( product_Pair @ A @ B @ ( G @ ( product_fst @ D @ C @ X ) ) @ ( F @ ( product_snd @ D @ C @ X ) ) ) ) ).
% apsnd_apfst
thf(fact_2537_eq__snd__iff,axiom,
! [A: $tType,B: $tType,B2: A,P5: product_prod @ B @ A] :
( ( B2
= ( product_snd @ B @ A @ P5 ) )
= ( ? [A7: B] :
( P5
= ( product_Pair @ B @ A @ A7 @ B2 ) ) ) ) ).
% eq_snd_iff
thf(fact_2538_SigmaI,axiom,
! [B: $tType,A: $tType,A3: A,A4: set @ A,B2: B,B6: A > ( set @ B )] :
( ( member @ A @ A3 @ A4 )
=> ( ( member @ B @ B2 @ ( B6 @ A3 ) )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( product_Sigma @ A @ B @ A4 @ B6 ) ) ) ) ).
% SigmaI
thf(fact_2539_mem__Sigma__iff,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,A4: set @ A,B6: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( product_Sigma @ A @ B @ A4 @ B6 ) )
= ( ( member @ A @ A3 @ A4 )
& ( member @ B @ B2 @ ( B6 @ A3 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_2540_gcd__add1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [M2: A,N: A] :
( ( gcd_gcd @ A @ ( plus_plus @ A @ M2 @ N ) @ N )
= ( gcd_gcd @ A @ M2 @ N ) ) ) ).
% gcd_add1
thf(fact_2541_gcd__add2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [M2: A,N: A] :
( ( gcd_gcd @ A @ M2 @ ( plus_plus @ A @ M2 @ N ) )
= ( gcd_gcd @ A @ M2 @ N ) ) ) ).
% gcd_add2
thf(fact_2542_gcd_Obottom__left__bottom,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A] :
( ( gcd_gcd @ A @ ( one_one @ A ) @ A3 )
= ( one_one @ A ) ) ) ).
% gcd.bottom_left_bottom
thf(fact_2543_gcd_Obottom__right__bottom,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A] :
( ( gcd_gcd @ A @ A3 @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% gcd.bottom_right_bottom
thf(fact_2544_apfst__conv,axiom,
! [C: $tType,A: $tType,B: $tType,F: C > A,X: C,Y: B] :
( ( product_apfst @ C @ A @ B @ F @ ( product_Pair @ C @ B @ X @ Y ) )
= ( product_Pair @ A @ B @ ( F @ X ) @ Y ) ) ).
% apfst_conv
thf(fact_2545_apfst__eq__conv,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > A,X: product_prod @ C @ B,G: C > A] :
( ( ( product_apfst @ C @ A @ B @ F @ X )
= ( product_apfst @ C @ A @ B @ G @ X ) )
= ( ( F @ ( product_fst @ C @ B @ X ) )
= ( G @ ( product_fst @ C @ B @ X ) ) ) ) ).
% apfst_eq_conv
thf(fact_2546_fst__apfst,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > A,X: product_prod @ C @ B] :
( ( product_fst @ A @ B @ ( product_apfst @ C @ A @ B @ F @ X ) )
= ( F @ ( product_fst @ C @ B @ X ) ) ) ).
% fst_apfst
thf(fact_2547_snd__apfst,axiom,
! [B: $tType,A: $tType,C: $tType,F: C > B,X: product_prod @ C @ A] :
( ( product_snd @ B @ A @ ( product_apfst @ C @ B @ A @ F @ X ) )
= ( product_snd @ C @ A @ X ) ) ).
% snd_apfst
thf(fact_2548_Collect__case__prod,axiom,
! [B: $tType,A: $tType,P: A > $o,Q: B > $o] :
( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A7: A,B5: B] :
( ( P @ A7 )
& ( Q @ B5 ) ) ) )
= ( product_Sigma @ A @ B @ ( collect @ A @ P )
@ ^ [Uu: A] : ( collect @ B @ Q ) ) ) ).
% Collect_case_prod
thf(fact_2549_is__unit__gcd__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ ( gcd_gcd @ A @ A3 @ B2 ) @ ( one_one @ A ) )
= ( ( gcd_gcd @ A @ A3 @ B2 )
= ( one_one @ A ) ) ) ) ).
% is_unit_gcd_iff
thf(fact_2550_Sigma__empty1,axiom,
! [B: $tType,A: $tType,B6: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ ( bot_bot @ ( set @ A ) ) @ B6 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% Sigma_empty1
thf(fact_2551_Sigma__empty2,axiom,
! [B: $tType,A: $tType,A4: set @ A] :
( ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : ( bot_bot @ ( set @ B ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% Sigma_empty2
thf(fact_2552_Times__empty,axiom,
! [A: $tType,B: $tType,A4: set @ A,B6: set @ B] :
( ( ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : B6 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( ( A4
= ( bot_bot @ ( set @ A ) ) )
| ( B6
= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% Times_empty
thf(fact_2553_Sigma__UNIV__cancel,axiom,
! [B: $tType,A: $tType,A4: set @ A,X10: set @ B] :
( ( minus_minus @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : X10 )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : ( top_top @ ( set @ B ) ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% Sigma_UNIV_cancel
thf(fact_2554_Compl__Times__UNIV2,axiom,
! [B: $tType,A: $tType,A4: set @ A] :
( ( uminus_uminus @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : ( top_top @ ( set @ B ) ) ) )
= ( product_Sigma @ A @ B @ ( uminus_uminus @ ( set @ A ) @ A4 )
@ ^ [Uu: A] : ( top_top @ ( set @ B ) ) ) ) ).
% Compl_Times_UNIV2
thf(fact_2555_Compl__Times__UNIV1,axiom,
! [B: $tType,A: $tType,A4: set @ B] :
( ( uminus_uminus @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
@ ^ [Uu: A] : A4 ) )
= ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
@ ^ [Uu: A] : ( uminus_uminus @ ( set @ B ) @ A4 ) ) ) ).
% Compl_Times_UNIV1
thf(fact_2556_snd__comp__apfst,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > C] :
( ( comp @ ( product_prod @ C @ B ) @ B @ ( product_prod @ A @ B ) @ ( product_snd @ C @ B ) @ ( product_apfst @ A @ C @ B @ F ) )
= ( product_snd @ A @ B ) ) ).
% snd_comp_apfst
thf(fact_2557_UNIV__Times__UNIV,axiom,
! [B: $tType,A: $tType] :
( ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
@ ^ [Uu: A] : ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% UNIV_Times_UNIV
thf(fact_2558_apfst__id,axiom,
! [B: $tType,A: $tType] :
( ( product_apfst @ A @ A @ B @ ( id @ A ) )
= ( id @ ( product_prod @ A @ B ) ) ) ).
% apfst_id
thf(fact_2559_pairself__image__cart,axiom,
! [B: $tType,A: $tType,F: B > A,A4: set @ B,B6: set @ B] :
( ( image2 @ ( product_prod @ B @ B ) @ ( product_prod @ A @ A ) @ ( pairself @ B @ A @ F )
@ ( product_Sigma @ B @ B @ A4
@ ^ [Uu: B] : B6 ) )
= ( product_Sigma @ A @ A @ ( image2 @ B @ A @ F @ A4 )
@ ^ [Uu: A] : ( image2 @ B @ A @ F @ B6 ) ) ) ).
% pairself_image_cart
thf(fact_2560_fst__image__times,axiom,
! [B: $tType,A: $tType,B6: set @ B,A4: set @ A] :
( ( ( B6
= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : B6 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( B6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : B6 ) )
= A4 ) ) ) ).
% fst_image_times
thf(fact_2561_snd__image__times,axiom,
! [B: $tType,A: $tType,A4: set @ B,B6: set @ A] :
( ( ( A4
= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A )
@ ( product_Sigma @ B @ A @ A4
@ ^ [Uu: B] : B6 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A )
@ ( product_Sigma @ B @ A @ A4
@ ^ [Uu: B] : B6 ) )
= B6 ) ) ) ).
% snd_image_times
thf(fact_2562_fst__comp__apfst,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > C] :
( ( comp @ ( product_prod @ C @ B ) @ C @ ( product_prod @ A @ B ) @ ( product_fst @ C @ B ) @ ( product_apfst @ A @ C @ B @ F ) )
= ( comp @ A @ C @ ( product_prod @ A @ B ) @ F @ ( product_fst @ A @ B ) ) ) ).
% fst_comp_apfst
thf(fact_2563_fun_Omap__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,G3: $tType,F5: $tType,Rb2: A > F5 > $o,Sd: B > G3 > $o] :
( bNF_rel_fun @ ( A > B ) @ ( F5 > G3 ) @ ( ( D > A ) > D > B ) @ ( ( D > F5 ) > D > G3 ) @ ( bNF_rel_fun @ A @ F5 @ B @ G3 @ Rb2 @ Sd )
@ ( bNF_rel_fun @ ( D > A ) @ ( D > F5 ) @ ( D > B ) @ ( D > G3 )
@ ( bNF_rel_fun @ D @ D @ A @ F5
@ ^ [Y5: D,Z4: D] : Y5 = Z4
@ Rb2 )
@ ( bNF_rel_fun @ D @ D @ B @ G3
@ ^ [Y5: D,Z4: D] : Y5 = Z4
@ Sd ) )
@ ( comp @ A @ B @ D )
@ ( comp @ F5 @ G3 @ D ) ) ).
% fun.map_transfer
thf(fact_2564_comp__transfer,axiom,
! [A: $tType,B: $tType,E: $tType,F5: $tType,D: $tType,C: $tType,B6: A > C > $o,C5: B > D > $o,A4: E > F5 > $o] : ( bNF_rel_fun @ ( A > B ) @ ( C > D ) @ ( ( E > A ) > E > B ) @ ( ( F5 > C ) > F5 > D ) @ ( bNF_rel_fun @ A @ C @ B @ D @ B6 @ C5 ) @ ( bNF_rel_fun @ ( E > A ) @ ( F5 > C ) @ ( E > B ) @ ( F5 > D ) @ ( bNF_rel_fun @ E @ F5 @ A @ C @ A4 @ B6 ) @ ( bNF_rel_fun @ E @ F5 @ B @ D @ A4 @ C5 ) ) @ ( comp @ A @ B @ E ) @ ( comp @ C @ D @ F5 ) ) ).
% comp_transfer
thf(fact_2565_power__transfer,axiom,
! [A: $tType,B: $tType] :
( ( ( power @ B )
& ( power @ A ) )
=> ! [R5: A > B > $o] :
( ( R5 @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R5 @ ( bNF_rel_fun @ A @ B @ A @ B @ R5 @ R5 ) @ ( times_times @ A ) @ ( times_times @ B ) )
=> ( bNF_rel_fun @ A @ B @ ( nat > A ) @ ( nat > B ) @ R5
@ ( bNF_rel_fun @ nat @ nat @ A @ B
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ R5 )
@ ( power_power @ A )
@ ( power_power @ B ) ) ) ) ) ).
% power_transfer
thf(fact_2566_times__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ^ [Y5: int,Z4: int] : Y5 = Z4 )
@ ( times_times @ int )
@ ( times_times @ int ) ) ).
% times_integer.rsp
thf(fact_2567_times__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4 )
@ ( times_times @ nat )
@ ( times_times @ nat ) ) ).
% times_natural.rsp
thf(fact_2568_Times__eq__cancel2,axiom,
! [A: $tType,B: $tType,X: A,C5: set @ A,A4: set @ B,B6: set @ B] :
( ( member @ A @ X @ C5 )
=> ( ( ( product_Sigma @ B @ A @ A4
@ ^ [Uu: B] : C5 )
= ( product_Sigma @ B @ A @ B6
@ ^ [Uu: B] : C5 ) )
= ( A4 = B6 ) ) ) ).
% Times_eq_cancel2
thf(fact_2569_Sigma__cong,axiom,
! [B: $tType,A: $tType,A4: set @ A,B6: set @ A,C5: A > ( set @ B ),D4: A > ( set @ B )] :
( ( A4 = B6 )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ B6 )
=> ( ( C5 @ X2 )
= ( D4 @ X2 ) ) )
=> ( ( product_Sigma @ A @ B @ A4 @ C5 )
= ( product_Sigma @ A @ B @ B6 @ D4 ) ) ) ) ).
% Sigma_cong
thf(fact_2570_divide__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4 )
@ ( divide_divide @ nat )
@ ( divide_divide @ nat ) ) ).
% divide_natural.rsp
thf(fact_2571_divide__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ^ [Y5: int,Z4: int] : Y5 = Z4 )
@ ( divide_divide @ int )
@ ( divide_divide @ int ) ) ).
% divide_integer.rsp
thf(fact_2572_or__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4 )
@ ( bit_se1065995026697491101ons_or @ nat )
@ ( bit_se1065995026697491101ons_or @ nat ) ) ).
% or_natural.rsp
thf(fact_2573_or__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ^ [Y5: int,Z4: int] : Y5 = Z4 )
@ ( bit_se1065995026697491101ons_or @ int )
@ ( bit_se1065995026697491101ons_or @ int ) ) ).
% or_integer.rsp
thf(fact_2574_set__bit__integer_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( int > int )
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ^ [Y5: int,Z4: int] : Y5 = Z4 )
@ ( bit_se5668285175392031749et_bit @ int )
@ ( bit_se5668285175392031749et_bit @ int ) ) ).
% set_bit_integer.rsp
thf(fact_2575_set__bit__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4 )
@ ( bit_se5668285175392031749et_bit @ nat )
@ ( bit_se5668285175392031749et_bit @ nat ) ) ).
% set_bit_natural.rsp
thf(fact_2576_xor__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ^ [Y5: int,Z4: int] : Y5 = Z4 )
@ ( bit_se5824344971392196577ns_xor @ int )
@ ( bit_se5824344971392196577ns_xor @ int ) ) ).
% xor_integer.rsp
thf(fact_2577_xor__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4 )
@ ( bit_se5824344971392196577ns_xor @ nat )
@ ( bit_se5824344971392196577ns_xor @ nat ) ) ).
% xor_natural.rsp
thf(fact_2578_and__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ^ [Y5: int,Z4: int] : Y5 = Z4 )
@ ( bit_se5824344872417868541ns_and @ int )
@ ( bit_se5824344872417868541ns_and @ int ) ) ).
% and_integer.rsp
thf(fact_2579_and__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4 )
@ ( bit_se5824344872417868541ns_and @ nat )
@ ( bit_se5824344872417868541ns_and @ nat ) ) ).
% and_natural.rsp
thf(fact_2580_modulo__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ^ [Y5: int,Z4: int] : Y5 = Z4 )
@ ( modulo_modulo @ int )
@ ( modulo_modulo @ int ) ) ).
% modulo_integer.rsp
thf(fact_2581_modulo__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4 )
@ ( modulo_modulo @ nat )
@ ( modulo_modulo @ nat ) ) ).
% modulo_natural.rsp
thf(fact_2582_less__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > $o ) @ ( int > $o )
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ( bNF_rel_fun @ int @ int @ $o @ $o
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 )
@ ( ord_less @ int )
@ ( ord_less @ int ) ) ).
% less_integer.rsp
thf(fact_2583_less__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > $o ) @ ( nat > $o )
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ( bNF_rel_fun @ nat @ nat @ $o @ $o
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 )
@ ( ord_less @ nat )
@ ( ord_less @ nat ) ) ).
% less_natural.rsp
thf(fact_2584_unset__bit__integer_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( int > int )
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ^ [Y5: int,Z4: int] : Y5 = Z4 )
@ ( bit_se2638667681897837118et_bit @ int )
@ ( bit_se2638667681897837118et_bit @ int ) ) ).
% unset_bit_integer.rsp
thf(fact_2585_take__bit__integer_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( int > int )
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ^ [Y5: int,Z4: int] : Y5 = Z4 )
@ ( bit_se2584673776208193580ke_bit @ int )
@ ( bit_se2584673776208193580ke_bit @ int ) ) ).
% take_bit_integer.rsp
thf(fact_2586_flip__bit__integer_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( int > int )
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ^ [Y5: int,Z4: int] : Y5 = Z4 )
@ ( bit_se8732182000553998342ip_bit @ int )
@ ( bit_se8732182000553998342ip_bit @ int ) ) ).
% flip_bit_integer.rsp
thf(fact_2587_flip__bit__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4 )
@ ( bit_se8732182000553998342ip_bit @ nat )
@ ( bit_se8732182000553998342ip_bit @ nat ) ) ).
% flip_bit_natural.rsp
thf(fact_2588_push__bit__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4 )
@ ( bit_se4730199178511100633sh_bit @ nat )
@ ( bit_se4730199178511100633sh_bit @ nat ) ) ).
% push_bit_natural.rsp
thf(fact_2589_push__bit__integer_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( int > int )
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ^ [Y5: int,Z4: int] : Y5 = Z4 )
@ ( bit_se4730199178511100633sh_bit @ int )
@ ( bit_se4730199178511100633sh_bit @ int ) ) ).
% push_bit_integer.rsp
thf(fact_2590_plus__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ^ [Y5: int,Z4: int] : Y5 = Z4 )
@ ( plus_plus @ int )
@ ( plus_plus @ int ) ) ).
% plus_integer.rsp
thf(fact_2591_plus__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4 )
@ ( plus_plus @ nat )
@ ( plus_plus @ nat ) ) ).
% plus_natural.rsp
thf(fact_2592_drop__bit__integer_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( int > int )
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ^ [Y5: int,Z4: int] : Y5 = Z4 )
@ ( bit_se4197421643247451524op_bit @ int )
@ ( bit_se4197421643247451524op_bit @ int ) ) ).
% drop_bit_integer.rsp
thf(fact_2593_minus__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4 )
@ ( minus_minus @ nat )
@ ( minus_minus @ nat ) ) ).
% minus_natural.rsp
thf(fact_2594_minus__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ^ [Y5: int,Z4: int] : Y5 = Z4 )
@ ( minus_minus @ int )
@ ( minus_minus @ int ) ) ).
% minus_integer.rsp
thf(fact_2595_less__eq__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > $o ) @ ( int > $o )
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ( bNF_rel_fun @ int @ int @ $o @ $o
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 )
@ ( ord_less_eq @ int )
@ ( ord_less_eq @ int ) ) ).
% less_eq_integer.rsp
thf(fact_2596_less__eq__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > $o ) @ ( nat > $o )
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ( bNF_rel_fun @ nat @ nat @ $o @ $o
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 )
@ ( ord_less_eq @ nat )
@ ( ord_less_eq @ nat ) ) ).
% less_eq_natural.rsp
thf(fact_2597_SigmaE,axiom,
! [A: $tType,B: $tType,C2: product_prod @ A @ B,A4: set @ A,B6: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ C2 @ ( product_Sigma @ A @ B @ A4 @ B6 ) )
=> ~ ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ! [Y4: B] :
( ( member @ B @ Y4 @ ( B6 @ X2 ) )
=> ( C2
!= ( product_Pair @ A @ B @ X2 @ Y4 ) ) ) ) ) ).
% SigmaE
thf(fact_2598_SigmaD1,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,A4: set @ A,B6: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( product_Sigma @ A @ B @ A4 @ B6 ) )
=> ( member @ A @ A3 @ A4 ) ) ).
% SigmaD1
thf(fact_2599_SigmaD2,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,A4: set @ A,B6: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( product_Sigma @ A @ B @ A4 @ B6 ) )
=> ( member @ B @ B2 @ ( B6 @ A3 ) ) ) ).
% SigmaD2
thf(fact_2600_SigmaE2,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,A4: set @ A,B6: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( product_Sigma @ A @ B @ A4 @ B6 ) )
=> ~ ( ( member @ A @ A3 @ A4 )
=> ~ ( member @ B @ B2 @ ( B6 @ A3 ) ) ) ) ).
% SigmaE2
thf(fact_2601_Suc_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ suc
@ suc ) ).
% Suc.rsp
thf(fact_2602_sub_Orsp,axiom,
( bNF_rel_fun @ num @ num @ ( num > int ) @ ( num > int )
@ ^ [Y5: num,Z4: num] : Y5 = Z4
@ ( bNF_rel_fun @ num @ num @ int @ int
@ ^ [Y5: num,Z4: num] : Y5 = Z4
@ ^ [Y5: int,Z4: int] : Y5 = Z4 )
@ ^ [M: num,N5: num] : ( minus_minus @ int @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N5 ) )
@ ^ [M: num,N5: num] : ( minus_minus @ int @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N5 ) ) ) ).
% sub.rsp
thf(fact_2603_transfer__rule__numeral,axiom,
! [A: $tType,B: $tType] :
( ( ( monoid_add @ B )
& ( semiring_numeral @ B )
& ( monoid_add @ A )
& ( semiring_numeral @ A ) )
=> ! [R5: A > B > $o] :
( ( R5 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( R5 @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R5 @ ( bNF_rel_fun @ A @ B @ A @ B @ R5 @ R5 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
=> ( bNF_rel_fun @ num @ num @ A @ B
@ ^ [Y5: num,Z4: num] : Y5 = Z4
@ R5
@ ( numeral_numeral @ A )
@ ( numeral_numeral @ B ) ) ) ) ) ) ).
% transfer_rule_numeral
thf(fact_2604_uminus__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ( uminus_uminus @ int )
@ ( uminus_uminus @ int ) ) ).
% uminus_integer.rsp
thf(fact_2605_integer__of__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ int @ int
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ( semiring_1_of_nat @ int )
@ ( semiring_1_of_nat @ int ) ) ).
% integer_of_natural.rsp
thf(fact_2606_Sigma__Diff__distrib1,axiom,
! [B: $tType,A: $tType,I4: set @ A,J4: set @ A,C5: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ ( minus_minus @ ( set @ A ) @ I4 @ J4 ) @ C5 )
= ( minus_minus @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I4 @ C5 ) @ ( product_Sigma @ A @ B @ J4 @ C5 ) ) ) ).
% Sigma_Diff_distrib1
thf(fact_2607_natural__of__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ nat @ nat
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ nat2
@ nat2 ) ).
% natural_of_integer.rsp
thf(fact_2608_abs__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ( abs_abs @ int )
@ ( abs_abs @ int ) ) ).
% abs_integer.rsp
thf(fact_2609_bit__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( nat > $o ) @ ( nat > $o )
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ^ [Y5: nat > $o,Z4: nat > $o] : Y5 = Z4
@ ( bit_se5641148757651400278ts_bit @ int )
@ ( bit_se5641148757651400278ts_bit @ int ) ) ).
% bit_integer.rsp
thf(fact_2610_not__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ( bit_ri4277139882892585799ns_not @ int )
@ ( bit_ri4277139882892585799ns_not @ int ) ) ).
% not_integer.rsp
thf(fact_2611_sgn__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ( sgn_sgn @ int )
@ ( sgn_sgn @ int ) ) ).
% sgn_integer.rsp
thf(fact_2612_mask__integer_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ int @ int
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ( bit_se2239418461657761734s_mask @ int )
@ ( bit_se2239418461657761734s_mask @ int ) ) ).
% mask_integer.rsp
thf(fact_2613_mask__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ( bit_se2239418461657761734s_mask @ nat )
@ ( bit_se2239418461657761734s_mask @ nat ) ) ).
% mask_natural.rsp
thf(fact_2614_transfer__rule__of__int,axiom,
! [A: $tType,B: $tType] :
( ( ( ring_1 @ B )
& ( ring_1 @ A ) )
=> ! [R5: A > B > $o] :
( ( R5 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( R5 @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R5 @ ( bNF_rel_fun @ A @ B @ A @ B @ R5 @ R5 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ A @ B @ R5 @ R5 @ ( uminus_uminus @ A ) @ ( uminus_uminus @ B ) )
=> ( bNF_rel_fun @ int @ int @ A @ B
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ R5
@ ( ring_1_of_int @ A )
@ ( ring_1_of_int @ B ) ) ) ) ) ) ) ).
% transfer_rule_of_int
thf(fact_2615_fun_Orel__map_I1_J,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,Sb: C > B > $o,I: A > C,X: D > A,Y: D > B] :
( ( bNF_rel_fun @ D @ D @ C @ B
@ ^ [Y5: D,Z4: D] : Y5 = Z4
@ Sb
@ ( comp @ A @ C @ D @ I @ X )
@ Y )
= ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y5: D,Z4: D] : Y5 = Z4
@ ^ [X3: A] : ( Sb @ ( I @ X3 ) )
@ X
@ Y ) ) ).
% fun.rel_map(1)
thf(fact_2616_fun_Orel__map_I2_J,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,Sa: A > C > $o,X: D > A,G: B > C,Y: D > B] :
( ( bNF_rel_fun @ D @ D @ A @ C
@ ^ [Y5: D,Z4: D] : Y5 = Z4
@ Sa
@ X
@ ( comp @ B @ C @ D @ G @ Y ) )
= ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y5: D,Z4: D] : Y5 = Z4
@ ^ [X3: A,Y3: B] : ( Sa @ X3 @ ( G @ Y3 ) )
@ X
@ Y ) ) ).
% fun.rel_map(2)
thf(fact_2617_times__eq__iff,axiom,
! [A: $tType,B: $tType,A4: set @ A,B6: set @ B,C5: set @ A,D4: set @ B] :
( ( ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : B6 )
= ( product_Sigma @ A @ B @ C5
@ ^ [Uu: A] : D4 ) )
= ( ( ( A4 = C5 )
& ( B6 = D4 ) )
| ( ( ( A4
= ( bot_bot @ ( set @ A ) ) )
| ( B6
= ( bot_bot @ ( set @ B ) ) ) )
& ( ( C5
= ( bot_bot @ ( set @ A ) ) )
| ( D4
= ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ).
% times_eq_iff
thf(fact_2618_dup_Orsp,axiom,
( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ^ [K4: int] : ( plus_plus @ int @ K4 @ K4 )
@ ^ [K4: int] : ( plus_plus @ int @ K4 @ K4 ) ) ).
% dup.rsp
thf(fact_2619_Sigma__Diff__distrib2,axiom,
! [B: $tType,A: $tType,I4: set @ A,A4: A > ( set @ B ),B6: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ I4
@ ^ [I3: A] : ( minus_minus @ ( set @ B ) @ ( A4 @ I3 ) @ ( B6 @ I3 ) ) )
= ( minus_minus @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I4 @ A4 ) @ ( product_Sigma @ A @ B @ I4 @ B6 ) ) ) ).
% Sigma_Diff_distrib2
thf(fact_2620_Times__Diff__distrib1,axiom,
! [B: $tType,A: $tType,A4: set @ A,B6: set @ A,C5: set @ B] :
( ( product_Sigma @ A @ B @ ( minus_minus @ ( set @ A ) @ A4 @ B6 )
@ ^ [Uu: A] : C5 )
= ( minus_minus @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : C5 )
@ ( product_Sigma @ A @ B @ B6
@ ^ [Uu: A] : C5 ) ) ) ).
% Times_Diff_distrib1
thf(fact_2621_Sigma__Union,axiom,
! [B: $tType,A: $tType,X10: set @ ( set @ A ),B6: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ ( complete_Sup_Sup @ ( set @ A ) @ X10 ) @ B6 )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ ( set @ A ) @ ( set @ ( product_prod @ A @ B ) )
@ ^ [A9: set @ A] : ( product_Sigma @ A @ B @ A9 @ B6 )
@ X10 ) ) ) ).
% Sigma_Union
thf(fact_2622_Collect__case__prod__Sigma,axiom,
! [B: $tType,A: $tType,P: A > $o,Q: A > B > $o] :
( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X3: A,Y3: B] :
( ( P @ X3 )
& ( Q @ X3 @ Y3 ) ) ) )
= ( product_Sigma @ A @ B @ ( collect @ A @ P )
@ ^ [X3: A] : ( collect @ B @ ( Q @ X3 ) ) ) ) ).
% Collect_case_prod_Sigma
thf(fact_2623_apfst__compose,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F: C > A,G: D > C,X: product_prod @ D @ B] :
( ( product_apfst @ C @ A @ B @ F @ ( product_apfst @ D @ C @ B @ G @ X ) )
= ( product_apfst @ D @ A @ B @ ( comp @ C @ A @ D @ F @ G ) @ X ) ) ).
% apfst_compose
thf(fact_2624_apsnd__apfst__commute,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,F: C > B,G: D > A,P5: product_prod @ D @ C] :
( ( product_apsnd @ C @ B @ A @ F @ ( product_apfst @ D @ A @ C @ G @ P5 ) )
= ( product_apfst @ D @ A @ B @ G @ ( product_apsnd @ C @ B @ D @ F @ P5 ) ) ) ).
% apsnd_apfst_commute
thf(fact_2625_gcd__add__mult,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [M2: A,K: A,N: A] :
( ( gcd_gcd @ A @ M2 @ ( plus_plus @ A @ ( times_times @ A @ K @ M2 ) @ N ) )
= ( gcd_gcd @ A @ M2 @ N ) ) ) ).
% gcd_add_mult
thf(fact_2626_Sigma__mono,axiom,
! [B: $tType,A: $tType,A4: set @ A,C5: set @ A,B6: A > ( set @ B ),D4: A > ( set @ B )] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ C5 )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( ord_less_eq @ ( set @ B ) @ ( B6 @ X2 ) @ ( D4 @ X2 ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ A4 @ B6 ) @ ( product_Sigma @ A @ B @ C5 @ D4 ) ) ) ) ).
% Sigma_mono
thf(fact_2627_bezout__int,axiom,
! [X: int,Y: int] :
? [U3: int,V4: int] :
( ( plus_plus @ int @ ( times_times @ int @ U3 @ X ) @ ( times_times @ int @ V4 @ Y ) )
= ( gcd_gcd @ int @ X @ Y ) ) ).
% bezout_int
thf(fact_2628_Times__subset__cancel2,axiom,
! [A: $tType,B: $tType,X: A,C5: set @ A,A4: set @ B,B6: set @ B] :
( ( member @ A @ X @ C5 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) )
@ ( product_Sigma @ B @ A @ A4
@ ^ [Uu: B] : C5 )
@ ( product_Sigma @ B @ A @ B6
@ ^ [Uu: B] : C5 ) )
= ( ord_less_eq @ ( set @ B ) @ A4 @ B6 ) ) ) ).
% Times_subset_cancel2
thf(fact_2629_UN__Times__distrib,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,E4: C > ( set @ A ),F6: D > ( set @ B ),A4: set @ C,B6: set @ D] :
( ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ ( product_prod @ C @ D ) @ ( set @ ( product_prod @ A @ B ) )
@ ( product_case_prod @ C @ D @ ( set @ ( product_prod @ A @ B ) )
@ ^ [A7: C,B5: D] :
( product_Sigma @ A @ B @ ( E4 @ A7 )
@ ^ [Uu: A] : ( F6 @ B5 ) ) )
@ ( product_Sigma @ C @ D @ A4
@ ^ [Uu: C] : B6 ) ) )
= ( product_Sigma @ A @ B @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ C @ ( set @ A ) @ E4 @ A4 ) )
@ ^ [Uu: A] : ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ D @ ( set @ B ) @ F6 @ B6 ) ) ) ) ).
% UN_Times_distrib
thf(fact_2630_mem__Times__iff,axiom,
! [A: $tType,B: $tType,X: product_prod @ A @ B,A4: set @ A,B6: set @ B] :
( ( member @ ( product_prod @ A @ B ) @ X
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : B6 ) )
= ( ( member @ A @ ( product_fst @ A @ B @ X ) @ A4 )
& ( member @ B @ ( product_snd @ A @ B @ X ) @ B6 ) ) ) ).
% mem_Times_iff
thf(fact_2631_in__prod__fst__sndI,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B,A4: set @ A,B6: set @ B] :
( ( member @ A @ ( product_fst @ A @ B @ X ) @ A4 )
=> ( ( member @ B @ ( product_snd @ A @ B @ X ) @ B6 )
=> ( member @ ( product_prod @ A @ B ) @ X
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : B6 ) ) ) ) ).
% in_prod_fst_sndI
thf(fact_2632_swap__product,axiom,
! [B: $tType,A: $tType,A4: set @ B,B6: set @ A] :
( ( image2 @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ B @ A @ ( product_prod @ A @ B )
@ ^ [I3: B,J2: A] : ( product_Pair @ A @ B @ J2 @ I3 ) )
@ ( product_Sigma @ B @ A @ A4
@ ^ [Uu: B] : B6 ) )
= ( product_Sigma @ A @ B @ B6
@ ^ [Uu: A] : A4 ) ) ).
% swap_product
thf(fact_2633_Sigma__empty__iff,axiom,
! [B: $tType,A: $tType,I4: set @ A,X10: A > ( set @ B )] :
( ( ( product_Sigma @ A @ B @ I4 @ X10 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ I4 )
=> ( ( X10 @ X3 )
= ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% Sigma_empty_iff
thf(fact_2634_gcd__mult__unit2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( gcd_gcd @ A @ B2 @ ( times_times @ A @ C2 @ A3 ) )
= ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ).
% gcd_mult_unit2
thf(fact_2635_gcd__mult__unit1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( gcd_gcd @ A @ ( times_times @ A @ B2 @ A3 ) @ C2 )
= ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ).
% gcd_mult_unit1
thf(fact_2636_gcd__div__unit1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( gcd_gcd @ A @ ( divide_divide @ A @ B2 @ A3 ) @ C2 )
= ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ).
% gcd_div_unit1
thf(fact_2637_gcd__div__unit2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( gcd_gcd @ A @ B2 @ ( divide_divide @ A @ C2 @ A3 ) )
= ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ).
% gcd_div_unit2
thf(fact_2638_less__int_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( ( product_prod @ nat @ nat ) > $o ) @ ( int > $o ) @ pcr_int
@ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ $o @ $o @ pcr_int
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X3: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V2: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X3 @ V2 ) @ ( plus_plus @ nat @ U2 @ Y3 ) ) ) )
@ ( ord_less @ int ) ) ).
% less_int.transfer
thf(fact_2639_less__eq__int_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( ( product_prod @ nat @ nat ) > $o ) @ ( int > $o ) @ pcr_int
@ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ $o @ $o @ pcr_int
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X3: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V2: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X3 @ V2 ) @ ( plus_plus @ nat @ U2 @ Y3 ) ) ) )
@ ( ord_less_eq @ int ) ) ).
% less_eq_int.transfer
thf(fact_2640_plus__int_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( 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 @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X3: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X3 @ U2 ) @ ( plus_plus @ nat @ Y3 @ V2 ) ) ) )
@ ( plus_plus @ int ) ) ).
% plus_int.transfer
thf(fact_2641_minus__int_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( 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 @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X3: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X3 @ V2 ) @ ( plus_plus @ nat @ Y3 @ U2 ) ) ) )
@ ( minus_minus @ int ) ) ).
% minus_int.transfer
thf(fact_2642_times__subset__iff,axiom,
! [A: $tType,B: $tType,A4: set @ A,C5: set @ B,B6: set @ A,D4: set @ B] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : C5 )
@ ( product_Sigma @ A @ B @ B6
@ ^ [Uu: A] : D4 ) )
= ( ( A4
= ( bot_bot @ ( set @ A ) ) )
| ( C5
= ( bot_bot @ ( set @ B ) ) )
| ( ( ord_less_eq @ ( set @ A ) @ A4 @ B6 )
& ( ord_less_eq @ ( set @ B ) @ C5 @ D4 ) ) ) ) ).
% times_subset_iff
thf(fact_2643_image__paired__Times,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,F: C > A,G: D > B,A4: set @ C,B6: set @ D] :
( ( image2 @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ C @ D @ ( product_prod @ A @ B )
@ ^ [X3: C,Y3: D] : ( product_Pair @ A @ B @ ( F @ X3 ) @ ( G @ Y3 ) ) )
@ ( product_Sigma @ C @ D @ A4
@ ^ [Uu: C] : B6 ) )
= ( product_Sigma @ A @ B @ ( image2 @ C @ A @ F @ A4 )
@ ^ [Uu: A] : ( image2 @ D @ B @ G @ B6 ) ) ) ).
% image_paired_Times
thf(fact_2644_fst__image__Sigma,axiom,
! [B: $tType,A: $tType,A4: set @ A,B6: A > ( set @ B )] :
( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( product_Sigma @ A @ B @ A4 @ B6 ) )
= ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ A4 )
& ( ( B6 @ X3 )
!= ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% fst_image_Sigma
thf(fact_2645_Sigma__def,axiom,
! [B: $tType,A: $tType] :
( ( product_Sigma @ A @ B )
= ( ^ [A9: set @ A,B9: A > ( set @ B )] :
( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ A @ ( set @ ( product_prod @ A @ B ) )
@ ^ [X3: A] :
( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ B @ ( set @ ( product_prod @ A @ B ) )
@ ^ [Y3: B] : ( insert2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
@ ( B9 @ X3 ) ) )
@ A9 ) ) ) ) ).
% Sigma_def
thf(fact_2646_times__int_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( 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 @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X3: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X3 @ U2 ) @ ( times_times @ nat @ Y3 @ V2 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X3 @ V2 ) @ ( times_times @ nat @ Y3 @ U2 ) ) ) ) )
@ ( times_times @ int ) ) ).
% times_int.transfer
thf(fact_2647_num__of__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ num @ num
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ^ [Y5: num,Z4: num] : Y5 = Z4
@ ( comp @ nat @ num @ int @ num_of_nat @ nat2 )
@ ( comp @ nat @ num @ int @ num_of_nat @ nat2 ) ) ).
% num_of_integer.rsp
thf(fact_2648_snd__image__Sigma,axiom,
! [A: $tType,B: $tType,A4: set @ B,B6: B > ( set @ A )] :
( ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( product_Sigma @ B @ A @ A4 @ B6 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B6 @ A4 ) ) ) ).
% snd_image_Sigma
thf(fact_2649_subset__fst__snd,axiom,
! [B: $tType,A: $tType,A4: set @ ( product_prod @ A @ B )] :
( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A4
@ ( product_Sigma @ A @ B @ ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A4 )
@ ^ [Uu: A] : ( image2 @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A4 ) ) ) ).
% subset_fst_snd
thf(fact_2650_int__transfer,axiom,
( bNF_rel_fun @ nat @ nat @ ( product_prod @ nat @ nat ) @ int
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ pcr_int
@ ^ [N5: nat] : ( product_Pair @ nat @ nat @ N5 @ ( zero_zero @ nat ) )
@ ( semiring_1_of_nat @ int ) ) ).
% int_transfer
thf(fact_2651_uminus__int_Otransfer,axiom,
( 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 )
@ ^ [X3: nat,Y3: nat] : ( product_Pair @ nat @ nat @ Y3 @ X3 ) )
@ ( uminus_uminus @ int ) ) ).
% uminus_int.transfer
thf(fact_2652_eq__fst__iff,axiom,
! [A: $tType,B: $tType,A3: A,P5: product_prod @ A @ B] :
( ( A3
= ( product_fst @ A @ B @ P5 ) )
= ( ? [B5: B] :
( P5
= ( product_Pair @ A @ B @ A3 @ B5 ) ) ) ) ).
% eq_fst_iff
thf(fact_2653_transfer__rule__of__nat,axiom,
! [A: $tType,B: $tType] :
( ( ( semiring_1 @ B )
& ( semiring_1 @ A ) )
=> ! [R5: A > B > $o] :
( ( R5 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( R5 @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R5 @ ( bNF_rel_fun @ A @ B @ A @ B @ R5 @ R5 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
=> ( bNF_rel_fun @ nat @ nat @ A @ B
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ R5
@ ( semiring_1_of_nat @ A )
@ ( semiring_1_of_nat @ B ) ) ) ) ) ) ).
% transfer_rule_of_nat
thf(fact_2654_transfer__rule__of__bool,axiom,
! [A: $tType,B: $tType] :
( ( ( zero_neq_one @ B )
& ( zero_neq_one @ A ) )
=> ! [R5: A > B > $o] :
( ( R5 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( R5 @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( bNF_rel_fun @ $o @ $o @ A @ B
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4
@ R5
@ ( zero_neq_one_of_bool @ A )
@ ( zero_neq_one_of_bool @ B ) ) ) ) ) ).
% transfer_rule_of_bool
thf(fact_2655_plus__rat_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ ( ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ) @ ( rat > rat ) @ pcr_rat @ ( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ ( product_prod @ int @ int ) @ rat @ pcr_rat @ pcr_rat )
@ ^ [X3: product_prod @ int @ int,Y3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X3 ) @ ( product_snd @ int @ int @ Y3 ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ Y3 ) @ ( product_snd @ int @ int @ X3 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X3 ) @ ( product_snd @ int @ int @ Y3 ) ) )
@ ( plus_plus @ rat ) ) ).
% plus_rat.transfer
thf(fact_2656_inverse__rat_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ ( product_prod @ int @ int ) @ rat @ pcr_rat @ pcr_rat
@ ^ [X3: product_prod @ int @ int] :
( if @ ( product_prod @ int @ int )
@ ( ( product_fst @ int @ int @ X3 )
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ X3 ) @ ( product_fst @ int @ int @ X3 ) ) )
@ ( inverse_inverse @ rat ) ) ).
% inverse_rat.transfer
thf(fact_2657_Id__on__def,axiom,
! [A: $tType] :
( ( id_on @ A )
= ( ^ [A9: set @ A] :
( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ A @ ( set @ ( product_prod @ A @ A ) )
@ ^ [X3: A] : ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
@ A9 ) ) ) ) ).
% Id_on_def
thf(fact_2658_times__rat_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ ( ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ) @ ( rat > rat ) @ pcr_rat @ ( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ ( product_prod @ int @ int ) @ rat @ pcr_rat @ pcr_rat )
@ ^ [X3: product_prod @ int @ int,Y3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X3 ) @ ( product_fst @ int @ int @ Y3 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X3 ) @ ( product_snd @ int @ int @ Y3 ) ) )
@ ( times_times @ rat ) ) ).
% times_rat.transfer
thf(fact_2659_gcd__1__nat,axiom,
! [M2: nat] :
( ( gcd_gcd @ nat @ M2 @ ( one_one @ nat ) )
= ( one_one @ nat ) ) ).
% gcd_1_nat
thf(fact_2660_Id__onI,axiom,
! [A: $tType,A3: A,A4: set @ A] :
( ( member @ A @ A3 @ A4 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ ( id_on @ A @ A4 ) ) ) ).
% Id_onI
thf(fact_2661_unset__bit__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4 )
@ ( bit_se2638667681897837118et_bit @ nat )
@ ( bit_se2638667681897837118et_bit @ nat ) ) ).
% unset_bit_natural.rsp
thf(fact_2662_take__bit__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4 )
@ ( bit_se2584673776208193580ke_bit @ nat )
@ ( bit_se2584673776208193580ke_bit @ nat ) ) ).
% take_bit_natural.rsp
thf(fact_2663_bit__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > $o ) @ ( nat > $o )
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ^ [Y5: nat > $o,Z4: nat > $o] : Y5 = Z4
@ ( bit_se5641148757651400278ts_bit @ nat )
@ ( bit_se5641148757651400278ts_bit @ nat ) ) ).
% bit_natural.rsp
thf(fact_2664_drop__bit__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4 )
@ ( bit_se4197421643247451524op_bit @ nat )
@ ( bit_se4197421643247451524op_bit @ nat ) ) ).
% drop_bit_natural.rsp
thf(fact_2665_Id__on__iff,axiom,
! [A: $tType,X: A,Y: A,A4: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( id_on @ A @ A4 ) )
= ( ( X = Y )
& ( member @ A @ X @ A4 ) ) ) ).
% Id_on_iff
thf(fact_2666_Id__on__eqI,axiom,
! [A: $tType,A3: A,B2: A,A4: set @ A] :
( ( A3 = B2 )
=> ( ( member @ A @ A3 @ A4 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( id_on @ A @ A4 ) ) ) ) ).
% Id_on_eqI
thf(fact_2667_Id__onE,axiom,
! [A: $tType,C2: product_prod @ A @ A,A4: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ C2 @ ( id_on @ A @ A4 ) )
=> ~ ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( C2
!= ( product_Pair @ A @ A @ X2 @ X2 ) ) ) ) ).
% Id_onE
thf(fact_2668_bezout__nat,axiom,
! [A3: nat,B2: nat] :
( ( A3
!= ( zero_zero @ nat ) )
=> ? [X2: nat,Y4: nat] :
( ( times_times @ nat @ A3 @ X2 )
= ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y4 ) @ ( gcd_gcd @ nat @ A3 @ B2 ) ) ) ) ).
% bezout_nat
thf(fact_2669_one__rat_Otransfer,axiom,
pcr_rat @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) @ ( one_one @ rat ) ).
% one_rat.transfer
thf(fact_2670_zero__rat_Otransfer,axiom,
pcr_rat @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) @ ( zero_zero @ rat ) ).
% zero_rat.transfer
thf(fact_2671_bezw__aux,axiom,
! [X: nat,Y: nat] :
( ( semiring_1_of_nat @ int @ ( gcd_gcd @ nat @ X @ Y ) )
= ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ ( bezw @ X @ Y ) ) @ ( semiring_1_of_nat @ int @ X ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ X @ Y ) ) @ ( semiring_1_of_nat @ int @ Y ) ) ) ) ).
% bezw_aux
thf(fact_2672_uminus__rat_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ ( product_prod @ int @ int ) @ rat @ pcr_rat @ pcr_rat
@ ^ [X3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( product_fst @ int @ int @ X3 ) ) @ ( product_snd @ int @ int @ X3 ) )
@ ( uminus_uminus @ rat ) ) ).
% uminus_rat.transfer
thf(fact_2673_gcd__nat_Opelims,axiom,
! [X: nat,Xa: nat,Y: nat] :
( ( ( gcd_gcd @ nat @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ nat @ nat ) @ gcd_nat_rel @ ( product_Pair @ nat @ nat @ X @ Xa ) )
=> ~ ( ( ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y = X ) )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( Y
= ( gcd_gcd @ nat @ Xa @ ( modulo_modulo @ nat @ X @ Xa ) ) ) ) )
=> ~ ( accp @ ( product_prod @ nat @ nat ) @ gcd_nat_rel @ ( product_Pair @ nat @ nat @ X @ Xa ) ) ) ) ) ).
% gcd_nat.pelims
thf(fact_2674_o__rsp_I1_J,axiom,
! [A: $tType,B: $tType,E: $tType,F5: $tType,D: $tType,C: $tType,R23: A > C > $o,R32: B > D > $o,R12: E > F5 > $o] : ( bNF_rel_fun @ ( A > B ) @ ( C > D ) @ ( ( E > A ) > E > B ) @ ( ( F5 > C ) > F5 > D ) @ ( bNF_rel_fun @ A @ C @ B @ D @ R23 @ R32 ) @ ( bNF_rel_fun @ ( E > A ) @ ( F5 > C ) @ ( E > B ) @ ( F5 > D ) @ ( bNF_rel_fun @ E @ F5 @ A @ C @ R12 @ R23 ) @ ( bNF_rel_fun @ E @ F5 @ B @ D @ R12 @ R32 ) ) @ ( comp @ A @ B @ E ) @ ( comp @ C @ D @ F5 ) ) ).
% o_rsp(1)
thf(fact_2675_o__rsp_I2_J,axiom,
! [E: $tType,F5: $tType,H8: $tType,G3: $tType,R12: E > F5 > $o] :
( bNF_rel_fun @ ( G3 > H8 ) @ ( G3 > H8 ) @ ( ( E > G3 ) > E > H8 ) @ ( ( F5 > G3 ) > F5 > H8 )
@ ^ [Y5: G3 > H8,Z4: G3 > H8] : Y5 = Z4
@ ( bNF_rel_fun @ ( E > G3 ) @ ( F5 > G3 ) @ ( E > H8 ) @ ( F5 > H8 )
@ ( bNF_rel_fun @ E @ F5 @ G3 @ G3 @ R12
@ ^ [Y5: G3,Z4: G3] : Y5 = Z4 )
@ ( bNF_rel_fun @ E @ F5 @ H8 @ H8 @ R12
@ ^ [Y5: H8,Z4: H8] : Y5 = Z4 ) )
@ ( comp @ G3 @ H8 @ E )
@ ( comp @ G3 @ H8 @ F5 ) ) ).
% o_rsp(2)
thf(fact_2676_apfst__convE,axiom,
! [C: $tType,A: $tType,B: $tType,Q3: product_prod @ A @ B,F: C > A,P5: product_prod @ C @ B] :
( ( Q3
= ( product_apfst @ C @ A @ B @ F @ P5 ) )
=> ~ ! [X2: C,Y4: B] :
( ( P5
= ( product_Pair @ C @ B @ X2 @ Y4 ) )
=> ( Q3
!= ( product_Pair @ A @ B @ ( F @ X2 ) @ Y4 ) ) ) ) ).
% apfst_convE
thf(fact_2677_UN__le__eq__Un0,axiom,
! [A: $tType,M3: nat > ( set @ A ),N: nat] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ M3 @ ( set_ord_atMost @ nat @ N ) ) )
= ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ M3 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N ) ) ) @ ( M3 @ ( zero_zero @ nat ) ) ) ) ).
% UN_le_eq_Un0
thf(fact_2678_times__int_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( ( 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 @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X3: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X3 @ U2 ) @ ( times_times @ nat @ Y3 @ V2 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X3 @ V2 ) @ ( times_times @ nat @ Y3 @ U2 ) ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X3: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X3 @ U2 ) @ ( times_times @ nat @ Y3 @ V2 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X3 @ V2 ) @ ( times_times @ nat @ Y3 @ U2 ) ) ) ) ) ) ).
% times_int.rsp
thf(fact_2679_sup__Some,axiom,
! [A: $tType] :
( ( sup @ A )
=> ! [X: A,Y: A] :
( ( sup_sup @ ( option @ A ) @ ( some @ A @ X ) @ ( some @ A @ Y ) )
= ( some @ A @ ( sup_sup @ A @ X @ Y ) ) ) ) ).
% sup_Some
thf(fact_2680_intrel__iff,axiom,
! [X: nat,Y: nat,U: nat,V: nat] :
( ( intrel @ ( product_Pair @ nat @ nat @ X @ Y ) @ ( product_Pair @ nat @ nat @ U @ V ) )
= ( ( plus_plus @ nat @ X @ V )
= ( plus_plus @ nat @ U @ Y ) ) ) ).
% intrel_iff
thf(fact_2681_insert__Times__insert,axiom,
! [B: $tType,A: $tType,A3: A,A4: set @ A,B2: B,B6: set @ B] :
( ( product_Sigma @ A @ B @ ( insert2 @ A @ A3 @ A4 )
@ ^ [Uu: A] : ( insert2 @ B @ B2 @ B6 ) )
= ( insert2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 )
@ ( sup_sup @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : ( insert2 @ B @ B2 @ B6 ) )
@ ( product_Sigma @ A @ B @ ( insert2 @ A @ A3 @ A4 )
@ ^ [Uu: A] : B6 ) ) ) ) ).
% insert_Times_insert
thf(fact_2682_Sigma__Un__distrib1,axiom,
! [B: $tType,A: $tType,I4: set @ A,J4: set @ A,C5: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ ( sup_sup @ ( set @ A ) @ I4 @ J4 ) @ C5 )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I4 @ C5 ) @ ( product_Sigma @ A @ B @ J4 @ C5 ) ) ) ).
% Sigma_Un_distrib1
thf(fact_2683_Sigma__Un__distrib2,axiom,
! [B: $tType,A: $tType,I4: set @ A,A4: A > ( set @ B ),B6: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ I4
@ ^ [I3: A] : ( sup_sup @ ( set @ B ) @ ( A4 @ I3 ) @ ( B6 @ I3 ) ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I4 @ A4 ) @ ( product_Sigma @ A @ B @ I4 @ B6 ) ) ) ).
% Sigma_Un_distrib2
thf(fact_2684_Times__Un__distrib1,axiom,
! [B: $tType,A: $tType,A4: set @ A,B6: set @ A,C5: set @ B] :
( ( product_Sigma @ A @ B @ ( sup_sup @ ( set @ A ) @ A4 @ B6 )
@ ^ [Uu: A] : C5 )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : C5 )
@ ( product_Sigma @ A @ B @ B6
@ ^ [Uu: A] : C5 ) ) ) ).
% Times_Un_distrib1
thf(fact_2685_set__diff__diff__left,axiom,
! [A: $tType,A4: set @ A,B6: set @ A,C5: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B6 ) @ C5 )
= ( minus_minus @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B6 @ C5 ) ) ) ).
% set_diff_diff_left
thf(fact_2686_sup__option__def,axiom,
! [A: $tType] :
( ( sup @ A )
=> ( ( sup_sup @ ( option @ A ) )
= ( ^ [X3: option @ A,Y3: option @ A] :
( case_option @ ( option @ A ) @ A @ Y3
@ ^ [X9: A] :
( case_option @ ( option @ A ) @ A @ X3
@ ^ [Z5: A] : ( some @ A @ ( sup_sup @ A @ X9 @ Z5 ) )
@ Y3 )
@ X3 ) ) ) ) ).
% sup_option_def
thf(fact_2687_ivl__disj__un__two__touch_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M2: A,U: A] :
( ( ord_less_eq @ A @ L @ M2 )
=> ( ( ord_less_eq @ A @ M2 @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M2 ) @ ( set_or1337092689740270186AtMost @ A @ M2 @ U ) )
= ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two_touch(4)
thf(fact_2688_ivl__disj__un__two_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M2: A,U: A] :
( ( ord_less_eq @ A @ L @ M2 )
=> ( ( ord_less_eq @ A @ M2 @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ M2 ) @ ( set_or7035219750837199246ssThan @ A @ M2 @ U ) )
= ( set_or7035219750837199246ssThan @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(3)
thf(fact_2689_max__ext__additive,axiom,
! [A: $tType,A4: set @ A,B6: set @ A,R5: set @ ( product_prod @ A @ A ),C5: set @ A,D4: set @ A] :
( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A4 @ B6 ) @ ( max_ext @ A @ R5 ) )
=> ( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ C5 @ D4 ) @ ( max_ext @ A @ R5 ) )
=> ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ C5 ) @ ( sup_sup @ ( set @ A ) @ B6 @ D4 ) ) @ ( max_ext @ A @ R5 ) ) ) ) ).
% max_ext_additive
thf(fact_2690_zero__int_Orsp,axiom,
intrel @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) ).
% zero_int.rsp
thf(fact_2691_ivl__disj__un__two_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M2: A,U: A] :
( ( ord_less_eq @ A @ L @ M2 )
=> ( ( ord_less_eq @ A @ M2 @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ M2 ) @ ( set_or1337092689740270186AtMost @ A @ M2 @ U ) )
= ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(7)
thf(fact_2692_card__Un__le,axiom,
! [A: $tType,A4: set @ A,B6: set @ A] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B6 ) ) @ ( plus_plus @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ A @ B6 ) ) ) ).
% card_Un_le
thf(fact_2693_ivl__disj__un__one_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less_eq @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_ord_lessThan @ A @ L ) @ ( set_or7035219750837199246ssThan @ A @ L @ U ) )
= ( set_ord_lessThan @ A @ U ) ) ) ) ).
% ivl_disj_un_one(2)
thf(fact_2694_atLeastLessThan__add__Un,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( set_or7035219750837199246ssThan @ nat @ I @ ( plus_plus @ nat @ J @ K ) )
= ( sup_sup @ ( set @ nat ) @ ( set_or7035219750837199246ssThan @ nat @ I @ J ) @ ( set_or7035219750837199246ssThan @ nat @ J @ ( plus_plus @ nat @ J @ K ) ) ) ) ) ).
% atLeastLessThan_add_Un
thf(fact_2695_uminus__int_Orsp,axiom,
( 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 )
@ ^ [X3: nat,Y3: nat] : ( product_Pair @ nat @ nat @ Y3 @ X3 ) )
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [X3: nat,Y3: nat] : ( product_Pair @ nat @ nat @ Y3 @ X3 ) ) ) ).
% uminus_int.rsp
thf(fact_2696_one__int_Orsp,axiom,
intrel @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) ).
% one_int.rsp
thf(fact_2697_ivl__disj__un__two__touch_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M2: A,U: A] :
( ( ord_less_eq @ A @ L @ M2 )
=> ( ( ord_less @ A @ M2 @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M2 ) @ ( set_or7035219750837199246ssThan @ A @ M2 @ U ) )
= ( set_or7035219750837199246ssThan @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two_touch(2)
thf(fact_2698_ivl__disj__un__two_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M2: A,U: A] :
( ( ord_less @ A @ L @ M2 )
=> ( ( ord_less_eq @ A @ M2 @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M2 ) @ ( set_or7035219750837199246ssThan @ A @ M2 @ U ) )
= ( set_or5935395276787703475ssThan @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(1)
thf(fact_2699_ivl__disj__un__singleton_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [U: A] :
( ( sup_sup @ ( set @ A ) @ ( set_ord_lessThan @ A @ U ) @ ( insert2 @ A @ U @ ( bot_bot @ ( set @ A ) ) ) )
= ( set_ord_atMost @ A @ U ) ) ) ).
% ivl_disj_un_singleton(2)
thf(fact_2700_ivl__disj__un__one_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less_eq @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_ord_lessThan @ A @ L ) @ ( set_or1337092689740270186AtMost @ A @ L @ U ) )
= ( set_ord_atMost @ A @ U ) ) ) ) ).
% ivl_disj_un_one(4)
thf(fact_2701_ivl__disj__un__one_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_ord_atMost @ A @ L ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) )
= ( set_ord_lessThan @ A @ U ) ) ) ) ).
% ivl_disj_un_one(1)
thf(fact_2702_intrel__def,axiom,
( intrel
= ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X3: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V2: nat] :
( ( plus_plus @ nat @ X3 @ V2 )
= ( plus_plus @ nat @ U2 @ Y3 ) ) ) ) ) ).
% intrel_def
thf(fact_2703_ivl__disj__un__singleton_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less_eq @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ U ) @ ( insert2 @ A @ U @ ( bot_bot @ ( set @ A ) ) ) )
= ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ).
% ivl_disj_un_singleton(6)
thf(fact_2704_ivl__disj__un__singleton_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( insert2 @ A @ L @ ( bot_bot @ ( set @ A ) ) ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) )
= ( set_or7035219750837199246ssThan @ A @ L @ U ) ) ) ) ).
% ivl_disj_un_singleton(3)
thf(fact_2705_ivl__disj__un__two_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M2: A,U: A] :
( ( ord_less_eq @ A @ L @ M2 )
=> ( ( ord_less @ A @ M2 @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M2 ) @ ( set_or5935395276787703475ssThan @ A @ M2 @ U ) )
= ( set_or7035219750837199246ssThan @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(4)
thf(fact_2706_less__int_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > $o ) @ ( ( product_prod @ nat @ nat ) > $o ) @ intrel
@ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ $o @ $o @ intrel
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X3: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V2: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X3 @ V2 ) @ ( plus_plus @ nat @ U2 @ Y3 ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X3: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V2: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X3 @ V2 ) @ ( plus_plus @ nat @ U2 @ Y3 ) ) ) ) ) ).
% less_int.rsp
thf(fact_2707_less__eq__int_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > $o ) @ ( ( product_prod @ nat @ nat ) > $o ) @ intrel
@ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ $o @ $o @ intrel
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X3: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V2: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X3 @ V2 ) @ ( plus_plus @ nat @ U2 @ Y3 ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X3: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V2: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X3 @ V2 ) @ ( plus_plus @ nat @ U2 @ Y3 ) ) ) ) ) ).
% less_eq_int.rsp
thf(fact_2708_plus__int_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( ( 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 @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X3: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X3 @ U2 ) @ ( plus_plus @ nat @ Y3 @ V2 ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X3: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X3 @ U2 ) @ ( plus_plus @ nat @ Y3 @ V2 ) ) ) ) ) ).
% plus_int.rsp
thf(fact_2709_minus__int_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( ( 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 @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X3: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X3 @ V2 ) @ ( plus_plus @ nat @ Y3 @ U2 ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X3: nat,Y3: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X3 @ V2 ) @ ( plus_plus @ nat @ Y3 @ U2 ) ) ) ) ) ).
% minus_int.rsp
thf(fact_2710_max__weak__def,axiom,
( fun_max_weak
= ( sup_sup @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ ( max_ext @ ( product_prod @ nat @ nat ) @ fun_pair_leq ) @ ( insert2 @ ( 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
thf(fact_2711_min__weak__def,axiom,
( fun_min_weak
= ( sup_sup @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ ( min_ext @ ( product_prod @ nat @ nat ) @ fun_pair_leq ) @ ( insert2 @ ( 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
thf(fact_2712_fold__union__pair,axiom,
! [B: $tType,A: $tType,B6: set @ A,X: B,A4: set @ ( product_prod @ B @ A )] :
( ( finite_finite2 @ A @ B6 )
=> ( ( sup_sup @ ( set @ ( product_prod @ B @ A ) )
@ ( complete_Sup_Sup @ ( set @ ( product_prod @ B @ A ) )
@ ( image2 @ A @ ( set @ ( product_prod @ B @ A ) )
@ ^ [Y3: A] : ( insert2 @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y3 ) @ ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) )
@ B6 ) )
@ A4 )
= ( finite_fold @ A @ ( set @ ( product_prod @ B @ A ) )
@ ^ [Y3: A] : ( insert2 @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y3 ) )
@ A4
@ B6 ) ) ) ).
% fold_union_pair
thf(fact_2713_Field__insert,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A )] :
( ( field2 @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R ) )
= ( sup_sup @ ( set @ A ) @ ( insert2 @ A @ A3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( field2 @ A @ R ) ) ) ).
% Field_insert
thf(fact_2714_set__to__map__insert,axiom,
! [B: $tType,A: $tType,Kv: product_prod @ A @ B,S2: set @ ( product_prod @ A @ B )] :
( ~ ( member @ A @ ( product_fst @ A @ B @ Kv ) @ ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ S2 ) )
=> ( ( set_to_map @ A @ B @ ( insert2 @ ( product_prod @ A @ B ) @ Kv @ S2 ) )
= ( fun_upd @ A @ ( option @ B ) @ ( set_to_map @ A @ B @ S2 ) @ ( product_fst @ A @ B @ Kv ) @ ( some @ B @ ( product_snd @ A @ B @ Kv ) ) ) ) ) ).
% set_to_map_insert
thf(fact_2715_plus__rat_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ) @ ( ( 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 )
@ ^ [X3: product_prod @ int @ int,Y3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X3 ) @ ( product_snd @ int @ int @ Y3 ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ Y3 ) @ ( product_snd @ int @ int @ X3 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X3 ) @ ( product_snd @ int @ int @ Y3 ) ) )
@ ^ [X3: product_prod @ int @ int,Y3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X3 ) @ ( product_snd @ int @ int @ Y3 ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ Y3 ) @ ( product_snd @ int @ int @ X3 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X3 ) @ ( product_snd @ int @ int @ Y3 ) ) ) ) ).
% plus_rat.rsp
thf(fact_2716_inverse__rat_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ratrel @ ratrel
@ ^ [X3: product_prod @ int @ int] :
( if @ ( product_prod @ int @ int )
@ ( ( product_fst @ int @ int @ X3 )
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ X3 ) @ ( product_fst @ int @ int @ X3 ) ) )
@ ^ [X3: product_prod @ int @ int] :
( if @ ( product_prod @ int @ int )
@ ( ( product_fst @ int @ int @ X3 )
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ X3 ) @ ( product_fst @ int @ int @ X3 ) ) ) ) ).
% inverse_rat.rsp
thf(fact_2717_fun__upd__upd,axiom,
! [A: $tType,B: $tType,F: A > B,X: A,Y: B,Z2: B] :
( ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ F @ X @ Y ) @ X @ Z2 )
= ( fun_upd @ A @ B @ F @ X @ Z2 ) ) ).
% fun_upd_upd
thf(fact_2718_fun__upd__triv,axiom,
! [B: $tType,A: $tType,F: A > B,X: A] :
( ( fun_upd @ A @ B @ F @ X @ ( F @ X ) )
= F ) ).
% fun_upd_triv
thf(fact_2719_fun__upd__apply,axiom,
! [A: $tType,B: $tType] :
( ( fun_upd @ B @ A )
= ( ^ [F2: B > A,X3: B,Y3: A,Z5: B] : ( if @ A @ ( Z5 = X3 ) @ Y3 @ ( F2 @ Z5 ) ) ) ) ).
% fun_upd_apply
thf(fact_2720_image__update,axiom,
! [B: $tType,A: $tType,X: A,A4: set @ A,F: A > B,N: B] :
( ~ ( member @ A @ X @ A4 )
=> ( ( image2 @ A @ B @ ( fun_upd @ A @ B @ F @ X @ N ) @ A4 )
= ( image2 @ A @ B @ F @ A4 ) ) ) ).
% image_update
thf(fact_2721_empty__upd__none,axiom,
! [B: $tType,A: $tType,X: A] :
( ( fun_upd @ A @ ( option @ B )
@ ^ [X3: A] : ( none @ B )
@ X
@ ( none @ B ) )
= ( ^ [X3: A] : ( none @ B ) ) ) ).
% empty_upd_none
thf(fact_2722_sup__None__1,axiom,
! [A: $tType] :
( ( sup @ A )
=> ! [Y: option @ A] :
( ( sup_sup @ ( option @ A ) @ ( none @ A ) @ Y )
= Y ) ) ).
% sup_None_1
thf(fact_2723_sup__None__2,axiom,
! [A: $tType] :
( ( sup @ A )
=> ! [X: option @ A] :
( ( sup_sup @ ( option @ A ) @ X @ ( none @ A ) )
= X ) ) ).
% sup_None_2
thf(fact_2724_finite__Field__eq__finite,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ A @ ( field2 @ A @ R5 ) )
= ( finite_finite2 @ ( product_prod @ A @ A ) @ R5 ) ) ).
% finite_Field_eq_finite
thf(fact_2725_image__map__upd,axiom,
! [B: $tType,A: $tType,X: A,A4: set @ A,M2: A > ( option @ B ),Y: B] :
( ~ ( member @ A @ X @ A4 )
=> ( ( image2 @ A @ ( option @ B ) @ ( fun_upd @ A @ ( option @ B ) @ M2 @ X @ ( some @ B @ Y ) ) @ A4 )
= ( image2 @ A @ ( option @ B ) @ M2 @ A4 ) ) ) ).
% image_map_upd
thf(fact_2726_map__option__o__map__upd,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > B,M2: A > ( option @ C ),A3: A,B2: C] :
( ( comp @ ( option @ C ) @ ( option @ B ) @ A @ ( map_option @ C @ B @ F ) @ ( fun_upd @ A @ ( option @ C ) @ M2 @ A3 @ ( some @ C @ B2 ) ) )
= ( fun_upd @ A @ ( option @ B ) @ ( comp @ ( option @ C ) @ ( option @ B ) @ A @ ( map_option @ C @ B @ F ) @ M2 ) @ A3 @ ( some @ B @ ( F @ B2 ) ) ) ) ).
% map_option_o_map_upd
thf(fact_2727_fun__upd__def,axiom,
! [B: $tType,A: $tType] :
( ( fun_upd @ A @ B )
= ( ^ [F2: A > B,A7: A,B5: B,X3: A] : ( if @ B @ ( X3 = A7 ) @ B5 @ ( F2 @ X3 ) ) ) ) ).
% fun_upd_def
thf(fact_2728_fun__upd__eqD,axiom,
! [A: $tType,B: $tType,F: A > B,X: A,Y: B,G: A > B,Z2: B] :
( ( ( fun_upd @ A @ B @ F @ X @ Y )
= ( fun_upd @ A @ B @ G @ X @ Z2 ) )
=> ( Y = Z2 ) ) ).
% fun_upd_eqD
thf(fact_2729_fun__upd__idem,axiom,
! [A: $tType,B: $tType,F: B > A,X: B,Y: A] :
( ( ( F @ X )
= Y )
=> ( ( fun_upd @ B @ A @ F @ X @ Y )
= F ) ) ).
% fun_upd_idem
thf(fact_2730_fun__upd__same,axiom,
! [B: $tType,A: $tType,F: B > A,X: B,Y: A] :
( ( fun_upd @ B @ A @ F @ X @ Y @ X )
= Y ) ).
% fun_upd_same
thf(fact_2731_fun__upd__other,axiom,
! [B: $tType,A: $tType,Z2: A,X: A,F: A > B,Y: B] :
( ( Z2 != X )
=> ( ( fun_upd @ A @ B @ F @ X @ Y @ Z2 )
= ( F @ Z2 ) ) ) ).
% fun_upd_other
thf(fact_2732_fun__upd__twist,axiom,
! [A: $tType,B: $tType,A3: A,C2: A,M2: A > B,B2: B,D3: B] :
( ( A3 != C2 )
=> ( ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ M2 @ A3 @ B2 ) @ C2 @ D3 )
= ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ M2 @ C2 @ D3 ) @ A3 @ B2 ) ) ) ).
% fun_upd_twist
thf(fact_2733_fun__upd__idem__iff,axiom,
! [A: $tType,B: $tType,F: A > B,X: A,Y: B] :
( ( ( fun_upd @ A @ B @ F @ X @ Y )
= F )
= ( ( F @ X )
= Y ) ) ).
% fun_upd_idem_iff
thf(fact_2734_map__upd__Some__unfold,axiom,
! [B: $tType,A: $tType,M2: B > ( option @ A ),A3: B,B2: A,X: B,Y: A] :
( ( ( fun_upd @ B @ ( option @ A ) @ M2 @ A3 @ ( some @ A @ B2 ) @ X )
= ( some @ A @ Y ) )
= ( ( ( X = A3 )
& ( B2 = Y ) )
| ( ( X != A3 )
& ( ( M2 @ X )
= ( some @ A @ Y ) ) ) ) ) ).
% map_upd_Some_unfold
thf(fact_2735_map__upd__triv,axiom,
! [A: $tType,B: $tType,T2: B > ( option @ A ),K: B,X: A] :
( ( ( T2 @ K )
= ( some @ A @ X ) )
=> ( ( fun_upd @ B @ ( option @ A ) @ T2 @ K @ ( some @ A @ X ) )
= T2 ) ) ).
% map_upd_triv
thf(fact_2736_map__upd__eqD1,axiom,
! [A: $tType,B: $tType,M2: A > ( option @ B ),A3: A,X: B,N: A > ( option @ B ),Y: B] :
( ( ( fun_upd @ A @ ( option @ B ) @ M2 @ A3 @ ( some @ B @ X ) )
= ( fun_upd @ A @ ( option @ B ) @ N @ A3 @ ( some @ B @ Y ) ) )
=> ( X = Y ) ) ).
% map_upd_eqD1
thf(fact_2737_FieldI2,axiom,
! [A: $tType,I: A,J: A,R5: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J ) @ R5 )
=> ( member @ A @ J @ ( field2 @ A @ R5 ) ) ) ).
% FieldI2
thf(fact_2738_FieldI1,axiom,
! [A: $tType,I: A,J: A,R5: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J ) @ R5 )
=> ( member @ A @ I @ ( field2 @ A @ R5 ) ) ) ).
% FieldI1
thf(fact_2739_sup__Un__eq2,axiom,
! [B: $tType,A: $tType,R5: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
( ( sup_sup @ ( A > B > $o )
@ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R5 )
@ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ S2 ) )
= ( ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ R5 @ S2 ) ) ) ) ).
% sup_Un_eq2
thf(fact_2740_fun__upd__comp,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > B,G: A > C,X: A,Y: C] :
( ( comp @ C @ B @ A @ F @ ( fun_upd @ A @ C @ G @ X @ Y ) )
= ( fun_upd @ A @ B @ ( comp @ C @ B @ A @ F @ G ) @ X @ ( F @ Y ) ) ) ).
% fun_upd_comp
thf(fact_2741_map__upd__nonempty,axiom,
! [B: $tType,A: $tType,T2: A > ( option @ B ),K: A,X: B] :
( ( fun_upd @ A @ ( option @ B ) @ T2 @ K @ ( some @ B @ X ) )
!= ( ^ [X3: A] : ( none @ B ) ) ) ).
% map_upd_nonempty
thf(fact_2742_one__rat_Orsp,axiom,
ratrel @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) ).
% one_rat.rsp
thf(fact_2743_R__subset__Field,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R5
@ ( product_Sigma @ A @ A @ ( field2 @ A @ R5 )
@ ^ [Uu: A] : ( field2 @ A @ R5 ) ) ) ).
% R_subset_Field
thf(fact_2744_fst__in__Field,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A )] : ( ord_less_eq @ ( set @ A ) @ ( image2 @ ( product_prod @ A @ A ) @ A @ ( product_fst @ A @ A ) @ R5 ) @ ( field2 @ A @ R5 ) ) ).
% fst_in_Field
thf(fact_2745_snd__in__Field,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A )] : ( ord_less_eq @ ( set @ A ) @ ( image2 @ ( product_prod @ A @ A ) @ A @ ( product_snd @ A @ A ) @ R5 ) @ ( field2 @ A @ R5 ) ) ).
% snd_in_Field
thf(fact_2746_zero__rat_Orsp,axiom,
ratrel @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ).
% zero_rat.rsp
thf(fact_2747_Fract_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > ( product_prod @ int @ int ) ) @ ( int > ( product_prod @ int @ int ) )
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ( bNF_rel_fun @ int @ int @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int )
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ratrel )
@ ^ [A7: int,B5: int] :
( if @ ( product_prod @ int @ int )
@ ( B5
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ A7 @ B5 ) )
@ ^ [A7: int,B5: int] :
( if @ ( product_prod @ int @ int )
@ ( B5
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ A7 @ B5 ) ) ) ).
% Fract.rsp
thf(fact_2748_sum_Oeq__fold,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ( ( groups7311177749621191930dd_sum @ B @ A )
= ( ^ [G2: B > A] : ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( plus_plus @ A ) @ G2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% sum.eq_fold
thf(fact_2749_prod_Oeq__fold,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ( ( groups7121269368397514597t_prod @ B @ A )
= ( ^ [G2: B > A] : ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( times_times @ A ) @ G2 ) @ ( one_one @ A ) ) ) ) ) ).
% prod.eq_fold
thf(fact_2750_finite__range__updI,axiom,
! [A: $tType,B: $tType,F: B > ( option @ A ),A3: B,B2: A] :
( ( finite_finite2 @ ( option @ A ) @ ( image2 @ B @ ( option @ A ) @ F @ ( top_top @ ( set @ B ) ) ) )
=> ( finite_finite2 @ ( option @ A ) @ ( image2 @ B @ ( option @ A ) @ ( fun_upd @ B @ ( option @ A ) @ F @ A3 @ ( some @ A @ B2 ) ) @ ( top_top @ ( set @ B ) ) ) ) ) ).
% finite_range_updI
thf(fact_2751_fun__upd__image,axiom,
! [A: $tType,B: $tType,X: B,A4: set @ B,F: B > A,Y: A] :
( ( ( member @ B @ X @ A4 )
=> ( ( image2 @ B @ A @ ( fun_upd @ B @ A @ F @ X @ Y ) @ A4 )
= ( insert2 @ A @ Y @ ( image2 @ B @ A @ F @ ( minus_minus @ ( set @ B ) @ A4 @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member @ B @ X @ A4 )
=> ( ( image2 @ B @ A @ ( fun_upd @ B @ A @ F @ X @ Y ) @ A4 )
= ( image2 @ B @ A @ F @ A4 ) ) ) ) ).
% fun_upd_image
thf(fact_2752_times__rat_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ) @ ( ( 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 )
@ ^ [X3: product_prod @ int @ int,Y3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X3 ) @ ( product_fst @ int @ int @ Y3 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X3 ) @ ( product_snd @ int @ int @ Y3 ) ) )
@ ^ [X3: product_prod @ int @ int,Y3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X3 ) @ ( product_fst @ int @ int @ Y3 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X3 ) @ ( product_snd @ int @ int @ Y3 ) ) ) ) ).
% times_rat.rsp
thf(fact_2753_uminus__rat_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ratrel @ ratrel
@ ^ [X3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( product_fst @ int @ int @ X3 ) ) @ ( product_snd @ int @ int @ X3 ) )
@ ^ [X3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( product_fst @ int @ int @ X3 ) ) @ ( product_snd @ int @ int @ X3 ) ) ) ).
% uminus_rat.rsp
thf(fact_2754_product__fold,axiom,
! [B: $tType,A: $tType,A4: set @ A,B6: set @ B] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ B @ B6 )
=> ( ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : B6 )
= ( finite_fold @ A @ ( set @ ( product_prod @ A @ B ) )
@ ^ [X3: A,Z5: set @ ( product_prod @ A @ B )] :
( finite_fold @ B @ ( set @ ( product_prod @ A @ B ) )
@ ^ [Y3: B] : ( insert2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) )
@ Z5
@ B6 )
@ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) )
@ A4 ) ) ) ) ).
% product_fold
thf(fact_2755_Id__on__fold,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( id_on @ A @ A4 )
= ( finite_fold @ A @ ( set @ ( product_prod @ A @ A ) )
@ ^ [X3: A] : ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) )
@ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) )
@ A4 ) ) ) ).
% Id_on_fold
thf(fact_2756_plus__rat_Oabs__eq,axiom,
! [Xa: product_prod @ int @ int,X: product_prod @ int @ int] :
( ( ratrel @ Xa @ Xa )
=> ( ( ratrel @ X @ X )
=> ( ( plus_plus @ rat @ ( abs_Rat @ Xa ) @ ( abs_Rat @ X ) )
= ( abs_Rat @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ Xa ) @ ( product_snd @ int @ int @ X ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ X ) @ ( product_snd @ int @ int @ Xa ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ Xa ) @ ( product_snd @ int @ int @ X ) ) ) ) ) ) ) ).
% plus_rat.abs_eq
thf(fact_2757_inverse__rat_Oabs__eq,axiom,
! [X: product_prod @ int @ int] :
( ( ratrel @ X @ X )
=> ( ( inverse_inverse @ rat @ ( abs_Rat @ X ) )
= ( abs_Rat
@ ( if @ ( product_prod @ int @ int )
@ ( ( product_fst @ int @ int @ X )
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ X ) @ ( product_fst @ int @ int @ X ) ) ) ) ) ) ).
% inverse_rat.abs_eq
thf(fact_2758_times__rat_Oabs__eq,axiom,
! [Xa: product_prod @ int @ int,X: product_prod @ int @ int] :
( ( ratrel @ Xa @ Xa )
=> ( ( ratrel @ X @ X )
=> ( ( times_times @ rat @ ( abs_Rat @ Xa ) @ ( abs_Rat @ X ) )
= ( abs_Rat @ ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ Xa ) @ ( product_fst @ int @ int @ X ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ Xa ) @ ( product_snd @ int @ int @ X ) ) ) ) ) ) ) ).
% times_rat.abs_eq
thf(fact_2759_uminus__rat_Oabs__eq,axiom,
! [X: product_prod @ int @ int] :
( ( ratrel @ X @ X )
=> ( ( uminus_uminus @ rat @ ( abs_Rat @ X ) )
= ( abs_Rat @ ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( product_fst @ int @ int @ X ) ) @ ( product_snd @ int @ int @ X ) ) ) ) ) ).
% uminus_rat.abs_eq
thf(fact_2760_graph__map__upd,axiom,
! [A: $tType,B: $tType,M2: A > ( option @ B ),K: A,V: B] :
( ( graph @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ K @ ( some @ B @ V ) ) )
= ( insert2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V ) @ ( graph @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ K @ ( none @ B ) ) ) ) ) ).
% graph_map_upd
thf(fact_2761_insert__relcomp__union__fold,axiom,
! [C: $tType,B: $tType,A: $tType,S2: set @ ( product_prod @ A @ B ),X: product_prod @ C @ A,X10: set @ ( product_prod @ C @ B )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ S2 )
=> ( ( sup_sup @ ( set @ ( product_prod @ C @ B ) ) @ ( relcomp @ C @ A @ B @ ( insert2 @ ( product_prod @ C @ A ) @ X @ ( bot_bot @ ( set @ ( product_prod @ C @ A ) ) ) ) @ S2 ) @ X10 )
= ( finite_fold @ ( product_prod @ A @ B ) @ ( set @ ( product_prod @ C @ B ) )
@ ( product_case_prod @ A @ B @ ( ( set @ ( product_prod @ C @ B ) ) > ( set @ ( product_prod @ C @ B ) ) )
@ ^ [W2: A,Z5: B,A10: set @ ( product_prod @ C @ B )] :
( if @ ( set @ ( product_prod @ C @ B ) )
@ ( ( product_snd @ C @ A @ X )
= W2 )
@ ( insert2 @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ ( product_fst @ C @ A @ X ) @ Z5 ) @ A10 )
@ A10 ) )
@ X10
@ S2 ) ) ) ).
% insert_relcomp_union_fold
thf(fact_2762_graph__empty,axiom,
! [B: $tType,A: $tType] :
( ( graph @ A @ B
@ ^ [X3: A] : ( none @ B ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% graph_empty
thf(fact_2763_relcomp_Ocases,axiom,
! [A: $tType,C: $tType,B: $tType,A1: A,A22: C,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ B @ C )] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A1 @ A22 ) @ ( relcomp @ A @ B @ C @ R @ S ) )
=> ~ ! [B3: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A1 @ B3 ) @ R )
=> ~ ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B3 @ A22 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_2764_relcomp_Osimps,axiom,
! [B: $tType,C: $tType,A: $tType,A1: A,A22: C,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ B @ C )] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A1 @ A22 ) @ ( relcomp @ A @ B @ C @ R @ S ) )
= ( ? [A7: A,B5: B,C3: C] :
( ( A1 = A7 )
& ( A22 = C3 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A7 @ B5 ) @ R )
& ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B5 @ C3 ) @ S ) ) ) ) ).
% relcomp.simps
thf(fact_2765_relcomp_OrelcompI,axiom,
! [A: $tType,C: $tType,B: $tType,A3: A,B2: B,R: set @ ( product_prod @ A @ B ),C2: C,S: set @ ( product_prod @ B @ C )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R )
=> ( ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B2 @ C2 ) @ S )
=> ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A3 @ C2 ) @ ( relcomp @ A @ B @ C @ R @ S ) ) ) ) ).
% relcomp.relcompI
thf(fact_2766_relcompE,axiom,
! [A: $tType,B: $tType,C: $tType,Xz: product_prod @ A @ B,R: set @ ( product_prod @ A @ C ),S: set @ ( product_prod @ C @ B )] :
( ( member @ ( product_prod @ A @ B ) @ Xz @ ( relcomp @ A @ C @ B @ R @ S ) )
=> ~ ! [X2: A,Y4: C,Z3: B] :
( ( Xz
= ( product_Pair @ A @ B @ X2 @ Z3 ) )
=> ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X2 @ Y4 ) @ R )
=> ~ ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ Y4 @ Z3 ) @ S ) ) ) ) ).
% relcompE
thf(fact_2767_relcompEpair,axiom,
! [A: $tType,B: $tType,C: $tType,A3: A,C2: B,R: set @ ( product_prod @ A @ C ),S: set @ ( product_prod @ C @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ C2 ) @ ( relcomp @ A @ C @ B @ R @ S ) )
=> ~ ! [B3: C] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A3 @ B3 ) @ R )
=> ~ ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ B3 @ C2 ) @ S ) ) ) ).
% relcompEpair
thf(fact_2768_in__graphI,axiom,
! [A: $tType,B: $tType,M2: B > ( option @ A ),K: B,V: A] :
( ( ( M2 @ K )
= ( some @ A @ V ) )
=> ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ K @ V ) @ ( graph @ B @ A @ M2 ) ) ) ).
% in_graphI
thf(fact_2769_in__graphD,axiom,
! [A: $tType,B: $tType,K: A,V: B,M2: A > ( option @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V ) @ ( graph @ A @ B @ M2 ) )
=> ( ( M2 @ K )
= ( some @ B @ V ) ) ) ).
% in_graphD
thf(fact_2770_max__ext__compat,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R5 @ S2 ) @ R5 )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( relcomp @ ( set @ A ) @ ( set @ A ) @ ( set @ A ) @ ( max_ext @ A @ R5 ) @ ( sup_sup @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( max_ext @ A @ S2 ) @ ( insert2 @ ( 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 @ R5 ) ) ) ).
% max_ext_compat
thf(fact_2771_min__ext__compat,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R5 @ S2 ) @ R5 )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( relcomp @ ( set @ A ) @ ( set @ A ) @ ( set @ A ) @ ( min_ext @ A @ R5 ) @ ( sup_sup @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( min_ext @ A @ S2 ) @ ( insert2 @ ( 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 @ R5 ) ) ) ).
% min_ext_compat
thf(fact_2772_relcomp__fold,axiom,
! [C: $tType,B: $tType,A: $tType,R5: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ B @ C )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ R5 )
=> ( ( finite_finite2 @ ( product_prod @ B @ C ) @ S2 )
=> ( ( relcomp @ A @ B @ C @ R5 @ S2 )
= ( finite_fold @ ( product_prod @ A @ B ) @ ( set @ ( product_prod @ A @ C ) )
@ ( product_case_prod @ A @ B @ ( ( set @ ( product_prod @ A @ C ) ) > ( set @ ( product_prod @ A @ C ) ) )
@ ^ [X3: A,Y3: B,A9: set @ ( product_prod @ A @ C )] :
( finite_fold @ ( product_prod @ B @ C ) @ ( set @ ( product_prod @ A @ C ) )
@ ( product_case_prod @ B @ C @ ( ( set @ ( product_prod @ A @ C ) ) > ( set @ ( product_prod @ A @ C ) ) )
@ ^ [W2: B,Z5: C,A10: set @ ( product_prod @ A @ C )] : ( if @ ( set @ ( product_prod @ A @ C ) ) @ ( Y3 = W2 ) @ ( insert2 @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X3 @ Z5 ) @ A10 ) @ A10 ) )
@ A9
@ S2 ) )
@ ( bot_bot @ ( set @ ( product_prod @ A @ C ) ) )
@ R5 ) ) ) ) ).
% relcomp_fold
thf(fact_2773_one__rat__def,axiom,
( ( one_one @ rat )
= ( abs_Rat @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) ) ) ).
% one_rat_def
thf(fact_2774_graph__fun__upd__None,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B ),K: A] :
( ( graph @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ K @ ( none @ B ) ) )
= ( collect @ ( product_prod @ A @ B )
@ ^ [E5: product_prod @ A @ B] :
( ( member @ ( product_prod @ A @ B ) @ E5 @ ( graph @ A @ B @ M2 ) )
& ( ( product_fst @ A @ B @ E5 )
!= K ) ) ) ) ).
% graph_fun_upd_None
thf(fact_2775_insert__relcomp__fold,axiom,
! [C: $tType,B: $tType,A: $tType,S2: set @ ( product_prod @ A @ B ),X: product_prod @ C @ A,R5: set @ ( product_prod @ C @ A )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ S2 )
=> ( ( relcomp @ C @ A @ B @ ( insert2 @ ( product_prod @ C @ A ) @ X @ R5 ) @ S2 )
= ( finite_fold @ ( product_prod @ A @ B ) @ ( set @ ( product_prod @ C @ B ) )
@ ( product_case_prod @ A @ B @ ( ( set @ ( product_prod @ C @ B ) ) > ( set @ ( product_prod @ C @ B ) ) )
@ ^ [W2: A,Z5: B,A10: set @ ( product_prod @ C @ B )] :
( if @ ( set @ ( product_prod @ C @ B ) )
@ ( ( product_snd @ C @ A @ X )
= W2 )
@ ( insert2 @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ ( product_fst @ C @ A @ X ) @ Z5 ) @ A10 )
@ A10 ) )
@ ( relcomp @ C @ A @ B @ R5 @ S2 )
@ S2 ) ) ) ).
% insert_relcomp_fold
thf(fact_2776_zero__rat__def,axiom,
( ( zero_zero @ rat )
= ( abs_Rat @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ) ).
% zero_rat_def
thf(fact_2777_plus__rat__def,axiom,
( ( plus_plus @ rat )
= ( map_fun @ rat @ ( product_prod @ int @ int ) @ ( ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ) @ ( rat > rat ) @ rep_Rat @ ( map_fun @ rat @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ rat @ rep_Rat @ abs_Rat )
@ ^ [X3: product_prod @ int @ int,Y3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X3 ) @ ( product_snd @ int @ int @ Y3 ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ Y3 ) @ ( product_snd @ int @ int @ X3 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X3 ) @ ( product_snd @ int @ int @ Y3 ) ) ) ) ) ).
% plus_rat_def
thf(fact_2778_inverse__rat__def,axiom,
( ( inverse_inverse @ rat )
= ( map_fun @ rat @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ rat @ rep_Rat @ abs_Rat
@ ^ [X3: product_prod @ int @ int] :
( if @ ( product_prod @ int @ int )
@ ( ( product_fst @ int @ int @ X3 )
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ X3 ) @ ( product_fst @ int @ int @ X3 ) ) ) ) ) ).
% inverse_rat_def
thf(fact_2779_times__rat__def,axiom,
( ( times_times @ rat )
= ( map_fun @ rat @ ( product_prod @ int @ int ) @ ( ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ) @ ( rat > rat ) @ rep_Rat @ ( map_fun @ rat @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ rat @ rep_Rat @ abs_Rat )
@ ^ [X3: product_prod @ int @ int,Y3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X3 ) @ ( product_fst @ int @ int @ Y3 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X3 ) @ ( product_snd @ int @ int @ Y3 ) ) ) ) ) ).
% times_rat_def
thf(fact_2780_uminus__rat__def,axiom,
( ( uminus_uminus @ rat )
= ( map_fun @ rat @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ rat @ rep_Rat @ abs_Rat
@ ^ [X3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( product_fst @ int @ int @ X3 ) ) @ ( product_snd @ int @ int @ X3 ) ) ) ) ).
% uminus_rat_def
thf(fact_2781_comp__fun__commute__relcomp__fold,axiom,
! [A: $tType,B: $tType,C: $tType,S2: set @ ( product_prod @ A @ B )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ S2 )
=> ( finite6289374366891150609ommute @ ( product_prod @ C @ A ) @ ( set @ ( product_prod @ C @ B ) )
@ ( product_case_prod @ C @ A @ ( ( set @ ( product_prod @ C @ B ) ) > ( set @ ( product_prod @ C @ B ) ) )
@ ^ [X3: C,Y3: A,A9: set @ ( product_prod @ C @ B )] :
( finite_fold @ ( product_prod @ A @ B ) @ ( set @ ( product_prod @ C @ B ) )
@ ( product_case_prod @ A @ B @ ( ( set @ ( product_prod @ C @ B ) ) > ( set @ ( product_prod @ C @ B ) ) )
@ ^ [W2: A,Z5: B,A10: set @ ( product_prod @ C @ B )] : ( if @ ( set @ ( product_prod @ C @ B ) ) @ ( Y3 = W2 ) @ ( insert2 @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ X3 @ Z5 ) @ A10 ) @ A10 ) )
@ A9
@ S2 ) ) ) ) ).
% comp_fun_commute_relcomp_fold
thf(fact_2782_restrict__upd__same,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B ),X: A,Y: B] :
( ( restrict_map @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ X @ ( some @ B @ Y ) ) @ ( uminus_uminus @ ( set @ A ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( restrict_map @ A @ B @ M2 @ ( uminus_uminus @ ( set @ A ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% restrict_upd_same
thf(fact_2783_restrict__map__UNIV,axiom,
! [B: $tType,A: $tType,F: A > ( option @ B )] :
( ( restrict_map @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
= F ) ).
% restrict_map_UNIV
thf(fact_2784_restrict__out,axiom,
! [A: $tType,B: $tType,X: A,A4: set @ A,M2: A > ( option @ B )] :
( ~ ( member @ A @ X @ A4 )
=> ( ( restrict_map @ A @ B @ M2 @ A4 @ X )
= ( none @ B ) ) ) ).
% restrict_out
thf(fact_2785_restrict__map__empty,axiom,
! [B: $tType,A: $tType,D4: set @ A] :
( ( restrict_map @ A @ B
@ ^ [X3: A] : ( none @ B )
@ D4 )
= ( ^ [X3: A] : ( none @ B ) ) ) ).
% restrict_map_empty
thf(fact_2786_restrict__map__to__empty,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B )] :
( ( restrict_map @ A @ B @ M2 @ ( bot_bot @ ( set @ A ) ) )
= ( ^ [X3: A] : ( none @ B ) ) ) ).
% restrict_map_to_empty
thf(fact_2787_fun__upd__None__restrict,axiom,
! [B: $tType,A: $tType,X: A,D4: set @ A,M2: A > ( option @ B )] :
( ( ( member @ A @ X @ D4 )
=> ( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M2 @ D4 ) @ X @ ( none @ B ) )
= ( restrict_map @ A @ B @ M2 @ ( minus_minus @ ( set @ A ) @ D4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) )
& ( ~ ( member @ A @ X @ D4 )
=> ( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M2 @ D4 ) @ X @ ( none @ B ) )
= ( restrict_map @ A @ B @ M2 @ D4 ) ) ) ) ).
% fun_upd_None_restrict
thf(fact_2788_le__map__restrict,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [M2: A > ( option @ B ),X10: set @ A] : ( ord_less_eq @ ( A > ( option @ B ) ) @ ( restrict_map @ A @ B @ M2 @ X10 ) @ M2 ) ) ).
% le_map_restrict
thf(fact_2789_restrict__map__subset__eq,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B ),R5: set @ A,M7: A > ( option @ B ),R7: set @ A] :
( ( ( restrict_map @ A @ B @ M2 @ R5 )
= M7 )
=> ( ( ord_less_eq @ ( set @ A ) @ R7 @ R5 )
=> ( ( restrict_map @ A @ B @ M2 @ R7 )
= ( restrict_map @ A @ B @ M7 @ R7 ) ) ) ) ).
% restrict_map_subset_eq
thf(fact_2790_restrict__map__eq_I2_J,axiom,
! [A: $tType,B: $tType,M2: B > ( option @ A ),A4: set @ B,K: B,V: A] :
( ( ( restrict_map @ B @ A @ M2 @ A4 @ K )
= ( some @ A @ V ) )
= ( ( ( M2 @ K )
= ( some @ A @ V ) )
& ( member @ B @ K @ A4 ) ) ) ).
% restrict_map_eq(2)
thf(fact_2791_restrict__map__def,axiom,
! [B: $tType,A: $tType] :
( ( restrict_map @ A @ B )
= ( ^ [M: A > ( option @ B ),A9: set @ A,X3: A] : ( if @ ( option @ B ) @ ( member @ A @ X3 @ A9 ) @ ( M @ X3 ) @ ( none @ B ) ) ) ) ).
% restrict_map_def
thf(fact_2792_comp__fun__commute_Ointro,axiom,
! [B: $tType,A: $tType,F: A > B > B] :
( ! [Y4: A,X2: A] :
( ( comp @ B @ B @ B @ ( F @ Y4 ) @ ( F @ X2 ) )
= ( comp @ B @ B @ B @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( finite6289374366891150609ommute @ A @ B @ F ) ) ).
% comp_fun_commute.intro
thf(fact_2793_comp__fun__commute_Ocomp__fun__commute,axiom,
! [B: $tType,A: $tType,F: A > B > B,Y: A,X: A] :
( ( finite6289374366891150609ommute @ A @ B @ F )
=> ( ( comp @ B @ B @ B @ ( F @ Y ) @ ( F @ X ) )
= ( comp @ B @ B @ B @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% comp_fun_commute.comp_fun_commute
thf(fact_2794_comp__fun__commute__def,axiom,
! [B: $tType,A: $tType] :
( ( finite6289374366891150609ommute @ A @ B )
= ( ^ [F2: A > B > B] :
! [Y3: A,X3: A] :
( ( comp @ B @ B @ B @ ( F2 @ Y3 ) @ ( F2 @ X3 ) )
= ( comp @ B @ B @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) ) ) ) ).
% comp_fun_commute_def
thf(fact_2795_graph__restrictD_I1_J,axiom,
! [B: $tType,A: $tType,K: A,V: B,M2: A > ( option @ B ),A4: set @ A] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V ) @ ( graph @ A @ B @ ( restrict_map @ A @ B @ M2 @ A4 ) ) )
=> ( member @ A @ K @ A4 ) ) ).
% graph_restrictD(1)
thf(fact_2796_map__restrict__insert__none__simp,axiom,
! [A: $tType,B: $tType,M2: B > ( option @ A ),X: B,S: set @ B] :
( ( ( M2 @ X )
= ( none @ A ) )
=> ( ( restrict_map @ B @ A @ M2 @ ( uminus_uminus @ ( set @ B ) @ ( insert2 @ B @ X @ S ) ) )
= ( restrict_map @ B @ A @ M2 @ ( uminus_uminus @ ( set @ B ) @ S ) ) ) ) ).
% map_restrict_insert_none_simp
thf(fact_2797_restrict__map__upd,axiom,
! [B: $tType,A: $tType,F: A > ( option @ B ),S2: set @ A,K: A,V: B] :
( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ F @ S2 ) @ K @ ( some @ B @ V ) )
= ( restrict_map @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F @ K @ ( some @ B @ V ) ) @ ( insert2 @ A @ K @ S2 ) ) ) ).
% restrict_map_upd
thf(fact_2798_graph__restrictD_I2_J,axiom,
! [A: $tType,B: $tType,K: A,V: B,M2: A > ( option @ B ),A4: set @ A] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V ) @ ( graph @ A @ B @ ( restrict_map @ A @ B @ M2 @ A4 ) ) )
=> ( ( M2 @ K )
= ( some @ B @ V ) ) ) ).
% graph_restrictD(2)
thf(fact_2799_restrict__complement__singleton__eq,axiom,
! [A: $tType,B: $tType,F: A > ( option @ B ),X: A] :
( ( restrict_map @ A @ B @ F @ ( uminus_uminus @ ( set @ A ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( fun_upd @ A @ ( option @ B ) @ F @ X @ ( none @ B ) ) ) ).
% restrict_complement_singleton_eq
thf(fact_2800_map__upd__eq__restrict,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B ),X: A] :
( ( fun_upd @ A @ ( option @ B ) @ M2 @ X @ ( none @ B ) )
= ( restrict_map @ A @ B @ M2 @ ( uminus_uminus @ ( set @ A ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% map_upd_eq_restrict
thf(fact_2801_comp__fun__commute__product__fold,axiom,
! [A: $tType,B: $tType,B6: set @ A] :
( ( finite_finite2 @ A @ B6 )
=> ( finite6289374366891150609ommute @ B @ ( set @ ( product_prod @ B @ A ) )
@ ^ [X3: B,Z5: set @ ( product_prod @ B @ A )] :
( finite_fold @ A @ ( set @ ( product_prod @ B @ A ) )
@ ^ [Y3: A] : ( insert2 @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 @ Y3 ) )
@ Z5
@ B6 ) ) ) ).
% comp_fun_commute_product_fold
thf(fact_2802_Sup__fin_Oeq__fold_H,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( lattic5882676163264333800up_fin @ A )
= ( ^ [A9: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X3: A,Y3: option @ A] : ( some @ A @ ( case_option @ A @ A @ X3 @ ( sup_sup @ A @ X3 ) @ Y3 ) )
@ ( none @ A )
@ A9 ) ) ) ) ) ).
% Sup_fin.eq_fold'
thf(fact_2803_ran__map__upd,axiom,
! [A: $tType,B: $tType,M2: B > ( option @ A ),A3: B,B2: A] :
( ( ( M2 @ A3 )
= ( none @ A ) )
=> ( ( ran @ B @ A @ ( fun_upd @ B @ ( option @ A ) @ M2 @ A3 @ ( some @ A @ B2 ) ) )
= ( insert2 @ A @ B2 @ ( ran @ B @ A @ M2 ) ) ) ) ).
% ran_map_upd
thf(fact_2804_Gcd__fin_Oeq__fold,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( semiring_gcd_Gcd_fin @ A )
= ( ^ [A9: set @ A] : ( if @ A @ ( finite_finite2 @ A @ A9 ) @ ( finite_fold @ A @ A @ ( gcd_gcd @ A ) @ ( zero_zero @ A ) @ A9 ) @ ( one_one @ A ) ) ) ) ) ).
% Gcd_fin.eq_fold
thf(fact_2805_Max_Oeq__fold_H,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattic643756798349783984er_Max @ A )
= ( ^ [A9: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X3: A,Y3: option @ A] : ( some @ A @ ( case_option @ A @ A @ X3 @ ( ord_max @ A @ X3 ) @ Y3 ) )
@ ( none @ A )
@ A9 ) ) ) ) ) ).
% Max.eq_fold'
thf(fact_2806_Min_Oeq__fold_H,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattic643756798350308766er_Min @ A )
= ( ^ [A9: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X3: A,Y3: option @ A] : ( some @ A @ ( case_option @ A @ A @ X3 @ ( ord_min @ A @ X3 ) @ Y3 ) )
@ ( none @ A )
@ A9 ) ) ) ) ) ).
% Min.eq_fold'
thf(fact_2807_pairself__image__eq,axiom,
! [B: $tType,A: $tType,F: B > A,P: B > B > $o] :
( ( image2 @ ( product_prod @ B @ B ) @ ( product_prod @ A @ A ) @ ( pairself @ B @ A @ F ) @ ( collect @ ( product_prod @ B @ B ) @ ( product_case_prod @ B @ B @ $o @ P ) ) )
= ( collect @ ( product_prod @ A @ A )
@ ^ [Uu: product_prod @ A @ A] :
? [A7: B,B5: B] :
( ( Uu
= ( product_Pair @ A @ A @ ( F @ A7 ) @ ( F @ B5 ) ) )
& ( P @ A7 @ B5 ) ) ) ) ).
% pairself_image_eq
thf(fact_2808_eq__or__mem__image__simp,axiom,
! [B: $tType,A: $tType,F: B > A,A3: B,B6: set @ B] :
( ( collect @ A
@ ^ [Uu: A] :
? [L3: B] :
( ( Uu
= ( F @ L3 ) )
& ( ( L3 = A3 )
| ( member @ B @ L3 @ B6 ) ) ) )
= ( insert2 @ A @ ( F @ A3 )
@ ( collect @ A
@ ^ [Uu: A] :
? [L3: B] :
( ( Uu
= ( F @ L3 ) )
& ( member @ B @ L3 @ B6 ) ) ) ) ) ).
% eq_or_mem_image_simp
thf(fact_2809_Gcd__fin_Oinfinite,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: set @ A] :
( ~ ( finite_finite2 @ A @ A4 )
=> ( ( semiring_gcd_Gcd_fin @ A @ A4 )
= ( one_one @ A ) ) ) ) ).
% Gcd_fin.infinite
thf(fact_2810_is__unit__Gcd__fin__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: set @ A] :
( ( dvd_dvd @ A @ ( semiring_gcd_Gcd_fin @ A @ A4 ) @ ( one_one @ A ) )
= ( ( semiring_gcd_Gcd_fin @ A @ A4 )
= ( one_one @ A ) ) ) ) ).
% is_unit_Gcd_fin_iff
thf(fact_2811_Eps__Opt__eq__None,axiom,
! [A: $tType,P: A > $o] :
( ( ( eps_Opt @ A @ P )
= ( none @ A ) )
= ( ~ ? [X8: A] : ( P @ X8 ) ) ) ).
% Eps_Opt_eq_None
thf(fact_2812_ran__empty,axiom,
! [B: $tType,A: $tType] :
( ( ran @ B @ A
@ ^ [X3: B] : ( none @ A ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% ran_empty
thf(fact_2813_map__update__eta__repair_I2_J,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B ),K: A,V: B] :
( ( ( M2 @ K )
= ( none @ B ) )
=> ( ( ran @ A @ B
@ ^ [X3: A] : ( if @ ( option @ B ) @ ( X3 = K ) @ ( some @ B @ V ) @ ( M2 @ X3 ) ) )
= ( insert2 @ B @ V @ ( ran @ A @ B @ M2 ) ) ) ) ).
% map_update_eta_repair(2)
thf(fact_2814_ran__def,axiom,
! [B: $tType,A: $tType] :
( ( ran @ A @ B )
= ( ^ [M: A > ( option @ B )] :
( collect @ B
@ ^ [B5: B] :
? [A7: A] :
( ( M @ A7 )
= ( some @ B @ B5 ) ) ) ) ) ).
% ran_def
thf(fact_2815_ranI,axiom,
! [A: $tType,B: $tType,M2: B > ( option @ A ),A3: B,B2: A] :
( ( ( M2 @ A3 )
= ( some @ A @ B2 ) )
=> ( member @ A @ B2 @ ( ran @ B @ A @ M2 ) ) ) ).
% ranI
thf(fact_2816_fs__contract,axiom,
! [B: $tType,C: $tType,A: $tType,F: A > B > C,S2: set @ C] :
( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
@ ( collect @ ( product_prod @ A @ B )
@ ^ [Uu: product_prod @ A @ B] :
? [P6: product_prod @ A @ B] :
( ( Uu = P6 )
& ( member @ C @ ( F @ ( product_fst @ A @ B @ P6 ) @ ( product_snd @ A @ B @ P6 ) ) @ S2 ) ) ) )
= ( collect @ A
@ ^ [A7: A] :
? [B5: B] : ( member @ C @ ( F @ A7 @ B5 ) @ S2 ) ) ) ).
% fs_contract
thf(fact_2817_ran__restrictD,axiom,
! [B: $tType,A: $tType,Y: A,M2: B > ( option @ A ),A4: set @ B] :
( ( member @ A @ Y @ ( ran @ B @ A @ ( restrict_map @ B @ A @ M2 @ A4 ) ) )
=> ? [X2: B] :
( ( member @ B @ X2 @ A4 )
& ( ( M2 @ X2 )
= ( some @ A @ Y ) ) ) ) ).
% ran_restrictD
thf(fact_2818_Un__interval,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [B1: A,B22: A,B32: A,F: A > B] :
( ( ord_less_eq @ A @ B1 @ B22 )
=> ( ( ord_less_eq @ A @ B22 @ B32 )
=> ( ( sup_sup @ ( set @ B )
@ ( collect @ B
@ ^ [Uu: B] :
? [I3: A] :
( ( Uu
= ( F @ I3 ) )
& ( ord_less_eq @ A @ B1 @ I3 )
& ( ord_less @ A @ I3 @ B22 ) ) )
@ ( collect @ B
@ ^ [Uu: B] :
? [I3: A] :
( ( Uu
= ( F @ I3 ) )
& ( ord_less_eq @ A @ B22 @ I3 )
& ( ord_less @ A @ I3 @ B32 ) ) ) )
= ( collect @ B
@ ^ [Uu: B] :
? [I3: A] :
( ( Uu
= ( F @ I3 ) )
& ( ord_less_eq @ A @ B1 @ I3 )
& ( ord_less @ A @ I3 @ B32 ) ) ) ) ) ) ) ).
% Un_interval
thf(fact_2819_fold__atLeastAtMost__nat,axiom,
! [A: $tType,F: nat > A > A,A3: nat,B2: nat,Acc2: A] :
( ( finite6289374366891150609ommute @ nat @ A @ F )
=> ( ( set_fo6178422350223883121st_nat @ A @ F @ A3 @ B2 @ Acc2 )
= ( finite_fold @ nat @ A @ F @ Acc2 @ ( set_or1337092689740270186AtMost @ nat @ A3 @ B2 ) ) ) ) ).
% fold_atLeastAtMost_nat
thf(fact_2820_Min_Oinfinite,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A] :
( ~ ( finite_finite2 @ A @ A4 )
=> ( ( lattic643756798350308766er_Min @ A @ A4 )
= ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% Min.infinite
thf(fact_2821_relcomp__unfold,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( relcomp @ A @ C @ B )
= ( ^ [R3: set @ ( product_prod @ A @ C ),S4: set @ ( product_prod @ C @ B )] :
( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X3: A,Z5: B] :
? [Y3: C] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X3 @ Y3 ) @ R3 )
& ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ Y3 @ Z5 ) @ S4 ) ) ) ) ) ) ).
% relcomp_unfold
thf(fact_2822_Max_Oinfinite,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A] :
( ~ ( finite_finite2 @ A @ A4 )
=> ( ( lattic643756798349783984er_Max @ A @ A4 )
= ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% Max.infinite
thf(fact_2823_graph__def,axiom,
! [B: $tType,A: $tType] :
( ( graph @ A @ B )
= ( ^ [M: A > ( option @ B )] :
( collect @ ( product_prod @ A @ B )
@ ^ [Uu: product_prod @ A @ B] :
? [A7: A,B5: B] :
( ( Uu
= ( product_Pair @ A @ B @ A7 @ B5 ) )
& ( ( M @ A7 )
= ( some @ B @ B5 ) ) ) ) ) ) ).
% graph_def
thf(fact_2824_Sup__fin_Oinfinite,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: set @ A] :
( ~ ( finite_finite2 @ A @ A4 )
=> ( ( lattic5882676163264333800up_fin @ A @ A4 )
= ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% Sup_fin.infinite
thf(fact_2825_Min__add__commute,axiom,
! [B: $tType,A: $tType] :
( ( linord4140545234300271783up_add @ A )
=> ! [S2: set @ B,F: B > A,K: A] :
( ( finite_finite2 @ B @ S2 )
=> ( ( S2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798350308766er_Min @ A
@ ( image2 @ B @ A
@ ^ [X3: B] : ( plus_plus @ A @ ( F @ X3 ) @ K )
@ S2 ) )
= ( plus_plus @ A @ ( lattic643756798350308766er_Min @ A @ ( image2 @ B @ A @ F @ S2 ) ) @ K ) ) ) ) ) ).
% Min_add_commute
thf(fact_2826_card__le__Suc__Max,axiom,
! [S2: set @ nat] :
( ( finite_finite2 @ nat @ S2 )
=> ( ord_less_eq @ nat @ ( finite_card @ nat @ S2 ) @ ( suc @ ( lattic643756798349783984er_Max @ nat @ S2 ) ) ) ) ).
% card_le_Suc_Max
thf(fact_2827_Max__add__commute,axiom,
! [B: $tType,A: $tType] :
( ( linord4140545234300271783up_add @ A )
=> ! [S2: set @ B,F: B > A,K: A] :
( ( finite_finite2 @ B @ S2 )
=> ( ( S2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798349783984er_Max @ A
@ ( image2 @ B @ A
@ ^ [X3: B] : ( plus_plus @ A @ ( F @ X3 ) @ K )
@ S2 ) )
= ( plus_plus @ A @ ( lattic643756798349783984er_Max @ A @ ( image2 @ B @ A @ F @ S2 ) ) @ K ) ) ) ) ) ).
% Max_add_commute
thf(fact_2828_set__to__map__ran,axiom,
! [A: $tType,B: $tType,S2: set @ ( product_prod @ B @ A )] : ( ord_less_eq @ ( set @ A ) @ ( ran @ B @ A @ ( set_to_map @ B @ A @ S2 ) ) @ ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ S2 ) ) ).
% set_to_map_ran
thf(fact_2829_map__to__set__upd,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B ),K: A,V: B] :
( ( map_to_set @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ K @ ( some @ B @ V ) ) )
= ( insert2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V )
@ ( minus_minus @ ( set @ ( product_prod @ A @ B ) ) @ ( map_to_set @ A @ B @ M2 )
@ ( collect @ ( product_prod @ A @ B )
@ ^ [Uu: product_prod @ A @ B] :
? [V5: B] :
( Uu
= ( product_Pair @ A @ B @ K @ V5 ) ) ) ) ) ) ).
% map_to_set_upd
thf(fact_2830_brk__rel__def,axiom,
! [B: $tType,A: $tType] :
( ( brk_rel @ A @ B )
= ( ^ [R6: set @ ( product_prod @ A @ B )] :
( sup_sup @ ( set @ ( product_prod @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B ) ) )
@ ( collect @ ( product_prod @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B ) )
@ ^ [Uu: product_prod @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B )] :
? [X3: A,Y3: B] :
( ( Uu
= ( product_Pair @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B ) @ ( product_Pair @ $o @ A @ $false @ X3 ) @ ( product_Pair @ $o @ B @ $false @ Y3 ) ) )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R6 ) ) )
@ ( collect @ ( product_prod @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B ) )
@ ^ [Uu: product_prod @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B )] :
? [X3: A,Y3: B] :
( Uu
= ( product_Pair @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B ) @ ( product_Pair @ $o @ A @ $true @ X3 ) @ ( product_Pair @ $o @ B @ $false @ Y3 ) ) ) ) ) ) ) ).
% brk_rel_def
thf(fact_2831_image2__def,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( bNF_Greatest_image2 @ C @ A @ B )
= ( ^ [A9: set @ C,F2: C > A,G2: C > B] :
( collect @ ( product_prod @ A @ B )
@ ^ [Uu: product_prod @ A @ B] :
? [A7: C] :
( ( Uu
= ( product_Pair @ A @ B @ ( F2 @ A7 ) @ ( G2 @ A7 ) ) )
& ( member @ C @ A7 @ A9 ) ) ) ) ) ).
% image2_def
thf(fact_2832_greaterThan__Suc,axiom,
! [K: nat] :
( ( set_ord_greaterThan @ nat @ ( suc @ K ) )
= ( minus_minus @ ( set @ nat ) @ ( set_ord_greaterThan @ nat @ K ) @ ( insert2 @ nat @ ( suc @ K ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% greaterThan_Suc
thf(fact_2833_refl__on__singleton,axiom,
! [A: $tType,X: A] : ( refl_on @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% refl_on_singleton
thf(fact_2834_time__refines,axiom,
! [A: $tType,C2: heap_Time_Heap @ A,C6: heap_Time_Heap @ A,H3: heap_ext @ product_unit] :
( ( refine_Imp_refines @ A @ C2 @ C6 )
=> ( ~ ( time_fails @ A @ C6 @ H3 )
=> ( ord_less_eq @ nat @ ( time_time @ A @ C2 @ H3 ) @ ( time_time @ A @ C6 @ H3 ) ) ) ) ).
% time_refines
thf(fact_2835_greaterThan__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ( set_ord_greaterThan @ A @ X )
= ( set_ord_greaterThan @ A @ Y ) )
= ( X = Y ) ) ) ).
% greaterThan_eq_iff
thf(fact_2836_greaterThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,K: A] :
( ( member @ A @ I @ ( set_ord_greaterThan @ A @ K ) )
= ( ord_less @ A @ K @ I ) ) ) ).
% greaterThan_iff
thf(fact_2837_Inf__greaterThan,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X: A] :
( ( complete_Inf_Inf @ A @ ( set_ord_greaterThan @ A @ X ) )
= X ) ) ).
% Inf_greaterThan
thf(fact_2838_greaterThan__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_greaterThan @ A @ X ) @ ( set_ord_greaterThan @ A @ Y ) )
= ( ord_less_eq @ A @ Y @ X ) ) ) ).
% greaterThan_subset_iff
thf(fact_2839_Compl__greaterThan,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [K: A] :
( ( uminus_uminus @ ( set @ A ) @ ( set_ord_greaterThan @ A @ K ) )
= ( set_ord_atMost @ A @ K ) ) ) ).
% Compl_greaterThan
thf(fact_2840_Compl__atMost,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [K: A] :
( ( uminus_uminus @ ( set @ A ) @ ( set_ord_atMost @ A @ K ) )
= ( set_ord_greaterThan @ A @ K ) ) ) ).
% Compl_atMost
thf(fact_2841_map__to__set__empty,axiom,
! [B: $tType,A: $tType] :
( ( map_to_set @ A @ B
@ ^ [X3: A] : ( none @ B ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% map_to_set_empty
thf(fact_2842_Sup__greaterThanAtLeast,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A] :
( ( ord_less @ A @ X @ ( top_top @ A ) )
=> ( ( complete_Sup_Sup @ A @ ( set_ord_greaterThan @ A @ X ) )
= ( top_top @ A ) ) ) ) ).
% Sup_greaterThanAtLeast
thf(fact_2843_image__uminus__greaterThan,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A] :
( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_ord_greaterThan @ A @ X ) )
= ( set_ord_lessThan @ A @ ( uminus_uminus @ A @ X ) ) ) ) ).
% image_uminus_greaterThan
thf(fact_2844_image__uminus__lessThan,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A] :
( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_ord_lessThan @ A @ X ) )
= ( set_ord_greaterThan @ A @ ( uminus_uminus @ A @ X ) ) ) ) ).
% image_uminus_lessThan
thf(fact_2845_refl__onD2,axiom,
! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( refl_on @ A @ A4 @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
=> ( member @ A @ Y @ A4 ) ) ) ).
% refl_onD2
thf(fact_2846_refl__onD1,axiom,
! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( refl_on @ A @ A4 @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
=> ( member @ A @ X @ A4 ) ) ) ).
% refl_onD1
thf(fact_2847_refl__onD,axiom,
! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),A3: A] :
( ( refl_on @ A @ A4 @ R )
=> ( ( member @ A @ A3 @ A4 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ R ) ) ) ).
% refl_onD
thf(fact_2848_greaterThan__non__empty,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [X: A] :
( ( set_ord_greaterThan @ A @ X )
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% greaterThan_non_empty
thf(fact_2849_infinite__Ioi,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_top @ A ) )
=> ! [A3: A] :
~ ( finite_finite2 @ A @ ( set_ord_greaterThan @ A @ A3 ) ) ) ).
% infinite_Ioi
thf(fact_2850_greaterThan__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_greaterThan @ A )
= ( ^ [L3: A] : ( collect @ A @ ( ord_less @ A @ L3 ) ) ) ) ) ).
% greaterThan_def
thf(fact_2851_fails__refines,axiom,
! [A: $tType,C2: heap_Time_Heap @ A,C6: heap_Time_Heap @ A,H3: heap_ext @ product_unit] :
( ( refine_Imp_refines @ A @ C2 @ C6 )
=> ( ( time_fails @ A @ C2 @ H3 )
=> ( time_fails @ A @ C6 @ H3 ) ) ) ).
% fails_refines
thf(fact_2852_map__to__set__inverse,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B )] :
( ( set_to_map @ A @ B @ ( map_to_set @ A @ B @ M2 ) )
= M2 ) ).
% map_to_set_inverse
thf(fact_2853_map__to__set__empty__iff_I1_J,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B )] :
( ( ( map_to_set @ A @ B @ M2 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( M2
= ( ^ [X3: A] : ( none @ B ) ) ) ) ).
% map_to_set_empty_iff(1)
thf(fact_2854_map__to__set__empty__iff_I2_J,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B )] :
( ( ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) )
= ( map_to_set @ A @ B @ M2 ) )
= ( M2
= ( ^ [X3: A] : ( none @ B ) ) ) ) ).
% map_to_set_empty_iff(2)
thf(fact_2855_image2__eqI,axiom,
! [A: $tType,C: $tType,B: $tType,B2: A,F: B > A,X: B,C2: C,G: B > C,A4: set @ B] :
( ( B2
= ( F @ X ) )
=> ( ( C2
= ( G @ X ) )
=> ( ( member @ B @ X @ A4 )
=> ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ B2 @ C2 ) @ ( bNF_Greatest_image2 @ B @ A @ C @ A4 @ F @ G ) ) ) ) ) ).
% image2_eqI
thf(fact_2856_refl__on__def,axiom,
! [A: $tType] :
( ( refl_on @ A )
= ( ^ [A9: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A9
@ ^ [Uu: A] : A9 ) )
& ! [X3: A] :
( ( member @ A @ X3 @ A9 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ R3 ) ) ) ) ) ).
% refl_on_def
thf(fact_2857_refl__onI,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A4: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu: A] : A4 ) )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ R ) )
=> ( refl_on @ A @ A4 @ R ) ) ) ).
% refl_onI
thf(fact_2858_map__to__set__def,axiom,
! [B: $tType,A: $tType] :
( ( map_to_set @ A @ B )
= ( ^ [M: A > ( option @ B )] :
( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [K4: A,V2: B] :
( ( M @ K4 )
= ( some @ B @ V2 ) ) ) ) ) ) ).
% map_to_set_def
thf(fact_2859_INT__greaterThan__UNIV,axiom,
( ( complete_Inf_Inf @ ( set @ nat ) @ ( image2 @ nat @ ( set @ nat ) @ ( set_ord_greaterThan @ nat ) @ ( top_top @ ( set @ nat ) ) ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% INT_greaterThan_UNIV
thf(fact_2860_map__to__set__ran,axiom,
! [A: $tType,B: $tType] :
( ( ran @ B @ A )
= ( ^ [M: B > ( option @ A )] : ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( map_to_set @ B @ A @ M ) ) ) ) ).
% map_to_set_ran
thf(fact_2861_greaterThan__0,axiom,
( ( set_ord_greaterThan @ nat @ ( zero_zero @ nat ) )
= ( image2 @ nat @ nat @ suc @ ( top_top @ ( set @ nat ) ) ) ) ).
% greaterThan_0
thf(fact_2862_refines__def,axiom,
! [A: $tType] :
( ( refine_Imp_refines @ A )
= ( ^ [P6: heap_Time_Heap @ A,Q5: heap_Time_Heap @ A] :
! [H: heap_ext @ product_unit] :
( case_option @ $o @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ $true
@ ( product_case_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ $o
@ ^ [R3: A] :
( product_case_prod @ ( heap_ext @ product_unit ) @ nat @ $o
@ ^ [H7: heap_ext @ product_unit,T3: nat] :
( ( heap_Time_execute @ A @ P6 @ H )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H7 @ T3 ) ) ) ) ) )
@ ( heap_Time_execute @ A @ Q5 @ H ) ) ) ) ).
% refines_def
thf(fact_2863_refines__empty,axiom,
! [A: $tType,M2: heap_Time_Heap @ A] :
( refine_Imp_refines @ A @ M2
@ ( heap_Time_Heap2 @ A
@ ^ [X3: heap_ext @ product_unit] : ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ).
% refines_empty
thf(fact_2864_refines__bind,axiom,
! [B: $tType,A: $tType,M2: heap_Time_Heap @ A,M7: heap_Time_Heap @ A,F: A > ( heap_Time_Heap @ B ),F3: A > ( heap_Time_Heap @ B )] :
( ( refine_Imp_refines @ A @ M2 @ M7 )
=> ( ! [X2: A] : ( refine_Imp_refines @ B @ ( F @ X2 ) @ ( F3 @ X2 ) )
=> ( refine_Imp_refines @ B @ ( heap_Time_bind @ A @ B @ M2 @ F ) @ ( heap_Time_bind @ A @ B @ M7 @ F3 ) ) ) ) ).
% refines_bind
thf(fact_2865_refl__on__domain,axiom,
! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( refl_on @ A @ A4 @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
=> ( ( member @ A @ A3 @ A4 )
& ( member @ A @ B2 @ A4 ) ) ) ) ).
% refl_on_domain
thf(fact_2866_refines__option,axiom,
! [B: $tType,A: $tType,A3: option @ A,A6: option @ A,M1: heap_Time_Heap @ B,M12: heap_Time_Heap @ B,M22: A > ( heap_Time_Heap @ B ),M23: A > ( heap_Time_Heap @ B )] :
( ( A3 = A6 )
=> ( ( refine_Imp_refines @ B @ M1 @ M12 )
=> ( ! [X2: A] : ( refine_Imp_refines @ B @ ( M22 @ X2 ) @ ( M23 @ X2 ) )
=> ( refine_Imp_refines @ B @ ( case_option @ ( heap_Time_Heap @ B ) @ A @ M1 @ M22 @ A3 ) @ ( case_option @ ( heap_Time_Heap @ B ) @ A @ M12 @ M23 @ A6 ) ) ) ) ) ).
% refines_option
thf(fact_2867_linear__order__on__singleton,axiom,
! [A: $tType,X: A] : ( order_679001287576687338der_on @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% linear_order_on_singleton
thf(fact_2868_Total__subset__Id,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( total_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( id2 @ A ) )
=> ( ( R
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
| ? [A5: A] :
( R
= ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ A5 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ) ) ) ).
% Total_subset_Id
thf(fact_2869_relation__of__def,axiom,
! [A: $tType] :
( ( order_relation_of @ A )
= ( ^ [P4: A > A > $o,A9: set @ A] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [A7: A,B5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B5 )
@ ( product_Sigma @ A @ A @ A9
@ ^ [Uu: A] : A9 ) )
& ( P4 @ A7 @ B5 ) ) ) ) ) ) ).
% relation_of_def
thf(fact_2870_euclidean__size__times__nonunit,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ~ ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ B2 ) @ ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ) ) ).
% euclidean_size_times_nonunit
thf(fact_2871_surj__swap,axiom,
! [B: $tType,A: $tType] :
( ( image2 @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
= ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% surj_swap
thf(fact_2872_euclidean__size__integer_Orep__eq,axiom,
( ( euclid6346220572633701492n_size @ code_integer )
= ( ^ [X3: code_integer] : ( euclid6346220572633701492n_size @ int @ ( code_int_of_integer @ X3 ) ) ) ) ).
% euclidean_size_integer.rep_eq
thf(fact_2873_swap__swap,axiom,
! [B: $tType,A: $tType,P5: product_prod @ A @ B] :
( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P5 ) )
= P5 ) ).
% swap_swap
thf(fact_2874_pair__in__Id__conv,axiom,
! [A: $tType,A3: A,B2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( id2 @ A ) )
= ( A3 = B2 ) ) ).
% pair_in_Id_conv
thf(fact_2875_IdI,axiom,
! [A: $tType,A3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ ( id2 @ A ) ) ).
% IdI
thf(fact_2876_swap__simp,axiom,
! [A: $tType,B: $tType,X: B,Y: A] :
( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
= ( product_Pair @ A @ B @ Y @ X ) ) ).
% swap_simp
thf(fact_2877_case__swap,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > B > A,P5: product_prod @ C @ B] :
( ( product_case_prod @ B @ C @ A
@ ^ [Y3: B,X3: C] : ( F @ X3 @ Y3 )
@ ( product_swap @ C @ B @ P5 ) )
= ( product_case_prod @ C @ B @ A @ F @ P5 ) ) ).
% case_swap
thf(fact_2878_swap__comp__swap,axiom,
! [B: $tType,A: $tType] :
( ( comp @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ ( product_swap @ A @ B ) )
= ( id @ ( product_prod @ A @ B ) ) ) ).
% swap_comp_swap
thf(fact_2879_bijective__Id,axiom,
! [A: $tType] : ( bijective @ A @ A @ ( id2 @ A ) ) ).
% bijective_Id
thf(fact_2880_euclidean__size__1,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ( ( euclid6346220572633701492n_size @ A @ ( one_one @ A ) )
= ( one_one @ nat ) ) ) ).
% euclidean_size_1
thf(fact_2881_fst__swap,axiom,
! [A: $tType,B: $tType,X: product_prod @ B @ A] :
( ( product_fst @ A @ B @ ( product_swap @ B @ A @ X ) )
= ( product_snd @ B @ A @ X ) ) ).
% fst_swap
thf(fact_2882_snd__swap,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B] :
( ( product_snd @ B @ A @ ( product_swap @ A @ B @ X ) )
= ( product_fst @ A @ B @ X ) ) ).
% snd_swap
thf(fact_2883_pair__in__swap__image,axiom,
! [A: $tType,B: $tType,Y: A,X: B,A4: set @ ( product_prod @ B @ A )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y @ X ) @ ( image2 @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ A4 ) )
= ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y ) @ A4 ) ) ).
% pair_in_swap_image
thf(fact_2884_euclidean__size__integer_Oabs__eq,axiom,
! [X: int] :
( ( euclid6346220572633701492n_size @ code_integer @ ( code_integer_of_int @ X ) )
= ( euclid6346220572633701492n_size @ int @ X ) ) ).
% euclidean_size_integer.abs_eq
thf(fact_2885_BNF__Greatest__Fixpoint_OIdD,axiom,
! [A: $tType,A3: A,B2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( id2 @ A ) )
=> ( A3 = B2 ) ) ).
% BNF_Greatest_Fixpoint.IdD
thf(fact_2886_IdE,axiom,
! [A: $tType,P5: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ P5 @ ( id2 @ A ) )
=> ~ ! [X2: A] :
( P5
!= ( product_Pair @ A @ A @ X2 @ X2 ) ) ) ).
% IdE
thf(fact_2887_euclidean__size__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ( euclid6346220572633701492n_size @ nat )
@ ( euclid6346220572633701492n_size @ nat ) ) ).
% euclidean_size_natural.rsp
thf(fact_2888_Linear__order__in__diff__Id,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R ) )
=> ( ( member @ A @ B2 @ ( field2 @ A @ R ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
= ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( id2 @ A ) ) ) ) ) ) ) ) ).
% Linear_order_in_diff_Id
thf(fact_2889_euclidean__size__unit,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( euclid6346220572633701492n_size @ A @ A3 )
= ( euclid6346220572633701492n_size @ A @ ( one_one @ A ) ) ) ) ) ).
% euclidean_size_unit
thf(fact_2890_euclidean__size__integer__def,axiom,
( ( euclid6346220572633701492n_size @ code_integer )
= ( map_fun @ code_integer @ int @ nat @ nat @ code_int_of_integer @ ( id @ nat ) @ ( euclid6346220572633701492n_size @ int ) ) ) ).
% euclidean_size_integer_def
thf(fact_2891_euclidean__size__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ nat @ nat
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ( euclid6346220572633701492n_size @ int )
@ ( euclid6346220572633701492n_size @ int ) ) ).
% euclidean_size_integer.rsp
thf(fact_2892_Id__def,axiom,
! [A: $tType] :
( ( id2 @ A )
= ( collect @ ( product_prod @ A @ A )
@ ^ [P6: product_prod @ A @ A] :
? [X3: A] :
( P6
= ( product_Pair @ A @ A @ X3 @ X3 ) ) ) ) ).
% Id_def
thf(fact_2893_unit__iff__euclidean__size,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
= ( ( ( euclid6346220572633701492n_size @ A @ A3 )
= ( euclid6346220572633701492n_size @ A @ ( one_one @ A ) ) )
& ( A3
!= ( zero_zero @ A ) ) ) ) ) ).
% unit_iff_euclidean_size
thf(fact_2894_euclidean__size__times__unit,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( euclid6346220572633701492n_size @ A @ B2 ) ) ) ) ).
% euclidean_size_times_unit
thf(fact_2895_product__swap,axiom,
! [B: $tType,A: $tType,A4: set @ B,B6: set @ A] :
( ( image2 @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A )
@ ( product_Sigma @ B @ A @ A4
@ ^ [Uu: B] : B6 ) )
= ( product_Sigma @ A @ B @ B6
@ ^ [Uu: A] : A4 ) ) ).
% product_swap
thf(fact_2896_prod_Oswap__def,axiom,
! [B: $tType,A: $tType] :
( ( product_swap @ A @ B )
= ( ^ [P6: product_prod @ A @ B] : ( product_Pair @ B @ A @ ( product_snd @ A @ B @ P6 ) @ ( product_fst @ A @ B @ P6 ) ) ) ) ).
% prod.swap_def
thf(fact_2897_reflcl__set__eq,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( sup_sup @ ( A > A > $o )
@ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R )
@ ^ [Y5: A,Z4: A] : Y5 = Z4 )
= ( ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( id2 @ A ) ) ) ) ) ).
% reflcl_set_eq
thf(fact_2898_divmod__cases,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B2: A,A3: A] :
( ( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( modulo_modulo @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
=> ( A3
!= ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) ) ) )
=> ( ( ( B2
!= ( zero_zero @ A ) )
=> ! [Q2: A,R2: A] :
( ( ( euclid7384307370059645450egment @ A @ R2 )
= ( euclid7384307370059645450egment @ A @ B2 ) )
=> ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R2 ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
=> ( ( R2
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ A3 @ B2 )
= Q2 )
=> ( ( ( modulo_modulo @ A @ A3 @ B2 )
= R2 )
=> ( A3
!= ( plus_plus @ A @ ( times_times @ A @ Q2 @ B2 ) @ R2 ) ) ) ) ) ) ) )
=> ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% divmod_cases
thf(fact_2899_mod__eqI,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B2: A,R: A,Q3: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( euclid7384307370059645450egment @ A @ R )
= ( euclid7384307370059645450egment @ A @ B2 ) )
=> ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
=> ( ( ( plus_plus @ A @ ( times_times @ A @ Q3 @ B2 ) @ R )
= A3 )
=> ( ( modulo_modulo @ A @ A3 @ B2 )
= R ) ) ) ) ) ) ).
% mod_eqI
thf(fact_2900_div__bounded,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B2: A,R: A,Q3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( euclid7384307370059645450egment @ A @ R )
= ( euclid7384307370059645450egment @ A @ B2 ) )
=> ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ Q3 @ B2 ) @ R ) @ B2 )
= Q3 ) ) ) ) ) ).
% div_bounded
thf(fact_2901_div__eqI,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B2: A,R: A,Q3: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( euclid7384307370059645450egment @ A @ R )
= ( euclid7384307370059645450egment @ A @ B2 ) )
=> ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
=> ( ( ( plus_plus @ A @ ( times_times @ A @ Q3 @ B2 ) @ R )
= A3 )
=> ( ( divide_divide @ A @ A3 @ B2 )
= Q3 ) ) ) ) ) ) ).
% div_eqI
thf(fact_2902_division__segment__integer_Orep__eq,axiom,
! [X: code_integer] :
( ( code_int_of_integer @ ( euclid7384307370059645450egment @ code_integer @ X ) )
= ( euclid7384307370059645450egment @ int @ ( code_int_of_integer @ X ) ) ) ).
% division_segment_integer.rep_eq
thf(fact_2903_division__segment__1,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ( ( euclid7384307370059645450egment @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% division_segment_1
thf(fact_2904_division__segment__numeral,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [K: num] :
( ( euclid7384307370059645450egment @ A @ ( numeral_numeral @ A @ K ) )
= ( one_one @ A ) ) ) ).
% division_segment_numeral
thf(fact_2905_division__segment__of__nat,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N: nat] :
( ( euclid7384307370059645450egment @ A @ ( semiring_1_of_nat @ A @ N ) )
= ( one_one @ A ) ) ) ).
% division_segment_of_nat
thf(fact_2906_division__segment__integer_Oabs__eq,axiom,
! [X: int] :
( ( euclid7384307370059645450egment @ code_integer @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( euclid7384307370059645450egment @ int @ X ) ) ) ).
% division_segment_integer.abs_eq
thf(fact_2907_division__segment__nat__def,axiom,
( ( euclid7384307370059645450egment @ nat )
= ( ^ [N5: nat] : ( one_one @ nat ) ) ) ).
% division_segment_nat_def
thf(fact_2908_division__segment__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ( euclid7384307370059645450egment @ int )
@ ( euclid7384307370059645450egment @ int ) ) ).
% division_segment_integer.rsp
thf(fact_2909_division__segment__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ( euclid7384307370059645450egment @ nat )
@ ( euclid7384307370059645450egment @ nat ) ) ).
% division_segment_natural.rsp
thf(fact_2910_division__segment__integer__def,axiom,
( ( euclid7384307370059645450egment @ code_integer )
= ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int @ ( euclid7384307370059645450egment @ int ) ) ) ).
% division_segment_integer_def
thf(fact_2911_is__unit__division__segment,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A3: A] : ( dvd_dvd @ A @ ( euclid7384307370059645450egment @ A @ A3 ) @ ( one_one @ A ) ) ) ).
% is_unit_division_segment
thf(fact_2912_Linear__order__wf__diff__Id,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( wf @ A @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( id2 @ A ) ) )
= ( ! [A9: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A9 @ ( field2 @ A @ R ) )
=> ( ( A9
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ A9 )
& ! [Y3: A] :
( ( member @ A @ Y3 @ A9 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R ) ) ) ) ) ) ) ) ).
% Linear_order_wf_diff_Id
thf(fact_2913_bsqr__def,axiom,
! [A: $tType] :
( ( bNF_Wellorder_bsqr @ A )
= ( ^ [R3: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) )
@ ( product_case_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) @ $o
@ ( product_case_prod @ A @ A @ ( ( product_prod @ A @ A ) > $o )
@ ^ [A12: A,A23: A] :
( product_case_prod @ A @ A @ $o
@ ^ [B12: A,B23: A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ A12 @ ( insert2 @ A @ A23 @ ( insert2 @ A @ B12 @ ( insert2 @ A @ B23 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) @ ( field2 @ A @ R3 ) )
& ( ( ( A12 = B12 )
& ( A23 = B23 ) )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( bNF_We1388413361240627857o_max2 @ A @ R3 @ A12 @ A23 ) @ ( bNF_We1388413361240627857o_max2 @ A @ R3 @ B12 @ B23 ) ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( id2 @ A ) ) )
| ( ( ( bNF_We1388413361240627857o_max2 @ A @ R3 @ A12 @ A23 )
= ( bNF_We1388413361240627857o_max2 @ A @ R3 @ B12 @ B23 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A12 @ B12 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( id2 @ A ) ) ) )
| ( ( ( bNF_We1388413361240627857o_max2 @ A @ R3 @ A12 @ A23 )
= ( bNF_We1388413361240627857o_max2 @ A @ R3 @ B12 @ B23 ) )
& ( A12 = B12 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A23 @ B23 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( id2 @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% bsqr_def
thf(fact_2914_wo__rel_Ocases__Total3,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B2: A,Phi: A > A > $o] :
( ( bNF_Wellorder_wo_rel @ A @ R )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ A3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( field2 @ A @ R ) )
=> ( ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( id2 @ A ) ) )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( id2 @ A ) ) ) )
=> ( Phi @ A3 @ B2 ) )
=> ( ( ( A3 = B2 )
=> ( Phi @ A3 @ B2 ) )
=> ( Phi @ A3 @ B2 ) ) ) ) ) ).
% wo_rel.cases_Total3
thf(fact_2915_ivl__disj__un__two_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M2: A,U: A] :
( ( ord_less @ A @ L @ M2 )
=> ( ( ord_less_eq @ A @ M2 @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M2 ) @ ( set_or1337092689740270186AtMost @ A @ M2 @ U ) )
= ( set_or3652927894154168847AtMost @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(5)
thf(fact_2916_ivl__disj__un__singleton_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) @ ( insert2 @ A @ U @ ( bot_bot @ ( set @ A ) ) ) )
= ( set_or3652927894154168847AtMost @ A @ L @ U ) ) ) ) ).
% ivl_disj_un_singleton(4)
thf(fact_2917_finite__greaterThanAtMost,axiom,
! [L: nat,U: nat] : ( finite_finite2 @ nat @ ( set_or3652927894154168847AtMost @ nat @ L @ U ) ) ).
% finite_greaterThanAtMost
thf(fact_2918_finite__greaterThanAtMost__int,axiom,
! [L: int,U: int] : ( finite_finite2 @ int @ ( set_or3652927894154168847AtMost @ int @ L @ U ) ) ).
% finite_greaterThanAtMost_int
thf(fact_2919_finite__greaterThanAtMost__integer,axiom,
! [L: code_integer,U: code_integer] : ( finite_finite2 @ code_integer @ ( set_or3652927894154168847AtMost @ code_integer @ L @ U ) ) ).
% finite_greaterThanAtMost_integer
thf(fact_2920_card__greaterThanAtMost,axiom,
! [L: nat,U: nat] :
( ( finite_card @ nat @ ( set_or3652927894154168847AtMost @ nat @ L @ U ) )
= ( minus_minus @ nat @ U @ L ) ) ).
% card_greaterThanAtMost
thf(fact_2921_greaterThanAtMost__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,L: A,U: A] :
( ( member @ A @ I @ ( set_or3652927894154168847AtMost @ A @ L @ U ) )
= ( ( ord_less @ A @ L @ I )
& ( ord_less_eq @ A @ I @ U ) ) ) ) ).
% greaterThanAtMost_iff
thf(fact_2922_greaterThanAtMost__empty,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,K: A] :
( ( ord_less_eq @ A @ L @ K )
=> ( ( set_or3652927894154168847AtMost @ A @ K @ L )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% greaterThanAtMost_empty
thf(fact_2923_greaterThanAtMost__empty__iff2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [K: A,L: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or3652927894154168847AtMost @ A @ K @ L ) )
= ( ~ ( ord_less @ A @ K @ L ) ) ) ) ).
% greaterThanAtMost_empty_iff2
thf(fact_2924_greaterThanAtMost__empty__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [K: A,L: A] :
( ( ( set_or3652927894154168847AtMost @ A @ K @ L )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( ord_less @ A @ K @ L ) ) ) ) ).
% greaterThanAtMost_empty_iff
thf(fact_2925_infinite__Ioc__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A] :
( ( ~ ( finite_finite2 @ A @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ).
% infinite_Ioc_iff
thf(fact_2926_image__add__greaterThanAtMost,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ C2 ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) )
= ( set_or3652927894154168847AtMost @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% image_add_greaterThanAtMost
thf(fact_2927_Sup__greaterThanAtMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( complete_Sup_Sup @ A @ ( set_or3652927894154168847AtMost @ A @ X @ Y ) )
= Y ) ) ) ).
% Sup_greaterThanAtMost
thf(fact_2928_Inf__greaterThanAtMost,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( complete_Inf_Inf @ A @ ( set_or3652927894154168847AtMost @ A @ X @ Y ) )
= X ) ) ) ).
% Inf_greaterThanAtMost
thf(fact_2929_brk__rel__wf,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ R5 )
=> ( wf @ ( product_prod @ $o @ A ) @ ( brk_rel @ A @ A @ R5 ) ) ) ).
% brk_rel_wf
thf(fact_2930_card__greaterThanAtMost__int,axiom,
! [L: int,U: int] :
( ( finite_card @ int @ ( set_or3652927894154168847AtMost @ int @ L @ U ) )
= ( nat2 @ ( minus_minus @ int @ U @ L ) ) ) ).
% card_greaterThanAtMost_int
thf(fact_2931_image__minus__const__greaterThanAtMost,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( image2 @ A @ A @ ( minus_minus @ A @ C2 ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) )
= ( set_or7035219750837199246ssThan @ A @ ( minus_minus @ A @ C2 @ B2 ) @ ( minus_minus @ A @ C2 @ A3 ) ) ) ) ).
% image_minus_const_greaterThanAtMost
thf(fact_2932_image__diff__atLeastLessThan,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( image2 @ A @ A @ ( minus_minus @ A @ C2 ) @ ( set_or7035219750837199246ssThan @ A @ A3 @ B2 ) )
= ( set_or3652927894154168847AtMost @ A @ ( minus_minus @ A @ C2 @ B2 ) @ ( minus_minus @ A @ C2 @ A3 ) ) ) ) ).
% image_diff_atLeastLessThan
thf(fact_2933_image__uminus__greaterThanAtMost,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A,Y: A] :
( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_or3652927894154168847AtMost @ A @ X @ Y ) )
= ( set_or7035219750837199246ssThan @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% image_uminus_greaterThanAtMost
thf(fact_2934_image__uminus__atLeastLessThan,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A,Y: A] :
( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_or7035219750837199246ssThan @ A @ X @ Y ) )
= ( set_or3652927894154168847AtMost @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% image_uminus_atLeastLessThan
thf(fact_2935_reduction__pairI,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ R5 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R5 @ S2 ) @ R5 )
=> ( fun_reduction_pair @ A @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R5 @ S2 ) ) ) ) ).
% reduction_pairI
thf(fact_2936_wo__rel_Omax2__equals1,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R ) )
=> ( ( member @ A @ B2 @ ( field2 @ A @ R ) )
=> ( ( ( bNF_We1388413361240627857o_max2 @ A @ R @ A3 @ B2 )
= A3 )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ R ) ) ) ) ) ).
% wo_rel.max2_equals1
thf(fact_2937_wo__rel_Omax2__equals2,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R ) )
=> ( ( member @ A @ B2 @ ( field2 @ A @ R ) )
=> ( ( ( bNF_We1388413361240627857o_max2 @ A @ R @ A3 @ B2 )
= B2 )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R ) ) ) ) ) ).
% wo_rel.max2_equals2
thf(fact_2938_wo__rel_Omax2__greater,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R ) )
=> ( ( member @ A @ B2 @ ( field2 @ A @ R ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ ( bNF_We1388413361240627857o_max2 @ A @ R @ A3 @ B2 ) ) @ R )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ ( bNF_We1388413361240627857o_max2 @ A @ R @ A3 @ B2 ) ) @ R ) ) ) ) ) ).
% wo_rel.max2_greater
thf(fact_2939_wf__induct__rule,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),P: A > $o,A3: A] :
( ( wf @ A @ R )
=> ( ! [X2: A] :
( ! [Y6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X2 ) @ R )
=> ( P @ Y6 ) )
=> ( P @ X2 ) )
=> ( P @ A3 ) ) ) ).
% wf_induct_rule
thf(fact_2940_wf__eq__minimal,axiom,
! [A: $tType] :
( ( wf @ A )
= ( ^ [R3: set @ ( product_prod @ A @ A )] :
! [Q6: set @ A] :
( ? [X3: A] : ( member @ A @ X3 @ Q6 )
=> ? [X3: A] :
( ( member @ A @ X3 @ Q6 )
& ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R3 )
=> ~ ( member @ A @ Y3 @ Q6 ) ) ) ) ) ) ).
% wf_eq_minimal
thf(fact_2941_wf__not__refl,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A] :
( ( wf @ A @ R )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ R ) ) ).
% wf_not_refl
thf(fact_2942_wf__not__sym,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,X: A] :
( ( wf @ A @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ X ) @ R )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ A3 ) @ R ) ) ) ).
% wf_not_sym
thf(fact_2943_wf__irrefl,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A] :
( ( wf @ A @ R )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ R ) ) ).
% wf_irrefl
thf(fact_2944_wf__induct,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),P: A > $o,A3: A] :
( ( wf @ A @ R )
=> ( ! [X2: A] :
( ! [Y6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X2 ) @ R )
=> ( P @ Y6 ) )
=> ( P @ X2 ) )
=> ( P @ A3 ) ) ) ).
% wf_induct
thf(fact_2945_wf__asym,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,X: A] :
( ( wf @ A @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ X ) @ R )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ A3 ) @ R ) ) ) ).
% wf_asym
thf(fact_2946_wfUNIVI,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ! [P7: A > $o,X2: A] :
( ! [Xa2: A] :
( ! [Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Xa2 ) @ R )
=> ( P7 @ Y4 ) )
=> ( P7 @ Xa2 ) )
=> ( P7 @ X2 ) )
=> ( wf @ A @ R ) ) ).
% wfUNIVI
thf(fact_2947_wfI__min,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
( ! [X2: A,Q7: set @ A] :
( ( member @ A @ X2 @ Q7 )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ Q7 )
& ! [Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Xa2 ) @ R5 )
=> ~ ( member @ A @ Y4 @ Q7 ) ) ) )
=> ( wf @ A @ R5 ) ) ).
% wfI_min
thf(fact_2948_wfE__min,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A ),X: A,Q: set @ A] :
( ( wf @ A @ R5 )
=> ( ( member @ A @ X @ Q )
=> ~ ! [Z3: A] :
( ( member @ A @ Z3 @ Q )
=> ~ ! [Y6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ Z3 ) @ R5 )
=> ~ ( member @ A @ Y6 @ Q ) ) ) ) ) ).
% wfE_min
thf(fact_2949_wf__def,axiom,
! [A: $tType] :
( ( wf @ A )
= ( ^ [R3: set @ ( product_prod @ A @ A )] :
! [P4: A > $o] :
( ! [X3: A] :
( ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R3 )
=> ( P4 @ Y3 ) )
=> ( P4 @ X3 ) )
=> ! [X8: A] : ( P4 @ X8 ) ) ) ) ).
% wf_def
thf(fact_2950_wo__rel_Omax2__def,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R )
=> ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
=> ( ( bNF_We1388413361240627857o_max2 @ A @ R @ A3 @ B2 )
= B2 ) )
& ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
=> ( ( bNF_We1388413361240627857o_max2 @ A @ R @ A3 @ B2 )
= A3 ) ) ) ) ).
% wo_rel.max2_def
thf(fact_2951_wo__rel_Owell__order__induct,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),P: A > $o,A3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R )
=> ( ! [X2: A] :
( ! [Y6: A] :
( ( ( Y6 != X2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X2 ) @ R ) )
=> ( P @ Y6 ) )
=> ( P @ X2 ) )
=> ( P @ A3 ) ) ) ).
% wo_rel.well_order_induct
thf(fact_2952_Ioc__inj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ( set_or3652927894154168847AtMost @ A @ A3 @ B2 )
= ( set_or3652927894154168847AtMost @ A @ C2 @ D3 ) )
= ( ( ( ord_less_eq @ A @ B2 @ A3 )
& ( ord_less_eq @ A @ D3 @ C2 ) )
| ( ( A3 = C2 )
& ( B2 = D3 ) ) ) ) ) ).
% Ioc_inj
thf(fact_2953_wo__rel_OTOTALS,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( bNF_Wellorder_wo_rel @ A @ R )
=> ! [X7: A] :
( ( member @ A @ X7 @ ( field2 @ A @ R ) )
=> ! [Xa2: A] :
( ( member @ A @ Xa2 @ ( field2 @ A @ R ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X7 @ Xa2 ) @ R )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Xa2 @ X7 ) @ R ) ) ) ) ) ).
% wo_rel.TOTALS
thf(fact_2954_wo__rel_Omax2__greater__among,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R ) )
=> ( ( member @ A @ B2 @ ( field2 @ A @ R ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ ( bNF_We1388413361240627857o_max2 @ A @ R @ A3 @ B2 ) ) @ R )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ ( bNF_We1388413361240627857o_max2 @ A @ R @ A3 @ B2 ) ) @ R )
& ( member @ A @ ( bNF_We1388413361240627857o_max2 @ A @ R @ A3 @ B2 ) @ ( insert2 @ A @ A3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% wo_rel.max2_greater_among
thf(fact_2955_wfE__min_H,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A ),Q: set @ A] :
( ( wf @ A @ R5 )
=> ( ( Q
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [Z3: A] :
( ( member @ A @ Z3 @ Q )
=> ~ ! [Y6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ Z3 ) @ R5 )
=> ~ ( member @ A @ Y6 @ Q ) ) ) ) ) ).
% wfE_min'
thf(fact_2956_infinite__Ioc,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ~ ( finite_finite2 @ A @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) ) ) ) ).
% infinite_Ioc
thf(fact_2957_Ioc__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C2 @ D3 ) )
= ( ( ord_less_eq @ A @ B2 @ A3 )
| ( ( ord_less_eq @ A @ C2 @ A3 )
& ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% Ioc_subset_iff
thf(fact_2958_atLeastSucAtMost__greaterThanAtMost,axiom,
! [L: nat,U: nat] :
( ( set_or1337092689740270186AtMost @ nat @ ( suc @ L ) @ U )
= ( set_or3652927894154168847AtMost @ nat @ L @ U ) ) ).
% atLeastSucAtMost_greaterThanAtMost
thf(fact_2959_ivl__disj__un__two_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M2: A,U: A] :
( ( ord_less_eq @ A @ L @ M2 )
=> ( ( ord_less_eq @ A @ M2 @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M2 ) @ ( set_or3652927894154168847AtMost @ A @ M2 @ U ) )
= ( set_or3652927894154168847AtMost @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(6)
thf(fact_2960_well__order__induct__imp,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),P: A > $o,A3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R )
=> ( ! [X2: A] :
( ! [Y6: A] :
( ( ( Y6 != X2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X2 ) @ R ) )
=> ( ( member @ A @ Y6 @ ( field2 @ A @ R ) )
=> ( P @ Y6 ) ) )
=> ( ( member @ A @ X2 @ ( field2 @ A @ R ) )
=> ( P @ X2 ) ) )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R ) )
=> ( P @ A3 ) ) ) ) ).
% well_order_induct_imp
thf(fact_2961_wf__bounded__measure,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),Ub: A > nat,F: A > nat] :
( ! [A5: A,B3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A5 ) @ R )
=> ( ( ord_less_eq @ nat @ ( Ub @ B3 ) @ ( Ub @ A5 ) )
& ( ord_less_eq @ nat @ ( F @ B3 ) @ ( Ub @ A5 ) )
& ( ord_less @ nat @ ( F @ A5 ) @ ( F @ B3 ) ) ) )
=> ( wf @ A @ R ) ) ).
% wf_bounded_measure
thf(fact_2962_wfI,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A4: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu: A] : B6 ) )
=> ( ! [X2: A,P7: A > $o] :
( ! [Xa2: A] :
( ! [Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Xa2 ) @ R )
=> ( P7 @ Y4 ) )
=> ( P7 @ Xa2 ) )
=> ( ( member @ A @ X2 @ A4 )
=> ( ( member @ A @ X2 @ B6 )
=> ( P7 @ X2 ) ) ) )
=> ( wf @ A @ R ) ) ) ).
% wfI
thf(fact_2963_ivl__disj__un__two_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M2: A,U: A] :
( ( ord_less_eq @ A @ L @ M2 )
=> ( ( ord_less_eq @ A @ M2 @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M2 ) @ ( set_or3652927894154168847AtMost @ A @ M2 @ U ) )
= ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(8)
thf(fact_2964_ivl__disj__un__one_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less_eq @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_ord_atMost @ A @ L ) @ ( set_or3652927894154168847AtMost @ A @ L @ U ) )
= ( set_ord_atMost @ A @ U ) ) ) ) ).
% ivl_disj_un_one(3)
thf(fact_2965_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
! [L: int,U: int] :
( ( set_or1337092689740270186AtMost @ int @ ( plus_plus @ int @ L @ ( one_one @ int ) ) @ U )
= ( set_or3652927894154168847AtMost @ int @ L @ U ) ) ).
% atLeastPlusOneAtMost_greaterThanAtMost_int
thf(fact_2966_ivl__disj__un__one_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less_eq @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ U ) @ ( set_ord_greaterThan @ A @ U ) )
= ( set_ord_greaterThan @ A @ L ) ) ) ) ).
% ivl_disj_un_one(5)
thf(fact_2967_wf__eq__minimal2,axiom,
! [A: $tType] :
( ( wf @ A )
= ( ^ [R3: set @ ( product_prod @ A @ A )] :
! [A9: set @ A] :
( ( ( ord_less_eq @ ( set @ A ) @ A9 @ ( field2 @ A @ R3 ) )
& ( A9
!= ( bot_bot @ ( set @ A ) ) ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ A9 )
& ! [Y3: A] :
( ( member @ A @ Y3 @ A9 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R3 ) ) ) ) ) ) ).
% wf_eq_minimal2
thf(fact_2968_wf__bounded__set,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),Ub: A > ( set @ B ),F: A > ( set @ B )] :
( ! [A5: A,B3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A5 ) @ R )
=> ( ( finite_finite2 @ B @ ( Ub @ A5 ) )
& ( ord_less_eq @ ( set @ B ) @ ( Ub @ B3 ) @ ( Ub @ A5 ) )
& ( ord_less_eq @ ( set @ B ) @ ( F @ B3 ) @ ( Ub @ A5 ) )
& ( ord_less @ ( set @ B ) @ ( F @ A5 ) @ ( F @ B3 ) ) ) )
=> ( wf @ A @ R ) ) ).
% wf_bounded_set
thf(fact_2969_wo__rel_Ocases__Total,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B2: A,Phi: A > A > $o] :
( ( bNF_Wellorder_wo_rel @ A @ R )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ A3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( field2 @ A @ R ) )
=> ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
=> ( Phi @ A3 @ B2 ) )
=> ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ R )
=> ( Phi @ A3 @ B2 ) )
=> ( Phi @ A3 @ B2 ) ) ) ) ) ).
% wo_rel.cases_Total
thf(fact_2970_sum_Ohead,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M2: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( plus_plus @ A @ ( G @ M2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or3652927894154168847AtMost @ nat @ M2 @ N ) ) ) ) ) ) ).
% sum.head
thf(fact_2971_prod_Ohead,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M2: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( times_times @ A @ ( G @ M2 ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or3652927894154168847AtMost @ nat @ M2 @ N ) ) ) ) ) ) ).
% prod.head
thf(fact_2972_greaterThanAtMost__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D3 ) )
= ( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A3 )
& ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% greaterThanAtMost_subseteq_atLeastAtMost_iff
thf(fact_2973_greaterThanAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D3 ) )
= ( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A3 )
& ( ord_less @ A @ B2 @ D3 ) ) ) ) ) ).
% greaterThanAtMost_subseteq_atLeastLessThan_iff
thf(fact_2974_ivl__disj__un__two__touch_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M2: A,U: A] :
( ( ord_less @ A @ L @ M2 )
=> ( ( ord_less_eq @ A @ M2 @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M2 ) @ ( set_or1337092689740270186AtMost @ A @ M2 @ U ) )
= ( set_or3652927894154168847AtMost @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two_touch(3)
thf(fact_2975_greaterThanAtMost__eq__atLeastAtMost__diff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( set_or3652927894154168847AtMost @ A )
= ( ^ [A7: A,B5: A] : ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A7 @ B5 ) @ ( insert2 @ A @ A7 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% greaterThanAtMost_eq_atLeastAtMost_diff
thf(fact_2976_greaterThanLessThan__subseteq__greaterThanAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C2 @ D3 ) )
= ( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A3 )
& ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_greaterThanAtMost_iff
thf(fact_2977_ivl__disj__un__two_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M2: A,U: A] :
( ( ord_less_eq @ A @ L @ M2 )
=> ( ( ord_less @ A @ M2 @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M2 ) @ ( set_or5935395276787703475ssThan @ A @ M2 @ U ) )
= ( set_or5935395276787703475ssThan @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(2)
thf(fact_2978_ivl__disj__un__two__touch_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M2: A,U: A] :
( ( ord_less @ A @ L @ M2 )
=> ( ( ord_less @ A @ M2 @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M2 ) @ ( set_or7035219750837199246ssThan @ A @ M2 @ U ) )
= ( set_or5935395276787703475ssThan @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two_touch(1)
thf(fact_2979_wf__bounded__supset,axiom,
! [A: $tType,S2: set @ A] :
( ( finite_finite2 @ A @ S2 )
=> ( wf @ ( set @ A )
@ ( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
@ ^ [Q8: set @ A,Q6: set @ A] :
( ( ord_less @ ( set @ A ) @ Q6 @ Q8 )
& ( ord_less_eq @ ( set @ A ) @ Q8 @ S2 ) ) ) ) ) ) ).
% wf_bounded_supset
thf(fact_2980_atLeastPlusOneAtMost__greaterThanAtMost__integer,axiom,
! [L: code_integer,U: code_integer] :
( ( set_or1337092689740270186AtMost @ code_integer @ ( plus_plus @ code_integer @ L @ ( one_one @ code_integer ) ) @ U )
= ( set_or3652927894154168847AtMost @ code_integer @ L @ U ) ) ).
% atLeastPlusOneAtMost_greaterThanAtMost_integer
thf(fact_2981_ivl__disj__un__singleton_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less_eq @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( insert2 @ A @ L @ ( bot_bot @ ( set @ A ) ) ) @ ( set_or3652927894154168847AtMost @ A @ L @ U ) )
= ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ).
% ivl_disj_un_singleton(5)
thf(fact_2982_wf__iff__no__infinite__down__chain,axiom,
! [A: $tType] :
( ( wf @ A )
= ( ^ [R3: set @ ( product_prod @ A @ A )] :
~ ? [F2: nat > A] :
! [I3: nat] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F2 @ ( suc @ I3 ) ) @ ( F2 @ I3 ) ) @ R3 ) ) ) ).
% wf_iff_no_infinite_down_chain
thf(fact_2983_wf__no__infinite__down__chainE,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),F: nat > A] :
( ( wf @ A @ R )
=> ~ ! [K5: nat] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F @ ( suc @ K5 ) ) @ ( F @ K5 ) ) @ R ) ) ).
% wf_no_infinite_down_chainE
thf(fact_2984_wo__rel_Ominim__least,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),B6: set @ A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ ( field2 @ A @ R ) )
=> ( ( member @ A @ B2 @ B6 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( bNF_We6954850376910717587_minim @ A @ R @ B6 ) @ B2 ) @ R ) ) ) ) ).
% wo_rel.minim_least
thf(fact_2985_wo__rel_Oequals__minim,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),B6: set @ A,A3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ ( field2 @ A @ R ) )
=> ( ( member @ A @ A3 @ B6 )
=> ( ! [B3: A] :
( ( member @ A @ B3 @ B6 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ R ) )
=> ( A3
= ( bNF_We6954850376910717587_minim @ A @ R @ B6 ) ) ) ) ) ) ).
% wo_rel.equals_minim
thf(fact_2986_dependent__wf__choice,axiom,
! [B: $tType,A: $tType,R5: set @ ( product_prod @ A @ A ),P: ( A > B ) > A > B > $o] :
( ( wf @ A @ R5 )
=> ( ! [F4: A > B,G4: A > B,X2: A,R2: B] :
( ! [Z6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z6 @ X2 ) @ R5 )
=> ( ( F4 @ Z6 )
= ( G4 @ Z6 ) ) )
=> ( ( P @ F4 @ X2 @ R2 )
= ( P @ G4 @ X2 @ R2 ) ) )
=> ( ! [X2: A,F4: A > B] :
( ! [Y6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X2 ) @ R5 )
=> ( P @ F4 @ Y6 @ ( F4 @ Y6 ) ) )
=> ? [X_1: B] : ( P @ F4 @ X2 @ X_1 ) )
=> ? [F4: A > B] :
! [X7: A] : ( P @ F4 @ X7 @ ( F4 @ X7 ) ) ) ) ) ).
% dependent_wf_choice
thf(fact_2987_of__rat__neg__one,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( field_char_0_of_rat @ A @ ( uminus_uminus @ rat @ ( one_one @ rat ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% of_rat_neg_one
thf(fact_2988_Ref__Time_Oupdate__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( ref_update @ A )
= ( ^ [R3: ref @ A,V2: A] :
( heap_Time_heap @ product_unit
@ ^ [H: heap_ext @ product_unit] : ( product_Pair @ product_unit @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ product_Unity @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ ( ref_set @ A @ R3 @ V2 @ H ) @ ( one_one @ nat ) ) ) ) ) ) ) ).
% Ref_Time.update_def
thf(fact_2989_sum__div__partition,axiom,
! [B: $tType,A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A4: set @ B,F: B > A,B2: A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( divide_divide @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A4 ) @ B2 )
= ( plus_plus @ A
@ ( groups7311177749621191930dd_sum @ B @ A
@ ^ [A7: B] : ( divide_divide @ A @ ( F @ A7 ) @ B2 )
@ ( inf_inf @ ( set @ B ) @ A4
@ ( collect @ B
@ ^ [A7: B] : ( dvd_dvd @ A @ B2 @ ( F @ A7 ) ) ) ) )
@ ( divide_divide @ A
@ ( groups7311177749621191930dd_sum @ B @ A @ F
@ ( inf_inf @ ( set @ B ) @ A4
@ ( collect @ B
@ ^ [A7: B] :
~ ( dvd_dvd @ A @ B2 @ ( F @ A7 ) ) ) ) )
@ B2 ) ) ) ) ) ).
% sum_div_partition
thf(fact_2990_bsqr__max2,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A1: A,A22: A,B1: A,B22: A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( member @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) @ ( product_Pair @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ ( product_Pair @ A @ A @ B1 @ B22 ) ) @ ( bNF_Wellorder_bsqr @ A @ R ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( bNF_We1388413361240627857o_max2 @ A @ R @ A1 @ A22 ) @ ( bNF_We1388413361240627857o_max2 @ A @ R @ B1 @ B22 ) ) @ R ) ) ) ).
% bsqr_max2
thf(fact_2991_UN__atLeast__UNIV,axiom,
( ( complete_Sup_Sup @ ( set @ nat ) @ ( image2 @ nat @ ( set @ nat ) @ ( set_ord_atLeast @ nat ) @ ( top_top @ ( set @ nat ) ) ) )
= ( top_top @ ( set @ nat ) ) ) ).
% UN_atLeast_UNIV
thf(fact_2992_atLeast__eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ( set_ord_atLeast @ A @ X )
= ( set_ord_atLeast @ A @ Y ) )
= ( X = Y ) ) ) ).
% atLeast_eq_iff
thf(fact_2993_unit__abs__eta__conv,axiom,
! [A: $tType,F: product_unit > A] :
( ( ^ [U2: product_unit] : ( F @ product_Unity ) )
= F ) ).
% unit_abs_eta_conv
thf(fact_2994_inf__Some,axiom,
! [A: $tType] :
( ( inf @ A )
=> ! [X: A,Y: A] :
( ( inf_inf @ ( option @ A ) @ ( some @ A @ X ) @ ( some @ A @ Y ) )
= ( some @ A @ ( inf_inf @ A @ X @ Y ) ) ) ) ).
% inf_Some
thf(fact_2995_atLeast__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,K: A] :
( ( member @ A @ I @ ( set_ord_atLeast @ A @ K ) )
= ( ord_less_eq @ A @ K @ I ) ) ) ).
% atLeast_iff
thf(fact_2996_Inf__atLeast,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A] :
( ( complete_Inf_Inf @ A @ ( set_ord_atLeast @ A @ X ) )
= X ) ) ).
% Inf_atLeast
thf(fact_2997_atLeast__subset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ X ) @ ( set_ord_atLeast @ A @ Y ) )
= ( ord_less_eq @ A @ Y @ X ) ) ) ).
% atLeast_subset_iff
thf(fact_2998_image__add__atLeast,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: A,I: A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ K ) @ ( set_ord_atLeast @ A @ I ) )
= ( set_ord_atLeast @ A @ ( plus_plus @ A @ K @ I ) ) ) ) ).
% image_add_atLeast
thf(fact_2999_Sup__atLeast,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A] :
( ( complete_Sup_Sup @ A @ ( set_ord_atLeast @ A @ X ) )
= ( top_top @ A ) ) ) ).
% Sup_atLeast
thf(fact_3000_inter__compl__diff__conv,axiom,
! [A: $tType,A4: set @ A,B6: set @ A] :
( ( inf_inf @ ( set @ A ) @ A4 @ ( uminus_uminus @ ( set @ A ) @ B6 ) )
= ( minus_minus @ ( set @ A ) @ A4 @ B6 ) ) ).
% inter_compl_diff_conv
thf(fact_3001_Int__atMost,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_atMost @ A @ A3 ) @ ( set_ord_atMost @ A @ B2 ) )
= ( set_ord_atMost @ A @ ( ord_min @ A @ A3 @ B2 ) ) ) ) ).
% Int_atMost
thf(fact_3002_Compl__lessThan,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [K: A] :
( ( uminus_uminus @ ( set @ A ) @ ( set_ord_lessThan @ A @ K ) )
= ( set_ord_atLeast @ A @ K ) ) ) ).
% Compl_lessThan
thf(fact_3003_Compl__atLeast,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [K: A] :
( ( uminus_uminus @ ( set @ A ) @ ( set_ord_atLeast @ A @ K ) )
= ( set_ord_lessThan @ A @ K ) ) ) ).
% Compl_atLeast
thf(fact_3004_one__eq__of__rat__iff,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: rat] :
( ( ( one_one @ A )
= ( field_char_0_of_rat @ A @ A3 ) )
= ( ( one_one @ rat )
= A3 ) ) ) ).
% one_eq_of_rat_iff
thf(fact_3005_of__rat__eq__1__iff,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: rat] :
( ( ( field_char_0_of_rat @ A @ A3 )
= ( one_one @ A ) )
= ( A3
= ( one_one @ rat ) ) ) ) ).
% of_rat_eq_1_iff
thf(fact_3006_of__rat__1,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( field_char_0_of_rat @ A @ ( one_one @ rat ) )
= ( one_one @ A ) ) ) ).
% of_rat_1
thf(fact_3007_atLeast__0,axiom,
( ( set_ord_atLeast @ nat @ ( zero_zero @ nat ) )
= ( top_top @ ( set @ nat ) ) ) ).
% atLeast_0
thf(fact_3008_Icc__subset__Ici__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [L: A,H3: A,L4: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ H3 ) @ ( set_ord_atLeast @ A @ L4 ) )
= ( ~ ( ord_less_eq @ A @ L @ H3 )
| ( ord_less_eq @ A @ L4 @ L ) ) ) ) ).
% Icc_subset_Ici_iff
thf(fact_3009_Int__atLeastAtMost,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( ord_max @ A @ A3 @ C2 ) @ ( ord_min @ A @ B2 @ D3 ) ) ) ) ).
% Int_atLeastAtMost
thf(fact_3010_Int__atLeastLessThan,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ A3 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D3 ) )
= ( set_or7035219750837199246ssThan @ A @ ( ord_max @ A @ A3 @ C2 ) @ ( ord_min @ A @ B2 @ D3 ) ) ) ) ).
% Int_atLeastLessThan
thf(fact_3011_Int__atLeastAtMostR1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C2: A,D3: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_atMost @ A @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D3 ) )
= ( set_or1337092689740270186AtMost @ A @ C2 @ ( ord_min @ A @ B2 @ D3 ) ) ) ) ).
% Int_atLeastAtMostR1
thf(fact_3012_Int__atLeastAtMostL1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,D3: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ ( set_ord_atMost @ A @ D3 ) )
= ( set_or1337092689740270186AtMost @ A @ A3 @ ( ord_min @ A @ B2 @ D3 ) ) ) ) ).
% Int_atLeastAtMostL1
thf(fact_3013_image__minus__const__atLeast,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,A3: A] :
( ( image2 @ A @ A @ ( minus_minus @ A @ C2 ) @ ( set_ord_atLeast @ A @ A3 ) )
= ( set_ord_atMost @ A @ ( minus_minus @ A @ C2 @ A3 ) ) ) ) ).
% image_minus_const_atLeast
thf(fact_3014_image__minus__const__AtMost,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,B2: A] :
( ( image2 @ A @ A @ ( minus_minus @ A @ C2 ) @ ( set_ord_atMost @ A @ B2 ) )
= ( set_ord_atLeast @ A @ ( minus_minus @ A @ C2 @ B2 ) ) ) ) ).
% image_minus_const_AtMost
thf(fact_3015_image__uminus__atLeast,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A] :
( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_ord_atLeast @ A @ X ) )
= ( set_ord_atMost @ A @ ( uminus_uminus @ A @ X ) ) ) ) ).
% image_uminus_atLeast
thf(fact_3016_image__uminus__atMost,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A] :
( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_ord_atMost @ A @ X ) )
= ( set_ord_atLeast @ A @ ( uminus_uminus @ A @ X ) ) ) ) ).
% image_uminus_atMost
thf(fact_3017_Int__atLeastAtMostL2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ ( set_ord_atLeast @ A @ C2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( ord_max @ A @ A3 @ C2 ) @ B2 ) ) ) ).
% Int_atLeastAtMostL2
thf(fact_3018_Int__atLeastAtMostR2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,C2: A,D3: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_atLeast @ A @ A3 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( ord_max @ A @ A3 @ C2 ) @ D3 ) ) ) ).
% Int_atLeastAtMostR2
thf(fact_3019_Int__greaterThanAtMost,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C2 @ D3 ) )
= ( set_or3652927894154168847AtMost @ A @ ( ord_max @ A @ A3 @ C2 ) @ ( ord_min @ A @ B2 @ D3 ) ) ) ) ).
% Int_greaterThanAtMost
thf(fact_3020_Int__greaterThanLessThan,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) @ ( set_or5935395276787703475ssThan @ A @ C2 @ D3 ) )
= ( set_or5935395276787703475ssThan @ A @ ( ord_max @ A @ A3 @ C2 ) @ ( ord_min @ A @ B2 @ D3 ) ) ) ) ).
% Int_greaterThanLessThan
thf(fact_3021_one__le__of__rat__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R: rat] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ ( field_char_0_of_rat @ A @ R ) )
= ( ord_less_eq @ rat @ ( one_one @ rat ) @ R ) ) ) ).
% one_le_of_rat_iff
thf(fact_3022_of__rat__le__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R: rat] :
( ( ord_less_eq @ A @ ( field_char_0_of_rat @ A @ R ) @ ( one_one @ A ) )
= ( ord_less_eq @ rat @ R @ ( one_one @ rat ) ) ) ) ).
% of_rat_le_1_iff
thf(fact_3023_one__less__of__rat__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R: rat] :
( ( ord_less @ A @ ( one_one @ A ) @ ( field_char_0_of_rat @ A @ R ) )
= ( ord_less @ rat @ ( one_one @ rat ) @ R ) ) ) ).
% one_less_of_rat_iff
thf(fact_3024_of__rat__less__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R: rat] :
( ( ord_less @ A @ ( field_char_0_of_rat @ A @ R ) @ ( one_one @ A ) )
= ( ord_less @ rat @ R @ ( one_one @ rat ) ) ) ) ).
% of_rat_less_1_iff
thf(fact_3025_atLeastLessThan__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_or7035219750837199246ssThan @ A )
= ( ^ [L3: A,U2: A] : ( inf_inf @ ( set @ A ) @ ( set_ord_atLeast @ A @ L3 ) @ ( set_ord_lessThan @ A @ U2 ) ) ) ) ) ).
% atLeastLessThan_def
thf(fact_3026_well__order__on__domain,axiom,
! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( order_well_order_on @ A @ A4 @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
=> ( ( member @ A @ A3 @ A4 )
& ( member @ A @ B2 @ A4 ) ) ) ) ).
% well_order_on_domain
thf(fact_3027_Inf__unit__def,axiom,
( ( complete_Inf_Inf @ product_unit )
= ( ^ [Uu3: set @ product_unit] : product_Unity ) ) ).
% Inf_unit_def
thf(fact_3028_Sup__unit__def,axiom,
( ( complete_Sup_Sup @ product_unit )
= ( ^ [Uu3: set @ product_unit] : product_Unity ) ) ).
% Sup_unit_def
thf(fact_3029_Times__Int__distrib1,axiom,
! [B: $tType,A: $tType,A4: set @ A,B6: set @ A,C5: set @ B] :
( ( product_Sigma @ A @ B @ ( inf_inf @ ( set @ A ) @ A4 @ B6 )
@ ^ [Uu: A] : C5 )
= ( inf_inf @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : C5 )
@ ( product_Sigma @ A @ B @ B6
@ ^ [Uu: A] : C5 ) ) ) ).
% Times_Int_distrib1
thf(fact_3030_Sigma__Int__distrib2,axiom,
! [B: $tType,A: $tType,I4: set @ A,A4: A > ( set @ B ),B6: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ I4
@ ^ [I3: A] : ( inf_inf @ ( set @ B ) @ ( A4 @ I3 ) @ ( B6 @ I3 ) ) )
= ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I4 @ A4 ) @ ( product_Sigma @ A @ B @ I4 @ B6 ) ) ) ).
% Sigma_Int_distrib2
thf(fact_3031_Times__Int__Times,axiom,
! [B: $tType,A: $tType,A4: set @ A,B6: set @ B,C5: set @ A,D4: set @ B] :
( ( inf_inf @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : B6 )
@ ( product_Sigma @ A @ B @ C5
@ ^ [Uu: A] : D4 ) )
= ( product_Sigma @ A @ B @ ( inf_inf @ ( set @ A ) @ A4 @ C5 )
@ ^ [Uu: A] : ( inf_inf @ ( set @ B ) @ B6 @ D4 ) ) ) ).
% Times_Int_Times
thf(fact_3032_Sigma__Int__distrib1,axiom,
! [B: $tType,A: $tType,I4: set @ A,J4: set @ A,C5: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ ( inf_inf @ ( set @ A ) @ I4 @ J4 ) @ C5 )
= ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I4 @ C5 ) @ ( product_Sigma @ A @ B @ J4 @ C5 ) ) ) ).
% Sigma_Int_distrib1
thf(fact_3033_old_Ounit_Oexhaust,axiom,
! [Y: product_unit] : Y = product_Unity ).
% old.unit.exhaust
thf(fact_3034_top__unit__def,axiom,
( ( top_top @ product_unit )
= product_Unity ) ).
% top_unit_def
thf(fact_3035_uminus__unit__def,axiom,
( ( uminus_uminus @ product_unit )
= ( ^ [Uu3: product_unit] : product_Unity ) ) ).
% uminus_unit_def
thf(fact_3036_bot__unit__def,axiom,
( ( bot_bot @ product_unit )
= product_Unity ) ).
% bot_unit_def
thf(fact_3037_ivl__disj__int__one_I8_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ U ) @ ( set_ord_atLeast @ A @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_one(8)
thf(fact_3038_sup__unit__def,axiom,
( ( sup_sup @ product_unit )
= ( ^ [Uu3: product_unit,Uv2: product_unit] : product_Unity ) ) ).
% sup_unit_def
thf(fact_3039_atLeastAtMost__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_or1337092689740270186AtMost @ A )
= ( ^ [L3: A,U2: A] : ( inf_inf @ ( set @ A ) @ ( set_ord_atLeast @ A @ L3 ) @ ( set_ord_atMost @ A @ U2 ) ) ) ) ) ).
% atLeastAtMost_def
thf(fact_3040_ivl__disj__int__one_I6_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) @ ( set_ord_atLeast @ A @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_one(6)
thf(fact_3041_disjointI,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ~ ( member @ A @ X2 @ B2 ) )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% disjointI
thf(fact_3042_inter__eq__subsetI,axiom,
! [A: $tType,S2: set @ A,S7: set @ A,A4: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S2 @ S7 )
=> ( ( ( inf_inf @ ( set @ A ) @ A4 @ S7 )
= ( inf_inf @ ( set @ A ) @ B6 @ S7 ) )
=> ( ( inf_inf @ ( set @ A ) @ A4 @ S2 )
= ( inf_inf @ ( set @ A ) @ B6 @ S2 ) ) ) ) ).
% inter_eq_subsetI
thf(fact_3043_not__empty__eq__Ici__eq__empty,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [L: A] :
( ( bot_bot @ ( set @ A ) )
!= ( set_ord_atLeast @ A @ L ) ) ) ).
% not_empty_eq_Ici_eq_empty
thf(fact_3044_infinite__Ici,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_top @ A ) )
=> ! [A3: A] :
~ ( finite_finite2 @ A @ ( set_ord_atLeast @ A @ A3 ) ) ) ).
% infinite_Ici
thf(fact_3045_not__UNIV__eq__Ici,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [L4: A] :
( ( top_top @ ( set @ A ) )
!= ( set_ord_atLeast @ A @ L4 ) ) ) ).
% not_UNIV_eq_Ici
thf(fact_3046_not__Ici__eq__Icc,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [L4: A,L: A,H3: A] :
( ( set_ord_atLeast @ A @ L4 )
!= ( set_or1337092689740270186AtMost @ A @ L @ H3 ) ) ) ).
% not_Ici_eq_Icc
thf(fact_3047_not__Iic__eq__Ici,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [H3: A,L4: A] :
( ( set_ord_atMost @ A @ H3 )
!= ( set_ord_atLeast @ A @ L4 ) ) ) ).
% not_Iic_eq_Ici
thf(fact_3048_atMost__Int__atLeast,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [N: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_atMost @ A @ N ) @ ( set_ord_atLeast @ A @ N ) )
= ( insert2 @ A @ N @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% atMost_Int_atLeast
thf(fact_3049_atLeast__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_atLeast @ A )
= ( ^ [L3: A] : ( collect @ A @ ( ord_less_eq @ A @ L3 ) ) ) ) ) ).
% atLeast_def
thf(fact_3050_inf__option__def,axiom,
! [A: $tType] :
( ( inf @ A )
=> ( ( inf_inf @ ( option @ A ) )
= ( ^ [X3: option @ A,Y3: option @ A] :
( case_option @ ( option @ A ) @ A @ ( none @ A )
@ ^ [Z5: A] :
( case_option @ ( option @ A ) @ A @ ( none @ A )
@ ^ [Aa2: A] : ( some @ A @ ( inf_inf @ A @ Z5 @ Aa2 ) )
@ Y3 )
@ X3 ) ) ) ) ).
% inf_option_def
thf(fact_3051_translation__Int,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,S: set @ A,T2: set @ A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ ( inf_inf @ ( set @ A ) @ S @ T2 ) )
= ( inf_inf @ ( set @ A ) @ ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ S ) @ ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ T2 ) ) ) ) ).
% translation_Int
thf(fact_3052_disjoint__mono,axiom,
! [A: $tType,A3: set @ A,A6: set @ A,B2: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ A6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ B4 )
=> ( ( ( inf_inf @ ( set @ A ) @ A6 @ B4 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% disjoint_mono
thf(fact_3053_disjoint__alt__simp2,axiom,
! [A: $tType,A4: set @ A,B6: set @ A] :
( ( ( minus_minus @ ( set @ A ) @ A4 @ B6 )
!= A4 )
= ( ( inf_inf @ ( set @ A ) @ A4 @ B6 )
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% disjoint_alt_simp2
thf(fact_3054_disjoint__alt__simp1,axiom,
! [A: $tType,A4: set @ A,B6: set @ A] :
( ( ( minus_minus @ ( set @ A ) @ A4 @ B6 )
= A4 )
= ( ( inf_inf @ ( set @ A ) @ A4 @ B6 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% disjoint_alt_simp1
thf(fact_3055_ivl__disj__int__two_I3_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M2: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ M2 ) @ ( set_or7035219750837199246ssThan @ A @ M2 @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(3)
thf(fact_3056_ivl__disj__int__two_I6_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M2: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M2 ) @ ( set_or3652927894154168847AtMost @ A @ M2 @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(6)
thf(fact_3057_greaterThan__Int__greaterThan,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ A3 ) @ ( set_ord_lessThan @ A @ B2 ) )
= ( set_ord_lessThan @ A @ ( ord_min @ A @ A3 @ B2 ) ) ) ) ).
% greaterThan_Int_greaterThan
thf(fact_3058_atLeast__eq__UNIV__iff,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X: A] :
( ( ( set_ord_atLeast @ A @ X )
= ( top_top @ ( set @ A ) ) )
= ( X
= ( bot_bot @ A ) ) ) ) ).
% atLeast_eq_UNIV_iff
thf(fact_3059_not__UNIV__le__Ici,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [L: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ ( set_ord_atLeast @ A @ L ) ) ) ).
% not_UNIV_le_Ici
thf(fact_3060_UNIV__unit,axiom,
( ( top_top @ ( set @ product_unit ) )
= ( insert2 @ product_unit @ product_Unity @ ( bot_bot @ ( set @ product_unit ) ) ) ) ).
% UNIV_unit
thf(fact_3061_not__Ici__le__Icc,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [L: A,L4: A,H4: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ L ) @ ( set_or1337092689740270186AtMost @ A @ L4 @ H4 ) ) ) ).
% not_Ici_le_Icc
thf(fact_3062_lessThan__Int__lessThan,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_greaterThan @ A @ A3 ) @ ( set_ord_greaterThan @ A @ B2 ) )
= ( set_ord_greaterThan @ A @ ( ord_max @ A @ A3 @ B2 ) ) ) ) ).
% lessThan_Int_lessThan
thf(fact_3063_not__Iic__le__Ici,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [H3: A,L4: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ H3 ) @ ( set_ord_atLeast @ A @ L4 ) ) ) ).
% not_Iic_le_Ici
thf(fact_3064_not__Ici__le__Iic,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [L: A,H4: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ L ) @ ( set_ord_atMost @ A @ H4 ) ) ) ).
% not_Ici_le_Iic
thf(fact_3065_translation__subtract__Int,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,S: set @ A,T2: set @ A] :
( ( image2 @ A @ A
@ ^ [X3: A] : ( minus_minus @ A @ X3 @ A3 )
@ ( inf_inf @ ( set @ A ) @ S @ T2 ) )
= ( inf_inf @ ( set @ A )
@ ( image2 @ A @ A
@ ^ [X3: A] : ( minus_minus @ A @ X3 @ A3 )
@ S )
@ ( image2 @ A @ A
@ ^ [X3: A] : ( minus_minus @ A @ X3 @ A3 )
@ T2 ) ) ) ) ).
% translation_subtract_Int
thf(fact_3066_of__rat__add,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: rat,B2: rat] :
( ( field_char_0_of_rat @ A @ ( plus_plus @ rat @ A3 @ B2 ) )
= ( plus_plus @ A @ ( field_char_0_of_rat @ A @ A3 ) @ ( field_char_0_of_rat @ A @ B2 ) ) ) ) ).
% of_rat_add
thf(fact_3067_Ioi__le__Ico,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A] : ( ord_less_eq @ ( set @ A ) @ ( set_ord_greaterThan @ A @ A3 ) @ ( set_ord_atLeast @ A @ A3 ) ) ) ).
% Ioi_le_Ico
thf(fact_3068_atLeast__Suc__greaterThan,axiom,
! [K: nat] :
( ( set_ord_atLeast @ nat @ ( suc @ K ) )
= ( set_ord_greaterThan @ nat @ K ) ) ).
% atLeast_Suc_greaterThan
thf(fact_3069_ivl__disj__int__two_I7_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M2: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ M2 ) @ ( set_or1337092689740270186AtMost @ A @ M2 @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(7)
thf(fact_3070_disjoint__alt__simp3,axiom,
! [A: $tType,A4: set @ A,B6: set @ A] :
( ( ord_less @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B6 ) @ A4 )
= ( ( inf_inf @ ( set @ A ) @ A4 @ B6 )
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% disjoint_alt_simp3
thf(fact_3071_Ioc__disjoint,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C2 @ D3 ) )
= ( bot_bot @ ( set @ A ) ) )
= ( ( ord_less_eq @ A @ B2 @ A3 )
| ( ord_less_eq @ A @ D3 @ C2 )
| ( ord_less_eq @ A @ B2 @ C2 )
| ( ord_less_eq @ A @ D3 @ A3 ) ) ) ) ).
% Ioc_disjoint
thf(fact_3072_ivl__disj__int__one_I4_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ L ) @ ( set_or1337092689740270186AtMost @ A @ L @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_one(4)
thf(fact_3073_ivl__disj__int__one_I2_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ L ) @ ( set_or7035219750837199246ssThan @ A @ L @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_one(2)
thf(fact_3074_wait__def,axiom,
( heap_Time_wait
= ( ^ [N5: nat] :
( heap_Time_Heap2 @ product_unit
@ ^ [H: heap_ext @ product_unit] : ( some @ ( product_prod @ product_unit @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ product_unit @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ product_Unity @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H @ N5 ) ) ) ) ) ) ).
% wait_def
thf(fact_3075_ivl__disj__int__two_I8_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M2: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M2 ) @ ( set_or3652927894154168847AtMost @ A @ M2 @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(8)
thf(fact_3076_ivl__disj__int__two_I5_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M2: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M2 ) @ ( set_or1337092689740270186AtMost @ A @ M2 @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(5)
thf(fact_3077_ivl__disj__int__two_I4_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M2: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M2 ) @ ( set_or5935395276787703475ssThan @ A @ M2 @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(4)
thf(fact_3078_ivl__disj__int__two_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M2: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M2 ) @ ( set_or7035219750837199246ssThan @ A @ M2 @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(1)
thf(fact_3079_ivl__disj__int__one_I3_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_atMost @ A @ L ) @ ( set_or3652927894154168847AtMost @ A @ L @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_one(3)
thf(fact_3080_ivl__disj__int__one_I7_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ U ) @ ( set_ord_greaterThan @ A @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_one(7)
thf(fact_3081_ivl__disj__int__one_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_atMost @ A @ L ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_one(1)
thf(fact_3082_ivl__disj__int__two_I2_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M2: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M2 ) @ ( set_or5935395276787703475ssThan @ A @ M2 @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(2)
thf(fact_3083_ivl__disj__int__one_I5_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ U ) @ ( set_ord_greaterThan @ A @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_one(5)
thf(fact_3084_ivl__disj__un__one_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less_eq @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ U ) @ ( set_ord_atLeast @ A @ U ) )
= ( set_ord_atLeast @ A @ L ) ) ) ) ).
% ivl_disj_un_one(8)
thf(fact_3085_prod_Ointer__restrict,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,G: B > A,B6: set @ B] :
( ( finite_finite2 @ B @ A4 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A4 @ B6 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X3: B] : ( if @ A @ ( member @ B @ X3 @ B6 ) @ ( G @ X3 ) @ ( one_one @ A ) )
@ A4 ) ) ) ) ).
% prod.inter_restrict
thf(fact_3086_greaterThanAtMost__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_or3652927894154168847AtMost @ A )
= ( ^ [L3: A,U2: A] : ( inf_inf @ ( set @ A ) @ ( set_ord_greaterThan @ A @ L3 ) @ ( set_ord_atMost @ A @ U2 ) ) ) ) ) ).
% greaterThanAtMost_def
thf(fact_3087_greaterThanLessThan__eq,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_or5935395276787703475ssThan @ A )
= ( ^ [A7: A,B5: A] : ( inf_inf @ ( set @ A ) @ ( set_ord_greaterThan @ A @ A7 ) @ ( set_ord_lessThan @ A @ B5 ) ) ) ) ) ).
% greaterThanLessThan_eq
thf(fact_3088_greaterThanLessThan__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_or5935395276787703475ssThan @ A )
= ( ^ [L3: A,U2: A] : ( inf_inf @ ( set @ A ) @ ( set_ord_greaterThan @ A @ L3 ) @ ( set_ord_lessThan @ A @ U2 ) ) ) ) ) ).
% greaterThanLessThan_def
thf(fact_3089_Ici__subset__Ioi__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ A3 ) @ ( set_ord_greaterThan @ A @ B2 ) )
= ( ord_less @ A @ B2 @ A3 ) ) ) ).
% Ici_subset_Ioi_iff
thf(fact_3090_execute__assert_H,axiom,
! [P: $o,H3: heap_ext @ product_unit] :
( ( P
=> ( ( heap_Time_execute @ product_unit @ ( refine_Imp_assert @ P ) @ H3 )
= ( some @ ( product_prod @ product_unit @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ product_unit @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ product_Unity @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ ( zero_zero @ nat ) ) ) ) ) )
& ( ~ P
=> ( ( heap_Time_execute @ product_unit @ ( refine_Imp_assert @ P ) @ H3 )
= ( none @ ( product_prod @ product_unit @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ).
% execute_assert'
thf(fact_3091_sum_Ounion__inter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,B6: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( finite_finite2 @ B @ B6 )
=> ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A4 @ B6 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A4 @ B6 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ B6 ) ) ) ) ) ) ).
% sum.union_inter
thf(fact_3092_sum_OInt__Diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,G: B > A,B6: set @ B] :
( ( finite_finite2 @ B @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A4 @ B6 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A4 @ B6 ) ) ) ) ) ) ).
% sum.Int_Diff
thf(fact_3093_prod_Omono__neutral__cong,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T6: set @ B,S2: set @ B,H3: B > A,G: B > A] :
( ( finite_finite2 @ B @ T6 )
=> ( ( finite_finite2 @ B @ S2 )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ ( minus_minus @ ( set @ B ) @ T6 @ S2 ) )
=> ( ( H3 @ I2 )
= ( one_one @ A ) ) )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ ( minus_minus @ ( set @ B ) @ S2 @ T6 ) )
=> ( ( G @ I2 )
= ( one_one @ A ) ) )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ ( inf_inf @ ( set @ B ) @ S2 @ T6 ) )
=> ( ( G @ X2 )
= ( H3 @ X2 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ S2 )
= ( groups7121269368397514597t_prod @ B @ A @ H3 @ T6 ) ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong
thf(fact_3094_Iio__Int__singleton,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,K: A] :
( ( ( ord_less @ A @ X @ K )
=> ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ K ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( ord_less @ A @ X @ K )
=> ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ K ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Iio_Int_singleton
thf(fact_3095_card__Un__Int,axiom,
! [A: $tType,A4: set @ A,B6: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ A @ B6 )
=> ( ( plus_plus @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ A @ B6 ) )
= ( plus_plus @ nat @ ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B6 ) ) @ ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ A4 @ B6 ) ) ) ) ) ) ).
% card_Un_Int
thf(fact_3096_sum_OIf__cases,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,P: B > $o,H3: B > A,G: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X3: B] : ( if @ A @ ( P @ X3 ) @ ( H3 @ X3 ) @ ( G @ X3 ) )
@ A4 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ H3 @ ( inf_inf @ ( set @ B ) @ A4 @ ( collect @ B @ P ) ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A4 @ ( uminus_uminus @ ( set @ B ) @ ( collect @ B @ P ) ) ) ) ) ) ) ) ).
% sum.If_cases
thf(fact_3097_ivl__disj__un__singleton_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A] :
( ( sup_sup @ ( set @ A ) @ ( insert2 @ A @ L @ ( bot_bot @ ( set @ A ) ) ) @ ( set_ord_greaterThan @ A @ L ) )
= ( set_ord_atLeast @ A @ L ) ) ) ).
% ivl_disj_un_singleton(1)
thf(fact_3098_ivl__disj__un__one_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less_eq @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ U ) @ ( set_ord_greaterThan @ A @ U ) )
= ( set_ord_atLeast @ A @ L ) ) ) ) ).
% ivl_disj_un_one(7)
thf(fact_3099_ivl__disj__un__one_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) @ ( set_ord_atLeast @ A @ U ) )
= ( set_ord_greaterThan @ A @ L ) ) ) ) ).
% ivl_disj_un_one(6)
thf(fact_3100_Linear__order__Well__order__iff,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
= ( ! [A9: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A9 @ ( field2 @ A @ R ) )
=> ( ( A9
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ A9 )
& ! [Y3: A] :
( ( member @ A @ Y3 @ A9 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R ) ) ) ) ) ) ) ) ).
% Linear_order_Well_order_iff
thf(fact_3101_sum_Ounion__inter__neutral,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,B6: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( finite_finite2 @ B @ B6 )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ ( inf_inf @ ( set @ B ) @ A4 @ B6 ) )
=> ( ( G @ X2 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A4 @ B6 ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ B6 ) ) ) ) ) ) ) ).
% sum.union_inter_neutral
thf(fact_3102_sum__Un,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [A4: set @ B,B6: set @ B,F: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( finite_finite2 @ B @ B6 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ F @ ( sup_sup @ ( set @ B ) @ A4 @ B6 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A4 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F @ B6 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ F @ ( inf_inf @ ( set @ B ) @ A4 @ B6 ) ) ) ) ) ) ) ).
% sum_Un
thf(fact_3103_sum_Ounion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,B6: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( finite_finite2 @ B @ B6 )
=> ( ( ( inf_inf @ ( set @ B ) @ A4 @ B6 )
= ( bot_bot @ ( set @ B ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A4 @ B6 ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ B6 ) ) ) ) ) ) ) ).
% sum.union_disjoint
thf(fact_3104_prod_Ounion__inter__neutral,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,B6: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( finite_finite2 @ B @ B6 )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ ( inf_inf @ ( set @ B ) @ A4 @ B6 ) )
=> ( ( G @ X2 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A4 @ B6 ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ A4 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B6 ) ) ) ) ) ) ) ).
% prod.union_inter_neutral
thf(fact_3105_execute__update,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R: ref @ A,V: A,H3: heap_ext @ product_unit] :
( ( heap_Time_execute @ product_unit @ ( ref_update @ A @ R @ V ) @ H3 )
= ( some @ ( product_prod @ product_unit @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ product_unit @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ product_Unity @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ ( ref_set @ A @ R @ V @ H3 ) @ ( one_one @ nat ) ) ) ) ) ) ).
% execute_update
thf(fact_3106_sum_Ounion__diff2,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,B6: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( finite_finite2 @ B @ B6 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A4 @ B6 ) )
= ( plus_plus @ A @ ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A4 @ B6 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ B6 @ A4 ) ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A4 @ B6 ) ) ) ) ) ) ) ).
% sum.union_diff2
thf(fact_3107_sum__Un2,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ B )
=> ! [A4: set @ A,B6: set @ A,F: A > B] :
( ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B6 ) )
=> ( ( groups7311177749621191930dd_sum @ A @ B @ F @ ( sup_sup @ ( set @ A ) @ A4 @ B6 ) )
= ( plus_plus @ B @ ( plus_plus @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ A4 @ B6 ) ) @ ( groups7311177749621191930dd_sum @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ B6 @ A4 ) ) ) @ ( groups7311177749621191930dd_sum @ A @ B @ F @ ( inf_inf @ ( set @ A ) @ A4 @ B6 ) ) ) ) ) ) ).
% sum_Un2
thf(fact_3108_card__Un__disjoint,axiom,
! [A: $tType,A4: set @ A,B6: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ A @ B6 )
=> ( ( ( inf_inf @ ( set @ A ) @ A4 @ B6 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B6 ) )
= ( plus_plus @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ A @ B6 ) ) ) ) ) ) ).
% card_Un_disjoint
thf(fact_3109_sum__Un__nat,axiom,
! [A: $tType,A4: set @ A,B6: set @ A,F: A > nat] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ A @ B6 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ F @ ( sup_sup @ ( set @ A ) @ A4 @ B6 ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F @ A4 ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F @ B6 ) ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F @ ( inf_inf @ ( set @ A ) @ A4 @ B6 ) ) ) ) ) ) ).
% sum_Un_nat
thf(fact_3110_atLeast__Suc,axiom,
! [K: nat] :
( ( set_ord_atLeast @ nat @ ( suc @ K ) )
= ( minus_minus @ ( set @ nat ) @ ( set_ord_atLeast @ nat @ K ) @ ( insert2 @ nat @ K @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeast_Suc
thf(fact_3111_old_Ounit_Orec,axiom,
! [T: $tType,F1: T] :
( ( product_rec_unit @ T @ F1 @ product_Unity )
= F1 ) ).
% old.unit.rec
thf(fact_3112_default__unit__def,axiom,
( ( default_default @ product_unit )
= product_Unity ) ).
% default_unit_def
thf(fact_3113_CODE__ABORT__def,axiom,
! [A: $tType] :
( ( cODE_ABORT @ A )
= ( ^ [F2: product_unit > A] : ( F2 @ product_Unity ) ) ) ).
% CODE_ABORT_def
thf(fact_3114_Inf__fin_Oeq__fold_H,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( lattic7752659483105999362nf_fin @ A )
= ( ^ [A9: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X3: A,Y3: option @ A] : ( some @ A @ ( case_option @ A @ A @ X3 @ ( inf_inf @ A @ X3 ) @ Y3 ) )
@ ( none @ A )
@ A9 ) ) ) ) ) ).
% Inf_fin.eq_fold'
thf(fact_3115_image__split__eq__Sigma,axiom,
! [C: $tType,B: $tType,A: $tType,F: C > A,G: C > B,A4: set @ C] :
( ( image2 @ C @ ( product_prod @ A @ B )
@ ^ [X3: C] : ( product_Pair @ A @ B @ ( F @ X3 ) @ ( G @ X3 ) )
@ A4 )
= ( product_Sigma @ A @ B @ ( image2 @ C @ A @ F @ A4 )
@ ^ [X3: A] : ( image2 @ C @ B @ G @ ( inf_inf @ ( set @ C ) @ ( vimage @ C @ A @ F @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) @ A4 ) ) ) ) ).
% image_split_eq_Sigma
thf(fact_3116_atLeast__empty__triv,axiom,
! [A: $tType] :
( ( set_ord_atLeast @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) )
= ( top_top @ ( set @ ( set @ A ) ) ) ) ).
% atLeast_empty_triv
thf(fact_3117_inf__None__2,axiom,
! [A: $tType] :
( ( inf @ A )
=> ! [X: option @ A] :
( ( inf_inf @ ( option @ A ) @ X @ ( none @ A ) )
= ( none @ A ) ) ) ).
% inf_None_2
thf(fact_3118_inf__None__1,axiom,
! [A: $tType] :
( ( inf @ A )
=> ! [Y: option @ A] :
( ( inf_inf @ ( option @ A ) @ ( none @ A ) @ Y )
= ( none @ A ) ) ) ).
% inf_None_1
thf(fact_3119_vimage__id,axiom,
! [A: $tType] :
( ( vimage @ A @ A @ ( id @ A ) )
= ( id @ ( set @ A ) ) ) ).
% vimage_id
thf(fact_3120_set_Ocomp,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > B,G: B > A] :
( ( comp @ ( set @ B ) @ ( set @ C ) @ ( set @ A ) @ ( vimage @ C @ B @ F ) @ ( vimage @ B @ A @ G ) )
= ( vimage @ C @ A @ ( comp @ B @ A @ C @ G @ F ) ) ) ).
% set.comp
thf(fact_3121_set_Ocompositionality,axiom,
! [C: $tType,B: $tType,A: $tType,F: C > B,G: B > A,Set: set @ A] :
( ( vimage @ C @ B @ F @ ( vimage @ B @ A @ G @ Set ) )
= ( vimage @ C @ A @ ( comp @ B @ A @ C @ G @ F ) @ Set ) ) ).
% set.compositionality
thf(fact_3122_vimage__comp,axiom,
! [A: $tType,B: $tType,C: $tType,F: A > B,G: B > C,X: set @ C] :
( ( vimage @ A @ B @ F @ ( vimage @ B @ C @ G @ X ) )
= ( vimage @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ X ) ) ).
% vimage_comp
thf(fact_3123_inf__unit__def,axiom,
( ( inf_inf @ product_unit )
= ( ^ [Uu3: product_unit,Uv2: product_unit] : product_Unity ) ) ).
% inf_unit_def
thf(fact_3124_inf__Int__eq2,axiom,
! [B: $tType,A: $tType,R5: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
( ( inf_inf @ ( A > B > $o )
@ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R5 )
@ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ S2 ) )
= ( ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ R5 @ S2 ) ) ) ) ).
% inf_Int_eq2
thf(fact_3125_surj__image__vimage__eq,axiom,
! [B: $tType,A: $tType,F: B > A,A4: set @ A] :
( ( ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( image2 @ B @ A @ F @ ( vimage @ B @ A @ F @ A4 ) )
= A4 ) ) ).
% surj_image_vimage_eq
thf(fact_3126_set_Oidentity,axiom,
! [A: $tType] :
( ( vimage @ A @ A
@ ^ [X3: A] : X3 )
= ( id @ ( set @ A ) ) ) ).
% set.identity
thf(fact_3127_vimage__Times,axiom,
! [A: $tType,B: $tType,C: $tType,F: A > ( product_prod @ B @ C ),A4: set @ B,B6: set @ C] :
( ( vimage @ A @ ( product_prod @ B @ C ) @ F
@ ( product_Sigma @ B @ C @ A4
@ ^ [Uu: B] : B6 ) )
= ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ ( comp @ ( product_prod @ B @ C ) @ B @ A @ ( product_fst @ B @ C ) @ F ) @ A4 ) @ ( vimage @ A @ C @ ( comp @ ( product_prod @ B @ C ) @ C @ A @ ( product_snd @ B @ C ) @ F ) @ B6 ) ) ) ).
% vimage_Times
thf(fact_3128_Pair__vimage__Sigma,axiom,
! [B: $tType,A: $tType,X: B,A4: set @ B,F: B > ( set @ A )] :
( ( ( member @ B @ X @ A4 )
=> ( ( vimage @ A @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X ) @ ( product_Sigma @ B @ A @ A4 @ F ) )
= ( F @ X ) ) )
& ( ~ ( member @ B @ X @ A4 )
=> ( ( vimage @ A @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X ) @ ( product_Sigma @ B @ A @ A4 @ F ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Pair_vimage_Sigma
thf(fact_3129_surj__vimage__empty,axiom,
! [B: $tType,A: $tType,F: B > A,A4: set @ A] :
( ( ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( ( vimage @ B @ A @ F @ A4 )
= ( bot_bot @ ( set @ B ) ) )
= ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% surj_vimage_empty
thf(fact_3130_vimage__subsetD,axiom,
! [A: $tType,B: $tType,F: B > A,B6: set @ A,A4: set @ B] :
( ( ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ B ) @ ( vimage @ B @ A @ F @ B6 ) @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ B6 @ ( image2 @ B @ A @ F @ A4 ) ) ) ) ).
% vimage_subsetD
thf(fact_3131_vimage__fst,axiom,
! [B: $tType,A: $tType,A4: set @ A] :
( ( vimage @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A4 )
= ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : ( top_top @ ( set @ B ) ) ) ) ).
% vimage_fst
thf(fact_3132_vimage__snd,axiom,
! [B: $tType,A: $tType,A4: set @ B] :
( ( vimage @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A4 )
= ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
@ ^ [Uu: A] : A4 ) ) ).
% vimage_snd
thf(fact_3133_Inf__fin_Oinfinite,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: set @ A] :
( ~ ( finite_finite2 @ A @ A4 )
=> ( ( lattic7752659483105999362nf_fin @ A @ A4 )
= ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% Inf_fin.infinite
thf(fact_3134_rtrancl__last__visit__node,axiom,
! [A: $tType,S: A,S3: A,R5: set @ ( product_prod @ A @ A ),Sh: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ S @ S3 ) @ ( transitive_rtrancl @ A @ R5 ) )
=> ( ( ( S != Sh )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ S @ S3 )
@ ( transitive_rtrancl @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R5
@ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
@ ^ [Uu: A] : ( uminus_uminus @ ( set @ A ) @ ( insert2 @ A @ Sh @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) )
| ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ S @ Sh ) @ ( transitive_rtrancl @ A @ R5 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Sh @ S3 )
@ ( transitive_rtrancl @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R5
@ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
@ ^ [Uu: A] : ( uminus_uminus @ ( set @ A ) @ ( insert2 @ A @ Sh @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% rtrancl_last_visit_node
thf(fact_3135_floor__rat__def,axiom,
( ( archim6421214686448440834_floor @ rat )
= ( ^ [X3: rat] :
( the @ int
@ ^ [Z5: int] :
( ( ord_less_eq @ rat @ ( ring_1_of_int @ rat @ Z5 ) @ X3 )
& ( ord_less @ rat @ X3 @ ( ring_1_of_int @ rat @ ( plus_plus @ int @ Z5 @ ( one_one @ int ) ) ) ) ) ) ) ) ).
% floor_rat_def
thf(fact_3136_rat__floor__lemma,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq @ rat @ ( ring_1_of_int @ rat @ ( divide_divide @ int @ A3 @ B2 ) ) @ ( fract @ A3 @ B2 ) )
& ( ord_less @ rat @ ( fract @ A3 @ B2 ) @ ( ring_1_of_int @ rat @ ( plus_plus @ int @ ( divide_divide @ int @ A3 @ B2 ) @ ( one_one @ int ) ) ) ) ) ).
% rat_floor_lemma
thf(fact_3137_num__of__integer_Otransfer,axiom,
( bNF_rel_fun @ int @ code_integer @ num @ num @ code_pcr_integer
@ ^ [Y5: num,Z4: num] : Y5 = Z4
@ ( comp @ nat @ num @ int @ num_of_nat @ nat2 )
@ code_num_of_integer ) ).
% num_of_integer.transfer
thf(fact_3138_prod__decode__triangle__add,axiom,
! [K: nat,M2: nat] :
( ( nat_prod_decode @ ( plus_plus @ nat @ ( nat_triangle @ K ) @ M2 ) )
= ( nat_prod_decode_aux @ K @ M2 ) ) ).
% prod_decode_triangle_add
thf(fact_3139_wf__insert,axiom,
! [A: $tType,Y: A,X: A,R: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ R ) )
= ( ( wf @ A @ R )
& ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ).
% wf_insert
thf(fact_3140_add__rat,axiom,
! [B2: int,D3: int,A3: int,C2: int] :
( ( B2
!= ( zero_zero @ int ) )
=> ( ( D3
!= ( zero_zero @ int ) )
=> ( ( plus_plus @ rat @ ( fract @ A3 @ B2 ) @ ( fract @ C2 @ D3 ) )
= ( fract @ ( plus_plus @ int @ ( times_times @ int @ A3 @ D3 ) @ ( times_times @ int @ C2 @ B2 ) ) @ ( times_times @ int @ B2 @ D3 ) ) ) ) ) ).
% add_rat
thf(fact_3141_r__le__rtrancl,axiom,
! [A: $tType,S2: set @ ( product_prod @ A @ A )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ S2 @ ( transitive_rtrancl @ A @ S2 ) ) ).
% r_le_rtrancl
thf(fact_3142_rtrancl__mono__mp,axiom,
! [A: $tType,U4: set @ ( product_prod @ A @ A ),V6: set @ ( product_prod @ A @ A ),X: product_prod @ A @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ U4 @ V6 )
=> ( ( member @ ( product_prod @ A @ A ) @ X @ ( transitive_rtrancl @ A @ U4 ) )
=> ( member @ ( product_prod @ A @ A ) @ X @ ( transitive_rtrancl @ A @ V6 ) ) ) ) ).
% rtrancl_mono_mp
thf(fact_3143_rtrancl__mono__rightI,axiom,
! [A: $tType,S2: set @ ( product_prod @ A @ A ),S7: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ S2 @ S7 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ S2 @ ( transitive_rtrancl @ A @ S7 ) ) ) ).
% rtrancl_mono_rightI
thf(fact_3144_converse__rtranclE_H,axiom,
! [A: $tType,U: A,V: A,R5: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V ) @ ( transitive_rtrancl @ A @ R5 ) )
=> ( ( U != V )
=> ~ ! [Vh: A] :
( ( U != Vh )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ Vh ) @ R5 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Vh @ V ) @ ( transitive_rtrancl @ A @ R5 ) ) ) ) ) ) ).
% converse_rtranclE'
thf(fact_3145_rtrancl_Ocases,axiom,
! [A: $tType,A1: A,A22: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( A22 != A1 )
=> ~ ! [B3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ B3 ) @ ( transitive_rtrancl @ A @ R ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A22 ) @ R ) ) ) ) ).
% rtrancl.cases
thf(fact_3146_rtrancl_Osimps,axiom,
! [A: $tType,A1: A,A22: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ ( transitive_rtrancl @ A @ R ) )
= ( ? [A7: A] :
( ( A1 = A7 )
& ( A22 = A7 ) )
| ? [A7: A,B5: A,C3: A] :
( ( A1 = A7 )
& ( A22 = C3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B5 ) @ ( transitive_rtrancl @ A @ R ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ C3 ) @ R ) ) ) ) ).
% rtrancl.simps
thf(fact_3147_rtrancl_Ortrancl__refl,axiom,
! [A: $tType,A3: A,R: set @ ( product_prod @ A @ A )] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ ( transitive_rtrancl @ A @ R ) ) ).
% rtrancl.rtrancl_refl
thf(fact_3148_rtrancl_Ortrancl__into__rtrancl,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),C2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C2 ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C2 ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ).
% rtrancl.rtrancl_into_rtrancl
thf(fact_3149_rtranclE,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( A3 != B2 )
=> ~ ! [Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ Y4 ) @ ( transitive_rtrancl @ A @ R ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ B2 ) @ R ) ) ) ) ).
% rtranclE
thf(fact_3150_rtrancl__trans,axiom,
! [A: $tType,X: A,Y: A,R: set @ ( product_prod @ A @ A ),Z2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ).
% rtrancl_trans
thf(fact_3151_rtrancl__induct,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),P: A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( P @ A3 )
=> ( ! [Y4: A,Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ Y4 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z3 ) @ R )
=> ( ( P @ Y4 )
=> ( P @ Z3 ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% rtrancl_induct
thf(fact_3152_converse__rtranclE,axiom,
! [A: $tType,X: A,Z2: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( X != Z2 )
=> ~ ! [Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y4 ) @ R )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z2 ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ) ).
% converse_rtranclE
thf(fact_3153_converse__rtrancl__induct,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),P: A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( P @ B2 )
=> ( ! [Y4: A,Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z3 ) @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z3 @ B2 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( P @ Z3 )
=> ( P @ Y4 ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% converse_rtrancl_induct
thf(fact_3154_converse__rtrancl__into__rtrancl,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),C2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C2 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C2 ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ).
% converse_rtrancl_into_rtrancl
thf(fact_3155_in__rtrancl__insert,axiom,
! [A: $tType,X: product_prod @ A @ A,R5: set @ ( product_prod @ A @ A ),R: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X @ ( transitive_rtrancl @ A @ R5 ) )
=> ( member @ ( product_prod @ A @ A ) @ X @ ( transitive_rtrancl @ A @ ( insert2 @ ( product_prod @ A @ A ) @ R @ R5 ) ) ) ) ).
% in_rtrancl_insert
thf(fact_3156_integer_Orep__transfer,axiom,
( bNF_rel_fun @ int @ code_integer @ int @ int @ code_pcr_integer
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ^ [X3: int] : X3
@ code_int_of_integer ) ).
% integer.rep_transfer
thf(fact_3157_integer_Oid__abs__transfer,axiom,
( bNF_rel_fun @ int @ int @ int @ code_integer
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ code_pcr_integer
@ ^ [X3: int] : X3
@ code_integer_of_int ) ).
% integer.id_abs_transfer
thf(fact_3158_rtrancl__Un__separator__converseE,axiom,
! [A: $tType,A3: A,B2: A,P: set @ ( product_prod @ A @ A ),Q: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ P @ Q ) ) )
=> ( ! [X2: A,Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ B2 ) @ ( transitive_rtrancl @ A @ P ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X2 ) @ Q )
=> ( Y4 = X2 ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ P ) ) ) ) ).
% rtrancl_Un_separator_converseE
thf(fact_3159_rtrancl__Un__separatorE,axiom,
! [A: $tType,A3: A,B2: A,P: set @ ( product_prod @ A @ A ),Q: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ P @ Q ) ) )
=> ( ! [X2: A,Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ X2 ) @ ( transitive_rtrancl @ A @ P ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y4 ) @ Q )
=> ( X2 = Y4 ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ P ) ) ) ) ).
% rtrancl_Un_separatorE
thf(fact_3160_zero__integer_Otransfer,axiom,
code_pcr_integer @ ( zero_zero @ int ) @ ( zero_zero @ code_integer ) ).
% zero_integer.transfer
thf(fact_3161_converse__rtrancl__induct2,axiom,
! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R: set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ),P: A > B > $o] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R ) )
=> ( ( P @ Bx @ By )
=> ( ! [A5: A,B3: B,Aa3: A,Ba: B] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B3 ) @ ( product_Pair @ A @ B @ Aa3 @ Ba ) ) @ R )
=> ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Aa3 @ Ba ) @ ( product_Pair @ A @ B @ Bx @ By ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R ) )
=> ( ( P @ Aa3 @ Ba )
=> ( P @ A5 @ B3 ) ) ) )
=> ( P @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_3162_converse__rtranclE2,axiom,
! [B: $tType,A: $tType,Xa: A,Xb: B,Za2: A,Zb: B,R: set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) )] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Xa @ Xb ) @ ( product_Pair @ A @ B @ Za2 @ Zb ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R ) )
=> ( ( ( product_Pair @ A @ B @ Xa @ Xb )
!= ( product_Pair @ A @ B @ Za2 @ Zb ) )
=> ~ ! [A5: A,B3: B] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Xa @ Xb ) @ ( product_Pair @ A @ B @ A5 @ B3 ) ) @ R )
=> ~ ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B3 ) @ ( product_Pair @ A @ B @ Za2 @ Zb ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_3163_rtrancl__induct2,axiom,
! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R: set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ),P: A > B > $o] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R ) )
=> ( ( P @ Ax @ Ay )
=> ( ! [A5: A,B3: B,Aa3: A,Ba: B] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A5 @ B3 ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R ) )
=> ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B3 ) @ ( product_Pair @ A @ B @ Aa3 @ Ba ) ) @ R )
=> ( ( P @ A5 @ B3 )
=> ( P @ Aa3 @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_3164_rtrancl__sub__insert__rtrancl,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A ),X: product_prod @ A @ A] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_rtrancl @ A @ R5 ) @ ( transitive_rtrancl @ A @ ( insert2 @ ( product_prod @ A @ A ) @ X @ R5 ) ) ) ).
% rtrancl_sub_insert_rtrancl
thf(fact_3165_plus__integer_Otransfer,axiom,
bNF_rel_fun @ int @ code_integer @ ( int > int ) @ ( 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
thf(fact_3166_quotient__of__eq,axiom,
! [A3: int,B2: int,P5: int,Q3: int] :
( ( ( quotient_of @ ( fract @ A3 @ B2 ) )
= ( product_Pair @ int @ int @ P5 @ Q3 ) )
=> ( ( fract @ P5 @ Q3 )
= ( fract @ A3 @ B2 ) ) ) ).
% quotient_of_eq
thf(fact_3167_less__integer_Otransfer,axiom,
( bNF_rel_fun @ int @ code_integer @ ( int > $o ) @ ( code_integer > $o ) @ code_pcr_integer
@ ( bNF_rel_fun @ int @ code_integer @ $o @ $o @ code_pcr_integer
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 )
@ ( ord_less @ int )
@ ( ord_less @ code_integer ) ) ).
% less_integer.transfer
thf(fact_3168_one__integer_Otransfer,axiom,
code_pcr_integer @ ( one_one @ int ) @ ( one_one @ code_integer ) ).
% one_integer.transfer
thf(fact_3169_less__eq__integer_Otransfer,axiom,
( bNF_rel_fun @ int @ code_integer @ ( int > $o ) @ ( code_integer > $o ) @ code_pcr_integer
@ ( bNF_rel_fun @ int @ code_integer @ $o @ $o @ code_pcr_integer
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 )
@ ( ord_less_eq @ int )
@ ( ord_less_eq @ code_integer ) ) ).
% less_eq_integer.transfer
thf(fact_3170_uminus__integer_Otransfer,axiom,
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
thf(fact_3171_times__integer_Otransfer,axiom,
bNF_rel_fun @ int @ code_integer @ ( int > int ) @ ( 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
thf(fact_3172_minus__integer_Otransfer,axiom,
bNF_rel_fun @ int @ code_integer @ ( int > int ) @ ( 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
thf(fact_3173_abs__integer_Otransfer,axiom,
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
thf(fact_3174_nat__of__integer_Otransfer,axiom,
( bNF_rel_fun @ int @ code_integer @ nat @ nat @ code_pcr_integer
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ nat2
@ code_nat_of_integer ) ).
% nat_of_integer.transfer
thf(fact_3175_normalize__eq,axiom,
! [A3: int,B2: int,P5: int,Q3: int] :
( ( ( normalize @ ( product_Pair @ int @ int @ A3 @ B2 ) )
= ( product_Pair @ int @ int @ P5 @ Q3 ) )
=> ( ( fract @ P5 @ Q3 )
= ( fract @ A3 @ B2 ) ) ) ).
% normalize_eq
thf(fact_3176_integer__of__nat_Otransfer,axiom,
( bNF_rel_fun @ nat @ nat @ int @ code_integer
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ code_pcr_integer
@ ( semiring_1_of_nat @ int )
@ code_integer_of_nat ) ).
% integer_of_nat.transfer
thf(fact_3177_divide__integer_Otransfer,axiom,
bNF_rel_fun @ int @ code_integer @ ( int > int ) @ ( 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
thf(fact_3178_modulo__integer_Otransfer,axiom,
bNF_rel_fun @ int @ code_integer @ ( int > int ) @ ( 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
thf(fact_3179_sgn__integer_Otransfer,axiom,
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
thf(fact_3180_dup_Otransfer,axiom,
( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer
@ ^ [K4: int] : ( plus_plus @ int @ K4 @ K4 )
@ code_dup ) ).
% dup.transfer
thf(fact_3181_take__bit__integer_Otransfer,axiom,
( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer )
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ( 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
thf(fact_3182_not__integer_Otransfer,axiom,
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
thf(fact_3183_and__integer_Otransfer,axiom,
bNF_rel_fun @ int @ code_integer @ ( int > int ) @ ( 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
thf(fact_3184_bit__integer_Otransfer,axiom,
( bNF_rel_fun @ int @ code_integer @ ( nat > $o ) @ ( nat > $o ) @ code_pcr_integer
@ ^ [Y5: nat > $o,Z4: nat > $o] : Y5 = Z4
@ ( bit_se5641148757651400278ts_bit @ int )
@ ( bit_se5641148757651400278ts_bit @ code_integer ) ) ).
% bit_integer.transfer
thf(fact_3185_or__integer_Otransfer,axiom,
bNF_rel_fun @ int @ code_integer @ ( int > int ) @ ( 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
thf(fact_3186_xor__integer_Otransfer,axiom,
bNF_rel_fun @ int @ code_integer @ ( int > int ) @ ( 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
thf(fact_3187_drop__bit__integer_Otransfer,axiom,
( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer )
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ( 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
thf(fact_3188_push__bit__integer_Otransfer,axiom,
( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer )
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ( 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
thf(fact_3189_mask__integer_Otransfer,axiom,
( bNF_rel_fun @ nat @ nat @ int @ code_integer
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ code_pcr_integer
@ ( bit_se2239418461657761734s_mask @ int )
@ ( bit_se2239418461657761734s_mask @ code_integer ) ) ).
% mask_integer.transfer
thf(fact_3190_unset__bit__integer_Otransfer,axiom,
( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer )
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ( 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
thf(fact_3191_set__bit__integer_Otransfer,axiom,
( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer )
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ( 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
thf(fact_3192_division__segment__integer_Otransfer,axiom,
bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer @ ( euclid7384307370059645450egment @ int ) @ ( euclid7384307370059645450egment @ code_integer ) ).
% division_segment_integer.transfer
thf(fact_3193_flip__bit__integer_Otransfer,axiom,
( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer )
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ( 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
thf(fact_3194_euclidean__size__integer_Otransfer,axiom,
( bNF_rel_fun @ int @ code_integer @ nat @ nat @ code_pcr_integer
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ( euclid6346220572633701492n_size @ int )
@ ( euclid6346220572633701492n_size @ code_integer ) ) ).
% euclidean_size_integer.transfer
thf(fact_3195_trancl__subset__Sigma__aux,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),A4: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu: A] : A4 ) )
=> ( ( A3 = B2 )
| ( member @ A @ A3 @ A4 ) ) ) ) ).
% trancl_subset_Sigma_aux
thf(fact_3196_Restr__rtrancl__mono,axiom,
! [A: $tType,V: A,W: A,E4: set @ ( product_prod @ A @ A ),U4: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W )
@ ( transitive_rtrancl @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ E4
@ ( product_Sigma @ A @ A @ U4
@ ^ [Uu: A] : U4 ) ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W ) @ ( transitive_rtrancl @ A @ E4 ) ) ) ).
% Restr_rtrancl_mono
thf(fact_3197_quotient__of__Fract,axiom,
! [A3: int,B2: int] :
( ( quotient_of @ ( fract @ A3 @ B2 ) )
= ( normalize @ ( product_Pair @ int @ int @ A3 @ B2 ) ) ) ).
% quotient_of_Fract
thf(fact_3198_sub_Otransfer,axiom,
( bNF_rel_fun @ num @ num @ ( num > int ) @ ( num > code_integer )
@ ^ [Y5: num,Z4: num] : Y5 = Z4
@ ( bNF_rel_fun @ num @ num @ int @ code_integer
@ ^ [Y5: num,Z4: num] : Y5 = Z4
@ code_pcr_integer )
@ ^ [M: num,N5: num] : ( minus_minus @ int @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N5 ) )
@ code_sub ) ).
% sub.transfer
thf(fact_3199_rtrancl__mapI,axiom,
! [B: $tType,A: $tType,A3: A,B2: A,E4: set @ ( product_prod @ A @ A ),F: A > B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ E4 ) )
=> ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F @ A3 ) @ ( F @ B2 ) ) @ ( transitive_rtrancl @ B @ ( image2 @ ( product_prod @ A @ A ) @ ( product_prod @ B @ B ) @ ( pairself @ A @ B @ F ) @ E4 ) ) ) ) ).
% rtrancl_mapI
thf(fact_3200_Fract_Oabs__eq,axiom,
( fract
= ( ^ [Xa4: int,X3: int] :
( abs_Rat
@ ( if @ ( product_prod @ int @ int )
@ ( X3
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ Xa4 @ X3 ) ) ) ) ) ).
% Fract.abs_eq
thf(fact_3201_rtrancl__last__touch,axiom,
! [A: $tType,Q3: A,Q4: A,R5: set @ ( product_prod @ A @ A ),S2: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q3 @ Q4 ) @ ( transitive_rtrancl @ A @ R5 ) )
=> ( ( member @ A @ Q3 @ S2 )
=> ~ ! [Qt: A] :
( ( member @ A @ Qt @ S2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q3 @ Qt ) @ ( transitive_rtrancl @ A @ R5 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Qt @ Q4 )
@ ( transitive_rtrancl @ A
@ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R5
@ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
@ ^ [Uu: A] : S2 ) ) ) ) ) ) ) ) ).
% rtrancl_last_touch
thf(fact_3202_rtrancl__last__visit_H,axiom,
! [A: $tType,Q3: A,Q4: A,R5: set @ ( product_prod @ A @ A ),S2: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q3 @ Q4 ) @ ( transitive_rtrancl @ A @ R5 ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q3 @ Q4 )
@ ( transitive_rtrancl @ A
@ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R5
@ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
@ ^ [Uu: A] : S2 ) ) ) )
=> ~ ! [Qt: A] :
( ( member @ A @ Qt @ S2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q3 @ Qt ) @ ( transitive_rtrancl @ A @ R5 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Qt @ Q4 )
@ ( transitive_rtrancl @ A
@ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R5
@ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
@ ^ [Uu: A] : S2 ) ) ) ) ) ) ) ) ).
% rtrancl_last_visit'
thf(fact_3203_rtrancl__insert,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A )] :
( ( transitive_rtrancl @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_rtrancl @ A @ R )
@ ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X3: A,Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ A3 ) @ ( transitive_rtrancl @ A @ R ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ Y3 ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ) ) ) ).
% rtrancl_insert
thf(fact_3204_Fract_Otransfer,axiom,
( bNF_rel_fun @ int @ int @ ( int > ( product_prod @ int @ int ) ) @ ( int > rat )
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ ( bNF_rel_fun @ int @ int @ ( product_prod @ int @ int ) @ rat
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ pcr_rat )
@ ^ [A7: int,B5: int] :
( if @ ( product_prod @ int @ int )
@ ( B5
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ A7 @ B5 ) )
@ fract ) ).
% Fract.transfer
thf(fact_3205_Fract__add__one,axiom,
! [N: int,M2: int] :
( ( N
!= ( zero_zero @ int ) )
=> ( ( fract @ ( plus_plus @ int @ M2 @ N ) @ N )
= ( plus_plus @ rat @ ( fract @ M2 @ N ) @ ( one_one @ rat ) ) ) ) ).
% Fract_add_one
thf(fact_3206_Fract__def,axiom,
( fract
= ( map_fun @ int @ int @ ( int > ( product_prod @ int @ int ) ) @ ( int > rat ) @ ( id @ int ) @ ( map_fun @ int @ int @ ( product_prod @ int @ int ) @ rat @ ( id @ int ) @ abs_Rat )
@ ^ [A7: int,B5: int] :
( if @ ( product_prod @ int @ int )
@ ( B5
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ A7 @ B5 ) ) ) ) ).
% Fract_def
thf(fact_3207_old_Orec__unit__def,axiom,
! [T: $tType] :
( ( product_rec_unit @ T )
= ( ^ [F12: T,X3: product_unit] : ( the @ T @ ( product_rec_set_unit @ T @ F12 @ X3 ) ) ) ) ).
% old.rec_unit_def
thf(fact_3208_old_Orec__prod__def,axiom,
! [T: $tType,B: $tType,A: $tType] :
( ( product_rec_prod @ A @ B @ T )
= ( ^ [F12: A > B > T,X3: product_prod @ A @ B] : ( the @ T @ ( product_rec_set_prod @ A @ B @ T @ F12 @ X3 ) ) ) ) ).
% old.rec_prod_def
thf(fact_3209_list__decode_Opinduct,axiom,
! [A0: nat,P: nat > $o] :
( ( accp @ nat @ nat_list_decode_rel @ A0 )
=> ( ( ( accp @ nat @ nat_list_decode_rel @ ( zero_zero @ nat ) )
=> ( P @ ( zero_zero @ nat ) ) )
=> ( ! [N3: nat] :
( ( accp @ nat @ nat_list_decode_rel @ ( suc @ N3 ) )
=> ( ! [X7: nat,Y6: nat] :
( ( ( product_Pair @ nat @ nat @ X7 @ Y6 )
= ( nat_prod_decode @ N3 ) )
=> ( P @ Y6 ) )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% list_decode.pinduct
thf(fact_3210_rtrancl__restrictI,axiom,
! [A: $tType,U: A,V: A,E4: set @ ( product_prod @ A @ A ),R5: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V )
@ ( transitive_rtrancl @ A
@ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ E4
@ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
@ ^ [Uu: A] : R5 ) ) ) )
=> ( ~ ( member @ A @ U @ R5 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V ) @ ( transitive_rtrancl @ A @ ( rel_restrict @ A @ E4 @ R5 ) ) ) ) ) ).
% rtrancl_restrictI
thf(fact_3211_inj__on__vimage__singleton,axiom,
! [B: $tType,A: $tType,F: A > B,A4: set @ A,A3: B] :
( ( inj_on @ A @ B @ F @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) @ A4 )
@ ( insert2 @ A
@ ( the @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ A4 )
& ( ( F @ X3 )
= A3 ) ) )
@ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% inj_on_vimage_singleton
thf(fact_3212_inj__on__empty,axiom,
! [B: $tType,A: $tType,F: A > B] : ( inj_on @ A @ B @ F @ ( bot_bot @ ( set @ A ) ) ) ).
% inj_on_empty
thf(fact_3213_inj__uminus,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: set @ A] : ( inj_on @ A @ A @ ( uminus_uminus @ A ) @ A4 ) ) ).
% inj_uminus
thf(fact_3214_rel__restrict__empty,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
( ( rel_restrict @ A @ R5 @ ( bot_bot @ ( set @ A ) ) )
= R5 ) ).
% rel_restrict_empty
thf(fact_3215_The__split__eq,axiom,
! [A: $tType,B: $tType,X: A,Y: B] :
( ( the @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X9: A,Y8: B] :
( ( X = X9 )
& ( Y = Y8 ) ) ) )
= ( product_Pair @ A @ B @ X @ Y ) ) ).
% The_split_eq
thf(fact_3216_inj__apfst,axiom,
! [B: $tType,C: $tType,A: $tType,F: A > C] :
( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ C @ B ) @ ( product_apfst @ A @ C @ B @ F ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( inj_on @ A @ C @ F @ ( top_top @ ( set @ A ) ) ) ) ).
% inj_apfst
thf(fact_3217_inj__apsnd,axiom,
! [A: $tType,C: $tType,B: $tType,F: B > C] :
( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( inj_on @ B @ C @ F @ ( top_top @ ( set @ B ) ) ) ) ).
% inj_apsnd
thf(fact_3218_inj__on__apfst,axiom,
! [B: $tType,C: $tType,A: $tType,F: A > C,A4: set @ A] :
( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ C @ B ) @ ( product_apfst @ A @ C @ B @ F )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : ( top_top @ ( set @ B ) ) ) )
= ( inj_on @ A @ C @ F @ A4 ) ) ).
% inj_on_apfst
thf(fact_3219_inj__on__apsnd,axiom,
! [A: $tType,C: $tType,B: $tType,F: B > C,A4: set @ B] :
( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F )
@ ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
@ ^ [Uu: A] : A4 ) )
= ( inj_on @ B @ C @ F @ A4 ) ) ).
% inj_on_apsnd
thf(fact_3220_inj__on__insert,axiom,
! [B: $tType,A: $tType,F: A > B,A3: A,A4: set @ A] :
( ( inj_on @ A @ B @ F @ ( insert2 @ A @ A3 @ A4 ) )
= ( ( inj_on @ A @ B @ F @ A4 )
& ~ ( member @ B @ ( F @ A3 ) @ ( image2 @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_3221_inj__on__Int,axiom,
! [B: $tType,A: $tType,F: A > B,A4: set @ A,B6: set @ A] :
( ( ( inj_on @ A @ B @ F @ A4 )
| ( inj_on @ A @ B @ F @ B6 ) )
=> ( inj_on @ A @ B @ F @ ( inf_inf @ ( set @ A ) @ A4 @ B6 ) ) ) ).
% inj_on_Int
thf(fact_3222_fun_Oinj__map,axiom,
! [B: $tType,A: $tType,D: $tType,F: A > B] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( inj_on @ ( D > A ) @ ( D > B ) @ ( comp @ A @ B @ D @ F ) @ ( top_top @ ( set @ ( D > A ) ) ) ) ) ).
% fun.inj_map
thf(fact_3223_inj__on__imageI2,axiom,
! [B: $tType,C: $tType,A: $tType,F3: C > B,F: A > C,A4: set @ A] :
( ( inj_on @ A @ B @ ( comp @ C @ B @ A @ F3 @ F ) @ A4 )
=> ( inj_on @ A @ C @ F @ A4 ) ) ).
% inj_on_imageI2
thf(fact_3224_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
thf(fact_3225_inj__on__add,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A3: A,A4: set @ A] : ( inj_on @ A @ A @ ( plus_plus @ A @ A3 ) @ A4 ) ) ).
% inj_on_add
thf(fact_3226_inj__on__image__iff,axiom,
! [B: $tType,A: $tType,A4: set @ A,G: A > B,F: A > A] :
( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ! [Xa3: A] :
( ( member @ A @ Xa3 @ A4 )
=> ( ( ( G @ ( F @ X2 ) )
= ( G @ ( F @ Xa3 ) ) )
= ( ( G @ X2 )
= ( G @ Xa3 ) ) ) ) )
=> ( ( inj_on @ A @ A @ F @ A4 )
=> ( ( inj_on @ A @ B @ G @ ( image2 @ A @ A @ F @ A4 ) )
= ( inj_on @ A @ B @ G @ A4 ) ) ) ) ).
% inj_on_image_iff
thf(fact_3227_inj__on__id2,axiom,
! [A: $tType,A4: set @ A] :
( inj_on @ A @ A
@ ^ [X3: A] : X3
@ A4 ) ).
% inj_on_id2
thf(fact_3228_inj__on__inverseI,axiom,
! [B: $tType,A: $tType,A4: set @ A,G: B > A,F: A > B] :
( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( ( G @ ( F @ X2 ) )
= X2 ) )
=> ( inj_on @ A @ B @ F @ A4 ) ) ).
% inj_on_inverseI
thf(fact_3229_inj__on__contraD,axiom,
! [B: $tType,A: $tType,F: A > B,A4: set @ A,X: A,Y: A] :
( ( inj_on @ A @ B @ F @ A4 )
=> ( ( X != Y )
=> ( ( member @ A @ X @ A4 )
=> ( ( member @ A @ Y @ A4 )
=> ( ( F @ X )
!= ( F @ Y ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_3230_inj__on__eq__iff,axiom,
! [B: $tType,A: $tType,F: A > B,A4: set @ A,X: A,Y: A] :
( ( inj_on @ A @ B @ F @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( member @ A @ Y @ A4 )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_3231_inj__on__cong,axiom,
! [B: $tType,A: $tType,A4: set @ A,F: A > B,G: A > B] :
( ! [A5: A] :
( ( member @ A @ A5 @ A4 )
=> ( ( F @ A5 )
= ( G @ A5 ) ) )
=> ( ( inj_on @ A @ B @ F @ A4 )
= ( inj_on @ A @ B @ G @ A4 ) ) ) ).
% inj_on_cong
thf(fact_3232_inj__on__def,axiom,
! [B: $tType,A: $tType] :
( ( inj_on @ A @ B )
= ( ^ [F2: A > B,A9: set @ A] :
! [X3: A] :
( ( member @ A @ X3 @ A9 )
=> ! [Y3: A] :
( ( member @ A @ Y3 @ A9 )
=> ( ( ( F2 @ X3 )
= ( F2 @ Y3 ) )
=> ( X3 = Y3 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_3233_inj__onI,axiom,
! [B: $tType,A: $tType,A4: set @ A,F: A > B] :
( ! [X2: A,Y4: A] :
( ( member @ A @ X2 @ A4 )
=> ( ( member @ A @ Y4 @ A4 )
=> ( ( ( F @ X2 )
= ( F @ Y4 ) )
=> ( X2 = Y4 ) ) ) )
=> ( inj_on @ A @ B @ F @ A4 ) ) ).
% inj_onI
thf(fact_3234_inj__onD,axiom,
! [B: $tType,A: $tType,F: A > B,A4: set @ A,X: A,Y: A] :
( ( inj_on @ A @ B @ F @ A4 )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member @ A @ X @ A4 )
=> ( ( member @ A @ Y @ A4 )
=> ( X = Y ) ) ) ) ) ).
% inj_onD
thf(fact_3235_inj__on__diff,axiom,
! [B: $tType,A: $tType,F: A > B,A4: set @ A,B6: set @ A] :
( ( inj_on @ A @ B @ F @ A4 )
=> ( inj_on @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ A4 @ B6 ) ) ) ).
% inj_on_diff
thf(fact_3236_inj__on__id,axiom,
! [A: $tType,A4: set @ A] : ( inj_on @ A @ A @ ( id @ A ) @ A4 ) ).
% inj_on_id
thf(fact_3237_inj__Some,axiom,
! [A: $tType,A4: set @ A] : ( inj_on @ A @ ( option @ A ) @ ( some @ A ) @ A4 ) ).
% inj_Some
thf(fact_3238_inj__on__subset,axiom,
! [B: $tType,A: $tType,F: A > B,A4: set @ A,B6: set @ A] :
( ( inj_on @ A @ B @ F @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ A4 )
=> ( inj_on @ A @ B @ F @ B6 ) ) ) ).
% inj_on_subset
thf(fact_3239_subset__inj__on,axiom,
! [B: $tType,A: $tType,F: A > B,B6: set @ A,A4: set @ A] :
( ( inj_on @ A @ B @ F @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B6 )
=> ( inj_on @ A @ B @ F @ A4 ) ) ) ).
% subset_inj_on
thf(fact_3240_inj__fun,axiom,
! [B: $tType,C: $tType,A: $tType,F: A > B] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( inj_on @ A @ ( C > B )
@ ^ [X3: A,Y3: C] : ( F @ X3 )
@ ( top_top @ ( set @ A ) ) ) ) ).
% inj_fun
thf(fact_3241_inj__def,axiom,
! [B: $tType,A: $tType,F: A > B] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
= ( ! [X3: A,Y3: A] :
( ( ( F @ X3 )
= ( F @ Y3 ) )
=> ( X3 = Y3 ) ) ) ) ).
% inj_def
thf(fact_3242_inj__eq,axiom,
! [B: $tType,A: $tType,F: A > B,X: A,Y: A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ).
% inj_eq
thf(fact_3243_injI,axiom,
! [B: $tType,A: $tType,F: A > B] :
( ! [X2: A,Y4: A] :
( ( ( F @ X2 )
= ( F @ Y4 ) )
=> ( X2 = Y4 ) )
=> ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) ) ) ).
% injI
thf(fact_3244_injD,axiom,
! [B: $tType,A: $tType,F: A > B,X: A,Y: A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( X = Y ) ) ) ).
% injD
thf(fact_3245_inj__swap,axiom,
! [B: $tType,A: $tType,A4: set @ ( product_prod @ A @ B )] : ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A ) @ ( product_swap @ A @ B ) @ A4 ) ).
% inj_swap
thf(fact_3246_inj__on__convol__ident,axiom,
! [B: $tType,A: $tType,F: A > B,X10: set @ A] :
( inj_on @ A @ ( product_prod @ A @ B )
@ ^ [X3: A] : ( product_Pair @ A @ B @ X3 @ ( F @ X3 ) )
@ X10 ) ).
% inj_on_convol_ident
thf(fact_3247_inj__Pair_I1_J,axiom,
! [B: $tType,A: $tType,C2: A > B,S2: set @ A] :
( inj_on @ A @ ( product_prod @ A @ B )
@ ^ [X3: A] : ( product_Pair @ A @ B @ X3 @ ( C2 @ X3 ) )
@ S2 ) ).
% inj_Pair(1)
thf(fact_3248_inj__Pair_I2_J,axiom,
! [B: $tType,A: $tType,C2: A > B,S2: set @ A] :
( inj_on @ A @ ( product_prod @ B @ A )
@ ^ [X3: A] : ( product_Pair @ B @ A @ ( C2 @ X3 ) @ X3 )
@ S2 ) ).
% inj_Pair(2)
thf(fact_3249_inj__on__add_H,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A3: A,A4: set @ A] :
( inj_on @ A @ A
@ ^ [B5: A] : ( plus_plus @ A @ B5 @ A3 )
@ A4 ) ) ).
% inj_on_add'
thf(fact_3250_rel__restrict__lift,axiom,
! [A: $tType,X: A,Y: A,E4: set @ ( product_prod @ A @ A ),R5: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( rel_restrict @ A @ E4 @ R5 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ E4 ) ) ).
% rel_restrict_lift
thf(fact_3251_rel__restrictI,axiom,
! [A: $tType,X: A,R5: set @ A,Y: A,E4: set @ ( product_prod @ A @ A )] :
( ~ ( member @ A @ X @ R5 )
=> ( ~ ( member @ A @ Y @ R5 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ E4 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( rel_restrict @ A @ E4 @ R5 ) ) ) ) ) ).
% rel_restrictI
thf(fact_3252_rel__restrict__notR_I1_J,axiom,
! [A: $tType,X: A,Y: A,A4: set @ ( product_prod @ A @ A ),R5: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( rel_restrict @ A @ A4 @ R5 ) )
=> ~ ( member @ A @ X @ R5 ) ) ).
% rel_restrict_notR(1)
thf(fact_3253_rel__restrict__notR_I2_J,axiom,
! [A: $tType,X: A,Y: A,A4: set @ ( product_prod @ A @ A ),R5: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( rel_restrict @ A @ A4 @ R5 ) )
=> ~ ( member @ A @ Y @ R5 ) ) ).
% rel_restrict_notR(2)
thf(fact_3254_rel__restrict__union,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A ),A4: set @ A,B6: set @ A] :
( ( rel_restrict @ A @ R5 @ ( sup_sup @ ( set @ A ) @ A4 @ B6 ) )
= ( rel_restrict @ A @ ( rel_restrict @ A @ R5 @ A4 ) @ B6 ) ) ).
% rel_restrict_union
thf(fact_3255_rel__restrict__sub,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A ),A4: set @ A] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( rel_restrict @ A @ R5 @ A4 ) @ R5 ) ).
% rel_restrict_sub
thf(fact_3256_rel__restrict__mono,axiom,
! [A: $tType,A4: set @ ( product_prod @ A @ A ),B6: set @ ( product_prod @ A @ A ),R5: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ A4 @ B6 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( rel_restrict @ A @ A4 @ R5 ) @ ( rel_restrict @ A @ B6 @ R5 ) ) ) ).
% rel_restrict_mono
thf(fact_3257_finite__rel__restrict,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A ),A4: set @ A] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R5 )
=> ( finite_finite2 @ ( product_prod @ A @ A ) @ ( rel_restrict @ A @ R5 @ A4 ) ) ) ).
% finite_rel_restrict
thf(fact_3258_option_Oinj__map,axiom,
! [B: $tType,A: $tType,F: A > B] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( inj_on @ ( option @ A ) @ ( option @ B ) @ ( map_option @ A @ B @ F ) @ ( top_top @ ( set @ ( option @ A ) ) ) ) ) ).
% option.inj_map
thf(fact_3259_linorder__inj__onI,axiom,
! [B: $tType,A: $tType] :
( ( order @ A )
=> ! [A4: set @ A,F: A > B] :
( ! [X2: A,Y4: A] :
( ( ord_less @ A @ X2 @ Y4 )
=> ( ( member @ A @ X2 @ A4 )
=> ( ( member @ A @ Y4 @ A4 )
=> ( ( F @ X2 )
!= ( F @ Y4 ) ) ) ) )
=> ( ! [X2: A,Y4: A] :
( ( member @ A @ X2 @ A4 )
=> ( ( member @ A @ Y4 @ A4 )
=> ( ( ord_less_eq @ A @ X2 @ Y4 )
| ( ord_less_eq @ A @ Y4 @ X2 ) ) ) )
=> ( inj_on @ A @ B @ F @ A4 ) ) ) ) ).
% linorder_inj_onI
thf(fact_3260_linorder__injI,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [F: A > B] :
( ! [X2: A,Y4: A] :
( ( ord_less @ A @ X2 @ Y4 )
=> ( ( F @ X2 )
!= ( F @ Y4 ) ) )
=> ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) ) ) ) ).
% linorder_injI
thf(fact_3261_inj__add__left,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A3: A] : ( inj_on @ A @ A @ ( plus_plus @ A @ A3 ) @ ( top_top @ ( set @ A ) ) ) ) ).
% inj_add_left
thf(fact_3262_inj__on__image__eq__iff,axiom,
! [B: $tType,A: $tType,F: A > B,C5: set @ A,A4: set @ A,B6: set @ A] :
( ( inj_on @ A @ B @ F @ C5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ C5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ C5 )
=> ( ( ( image2 @ A @ B @ F @ A4 )
= ( image2 @ A @ B @ F @ B6 ) )
= ( A4 = B6 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_3263_inj__on__image__mem__iff,axiom,
! [B: $tType,A: $tType,F: A > B,B6: set @ A,A3: A,A4: set @ A] :
( ( inj_on @ A @ B @ F @ B6 )
=> ( ( member @ A @ A3 @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B6 )
=> ( ( member @ B @ ( F @ A3 ) @ ( image2 @ A @ B @ F @ A4 ) )
= ( member @ A @ A3 @ A4 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_3264_inj__img__insertE,axiom,
! [B: $tType,A: $tType,F: A > B,A4: set @ A,X: B,B6: set @ B] :
( ( inj_on @ A @ B @ F @ A4 )
=> ( ~ ( member @ B @ X @ B6 )
=> ( ( ( insert2 @ B @ X @ B6 )
= ( image2 @ A @ B @ F @ A4 ) )
=> ~ ! [X5: A,A11: set @ A] :
( ~ ( member @ A @ X5 @ A11 )
=> ( ( A4
= ( insert2 @ A @ X5 @ A11 ) )
=> ( ( X
= ( F @ X5 ) )
=> ( B6
!= ( image2 @ A @ B @ F @ A11 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_3265_range__ex1__eq,axiom,
! [B: $tType,A: $tType,F: A > B,B2: B] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( member @ B @ B2 @ ( image2 @ A @ B @ F @ ( top_top @ ( set @ A ) ) ) )
= ( ? [X3: A] :
( ( B2
= ( F @ X3 ) )
& ! [Y3: A] :
( ( B2
= ( F @ Y3 ) )
=> ( Y3 = X3 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_3266_inj__image__eq__iff,axiom,
! [B: $tType,A: $tType,F: A > B,A4: set @ A,B6: set @ A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( ( image2 @ A @ B @ F @ A4 )
= ( image2 @ A @ B @ F @ B6 ) )
= ( A4 = B6 ) ) ) ).
% inj_image_eq_iff
thf(fact_3267_inj__image__mem__iff,axiom,
! [B: $tType,A: $tType,F: A > B,A3: A,A4: set @ A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( member @ B @ ( F @ A3 ) @ ( image2 @ A @ B @ F @ A4 ) )
= ( member @ A @ A3 @ A4 ) ) ) ).
% inj_image_mem_iff
thf(fact_3268_comp__inj__on,axiom,
! [B: $tType,C: $tType,A: $tType,F: A > B,A4: set @ A,G: B > C] :
( ( inj_on @ A @ B @ F @ A4 )
=> ( ( inj_on @ B @ C @ G @ ( image2 @ A @ B @ F @ A4 ) )
=> ( inj_on @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ A4 ) ) ) ).
% comp_inj_on
thf(fact_3269_inj__on__imageI,axiom,
! [B: $tType,C: $tType,A: $tType,G: C > B,F: A > C,A4: set @ A] :
( ( inj_on @ A @ B @ ( comp @ C @ B @ A @ G @ F ) @ A4 )
=> ( inj_on @ C @ B @ G @ ( image2 @ A @ C @ F @ A4 ) ) ) ).
% inj_on_imageI
thf(fact_3270_comp__inj__on__iff,axiom,
! [B: $tType,C: $tType,A: $tType,F: A > B,A4: set @ A,F3: B > C] :
( ( inj_on @ A @ B @ F @ A4 )
=> ( ( inj_on @ B @ C @ F3 @ ( image2 @ A @ B @ F @ A4 ) )
= ( inj_on @ A @ C @ ( comp @ B @ C @ A @ F3 @ F ) @ A4 ) ) ) ).
% comp_inj_on_iff
thf(fact_3271_inj__on__Un__image__eq__iff,axiom,
! [B: $tType,A: $tType,F: A > B,A4: set @ A,B6: set @ A] :
( ( inj_on @ A @ B @ F @ ( sup_sup @ ( set @ A ) @ A4 @ B6 ) )
=> ( ( ( image2 @ A @ B @ F @ A4 )
= ( image2 @ A @ B @ F @ B6 ) )
= ( A4 = B6 ) ) ) ).
% inj_on_Un_image_eq_iff
thf(fact_3272_inj__compose,axiom,
! [A: $tType,B: $tType,C: $tType,F: A > B,G: C > A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( inj_on @ C @ A @ G @ ( top_top @ ( set @ C ) ) )
=> ( inj_on @ C @ B @ ( comp @ A @ B @ C @ F @ G ) @ ( top_top @ ( set @ C ) ) ) ) ) ).
% inj_compose
thf(fact_3273_inj__on__strict__subset,axiom,
! [B: $tType,A: $tType,F: A > B,B6: set @ A,A4: set @ A] :
( ( inj_on @ A @ B @ F @ B6 )
=> ( ( ord_less @ ( set @ A ) @ A4 @ B6 )
=> ( ord_less @ ( set @ B ) @ ( image2 @ A @ B @ F @ A4 ) @ ( image2 @ A @ B @ F @ B6 ) ) ) ) ).
% inj_on_strict_subset
thf(fact_3274_inj__on__fun__updI,axiom,
! [B: $tType,A: $tType,F: A > B,A4: set @ A,Y: B,X: A] :
( ( inj_on @ A @ B @ F @ A4 )
=> ( ~ ( member @ B @ Y @ ( image2 @ A @ B @ F @ A4 ) )
=> ( inj_on @ A @ B @ ( fun_upd @ A @ B @ F @ X @ Y ) @ A4 ) ) ) ).
% inj_on_fun_updI
thf(fact_3275_inj__on__fst__map__to__set,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B )] : ( inj_on @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( map_to_set @ A @ B @ M2 ) ) ).
% inj_on_fst_map_to_set
thf(fact_3276_inj__diff__right,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A] :
( inj_on @ A @ A
@ ^ [B5: A] : ( minus_minus @ A @ B5 @ A3 )
@ ( top_top @ ( set @ A ) ) ) ) ).
% inj_diff_right
thf(fact_3277_inj__singleton,axiom,
! [A: $tType,A4: set @ A] :
( inj_on @ A @ ( set @ A )
@ ^ [X3: A] : ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) )
@ A4 ) ).
% inj_singleton
thf(fact_3278_finite__Collect,axiom,
! [A: $tType,B: $tType,S2: set @ A,F: B > A] :
( ( finite_finite2 @ A @ S2 )
=> ( ( inj_on @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [A7: B] : ( member @ A @ ( F @ A7 ) @ S2 ) ) ) ) ) ).
% finite_Collect
thf(fact_3279_swap__inj__on,axiom,
! [B: $tType,A: $tType,A4: set @ ( product_prod @ A @ B )] :
( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A )
@ ( product_case_prod @ A @ B @ ( product_prod @ B @ A )
@ ^ [I3: A,J2: B] : ( product_Pair @ B @ A @ J2 @ I3 ) )
@ A4 ) ).
% swap_inj_on
thf(fact_3280_rel__restrict__mono2,axiom,
! [A: $tType,R5: set @ A,S2: set @ A,A4: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ A ) @ R5 @ S2 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( rel_restrict @ A @ A4 @ S2 ) @ ( rel_restrict @ A @ A4 @ R5 ) ) ) ).
% rel_restrict_mono2
thf(fact_3281_inj__image__subset__iff,axiom,
! [B: $tType,A: $tType,F: A > B,A4: set @ A,B6: set @ A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F @ A4 ) @ ( image2 @ A @ B @ F @ B6 ) )
= ( ord_less_eq @ ( set @ A ) @ A4 @ B6 ) ) ) ).
% inj_image_subset_iff
thf(fact_3282_inj__on__image__Int,axiom,
! [B: $tType,A: $tType,F: A > B,C5: set @ A,A4: set @ A,B6: set @ A] :
( ( inj_on @ A @ B @ F @ C5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ C5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ C5 )
=> ( ( image2 @ A @ B @ F @ ( inf_inf @ ( set @ A ) @ A4 @ B6 ) )
= ( inf_inf @ ( set @ B ) @ ( image2 @ A @ B @ F @ A4 ) @ ( image2 @ A @ B @ F @ B6 ) ) ) ) ) ) ).
% inj_on_image_Int
thf(fact_3283_image__Int,axiom,
! [B: $tType,A: $tType,F: A > B,A4: set @ A,B6: set @ A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( image2 @ A @ B @ F @ ( inf_inf @ ( set @ A ) @ A4 @ B6 ) )
= ( inf_inf @ ( set @ B ) @ ( image2 @ A @ B @ F @ A4 ) @ ( image2 @ A @ B @ F @ B6 ) ) ) ) ).
% image_Int
thf(fact_3284_inj__on__image__set__diff,axiom,
! [B: $tType,A: $tType,F: A > B,C5: set @ A,A4: set @ A,B6: set @ A] :
( ( inj_on @ A @ B @ F @ C5 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B6 ) @ C5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ C5 )
=> ( ( image2 @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ A4 @ B6 ) )
= ( minus_minus @ ( set @ B ) @ ( image2 @ A @ B @ F @ A4 ) @ ( image2 @ A @ B @ F @ B6 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_3285_sum_Oreindex,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [H3: B > C,A4: set @ B,G: C > A] :
( ( inj_on @ B @ C @ H3 @ A4 )
=> ( ( groups7311177749621191930dd_sum @ C @ A @ G @ ( image2 @ B @ C @ H3 @ A4 ) )
= ( groups7311177749621191930dd_sum @ B @ A @ ( comp @ C @ A @ B @ G @ H3 ) @ A4 ) ) ) ) ).
% sum.reindex
thf(fact_3286_image__set__diff,axiom,
! [B: $tType,A: $tType,F: A > B,A4: set @ A,B6: set @ A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( image2 @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ A4 @ B6 ) )
= ( minus_minus @ ( set @ B ) @ ( image2 @ A @ B @ F @ A4 ) @ ( image2 @ A @ B @ F @ B6 ) ) ) ) ).
% image_set_diff
thf(fact_3287_prod_Oreindex,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [H3: B > C,A4: set @ B,G: C > A] :
( ( inj_on @ B @ C @ H3 @ A4 )
=> ( ( groups7121269368397514597t_prod @ C @ A @ G @ ( image2 @ B @ C @ H3 @ A4 ) )
= ( groups7121269368397514597t_prod @ B @ A @ ( comp @ C @ A @ B @ G @ H3 ) @ A4 ) ) ) ) ).
% prod.reindex
thf(fact_3288_inj__vimage__image__eq,axiom,
! [B: $tType,A: $tType,F: A > B,A4: set @ A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( vimage @ A @ B @ F @ ( image2 @ A @ B @ F @ A4 ) )
= A4 ) ) ).
% inj_vimage_image_eq
thf(fact_3289_rel__restrict__def,axiom,
! [A: $tType] :
( ( rel_restrict @ A )
= ( ^ [R6: set @ ( product_prod @ A @ A ),A9: set @ A] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [V2: A,W2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V2 @ W2 ) @ R6 )
& ~ ( member @ A @ V2 @ A9 )
& ~ ( member @ A @ W2 @ A9 ) ) ) ) ) ) ).
% rel_restrict_def
thf(fact_3290_finite__imp__inj__to__nat__seg_H,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ~ ! [F4: A > nat] :
( ? [N3: nat] :
( ( image2 @ A @ nat @ F4 @ A4 )
= ( collect @ nat
@ ^ [I3: nat] : ( ord_less @ nat @ I3 @ N3 ) ) )
=> ~ ( inj_on @ A @ nat @ F4 @ A4 ) ) ) ).
% finite_imp_inj_to_nat_seg'
thf(fact_3291_set__to__map__inverse,axiom,
! [B: $tType,A: $tType,S2: set @ ( product_prod @ A @ B )] :
( ( inj_on @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ S2 )
=> ( ( map_to_set @ A @ B @ ( set_to_map @ A @ B @ S2 ) )
= S2 ) ) ).
% set_to_map_inverse
thf(fact_3292_rel__restrict__Int__empty,axiom,
! [A: $tType,A4: set @ A,R5: set @ ( product_prod @ A @ A )] :
( ( ( inf_inf @ ( set @ A ) @ A4 @ ( field2 @ A @ R5 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( rel_restrict @ A @ R5 @ A4 )
= R5 ) ) ).
% rel_restrict_Int_empty
thf(fact_3293_Field__rel__restrict,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A ),A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( field2 @ A @ ( rel_restrict @ A @ R5 @ A4 ) ) @ ( minus_minus @ ( set @ A ) @ ( field2 @ A @ R5 ) @ A4 ) ) ).
% Field_rel_restrict
thf(fact_3294_vimage__subsetI,axiom,
! [B: $tType,A: $tType,F: A > B,B6: set @ B,A4: set @ A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ B ) @ B6 @ ( image2 @ A @ B @ F @ A4 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( vimage @ A @ B @ F @ B6 ) @ A4 ) ) ) ).
% vimage_subsetI
thf(fact_3295_inj__image__Compl__subset,axiom,
! [B: $tType,A: $tType,F: A > B,A4: set @ A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F @ ( uminus_uminus @ ( set @ A ) @ A4 ) ) @ ( uminus_uminus @ ( set @ B ) @ ( image2 @ A @ B @ F @ A4 ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_3296_The__case__prod,axiom,
! [B: $tType,A: $tType,P: A > B > $o] :
( ( the @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P ) )
= ( the @ ( product_prod @ A @ B )
@ ^ [Xy: product_prod @ A @ B] : ( P @ ( product_fst @ A @ B @ Xy ) @ ( product_snd @ A @ B @ Xy ) ) ) ) ).
% The_case_prod
thf(fact_3297_inj__on__disjoint__Un,axiom,
! [B: $tType,A: $tType,F: A > B,A4: set @ A,G: A > B,B6: set @ A] :
( ( inj_on @ A @ B @ F @ A4 )
=> ( ( inj_on @ A @ B @ G @ B6 )
=> ( ( ( inf_inf @ ( set @ B ) @ ( image2 @ A @ B @ F @ A4 ) @ ( image2 @ A @ B @ G @ B6 ) )
= ( bot_bot @ ( set @ B ) ) )
=> ( inj_on @ A @ B
@ ^ [X3: A] : ( if @ B @ ( member @ A @ X3 @ A4 ) @ ( F @ X3 ) @ ( G @ X3 ) )
@ ( sup_sup @ ( set @ A ) @ A4 @ B6 ) ) ) ) ) ).
% inj_on_disjoint_Un
thf(fact_3298_set__to__map__simp,axiom,
! [B: $tType,A: $tType,S2: set @ ( product_prod @ A @ B ),K: A,V: B] :
( ( inj_on @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ S2 )
=> ( ( ( set_to_map @ A @ B @ S2 @ K )
= ( some @ B @ V ) )
= ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V ) @ S2 ) ) ) ).
% set_to_map_simp
thf(fact_3299_rel__restrict__compl,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A ),A4: set @ A] :
( ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ ( rel_restrict @ A @ R5 @ A4 ) @ ( rel_restrict @ A @ R5 @ ( uminus_uminus @ ( set @ A ) @ A4 ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% rel_restrict_compl
thf(fact_3300_homo__rel__restrict__mono,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A ),B6: set @ A,A4: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R5
@ ( product_Sigma @ A @ A @ B6
@ ^ [Uu: A] : B6 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( rel_restrict @ A @ R5 @ A4 )
@ ( product_Sigma @ A @ A @ ( minus_minus @ ( set @ A ) @ B6 @ A4 )
@ ^ [Uu: A] : ( minus_minus @ ( set @ A ) @ B6 @ A4 ) ) ) ) ).
% homo_rel_restrict_mono
thf(fact_3301_rel__restrict__alt__def,axiom,
! [A: $tType] :
( ( rel_restrict @ A )
= ( ^ [R6: set @ ( product_prod @ A @ A ),A9: set @ A] :
( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R6
@ ( product_Sigma @ A @ A @ ( uminus_uminus @ ( set @ A ) @ A9 )
@ ^ [Uu: A] : ( uminus_uminus @ ( set @ A ) @ A9 ) ) ) ) ) ).
% rel_restrict_alt_def
thf(fact_3302_inj__on__Un,axiom,
! [A: $tType,B: $tType,F: A > B,A4: set @ A,B6: set @ A] :
( ( inj_on @ A @ B @ F @ ( sup_sup @ ( set @ A ) @ A4 @ B6 ) )
= ( ( inj_on @ A @ B @ F @ A4 )
& ( inj_on @ A @ B @ F @ B6 )
& ( ( inf_inf @ ( set @ B ) @ ( image2 @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ A4 @ B6 ) ) @ ( image2 @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ B6 @ A4 ) ) )
= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% inj_on_Un
thf(fact_3303_inj__vimage__singleton,axiom,
! [B: $tType,A: $tType,F: A > B,A3: B] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( vimage @ A @ B @ F @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) )
@ ( insert2 @ A
@ ( the @ A
@ ^ [X3: A] :
( ( F @ X3 )
= A3 ) )
@ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% inj_vimage_singleton
thf(fact_3304_If__the__inv__into__f__f,axiom,
! [B: $tType,A: $tType,I: A,C5: set @ A,G: A > B,X: A] :
( ( member @ A @ I @ C5 )
=> ( ( inj_on @ A @ B @ G @ C5 )
=> ( ( comp @ B @ A @ A
@ ^ [I3: B] : ( if @ A @ ( member @ B @ I3 @ ( image2 @ A @ B @ G @ C5 ) ) @ ( the_inv_into @ A @ B @ C5 @ G @ I3 ) @ X )
@ G
@ I )
= ( id @ A @ I ) ) ) ) ).
% If_the_inv_into_f_f
thf(fact_3305_ran__map__upd__Some,axiom,
! [B: $tType,A: $tType,M2: B > ( option @ A ),X: B,Y: A,Z2: A] :
( ( ( M2 @ X )
= ( some @ A @ Y ) )
=> ( ( inj_on @ B @ ( option @ A ) @ M2 @ ( dom @ B @ A @ M2 ) )
=> ( ~ ( member @ A @ Z2 @ ( ran @ B @ A @ M2 ) )
=> ( ( ran @ B @ A @ ( fun_upd @ B @ ( option @ A ) @ M2 @ X @ ( some @ A @ Z2 ) ) )
= ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ ( ran @ B @ A @ M2 ) @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) @ ( insert2 @ A @ Z2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% ran_map_upd_Some
thf(fact_3306_trancl__multi__insert,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),X10: set @ A,M2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 )
@ ( transitive_trancl @ A
@ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ X10
@ ^ [Uu: A] : ( insert2 @ A @ M2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R ) )
=> ~ ! [X2: A] :
( ( member @ A @ X2 @ X10 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ X2 ) @ ( transitive_rtrancl @ A @ R ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ M2 @ B2 ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ) ) ).
% trancl_multi_insert
thf(fact_3307_trancl__multi__insert2,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),M2: A,X10: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 )
@ ( transitive_trancl @ A
@ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ ( insert2 @ A @ M2 @ ( bot_bot @ ( set @ A ) ) )
@ ^ [Uu: A] : X10 ) ) ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R ) )
=> ~ ! [X2: A] :
( ( member @ A @ X2 @ X10 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ M2 ) @ ( transitive_rtrancl @ A @ R ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ B2 ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ) ) ).
% trancl_multi_insert2
thf(fact_3308_the__inv__f__o__f__id,axiom,
! [B: $tType,A: $tType,F: A > B,Z2: A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( comp @ B @ A @ A @ ( the_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F ) @ F @ Z2 )
= ( id @ A @ Z2 ) ) ) ).
% the_inv_f_o_f_id
thf(fact_3309_restrict__map__self,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B )] :
( ( restrict_map @ A @ B @ M2 @ ( dom @ A @ B @ M2 ) )
= M2 ) ).
% restrict_map_self
thf(fact_3310_same__fst__trancl,axiom,
! [B: $tType,A: $tType,P: A > $o,R5: A > ( set @ ( product_prod @ B @ B ) )] :
( ( transitive_trancl @ ( product_prod @ A @ B ) @ ( same_fst @ A @ B @ P @ R5 ) )
= ( same_fst @ A @ B @ P
@ ^ [X3: A] : ( transitive_trancl @ B @ ( R5 @ X3 ) ) ) ) ).
% same_fst_trancl
thf(fact_3311_dom__eq__empty__conv,axiom,
! [B: $tType,A: $tType,F: A > ( option @ B )] :
( ( ( dom @ A @ B @ F )
= ( bot_bot @ ( set @ A ) ) )
= ( F
= ( ^ [X3: A] : ( none @ B ) ) ) ) ).
% dom_eq_empty_conv
thf(fact_3312_fun__upd__None__if__notin__dom,axiom,
! [B: $tType,A: $tType,K: A,M2: A > ( option @ B )] :
( ~ ( member @ A @ K @ ( dom @ A @ B @ M2 ) )
=> ( ( fun_upd @ A @ ( option @ B ) @ M2 @ K @ ( none @ B ) )
= M2 ) ) ).
% fun_upd_None_if_notin_dom
thf(fact_3313_the__inv__into__onto,axiom,
! [B: $tType,A: $tType,F: A > B,A4: set @ A] :
( ( inj_on @ A @ B @ F @ A4 )
=> ( ( image2 @ B @ A @ ( the_inv_into @ A @ B @ A4 @ F ) @ ( image2 @ A @ B @ F @ A4 ) )
= A4 ) ) ).
% the_inv_into_onto
thf(fact_3314_dom__empty,axiom,
! [B: $tType,A: $tType] :
( ( dom @ A @ B
@ ^ [X3: A] : ( none @ B ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% dom_empty
thf(fact_3315_map__update__eta__repair_I1_J,axiom,
! [B: $tType,A: $tType,K: A,V: B,M2: A > ( option @ B )] :
( ( dom @ A @ B
@ ^ [X3: A] : ( if @ ( option @ B ) @ ( X3 = K ) @ ( some @ B @ V ) @ ( M2 @ X3 ) ) )
= ( insert2 @ A @ K @ ( dom @ A @ B @ M2 ) ) ) ).
% map_update_eta_repair(1)
thf(fact_3316_dom__const_H,axiom,
! [B: $tType,A: $tType,F: A > B] :
( ( dom @ A @ B
@ ^ [X3: A] : ( some @ B @ ( F @ X3 ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% dom_const'
thf(fact_3317_dom__mmupd,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B ),K2: set @ A,V: B] :
( ( dom @ A @ B @ ( map_mmupd @ A @ B @ M2 @ K2 @ V ) )
= ( sup_sup @ ( set @ A ) @ ( dom @ A @ B @ M2 ) @ K2 ) ) ).
% dom_mmupd
thf(fact_3318_trancl__single,axiom,
! [A: $tType,A3: A,B2: A] :
( ( transitive_trancl @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) )
= ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% trancl_single
thf(fact_3319_restrict__map__inv,axiom,
! [B: $tType,A: $tType,F: A > ( option @ B )] :
( ( restrict_map @ A @ B @ F @ ( uminus_uminus @ ( set @ A ) @ ( dom @ A @ B @ F ) ) )
= ( ^ [X3: A] : ( none @ B ) ) ) ).
% restrict_map_inv
thf(fact_3320_dom__fun__upd,axiom,
! [B: $tType,A: $tType,Y: option @ B,F: A > ( option @ B ),X: A] :
( ( ( Y
= ( none @ B ) )
=> ( ( dom @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F @ X @ Y ) )
= ( minus_minus @ ( set @ A ) @ ( dom @ A @ B @ F ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) )
& ( ( Y
!= ( none @ B ) )
=> ( ( dom @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F @ X @ Y ) )
= ( insert2 @ A @ X @ ( dom @ A @ B @ F ) ) ) ) ) ).
% dom_fun_upd
thf(fact_3321_domI,axiom,
! [A: $tType,B: $tType,M2: B > ( option @ A ),A3: B,B2: A] :
( ( ( M2 @ A3 )
= ( some @ A @ B2 ) )
=> ( member @ B @ A3 @ ( dom @ B @ A @ M2 ) ) ) ).
% domI
thf(fact_3322_domD,axiom,
! [A: $tType,B: $tType,A3: A,M2: A > ( option @ B )] :
( ( member @ A @ A3 @ ( dom @ A @ B @ M2 ) )
=> ? [B3: B] :
( ( M2 @ A3 )
= ( some @ B @ B3 ) ) ) ).
% domD
thf(fact_3323_domIff,axiom,
! [A: $tType,B: $tType,A3: A,M2: A > ( option @ B )] :
( ( member @ A @ A3 @ ( dom @ A @ B @ M2 ) )
= ( ( M2 @ A3 )
!= ( none @ B ) ) ) ).
% domIff
thf(fact_3324_nempty__dom,axiom,
! [B: $tType,A: $tType,E2: A > ( option @ B )] :
( ( E2
!= ( ^ [X3: A] : ( none @ B ) ) )
=> ~ ! [M5: A] :
~ ( member @ A @ M5 @ ( dom @ A @ B @ E2 ) ) ) ).
% nempty_dom
thf(fact_3325_trancl__sub,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R5 @ ( transitive_trancl @ A @ R5 ) ) ).
% trancl_sub
thf(fact_3326_trancl__mono__mp,axiom,
! [A: $tType,U4: set @ ( product_prod @ A @ A ),V6: set @ ( product_prod @ A @ A ),X: product_prod @ A @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ U4 @ V6 )
=> ( ( member @ ( product_prod @ A @ A ) @ X @ ( transitive_trancl @ A @ U4 ) )
=> ( member @ ( product_prod @ A @ A ) @ X @ ( transitive_trancl @ A @ V6 ) ) ) ) ).
% trancl_mono_mp
thf(fact_3327_trancl_Ocases,axiom,
! [A: $tType,A1: A,A22: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ ( transitive_trancl @ A @ R ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ R )
=> ~ ! [B3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ B3 ) @ ( transitive_trancl @ A @ R ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A22 ) @ R ) ) ) ) ).
% trancl.cases
thf(fact_3328_trancl_Osimps,axiom,
! [A: $tType,A1: A,A22: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ ( transitive_trancl @ A @ R ) )
= ( ? [A7: A,B5: A] :
( ( A1 = A7 )
& ( A22 = B5 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B5 ) @ R ) )
| ? [A7: A,B5: A,C3: A] :
( ( A1 = A7 )
& ( A22 = C3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B5 ) @ ( transitive_trancl @ A @ R ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ C3 ) @ R ) ) ) ) ).
% trancl.simps
thf(fact_3329_trancl_Or__into__trancl,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R ) ) ) ).
% trancl.r_into_trancl
thf(fact_3330_tranclE,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
=> ~ ! [C4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C4 ) @ ( transitive_trancl @ A @ R ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ C4 @ B2 ) @ R ) ) ) ) ).
% tranclE
thf(fact_3331_trancl__trans,axiom,
! [A: $tType,X: A,Y: A,R: set @ ( product_prod @ A @ A ),Z2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ R ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ ( transitive_trancl @ A @ R ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( transitive_trancl @ A @ R ) ) ) ) ).
% trancl_trans
thf(fact_3332_trancl__induct,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),P: A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R ) )
=> ( ! [Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ Y4 ) @ R )
=> ( P @ Y4 ) )
=> ( ! [Y4: A,Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ Y4 ) @ ( transitive_trancl @ A @ R ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z3 ) @ R )
=> ( ( P @ Y4 )
=> ( P @ Z3 ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% trancl_induct
thf(fact_3333_r__r__into__trancl,axiom,
! [A: $tType,A3: A,B2: A,R5: set @ ( product_prod @ A @ A ),C2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R5 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C2 ) @ R5 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C2 ) @ ( transitive_trancl @ A @ R5 ) ) ) ) ).
% r_r_into_trancl
thf(fact_3334_converse__tranclE,axiom,
! [A: $tType,X: A,Z2: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( transitive_trancl @ A @ R ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ R )
=> ~ ! [Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y4 ) @ R )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z2 ) @ ( transitive_trancl @ A @ R ) ) ) ) ) ).
% converse_tranclE
thf(fact_3335_irrefl__trancl__rD,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ! [X2: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ ( transitive_trancl @ A @ R ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
=> ( X != Y ) ) ) ).
% irrefl_trancl_rD
thf(fact_3336_Transitive__Closure_Otrancl__into__trancl,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),C2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C2 ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C2 ) @ ( transitive_trancl @ A @ R ) ) ) ) ).
% Transitive_Closure.trancl_into_trancl
thf(fact_3337_trancl__into__trancl2,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),C2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C2 ) @ ( transitive_trancl @ A @ R ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C2 ) @ ( transitive_trancl @ A @ R ) ) ) ) ).
% trancl_into_trancl2
thf(fact_3338_trancl__trans__induct,axiom,
! [A: $tType,X: A,Y: A,R: set @ ( product_prod @ A @ A ),P: A > A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ R ) )
=> ( ! [X2: A,Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y4 ) @ R )
=> ( P @ X2 @ Y4 ) )
=> ( ! [X2: A,Y4: A,Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y4 ) @ ( transitive_trancl @ A @ R ) )
=> ( ( P @ X2 @ Y4 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z3 ) @ ( transitive_trancl @ A @ R ) )
=> ( ( P @ Y4 @ Z3 )
=> ( P @ X2 @ Z3 ) ) ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% trancl_trans_induct
thf(fact_3339_converse__trancl__induct,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),P: A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R ) )
=> ( ! [Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ B2 ) @ R )
=> ( P @ Y4 ) )
=> ( ! [Y4: A,Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z3 ) @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z3 @ B2 ) @ ( transitive_trancl @ A @ R ) )
=> ( ( P @ Z3 )
=> ( P @ Y4 ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% converse_trancl_induct
thf(fact_3340_dom__def,axiom,
! [B: $tType,A: $tType] :
( ( dom @ A @ B )
= ( ^ [M: A > ( option @ B )] :
( collect @ A
@ ^ [A7: A] :
( ( M @ A7 )
!= ( none @ B ) ) ) ) ) ).
% dom_def
thf(fact_3341_the__inv__into__f__f,axiom,
! [B: $tType,A: $tType,F: A > B,A4: set @ A,X: A] :
( ( inj_on @ A @ B @ F @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( the_inv_into @ A @ B @ A4 @ F @ ( F @ X ) )
= X ) ) ) ).
% the_inv_into_f_f
thf(fact_3342_the__inv__into__f__eq,axiom,
! [B: $tType,A: $tType,F: A > B,A4: set @ A,X: A,Y: B] :
( ( inj_on @ A @ B @ F @ A4 )
=> ( ( ( F @ X )
= Y )
=> ( ( member @ A @ X @ A4 )
=> ( ( the_inv_into @ A @ B @ A4 @ F @ Y )
= X ) ) ) ) ).
% the_inv_into_f_eq
thf(fact_3343_insert__dom,axiom,
! [A: $tType,B: $tType,F: B > ( option @ A ),X: B,Y: A] :
( ( ( F @ X )
= ( some @ A @ Y ) )
=> ( ( insert2 @ B @ X @ ( dom @ B @ A @ F ) )
= ( dom @ B @ A @ F ) ) ) ).
% insert_dom
thf(fact_3344_trancl__rtrancl__trancl,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),C2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C2 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C2 ) @ ( transitive_trancl @ A @ R ) ) ) ) ).
% trancl_rtrancl_trancl
thf(fact_3345_rtrancl__trancl__trancl,axiom,
! [A: $tType,X: A,Y: A,R: set @ ( product_prod @ A @ A ),Z2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ ( transitive_trancl @ A @ R ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( transitive_trancl @ A @ R ) ) ) ) ).
% rtrancl_trancl_trancl
thf(fact_3346_rtrancl__into__trancl2,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),C2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C2 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C2 ) @ ( transitive_trancl @ A @ R ) ) ) ) ).
% rtrancl_into_trancl2
thf(fact_3347_rtrancl__into__trancl1,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),C2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C2 ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C2 ) @ ( transitive_trancl @ A @ R ) ) ) ) ).
% rtrancl_into_trancl1
thf(fact_3348_rtrancl__eq__or__trancl,axiom,
! [A: $tType,X: A,Y: A,R5: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R5 ) )
= ( ( X = Y )
| ( ( X != Y )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ R5 ) ) ) ) ) ).
% rtrancl_eq_or_trancl
thf(fact_3349_trancl__into__rtrancl,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R ) ) ) ).
% trancl_into_rtrancl
thf(fact_3350_tranclD2,axiom,
! [A: $tType,X: A,Y: A,R5: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ R5 ) )
=> ? [Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z3 ) @ ( transitive_rtrancl @ A @ R5 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z3 @ Y ) @ R5 ) ) ) ).
% tranclD2
thf(fact_3351_rtranclD,axiom,
! [A: $tType,A3: A,B2: A,R5: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R5 ) )
=> ( ( A3 = B2 )
| ( ( A3 != B2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R5 ) ) ) ) ) ).
% rtranclD
thf(fact_3352_tranclD,axiom,
! [A: $tType,X: A,Y: A,R5: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ R5 ) )
=> ? [Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z3 ) @ R5 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z3 @ Y ) @ ( transitive_rtrancl @ A @ R5 ) ) ) ) ).
% tranclD
thf(fact_3353_map__dom__ran__finite,axiom,
! [B: $tType,A: $tType,M3: A > ( option @ B )] :
( ( finite_finite2 @ A @ ( dom @ A @ B @ M3 ) )
=> ( finite_finite2 @ B @ ( ran @ A @ B @ M3 ) ) ) ).
% map_dom_ran_finite
thf(fact_3354_rel__restrict__trancl__notR_I2_J,axiom,
! [A: $tType,V: A,W: A,E4: set @ ( product_prod @ A @ A ),R5: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ E4 @ R5 ) ) )
=> ~ ( member @ A @ W @ R5 ) ) ).
% rel_restrict_trancl_notR(2)
thf(fact_3355_rel__restrict__trancl__notR_I1_J,axiom,
! [A: $tType,V: A,W: A,E4: set @ ( product_prod @ A @ A ),R5: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ E4 @ R5 ) ) )
=> ~ ( member @ A @ V @ R5 ) ) ).
% rel_restrict_trancl_notR(1)
thf(fact_3356_rel__restrict__trancl__mem,axiom,
! [A: $tType,A3: A,B2: A,A4: set @ ( product_prod @ A @ A ),R5: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ A4 @ R5 ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( rel_restrict @ A @ ( transitive_trancl @ A @ A4 ) @ R5 ) ) ) ).
% rel_restrict_trancl_mem
thf(fact_3357_trancl__induct2,axiom,
! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R: set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ),P: A > B > $o] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) ) @ ( transitive_trancl @ ( product_prod @ A @ B ) @ R ) )
=> ( ! [A5: A,B3: B] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A5 @ B3 ) ) @ R )
=> ( P @ A5 @ B3 ) )
=> ( ! [A5: A,B3: B,Aa3: A,Ba: B] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A5 @ B3 ) ) @ ( transitive_trancl @ ( product_prod @ A @ B ) @ R ) )
=> ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B3 ) @ ( product_Pair @ A @ B @ Aa3 @ Ba ) ) @ R )
=> ( ( P @ A5 @ B3 )
=> ( P @ Aa3 @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% trancl_induct2
thf(fact_3358_card__map__to__set,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B )] :
( ( finite_card @ ( product_prod @ A @ B ) @ ( map_to_set @ A @ B @ M2 ) )
= ( finite_card @ A @ ( dom @ A @ B @ M2 ) ) ) ).
% card_map_to_set
thf(fact_3359_trancl__sub__insert__trancl,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A ),X: product_prod @ A @ A] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R5 ) @ ( transitive_trancl @ A @ ( insert2 @ ( product_prod @ A @ A ) @ X @ R5 ) ) ) ).
% trancl_sub_insert_trancl
thf(fact_3360_rel__restrict__trancl__sub,axiom,
! [A: $tType,A4: set @ ( product_prod @ A @ A ),R5: set @ A] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ A4 @ R5 ) ) @ ( rel_restrict @ A @ ( transitive_trancl @ A @ A4 ) @ R5 ) ) ).
% rel_restrict_trancl_sub
thf(fact_3361_le__map__dom__mono,axiom,
! [B: $tType,A: $tType] :
( ( preorder @ B )
=> ! [M2: A > ( option @ B ),M7: A > ( option @ B )] :
( ( ord_less_eq @ ( A > ( option @ B ) ) @ M2 @ M7 )
=> ( ord_less_eq @ ( set @ A ) @ ( dom @ A @ B @ M2 ) @ ( dom @ A @ B @ M7 ) ) ) ) ).
% le_map_dom_mono
thf(fact_3362_the__inv__into__def,axiom,
! [B: $tType,A: $tType] :
( ( the_inv_into @ A @ B )
= ( ^ [A9: set @ A,F2: A > B,X3: B] :
( the @ A
@ ^ [Y3: A] :
( ( member @ A @ Y3 @ A9 )
& ( ( F2 @ Y3 )
= X3 ) ) ) ) ) ).
% the_inv_into_def
thf(fact_3363_finite__map__freshness,axiom,
! [A: $tType,B: $tType,F: A > ( option @ B )] :
( ( finite_finite2 @ A @ ( dom @ A @ B @ F ) )
=> ( ~ ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
=> ? [X2: A] :
( ( F @ X2 )
= ( none @ B ) ) ) ) ).
% finite_map_freshness
thf(fact_3364_dom__minus,axiom,
! [A: $tType,B: $tType,F: B > ( option @ A ),X: B,A4: set @ B] :
( ( ( F @ X )
= ( none @ A ) )
=> ( ( minus_minus @ ( set @ B ) @ ( dom @ B @ A @ F ) @ ( insert2 @ B @ X @ A4 ) )
= ( minus_minus @ ( set @ B ) @ ( dom @ B @ A @ F ) @ A4 ) ) ) ).
% dom_minus
thf(fact_3365_restrict__map__eq_I1_J,axiom,
! [A: $tType,B: $tType,M2: B > ( option @ A ),A4: set @ B,K: B] :
( ( ( restrict_map @ B @ A @ M2 @ A4 @ K )
= ( none @ A ) )
= ( ~ ( member @ B @ K @ ( inf_inf @ ( set @ B ) @ ( dom @ B @ A @ M2 ) @ A4 ) ) ) ) ).
% restrict_map_eq(1)
thf(fact_3366_map__card__eq__iff,axiom,
! [B: $tType,A: $tType,M3: A > ( option @ B ),X: A,Y: A] :
( ( finite_finite2 @ A @ ( dom @ A @ B @ M3 ) )
=> ( ( ( finite_card @ A @ ( dom @ A @ B @ M3 ) )
= ( finite_card @ B @ ( ran @ A @ B @ M3 ) ) )
=> ( ( member @ A @ X @ ( dom @ A @ B @ M3 ) )
=> ( ( ( M3 @ X )
= ( M3 @ Y ) )
= ( X = Y ) ) ) ) ) ).
% map_card_eq_iff
thf(fact_3367_wf__linord__ex__has__least,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),P: B > $o,K: B,M2: B > A] :
( ( wf @ A @ R )
=> ( ! [X2: A,Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y4 ) @ ( transitive_trancl @ A @ R ) )
= ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X2 ) @ ( transitive_rtrancl @ A @ R ) ) ) )
=> ( ( P @ K )
=> ? [X2: B] :
( ( P @ X2 )
& ! [Y6: B] :
( ( P @ Y6 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( M2 @ X2 ) @ ( M2 @ Y6 ) ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ) ) ) ).
% wf_linord_ex_has_least
thf(fact_3368_trancl__union__outside,axiom,
! [A: $tType,V: A,W: A,E4: set @ ( product_prod @ A @ A ),U4: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W ) @ ( transitive_trancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ E4 @ U4 ) ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W ) @ ( transitive_trancl @ A @ E4 ) )
=> ? [X2: A,Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ X2 ) @ ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ E4 @ U4 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y4 ) @ U4 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ W ) @ ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ E4 @ U4 ) ) ) ) ) ) ).
% trancl_union_outside
thf(fact_3369_trancl__over__edgeE,axiom,
! [A: $tType,U: A,W: A,V1: A,V22: A,E4: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ W ) @ ( transitive_trancl @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V1 @ V22 ) @ E4 ) ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ W ) @ ( transitive_trancl @ A @ E4 ) )
=> ~ ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V1 ) @ ( transitive_rtrancl @ A @ E4 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V22 @ W ) @ ( transitive_rtrancl @ A @ E4 ) ) ) ) ) ).
% trancl_over_edgeE
thf(fact_3370_finite__map__to__set,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ ( map_to_set @ A @ B @ M2 ) )
= ( finite_finite2 @ A @ ( dom @ A @ B @ M2 ) ) ) ).
% finite_map_to_set
thf(fact_3371_map__to__set__dom,axiom,
! [B: $tType,A: $tType] :
( ( dom @ A @ B )
= ( ^ [M: A > ( option @ B )] : ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( map_to_set @ A @ B @ M ) ) ) ) ).
% map_to_set_dom
thf(fact_3372_inj__on__the__inv__into,axiom,
! [B: $tType,A: $tType,F: A > B,A4: set @ A] :
( ( inj_on @ A @ B @ F @ A4 )
=> ( inj_on @ B @ A @ ( the_inv_into @ A @ B @ A4 @ F ) @ ( image2 @ A @ B @ F @ A4 ) ) ) ).
% inj_on_the_inv_into
thf(fact_3373_f__the__inv__into__f,axiom,
! [A: $tType,B: $tType,F: A > B,A4: set @ A,Y: B] :
( ( inj_on @ A @ B @ F @ A4 )
=> ( ( member @ B @ Y @ ( image2 @ A @ B @ F @ A4 ) )
=> ( ( F @ ( the_inv_into @ A @ B @ A4 @ F @ Y ) )
= Y ) ) ) ).
% f_the_inv_into_f
thf(fact_3374_the__inv__f__f,axiom,
! [B: $tType,A: $tType,F: A > B,X: A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( the_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F @ ( F @ X ) )
= X ) ) ).
% the_inv_f_f
thf(fact_3375_set__to__map__dom,axiom,
! [B: $tType,A: $tType,S2: set @ ( product_prod @ A @ B )] :
( ( dom @ A @ B @ ( set_to_map @ A @ B @ S2 ) )
= ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ S2 ) ) ).
% set_to_map_dom
thf(fact_3376_Restr__trancl__mono,axiom,
! [A: $tType,V: A,W: A,E4: set @ ( product_prod @ A @ A ),U4: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W )
@ ( transitive_trancl @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ E4
@ ( product_Sigma @ A @ A @ U4
@ ^ [Uu: A] : U4 ) ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W ) @ ( transitive_trancl @ A @ E4 ) ) ) ).
% Restr_trancl_mono
thf(fact_3377_inj__on__map__the,axiom,
! [B: $tType,A: $tType,D4: set @ A,M2: A > ( option @ B )] :
( ( ord_less_eq @ ( set @ A ) @ D4 @ ( dom @ A @ B @ M2 ) )
=> ( ( inj_on @ A @ ( option @ B ) @ M2 @ D4 )
=> ( inj_on @ A @ B @ ( comp @ ( option @ B ) @ B @ A @ ( the2 @ B ) @ M2 ) @ D4 ) ) ) ).
% inj_on_map_the
thf(fact_3378_finite__Map__induct,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B ),P: ( A > ( option @ B ) ) > $o] :
( ( finite_finite2 @ A @ ( dom @ A @ B @ M2 ) )
=> ( ( P
@ ^ [X3: A] : ( none @ B ) )
=> ( ! [K5: A,V4: B,M5: A > ( option @ B )] :
( ( finite_finite2 @ A @ ( dom @ A @ B @ M5 ) )
=> ( ~ ( member @ A @ K5 @ ( dom @ A @ B @ M5 ) )
=> ( ( P @ M5 )
=> ( P @ ( fun_upd @ A @ ( option @ B ) @ M5 @ K5 @ ( some @ B @ V4 ) ) ) ) ) )
=> ( P @ M2 ) ) ) ) ).
% finite_Map_induct
thf(fact_3379_ran__is__image,axiom,
! [A: $tType,B: $tType] :
( ( ran @ B @ A )
= ( ^ [M8: B > ( option @ A )] : ( image2 @ B @ A @ ( comp @ ( option @ A ) @ A @ B @ ( the2 @ A ) @ M8 ) @ ( dom @ B @ A @ M8 ) ) ) ) ).
% ran_is_image
thf(fact_3380_graph__eq__to__snd__dom,axiom,
! [B: $tType,A: $tType] :
( ( graph @ A @ B )
= ( ^ [M: A > ( option @ B )] :
( image2 @ A @ ( product_prod @ A @ B )
@ ^ [X3: A] : ( product_Pair @ A @ B @ X3 @ ( the2 @ B @ ( M @ X3 ) ) )
@ ( dom @ A @ B @ M ) ) ) ) ).
% graph_eq_to_snd_dom
thf(fact_3381_trancl__insert2,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A )] :
( ( transitive_trancl @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R )
@ ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X3: A,Y3: A] :
( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ A3 ) @ ( transitive_trancl @ A @ R ) )
| ( X3 = A3 ) )
& ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ Y3 ) @ ( transitive_trancl @ A @ R ) )
| ( Y3 = B2 ) ) ) ) ) ) ) ).
% trancl_insert2
thf(fact_3382_rel__restrict__Sigma__sub,axiom,
! [A: $tType,A4: set @ A,R5: set @ A] :
( ord_less_eq @ ( set @ ( product_prod @ A @ A ) )
@ ( rel_restrict @ A
@ ( transitive_trancl @ A
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu: A] : A4 ) )
@ R5 )
@ ( transitive_trancl @ A
@ ( product_Sigma @ A @ A @ ( minus_minus @ ( set @ A ) @ A4 @ R5 )
@ ^ [Uu: A] : ( minus_minus @ ( set @ A ) @ A4 @ R5 ) ) ) ) ).
% rel_restrict_Sigma_sub
thf(fact_3383_the__inv__into__into,axiom,
! [B: $tType,A: $tType,F: A > B,A4: set @ A,X: B,B6: set @ A] :
( ( inj_on @ A @ B @ F @ A4 )
=> ( ( member @ B @ X @ ( image2 @ A @ B @ F @ A4 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B6 )
=> ( member @ A @ ( the_inv_into @ A @ B @ A4 @ F @ X ) @ B6 ) ) ) ) ).
% the_inv_into_into
thf(fact_3384_the__inv__into__comp,axiom,
! [A: $tType,C: $tType,B: $tType,F: A > B,G: C > A,A4: set @ C,X: B] :
( ( inj_on @ A @ B @ F @ ( image2 @ C @ A @ G @ A4 ) )
=> ( ( inj_on @ C @ A @ G @ A4 )
=> ( ( member @ B @ X @ ( image2 @ A @ B @ F @ ( image2 @ C @ A @ G @ A4 ) ) )
=> ( ( the_inv_into @ C @ B @ A4 @ ( comp @ A @ B @ C @ F @ G ) @ X )
= ( comp @ A @ C @ B @ ( the_inv_into @ C @ A @ A4 @ G ) @ ( the_inv_into @ A @ B @ ( image2 @ C @ A @ G @ A4 ) @ F ) @ X ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_3385_dom__eq__singleton__conv,axiom,
! [A: $tType,B: $tType,F: A > ( option @ B ),X: A] :
( ( ( dom @ A @ B @ F )
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( ? [V2: B] :
( F
= ( fun_upd @ A @ ( option @ B )
@ ^ [X3: A] : ( none @ B )
@ X
@ ( some @ B @ V2 ) ) ) ) ) ).
% dom_eq_singleton_conv
thf(fact_3386_rtrancl__last__visit,axiom,
! [A: $tType,Q3: A,Q4: A,R5: set @ ( product_prod @ A @ A ),S2: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q3 @ Q4 ) @ ( transitive_rtrancl @ A @ R5 ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q3 @ Q4 )
@ ( transitive_rtrancl @ A
@ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R5
@ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
@ ^ [Uu: A] : S2 ) ) ) )
=> ~ ! [Qt: A] :
( ( member @ A @ Qt @ S2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q3 @ Qt ) @ ( transitive_trancl @ A @ R5 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Qt @ Q4 )
@ ( transitive_rtrancl @ A
@ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R5
@ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
@ ^ [Uu: A] : S2 ) ) ) ) ) ) ) ) ).
% rtrancl_last_visit
thf(fact_3387_trancl__insert,axiom,
! [A: $tType,Y: A,X: A,R: set @ ( product_prod @ A @ A )] :
( ( transitive_trancl @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ R ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R )
@ ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [A7: A,B5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ Y ) @ ( transitive_rtrancl @ A @ R ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ B5 ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ) ) ) ).
% trancl_insert
thf(fact_3388_finite__trancl__ntranl,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R5 )
=> ( ( transitive_trancl @ A @ R5 )
= ( transitive_ntrancl @ A @ ( minus_minus @ nat @ ( finite_card @ ( product_prod @ A @ A ) @ R5 ) @ ( one_one @ nat ) ) @ R5 ) ) ) ).
% finite_trancl_ntranl
thf(fact_3389_ran__add,axiom,
! [B: $tType,A: $tType,F: A > ( option @ B ),G: A > ( option @ B )] :
( ( ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ F ) @ ( dom @ A @ B @ G ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( ran @ A @ B @ ( map_add @ A @ B @ F @ G ) )
= ( sup_sup @ ( set @ B ) @ ( ran @ A @ B @ F ) @ ( ran @ A @ B @ G ) ) ) ) ).
% ran_add
thf(fact_3390_eq__f__restr__ss__eq,axiom,
! [B: $tType,A: $tType,S: set @ A,F: ( A > ( option @ B ) ) > A > ( option @ B ),A4: A > ( option @ B )] :
( ( ord_less_eq @ ( set @ A ) @ S @ ( dom @ A @ B @ ( F @ A4 ) ) )
=> ( ( A4
= ( restrict_map @ A @ B @ ( F @ A4 ) @ ( uminus_uminus @ ( set @ A ) @ S ) ) )
= ( ( map_le @ A @ B @ A4 @ ( F @ A4 ) )
& ( S
= ( minus_minus @ ( set @ A ) @ ( dom @ A @ B @ ( F @ A4 ) ) @ ( dom @ A @ B @ A4 ) ) ) ) ) ) ).
% eq_f_restr_ss_eq
thf(fact_3391_eq__f__restr__conv,axiom,
! [B: $tType,A: $tType,S: set @ A,F: ( A > ( option @ B ) ) > A > ( option @ B ),A4: A > ( option @ B )] :
( ( ( ord_less_eq @ ( set @ A ) @ S @ ( dom @ A @ B @ ( F @ A4 ) ) )
& ( A4
= ( restrict_map @ A @ B @ ( F @ A4 ) @ ( uminus_uminus @ ( set @ A ) @ S ) ) ) )
= ( ( map_le @ A @ B @ A4 @ ( F @ A4 ) )
& ( S
= ( minus_minus @ ( set @ A ) @ ( dom @ A @ B @ ( F @ A4 ) ) @ ( dom @ A @ B @ A4 ) ) ) ) ) ).
% eq_f_restr_conv
thf(fact_3392_type__copy__set__map0,axiom,
! [A: $tType,B: $tType,D: $tType,E: $tType,C: $tType,F5: $tType,Rep: A > B,Abs: B > A,S2: B > ( set @ D ),M3: C > B,F: E > D,S7: C > ( set @ E ),G: F5 > C] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ( ( ( comp @ B @ ( set @ D ) @ C @ S2 @ M3 )
= ( comp @ ( set @ E ) @ ( set @ D ) @ C @ ( image2 @ E @ D @ F ) @ S7 ) )
=> ( ( comp @ A @ ( set @ D ) @ F5 @ ( comp @ B @ ( set @ D ) @ A @ S2 @ Rep ) @ ( comp @ C @ A @ F5 @ ( comp @ B @ A @ C @ Abs @ M3 ) @ G ) )
= ( comp @ ( set @ E ) @ ( set @ D ) @ F5 @ ( image2 @ E @ D @ F ) @ ( comp @ C @ ( set @ E ) @ F5 @ S7 @ G ) ) ) ) ) ).
% type_copy_set_map0
thf(fact_3393_map__add__find__right,axiom,
! [B: $tType,A: $tType,N: B > ( option @ A ),K: B,Xx: A,M2: B > ( option @ A )] :
( ( ( N @ K )
= ( some @ A @ Xx ) )
=> ( ( map_add @ B @ A @ M2 @ N @ K )
= ( some @ A @ Xx ) ) ) ).
% map_add_find_right
thf(fact_3394_map__add__eq__empty__iff,axiom,
! [B: $tType,A: $tType,F: A > ( option @ B ),G: A > ( option @ B )] :
( ( ( map_add @ A @ B @ F @ G )
= ( ^ [X3: A] : ( none @ B ) ) )
= ( ( F
= ( ^ [X3: A] : ( none @ B ) ) )
& ( G
= ( ^ [X3: A] : ( none @ B ) ) ) ) ) ).
% map_add_eq_empty_iff
thf(fact_3395_map__add__None,axiom,
! [B: $tType,A: $tType,M2: B > ( option @ A ),N: B > ( option @ A ),K: B] :
( ( ( map_add @ B @ A @ M2 @ N @ K )
= ( none @ A ) )
= ( ( ( N @ K )
= ( none @ A ) )
& ( ( M2 @ K )
= ( none @ A ) ) ) ) ).
% map_add_None
thf(fact_3396_empty__map__add,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B )] :
( ( map_add @ A @ B
@ ^ [X3: A] : ( none @ B )
@ M2 )
= M2 ) ).
% empty_map_add
thf(fact_3397_map__add__empty,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B )] :
( ( map_add @ A @ B @ M2
@ ^ [X3: A] : ( none @ B ) )
= M2 ) ).
% map_add_empty
thf(fact_3398_empty__eq__map__add__iff,axiom,
! [B: $tType,A: $tType,F: A > ( option @ B ),G: A > ( option @ B )] :
( ( ( ^ [X3: A] : ( none @ B ) )
= ( map_add @ A @ B @ F @ G ) )
= ( ( F
= ( ^ [X3: A] : ( none @ B ) ) )
& ( G
= ( ^ [X3: A] : ( none @ B ) ) ) ) ) ).
% empty_eq_map_add_iff
thf(fact_3399_map__add__upd,axiom,
! [A: $tType,B: $tType,F: A > ( option @ B ),G: A > ( option @ B ),X: A,Y: B] :
( ( map_add @ A @ B @ F @ ( fun_upd @ A @ ( option @ B ) @ G @ X @ ( some @ B @ Y ) ) )
= ( fun_upd @ A @ ( option @ B ) @ ( map_add @ A @ B @ F @ G ) @ X @ ( some @ B @ Y ) ) ) ).
% map_add_upd
thf(fact_3400_map__add__upd__left,axiom,
! [A: $tType,B: $tType,M2: A,E22: A > ( option @ B ),E1: A > ( option @ B ),U1: B] :
( ~ ( member @ A @ M2 @ ( dom @ A @ B @ E22 ) )
=> ( ( map_add @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ E1 @ M2 @ ( some @ B @ U1 ) ) @ E22 )
= ( fun_upd @ A @ ( option @ B ) @ ( map_add @ A @ B @ E1 @ E22 ) @ M2 @ ( some @ B @ U1 ) ) ) ) ).
% map_add_upd_left
thf(fact_3401_le__map__mmupd__not__dom,axiom,
! [A: $tType,B: $tType,M2: A > ( option @ B ),K2: set @ A,V: B] : ( map_le @ A @ B @ M2 @ ( map_mmupd @ A @ B @ M2 @ ( minus_minus @ ( set @ A ) @ K2 @ ( dom @ A @ B @ M2 ) ) @ V ) ) ).
% le_map_mmupd_not_dom
thf(fact_3402_map__add__find__left,axiom,
! [A: $tType,B: $tType,G: B > ( option @ A ),K: B,F: B > ( option @ A )] :
( ( ( G @ K )
= ( none @ A ) )
=> ( ( map_add @ B @ A @ F @ G @ K )
= ( F @ K ) ) ) ).
% map_add_find_left
thf(fact_3403_map__add__left__None,axiom,
! [A: $tType,B: $tType,F: B > ( option @ A ),K: B,G: B > ( option @ A )] :
( ( ( F @ K )
= ( none @ A ) )
=> ( ( map_add @ B @ A @ F @ G @ K )
= ( G @ K ) ) ) ).
% map_add_left_None
thf(fact_3404_map__leI,axiom,
! [B: $tType,A: $tType,M1: A > ( option @ B ),M22: A > ( option @ B )] :
( ! [X2: A,V4: B] :
( ( ( M1 @ X2 )
= ( some @ B @ V4 ) )
=> ( ( M22 @ X2 )
= ( some @ B @ V4 ) ) )
=> ( map_le @ A @ B @ M1 @ M22 ) ) ).
% map_leI
thf(fact_3405_map__leD,axiom,
! [A: $tType,B: $tType,M1: A > ( option @ B ),M22: A > ( option @ B ),K: A,V: B] :
( ( map_le @ A @ B @ M1 @ M22 )
=> ( ( ( M1 @ K )
= ( some @ B @ V ) )
=> ( ( M22 @ K )
= ( some @ B @ V ) ) ) ) ).
% map_leD
thf(fact_3406_map__add__first__le,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [M2: A > ( option @ B ),M7: A > ( option @ B ),N: A > ( option @ B )] :
( ( ord_less_eq @ ( A > ( option @ B ) ) @ M2 @ M7 )
=> ( ord_less_eq @ ( A > ( option @ B ) ) @ ( map_add @ A @ B @ M2 @ N ) @ ( map_add @ A @ B @ M7 @ N ) ) ) ) ).
% map_add_first_le
thf(fact_3407_map__le__empty,axiom,
! [B: $tType,A: $tType,G: A > ( option @ B )] :
( map_le @ A @ B
@ ^ [X3: A] : ( none @ B )
@ G ) ).
% map_le_empty
thf(fact_3408_map__add__Some__iff,axiom,
! [B: $tType,A: $tType,M2: B > ( option @ A ),N: B > ( option @ A ),K: B,X: A] :
( ( ( map_add @ B @ A @ M2 @ N @ K )
= ( some @ A @ X ) )
= ( ( ( N @ K )
= ( some @ A @ X ) )
| ( ( ( N @ K )
= ( none @ A ) )
& ( ( M2 @ K )
= ( some @ A @ X ) ) ) ) ) ).
% map_add_Some_iff
thf(fact_3409_map__add__SomeD,axiom,
! [B: $tType,A: $tType,M2: B > ( option @ A ),N: B > ( option @ A ),K: B,X: A] :
( ( ( map_add @ B @ A @ M2 @ N @ K )
= ( some @ A @ X ) )
=> ( ( ( N @ K )
= ( some @ A @ X ) )
| ( ( ( N @ K )
= ( none @ A ) )
& ( ( M2 @ K )
= ( some @ A @ X ) ) ) ) ) ).
% map_add_SomeD
thf(fact_3410_type__copy__map__comp0__undo,axiom,
! [E: $tType,A: $tType,C: $tType,B: $tType,D: $tType,F5: $tType,Rep: A > B,Abs: B > A,Rep2: C > D,Abs2: D > C,Rep3: E > F5,Abs3: F5 > E,M3: F5 > D,M13: B > D,M24: F5 > B] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ( ( type_definition @ C @ D @ Rep2 @ Abs2 @ ( top_top @ ( set @ D ) ) )
=> ( ( type_definition @ E @ F5 @ Rep3 @ Abs3 @ ( top_top @ ( set @ F5 ) ) )
=> ( ( ( comp @ F5 @ C @ E @ ( comp @ D @ C @ F5 @ Abs2 @ M3 ) @ Rep3 )
= ( comp @ A @ C @ E @ ( comp @ B @ C @ A @ ( comp @ D @ C @ B @ Abs2 @ M13 ) @ Rep ) @ ( comp @ F5 @ A @ E @ ( comp @ B @ A @ F5 @ Abs @ M24 ) @ Rep3 ) ) )
=> ( ( comp @ B @ D @ F5 @ M13 @ M24 )
= M3 ) ) ) ) ) ).
% type_copy_map_comp0_undo
thf(fact_3411_type__copy__map__comp0,axiom,
! [F5: $tType,D: $tType,B: $tType,A: $tType,C: $tType,E: $tType,Rep: A > B,Abs: B > A,M3: C > D,M13: B > D,M24: C > B,F: D > F5,G: E > C] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ( ( M3
= ( comp @ B @ D @ C @ M13 @ M24 ) )
=> ( ( comp @ C @ F5 @ E @ ( comp @ D @ F5 @ C @ F @ M3 ) @ G )
= ( comp @ A @ F5 @ E @ ( comp @ B @ F5 @ A @ ( comp @ D @ F5 @ B @ F @ M13 ) @ Rep ) @ ( comp @ C @ A @ E @ ( comp @ B @ A @ C @ Abs @ M24 ) @ G ) ) ) ) ) ).
% type_copy_map_comp0
thf(fact_3412_upd__None__map__le,axiom,
! [B: $tType,A: $tType,F: A > ( option @ B ),X: A] : ( map_le @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F @ X @ ( none @ B ) ) @ F ) ).
% upd_None_map_le
thf(fact_3413_map__add__def,axiom,
! [B: $tType,A: $tType] :
( ( map_add @ A @ B )
= ( ^ [M14: A > ( option @ B ),M25: A > ( option @ B ),X3: A] : ( case_option @ ( option @ B ) @ B @ ( M14 @ X3 ) @ ( some @ B ) @ ( M25 @ X3 ) ) ) ) ).
% map_add_def
thf(fact_3414_map__add__left__comm,axiom,
! [B: $tType,A: $tType,A4: A > ( option @ B ),B6: A > ( option @ B ),C5: A > ( option @ B )] :
( ( ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ A4 ) @ ( dom @ A @ B @ B6 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( map_add @ A @ B @ A4 @ ( map_add @ A @ B @ B6 @ C5 ) )
= ( map_add @ A @ B @ B6 @ ( map_add @ A @ B @ A4 @ C5 ) ) ) ) ).
% map_add_left_comm
thf(fact_3415_type__copy__map__id0,axiom,
! [B: $tType,A: $tType,Rep: A > B,Abs: B > A,M3: B > B] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ( ( M3
= ( id @ B ) )
=> ( ( comp @ B @ A @ A @ ( comp @ B @ A @ B @ Abs @ M3 ) @ Rep )
= ( id @ A ) ) ) ) ).
% type_copy_map_id0
thf(fact_3416_type__copy__Abs__o__Rep,axiom,
! [B: $tType,A: $tType,Rep: A > B,Abs: B > A] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ( ( comp @ B @ A @ A @ Abs @ Rep )
= ( id @ A ) ) ) ).
% type_copy_Abs_o_Rep
thf(fact_3417_type__copy__Rep__o__Abs,axiom,
! [A: $tType,B: $tType,Rep: A > B,Abs: B > A] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ( ( comp @ A @ B @ B @ Rep @ Abs )
= ( id @ B ) ) ) ).
% type_copy_Rep_o_Abs
thf(fact_3418_type__definition__integer,axiom,
type_definition @ code_integer @ int @ code_int_of_integer @ code_integer_of_int @ ( top_top @ ( set @ int ) ) ).
% type_definition_integer
thf(fact_3419_map__le__imp__upd__le,axiom,
! [A: $tType,B: $tType,M1: A > ( option @ B ),M22: A > ( option @ B ),X: A,Y: B] :
( ( map_le @ A @ B @ M1 @ M22 )
=> ( map_le @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M1 @ X @ ( none @ B ) ) @ ( fun_upd @ A @ ( option @ B ) @ M22 @ X @ ( some @ B @ Y ) ) ) ) ).
% map_le_imp_upd_le
thf(fact_3420_map__add__distinct__le,axiom,
! [B: $tType,A: $tType] :
( ( preorder @ B )
=> ! [M2: A > ( option @ B ),M7: A > ( option @ B ),N: A > ( option @ B ),N2: A > ( option @ B )] :
( ( ord_less_eq @ ( A > ( option @ B ) ) @ M2 @ M7 )
=> ( ( ord_less_eq @ ( A > ( option @ B ) ) @ N @ N2 )
=> ( ( ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ M7 ) @ ( dom @ A @ B @ N2 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ ( A > ( option @ B ) ) @ ( map_add @ A @ B @ M2 @ N ) @ ( map_add @ A @ B @ M7 @ N2 ) ) ) ) ) ) ).
% map_add_distinct_le
thf(fact_3421_map__mmupd__update__less,axiom,
! [A: $tType,B: $tType,K2: set @ A,K9: set @ A,M2: A > ( option @ B ),V: B] :
( ( ord_less_eq @ ( set @ A ) @ K2 @ K9 )
=> ( map_le @ A @ B @ ( map_mmupd @ A @ B @ M2 @ ( minus_minus @ ( set @ A ) @ K2 @ ( dom @ A @ B @ M2 ) ) @ V ) @ ( map_mmupd @ A @ B @ M2 @ ( minus_minus @ ( set @ A ) @ K9 @ ( dom @ A @ B @ M2 ) ) @ V ) ) ) ).
% map_mmupd_update_less
thf(fact_3422_trancl__set__ntrancl,axiom,
! [A: $tType,Xs: list @ ( product_prod @ A @ A )] :
( ( transitive_trancl @ A @ ( set2 @ ( product_prod @ A @ A ) @ Xs ) )
= ( transitive_ntrancl @ A @ ( minus_minus @ 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
thf(fact_3423_pred__nat__trancl__eq__le,axiom,
! [M2: nat,N: nat] :
( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ M2 @ N ) @ ( transitive_rtrancl @ nat @ pred_nat ) )
= ( ord_less_eq @ nat @ M2 @ N ) ) ).
% pred_nat_trancl_eq_le
thf(fact_3424_less__eq,axiom,
! [M2: nat,N: nat] :
( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ M2 @ N ) @ ( transitive_trancl @ nat @ pred_nat ) )
= ( ord_less @ nat @ M2 @ N ) ) ).
% less_eq
thf(fact_3425_Gcd__fin__def,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( semiring_gcd_Gcd_fin @ A )
= ( bounde2362111253966948842tice_F @ A @ ( gcd_gcd @ A ) @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ) ).
% Gcd_fin_def
thf(fact_3426_relImage__def,axiom,
! [A: $tType,B: $tType] :
( ( bNF_Gr4221423524335903396lImage @ B @ A )
= ( ^ [R6: set @ ( product_prod @ B @ B ),F2: B > A] :
( collect @ ( product_prod @ A @ A )
@ ^ [Uu: product_prod @ A @ A] :
? [A12: B,A23: B] :
( ( Uu
= ( product_Pair @ A @ A @ ( F2 @ A12 ) @ ( F2 @ A23 ) ) )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A12 @ A23 ) @ R6 ) ) ) ) ) ).
% relImage_def
thf(fact_3427_set__map__filter,axiom,
! [B: $tType,A: $tType,G: B > ( option @ A ),Xs: list @ B] :
( ( set2 @ A @ ( map_filter @ B @ A @ G @ Xs ) )
= ( collect @ A
@ ^ [Y3: A] :
? [X3: B] :
( ( member @ B @ X3 @ ( set2 @ B @ Xs ) )
& ( ( G @ X3 )
= ( some @ A @ Y3 ) ) ) ) ) ).
% set_map_filter
thf(fact_3428_ran__map__of,axiom,
! [A: $tType,B: $tType,Xs: list @ ( product_prod @ B @ A )] : ( ord_less_eq @ ( set @ A ) @ ( ran @ B @ A @ ( map_of @ B @ A @ Xs ) ) @ ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( set2 @ ( product_prod @ B @ A ) @ Xs ) ) ) ).
% ran_map_of
thf(fact_3429_trancl__restrict__reachable,axiom,
! [A: $tType,U: A,V: A,E4: set @ ( product_prod @ A @ A ),S2: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V ) @ ( transitive_trancl @ A @ E4 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ E4 @ S2 ) @ S2 )
=> ( ( member @ A @ U @ S2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V )
@ ( transitive_trancl @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ E4
@ ( product_Sigma @ A @ A @ S2
@ ^ [Uu: A] : S2 ) ) ) ) ) ) ) ).
% trancl_restrict_reachable
thf(fact_3430_pair__vimage__is__Image,axiom,
! [A: $tType,B: $tType,U: B,E4: set @ ( product_prod @ B @ A )] :
( ( vimage @ A @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ U ) @ E4 )
= ( image @ B @ A @ E4 @ ( insert2 @ B @ U @ ( bot_bot @ ( set @ B ) ) ) ) ) ).
% pair_vimage_is_Image
thf(fact_3431_set__bind__option,axiom,
! [A: $tType,B: $tType,X: option @ B,F: B > ( option @ A )] :
( ( set_option @ A @ ( bind @ B @ A @ X @ F ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ ( comp @ ( option @ A ) @ ( set @ A ) @ B @ ( set_option @ A ) @ F ) @ ( set_option @ B @ X ) ) ) ) ).
% set_bind_option
thf(fact_3432_ImageI,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,R: set @ ( product_prod @ A @ B ),A4: set @ A] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R )
=> ( ( member @ A @ A3 @ A4 )
=> ( member @ B @ B2 @ ( image @ A @ B @ R @ A4 ) ) ) ) ).
% ImageI
thf(fact_3433_elem__set,axiom,
! [A: $tType,X: A,Xo: option @ A] :
( ( member @ A @ X @ ( set_option @ A @ Xo ) )
= ( Xo
= ( some @ A @ X ) ) ) ).
% elem_set
thf(fact_3434_set__empty__eq,axiom,
! [A: $tType,Xo: option @ A] :
( ( ( set_option @ A @ Xo )
= ( bot_bot @ ( set @ A ) ) )
= ( Xo
= ( none @ A ) ) ) ).
% set_empty_eq
thf(fact_3435_Image__singleton__iff,axiom,
! [A: $tType,B: $tType,B2: A,R: set @ ( product_prod @ B @ A ),A3: B] :
( ( member @ A @ B2 @ ( image @ B @ A @ R @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A3 @ B2 ) @ R ) ) ).
% Image_singleton_iff
thf(fact_3436_rev__ImageI,axiom,
! [B: $tType,A: $tType,A3: A,A4: set @ A,B2: B,R: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A3 @ A4 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R )
=> ( member @ B @ B2 @ ( image @ A @ B @ R @ A4 ) ) ) ) ).
% rev_ImageI
thf(fact_3437_Image__iff,axiom,
! [A: $tType,B: $tType,B2: A,R: set @ ( product_prod @ B @ A ),A4: set @ B] :
( ( member @ A @ B2 @ ( image @ B @ A @ R @ A4 ) )
= ( ? [X3: B] :
( ( member @ B @ X3 @ A4 )
& ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 @ B2 ) @ R ) ) ) ) ).
% Image_iff
thf(fact_3438_ImageE,axiom,
! [A: $tType,B: $tType,B2: A,R: set @ ( product_prod @ B @ A ),A4: set @ B] :
( ( member @ A @ B2 @ ( image @ B @ A @ R @ A4 ) )
=> ~ ! [X2: B] :
( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X2 @ B2 ) @ R )
=> ~ ( member @ B @ X2 @ A4 ) ) ) ).
% ImageE
thf(fact_3439_ospec,axiom,
! [A: $tType,A4: option @ A,P: A > $o,X: A] :
( ! [X2: A] :
( ( member @ A @ X2 @ ( set_option @ A @ A4 ) )
=> ( P @ X2 ) )
=> ( ( A4
= ( some @ A @ X ) )
=> ( P @ X ) ) ) ).
% ospec
thf(fact_3440_option_Oset__intros,axiom,
! [A: $tType,X22: A] : ( member @ A @ X22 @ ( set_option @ A @ ( some @ A @ X22 ) ) ) ).
% option.set_intros
thf(fact_3441_option_Oset__cases,axiom,
! [A: $tType,E2: A,A3: option @ A] :
( ( member @ A @ E2 @ ( set_option @ A @ A3 ) )
=> ( A3
= ( some @ A @ E2 ) ) ) ).
% option.set_cases
thf(fact_3442_map__option__idI,axiom,
! [A: $tType,X: option @ A,F: A > A] :
( ! [Y4: A] :
( ( member @ A @ Y4 @ ( set_option @ A @ X ) )
=> ( ( F @ Y4 )
= Y4 ) )
=> ( ( map_option @ A @ A @ F @ X )
= X ) ) ).
% map_option_idI
thf(fact_3443_option_Oinj__map__strong,axiom,
! [B: $tType,A: $tType,X: option @ A,Xa: option @ A,F: A > B,Fa: A > B] :
( ! [Z3: A,Za: A] :
( ( member @ A @ Z3 @ ( set_option @ A @ X ) )
=> ( ( member @ A @ Za @ ( set_option @ A @ Xa ) )
=> ( ( ( F @ Z3 )
= ( Fa @ Za ) )
=> ( Z3 = Za ) ) ) )
=> ( ( ( map_option @ A @ B @ F @ X )
= ( map_option @ A @ B @ Fa @ Xa ) )
=> ( X = Xa ) ) ) ).
% option.inj_map_strong
thf(fact_3444_option_Omap__cong0,axiom,
! [B: $tType,A: $tType,X: option @ A,F: A > B,G: A > B] :
( ! [Z3: A] :
( ( member @ A @ Z3 @ ( set_option @ A @ X ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( map_option @ A @ B @ F @ X )
= ( map_option @ A @ B @ G @ X ) ) ) ).
% option.map_cong0
thf(fact_3445_option_Omap__cong,axiom,
! [B: $tType,A: $tType,X: option @ A,Ya: option @ A,F: A > B,G: A > B] :
( ( X = Ya )
=> ( ! [Z3: A] :
( ( member @ A @ Z3 @ ( set_option @ A @ Ya ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( map_option @ A @ B @ F @ X )
= ( map_option @ A @ B @ G @ Ya ) ) ) ) ).
% option.map_cong
thf(fact_3446_bind__option__cong,axiom,
! [B: $tType,A: $tType,X: option @ A,Y: option @ A,F: A > ( option @ B ),G: A > ( option @ B )] :
( ( X = Y )
=> ( ! [Z3: A] :
( ( member @ A @ Z3 @ ( set_option @ A @ Y ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( bind @ A @ B @ X @ F )
= ( bind @ A @ B @ Y @ G ) ) ) ) ).
% bind_option_cong
thf(fact_3447_rtrancl__image__advance__rtrancl,axiom,
! [A: $tType,Q3: A,R5: set @ ( product_prod @ A @ A ),Q0: set @ A,X: A] :
( ( member @ A @ Q3 @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R5 ) @ Q0 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q3 @ X ) @ ( transitive_rtrancl @ A @ R5 ) )
=> ( member @ A @ X @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R5 ) @ Q0 ) ) ) ) ).
% rtrancl_image_advance_rtrancl
thf(fact_3448_rtrancl__image__advance,axiom,
! [A: $tType,Q3: A,R5: set @ ( product_prod @ A @ A ),Q0: set @ A,X: A] :
( ( member @ A @ Q3 @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R5 ) @ Q0 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q3 @ X ) @ R5 )
=> ( member @ A @ X @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R5 ) @ Q0 ) ) ) ) ).
% rtrancl_image_advance
thf(fact_3449_finite__Image__subset,axiom,
! [A: $tType,B: $tType,A4: set @ ( product_prod @ B @ A ),B6: set @ B,C5: set @ ( product_prod @ B @ A )] :
( ( finite_finite2 @ A @ ( image @ B @ A @ A4 @ B6 ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) ) @ C5 @ A4 )
=> ( finite_finite2 @ A @ ( image @ B @ A @ C5 @ B6 ) ) ) ) ).
% finite_Image_subset
thf(fact_3450_rtrancl__image__unfold__right,axiom,
! [A: $tType,E4: set @ ( product_prod @ A @ A ),V6: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ E4 @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E4 ) @ V6 ) ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E4 ) @ V6 ) ) ).
% rtrancl_image_unfold_right
thf(fact_3451_rtrancl__reachable__induct,axiom,
! [A: $tType,I4: set @ A,INV: set @ A,E4: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ A ) @ I4 @ INV )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ E4 @ INV ) @ INV )
=> ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E4 ) @ I4 ) @ INV ) ) ) ).
% rtrancl_reachable_induct
thf(fact_3452_trancl__Image__unfold__right,axiom,
! [A: $tType,E4: set @ ( product_prod @ A @ A ),S2: set @ A] :
( ( image @ A @ A @ ( transitive_trancl @ A @ E4 ) @ S2 )
= ( image @ A @ A @ E4 @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E4 ) @ S2 ) ) ) ).
% trancl_Image_unfold_right
thf(fact_3453_trancl__Image__unfold__left,axiom,
! [A: $tType,E4: set @ ( product_prod @ A @ A ),S2: set @ A] :
( ( image @ A @ A @ ( transitive_trancl @ A @ E4 ) @ S2 )
= ( image @ A @ A @ ( transitive_rtrancl @ A @ E4 ) @ ( image @ A @ A @ E4 @ S2 ) ) ) ).
% trancl_Image_unfold_left
thf(fact_3454_option_Osimps_I14_J,axiom,
! [A: $tType] :
( ( set_option @ A @ ( none @ A ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% option.simps(14)
thf(fact_3455_option_Oset__map,axiom,
! [B: $tType,A: $tType,F: A > B,V: option @ A] :
( ( set_option @ B @ ( map_option @ A @ B @ F @ V ) )
= ( image2 @ A @ B @ F @ ( set_option @ A @ V ) ) ) ).
% option.set_map
thf(fact_3456_option_Oset__sel,axiom,
! [A: $tType,A3: option @ A] :
( ( A3
!= ( none @ A ) )
=> ( member @ A @ ( the2 @ A @ A3 ) @ ( set_option @ A @ A3 ) ) ) ).
% option.set_sel
thf(fact_3457_Image__empty__rtrancl__Image__id,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A ),V: A] :
( ( ( image @ A @ A @ R5 @ ( insert2 @ A @ V @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image @ A @ A @ ( transitive_rtrancl @ A @ R5 ) @ ( insert2 @ A @ V @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert2 @ A @ V @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Image_empty_rtrancl_Image_id
thf(fact_3458_Image__empty__trancl__Image__empty,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A ),V: A] :
( ( ( image @ A @ A @ R5 @ ( insert2 @ A @ V @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image @ A @ A @ ( transitive_trancl @ A @ R5 ) @ ( insert2 @ A @ V @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% Image_empty_trancl_Image_empty
thf(fact_3459_reachable__mono,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ A @ A ),X10: set @ A,X11: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R5 @ R7 )
=> ( ( ord_less_eq @ ( set @ A ) @ X10 @ X11 )
=> ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R5 ) @ X10 ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R7 ) @ X11 ) ) ) ) ).
% reachable_mono
thf(fact_3460_Image__subset__snd__image,axiom,
! [A: $tType,B: $tType,A4: set @ ( product_prod @ B @ A ),B6: set @ B] : ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ A4 @ B6 ) @ ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ A4 ) ) ).
% Image_subset_snd_image
thf(fact_3461_trancl__image__by__rtrancl,axiom,
! [A: $tType,E4: set @ ( product_prod @ A @ A ),Vi: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( image @ A @ A @ ( transitive_trancl @ A @ E4 ) @ Vi ) @ Vi )
= ( image @ A @ A @ ( transitive_rtrancl @ A @ E4 ) @ Vi ) ) ).
% trancl_image_by_rtrancl
thf(fact_3462_weak__map__of__SomeI,axiom,
! [A: $tType,B: $tType,K: A,X: B,L: list @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ X ) @ ( set2 @ ( product_prod @ A @ B ) @ L ) )
=> ? [X2: B] :
( ( map_of @ A @ B @ L @ K )
= ( some @ B @ X2 ) ) ) ).
% weak_map_of_SomeI
thf(fact_3463_map__of__SomeD,axiom,
! [A: $tType,B: $tType,Xs: list @ ( product_prod @ B @ A ),K: B,Y: A] :
( ( ( map_of @ B @ A @ Xs @ K )
= ( some @ A @ Y ) )
=> ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ K @ Y ) @ ( set2 @ ( product_prod @ B @ A ) @ Xs ) ) ) ).
% map_of_SomeD
thf(fact_3464_Image__singleton,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ B @ A ),A3: B] :
( ( image @ B @ A @ R @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) )
= ( collect @ A
@ ^ [B5: A] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A3 @ B5 ) @ R ) ) ) ).
% Image_singleton
thf(fact_3465_option_Osimps_I15_J,axiom,
! [A: $tType,X22: A] :
( ( set_option @ A @ ( some @ A @ X22 ) )
= ( insert2 @ A @ X22 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% option.simps(15)
thf(fact_3466_rtrancl__apply__insert,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A ),X: A,S2: set @ A] :
( ( image @ A @ A @ ( transitive_rtrancl @ A @ R5 ) @ ( insert2 @ A @ X @ S2 ) )
= ( insert2 @ A @ X @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R5 ) @ ( sup_sup @ ( set @ A ) @ S2 @ ( image @ A @ A @ R5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% rtrancl_apply_insert
thf(fact_3467_rtrancl__Image__in__Field,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A ),V6: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R5 ) @ V6 ) @ ( sup_sup @ ( set @ A ) @ ( field2 @ A @ R5 ) @ V6 ) ) ).
% rtrancl_Image_in_Field
thf(fact_3468_E__closed__restr__reach__cases,axiom,
! [A: $tType,U: A,V: A,E4: set @ ( product_prod @ A @ A ),R5: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V ) @ ( transitive_rtrancl @ A @ E4 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ E4 @ R5 ) @ R5 )
=> ( ~ ( member @ A @ V @ R5 )
=> ~ ( ~ ( member @ A @ U @ R5 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V ) @ ( transitive_rtrancl @ A @ ( rel_restrict @ A @ E4 @ R5 ) ) ) ) ) ) ) ).
% E_closed_restr_reach_cases
thf(fact_3469_rel__restrict__tranclI,axiom,
! [A: $tType,X: A,Y: A,E4: set @ ( product_prod @ A @ A ),R5: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ E4 ) )
=> ( ~ ( member @ A @ X @ R5 )
=> ( ~ ( member @ A @ Y @ R5 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ E4 @ R5 ) @ R5 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ E4 @ R5 ) ) ) ) ) ) ) ).
% rel_restrict_tranclI
thf(fact_3470_refl__on__reflcl__Image,axiom,
! [A: $tType,B6: set @ A,A4: set @ ( product_prod @ A @ A ),C5: set @ A] :
( ( refl_on @ A @ B6 @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ C5 @ B6 )
=> ( ( image @ A @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ A4 @ ( id2 @ A ) ) @ C5 )
= ( image @ A @ A @ A4 @ C5 ) ) ) ) ).
% refl_on_reflcl_Image
thf(fact_3471_map__of__eq__None__iff,axiom,
! [A: $tType,B: $tType,Xys: list @ ( product_prod @ B @ A ),X: B] :
( ( ( map_of @ B @ A @ Xys @ X )
= ( none @ A ) )
= ( ~ ( member @ B @ X @ ( image2 @ ( product_prod @ B @ A ) @ B @ ( product_fst @ B @ A ) @ ( set2 @ ( product_prod @ B @ A ) @ Xys ) ) ) ) ) ).
% map_of_eq_None_iff
thf(fact_3472_finite__reachable__advance,axiom,
! [A: $tType,E4: set @ ( product_prod @ A @ A ),V0: A,V: A] :
( ( finite_finite2 @ A @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E4 ) @ ( insert2 @ A @ V0 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V0 @ V ) @ ( transitive_rtrancl @ A @ E4 ) )
=> ( finite_finite2 @ A @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E4 ) @ ( insert2 @ A @ V @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% finite_reachable_advance
thf(fact_3473_rtrancl__Image__advance__ss,axiom,
! [A: $tType,U: A,V: A,E4: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V ) @ E4 )
=> ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E4 ) @ ( insert2 @ A @ V @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E4 ) @ ( insert2 @ A @ U @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% rtrancl_Image_advance_ss
thf(fact_3474_trancl__Image__advance__ss,axiom,
! [A: $tType,U: A,V: A,E4: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V ) @ E4 )
=> ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ ( transitive_trancl @ A @ E4 ) @ ( insert2 @ A @ V @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ ( transitive_trancl @ A @ E4 ) @ ( insert2 @ A @ U @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% trancl_Image_advance_ss
thf(fact_3475_subset__Image1__Image1__iff,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( order_preorder_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R ) )
=> ( ( member @ A @ B2 @ ( field2 @ A @ R ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ R ) ) ) ) ) ).
% subset_Image1_Image1_iff
thf(fact_3476_map__of__distinct__upd,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ ( product_prod @ A @ B ),Y: B] :
( ~ ( member @ A @ X @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) ) )
=> ( ( map_add @ A @ B
@ ( fun_upd @ A @ ( option @ B )
@ ^ [X3: A] : ( none @ B )
@ X
@ ( some @ B @ Y ) )
@ ( map_of @ A @ B @ Xs ) )
= ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ Xs ) @ X @ ( some @ B @ Y ) ) ) ) ).
% map_of_distinct_upd
thf(fact_3477_subset__Image__Image__iff,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A4: set @ A,B6: set @ A] :
( ( order_preorder_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( field2 @ A @ R ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ ( field2 @ A @ R ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R @ A4 ) @ ( image @ A @ A @ R @ B6 ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ? [Y3: A] :
( ( member @ A @ Y3 @ B6 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R ) ) ) ) ) ) ) ) ).
% subset_Image_Image_iff
thf(fact_3478_map__of__map__restrict,axiom,
! [B: $tType,A: $tType,F: A > B,Ks: list @ A] :
( ( map_of @ A @ B
@ ( map @ A @ ( product_prod @ A @ B )
@ ^ [K4: A] : ( product_Pair @ A @ B @ K4 @ ( F @ K4 ) )
@ Ks ) )
= ( restrict_map @ A @ B @ ( comp @ B @ ( option @ B ) @ A @ ( some @ B ) @ F ) @ ( set2 @ A @ Ks ) ) ) ).
% map_of_map_restrict
thf(fact_3479_map__of__mapk__SomeI,axiom,
! [A: $tType,B: $tType,C: $tType,F: A > B,T2: list @ ( product_prod @ A @ C ),K: A,X: C] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( ( map_of @ A @ C @ T2 @ K )
= ( some @ C @ X ) )
=> ( ( map_of @ B @ C
@ ( map @ ( product_prod @ A @ C ) @ ( product_prod @ B @ C )
@ ( product_case_prod @ A @ C @ ( product_prod @ B @ C )
@ ^ [K4: A] : ( product_Pair @ B @ C @ ( F @ K4 ) ) )
@ T2 )
@ ( F @ K ) )
= ( some @ C @ X ) ) ) ) ).
% map_of_mapk_SomeI
thf(fact_3480_map__comp__map,axiom,
! [B: $tType,C: $tType,A: $tType,F: C > B,G: A > C] :
( ( comp @ ( list @ C ) @ ( list @ B ) @ ( list @ A ) @ ( map @ C @ B @ F ) @ ( map @ A @ C @ G ) )
= ( map @ A @ B @ ( comp @ C @ B @ A @ F @ G ) ) ) ).
% map_comp_map
thf(fact_3481_map__map,axiom,
! [B: $tType,A: $tType,C: $tType,F: B > A,G: C > B,Xs: list @ C] :
( ( map @ B @ A @ F @ ( map @ C @ B @ G @ Xs ) )
= ( map @ C @ A @ ( comp @ B @ A @ C @ F @ G ) @ Xs ) ) ).
% map_map
thf(fact_3482_List_Omap_Ocompositionality,axiom,
! [B: $tType,C: $tType,A: $tType,F: B > C,G: A > B,List: list @ A] :
( ( map @ B @ C @ F @ ( map @ A @ B @ G @ List ) )
= ( map @ A @ C @ ( comp @ B @ C @ A @ F @ G ) @ List ) ) ).
% List.map.compositionality
thf(fact_3483_list_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,G: B > C,F: A > B,V: list @ A] :
( ( map @ B @ C @ G @ ( map @ A @ B @ F @ V ) )
= ( map @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ V ) ) ).
% list.map_comp
thf(fact_3484_List_Omap_Ocomp,axiom,
! [C: $tType,B: $tType,A: $tType,F: B > C,G: A > B] :
( ( comp @ ( list @ B ) @ ( list @ C ) @ ( list @ A ) @ ( map @ B @ C @ F ) @ ( map @ A @ B @ G ) )
= ( map @ A @ C @ ( comp @ B @ C @ A @ F @ G ) ) ) ).
% List.map.comp
thf(fact_3485_map__fst__mk__snd,axiom,
! [B: $tType,A: $tType,K: B,L: list @ A] :
( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
@ ( map @ A @ ( product_prod @ A @ B )
@ ^ [X3: A] : ( product_Pair @ A @ B @ X3 @ K )
@ L ) )
= L ) ).
% map_fst_mk_snd
thf(fact_3486_map__snd__mk__fst,axiom,
! [B: $tType,A: $tType,K: B,L: list @ A] :
( ( map @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( map @ A @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ K ) @ L ) )
= L ) ).
% map_snd_mk_fst
thf(fact_3487_set__oo__map__alt,axiom,
! [B: $tType,A: $tType,F: A > B] :
( ( comp @ ( list @ B ) @ ( set @ B ) @ ( list @ A ) @ ( set2 @ B ) @ ( map @ A @ B @ F ) )
= ( ^ [L3: list @ A] : ( image2 @ A @ B @ F @ ( set2 @ A @ L3 ) ) ) ) ).
% set_oo_map_alt
thf(fact_3488_Id__on__set,axiom,
! [A: $tType,Xs: list @ A] :
( ( id_on @ A @ ( set2 @ A @ Xs ) )
= ( set2 @ ( product_prod @ A @ A )
@ ( map @ A @ ( product_prod @ A @ A )
@ ^ [X3: A] : ( product_Pair @ A @ A @ X3 @ X3 )
@ Xs ) ) ) ).
% Id_on_set
thf(fact_3489_map__of__map__keys,axiom,
! [B: $tType,A: $tType,Xs: list @ A,M2: A > ( option @ B )] :
( ( ( set2 @ A @ Xs )
= ( dom @ A @ B @ M2 ) )
=> ( ( map_of @ A @ B
@ ( map @ A @ ( product_prod @ A @ B )
@ ^ [K4: A] : ( product_Pair @ A @ B @ K4 @ ( the2 @ B @ ( M2 @ K4 ) ) )
@ Xs ) )
= M2 ) ) ).
% map_of_map_keys
thf(fact_3490_map__of__map,axiom,
! [B: $tType,C: $tType,A: $tType,F: C > B,Xs: list @ ( product_prod @ A @ C )] :
( ( map_of @ A @ B
@ ( map @ ( product_prod @ A @ C ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ C @ ( product_prod @ A @ B )
@ ^ [K4: A,V2: C] : ( product_Pair @ A @ B @ K4 @ ( F @ V2 ) ) )
@ Xs ) )
= ( comp @ ( option @ C ) @ ( option @ B ) @ A @ ( map_option @ C @ B @ F ) @ ( map_of @ A @ C @ Xs ) ) ) ).
% map_of_map
thf(fact_3491_inj__on__map__inv__f,axiom,
! [B: $tType,A: $tType,L: list @ A,A4: set @ A,F: A > B] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ L ) @ A4 )
=> ( ( inj_on @ A @ B @ F @ A4 )
=> ( ( map @ B @ A @ ( inv_on @ A @ B @ F @ A4 ) @ ( map @ A @ B @ F @ L ) )
= L ) ) ) ).
% inj_on_map_inv_f
thf(fact_3492_Misc_Oran__distinct,axiom,
! [B: $tType,A: $tType,Al: list @ ( product_prod @ A @ B )] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Al ) )
=> ( ( ran @ A @ B @ ( map_of @ A @ B @ Al ) )
= ( image2 @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ ( set2 @ ( product_prod @ A @ B ) @ Al ) ) ) ) ).
% Misc.ran_distinct
thf(fact_3493_map__of__eq__Some__iff,axiom,
! [B: $tType,A: $tType,Xys: list @ ( product_prod @ A @ B ),X: A,Y: B] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xys ) )
=> ( ( ( map_of @ A @ B @ Xys @ X )
= ( some @ B @ Y ) )
= ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ Xys ) ) ) ) ).
% map_of_eq_Some_iff
thf(fact_3494_Some__eq__map__of__iff,axiom,
! [B: $tType,A: $tType,Xys: list @ ( product_prod @ A @ B ),Y: B,X: A] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xys ) )
=> ( ( ( some @ B @ Y )
= ( map_of @ A @ B @ Xys @ X ) )
= ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ Xys ) ) ) ) ).
% Some_eq_map_of_iff
thf(fact_3495_map__of__is__SomeI,axiom,
! [A: $tType,B: $tType,Xys: list @ ( product_prod @ A @ B ),X: A,Y: B] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xys ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ Xys ) )
=> ( ( map_of @ A @ B @ Xys @ X )
= ( some @ B @ Y ) ) ) ) ).
% map_of_is_SomeI
thf(fact_3496_inv__on__f__f,axiom,
! [B: $tType,A: $tType,F: A > B,A4: set @ A,X: A] :
( ( inj_on @ A @ B @ F @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( inv_on @ A @ B @ F @ A4 @ ( F @ X ) )
= X ) ) ) ).
% inv_on_f_f
thf(fact_3497_distinct__mapI,axiom,
! [A: $tType,B: $tType,F: B > A,L: list @ B] :
( ( distinct @ A @ ( map @ B @ A @ F @ L ) )
=> ( distinct @ B @ L ) ) ).
% distinct_mapI
thf(fact_3498_distinct__finite__set,axiom,
! [A: $tType,X: set @ A] :
( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Ys: list @ A] :
( ( ( set2 @ A @ Ys )
= X )
& ( distinct @ A @ Ys ) ) ) ) ).
% distinct_finite_set
thf(fact_3499_distinct__map__eq,axiom,
! [A: $tType,B: $tType,F: B > A,L: list @ B,X: B,Y: B] :
( ( distinct @ A @ ( map @ B @ A @ F @ L ) )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member @ B @ X @ ( set2 @ B @ L ) )
=> ( ( member @ B @ Y @ ( set2 @ B @ L ) )
=> ( X = Y ) ) ) ) ) ).
% distinct_map_eq
thf(fact_3500_finite__set__image,axiom,
! [A: $tType,A4: set @ ( list @ A )] :
( ( finite_finite2 @ ( set @ A ) @ ( image2 @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ A4 ) )
=> ( ! [Xs2: list @ A] :
( ( member @ ( list @ A ) @ Xs2 @ A4 )
=> ( distinct @ A @ Xs2 ) )
=> ( finite_finite2 @ ( list @ A ) @ A4 ) ) ) ).
% finite_set_image
thf(fact_3501_inv__on__f__range,axiom,
! [A: $tType,B: $tType,Y: A,F: B > A,A4: set @ B] :
( ( member @ A @ Y @ ( image2 @ B @ A @ F @ A4 ) )
=> ( member @ B @ ( inv_on @ B @ A @ F @ A4 @ Y ) @ A4 ) ) ).
% inv_on_f_range
thf(fact_3502_f__inv__on__f,axiom,
! [B: $tType,A: $tType,Y: A,F: B > A,A4: set @ B] :
( ( member @ A @ Y @ ( image2 @ B @ A @ F @ A4 ) )
=> ( ( F @ ( inv_on @ B @ A @ F @ A4 @ Y ) )
= Y ) ) ).
% f_inv_on_f
thf(fact_3503_distinct__finite__subset,axiom,
! [A: $tType,X: set @ A] :
( ( finite_finite2 @ A @ X )
=> ( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Ys: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Ys ) @ X )
& ( distinct @ A @ Ys ) ) ) ) ) ).
% distinct_finite_subset
thf(fact_3504_distinct__map__fstD,axiom,
! [A: $tType,B: $tType,Xs: list @ ( product_prod @ A @ B ),X: A,Y: B,Z2: B] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ Xs ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Z2 ) @ ( set2 @ ( product_prod @ A @ B ) @ Xs ) )
=> ( Y = Z2 ) ) ) ) ).
% distinct_map_fstD
thf(fact_3505_eq__key__imp__eq__value,axiom,
! [A: $tType,B: $tType,Xs: list @ ( product_prod @ A @ B ),K: A,V1: B,V22: B] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V1 ) @ ( set2 @ ( product_prod @ A @ B ) @ Xs ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V22 ) @ ( set2 @ ( product_prod @ A @ B ) @ Xs ) )
=> ( V1 = V22 ) ) ) ) ).
% eq_key_imp_eq_value
thf(fact_3506_map__to__set__map__of,axiom,
! [B: $tType,A: $tType,L: list @ ( product_prod @ A @ B )] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ L ) )
=> ( ( map_to_set @ A @ B @ ( map_of @ A @ B @ L ) )
= ( set2 @ ( product_prod @ A @ B ) @ L ) ) ) ).
% map_to_set_map_of
thf(fact_3507_map__of__map__to__set,axiom,
! [B: $tType,A: $tType,L: list @ ( product_prod @ A @ B ),M2: A > ( option @ B )] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ L ) )
=> ( ( ( map_of @ A @ B @ L )
= M2 )
= ( ( set2 @ ( product_prod @ A @ B ) @ L )
= ( map_to_set @ A @ B @ M2 ) ) ) ) ).
% map_of_map_to_set
thf(fact_3508_set__map__of__compr,axiom,
! [B: $tType,A: $tType,Xs: list @ ( product_prod @ A @ B )] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) )
=> ( ( set2 @ ( product_prod @ A @ B ) @ Xs )
= ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [K4: A,V2: B] :
( ( map_of @ A @ B @ Xs @ K4 )
= ( some @ B @ V2 ) ) ) ) ) ) ).
% set_map_of_compr
thf(fact_3509_card__lists__distinct__length__eq,axiom,
! [A: $tType,A4: set @ A,K: nat] :
( ( finite_finite2 @ A @ A4 )
=> ( ( ord_less_eq @ nat @ K @ ( finite_card @ A @ A4 ) )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= K )
& ( distinct @ A @ Xs3 )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A4 ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X3: nat] : X3
@ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A4 ) @ K ) @ ( one_one @ nat ) ) @ ( finite_card @ A @ A4 ) ) ) ) ) ) ).
% card_lists_distinct_length_eq
thf(fact_3510_card__lists__distinct__length__eq_H,axiom,
! [A: $tType,K: nat,A4: set @ A] :
( ( ord_less @ nat @ K @ ( finite_card @ A @ A4 ) )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= K )
& ( distinct @ A @ Xs3 )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A4 ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X3: nat] : X3
@ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A4 ) @ K ) @ ( one_one @ nat ) ) @ ( finite_card @ A @ A4 ) ) ) ) ) ).
% card_lists_distinct_length_eq'
thf(fact_3511_distinct__foldl__invar,axiom,
! [B: $tType,A: $tType,S2: list @ A,I4: ( set @ A ) > B > $o,Sigma_0: B,F: B > A > B] :
( ( distinct @ A @ S2 )
=> ( ( I4 @ ( set2 @ A @ S2 ) @ Sigma_0 )
=> ( ! [X2: A,It: set @ A,Sigma: B] :
( ( member @ A @ X2 @ It )
=> ( ( ord_less_eq @ ( set @ A ) @ It @ ( set2 @ A @ S2 ) )
=> ( ( I4 @ It @ Sigma )
=> ( I4 @ ( minus_minus @ ( set @ A ) @ It @ ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( F @ Sigma @ X2 ) ) ) ) )
=> ( I4 @ ( bot_bot @ ( set @ A ) ) @ ( foldl @ B @ A @ F @ Sigma_0 @ S2 ) ) ) ) ) ).
% distinct_foldl_invar
thf(fact_3512_map__of__Some__filter__not__in,axiom,
! [B: $tType,A: $tType,Xs: list @ ( product_prod @ B @ A ),K: B,V: A,P: ( product_prod @ B @ A ) > $o] :
( ( ( map_of @ B @ A @ Xs @ K )
= ( some @ A @ V ) )
=> ( ~ ( P @ ( product_Pair @ B @ A @ K @ V ) )
=> ( ( distinct @ B @ ( map @ ( product_prod @ B @ A ) @ B @ ( product_fst @ B @ A ) @ Xs ) )
=> ( ( map_of @ B @ A @ ( filter @ ( product_prod @ B @ A ) @ P @ Xs ) @ K )
= ( none @ A ) ) ) ) ) ).
% map_of_Some_filter_not_in
thf(fact_3513_map__add__map__of__foldr,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B ),Ps: list @ ( product_prod @ A @ B )] :
( ( map_add @ A @ B @ M2 @ ( map_of @ A @ B @ Ps ) )
= ( foldr @ ( product_prod @ A @ B ) @ ( A > ( option @ B ) )
@ ( product_case_prod @ A @ B @ ( ( A > ( option @ B ) ) > A > ( option @ B ) )
@ ^ [K4: A,V2: B,M: A > ( option @ B )] : ( fun_upd @ A @ ( option @ B ) @ M @ K4 @ ( some @ B @ V2 ) ) )
@ Ps
@ M2 ) ) ).
% map_add_map_of_foldr
thf(fact_3514_foldl__length,axiom,
! [A: $tType,L: list @ A] :
( ( foldl @ nat @ A
@ ^ [I3: nat,X3: A] : ( suc @ I3 )
@ ( zero_zero @ nat )
@ L )
= ( size_size @ ( list @ A ) @ L ) ) ).
% foldl_length
thf(fact_3515_foldl__length__aux,axiom,
! [A: $tType,A3: nat,L: list @ A] :
( ( foldl @ nat @ A
@ ^ [I3: nat,X3: A] : ( suc @ I3 )
@ A3
@ L )
= ( plus_plus @ nat @ A3 @ ( size_size @ ( list @ A ) @ L ) ) ) ).
% foldl_length_aux
thf(fact_3516_foldl__A1__eq,axiom,
! [A: $tType,F: A > A > A,N: A,I: A,Ww: list @ A] :
( ! [E3: A] :
( ( F @ N @ E3 )
= E3 )
=> ( ! [E3: A] :
( ( F @ E3 @ N )
= E3 )
=> ( ! [A5: A,B3: A,C4: A] :
( ( F @ A5 @ ( F @ B3 @ C4 ) )
= ( F @ ( F @ A5 @ B3 ) @ C4 ) )
=> ( ( foldl @ A @ A @ F @ I @ Ww )
= ( F @ I @ ( foldl @ A @ A @ F @ N @ Ww ) ) ) ) ) ) ).
% foldl_A1_eq
thf(fact_3517_sum__length__filter__compl,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ ( filter @ A @ P @ Xs ) )
@ ( size_size @ ( list @ A )
@ ( filter @ A
@ ^ [X3: A] :
~ ( P @ X3 )
@ Xs ) ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% sum_length_filter_compl
thf(fact_3518_comp__fun__commute_Ofoldr__conv__foldl,axiom,
! [B: $tType,A: $tType,F: A > B > B,Xs: list @ A,A3: B] :
( ( finite6289374366891150609ommute @ A @ B @ F )
=> ( ( foldr @ A @ B @ F @ Xs @ A3 )
= ( foldl @ B @ A
@ ^ [A7: B,B5: A] : ( F @ B5 @ A7 )
@ A3
@ Xs ) ) ) ).
% comp_fun_commute.foldr_conv_foldl
thf(fact_3519_comp__fun__commute_Ofoldl__f__commute,axiom,
! [B: $tType,A: $tType,F: A > B > B,A3: A,B2: B,Xs: list @ A] :
( ( finite6289374366891150609ommute @ A @ B @ F )
=> ( ( F @ A3
@ ( foldl @ B @ A
@ ^ [A7: B,B5: A] : ( F @ B5 @ A7 )
@ B2
@ Xs ) )
= ( foldl @ B @ A
@ ^ [A7: B,B5: A] : ( F @ B5 @ A7 )
@ ( F @ A3 @ B2 )
@ Xs ) ) ) ).
% comp_fun_commute.foldl_f_commute
thf(fact_3520_map__of__None__filterD,axiom,
! [B: $tType,A: $tType,Xs: list @ ( product_prod @ B @ A ),X: B,P: ( product_prod @ B @ A ) > $o] :
( ( ( map_of @ B @ A @ Xs @ X )
= ( none @ A ) )
=> ( ( map_of @ B @ A @ ( filter @ ( product_prod @ B @ A ) @ P @ Xs ) @ X )
= ( none @ A ) ) ) ).
% map_of_None_filterD
thf(fact_3521_fst__foldl,axiom,
! [B: $tType,A: $tType,C: $tType,F: A > C > A,G: A > B > C > B,A3: A,B2: B,Xs: list @ C] :
( ( product_fst @ A @ B
@ ( foldl @ ( product_prod @ A @ B ) @ C
@ ( product_case_prod @ A @ B @ ( C > ( product_prod @ A @ B ) )
@ ^ [A7: A,B5: B,X3: C] : ( product_Pair @ A @ B @ ( F @ A7 @ X3 ) @ ( G @ A7 @ B5 @ X3 ) ) )
@ ( product_Pair @ A @ B @ A3 @ B2 )
@ Xs ) )
= ( foldl @ A @ C @ F @ A3 @ Xs ) ) ).
% fst_foldl
thf(fact_3522_foldl__absorb1,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [X: A,Zs: list @ A] :
( ( times_times @ A @ X @ ( foldl @ A @ A @ ( times_times @ A ) @ ( one_one @ A ) @ Zs ) )
= ( foldl @ A @ A @ ( times_times @ A ) @ X @ Zs ) ) ) ).
% foldl_absorb1
thf(fact_3523_foldl__un__empty__eq,axiom,
! [A: $tType,I: set @ A,Ww: list @ ( set @ A )] :
( ( foldl @ ( set @ A ) @ ( set @ A ) @ ( sup_sup @ ( set @ A ) ) @ I @ Ww )
= ( sup_sup @ ( set @ A ) @ I @ ( foldl @ ( set @ A ) @ ( set @ A ) @ ( sup_sup @ ( set @ A ) ) @ ( bot_bot @ ( set @ A ) ) @ Ww ) ) ) ).
% foldl_un_empty_eq
thf(fact_3524_distinct__map__fst__filterI,axiom,
! [B: $tType,A: $tType,Xs: list @ ( product_prod @ A @ B ),P: ( product_prod @ A @ B ) > $o] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) )
=> ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( filter @ ( product_prod @ A @ B ) @ P @ Xs ) ) ) ) ).
% distinct_map_fst_filterI
thf(fact_3525_map__of__filter__in,axiom,
! [B: $tType,A: $tType,Xs: list @ ( product_prod @ B @ A ),K: B,Z2: A,P: B > A > $o] :
( ( ( map_of @ B @ A @ Xs @ K )
= ( some @ A @ Z2 ) )
=> ( ( P @ K @ Z2 )
=> ( ( map_of @ B @ A @ ( filter @ ( product_prod @ B @ A ) @ ( product_case_prod @ B @ A @ $o @ P ) @ Xs ) @ K )
= ( some @ A @ Z2 ) ) ) ) ).
% map_of_filter_in
thf(fact_3526_distinct__length__le,axiom,
! [A: $tType,Ys2: list @ A,Xs: list @ A] :
( ( distinct @ A @ Ys2 )
=> ( ( ( set2 @ A @ Ys2 )
= ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ys2 ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% distinct_length_le
thf(fact_3527_foldl__set,axiom,
! [A: $tType,L: list @ ( set @ A )] :
( ( foldl @ ( set @ A ) @ ( set @ A ) @ ( sup_sup @ ( set @ A ) ) @ ( bot_bot @ ( set @ A ) ) @ L )
= ( complete_Sup_Sup @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [X3: set @ A] : ( member @ ( set @ A ) @ X3 @ ( set2 @ ( set @ A ) @ L ) ) ) ) ) ).
% foldl_set
thf(fact_3528_map__filter__map__filter,axiom,
! [A: $tType,B: $tType,F: B > A,P: B > $o,Xs: list @ B] :
( ( map @ B @ A @ F @ ( filter @ B @ P @ Xs ) )
= ( map_filter @ B @ A
@ ^ [X3: B] : ( if @ ( option @ A ) @ ( P @ X3 ) @ ( some @ A @ ( F @ X3 ) ) @ ( none @ A ) )
@ Xs ) ) ).
% map_filter_map_filter
thf(fact_3529_map__filter__def,axiom,
! [B: $tType,A: $tType] :
( ( map_filter @ A @ B )
= ( ^ [F2: A > ( option @ B ),Xs3: list @ A] :
( map @ A @ B @ ( comp @ ( option @ B ) @ B @ A @ ( the2 @ B ) @ F2 )
@ ( filter @ A
@ ^ [X3: A] :
( ( F2 @ X3 )
!= ( none @ B ) )
@ Xs3 ) ) ) ) ).
% map_filter_def
thf(fact_3530_execute__of__list,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [Xs: list @ A,H3: heap_ext @ product_unit] :
( ( heap_Time_execute @ ( array @ A ) @ ( array_of_list @ A @ Xs ) @ H3 )
= ( some @ ( 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 ) )
@ ^ [R3: array @ A,H7: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H7 @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) )
@ ( array_alloc @ A @ Xs @ H3 ) ) ) ) ) ).
% execute_of_list
thf(fact_3531_mergesort__by__rel__split__length,axiom,
! [A: $tType,Xs1: list @ A,Xs22: list @ A,Xs: list @ A] :
( ( ( size_size @ ( list @ A ) @ ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) @ Xs ) ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs1 ) @ ( divide_divide @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( modulo_modulo @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( ( size_size @ ( list @ A ) @ ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) @ Xs ) ) )
= ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs22 ) @ ( divide_divide @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% mergesort_by_rel_split_length
thf(fact_3532_card__disjoint__shuffles,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys2 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_card @ ( list @ A ) @ ( shuffles @ A @ Xs @ Ys2 ) )
= ( binomial @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys2 ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% card_disjoint_shuffles
thf(fact_3533_of__list__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_of_list @ A )
= ( ^ [Xs3: list @ A] :
( heap_Time_heap @ ( array @ A )
@ ^ [H: heap_ext @ product_unit] :
( product_case_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
@ ^ [R3: array @ A,H7: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H7 @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( size_size @ ( list @ A ) @ Xs3 ) ) ) )
@ ( array_alloc @ A @ Xs3 @ H ) ) ) ) ) ) ).
% of_list_def
thf(fact_3534_slice__len,axiom,
! [A: $tType,From: nat,To: nat,Xs: list @ A] :
( ( ord_less_eq @ nat @ From @ To )
=> ( ( ord_less_eq @ nat @ To @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( size_size @ ( list @ A ) @ ( slice @ A @ From @ To @ Xs ) )
= ( minus_minus @ nat @ To @ From ) ) ) ) ).
% slice_len
thf(fact_3535_slice__complete,axiom,
! [A: $tType,Xs: list @ A] :
( ( slice @ A @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) @ Xs )
= Xs ) ).
% slice_complete
thf(fact_3536_foldr__length,axiom,
! [A: $tType,L: list @ A] :
( ( foldr @ A @ nat
@ ^ [X3: A] : suc
@ L
@ ( zero_zero @ nat ) )
= ( size_size @ ( list @ A ) @ L ) ) ).
% foldr_length
thf(fact_3537_length__shuffles,axiom,
! [A: $tType,Zs: list @ A,Xs: list @ A,Ys2: list @ A] :
( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys2 ) )
=> ( ( size_size @ ( list @ A ) @ Zs )
= ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys2 ) ) ) ) ).
% length_shuffles
thf(fact_3538_foldr__length__aux,axiom,
! [A: $tType,L: list @ A,A3: nat] :
( ( foldr @ A @ nat
@ ^ [X3: A] : suc
@ L
@ A3 )
= ( plus_plus @ nat @ A3 @ ( size_size @ ( list @ A ) @ L ) ) ) ).
% foldr_length_aux
thf(fact_3539_effect__of__listI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [A3: array @ A,H4: heap_ext @ product_unit,Xs: list @ A,H3: heap_ext @ product_unit] :
( ( ( product_Pair @ ( array @ A ) @ ( heap_ext @ product_unit ) @ A3 @ H4 )
= ( array_alloc @ A @ Xs @ H3 ) )
=> ( heap_Time_effect @ ( array @ A ) @ ( array_of_list @ A @ Xs ) @ H3 @ H4 @ A3 @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ).
% effect_of_listI
thf(fact_3540_execute__make,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [N: nat,F: nat > A,H3: heap_ext @ product_unit] :
( ( heap_Time_execute @ ( array @ A ) @ ( array_make @ A @ N @ F ) @ H3 )
= ( some @ ( 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 ) )
@ ^ [R3: array @ A,H7: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H7 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) )
@ ( array_alloc @ A @ ( map @ nat @ A @ F @ ( upt @ ( zero_zero @ nat ) @ N ) ) @ H3 ) ) ) ) ) ).
% execute_make
thf(fact_3541_execute__new,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [N: nat,X: A,H3: heap_ext @ product_unit] :
( ( heap_Time_execute @ ( array @ A ) @ ( array_new @ A @ N @ X ) @ H3 )
= ( some @ ( 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 ) )
@ ^ [R3: array @ A,H7: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H7 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) )
@ ( array_alloc @ A @ ( replicate @ A @ N @ X ) @ H3 ) ) ) ) ) ).
% execute_new
thf(fact_3542_Array__Time_Omake__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_make @ A )
= ( ^ [N5: nat,F2: nat > A] :
( heap_Time_heap @ ( array @ A )
@ ^ [H: heap_ext @ product_unit] :
( product_case_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
@ ^ [R3: array @ A,H7: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H7 @ ( plus_plus @ nat @ N5 @ ( one_one @ nat ) ) ) )
@ ( array_alloc @ A @ ( map @ nat @ A @ F2 @ ( upt @ ( zero_zero @ nat ) @ N5 ) ) @ H ) ) ) ) ) ) ).
% Array_Time.make_def
thf(fact_3543_new__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_new @ A )
= ( ^ [N5: nat,X3: A] :
( heap_Time_heap @ ( array @ A )
@ ^ [H: heap_ext @ product_unit] :
( product_case_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
@ ^ [R3: array @ A,H7: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H7 @ ( plus_plus @ nat @ N5 @ ( one_one @ nat ) ) ) )
@ ( array_alloc @ A @ ( replicate @ A @ N5 @ X3 ) @ H ) ) ) ) ) ) ).
% new_def
thf(fact_3544_upt__0__eq__Nil__conv,axiom,
! [J: nat] :
( ( ( upt @ ( zero_zero @ nat ) @ J )
= ( nil @ nat ) )
= ( J
= ( zero_zero @ nat ) ) ) ).
% upt_0_eq_Nil_conv
thf(fact_3545_map__of__eq__empty__iff,axiom,
! [B: $tType,A: $tType,Xys: list @ ( product_prod @ A @ B )] :
( ( ( map_of @ A @ B @ Xys )
= ( ^ [X3: A] : ( none @ B ) ) )
= ( Xys
= ( nil @ ( product_prod @ A @ B ) ) ) ) ).
% map_of_eq_empty_iff
thf(fact_3546_slice__Nil,axiom,
! [A: $tType,Begin: nat,End: nat] :
( ( slice @ A @ Begin @ End @ ( nil @ A ) )
= ( nil @ A ) ) ).
% slice_Nil
thf(fact_3547_slice__eq__bounds__empty,axiom,
! [A: $tType,I: nat,Xs: list @ A] :
( ( slice @ A @ I @ I @ Xs )
= ( nil @ A ) ) ).
% slice_eq_bounds_empty
thf(fact_3548_empty__eq__map__of__iff,axiom,
! [B: $tType,A: $tType,Xys: list @ ( product_prod @ A @ B )] :
( ( ( ^ [X3: A] : ( none @ B ) )
= ( map_of @ A @ B @ Xys ) )
= ( Xys
= ( nil @ ( product_prod @ A @ B ) ) ) ) ).
% empty_eq_map_of_iff
thf(fact_3549_length__ge__1__conv,axiom,
! [A: $tType,L: list @ A] :
( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( size_size @ ( list @ A ) @ L ) )
= ( L
!= ( nil @ A ) ) ) ).
% length_ge_1_conv
thf(fact_3550_map__fst__mk__fst,axiom,
! [B: $tType,A: $tType,K: A,L: list @ B] :
( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K ) @ L ) )
= ( replicate @ A @ ( size_size @ ( list @ B ) @ L ) @ K ) ) ).
% map_fst_mk_fst
thf(fact_3551_map__snd__mk__snd,axiom,
! [B: $tType,A: $tType,K: A,L: list @ B] :
( ( map @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A )
@ ( map @ B @ ( product_prod @ B @ A )
@ ^ [X3: B] : ( product_Pair @ B @ A @ X3 @ K )
@ L ) )
= ( replicate @ A @ ( size_size @ ( list @ B ) @ L ) @ K ) ) ).
% map_snd_mk_snd
thf(fact_3552_map__add__upt_H,axiom,
! [Ofs: nat,A3: nat,B2: nat] :
( ( map @ nat @ nat
@ ^ [I3: nat] : ( plus_plus @ nat @ I3 @ Ofs )
@ ( upt @ A3 @ B2 ) )
= ( upt @ ( plus_plus @ nat @ A3 @ Ofs ) @ ( plus_plus @ nat @ B2 @ Ofs ) ) ) ).
% map_add_upt'
thf(fact_3553_len__greater__imp__nonempty,axiom,
! [A: $tType,X: nat,L: list @ A] :
( ( ord_less @ nat @ X @ ( size_size @ ( list @ A ) @ L ) )
=> ( L
!= ( nil @ A ) ) ) ).
% len_greater_imp_nonempty
thf(fact_3554_map__add__upt,axiom,
! [N: nat,M2: nat] :
( ( map @ nat @ nat
@ ^ [I3: nat] : ( plus_plus @ nat @ I3 @ N )
@ ( upt @ ( zero_zero @ nat ) @ M2 ) )
= ( upt @ N @ ( plus_plus @ nat @ M2 @ N ) ) ) ).
% map_add_upt
thf(fact_3555_map__of__Cons__code_I1_J,axiom,
! [B: $tType,A: $tType,K: B] :
( ( map_of @ B @ A @ ( nil @ ( product_prod @ B @ A ) ) @ K )
= ( none @ A ) ) ).
% map_of_Cons_code(1)
thf(fact_3556_map__of_Osimps_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( map_of @ A @ B @ ( nil @ ( product_prod @ A @ B ) ) )
= ( ^ [X3: A] : ( none @ B ) ) ) ).
% map_of.simps(1)
thf(fact_3557_mergesort__by__rel__split_Osimps_I1_J,axiom,
! [A: $tType,Xs1: list @ A,Xs22: list @ A] :
( ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) @ ( nil @ A ) )
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) ) ).
% mergesort_by_rel_split.simps(1)
thf(fact_3558_upt__filter__extend,axiom,
! [U: nat,U5: nat,P: nat > $o] :
( ( ord_less_eq @ nat @ U @ U5 )
=> ( ! [I2: nat] :
( ( ( ord_less_eq @ nat @ U @ I2 )
& ( ord_less @ nat @ I2 @ U5 ) )
=> ~ ( P @ I2 ) )
=> ( ( filter @ nat @ P @ ( upt @ ( zero_zero @ nat ) @ U ) )
= ( filter @ nat @ P @ ( upt @ ( zero_zero @ nat ) @ U5 ) ) ) ) ) ).
% upt_filter_extend
thf(fact_3559_effect__newI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [A3: array @ A,H4: heap_ext @ product_unit,N: nat,X: A,H3: heap_ext @ product_unit] :
( ( ( product_Pair @ ( array @ A ) @ ( heap_ext @ product_unit ) @ A3 @ H4 )
= ( array_alloc @ A @ ( replicate @ A @ N @ X ) @ H3 ) )
=> ( heap_Time_effect @ ( array @ A ) @ ( array_new @ A @ N @ X ) @ H3 @ H4 @ A3 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% effect_newI
thf(fact_3560_effect__makeI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [A3: array @ A,H4: heap_ext @ product_unit,F: nat > A,N: nat,H3: heap_ext @ product_unit] :
( ( ( product_Pair @ ( array @ A ) @ ( heap_ext @ product_unit ) @ A3 @ H4 )
= ( array_alloc @ A @ ( map @ nat @ A @ F @ ( upt @ ( zero_zero @ nat ) @ N ) ) @ H3 ) )
=> ( heap_Time_effect @ ( array @ A ) @ ( array_make @ A @ N @ F ) @ H3 @ H4 @ A3 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% effect_makeI
thf(fact_3561_mergesort__by__rel_Opinduct,axiom,
! [A: $tType,A0: A > A > $o,A1: list @ A,P: ( A > A > $o ) > ( list @ A ) > $o] :
( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( mergesort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ A0 @ A1 ) )
=> ( ! [R8: A > A > $o,Xs2: list @ A] :
( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( mergesort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ R8 @ Xs2 ) )
=> ( ( ~ ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( P @ R8 @ ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xs2 ) ) ) )
=> ( ( ~ ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( P @ R8 @ ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xs2 ) ) ) )
=> ( P @ R8 @ Xs2 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% mergesort_by_rel.pinduct
thf(fact_3562_mergesort__by__rel_Oelims,axiom,
! [A: $tType,X: A > A > $o,Xa: list @ A,Y: list @ A] :
( ( ( mergesort_by_rel @ A @ X @ Xa )
= Y )
=> ( ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xa ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( Y = Xa ) )
& ( ~ ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xa ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( Y
= ( merges9089515139780605204_merge @ A @ X @ ( mergesort_by_rel @ A @ X @ ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xa ) ) ) @ ( mergesort_by_rel @ A @ X @ ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xa ) ) ) ) ) ) ) ) ).
% mergesort_by_rel.elims
thf(fact_3563_mergesort__by__rel_Osimps,axiom,
! [A: $tType] :
( ( mergesort_by_rel @ A )
= ( ^ [R6: A > A > $o,Xs3: list @ A] : ( if @ ( list @ A ) @ ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Xs3 @ ( merges9089515139780605204_merge @ A @ R6 @ ( mergesort_by_rel @ A @ R6 @ ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xs3 ) ) ) @ ( mergesort_by_rel @ A @ R6 @ ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xs3 ) ) ) ) ) ) ) ).
% mergesort_by_rel.simps
thf(fact_3564_enumerate__replicate__eq,axiom,
! [A: $tType,N: nat,M2: nat,A3: A] :
( ( enumerate @ A @ N @ ( replicate @ A @ M2 @ A3 ) )
= ( map @ nat @ ( product_prod @ nat @ A )
@ ^ [Q5: nat] : ( product_Pair @ nat @ A @ Q5 @ A3 )
@ ( upt @ N @ ( plus_plus @ nat @ N @ M2 ) ) ) ) ).
% enumerate_replicate_eq
thf(fact_3565_nth__step__trancl,axiom,
! [A: $tType,Xs: list @ A,R5: set @ ( product_prod @ A @ A ),N: nat,M2: nat] :
( ! [N3: nat] :
( ( ord_less @ nat @ N3 @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ ( suc @ N3 ) ) @ ( nth @ A @ Xs @ N3 ) ) @ R5 ) )
=> ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ M2 @ N )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ N ) @ ( nth @ A @ Xs @ M2 ) ) @ ( transitive_trancl @ A @ R5 ) ) ) ) ) ).
% nth_step_trancl
thf(fact_3566_mergesort__by__rel__simps_I1_J,axiom,
! [A: $tType,R5: A > A > $o] :
( ( mergesort_by_rel @ A @ R5 @ ( nil @ A ) )
= ( nil @ A ) ) ).
% mergesort_by_rel_simps(1)
thf(fact_3567_set__mergesort__by__rel,axiom,
! [A: $tType,R5: A > A > $o,Xs: list @ A] :
( ( set2 @ A @ ( mergesort_by_rel @ A @ R5 @ Xs ) )
= ( set2 @ A @ Xs ) ) ).
% set_mergesort_by_rel
thf(fact_3568_mergesort__by__rel__merge__simps_I3_J,axiom,
! [A: $tType,R5: A > A > $o,Ys2: list @ A] :
( ( merges9089515139780605204_merge @ A @ R5 @ ( nil @ A ) @ Ys2 )
= Ys2 ) ).
% mergesort_by_rel_merge_simps(3)
thf(fact_3569_nth__upt,axiom,
! [I: nat,K: nat,J: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ J )
=> ( ( nth @ nat @ ( upt @ I @ J ) @ K )
= ( plus_plus @ nat @ I @ K ) ) ) ).
% nth_upt
thf(fact_3570_set__mergesort__by__rel__merge,axiom,
! [A: $tType,R5: A > A > $o,Xs: list @ A,Ys2: list @ A] :
( ( set2 @ A @ ( merges9089515139780605204_merge @ A @ R5 @ Xs @ Ys2 ) )
= ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys2 ) ) ) ).
% set_mergesort_by_rel_merge
thf(fact_3571_map__fst__enumerate,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( map @ ( product_prod @ nat @ A ) @ nat @ ( product_fst @ nat @ A ) @ ( enumerate @ A @ N @ Xs ) )
= ( upt @ N @ ( plus_plus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% map_fst_enumerate
thf(fact_3572_nth__image__indices,axiom,
! [A: $tType,L: list @ A] :
( ( image2 @ nat @ A @ ( nth @ A @ L ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ L ) ) )
= ( set2 @ A @ L ) ) ).
% nth_image_indices
thf(fact_3573_ran__nth__set__encoding__conv,axiom,
! [A: $tType,L: list @ A] :
( ( ran @ nat @ A
@ ^ [I3: nat] : ( if @ ( option @ A ) @ ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ L ) ) @ ( some @ A @ ( nth @ A @ L @ I3 ) ) @ ( none @ A ) ) )
= ( set2 @ A @ L ) ) ).
% ran_nth_set_encoding_conv
thf(fact_3574_mergesort__by__rel_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( list @ A )] :
~ ! [R8: A > A > $o,Xs2: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ R8 @ Xs2 ) ) ).
% mergesort_by_rel.cases
thf(fact_3575_map__eq__nth__eq,axiom,
! [A: $tType,B: $tType,F: B > A,L: list @ B,L4: list @ B,I: nat] :
( ( ( map @ B @ A @ F @ L )
= ( map @ B @ A @ F @ L4 ) )
=> ( ( F @ ( nth @ B @ L @ I ) )
= ( F @ ( nth @ B @ L4 @ I ) ) ) ) ).
% map_eq_nth_eq
thf(fact_3576_nth__enumerate__eq,axiom,
! [A: $tType,M2: nat,Xs: list @ A,N: nat] :
( ( ord_less @ nat @ M2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N @ Xs ) @ M2 )
= ( product_Pair @ nat @ A @ ( plus_plus @ nat @ N @ M2 ) @ ( nth @ A @ Xs @ M2 ) ) ) ) ).
% nth_enumerate_eq
thf(fact_3577_mergesort__by__rel__merge__simps_I2_J,axiom,
! [A: $tType,R5: A > A > $o,Xs: list @ A] :
( ( merges9089515139780605204_merge @ A @ R5 @ Xs @ ( nil @ A ) )
= Xs ) ).
% mergesort_by_rel_merge_simps(2)
thf(fact_3578_obtain__list__from__elements,axiom,
! [A: $tType,N: nat,P: A > nat > $o] :
( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ N )
=> ? [Li: A] : ( P @ Li @ I2 ) )
=> ~ ! [L5: list @ A] :
( ( ( size_size @ ( list @ A ) @ L5 )
= N )
=> ~ ! [I6: nat] :
( ( ord_less @ nat @ I6 @ N )
=> ( P @ ( nth @ A @ L5 @ I6 ) @ I6 ) ) ) ) ).
% obtain_list_from_elements
thf(fact_3579_in__set__enumerate__eq,axiom,
! [A: $tType,P5: product_prod @ nat @ A,N: nat,Xs: list @ A] :
( ( member @ ( product_prod @ nat @ A ) @ P5 @ ( set2 @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N @ Xs ) ) )
= ( ( ord_less_eq @ nat @ N @ ( product_fst @ nat @ A @ P5 ) )
& ( ord_less @ nat @ ( product_fst @ nat @ A @ P5 ) @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) )
& ( ( nth @ A @ Xs @ ( minus_minus @ nat @ ( product_fst @ nat @ A @ P5 ) @ N ) )
= ( product_snd @ nat @ A @ P5 ) ) ) ) ).
% in_set_enumerate_eq
thf(fact_3580_all__set__conv__nth,axiom,
! [A: $tType,L: list @ A,P: A > $o] :
( ( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ L ) )
=> ( P @ X3 ) ) )
= ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ L ) )
=> ( P @ ( nth @ A @ L @ I3 ) ) ) ) ) ).
% all_set_conv_nth
thf(fact_3581_set__image__eq__pointwiseI,axiom,
! [B: $tType,A: $tType,L: list @ A,L4: list @ A,F: A > B] :
( ( ( size_size @ ( list @ A ) @ L )
= ( size_size @ ( list @ A ) @ L4 ) )
=> ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( F @ ( nth @ A @ L @ I2 ) )
= ( F @ ( nth @ A @ L4 @ I2 ) ) ) )
=> ( ( image2 @ A @ B @ F @ ( set2 @ A @ L ) )
= ( image2 @ A @ B @ F @ ( set2 @ A @ L4 ) ) ) ) ) ).
% set_image_eq_pointwiseI
thf(fact_3582_in__set__image__conv__nth,axiom,
! [B: $tType,A: $tType,F: B > A,X: B,L: list @ B] :
( ( member @ A @ ( F @ X ) @ ( image2 @ B @ A @ F @ ( set2 @ B @ L ) ) )
= ( ? [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ B ) @ L ) )
& ( ( F @ ( nth @ B @ L @ I3 ) )
= ( F @ X ) ) ) ) ) ).
% in_set_image_conv_nth
thf(fact_3583_distinct__idx,axiom,
! [B: $tType,A: $tType,F: B > A,L: list @ B,I: nat,J: nat] :
( ( distinct @ A @ ( map @ B @ A @ F @ L ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ B ) @ L ) )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ B ) @ L ) )
=> ( ( ( F @ ( nth @ B @ L @ I ) )
= ( F @ ( nth @ B @ L @ J ) ) )
=> ( I = J ) ) ) ) ) ).
% distinct_idx
thf(fact_3584_array__of__list__make,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_of_list @ A )
= ( ^ [Xs3: list @ A] : ( array_make @ A @ ( size_size @ ( list @ A ) @ Xs3 ) @ ( nth @ A @ Xs3 ) ) ) ) ) ).
% array_of_list_make
thf(fact_3585_nth__map__upt,axiom,
! [A: $tType,I: nat,N: nat,M2: nat,F: nat > A] :
( ( ord_less @ nat @ I @ ( minus_minus @ nat @ N @ M2 ) )
=> ( ( nth @ A @ ( map @ nat @ A @ F @ ( upt @ M2 @ N ) ) @ I )
= ( F @ ( plus_plus @ nat @ M2 @ I ) ) ) ) ).
% nth_map_upt
thf(fact_3586_mergesort__by__rel_Opsimps,axiom,
! [A: $tType,R5: A > A > $o,Xs: list @ A] :
( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( mergesort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ R5 @ Xs ) )
=> ( ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( ( mergesort_by_rel @ A @ R5 @ Xs )
= Xs ) )
& ( ~ ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( ( mergesort_by_rel @ A @ R5 @ Xs )
= ( merges9089515139780605204_merge @ A @ R5 @ ( mergesort_by_rel @ A @ R5 @ ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xs ) ) ) @ ( mergesort_by_rel @ A @ R5 @ ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xs ) ) ) ) ) ) ) ) ).
% mergesort_by_rel.psimps
thf(fact_3587_mergesort__by__rel_Opelims,axiom,
! [A: $tType,X: A > A > $o,Xa: list @ A,Y: list @ A] :
( ( ( mergesort_by_rel @ A @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( mergesort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ Xa ) )
=> ~ ( ( ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xa ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( Y = Xa ) )
& ( ~ ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xa ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( Y
= ( merges9089515139780605204_merge @ A @ X @ ( mergesort_by_rel @ A @ X @ ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xa ) ) ) @ ( mergesort_by_rel @ A @ X @ ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xa ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( mergesort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ Xa ) ) ) ) ) ).
% mergesort_by_rel.pelims
thf(fact_3588_enumerate__map__upt,axiom,
! [A: $tType,N: nat,F: nat > A,M2: nat] :
( ( enumerate @ A @ N @ ( map @ nat @ A @ F @ ( upt @ N @ M2 ) ) )
= ( map @ nat @ ( product_prod @ nat @ A )
@ ^ [K4: nat] : ( product_Pair @ nat @ A @ K4 @ ( F @ K4 ) )
@ ( upt @ N @ M2 ) ) ) ).
% enumerate_map_upt
thf(fact_3589_map__upt__eqI,axiom,
! [A: $tType,Xs: list @ A,N: nat,M2: nat,F: nat > A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( minus_minus @ nat @ N @ M2 ) )
=> ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ Xs @ I2 )
= ( F @ ( plus_plus @ nat @ M2 @ I2 ) ) ) )
=> ( ( map @ nat @ A @ F @ ( upt @ M2 @ N ) )
= Xs ) ) ) ).
% map_upt_eqI
thf(fact_3590_slice__nth,axiom,
! [A: $tType,From: nat,To: nat,Xs: list @ A,I: nat] :
( ( ord_less @ nat @ From @ To )
=> ( ( ord_less_eq @ nat @ To @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ I @ ( minus_minus @ nat @ To @ From ) )
=> ( ( nth @ A @ ( slice @ A @ From @ To @ Xs ) @ I )
= ( nth @ A @ Xs @ ( plus_plus @ nat @ From @ I ) ) ) ) ) ) ).
% slice_nth
thf(fact_3591_product__nth,axiom,
! [A: $tType,B: $tType,N: nat,Xs: list @ A,Ys2: list @ B] :
( ( ord_less @ nat @ N @ ( times_times @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys2 ) ) )
=> ( ( nth @ ( product_prod @ A @ B ) @ ( product @ A @ B @ Xs @ Ys2 ) @ N )
= ( product_Pair @ A @ B @ ( nth @ A @ Xs @ ( divide_divide @ nat @ N @ ( size_size @ ( list @ B ) @ Ys2 ) ) ) @ ( nth @ B @ Ys2 @ ( modulo_modulo @ nat @ N @ ( size_size @ ( list @ B ) @ Ys2 ) ) ) ) ) ) ).
% product_nth
thf(fact_3592_mergesort__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( mergesort @ A )
= ( mergesort_by_rel @ A @ ( ord_less_eq @ A ) ) ) ) ).
% mergesort_def
thf(fact_3593_sorted__in__between,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [I: nat,J: nat,L: list @ A,X: A] :
( ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ I )
=> ( ( ord_less @ nat @ I @ J )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L )
=> ( ( ord_less_eq @ A @ ( nth @ A @ L @ I ) @ X )
=> ( ( ord_less @ A @ X @ ( nth @ A @ L @ J ) )
=> ~ ! [K5: nat] :
( ( ord_less_eq @ nat @ I @ K5 )
=> ( ( ord_less @ nat @ K5 @ J )
=> ( ( ord_less_eq @ A @ ( nth @ A @ L @ K5 ) @ X )
=> ~ ( ord_less @ A @ X @ ( nth @ A @ L @ ( plus_plus @ nat @ K5 @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% sorted_in_between
thf(fact_3594_Un__set__drop__extend,axiom,
! [A: $tType,J: nat,L: list @ ( set @ A )] :
( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ J )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ ( set @ A ) ) @ L ) )
=> ( ( sup_sup @ ( set @ A ) @ ( nth @ ( set @ A ) @ L @ ( minus_minus @ nat @ J @ ( suc @ ( zero_zero @ nat ) ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( set2 @ ( set @ A ) @ ( drop @ ( set @ A ) @ J @ L ) ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( set2 @ ( set @ A ) @ ( drop @ ( set @ A ) @ ( minus_minus @ nat @ J @ ( suc @ ( zero_zero @ nat ) ) ) @ L ) ) ) ) ) ) ).
% Un_set_drop_extend
thf(fact_3595_mergesort__by__rel__simps_I3_J,axiom,
! [A: $tType,R5: A > A > $o,X1: A,X22: A,Xs: list @ A] :
( ( mergesort_by_rel @ A @ R5 @ ( cons @ A @ X1 @ ( cons @ A @ X22 @ Xs ) ) )
= ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ ( list @ A )
@ ^ [Xs12: list @ A,Xs23: list @ A] : ( merges9089515139780605204_merge @ A @ R5 @ ( mergesort_by_rel @ A @ R5 @ Xs12 ) @ ( mergesort_by_rel @ A @ R5 @ Xs23 ) )
@ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X1 @ ( nil @ A ) ) @ ( cons @ A @ X22 @ ( nil @ A ) ) ) @ Xs ) ) ) ).
% mergesort_by_rel_simps(3)
thf(fact_3596_drop__drop,axiom,
! [A: $tType,N: nat,M2: nat,Xs: list @ A] :
( ( drop @ A @ N @ ( drop @ A @ M2 @ Xs ) )
= ( drop @ A @ ( plus_plus @ nat @ N @ M2 ) @ Xs ) ) ).
% drop_drop
thf(fact_3597_sorted__wrt__mergesort__by__rel__merge,axiom,
! [A: $tType,R5: A > A > $o,Xs: list @ A,Ys2: list @ A] :
( ! [X2: A,Y4: A] :
( ( R5 @ X2 @ Y4 )
| ( R5 @ Y4 @ X2 ) )
=> ( ! [X2: A,Y4: A,Z3: A] :
( ( R5 @ X2 @ Y4 )
=> ( ( R5 @ Y4 @ Z3 )
=> ( R5 @ X2 @ Z3 ) ) )
=> ( ( sorted_wrt @ A @ R5 @ ( merges9089515139780605204_merge @ A @ R5 @ Xs @ Ys2 ) )
= ( ( sorted_wrt @ A @ R5 @ Xs )
& ( sorted_wrt @ A @ R5 @ Ys2 ) ) ) ) ) ).
% sorted_wrt_mergesort_by_rel_merge
thf(fact_3598_mergesort__by__rel__simps_I2_J,axiom,
! [A: $tType,R5: A > A > $o,X: A] :
( ( mergesort_by_rel @ A @ R5 @ ( cons @ A @ X @ ( nil @ A ) ) )
= ( cons @ A @ X @ ( nil @ A ) ) ) ).
% mergesort_by_rel_simps(2)
thf(fact_3599_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 ) @ ( product_Pair @ nat @ B @ N @ X ) @ ( enumerate @ B @ ( suc @ N ) @ Xs ) ) ) ).
% enumerate_simps(2)
thf(fact_3600_sorted__wrt__map__linord,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [L: list @ ( product_prod @ A @ B )] :
( ( sorted_wrt @ ( product_prod @ A @ B )
@ ^ [X3: product_prod @ A @ B,Y3: product_prod @ A @ B] : ( ord_less_eq @ A @ ( product_fst @ A @ B @ X3 ) @ ( product_fst @ A @ B @ Y3 ) )
@ L )
= ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ L ) ) ) ) ).
% sorted_wrt_map_linord
thf(fact_3601_nth__Cons__numeral,axiom,
! [A: $tType,X: A,Xs: list @ A,V: num] :
( ( nth @ A @ ( cons @ A @ X @ Xs ) @ ( numeral_numeral @ nat @ V ) )
= ( nth @ A @ Xs @ ( minus_minus @ nat @ ( numeral_numeral @ nat @ V ) @ ( one_one @ nat ) ) ) ) ).
% nth_Cons_numeral
thf(fact_3602_drop__Cons__numeral,axiom,
! [A: $tType,V: num,X: A,Xs: list @ A] :
( ( drop @ A @ ( numeral_numeral @ nat @ V ) @ ( cons @ A @ X @ Xs ) )
= ( drop @ A @ ( minus_minus @ nat @ ( numeral_numeral @ nat @ V ) @ ( one_one @ nat ) ) @ Xs ) ) ).
% drop_Cons_numeral
thf(fact_3603_nth__drop,axiom,
! [A: $tType,N: nat,Xs: list @ A,I: nat] :
( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( drop @ A @ N @ Xs ) @ I )
= ( nth @ A @ Xs @ ( plus_plus @ nat @ N @ I ) ) ) ) ).
% nth_drop
thf(fact_3604_nth__Cons__pos,axiom,
! [A: $tType,N: nat,X: A,Xs: list @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
= ( nth @ A @ Xs @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% nth_Cons_pos
thf(fact_3605_Misc_Ofoldr__Cons,axiom,
! [A: $tType,Xs: list @ A] :
( ( foldr @ A @ ( list @ A ) @ ( cons @ A ) @ Xs @ ( nil @ A ) )
= Xs ) ).
% Misc.foldr_Cons
thf(fact_3606_map__consI_I1_J,axiom,
! [A: $tType,B: $tType,W: list @ A,F: B > A,Ww: list @ B,A3: B] :
( ( W
= ( map @ B @ A @ F @ Ww ) )
=> ( ( cons @ A @ ( F @ A3 ) @ W )
= ( map @ B @ A @ F @ ( cons @ B @ A3 @ Ww ) ) ) ) ).
% map_consI(1)
thf(fact_3607_map__eq__consE,axiom,
! [B: $tType,A: $tType,F: B > A,Ls: list @ B,Fa: A,Fl: list @ A] :
( ( ( map @ B @ A @ F @ Ls )
= ( cons @ A @ Fa @ Fl ) )
=> ~ ! [A5: B,L5: list @ B] :
( ( Ls
= ( cons @ B @ A5 @ L5 ) )
=> ( ( ( F @ A5 )
= Fa )
=> ( ( map @ B @ A @ F @ L5 )
!= Fl ) ) ) ) ).
% map_eq_consE
thf(fact_3608_neq__NilE,axiom,
! [A: $tType,L: list @ A] :
( ( L
!= ( nil @ A ) )
=> ~ ! [X2: A,Xs2: list @ A] :
( L
!= ( cons @ A @ X2 @ Xs2 ) ) ) ).
% neq_NilE
thf(fact_3609_list__2pre__induct,axiom,
! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,W1: list @ A,W22: list @ B] :
( ( P @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [E3: A,W12: list @ A,W23: list @ B] :
( ( P @ W12 @ W23 )
=> ( P @ ( cons @ A @ E3 @ W12 ) @ W23 ) )
=> ( ! [E3: B,W13: list @ A,W24: list @ B] :
( ( P @ W13 @ W24 )
=> ( P @ W13 @ ( cons @ B @ E3 @ W24 ) ) )
=> ( P @ W1 @ W22 ) ) ) ) ).
% list_2pre_induct
thf(fact_3610_list__induct__first2,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ( ! [X13: A,X23: A,Xs2: list @ A] :
( ( P @ Xs2 )
=> ( P @ ( cons @ A @ X13 @ ( cons @ A @ X23 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_induct_first2
thf(fact_3611_mergesort__by__rel__merge__induct,axiom,
! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,R5: A > B > $o,Xs: list @ A,Ys2: list @ B] :
( ! [Xs2: list @ A] : ( P @ Xs2 @ ( nil @ B ) )
=> ( ! [X_12: list @ B] : ( P @ ( nil @ A ) @ X_12 )
=> ( ! [X2: A,Xs2: list @ A,Y4: B,Ys3: list @ B] :
( ( R5 @ X2 @ Y4 )
=> ( ( P @ Xs2 @ ( cons @ B @ Y4 @ Ys3 ) )
=> ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ B @ Y4 @ Ys3 ) ) ) )
=> ( ! [X2: A,Xs2: list @ A,Y4: B,Ys3: list @ B] :
( ~ ( R5 @ X2 @ Y4 )
=> ( ( P @ ( cons @ A @ X2 @ Xs2 ) @ Ys3 )
=> ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ B @ Y4 @ Ys3 ) ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ).
% mergesort_by_rel_merge_induct
thf(fact_3612_sorted__wrt__mergesort__by__rel,axiom,
! [X14: $tType,R5: X14 > X14 > $o,Xs: list @ X14] :
( ! [X2: X14,Y4: X14] :
( ( R5 @ X2 @ Y4 )
| ( R5 @ Y4 @ X2 ) )
=> ( ! [X2: X14,Y4: X14,Z3: X14] :
( ( R5 @ X2 @ Y4 )
=> ( ( R5 @ Y4 @ Z3 )
=> ( R5 @ X2 @ Z3 ) ) )
=> ( sorted_wrt @ X14 @ R5 @ ( mergesort_by_rel @ X14 @ R5 @ Xs ) ) ) ) ).
% sorted_wrt_mergesort_by_rel
thf(fact_3613_list__tail__coinc,axiom,
! [A: $tType,N12: A,R1: list @ A,N23: A,R22: list @ A] :
( ( ( cons @ A @ N12 @ R1 )
= ( cons @ A @ N23 @ R22 ) )
=> ( ( N12 = N23 )
& ( R1 = R22 ) ) ) ).
% list_tail_coinc
thf(fact_3614_drop__eq__ConsD,axiom,
! [A: $tType,N: nat,Xs: list @ A,X: A,Xs4: list @ A] :
( ( ( drop @ A @ N @ Xs )
= ( cons @ A @ X @ Xs4 ) )
=> ( ( drop @ A @ ( suc @ N ) @ Xs )
= Xs4 ) ) ).
% drop_eq_ConsD
thf(fact_3615_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 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ Xs ) ) ) ) ).
% drop_Cons'
thf(fact_3616_mergesort__by__rel__merge__simps_I1_J,axiom,
! [A: $tType,R5: A > A > $o,X: A,Y: A,Xs: list @ A,Ys2: list @ A] :
( ( ( R5 @ X @ Y )
=> ( ( merges9089515139780605204_merge @ A @ R5 @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys2 ) )
= ( cons @ A @ X @ ( merges9089515139780605204_merge @ A @ R5 @ Xs @ ( cons @ A @ Y @ Ys2 ) ) ) ) )
& ( ~ ( R5 @ X @ Y )
=> ( ( merges9089515139780605204_merge @ A @ R5 @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys2 ) )
= ( cons @ A @ Y @ ( merges9089515139780605204_merge @ A @ R5 @ ( cons @ A @ X @ Xs ) @ Ys2 ) ) ) ) ) ).
% mergesort_by_rel_merge_simps(1)
thf(fact_3617_sorted__filter_H,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: list @ A,P: A > $o] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( filter @ A @ P @ L ) ) ) ) ).
% sorted_filter'
thf(fact_3618_sorted__mergesort__by__rel,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( mergesort_by_rel @ A @ ( ord_less_eq @ A ) @ Xs ) ) ) ).
% sorted_mergesort_by_rel
thf(fact_3619_map__filter__simps_I1_J,axiom,
! [A: $tType,B: $tType,F: B > ( option @ A ),X: B,Xs: list @ B] :
( ( map_filter @ B @ A @ F @ ( cons @ B @ X @ Xs ) )
= ( case_option @ ( list @ A ) @ A @ ( map_filter @ B @ A @ F @ Xs )
@ ^ [Y3: A] : ( cons @ A @ Y3 @ ( map_filter @ B @ A @ F @ Xs ) )
@ ( F @ X ) ) ) ).
% map_filter_simps(1)
thf(fact_3620_zipf_Ocases,axiom,
! [C: $tType,A: $tType,B: $tType,X: product_prod @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) )] :
( ! [F4: A > B > C] :
( X
!= ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F4 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( nil @ B ) ) ) )
=> ( ! [F4: A > B > C,A5: A,As: list @ A,B3: B,Bs: list @ B] :
( X
!= ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F4 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A5 @ As ) @ ( cons @ B @ B3 @ Bs ) ) ) )
=> ( ! [A5: A > B > C,V4: A,Va: list @ A] :
( X
!= ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ A5 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ V4 @ Va ) @ ( nil @ B ) ) ) )
=> ~ ! [A5: A > B > C,V4: B,Va: list @ B] :
( X
!= ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ A5 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( cons @ B @ V4 @ Va ) ) ) ) ) ) ) ).
% zipf.cases
thf(fact_3621_merge_Ocases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: product_prod @ ( list @ A ) @ ( list @ A )] :
( ! [L22: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ L22 ) )
=> ( ! [V4: A,Va: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ V4 @ Va ) @ ( nil @ A ) ) )
=> ~ ! [X13: A,L12: list @ A,X23: A,L22: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X13 @ L12 ) @ ( cons @ A @ X23 @ L22 ) ) ) ) ) ) ).
% merge.cases
thf(fact_3622_list__all__zip_Ocases,axiom,
! [A: $tType,B: $tType,X: product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) )] :
( ! [P7: A > B > $o] :
( X
!= ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ P7 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( nil @ B ) ) ) )
=> ( ! [P7: A > B > $o,A5: A,As: list @ A,B3: B,Bs: list @ B] :
( X
!= ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ P7 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A5 @ As ) @ ( cons @ B @ B3 @ Bs ) ) ) )
=> ( ! [P7: A > B > $o,V4: A,Va: list @ A] :
( X
!= ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ P7 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ V4 @ Va ) @ ( nil @ B ) ) ) )
=> ~ ! [P7: A > B > $o,V4: B,Va: list @ B] :
( X
!= ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ P7 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( cons @ B @ V4 @ Va ) ) ) ) ) ) ) ).
% list_all_zip.cases
thf(fact_3623_partition__rev_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) )] :
( ! [P7: A > $o,Yes: list @ A,No: list @ A] :
( X
!= ( product_Pair @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ P7 @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) @ ( nil @ A ) ) ) )
=> ~ ! [P7: A > $o,Yes: list @ A,No: list @ A,X2: A,Xs2: list @ A] :
( X
!= ( product_Pair @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ P7 @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) @ ( cons @ A @ X2 @ Xs2 ) ) ) ) ) ).
% partition_rev.cases
thf(fact_3624_quicksort__by__rel_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) )] :
( ! [R8: A > A > $o,Sl: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R8 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Sl @ ( nil @ A ) ) ) )
=> ~ ! [R8: A > A > $o,Sl: list @ A,X2: A,Xs2: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R8 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Sl @ ( cons @ A @ X2 @ Xs2 ) ) ) ) ) ).
% quicksort_by_rel.cases
thf(fact_3625_mergesort__by__rel__merge_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) )] :
( ! [R8: A > A > $o,X2: A,Xs2: list @ A,Y4: A,Ys3: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R8 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ A @ Y4 @ Ys3 ) ) ) )
=> ( ! [R8: A > A > $o,Xs2: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R8 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ ( nil @ A ) ) ) )
=> ~ ! [R8: A > A > $o,V4: A,Va: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R8 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( cons @ A @ V4 @ Va ) ) ) ) ) ) ).
% mergesort_by_rel_merge.cases
thf(fact_3626_mergesort__by__rel__split_Ocases,axiom,
! [A: $tType,X: product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A )] :
( ! [Xs13: list @ A,Xs24: list @ A] :
( X
!= ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs24 ) @ ( nil @ A ) ) )
=> ( ! [Xs13: list @ A,Xs24: list @ A,X2: A] :
( X
!= ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs24 ) @ ( cons @ A @ X2 @ ( nil @ A ) ) ) )
=> ~ ! [Xs13: list @ A,Xs24: list @ A,X13: A,X23: A,Xs2: list @ A] :
( X
!= ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs24 ) @ ( cons @ A @ X13 @ ( cons @ A @ X23 @ Xs2 ) ) ) ) ) ) ).
% mergesort_by_rel_split.cases
thf(fact_3627_shuffles_Ocases,axiom,
! [A: $tType,X: product_prod @ ( list @ A ) @ ( list @ A )] :
( ! [Ys3: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys3 ) )
=> ( ! [Xs2: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ ( nil @ A ) ) )
=> ~ ! [X2: A,Xs2: list @ A,Y4: A,Ys3: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ A @ Y4 @ Ys3 ) ) ) ) ) ).
% shuffles.cases
thf(fact_3628_map__tailrec__rev_Ocases,axiom,
! [A: $tType,B: $tType,X: product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) )] :
( ! [F4: A > B,Bs: list @ B] :
( X
!= ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F4 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ Bs ) ) )
=> ~ ! [F4: A > B,A5: A,As: list @ A,Bs: list @ B] :
( X
!= ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F4 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A5 @ As ) @ Bs ) ) ) ) ).
% map_tailrec_rev.cases
thf(fact_3629_foldr_Ofoldr__Cons,axiom,
! [B: $tType,A: $tType,F: A > B > B,X: A,Xs: list @ A] :
( ( foldr @ A @ B @ F @ ( cons @ A @ X @ Xs ) )
= ( comp @ B @ B @ B @ ( F @ X ) @ ( foldr @ A @ B @ F @ Xs ) ) ) ).
% foldr.foldr_Cons
thf(fact_3630_arg__min__list_Ocases,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [X: product_prod @ ( A > B ) @ ( list @ A )] :
( ! [F4: A > B,X2: A] :
( X
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ F4 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) )
=> ( ! [F4: A > B,X2: A,Y4: A,Zs2: list @ A] :
( X
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ F4 @ ( cons @ A @ X2 @ ( cons @ A @ Y4 @ Zs2 ) ) ) )
=> ~ ! [A5: A > B] :
( X
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ A5 @ ( nil @ A ) ) ) ) ) ) ).
% arg_min_list.cases
thf(fact_3631_successively_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( list @ A )] :
( ! [P7: A > A > $o] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P7 @ ( nil @ A ) ) )
=> ( ! [P7: A > A > $o,X2: A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P7 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) )
=> ~ ! [P7: A > A > $o,X2: A,Y4: A,Xs2: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P7 @ ( cons @ A @ X2 @ ( cons @ A @ Y4 @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_3632_sorted__wrt_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( list @ A )] :
( ! [P7: A > A > $o] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P7 @ ( nil @ A ) ) )
=> ~ ! [P7: A > A > $o,X2: A,Ys3: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P7 @ ( cons @ A @ X2 @ Ys3 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_3633_mergesort__by__rel__merge_Oelims,axiom,
! [A: $tType,X: A > A > $o,Xa: list @ A,Xb: list @ A,Y: list @ A] :
( ( ( merges9089515139780605204_merge @ A @ X @ Xa @ Xb )
= Y )
=> ( ! [X2: A,Xs2: list @ A] :
( ( Xa
= ( cons @ A @ X2 @ Xs2 ) )
=> ! [Y4: A,Ys3: list @ A] :
( ( Xb
= ( cons @ A @ Y4 @ Ys3 ) )
=> ~ ( ( ( X @ X2 @ Y4 )
=> ( Y
= ( cons @ A @ X2 @ ( merges9089515139780605204_merge @ A @ X @ Xs2 @ ( cons @ A @ Y4 @ Ys3 ) ) ) ) )
& ( ~ ( X @ X2 @ Y4 )
=> ( Y
= ( cons @ A @ Y4 @ ( merges9089515139780605204_merge @ A @ X @ ( cons @ A @ X2 @ Xs2 ) @ Ys3 ) ) ) ) ) ) )
=> ( ( ( Xb
= ( nil @ A ) )
=> ( Y != Xa ) )
=> ~ ( ( Xa
= ( nil @ A ) )
=> ! [V4: A,Va: list @ A] :
( ( Xb
= ( cons @ A @ V4 @ Va ) )
=> ( Y
!= ( cons @ A @ V4 @ Va ) ) ) ) ) ) ) ).
% mergesort_by_rel_merge.elims
thf(fact_3634_mergesort__by__rel__merge_Osimps_I3_J,axiom,
! [A: $tType,R5: A > A > $o,V: A,Va2: list @ A] :
( ( merges9089515139780605204_merge @ A @ R5 @ ( nil @ A ) @ ( cons @ A @ V @ Va2 ) )
= ( cons @ A @ V @ Va2 ) ) ).
% mergesort_by_rel_merge.simps(3)
thf(fact_3635_mergesort__by__rel__split_Osimps_I3_J,axiom,
! [A: $tType,Xs1: list @ A,Xs22: list @ A,X1: A,X22: A,Xs: list @ A] :
( ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) @ ( cons @ A @ X1 @ ( cons @ A @ X22 @ Xs ) ) )
= ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X1 @ Xs1 ) @ ( cons @ A @ X22 @ Xs22 ) ) @ Xs ) ) ).
% mergesort_by_rel_split.simps(3)
thf(fact_3636_filter__conv__foldr,axiom,
! [A: $tType] :
( ( filter @ A )
= ( ^ [P4: A > $o,Xs3: list @ A] :
( foldr @ A @ ( list @ A )
@ ^ [X3: A,Xt: list @ A] : ( if @ ( list @ A ) @ ( P4 @ X3 ) @ ( cons @ A @ X3 @ Xt ) @ Xt )
@ Xs3
@ ( nil @ A ) ) ) ) ).
% filter_conv_foldr
thf(fact_3637_sorted__wrt01,axiom,
! [A: $tType,Xs: list @ A,P: A > A > $o] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) )
=> ( sorted_wrt @ A @ P @ Xs ) ) ).
% sorted_wrt01
thf(fact_3638_length__compl__induct,axiom,
! [A: $tType,P: ( list @ A ) > $o,L: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [E3: A,L5: list @ A] :
( ! [Ll: list @ A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ll ) @ ( size_size @ ( list @ A ) @ L5 ) )
=> ( P @ Ll ) )
=> ( P @ ( cons @ A @ E3 @ L5 ) ) )
=> ( P @ L ) ) ) ).
% length_compl_induct
thf(fact_3639_list__decomp__1,axiom,
! [A: $tType,L: list @ A] :
( ( ( size_size @ ( list @ A ) @ L )
= ( one_one @ nat ) )
=> ? [A5: A] :
( L
= ( cons @ A @ A5 @ ( nil @ A ) ) ) ) ).
% list_decomp_1
thf(fact_3640_map__of__Cons__code_I2_J,axiom,
! [C: $tType,B: $tType,L: B,K: B,V: C,Ps: list @ ( product_prod @ B @ C )] :
( ( ( L = K )
=> ( ( map_of @ B @ C @ ( cons @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ L @ V ) @ Ps ) @ K )
= ( some @ C @ V ) ) )
& ( ( L != K )
=> ( ( map_of @ B @ C @ ( cons @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ L @ V ) @ Ps ) @ K )
= ( map_of @ B @ C @ Ps @ K ) ) ) ) ).
% map_of_Cons_code(2)
thf(fact_3641_mergesort__by__rel__split_Osimps_I2_J,axiom,
! [A: $tType,Xs1: list @ A,Xs22: list @ A,X: A] :
( ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) @ ( cons @ A @ X @ ( nil @ A ) ) )
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs1 ) @ Xs22 ) ) ).
% mergesort_by_rel_split.simps(2)
thf(fact_3642_mergesort__by__rel__split_Oelims,axiom,
! [A: $tType,X: product_prod @ ( list @ A ) @ ( list @ A ),Xa: list @ A,Y: product_prod @ ( list @ A ) @ ( list @ A )] :
( ( ( merges295452479951948502_split @ A @ X @ Xa )
= Y )
=> ( ! [Xs13: list @ A,Xs24: list @ A] :
( ( X
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs24 ) )
=> ( ( Xa
= ( nil @ A ) )
=> ( Y
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs24 ) ) ) )
=> ( ! [Xs13: list @ A,Xs24: list @ A] :
( ( X
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs24 ) )
=> ! [X2: A] :
( ( Xa
= ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ( Y
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs13 ) @ Xs24 ) ) ) )
=> ~ ! [Xs13: list @ A,Xs24: list @ A] :
( ( X
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs24 ) )
=> ! [X13: A,X23: A,Xs2: list @ A] :
( ( Xa
= ( cons @ A @ X13 @ ( cons @ A @ X23 @ Xs2 ) ) )
=> ( Y
!= ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X13 @ Xs13 ) @ ( cons @ A @ X23 @ Xs24 ) ) @ Xs2 ) ) ) ) ) ) ) ).
% mergesort_by_rel_split.elims
thf(fact_3643_sorted01,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs ) ) ) ).
% sorted01
thf(fact_3644_list_Osize_I4_J,axiom,
! [A: $tType,X21: A,X222: list @ A] :
( ( size_size @ ( list @ A ) @ ( cons @ A @ X21 @ X222 ) )
= ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ X222 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% list.size(4)
thf(fact_3645_nth__Cons_H,axiom,
! [A: $tType,N: nat,X: A,Xs: list @ A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
= X ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
= ( nth @ A @ Xs @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ).
% nth_Cons'
thf(fact_3646_upt__eq__Cons__conv,axiom,
! [I: nat,J: nat,X: nat,Xs: list @ nat] :
( ( ( upt @ I @ J )
= ( cons @ nat @ X @ Xs ) )
= ( ( ord_less @ nat @ I @ J )
& ( I = X )
& ( ( upt @ ( plus_plus @ nat @ I @ ( one_one @ nat ) ) @ J )
= Xs ) ) ) ).
% upt_eq_Cons_conv
thf(fact_3647_in__set__drop__conv__nth,axiom,
! [A: $tType,X: A,N: nat,L: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ ( drop @ A @ N @ L ) ) )
= ( ? [I3: nat] :
( ( ord_less_eq @ nat @ N @ I3 )
& ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ L ) )
& ( X
= ( nth @ A @ L @ I3 ) ) ) ) ) ).
% in_set_drop_conv_nth
thf(fact_3648_set__drop__conv,axiom,
! [A: $tType,N: nat,L: list @ A] :
( ( set2 @ A @ ( drop @ A @ N @ L ) )
= ( collect @ A
@ ^ [Uu: A] :
? [I3: nat] :
( ( Uu
= ( nth @ A @ L @ I3 ) )
& ( ord_less_eq @ nat @ N @ I3 )
& ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ L ) ) ) ) ) ).
% set_drop_conv
thf(fact_3649_list__decomp__2,axiom,
! [A: $tType,L: list @ A] :
( ( ( size_size @ ( list @ A ) @ L )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ? [A5: A,B3: A] :
( L
= ( cons @ A @ A5 @ ( cons @ A @ B3 @ ( nil @ A ) ) ) ) ) ).
% list_decomp_2
thf(fact_3650_nth__non__equal__first__eq,axiom,
! [A: $tType,X: A,Y: A,Xs: list @ A,N: nat] :
( ( X != Y )
=> ( ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
= Y )
= ( ( ( nth @ A @ Xs @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
= Y )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_3651_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 )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
& ( Xs
= ( replicate @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ X ) ) ) ) ).
% Cons_replicate_eq
thf(fact_3652_distinct__sorted__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: list @ A,I: nat,J: nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L )
=> ( ( distinct @ A @ L )
=> ( ( ord_less @ nat @ I @ J )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ L ) )
=> ( ord_less @ A @ ( nth @ A @ L @ I ) @ ( nth @ A @ L @ J ) ) ) ) ) ) ) ).
% distinct_sorted_mono
thf(fact_3653_distinct__sorted__strict__mono__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: list @ A,I: nat,J: nat] :
( ( distinct @ A @ L )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( ord_less @ A @ ( nth @ A @ L @ I ) @ ( nth @ A @ L @ J ) )
= ( ord_less @ nat @ I @ J ) ) ) ) ) ) ) ).
% distinct_sorted_strict_mono_iff
thf(fact_3654_distinct__sorted__mono__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: list @ A,I: nat,J: nat] :
( ( distinct @ A @ L )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( ord_less_eq @ A @ ( nth @ A @ L @ I ) @ ( nth @ A @ L @ J ) )
= ( ord_less_eq @ nat @ I @ J ) ) ) ) ) ) ) ).
% distinct_sorted_mono_iff
thf(fact_3655_map__of_Osimps_I2_J,axiom,
! [B: $tType,A: $tType,P5: product_prod @ A @ B,Ps: list @ ( product_prod @ A @ B )] :
( ( map_of @ A @ B @ ( cons @ ( product_prod @ A @ B ) @ P5 @ Ps ) )
= ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ Ps ) @ ( product_fst @ A @ B @ P5 ) @ ( some @ B @ ( product_snd @ A @ B @ P5 ) ) ) ) ).
% map_of.simps(2)
thf(fact_3656_slice__Cons,axiom,
! [A: $tType,Begin: nat,End: nat,X: A,Xs: list @ A] :
( ( ( ( Begin
= ( zero_zero @ nat ) )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ End ) )
=> ( ( slice @ A @ Begin @ End @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X @ ( slice @ A @ Begin @ ( minus_minus @ nat @ End @ ( one_one @ nat ) ) @ Xs ) ) ) )
& ( ~ ( ( Begin
= ( zero_zero @ nat ) )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ End ) )
=> ( ( slice @ A @ Begin @ End @ ( cons @ A @ X @ Xs ) )
= ( slice @ A @ ( minus_minus @ nat @ Begin @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ End @ ( one_one @ nat ) ) @ Xs ) ) ) ) ).
% slice_Cons
thf(fact_3657_mergesort__by__rel__split_Opelims,axiom,
! [A: $tType,X: product_prod @ ( list @ A ) @ ( list @ A ),Xa: list @ A,Y: product_prod @ ( list @ A ) @ ( list @ A )] :
( ( ( merges295452479951948502_split @ A @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ ( merges7066485432131860899it_rel @ A ) @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ X @ Xa ) )
=> ( ! [Xs13: list @ A,Xs24: list @ A] :
( ( X
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs24 ) )
=> ( ( Xa
= ( nil @ A ) )
=> ( ( Y
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs24 ) )
=> ~ ( accp @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ ( merges7066485432131860899it_rel @ A ) @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs24 ) @ ( nil @ A ) ) ) ) ) )
=> ( ! [Xs13: list @ A,Xs24: list @ A] :
( ( X
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs24 ) )
=> ! [X2: A] :
( ( Xa
= ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ( ( Y
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs13 ) @ Xs24 ) )
=> ~ ( accp @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ ( merges7066485432131860899it_rel @ A ) @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs24 ) @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) ) ) )
=> ~ ! [Xs13: list @ A,Xs24: list @ A] :
( ( X
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs24 ) )
=> ! [X13: A,X23: A,Xs2: list @ A] :
( ( Xa
= ( cons @ A @ X13 @ ( cons @ A @ X23 @ Xs2 ) ) )
=> ( ( Y
= ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X13 @ Xs13 ) @ ( cons @ A @ X23 @ Xs24 ) ) @ Xs2 ) )
=> ~ ( accp @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ ( merges7066485432131860899it_rel @ A ) @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs24 ) @ ( cons @ A @ X13 @ ( cons @ A @ X23 @ Xs2 ) ) ) ) ) ) ) ) ) ) ) ).
% mergesort_by_rel_split.pelims
thf(fact_3658_list__decode_Opelims,axiom,
! [X: nat,Y: 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
= ( suc @ N3 ) )
=> ( ( Y
= ( product_case_prod @ nat @ nat @ ( list @ nat )
@ ^ [X3: nat,Y3: nat] : ( cons @ nat @ X3 @ ( nat_list_decode @ Y3 ) )
@ ( nat_prod_decode @ N3 ) ) )
=> ~ ( accp @ nat @ nat_list_decode_rel @ ( suc @ N3 ) ) ) ) ) ) ) ).
% list_decode.pelims
thf(fact_3659_mergesort__by__rel__merge_Opelims,axiom,
! [A: $tType,X: A > A > $o,Xa: list @ A,Xb: list @ A,Y: list @ A] :
( ( ( merges9089515139780605204_merge @ A @ X @ Xa @ Xb )
= Y )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( merges2244889521215249637ge_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xa @ Xb ) ) )
=> ( ! [X2: A,Xs2: list @ A] :
( ( Xa
= ( cons @ A @ X2 @ Xs2 ) )
=> ! [Y4: A,Ys3: list @ A] :
( ( Xb
= ( cons @ A @ Y4 @ Ys3 ) )
=> ( ( ( ( X @ X2 @ Y4 )
=> ( Y
= ( cons @ A @ X2 @ ( merges9089515139780605204_merge @ A @ X @ Xs2 @ ( cons @ A @ Y4 @ Ys3 ) ) ) ) )
& ( ~ ( X @ X2 @ Y4 )
=> ( Y
= ( cons @ A @ Y4 @ ( merges9089515139780605204_merge @ A @ X @ ( cons @ A @ X2 @ Xs2 ) @ Ys3 ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( merges2244889521215249637ge_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ A @ Y4 @ Ys3 ) ) ) ) ) ) )
=> ( ( ( Xb
= ( nil @ A ) )
=> ( ( Y = Xa )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( merges2244889521215249637ge_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xa @ ( nil @ A ) ) ) ) ) )
=> ~ ( ( Xa
= ( nil @ A ) )
=> ! [V4: A,Va: list @ A] :
( ( Xb
= ( cons @ A @ V4 @ Va ) )
=> ( ( Y
= ( cons @ A @ V4 @ Va ) )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( merges2244889521215249637ge_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( cons @ A @ V4 @ Va ) ) ) ) ) ) ) ) ) ) ) ).
% mergesort_by_rel_merge.pelims
thf(fact_3660_subset__eq__mset__impl_Ocases,axiom,
! [A: $tType,X: product_prod @ ( list @ A ) @ ( list @ A )] :
( ! [Ys3: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys3 ) )
=> ~ ! [X2: A,Xs2: list @ A,Ys3: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ Ys3 ) ) ) ).
% subset_eq_mset_impl.cases
thf(fact_3661_drop__upt,axiom,
! [M2: nat,I: nat,J: nat] :
( ( drop @ nat @ M2 @ ( upt @ I @ J ) )
= ( upt @ ( plus_plus @ nat @ I @ M2 ) @ J ) ) ).
% drop_upt
thf(fact_3662_natural__zero__minus__one,axiom,
( ( minus_minus @ code_natural @ ( zero_zero @ code_natural ) @ ( one_one @ code_natural ) )
= ( zero_zero @ code_natural ) ) ).
% natural_zero_minus_one
thf(fact_3663_list__decode_Opsimps_I1_J,axiom,
( ( accp @ nat @ nat_list_decode_rel @ ( zero_zero @ nat ) )
=> ( ( nat_list_decode @ ( zero_zero @ nat ) )
= ( nil @ nat ) ) ) ).
% list_decode.psimps(1)
thf(fact_3664_list__decode_Opsimps_I2_J,axiom,
! [N: nat] :
( ( accp @ nat @ nat_list_decode_rel @ ( suc @ N ) )
=> ( ( nat_list_decode @ ( suc @ N ) )
= ( product_case_prod @ nat @ nat @ ( list @ nat )
@ ^ [X3: nat,Y3: nat] : ( cons @ nat @ X3 @ ( nat_list_decode @ Y3 ) )
@ ( nat_prod_decode @ N ) ) ) ) ).
% list_decode.psimps(2)
thf(fact_3665_list__encode_Oelims,axiom,
! [X: list @ nat,Y: nat] :
( ( ( nat_list_encode @ X )
= Y )
=> ( ( ( X
= ( nil @ nat ) )
=> ( Y
!= ( zero_zero @ nat ) ) )
=> ~ ! [X2: nat,Xs2: list @ nat] :
( ( X
= ( cons @ nat @ X2 @ Xs2 ) )
=> ( Y
!= ( suc @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ X2 @ ( nat_list_encode @ Xs2 ) ) ) ) ) ) ) ) ).
% list_encode.elims
thf(fact_3666_mergesort__remdups__correct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: list @ A] :
( ( distinct @ A @ ( mergesort_remdups @ A @ L ) )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( mergesort_remdups @ A @ L ) )
& ( ( set2 @ A @ ( mergesort_remdups @ A @ L ) )
= ( set2 @ A @ L ) ) ) ) ).
% mergesort_remdups_correct
thf(fact_3667_filter__upt__last,axiom,
! [A: $tType,P: A > $o,L: list @ A,Js: list @ nat,J: nat,I: nat] :
( ( ( filter @ nat
@ ^ [K4: nat] : ( P @ ( nth @ A @ L @ K4 ) )
@ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ L ) ) )
= ( append @ nat @ Js @ ( cons @ nat @ J @ ( nil @ nat ) ) ) )
=> ( ( ord_less @ nat @ J @ I )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ~ ( P @ ( nth @ A @ L @ I ) ) ) ) ) ).
% filter_upt_last
thf(fact_3668_empty__append__eq__id,axiom,
! [A: $tType] :
( ( append @ A @ ( nil @ A ) )
= ( ^ [X3: list @ A] : X3 ) ) ).
% empty_append_eq_id
thf(fact_3669_op__conc__empty__img__id,axiom,
! [A: $tType,L6: set @ ( list @ A )] :
( ( image2 @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ ( nil @ A ) ) @ L6 )
= L6 ) ).
% op_conc_empty_img_id
thf(fact_3670_list__e__eq__lel_I2_J,axiom,
! [A: $tType,L1: list @ A,E6: A,L2: list @ A,E2: A] :
( ( ( append @ A @ L1 @ ( cons @ A @ E6 @ L2 ) )
= ( cons @ A @ E2 @ ( nil @ A ) ) )
= ( ( L1
= ( nil @ A ) )
& ( E6 = E2 )
& ( L2
= ( nil @ A ) ) ) ) ).
% list_e_eq_lel(2)
thf(fact_3671_list__e__eq__lel_I1_J,axiom,
! [A: $tType,E2: A,L1: list @ A,E6: A,L2: list @ A] :
( ( ( cons @ A @ E2 @ ( nil @ A ) )
= ( append @ A @ L1 @ ( cons @ A @ E6 @ L2 ) ) )
= ( ( L1
= ( nil @ A ) )
& ( E6 = E2 )
& ( L2
= ( nil @ A ) ) ) ) ).
% list_e_eq_lel(1)
thf(fact_3672_list__se__match_I4_J,axiom,
! [A: $tType,L2: list @ A,A3: A,L1: list @ A] :
( ( L2
!= ( nil @ A ) )
=> ( ( ( cons @ A @ A3 @ ( nil @ A ) )
= ( append @ A @ L1 @ L2 ) )
= ( ( L1
= ( nil @ A ) )
& ( L2
= ( cons @ A @ A3 @ ( nil @ A ) ) ) ) ) ) ).
% list_se_match(4)
thf(fact_3673_list__se__match_I3_J,axiom,
! [A: $tType,L1: list @ A,A3: A,L2: list @ A] :
( ( L1
!= ( nil @ A ) )
=> ( ( ( cons @ A @ A3 @ ( nil @ A ) )
= ( append @ A @ L1 @ L2 ) )
= ( ( L1
= ( cons @ A @ A3 @ ( nil @ A ) ) )
& ( L2
= ( nil @ A ) ) ) ) ) ).
% list_se_match(3)
thf(fact_3674_list__se__match_I2_J,axiom,
! [A: $tType,L2: list @ A,L1: list @ A,A3: A] :
( ( L2
!= ( nil @ A ) )
=> ( ( ( append @ A @ L1 @ L2 )
= ( cons @ A @ A3 @ ( nil @ A ) ) )
= ( ( L1
= ( nil @ A ) )
& ( L2
= ( cons @ A @ A3 @ ( nil @ A ) ) ) ) ) ) ).
% list_se_match(2)
thf(fact_3675_list__se__match_I1_J,axiom,
! [A: $tType,L1: list @ A,L2: list @ A,A3: A] :
( ( L1
!= ( nil @ A ) )
=> ( ( ( append @ A @ L1 @ L2 )
= ( cons @ A @ A3 @ ( nil @ A ) ) )
= ( ( L1
= ( cons @ A @ A3 @ ( nil @ A ) ) )
& ( L2
= ( nil @ A ) ) ) ) ) ).
% list_se_match(1)
thf(fact_3676_list__ee__eq__leel_I2_J,axiom,
! [A: $tType,L1: list @ A,E12: A,E23: A,L2: list @ A,E1: A,E22: A] :
( ( ( append @ A @ L1 @ ( cons @ A @ E12 @ ( cons @ A @ E23 @ L2 ) ) )
= ( cons @ A @ E1 @ ( cons @ A @ E22 @ ( nil @ A ) ) ) )
= ( ( L1
= ( nil @ A ) )
& ( E1 = E12 )
& ( E22 = E23 )
& ( L2
= ( nil @ A ) ) ) ) ).
% list_ee_eq_leel(2)
thf(fact_3677_list__ee__eq__leel_I1_J,axiom,
! [A: $tType,E1: A,E22: A,L1: list @ A,E12: A,E23: A,L2: list @ A] :
( ( ( cons @ A @ E1 @ ( cons @ A @ E22 @ ( nil @ A ) ) )
= ( append @ A @ L1 @ ( cons @ A @ E12 @ ( cons @ A @ E23 @ L2 ) ) ) )
= ( ( L1
= ( nil @ A ) )
& ( E1 = E12 )
& ( E22 = E23 )
& ( L2
= ( nil @ A ) ) ) ) ).
% list_ee_eq_leel(1)
thf(fact_3678_length__append,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( size_size @ ( list @ A ) @ ( append @ A @ Xs @ Ys2 ) )
= ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys2 ) ) ) ).
% length_append
thf(fact_3679_upt__merge,axiom,
! [I: nat,J: nat,K: nat] :
( ( ( ord_less_eq @ nat @ I @ J )
& ( ord_less_eq @ nat @ J @ K ) )
=> ( ( append @ nat @ ( upt @ I @ J ) @ ( upt @ J @ K ) )
= ( upt @ I @ K ) ) ) ).
% upt_merge
thf(fact_3680_nth__append__first,axiom,
! [A: $tType,I: nat,L: list @ A,L4: list @ A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( nth @ A @ ( append @ A @ L @ L4 ) @ I )
= ( nth @ A @ L @ I ) ) ) ).
% nth_append_first
thf(fact_3681_nth__append__length__plus,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A,N: nat] :
( ( nth @ A @ ( append @ A @ Xs @ Ys2 ) @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) )
= ( nth @ A @ Ys2 @ N ) ) ).
% nth_append_length_plus
thf(fact_3682_list__match__lel__lel,axiom,
! [A: $tType,C1: list @ A,Qs: A,C22: list @ A,C12: list @ A,Qs2: A,C23: list @ A] :
( ( ( append @ A @ C1 @ ( cons @ A @ Qs @ C22 ) )
= ( append @ A @ C12 @ ( cons @ A @ Qs2 @ C23 ) ) )
=> ( ! [C21: list @ A] :
( ( C1
= ( append @ A @ C12 @ ( cons @ A @ Qs2 @ C21 ) ) )
=> ( C23
!= ( append @ A @ C21 @ ( cons @ A @ Qs @ C22 ) ) ) )
=> ( ( ( C12 = C1 )
=> ( ( Qs2 = Qs )
=> ( C23 != C22 ) ) )
=> ~ ! [C212: list @ A] :
( ( C12
= ( append @ A @ C1 @ ( cons @ A @ Qs @ C212 ) ) )
=> ( C22
!= ( append @ A @ C212 @ ( cons @ A @ Qs2 @ C23 ) ) ) ) ) ) ) ).
% list_match_lel_lel
thf(fact_3683_foldl__conc__empty__eq,axiom,
! [A: $tType,I: list @ A,Ww: list @ ( list @ A )] :
( ( foldl @ ( list @ A ) @ ( list @ A ) @ ( append @ A ) @ I @ Ww )
= ( append @ A @ I @ ( foldl @ ( list @ A ) @ ( list @ A ) @ ( append @ A ) @ ( nil @ A ) @ Ww ) ) ) ).
% foldl_conc_empty_eq
thf(fact_3684_Misc_Omap__eq__append__conv,axiom,
! [A: $tType,B: $tType,F: B > A,Ls: list @ B,Fl: list @ A,Fl2: list @ A] :
( ( ( map @ B @ A @ F @ Ls )
= ( append @ A @ Fl @ Fl2 ) )
= ( ? [L3: list @ B,L7: list @ B] :
( ( Ls
= ( append @ B @ L3 @ L7 ) )
& ( ( map @ B @ A @ F @ L3 )
= Fl )
& ( ( map @ B @ A @ F @ L7 )
= Fl2 ) ) ) ) ).
% Misc.map_eq_append_conv
thf(fact_3685_Misc_Oappend__eq__map__conv,axiom,
! [A: $tType,B: $tType,Fl: list @ A,Fl2: list @ A,F: B > A,Ls: list @ B] :
( ( ( append @ A @ Fl @ Fl2 )
= ( map @ B @ A @ F @ Ls ) )
= ( ? [L3: list @ B,L7: list @ B] :
( ( Ls
= ( append @ B @ L3 @ L7 ) )
& ( ( map @ B @ A @ F @ L3 )
= Fl )
& ( ( map @ B @ A @ F @ L7 )
= Fl2 ) ) ) ) ).
% Misc.append_eq_map_conv
thf(fact_3686_map__eq__appendE,axiom,
! [B: $tType,A: $tType,F: B > A,Ls: list @ B,Fl: list @ A,Fl2: list @ A] :
( ( ( map @ B @ A @ F @ Ls )
= ( append @ A @ Fl @ Fl2 ) )
=> ~ ! [L5: list @ B,L8: list @ B] :
( ( Ls
= ( append @ B @ L5 @ L8 ) )
=> ( ( ( map @ B @ A @ F @ L5 )
= Fl )
=> ( ( map @ B @ A @ F @ L8 )
!= Fl2 ) ) ) ) ).
% map_eq_appendE
thf(fact_3687_append__eq__mapE,axiom,
! [B: $tType,A: $tType,Fl: list @ A,Fl2: list @ A,F: B > A,Ls: list @ B] :
( ( ( append @ A @ Fl @ Fl2 )
= ( map @ B @ A @ F @ Ls ) )
=> ~ ! [L5: list @ B,L8: list @ B] :
( ( Ls
= ( append @ B @ L5 @ L8 ) )
=> ( ( ( map @ B @ A @ F @ L5 )
= Fl )
=> ( ( map @ B @ A @ F @ L8 )
!= Fl2 ) ) ) ) ).
% append_eq_mapE
thf(fact_3688_enumerate__append__eq,axiom,
! [A: $tType,N: nat,Xs: list @ A,Ys2: list @ A] :
( ( enumerate @ A @ N @ ( append @ A @ Xs @ Ys2 ) )
= ( append @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N @ Xs ) @ ( enumerate @ A @ ( plus_plus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) @ Ys2 ) ) ) ).
% enumerate_append_eq
thf(fact_3689_neq__Nil__revE,axiom,
! [A: $tType,L: list @ A] :
( ( L
!= ( nil @ A ) )
=> ~ ! [Ll2: list @ A,E3: A] :
( L
!= ( append @ A @ Ll2 @ ( cons @ A @ E3 @ ( nil @ A ) ) ) ) ) ).
% neq_Nil_revE
thf(fact_3690_rev__induct2_H,axiom,
! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,Xs: list @ A,Ys2: list @ B] :
( ( P @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X2: A,Xs2: list @ A] : ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) @ ( nil @ B ) )
=> ( ! [Y4: B,Ys3: list @ B] : ( P @ ( nil @ A ) @ ( append @ B @ Ys3 @ ( cons @ B @ Y4 @ ( nil @ B ) ) ) )
=> ( ! [X2: A,Xs2: list @ A,Y4: B,Ys3: list @ B] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) @ ( append @ B @ Ys3 @ ( cons @ B @ Y4 @ ( nil @ B ) ) ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ).
% rev_induct2'
thf(fact_3691_neq__Nil__rev__conv,axiom,
! [A: $tType,L: list @ A] :
( ( L
!= ( nil @ A ) )
= ( ? [Xs3: list @ A,X3: A] :
( L
= ( append @ A @ Xs3 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ) ) ).
% neq_Nil_rev_conv
thf(fact_3692_rev__nonempty__induct2_H,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B,P: ( list @ A ) > ( list @ B ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ( Ys2
!= ( nil @ B ) )
=> ( ! [X2: A,Y4: B] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) @ ( cons @ B @ Y4 @ ( nil @ B ) ) )
=> ( ! [X2: A,Xs2: list @ A,Y4: B] :
( ( Xs2
!= ( nil @ A ) )
=> ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) @ ( cons @ B @ Y4 @ ( nil @ B ) ) ) )
=> ( ! [X2: A,Y4: B,Ys3: list @ B] :
( ( Ys3
!= ( nil @ B ) )
=> ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) @ ( append @ B @ Ys3 @ ( cons @ B @ Y4 @ ( nil @ B ) ) ) ) )
=> ( ! [X2: A,Xs2: list @ A,Y4: B,Ys3: list @ B] :
( ( P @ Xs2 @ Ys3 )
=> ( ( Xs2
!= ( nil @ A ) )
=> ( ( Ys3
!= ( nil @ B ) )
=> ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) @ ( append @ B @ Ys3 @ ( cons @ B @ Y4 @ ( nil @ B ) ) ) ) ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ) ) ).
% rev_nonempty_induct2'
thf(fact_3693_list__Cons__eq__append__cases,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys2: list @ A,Zs: list @ A] :
( ( ( cons @ A @ X @ Xs )
= ( append @ A @ Ys2 @ Zs ) )
=> ( ( ( Ys2
= ( nil @ A ) )
=> ( Zs
!= ( cons @ A @ X @ Xs ) ) )
=> ~ ! [Ys4: list @ A] :
( ( Ys2
= ( cons @ A @ X @ Ys4 ) )
=> ( ( append @ A @ Ys4 @ Zs )
!= Xs ) ) ) ) ).
% list_Cons_eq_append_cases
thf(fact_3694_list__append__eq__Cons__cases,axiom,
! [A: $tType,Ys2: list @ A,Zs: list @ A,X: A,Xs: list @ A] :
( ( ( append @ A @ Ys2 @ Zs )
= ( cons @ A @ X @ Xs ) )
=> ( ( ( Ys2
= ( nil @ A ) )
=> ( Zs
!= ( cons @ A @ X @ Xs ) ) )
=> ~ ! [Ys4: list @ A] :
( ( Ys2
= ( cons @ A @ X @ Ys4 ) )
=> ( ( append @ A @ Ys4 @ Zs )
!= Xs ) ) ) ) ).
% list_append_eq_Cons_cases
thf(fact_3695_in__set__list__format,axiom,
! [A: $tType,E2: A,L: list @ A] :
( ( member @ A @ E2 @ ( set2 @ A @ L ) )
=> ~ ! [L12: list @ A,L22: list @ A] :
( L
!= ( append @ A @ L12 @ ( cons @ A @ E2 @ L22 ) ) ) ) ).
% in_set_list_format
thf(fact_3696_xy__in__set__cases,axiom,
! [A: $tType,X: A,L: list @ A,Y: A] :
( ( member @ A @ X @ ( set2 @ A @ L ) )
=> ( ( member @ A @ Y @ ( set2 @ A @ L ) )
=> ( ( ( X = Y )
=> ! [L12: list @ A,L22: list @ A] :
( L
!= ( append @ A @ L12 @ ( cons @ A @ Y @ L22 ) ) ) )
=> ( ( ( X != Y )
=> ! [L12: list @ A,L22: list @ A,L32: list @ A] :
( L
!= ( append @ A @ L12 @ ( cons @ A @ X @ ( append @ A @ L22 @ ( cons @ A @ Y @ L32 ) ) ) ) ) )
=> ~ ( ( X != Y )
=> ! [L12: list @ A,L22: list @ A,L32: list @ A] :
( L
!= ( append @ A @ L12 @ ( cons @ A @ Y @ ( append @ A @ L22 @ ( cons @ A @ X @ L32 ) ) ) ) ) ) ) ) ) ) ).
% xy_in_set_cases
thf(fact_3697_map__consI_I2_J,axiom,
! [B: $tType,A: $tType,W: list @ A,L: list @ A,F: B > A,Ww: list @ B,A3: B] :
( ( ( append @ A @ W @ L )
= ( append @ A @ ( map @ B @ A @ F @ Ww ) @ L ) )
=> ( ( cons @ A @ ( F @ A3 ) @ ( append @ A @ W @ L ) )
= ( append @ A @ ( map @ B @ A @ F @ ( cons @ B @ A3 @ Ww ) ) @ L ) ) ) ).
% map_consI(2)
thf(fact_3698_list__rest__coinc,axiom,
! [A: $tType,S22: list @ A,S1: list @ A,R1: list @ A,R22: list @ A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ S22 ) @ ( size_size @ ( list @ A ) @ S1 ) )
=> ( ( ( append @ A @ S1 @ R1 )
= ( append @ A @ S22 @ R22 ) )
=> ? [R1p: list @ A] :
( R22
= ( append @ A @ R1p @ R1 ) ) ) ) ).
% list_rest_coinc
thf(fact_3699_set__union__code,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys2 ) )
= ( set2 @ A @ ( append @ A @ Xs @ Ys2 ) ) ) ).
% set_union_code
thf(fact_3700_distinct__match,axiom,
! [A: $tType,Al: list @ A,E2: A,Bl: list @ A,Al2: list @ A,Bl2: list @ A] :
( ( distinct @ A @ ( append @ A @ Al @ ( cons @ A @ E2 @ Bl ) ) )
=> ( ( ( append @ A @ Al @ ( cons @ A @ E2 @ Bl ) )
= ( append @ A @ Al2 @ ( cons @ A @ E2 @ Bl2 ) ) )
= ( ( Al = Al2 )
& ( Bl = Bl2 ) ) ) ) ).
% distinct_match
thf(fact_3701_upt__eq__append__conv,axiom,
! [I: nat,J: nat,Xs: list @ nat,Ys2: list @ nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ( upt @ I @ J )
= ( append @ nat @ Xs @ Ys2 ) )
= ( ? [K4: nat] :
( ( ord_less_eq @ nat @ I @ K4 )
& ( ord_less_eq @ nat @ K4 @ J )
& ( ( upt @ I @ K4 )
= Xs )
& ( ( upt @ K4 @ J )
= Ys2 ) ) ) ) ) ).
% upt_eq_append_conv
thf(fact_3702_replicate__add,axiom,
! [A: $tType,N: nat,M2: nat,X: A] :
( ( replicate @ A @ ( plus_plus @ nat @ N @ M2 ) @ X )
= ( append @ A @ ( replicate @ A @ N @ X ) @ ( replicate @ A @ M2 @ X ) ) ) ).
% replicate_add
thf(fact_3703_upt__append,axiom,
! [I: nat,J: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( append @ nat @ ( upt @ ( zero_zero @ nat ) @ I ) @ ( upt @ I @ J ) )
= ( upt @ ( zero_zero @ nat ) @ J ) ) ) ).
% upt_append
thf(fact_3704_upt__add__eq__append,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( upt @ I @ ( plus_plus @ nat @ J @ K ) )
= ( append @ nat @ ( upt @ I @ J ) @ ( upt @ J @ ( plus_plus @ nat @ J @ K ) ) ) ) ) ).
% upt_add_eq_append
thf(fact_3705_foldl__rule,axiom,
! [Sigma2: $tType,A: $tType,I4: Sigma2 > ( list @ A ) > ( list @ A ) > $o,Sigma_0: Sigma2,L0: list @ A,F: Sigma2 > A > Sigma2] :
( ( I4 @ Sigma_0 @ ( nil @ A ) @ L0 )
=> ( ! [L12: list @ A,L22: list @ A,X2: A,Sigma: Sigma2] :
( ( L0
= ( append @ A @ L12 @ ( cons @ A @ X2 @ L22 ) ) )
=> ( ( I4 @ Sigma @ L12 @ ( cons @ A @ X2 @ L22 ) )
=> ( I4 @ ( F @ Sigma @ X2 ) @ ( append @ A @ L12 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) @ L22 ) ) )
=> ( I4 @ ( foldl @ Sigma2 @ A @ F @ Sigma_0 @ L0 ) @ L0 @ ( nil @ A ) ) ) ) ).
% foldl_rule
thf(fact_3706_foldl__rule__P,axiom,
! [Sigma2: $tType,A: $tType,I4: Sigma2 > ( list @ A ) > ( list @ A ) > $o,Sigma_0: Sigma2,L0: list @ A,F: Sigma2 > A > Sigma2,P: Sigma2 > $o] :
( ( I4 @ Sigma_0 @ ( nil @ A ) @ L0 )
=> ( ! [L12: list @ A,L22: list @ A,X2: A,Sigma: Sigma2] :
( ( L0
= ( append @ A @ L12 @ ( cons @ A @ X2 @ L22 ) ) )
=> ( ( I4 @ Sigma @ L12 @ ( cons @ A @ X2 @ L22 ) )
=> ( I4 @ ( F @ Sigma @ X2 ) @ ( append @ A @ L12 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) @ L22 ) ) )
=> ( ! [Sigma: Sigma2] :
( ( I4 @ Sigma @ L0 @ ( nil @ A ) )
=> ( P @ Sigma ) )
=> ( P @ ( foldl @ Sigma2 @ A @ F @ Sigma_0 @ L0 ) ) ) ) ) ).
% foldl_rule_P
thf(fact_3707_foldl__rule__aux,axiom,
! [Sigma2: $tType,A: $tType,I4: Sigma2 > ( list @ A ) > $o,Sigma_0: Sigma2,L0: list @ A,F: Sigma2 > A > Sigma2] :
( ( I4 @ Sigma_0 @ L0 )
=> ( ! [L12: list @ A,L22: list @ A,X2: A,Sigma: Sigma2] :
( ( L0
= ( append @ A @ L12 @ ( cons @ A @ X2 @ L22 ) ) )
=> ( ( I4 @ Sigma @ ( cons @ A @ X2 @ L22 ) )
=> ( I4 @ ( F @ Sigma @ X2 ) @ L22 ) ) )
=> ( I4 @ ( foldl @ Sigma2 @ A @ F @ Sigma_0 @ L0 ) @ ( nil @ A ) ) ) ) ).
% foldl_rule_aux
thf(fact_3708_foldl__rule__aux__P,axiom,
! [Sigma2: $tType,A: $tType,I4: Sigma2 > ( list @ A ) > $o,Sigma_0: Sigma2,L0: list @ A,F: Sigma2 > A > Sigma2,P: Sigma2 > $o] :
( ( I4 @ Sigma_0 @ L0 )
=> ( ! [L12: list @ A,L22: list @ A,X2: A,Sigma: Sigma2] :
( ( L0
= ( append @ A @ L12 @ ( cons @ A @ X2 @ L22 ) ) )
=> ( ( I4 @ Sigma @ ( cons @ A @ X2 @ L22 ) )
=> ( I4 @ ( F @ Sigma @ X2 ) @ L22 ) ) )
=> ( ! [Sigma: Sigma2] :
( ( I4 @ Sigma @ ( nil @ A ) )
=> ( P @ Sigma ) )
=> ( P @ ( foldl @ Sigma2 @ A @ F @ Sigma_0 @ L0 ) ) ) ) ) ).
% foldl_rule_aux_P
thf(fact_3709_map__by__foldl,axiom,
! [B: $tType,A: $tType,F: A > B,L: list @ A] :
( ( foldl @ ( list @ B ) @ A
@ ^ [L3: list @ B,X3: A] : ( append @ B @ L3 @ ( cons @ B @ ( F @ X3 ) @ ( nil @ B ) ) )
@ ( nil @ B )
@ L )
= ( map @ A @ B @ F @ L ) ) ).
% map_by_foldl
thf(fact_3710_length__Suc__rev__conv,axiom,
! [A: $tType,Xs: list @ A,N: nat] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( suc @ N ) )
= ( ? [Ys: list @ A,Y3: A] :
( ( Xs
= ( append @ A @ Ys @ ( cons @ A @ Y3 @ ( nil @ A ) ) ) )
& ( ( size_size @ ( list @ A ) @ Ys )
= N ) ) ) ) ).
% length_Suc_rev_conv
thf(fact_3711_length__compl__rev__induct,axiom,
! [A: $tType,P: ( list @ A ) > $o,L: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [L5: list @ A,E3: A] :
( ! [Ll: list @ A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ll ) @ ( size_size @ ( list @ A ) @ L5 ) )
=> ( P @ Ll ) )
=> ( P @ ( append @ A @ L5 @ ( cons @ A @ E3 @ ( nil @ A ) ) ) ) )
=> ( P @ L ) ) ) ).
% length_compl_rev_induct
thf(fact_3712_not__distinct__split__distinct,axiom,
! [A: $tType,Xs: list @ A] :
( ~ ( distinct @ A @ Xs )
=> ~ ! [Y4: A,Ys3: list @ A] :
( ( distinct @ A @ Ys3 )
=> ( ( member @ A @ Y4 @ ( set2 @ A @ Ys3 ) )
=> ! [Zs2: list @ A] :
( Xs
!= ( append @ A @ Ys3 @ ( append @ A @ ( cons @ A @ Y4 @ ( nil @ A ) ) @ Zs2 ) ) ) ) ) ) ).
% not_distinct_split_distinct
thf(fact_3713_filter__eq__snocD,axiom,
! [A: $tType,P: A > $o,L: list @ A,L4: list @ A,X: A] :
( ( ( filter @ A @ P @ L )
= ( append @ A @ L4 @ ( cons @ A @ X @ ( nil @ A ) ) ) )
=> ( ( member @ A @ X @ ( set2 @ A @ L ) )
& ( P @ X ) ) ) ).
% filter_eq_snocD
thf(fact_3714_replicate__Suc__conv__snoc,axiom,
! [A: $tType,N: nat,X: A] :
( ( replicate @ A @ ( suc @ N ) @ X )
= ( append @ A @ ( replicate @ A @ N @ X ) @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ).
% replicate_Suc_conv_snoc
thf(fact_3715_slice__prepend,axiom,
! [A: $tType,I: nat,K: nat,Xs: list @ A,Ys2: list @ A] :
( ( ord_less_eq @ nat @ I @ K )
=> ( ( ord_less_eq @ nat @ K @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( slice @ A @ I @ K @ Xs )
= ( slice @ A @ ( plus_plus @ nat @ I @ ( size_size @ ( list @ A ) @ Ys2 ) ) @ ( plus_plus @ nat @ K @ ( size_size @ ( list @ A ) @ Ys2 ) ) @ ( append @ A @ Ys2 @ Xs ) ) ) ) ) ).
% slice_prepend
thf(fact_3716_sorted__append__bigger,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Y: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ X2 @ Y ) )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( append @ A @ Xs @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) ) ) ) ).
% sorted_append_bigger
thf(fact_3717_upt__eq__lel__conv,axiom,
! [L: nat,H3: nat,Is1: list @ nat,I: nat,Is2: list @ nat] :
( ( ( upt @ L @ H3 )
= ( append @ nat @ Is1 @ ( cons @ nat @ I @ Is2 ) ) )
= ( ( Is1
= ( upt @ L @ I ) )
& ( Is2
= ( upt @ ( suc @ I ) @ H3 ) )
& ( ord_less_eq @ nat @ L @ I )
& ( ord_less @ nat @ I @ H3 ) ) ) ).
% upt_eq_lel_conv
thf(fact_3718_list__encode_Osimps_I2_J,axiom,
! [X: nat,Xs: list @ nat] :
( ( nat_list_encode @ ( cons @ nat @ X @ Xs ) )
= ( suc @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ X @ ( nat_list_encode @ Xs ) ) ) ) ) ).
% list_encode.simps(2)
thf(fact_3719_list__encode_Opelims,axiom,
! [X: list @ nat,Y: 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
= ( suc @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ X2 @ ( nat_list_encode @ Xs2 ) ) ) ) )
=> ~ ( accp @ ( list @ nat ) @ nat_list_encode_rel @ ( cons @ nat @ X2 @ Xs2 ) ) ) ) ) ) ) ).
% list_encode.pelims
thf(fact_3720_map__upds__append1,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys2: list @ B,M2: A > ( option @ B ),X: A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( map_upds @ A @ B @ M2 @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) @ Ys2 )
= ( fun_upd @ A @ ( option @ B ) @ ( map_upds @ A @ B @ M2 @ Xs @ Ys2 ) @ X @ ( some @ B @ ( nth @ B @ Ys2 @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ).
% map_upds_append1
thf(fact_3721_lexn__conv,axiom,
! [A: $tType] :
( ( lexn @ A )
= ( ^ [R3: set @ ( product_prod @ A @ A ),N5: nat] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs3: list @ A,Ys: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= N5 )
& ( ( size_size @ ( list @ A ) @ Ys )
= N5 )
& ? [Xys2: list @ A,X3: A,Y3: A,Xs5: list @ A,Ys5: list @ A] :
( ( Xs3
= ( append @ A @ Xys2 @ ( cons @ A @ X3 @ Xs5 ) ) )
& ( Ys
= ( append @ A @ Xys2 @ ( cons @ A @ Y3 @ Ys5 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 ) ) ) ) ) ) ) ).
% lexn_conv
thf(fact_3722_drop__last__conv,axiom,
! [A: $tType,L: list @ A] :
( ( L
!= ( nil @ A ) )
=> ( ( drop @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ L ) @ ( suc @ ( zero_zero @ nat ) ) ) @ L )
= ( cons @ A @ ( last @ A @ L ) @ ( nil @ A ) ) ) ) ).
% drop_last_conv
thf(fact_3723_Misc_Olast__in__set,axiom,
! [A: $tType,L: list @ A] :
( ( L
!= ( nil @ A ) )
=> ( member @ A @ ( last @ A @ L ) @ ( set2 @ A @ L ) ) ) ).
% Misc.last_in_set
thf(fact_3724_last__upt,axiom,
! [I: nat,J: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( last @ nat @ ( upt @ I @ J ) )
= ( minus_minus @ nat @ J @ ( one_one @ nat ) ) ) ) ).
% last_upt
thf(fact_3725_map__upds__Cons,axiom,
! [A: $tType,B: $tType,M2: A > ( option @ B ),A3: A,As2: list @ A,B2: B,Bs2: list @ B] :
( ( map_upds @ A @ B @ M2 @ ( cons @ A @ A3 @ As2 ) @ ( cons @ B @ B2 @ Bs2 ) )
= ( map_upds @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ A3 @ ( some @ B @ B2 ) ) @ As2 @ Bs2 ) ) ).
% map_upds_Cons
thf(fact_3726_map__upds__twist,axiom,
! [A: $tType,B: $tType,A3: A,As2: list @ A,M2: A > ( option @ B ),B2: B,Bs2: list @ B] :
( ~ ( member @ A @ A3 @ ( set2 @ A @ As2 ) )
=> ( ( map_upds @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ A3 @ ( some @ B @ B2 ) ) @ As2 @ Bs2 )
= ( fun_upd @ A @ ( option @ B ) @ ( map_upds @ A @ B @ M2 @ As2 @ Bs2 ) @ A3 @ ( some @ B @ B2 ) ) ) ) ).
% map_upds_twist
thf(fact_3727_last__filter,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ( P @ ( last @ A @ Xs ) )
=> ( ( last @ A @ ( filter @ A @ P @ Xs ) )
= ( last @ A @ Xs ) ) ) ) ).
% last_filter
thf(fact_3728_lexn__length,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A ),N: nat] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( lexn @ A @ R @ N ) )
=> ( ( ( size_size @ ( list @ A ) @ Xs )
= N )
& ( ( size_size @ ( list @ A ) @ Ys2 )
= N ) ) ) ).
% lexn_length
thf(fact_3729_product_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ A,Ys2: list @ B] :
( ( product @ A @ B @ ( cons @ A @ X @ Xs ) @ Ys2 )
= ( append @ ( product_prod @ A @ B ) @ ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X ) @ Ys2 ) @ ( product @ A @ B @ Xs @ Ys2 ) ) ) ).
% product.simps(2)
thf(fact_3730_map__of__Some__split,axiom,
! [B: $tType,A: $tType,Xs: list @ ( product_prod @ B @ A ),K: B,V: A] :
( ( ( map_of @ B @ A @ Xs @ K )
= ( some @ A @ V ) )
=> ? [Ys3: list @ ( product_prod @ B @ A ),Zs2: list @ ( product_prod @ B @ A )] :
( ( Xs
= ( append @ ( product_prod @ B @ A ) @ Ys3 @ ( cons @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ K @ V ) @ Zs2 ) ) )
& ( ( map_of @ B @ A @ Ys3 @ K )
= ( none @ A ) ) ) ) ).
% map_of_Some_split
thf(fact_3731_last__conv__nth,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( last @ A @ Xs )
= ( nth @ A @ Xs @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) ) ) ) ).
% last_conv_nth
thf(fact_3732_map__of__distinct__lookup,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ ( product_prod @ A @ B ),Ys2: list @ ( product_prod @ A @ B ),Y: B] :
( ~ ( member @ A @ X @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) ) )
=> ( ~ ( member @ A @ X @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Ys2 ) ) )
=> ( ( map_of @ A @ B @ ( append @ ( product_prod @ A @ B ) @ Xs @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ Ys2 ) ) @ X )
= ( some @ B @ Y ) ) ) ) ).
% map_of_distinct_lookup
thf(fact_3733_map__of__distinct__upd3,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ ( product_prod @ A @ B ),Ys2: list @ ( product_prod @ A @ B ),Y: B,Y7: B] :
( ~ ( member @ A @ X @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) ) )
=> ( ~ ( member @ A @ X @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Ys2 ) ) )
=> ( ( map_of @ A @ B @ ( append @ ( product_prod @ A @ B ) @ Xs @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ Ys2 ) ) )
= ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ ( append @ ( product_prod @ A @ B ) @ Xs @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y7 ) @ Ys2 ) ) ) @ X @ ( some @ B @ Y ) ) ) ) ) ).
% map_of_distinct_upd3
thf(fact_3734_map__of__distinct__upd2,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ ( product_prod @ A @ B ),Ys2: list @ ( product_prod @ A @ B ),Y: B] :
( ~ ( member @ A @ X @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) ) )
=> ( ~ ( member @ A @ X @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Ys2 ) ) )
=> ( ( map_of @ A @ B @ ( append @ ( product_prod @ A @ B ) @ Xs @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ Ys2 ) ) )
= ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ ( append @ ( product_prod @ A @ B ) @ Xs @ Ys2 ) ) @ X @ ( some @ B @ Y ) ) ) ) ) ).
% map_of_distinct_upd2
thf(fact_3735_map__of__distinct__upd4,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ ( product_prod @ A @ B ),Ys2: list @ ( product_prod @ A @ B ),Y: B] :
( ~ ( member @ A @ X @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) ) )
=> ( ~ ( member @ A @ X @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Ys2 ) ) )
=> ( ( map_of @ A @ B @ ( append @ ( product_prod @ A @ B ) @ Xs @ Ys2 ) )
= ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ ( append @ ( product_prod @ A @ B ) @ Xs @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ Ys2 ) ) ) @ X @ ( none @ B ) ) ) ) ) ).
% map_of_distinct_upd4
thf(fact_3736_lex__conv,axiom,
! [A: $tType] :
( ( lex @ A )
= ( ^ [R3: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs3: list @ A,Ys: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ A ) @ Ys ) )
& ? [Xys2: list @ A,X3: A,Y3: A,Xs5: list @ A,Ys5: list @ A] :
( ( Xs3
= ( append @ A @ Xys2 @ ( cons @ A @ X3 @ Xs5 ) ) )
& ( Ys
= ( append @ A @ Xys2 @ ( cons @ A @ Y3 @ Ys5 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 ) ) ) ) ) ) ) ).
% lex_conv
thf(fact_3737_shuffles_Opelims,axiom,
! [A: $tType,X: list @ A,Xa: list @ A,Y: set @ ( list @ A )] :
( ( ( shuffles @ A @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Xa ) )
=> ( ( ( X
= ( nil @ A ) )
=> ( ( Y
= ( insert2 @ ( list @ A ) @ Xa @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xa ) ) ) )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( ( Y
= ( insert2 @ ( list @ A ) @ X @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ ( nil @ A ) ) ) ) )
=> ~ ! [X2: A,Xs2: list @ A] :
( ( X
= ( cons @ A @ X2 @ Xs2 ) )
=> ! [Y4: A,Ys3: list @ A] :
( ( Xa
= ( cons @ A @ Y4 @ Ys3 ) )
=> ( ( Y
= ( sup_sup @ ( set @ ( list @ A ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 ) @ ( shuffles @ A @ Xs2 @ ( cons @ A @ Y4 @ Ys3 ) ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y4 ) @ ( shuffles @ A @ ( cons @ A @ X2 @ Xs2 ) @ Ys3 ) ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ A @ Y4 @ Ys3 ) ) ) ) ) ) ) ) ) ) ).
% shuffles.pelims
thf(fact_3738_set__relcomp,axiom,
! [B: $tType,C: $tType,A: $tType,Xys: list @ ( product_prod @ A @ C ),Yzs: list @ ( product_prod @ C @ B )] :
( ( relcomp @ A @ C @ B @ ( set2 @ ( product_prod @ A @ C ) @ Xys ) @ ( set2 @ ( product_prod @ C @ B ) @ Yzs ) )
= ( set2 @ ( product_prod @ A @ B )
@ ( concat @ ( product_prod @ A @ B )
@ ( map @ ( product_prod @ A @ C ) @ ( list @ ( product_prod @ A @ B ) )
@ ^ [Xy: product_prod @ A @ C] :
( concat @ ( product_prod @ A @ B )
@ ( map @ ( product_prod @ C @ B ) @ ( list @ ( product_prod @ A @ B ) )
@ ^ [Yz: product_prod @ C @ B] :
( if @ ( list @ ( product_prod @ A @ B ) )
@ ( ( product_snd @ A @ C @ Xy )
= ( product_fst @ C @ B @ Yz ) )
@ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ ( product_fst @ A @ C @ Xy ) @ ( product_snd @ C @ B @ Yz ) ) @ ( nil @ ( product_prod @ A @ B ) ) )
@ ( nil @ ( product_prod @ A @ B ) ) )
@ Yzs ) )
@ Xys ) ) ) ) ).
% set_relcomp
thf(fact_3739_natural_Osimps_I4_J,axiom,
! [T: $tType,F1: T,F22: code_natural > T] :
( ( code_case_natural @ T @ F1 @ F22 @ ( zero_zero @ code_natural ) )
= F1 ) ).
% natural.simps(4)
thf(fact_3740_Cons__in__lex,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: A,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys2 ) ) @ ( lex @ A @ R ) )
= ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
& ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys2 ) ) )
| ( ( X = Y )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( lex @ A @ R ) ) ) ) ) ).
% Cons_in_lex
thf(fact_3741_concat__filter__neq__Nil,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( concat @ A
@ ( filter @ ( list @ A )
@ ^ [Ys: list @ A] :
( Ys
!= ( nil @ A ) )
@ Xs ) )
= ( concat @ A @ Xs ) ) ).
% concat_filter_neq_Nil
thf(fact_3742_Nil__notin__lex,axiom,
! [A: $tType,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys2 ) @ ( lex @ A @ R ) ) ).
% Nil_notin_lex
thf(fact_3743_Nil2__notin__lex,axiom,
! [A: $tType,Xs: list @ A,R: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ ( nil @ A ) ) @ ( lex @ A @ R ) ) ).
% Nil2_notin_lex
thf(fact_3744_lex__append__leftI,axiom,
! [A: $tType,Ys2: list @ A,Zs: list @ A,R: set @ ( product_prod @ A @ A ),Xs: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys2 @ Zs ) @ ( lex @ A @ R ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys2 ) @ ( append @ A @ Xs @ Zs ) ) @ ( lex @ A @ R ) ) ) ).
% lex_append_leftI
thf(fact_3745_foldl__foldl__conv__concat,axiom,
! [A: $tType,B: $tType,F: A > B > A,A3: A,Xs: list @ ( list @ B )] :
( ( foldl @ A @ ( list @ B ) @ ( foldl @ A @ B @ F ) @ A3 @ Xs )
= ( foldl @ A @ B @ F @ A3 @ ( concat @ B @ Xs ) ) ) ).
% foldl_foldl_conv_concat
thf(fact_3746_product__concat__map,axiom,
! [B: $tType,A: $tType] :
( ( product @ A @ B )
= ( ^ [Xs3: list @ A,Ys: list @ B] :
( concat @ ( product_prod @ A @ B )
@ ( map @ A @ ( list @ ( product_prod @ A @ B ) )
@ ^ [X3: A] : ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 ) @ Ys )
@ Xs3 ) ) ) ) ).
% product_concat_map
thf(fact_3747_lex__append__left__iff,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys2: list @ A,Zs: list @ A] :
( ! [X2: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ R )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys2 ) @ ( append @ A @ Xs @ Zs ) ) @ ( lex @ A @ R ) )
= ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys2 @ Zs ) @ ( lex @ A @ R ) ) ) ) ).
% lex_append_left_iff
thf(fact_3748_lex__append__leftD,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys2: list @ A,Zs: list @ A] :
( ! [X2: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ R )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys2 ) @ ( append @ A @ Xs @ Zs ) ) @ ( lex @ A @ R ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys2 @ Zs ) @ ( lex @ A @ R ) ) ) ) ).
% lex_append_leftD
thf(fact_3749_lex__append__rightI,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A ),Vs: list @ A,Us: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( lex @ A @ R ) )
=> ( ( ( size_size @ ( list @ A ) @ Vs )
= ( size_size @ ( list @ A ) @ Us ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Us ) @ ( append @ A @ Ys2 @ Vs ) ) @ ( lex @ A @ R ) ) ) ) ).
% lex_append_rightI
thf(fact_3750_map__of__concat,axiom,
! [B: $tType,A: $tType,Xss: list @ ( list @ ( product_prod @ A @ B ) )] :
( ( map_of @ A @ B @ ( concat @ ( product_prod @ A @ B ) @ Xss ) )
= ( foldr @ ( list @ ( product_prod @ A @ B ) ) @ ( A > ( option @ B ) )
@ ^ [Xs3: list @ ( product_prod @ A @ B ),F2: A > ( option @ B )] : ( map_add @ A @ B @ F2 @ ( map_of @ A @ B @ Xs3 ) )
@ Xss
@ ^ [X3: A] : ( none @ B ) ) ) ).
% map_of_concat
thf(fact_3751_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] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N3 )
& ( ( concat @ A @ ( replicate @ ( list @ A ) @ N3 @ Zs2 ) )
= ( append @ A @ Xs @ Ys2 ) ) ) ) ) ) ).
% comm_append_is_replicate
thf(fact_3752_shuffles_Opinduct,axiom,
! [A: $tType,A0: list @ A,A1: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ A0 @ A1 ) )
=> ( ! [Ys3: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys3 ) )
=> ( P @ ( nil @ A ) @ Ys3 ) )
=> ( ! [Xs2: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ ( nil @ A ) ) )
=> ( P @ Xs2 @ ( nil @ A ) ) )
=> ( ! [X2: A,Xs2: list @ A,Y4: A,Ys3: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ A @ Y4 @ Ys3 ) ) )
=> ( ( P @ Xs2 @ ( cons @ A @ Y4 @ Ys3 ) )
=> ( ( P @ ( cons @ A @ X2 @ Xs2 ) @ Ys3 )
=> ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ A @ Y4 @ Ys3 ) ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ) ).
% shuffles.pinduct
thf(fact_3753_shuffles_Opsimps_I1_J,axiom,
! [A: $tType,Ys2: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys2 ) )
=> ( ( shuffles @ A @ ( nil @ A ) @ Ys2 )
= ( insert2 @ ( list @ A ) @ Ys2 @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ) ).
% shuffles.psimps(1)
thf(fact_3754_shuffles_Opsimps_I2_J,axiom,
! [A: $tType,Xs: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ ( nil @ A ) ) )
=> ( ( shuffles @ A @ Xs @ ( nil @ A ) )
= ( insert2 @ ( list @ A ) @ Xs @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ) ).
% shuffles.psimps(2)
thf(fact_3755_distinct__concat_H,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( distinct @ ( list @ A )
@ ( filter @ ( list @ A )
@ ^ [Ys: list @ A] :
( Ys
!= ( nil @ A ) )
@ Xs ) )
=> ( ! [Ys3: list @ A] :
( ( member @ ( list @ A ) @ Ys3 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( distinct @ A @ Ys3 ) )
=> ( ! [Ys3: list @ A,Zs2: list @ A] :
( ( member @ ( list @ A ) @ Ys3 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( ( member @ ( list @ A ) @ Zs2 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( ( Ys3 != Zs2 )
=> ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Ys3 ) @ ( set2 @ A @ Zs2 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( distinct @ A @ ( concat @ A @ Xs ) ) ) ) ) ).
% distinct_concat'
thf(fact_3756_shuffles_Opsimps_I3_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: A,Ys2: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys2 ) ) )
=> ( ( shuffles @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys2 ) )
= ( sup_sup @ ( set @ ( list @ A ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X ) @ ( shuffles @ A @ Xs @ ( cons @ A @ Y @ Ys2 ) ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y ) @ ( shuffles @ A @ ( cons @ A @ X @ Xs ) @ Ys2 ) ) ) ) ) ).
% shuffles.psimps(3)
thf(fact_3757_product__code,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys2: list @ B] :
( ( product_product @ A @ B @ ( set2 @ A @ Xs ) @ ( set2 @ B @ Ys2 ) )
= ( set2 @ ( product_prod @ A @ B )
@ ( concat @ ( product_prod @ A @ B )
@ ( map @ A @ ( list @ ( product_prod @ A @ B ) )
@ ^ [X3: A] : ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 ) @ Ys2 )
@ Xs ) ) ) ) ).
% product_code
thf(fact_3758_lenlex__conv,axiom,
! [A: $tType] :
( ( lenlex @ A )
= ( ^ [R3: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs3: list @ A,Ys: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ ( size_size @ ( list @ A ) @ Ys ) )
| ( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ A ) @ Ys ) )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs3 @ Ys ) @ ( lex @ A @ R3 ) ) ) ) ) ) ) ) ).
% lenlex_conv
thf(fact_3759_nth__sorted__list__of__set__greaterThanLessThan,axiom,
! [N: nat,J: nat,I: nat] :
( ( ord_less @ nat @ N @ ( minus_minus @ nat @ J @ ( suc @ I ) ) )
=> ( ( nth @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_or5935395276787703475ssThan @ nat @ I @ J ) ) @ N )
= ( suc @ ( plus_plus @ nat @ I @ N ) ) ) ) ).
% nth_sorted_list_of_set_greaterThanLessThan
thf(fact_3760_splice_Opinduct,axiom,
! [A: $tType,A0: list @ A,A1: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ A0 @ A1 ) )
=> ( ! [Ys3: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys3 ) )
=> ( P @ ( nil @ A ) @ Ys3 ) )
=> ( ! [X2: A,Xs2: list @ A,Ys3: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ Ys3 ) )
=> ( ( P @ Ys3 @ Xs2 )
=> ( P @ ( cons @ A @ X2 @ Xs2 ) @ Ys3 ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% splice.pinduct
thf(fact_3761_Nil__lenlex__iff1,axiom,
! [A: $tType,Ns: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ns ) @ ( lenlex @ A @ R ) )
= ( Ns
!= ( nil @ A ) ) ) ).
% Nil_lenlex_iff1
thf(fact_3762_lenlex__irreflexive,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),Xs: list @ A] :
( ! [X2: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ R )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Xs ) @ ( lenlex @ A @ R ) ) ) ).
% lenlex_irreflexive
thf(fact_3763_Nil__lenlex__iff2,axiom,
! [A: $tType,Ns: list @ A,R: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ns @ ( nil @ A ) ) @ ( lenlex @ A @ R ) ) ).
% Nil_lenlex_iff2
thf(fact_3764_lenlex__length,axiom,
! [A: $tType,Ms: list @ A,Ns: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ms @ Ns ) @ ( lenlex @ A @ R ) )
=> ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ms ) @ ( size_size @ ( list @ A ) @ Ns ) ) ) ).
% lenlex_length
thf(fact_3765_lenlex__append1,axiom,
! [A: $tType,Us: list @ A,Xs: list @ A,R5: set @ ( product_prod @ A @ A ),Vs: list @ A,Ys2: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Us @ Xs ) @ ( lenlex @ A @ R5 ) )
=> ( ( ( size_size @ ( list @ A ) @ Vs )
= ( size_size @ ( list @ A ) @ Ys2 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Us @ Vs ) @ ( append @ A @ Xs @ Ys2 ) ) @ ( lenlex @ A @ R5 ) ) ) ) ).
% lenlex_append1
thf(fact_3766_member__product,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B,A4: set @ A,B6: set @ B] :
( ( member @ ( product_prod @ A @ B ) @ X @ ( product_product @ A @ B @ A4 @ B6 ) )
= ( member @ ( product_prod @ A @ B ) @ X
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : B6 ) ) ) ).
% member_product
thf(fact_3767_Product__Type_Oproduct__def,axiom,
! [B: $tType,A: $tType] :
( ( product_product @ A @ B )
= ( ^ [A9: set @ A,B9: set @ B] :
( product_Sigma @ A @ B @ A9
@ ^ [Uu: A] : B9 ) ) ) ).
% Product_Type.product_def
thf(fact_3768_lenlex__def,axiom,
! [A: $tType] :
( ( lenlex @ A )
= ( ^ [R3: set @ ( product_prod @ A @ A )] :
( inv_image @ ( product_prod @ nat @ ( list @ A ) ) @ ( list @ A ) @ ( lex_prod @ nat @ ( list @ A ) @ less_than @ ( lex @ A @ R3 ) )
@ ^ [Xs3: list @ A] : ( product_Pair @ nat @ ( list @ A ) @ ( size_size @ ( list @ A ) @ Xs3 ) @ Xs3 ) ) ) ) ).
% lenlex_def
thf(fact_3769_Cons__lenlex__iff,axiom,
! [A: $tType,M2: A,Ms: list @ A,N: A,Ns: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ M2 @ Ms ) @ ( cons @ A @ N @ Ns ) ) @ ( lenlex @ A @ R ) )
= ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ms ) @ ( size_size @ ( list @ A ) @ Ns ) )
| ( ( ( size_size @ ( list @ A ) @ Ms )
= ( size_size @ ( list @ A ) @ Ns ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ M2 @ N ) @ R ) )
| ( ( M2 = N )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ms @ Ns ) @ ( lenlex @ A @ R ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_3770_nth__sorted__list__of__set__greaterThanAtMost,axiom,
! [N: nat,J: nat,I: nat] :
( ( ord_less @ nat @ N @ ( minus_minus @ nat @ J @ I ) )
=> ( ( nth @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_or3652927894154168847AtMost @ nat @ I @ J ) ) @ N )
= ( suc @ ( plus_plus @ nat @ I @ N ) ) ) ) ).
% nth_sorted_list_of_set_greaterThanAtMost
thf(fact_3771_splice_Opelims,axiom,
! [A: $tType,X: list @ A,Xa: list @ A,Y: list @ A] :
( ( ( splice @ A @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Xa ) )
=> ( ( ( X
= ( nil @ A ) )
=> ( ( Y = Xa )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xa ) ) ) )
=> ~ ! [X2: A,Xs2: list @ A] :
( ( X
= ( cons @ A @ X2 @ Xs2 ) )
=> ( ( Y
= ( cons @ A @ X2 @ ( splice @ A @ Xa @ Xs2 ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ Xa ) ) ) ) ) ) ) ).
% splice.pelims
thf(fact_3772_sorted__find__Min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,P: A > $o] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ? [X7: A] :
( ( member @ A @ X7 @ ( set2 @ A @ Xs ) )
& ( P @ X7 ) )
=> ( ( find @ A @ P @ Xs )
= ( some @ A
@ ( lattic643756798350308766er_Min @ A
@ ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
& ( P @ X3 ) ) ) ) ) ) ) ) ) ).
% sorted_find_Min
thf(fact_3773_part__def,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ( ( linorder_part @ B @ A )
= ( ^ [F2: B > A,Pivot: A,Xs3: list @ B] :
( product_Pair @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) )
@ ( filter @ B
@ ^ [X3: B] : ( ord_less @ A @ ( F2 @ X3 ) @ Pivot )
@ Xs3 )
@ ( product_Pair @ ( list @ B ) @ ( list @ B )
@ ( filter @ B
@ ^ [X3: B] :
( ( F2 @ X3 )
= Pivot )
@ Xs3 )
@ ( filter @ B
@ ^ [X3: B] : ( ord_less @ A @ Pivot @ ( F2 @ X3 ) )
@ Xs3 ) ) ) ) ) ) ).
% part_def
thf(fact_3774_merge__list__correct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Ls: list @ ( list @ A ),As2: list @ ( list @ A )] :
( ! [L5: list @ A] :
( ( member @ ( list @ A ) @ L5 @ ( set2 @ ( list @ A ) @ Ls ) )
=> ( ( distinct @ A @ L5 )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L5 ) ) )
=> ( ! [L5: list @ A] :
( ( member @ ( list @ A ) @ L5 @ ( set2 @ ( list @ A ) @ As2 ) )
=> ( ( distinct @ A @ L5 )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L5 ) ) )
=> ( ( distinct @ A @ ( merge_list @ A @ As2 @ Ls ) )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( merge_list @ A @ As2 @ Ls ) )
& ( ( set2 @ A @ ( merge_list @ A @ As2 @ Ls ) )
= ( set2 @ A @ ( concat @ A @ ( append @ ( list @ A ) @ As2 @ Ls ) ) ) ) ) ) ) ) ).
% merge_list_correct
thf(fact_3775_length__splice,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( size_size @ ( list @ A ) @ ( splice @ A @ Xs @ Ys2 ) )
= ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys2 ) ) ) ).
% length_splice
thf(fact_3776_splice__replicate,axiom,
! [A: $tType,M2: nat,X: A,N: nat] :
( ( splice @ A @ ( replicate @ A @ M2 @ X ) @ ( replicate @ A @ N @ X ) )
= ( replicate @ A @ ( plus_plus @ nat @ M2 @ N ) @ X ) ) ).
% splice_replicate
thf(fact_3777_merge__list_Osimps_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [La: list @ A,Acc22: list @ ( list @ A ),L: list @ A] :
( ( merge_list @ A @ ( cons @ ( list @ A ) @ La @ Acc22 ) @ ( cons @ ( list @ A ) @ L @ ( nil @ ( list @ A ) ) ) )
= ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L @ ( cons @ ( list @ A ) @ La @ Acc22 ) ) ) ) ) ).
% merge_list.simps(4)
thf(fact_3778_merge__list_Osimps_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [La: list @ A,Acc22: list @ ( list @ A )] :
( ( merge_list @ A @ ( cons @ ( list @ A ) @ La @ Acc22 ) @ ( nil @ ( list @ A ) ) )
= ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La @ Acc22 ) ) ) ) ).
% merge_list.simps(3)
thf(fact_3779_merge__list_Osimps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: list @ A] :
( ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L @ ( nil @ ( list @ A ) ) ) )
= L ) ) ).
% merge_list.simps(2)
thf(fact_3780_merge__list_Osimps_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) )
= ( nil @ A ) ) ) ).
% merge_list.simps(1)
thf(fact_3781_find__SomeD_I1_J,axiom,
! [A: $tType,P: A > $o,Xs: list @ A,X: A] :
( ( ( find @ A @ P @ Xs )
= ( some @ A @ X ) )
=> ( P @ X ) ) ).
% find_SomeD(1)
thf(fact_3782_find_Osimps_I2_J,axiom,
! [A: $tType,P: A > $o,X: A,Xs: list @ A] :
( ( ( P @ X )
=> ( ( find @ A @ P @ ( cons @ A @ X @ Xs ) )
= ( some @ A @ X ) ) )
& ( ~ ( P @ X )
=> ( ( find @ A @ P @ ( cons @ A @ X @ Xs ) )
= ( find @ A @ P @ Xs ) ) ) ) ).
% find.simps(2)
thf(fact_3783_find_Osimps_I1_J,axiom,
! [A: $tType,Uu2: A > $o] :
( ( find @ A @ Uu2 @ ( nil @ A ) )
= ( none @ A ) ) ).
% find.simps(1)
thf(fact_3784_find__SomeD_I2_J,axiom,
! [A: $tType,P: A > $o,Xs: list @ A,X: A] :
( ( ( find @ A @ P @ Xs )
= ( some @ A @ X ) )
=> ( member @ A @ X @ ( set2 @ A @ Xs ) ) ) ).
% find_SomeD(2)
thf(fact_3785_find__None__iff2,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( ( none @ A )
= ( find @ A @ P @ Xs ) )
= ( ~ ? [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
& ( P @ X3 ) ) ) ) ).
% find_None_iff2
thf(fact_3786_find__None__iff,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( ( find @ A @ P @ Xs )
= ( none @ A ) )
= ( ~ ? [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
& ( P @ X3 ) ) ) ) ).
% find_None_iff
thf(fact_3787_part__code_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,Pivot2: A] :
( ( linorder_part @ B @ A @ F @ Pivot2 @ ( nil @ B ) )
= ( product_Pair @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) @ ( nil @ B ) @ ( product_Pair @ ( list @ B ) @ ( list @ B ) @ ( nil @ B ) @ ( nil @ B ) ) ) ) ) ).
% part_code(1)
thf(fact_3788_find__Some__iff,axiom,
! [A: $tType,P: A > $o,Xs: list @ A,X: A] :
( ( ( find @ A @ P @ Xs )
= ( some @ A @ X ) )
= ( ? [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( P @ ( nth @ A @ Xs @ I3 ) )
& ( X
= ( nth @ A @ Xs @ I3 ) )
& ! [J2: nat] :
( ( ord_less @ nat @ J2 @ I3 )
=> ~ ( P @ ( nth @ A @ Xs @ J2 ) ) ) ) ) ) ).
% find_Some_iff
thf(fact_3789_find__Some__iff2,axiom,
! [A: $tType,X: A,P: A > $o,Xs: list @ A] :
( ( ( some @ A @ X )
= ( find @ A @ P @ Xs ) )
= ( ? [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( P @ ( nth @ A @ Xs @ I3 ) )
& ( X
= ( nth @ A @ Xs @ I3 ) )
& ! [J2: nat] :
( ( ord_less @ nat @ J2 @ I3 )
=> ~ ( P @ ( nth @ A @ Xs @ J2 ) ) ) ) ) ) ).
% find_Some_iff2
thf(fact_3790_mergesort__remdups__def,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( mergesort_remdups @ A )
= ( ^ [Xs3: list @ A] :
( merge_list @ A @ ( nil @ ( list @ A ) )
@ ( map @ A @ ( list @ A )
@ ^ [X3: A] : ( cons @ A @ X3 @ ( nil @ A ) )
@ Xs3 ) ) ) ) ) ).
% mergesort_remdups_def
thf(fact_3791_splice_Opsimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys2: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ Ys2 ) )
=> ( ( splice @ A @ ( cons @ A @ X @ Xs ) @ Ys2 )
= ( cons @ A @ X @ ( splice @ A @ Ys2 @ Xs ) ) ) ) ).
% splice.psimps(2)
thf(fact_3792_splice_Opsimps_I1_J,axiom,
! [A: $tType,Ys2: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys2 ) )
=> ( ( splice @ A @ ( nil @ A ) @ Ys2 )
= Ys2 ) ) ).
% splice.psimps(1)
thf(fact_3793_part__code_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,Pivot2: A,X: B,Xs: list @ B] :
( ( linorder_part @ B @ A @ F @ Pivot2 @ ( cons @ B @ X @ Xs ) )
= ( product_case_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) @ ( product_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) )
@ ^ [Lts: list @ B] :
( product_case_prod @ ( list @ B ) @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) )
@ ^ [Eqs: list @ B,Gts: list @ B] : ( if @ ( product_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) ) @ ( ord_less @ A @ ( F @ X ) @ Pivot2 ) @ ( product_Pair @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) @ ( cons @ B @ X @ Lts ) @ ( product_Pair @ ( list @ B ) @ ( list @ B ) @ Eqs @ Gts ) ) @ ( if @ ( product_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) ) @ ( ord_less @ A @ Pivot2 @ ( F @ X ) ) @ ( product_Pair @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) @ Lts @ ( product_Pair @ ( list @ B ) @ ( list @ B ) @ Eqs @ ( cons @ B @ X @ Gts ) ) ) @ ( product_Pair @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) @ Lts @ ( product_Pair @ ( list @ B ) @ ( list @ B ) @ ( cons @ B @ X @ Eqs ) @ Gts ) ) ) ) )
@ ( linorder_part @ B @ A @ F @ Pivot2 @ Xs ) ) ) ) ).
% part_code(2)
thf(fact_3794_extract__SomeE,axiom,
! [A: $tType,P: A > $o,Xs: list @ A,Ys2: list @ A,Y: A,Zs: list @ A] :
( ( ( extract @ A @ P @ Xs )
= ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ Ys2 @ ( product_Pair @ A @ ( list @ A ) @ Y @ Zs ) ) ) )
=> ( ( Xs
= ( append @ A @ Ys2 @ ( cons @ A @ Y @ Zs ) ) )
& ( P @ Y )
& ~ ? [X7: A] :
( ( member @ A @ X7 @ ( set2 @ A @ Ys2 ) )
& ( P @ X7 ) ) ) ) ).
% extract_SomeE
thf(fact_3795_extract__Some__iff,axiom,
! [A: $tType,P: A > $o,Xs: list @ A,Ys2: list @ A,Y: A,Zs: list @ A] :
( ( ( extract @ A @ P @ Xs )
= ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ Ys2 @ ( product_Pair @ A @ ( list @ A ) @ Y @ Zs ) ) ) )
= ( ( Xs
= ( append @ A @ Ys2 @ ( cons @ A @ Y @ Zs ) ) )
& ( P @ Y )
& ~ ? [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Ys2 ) )
& ( P @ X3 ) ) ) ) ).
% extract_Some_iff
thf(fact_3796_extract__Cons__code,axiom,
! [A: $tType,P: A > $o,X: A,Xs: list @ A] :
( ( ( P @ X )
=> ( ( extract @ A @ P @ ( cons @ A @ X @ Xs ) )
= ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( nil @ A ) @ ( product_Pair @ A @ ( list @ A ) @ X @ Xs ) ) ) ) )
& ( ~ ( P @ X )
=> ( ( extract @ A @ P @ ( cons @ A @ X @ Xs ) )
= ( case_option @ ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( none @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) )
@ ( product_case_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) )
@ ^ [Ys: list @ A] :
( product_case_prod @ A @ ( list @ A ) @ ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) )
@ ^ [Y3: A,Zs3: list @ A] : ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( cons @ A @ X @ Ys ) @ ( product_Pair @ A @ ( list @ A ) @ Y3 @ Zs3 ) ) ) ) )
@ ( extract @ A @ P @ Xs ) ) ) ) ) ).
% extract_Cons_code
thf(fact_3797_merge__list_Ocases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) )] :
( ( X
!= ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) ) )
=> ( ! [L5: list @ A] :
( X
!= ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L5 @ ( nil @ ( list @ A ) ) ) ) )
=> ( ! [La2: list @ A,Acc23: list @ ( list @ A )] :
( X
!= ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La2 @ Acc23 ) @ ( nil @ ( list @ A ) ) ) )
=> ( ! [La2: list @ A,Acc23: list @ ( list @ A ),L5: list @ A] :
( X
!= ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La2 @ Acc23 ) @ ( cons @ ( list @ A ) @ L5 @ ( nil @ ( list @ A ) ) ) ) )
=> ~ ! [Acc23: list @ ( list @ A ),L12: list @ A,L22: list @ A,Ls2: list @ ( list @ A )] :
( X
!= ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ Acc23 @ ( cons @ ( list @ A ) @ L12 @ ( cons @ ( list @ A ) @ L22 @ Ls2 ) ) ) ) ) ) ) ) ) ).
% merge_list.cases
thf(fact_3798_subset__eq__mset__impl_Oelims,axiom,
! [A: $tType,X: list @ A,Xa: list @ A,Y: option @ $o] :
( ( ( subset_eq_mset_impl @ A @ X @ Xa )
= Y )
=> ( ( ( X
= ( nil @ A ) )
=> ( Y
!= ( some @ $o
@ ( Xa
!= ( nil @ A ) ) ) ) )
=> ~ ! [X2: A,Xs2: list @ A] :
( ( X
= ( cons @ A @ X2 @ Xs2 ) )
=> ( Y
!= ( case_option @ ( option @ $o ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( none @ $o )
@ ( product_case_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( option @ $o )
@ ^ [Ys1: list @ A] :
( product_case_prod @ A @ ( list @ A ) @ ( option @ $o )
@ ^ [Y3: A,Ys22: list @ A] : ( subset_eq_mset_impl @ A @ Xs2 @ ( append @ A @ Ys1 @ Ys22 ) ) ) )
@ ( extract @ A
@ ( ^ [Y5: A,Z4: A] : Y5 = Z4
@ X2 )
@ Xa ) ) ) ) ) ) ).
% subset_eq_mset_impl.elims
thf(fact_3799_lexord__def,axiom,
! [A: $tType] :
( ( lexord @ A )
= ( ^ [R3: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [X3: list @ A,Y3: list @ A] :
? [A7: A,V2: list @ A] :
( ( Y3
= ( append @ A @ X3 @ ( cons @ A @ A7 @ V2 ) ) )
| ? [U2: list @ A,B5: A,C3: A,W2: list @ A,Z5: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ C3 ) @ R3 )
& ( X3
= ( append @ A @ U2 @ ( cons @ A @ B5 @ W2 ) ) )
& ( Y3
= ( append @ A @ U2 @ ( cons @ A @ C3 @ Z5 ) ) ) ) ) ) ) ) ) ).
% lexord_def
thf(fact_3800_list__collect__set__alt,axiom,
! [A: $tType,B: $tType] :
( ( list_collect_set @ B @ A )
= ( ^ [F2: B > ( set @ A ),L3: list @ B] :
( complete_Sup_Sup @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu: set @ A] :
? [I3: nat] :
( ( Uu
= ( F2 @ ( nth @ B @ L3 @ I3 ) ) )
& ( ord_less @ nat @ I3 @ ( size_size @ ( list @ B ) @ L3 ) ) ) ) ) ) ) ).
% list_collect_set_alt
thf(fact_3801_subset__eq__mset__impl_Opelims,axiom,
! [A: $tType,X: list @ A,Xa: list @ A,Y: option @ $o] :
( ( ( subset_eq_mset_impl @ A @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( subset751672762298770561pl_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Xa ) )
=> ( ( ( X
= ( nil @ A ) )
=> ( ( Y
= ( some @ $o
@ ( Xa
!= ( nil @ A ) ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( subset751672762298770561pl_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xa ) ) ) )
=> ~ ! [X2: A,Xs2: list @ A] :
( ( X
= ( cons @ A @ X2 @ Xs2 ) )
=> ( ( Y
= ( case_option @ ( option @ $o ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( none @ $o )
@ ( product_case_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( option @ $o )
@ ^ [Ys1: list @ A] :
( product_case_prod @ A @ ( list @ A ) @ ( option @ $o )
@ ^ [Y3: A,Ys22: list @ A] : ( subset_eq_mset_impl @ A @ Xs2 @ ( append @ A @ Ys1 @ Ys22 ) ) ) )
@ ( extract @ A
@ ( ^ [Y5: A,Z4: A] : Y5 = Z4
@ X2 )
@ Xa ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( subset751672762298770561pl_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ Xa ) ) ) ) ) ) ) ).
% subset_eq_mset_impl.pelims
thf(fact_3802_list__collect__set__simps_I2_J,axiom,
! [A: $tType,B: $tType,F: B > ( set @ A ),A3: B] :
( ( list_collect_set @ B @ A @ F @ ( cons @ B @ A3 @ ( nil @ B ) ) )
= ( F @ A3 ) ) ).
% list_collect_set_simps(2)
thf(fact_3803_list__collect__set__simps_I1_J,axiom,
! [B: $tType,A: $tType,F: B > ( set @ A )] :
( ( list_collect_set @ B @ A @ F @ ( nil @ B ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% list_collect_set_simps(1)
thf(fact_3804_list__collect__set__simps_I3_J,axiom,
! [A: $tType,B: $tType,F: B > ( set @ A ),A3: B,L: list @ B] :
( ( list_collect_set @ B @ A @ F @ ( cons @ B @ A3 @ L ) )
= ( sup_sup @ ( set @ A ) @ ( F @ A3 ) @ ( list_collect_set @ B @ A @ F @ L ) ) ) ).
% list_collect_set_simps(3)
thf(fact_3805_list__collect__set__simps_I4_J,axiom,
! [A: $tType,B: $tType,F: B > ( set @ A ),L: list @ B,L4: list @ B] :
( ( list_collect_set @ B @ A @ F @ ( append @ B @ L @ L4 ) )
= ( sup_sup @ ( set @ A ) @ ( list_collect_set @ B @ A @ F @ L ) @ ( list_collect_set @ B @ A @ F @ L4 ) ) ) ).
% list_collect_set_simps(4)
thf(fact_3806_lexord__cons__cons,axiom,
! [A: $tType,A3: A,X: list @ A,B2: A,Y: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ A3 @ X ) @ ( cons @ A @ B2 @ Y ) ) @ ( lexord @ A @ R ) )
= ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
| ( ( A3 = B2 )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lexord @ A @ R ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_3807_lexord__Nil__left,axiom,
! [A: $tType,Y: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Y ) @ ( lexord @ A @ R ) )
= ( ? [A7: A,X3: list @ A] :
( Y
= ( cons @ A @ A7 @ X3 ) ) ) ) ).
% lexord_Nil_left
thf(fact_3808_list__collect__set__map__simps_I2_J,axiom,
! [A: $tType,B: $tType,C: $tType,F: B > ( set @ A ),X: C > B,A3: C] :
( ( list_collect_set @ B @ A @ F @ ( map @ C @ B @ X @ ( cons @ C @ A3 @ ( nil @ C ) ) ) )
= ( F @ ( X @ A3 ) ) ) ).
% list_collect_set_map_simps(2)
thf(fact_3809_list__collect__set__map__simps_I1_J,axiom,
! [C: $tType,B: $tType,A: $tType,F: B > ( set @ A ),X: C > B] :
( ( list_collect_set @ B @ A @ F @ ( map @ C @ B @ X @ ( nil @ C ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% list_collect_set_map_simps(1)
thf(fact_3810_list__collect__set__map__simps_I3_J,axiom,
! [A: $tType,B: $tType,C: $tType,F: B > ( set @ A ),X: C > B,A3: C,L: list @ C] :
( ( list_collect_set @ B @ A @ F @ ( map @ C @ B @ X @ ( cons @ C @ A3 @ L ) ) )
= ( sup_sup @ ( set @ A ) @ ( F @ ( X @ A3 ) ) @ ( list_collect_set @ B @ A @ F @ ( map @ C @ B @ X @ L ) ) ) ) ).
% list_collect_set_map_simps(3)
thf(fact_3811_list__collect__set__map__simps_I4_J,axiom,
! [A: $tType,B: $tType,C: $tType,F: B > ( set @ A ),X: C > B,L: list @ C,L4: list @ C] :
( ( list_collect_set @ B @ A @ F @ ( map @ C @ B @ X @ ( append @ C @ L @ L4 ) ) )
= ( sup_sup @ ( set @ A ) @ ( list_collect_set @ B @ A @ F @ ( map @ C @ B @ X @ L ) ) @ ( list_collect_set @ B @ A @ F @ ( map @ C @ B @ X @ L4 ) ) ) ) ).
% list_collect_set_map_simps(4)
thf(fact_3812_lexord__irreflexive,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),Xs: list @ A] :
( ! [X2: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ R )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Xs ) @ ( lexord @ A @ R ) ) ) ).
% lexord_irreflexive
thf(fact_3813_lexord__linear,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),X: list @ A,Y: list @ A] :
( ! [A5: A,B3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B3 ) @ R )
| ( A5 = B3 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A5 ) @ R ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lexord @ A @ R ) )
| ( X = Y )
| ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Y @ X ) @ ( lexord @ A @ R ) ) ) ) ).
% lexord_linear
thf(fact_3814_lexord__Nil__right,axiom,
! [A: $tType,X: list @ A,R: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ ( nil @ A ) ) @ ( lexord @ A @ R ) ) ).
% lexord_Nil_right
thf(fact_3815_lexord__append__leftI,axiom,
! [A: $tType,U: list @ A,V: list @ A,R: set @ ( product_prod @ A @ A ),X: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U @ V ) @ ( lexord @ A @ R ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ X @ U ) @ ( append @ A @ X @ V ) ) @ ( lexord @ A @ R ) ) ) ).
% lexord_append_leftI
thf(fact_3816_lexord__partial__trans,axiom,
! [A: $tType,Xs: list @ A,R: set @ ( product_prod @ A @ A ),Ys2: list @ A,Zs: list @ A] :
( ! [X2: A,Y4: A,Z3: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y4 ) @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z3 ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Z3 ) @ R ) ) ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( lexord @ A @ R ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys2 @ Zs ) @ ( lexord @ A @ R ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Zs ) @ ( lexord @ A @ R ) ) ) ) ) ).
% lexord_partial_trans
thf(fact_3817_lexord__append__leftD,axiom,
! [A: $tType,X: list @ A,U: list @ A,V: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ X @ U ) @ ( append @ A @ X @ V ) ) @ ( lexord @ A @ R ) )
=> ( ! [A5: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ A5 ) @ R )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U @ V ) @ ( lexord @ A @ R ) ) ) ) ).
% lexord_append_leftD
thf(fact_3818_lexord__append__rightI,axiom,
! [A: $tType,Y: list @ A,X: list @ A,R: set @ ( product_prod @ A @ A )] :
( ? [B8: A,Z6: list @ A] :
( Y
= ( cons @ A @ B8 @ Z6 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ ( append @ A @ X @ Y ) ) @ ( lexord @ A @ R ) ) ) ).
% lexord_append_rightI
thf(fact_3819_lexord__sufE,axiom,
! [A: $tType,Xs: list @ A,Zs: list @ A,Ys2: list @ A,Qs: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Zs ) @ ( append @ A @ Ys2 @ Qs ) ) @ ( lexord @ A @ R ) )
=> ( ( Xs != Ys2 )
=> ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys2 ) )
=> ( ( ( size_size @ ( list @ A ) @ Zs )
= ( size_size @ ( list @ A ) @ Qs ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( lexord @ A @ R ) ) ) ) ) ) ).
% lexord_sufE
thf(fact_3820_list__collect__set__as__map,axiom,
! [A: $tType,B: $tType] :
( ( list_collect_set @ B @ A )
= ( ^ [F2: B > ( set @ A ),L3: list @ B] : ( complete_Sup_Sup @ ( set @ A ) @ ( set2 @ ( set @ A ) @ ( map @ B @ ( set @ A ) @ F2 @ L3 ) ) ) ) ) ).
% list_collect_set_as_map
thf(fact_3821_lexord__lex,axiom,
! [A: $tType,X: list @ A,Y: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lex @ A @ R ) )
= ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lexord @ A @ R ) )
& ( ( size_size @ ( list @ A ) @ X )
= ( size_size @ ( list @ A ) @ Y ) ) ) ) ).
% lexord_lex
thf(fact_3822_lexord__append__left__rightI,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),U: list @ A,X: list @ A,Y: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ U @ ( cons @ A @ A3 @ X ) ) @ ( append @ A @ U @ ( cons @ A @ B2 @ Y ) ) ) @ ( lexord @ A @ R ) ) ) ).
% lexord_append_left_rightI
thf(fact_3823_lexord__same__pref__iff,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A,Zs: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys2 ) @ ( append @ A @ Xs @ Zs ) ) @ ( lexord @ A @ R ) )
= ( ? [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ R ) )
| ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys2 @ Zs ) @ ( lexord @ A @ R ) ) ) ) ).
% lexord_same_pref_iff
thf(fact_3824_lexord__sufI,axiom,
! [A: $tType,U: list @ A,W: list @ A,R: set @ ( product_prod @ A @ A ),V: list @ A,Z2: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U @ W ) @ ( lexord @ A @ R ) )
=> ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ W ) @ ( size_size @ ( list @ A ) @ U ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ U @ V ) @ ( append @ A @ W @ Z2 ) ) @ ( lexord @ A @ R ) ) ) ) ).
% lexord_sufI
thf(fact_3825_list__collect__set__def,axiom,
! [A: $tType,B: $tType] :
( ( list_collect_set @ B @ A )
= ( ^ [F2: B > ( set @ A ),L3: list @ B] :
( complete_Sup_Sup @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu: set @ A] :
? [A7: B] :
( ( Uu
= ( F2 @ A7 ) )
& ( member @ B @ A7 @ ( set2 @ B @ L3 ) ) ) ) ) ) ) ).
% list_collect_set_def
thf(fact_3826_subset__eq__mset__impl_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys2: list @ A] :
( ( subset_eq_mset_impl @ A @ ( cons @ A @ X @ Xs ) @ Ys2 )
= ( case_option @ ( option @ $o ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( none @ $o )
@ ( product_case_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( option @ $o )
@ ^ [Ys1: list @ A] :
( product_case_prod @ A @ ( list @ A ) @ ( option @ $o )
@ ^ [X3: A,Ys22: list @ A] : ( subset_eq_mset_impl @ A @ Xs @ ( append @ A @ Ys1 @ Ys22 ) ) ) )
@ ( extract @ A
@ ( ^ [Y5: A,Z4: A] : Y5 = Z4
@ X )
@ Ys2 ) ) ) ).
% subset_eq_mset_impl.simps(2)
thf(fact_3827_List_Olexordp__def,axiom,
! [A: $tType] :
( ( lexordp @ A )
= ( ^ [R3: A > A > $o,Xs3: list @ A,Ys: list @ A] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs3 @ Ys ) @ ( lexord @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R3 ) ) ) ) ) ) ).
% List.lexordp_def
thf(fact_3828_upto_Opelims,axiom,
! [X: int,Xa: int,Y: list @ int] :
( ( ( upto @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ X @ Xa ) )
=> ~ ( ( ( ( ord_less_eq @ int @ X @ Xa )
=> ( Y
= ( cons @ int @ X @ ( upto @ ( plus_plus @ int @ X @ ( one_one @ int ) ) @ Xa ) ) ) )
& ( ~ ( ord_less_eq @ int @ X @ Xa )
=> ( Y
= ( nil @ int ) ) ) )
=> ~ ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ X @ Xa ) ) ) ) ) ).
% upto.pelims
thf(fact_3829_upto_Opsimps,axiom,
! [I: int,J: int] :
( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ I @ J ) )
=> ( ( ( ord_less_eq @ int @ I @ J )
=> ( ( upto @ I @ J )
= ( cons @ int @ I @ ( upto @ ( plus_plus @ int @ I @ ( one_one @ int ) ) @ J ) ) ) )
& ( ~ ( ord_less_eq @ int @ I @ J )
=> ( ( upto @ I @ J )
= ( nil @ int ) ) ) ) ) ).
% upto.psimps
thf(fact_3830_listrel1__def,axiom,
! [A: $tType] :
( ( listrel1 @ A )
= ( ^ [R3: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs3: list @ A,Ys: list @ A] :
? [Us2: list @ A,Z5: A,Z8: A,Vs2: list @ A] :
( ( Xs3
= ( append @ A @ Us2 @ ( cons @ A @ Z5 @ Vs2 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z5 @ Z8 ) @ R3 )
& ( Ys
= ( append @ A @ Us2 @ ( cons @ A @ Z8 @ Vs2 ) ) ) ) ) ) ) ) ).
% listrel1_def
thf(fact_3831_Cons__listrel1__Cons,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: A,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys2 ) ) @ ( listrel1 @ A @ R ) )
= ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
& ( Xs = Ys2 ) )
| ( ( X = Y )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( listrel1 @ A @ R ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_3832_nth__upto,axiom,
! [I: int,K: nat,J: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ I @ ( semiring_1_of_nat @ int @ K ) ) @ J )
=> ( ( nth @ int @ ( upto @ I @ J ) @ K )
= ( plus_plus @ int @ I @ ( semiring_1_of_nat @ int @ K ) ) ) ) ).
% nth_upto
thf(fact_3833_length__upto,axiom,
! [I: int,J: int] :
( ( size_size @ ( list @ int ) @ ( upto @ I @ J ) )
= ( nat2 @ ( plus_plus @ int @ ( minus_minus @ int @ J @ I ) @ ( one_one @ int ) ) ) ) ).
% length_upto
thf(fact_3834_upto__rec__numeral_I1_J,axiom,
! [M2: num,N: num] :
( ( ( ord_less_eq @ int @ ( numeral_numeral @ int @ M2 ) @ ( numeral_numeral @ int @ N ) )
=> ( ( upto @ ( numeral_numeral @ int @ M2 ) @ ( numeral_numeral @ int @ N ) )
= ( cons @ int @ ( numeral_numeral @ int @ M2 ) @ ( upto @ ( plus_plus @ int @ ( numeral_numeral @ int @ M2 ) @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ N ) ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( numeral_numeral @ int @ M2 ) @ ( numeral_numeral @ int @ N ) )
=> ( ( upto @ ( numeral_numeral @ int @ M2 ) @ ( numeral_numeral @ int @ N ) )
= ( nil @ int ) ) ) ) ).
% upto_rec_numeral(1)
thf(fact_3835_upto__rec__numeral_I2_J,axiom,
! [M2: num,N: num] :
( ( ( ord_less_eq @ int @ ( numeral_numeral @ int @ M2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
=> ( ( upto @ ( numeral_numeral @ int @ M2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( cons @ int @ ( numeral_numeral @ int @ M2 ) @ ( upto @ ( plus_plus @ int @ ( numeral_numeral @ int @ M2 ) @ ( one_one @ int ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( numeral_numeral @ int @ M2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
=> ( ( upto @ ( numeral_numeral @ int @ M2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( nil @ int ) ) ) ) ).
% upto_rec_numeral(2)
thf(fact_3836_upto__rec__numeral_I3_J,axiom,
! [M2: num,N: num] :
( ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( numeral_numeral @ int @ N ) )
=> ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( numeral_numeral @ int @ N ) )
= ( cons @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( upto @ ( plus_plus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ N ) ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( numeral_numeral @ int @ N ) )
=> ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( numeral_numeral @ int @ N ) )
= ( nil @ int ) ) ) ) ).
% upto_rec_numeral(3)
thf(fact_3837_upto__rec__numeral_I4_J,axiom,
! [M2: num,N: num] :
( ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
=> ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( cons @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( upto @ ( plus_plus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( one_one @ int ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
=> ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( nil @ int ) ) ) ) ).
% upto_rec_numeral(4)
thf(fact_3838_listrel1I2,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A ),X: A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( listrel1 @ A @ R ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ X @ Ys2 ) ) @ ( listrel1 @ A @ R ) ) ) ).
% listrel1I2
thf(fact_3839_rtrancl__listrel1__ConsI1,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A ),X: A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R ) ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ X @ Ys2 ) ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R ) ) ) ) ).
% rtrancl_listrel1_ConsI1
thf(fact_3840_not__listrel1__Nil,axiom,
! [A: $tType,Xs: list @ A,R: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ ( nil @ A ) ) @ ( listrel1 @ A @ R ) ) ).
% not_listrel1_Nil
thf(fact_3841_not__Nil__listrel1,axiom,
! [A: $tType,Xs: list @ A,R: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xs ) @ ( listrel1 @ A @ R ) ) ).
% not_Nil_listrel1
thf(fact_3842_rtrancl__listrel1__eq__len,axiom,
! [A: $tType,X: list @ A,Y: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R ) ) )
=> ( ( size_size @ ( list @ A ) @ X )
= ( size_size @ ( list @ A ) @ Y ) ) ) ).
% rtrancl_listrel1_eq_len
thf(fact_3843_listrel1__eq__len,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( listrel1 @ A @ R ) )
=> ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys2 ) ) ) ).
% listrel1_eq_len
thf(fact_3844_append__listrel1I,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A ),Us: list @ A,Vs: list @ A] :
( ( ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( listrel1 @ A @ R ) )
& ( Us = Vs ) )
| ( ( Xs = Ys2 )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Us @ Vs ) @ ( listrel1 @ A @ R ) ) ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Us ) @ ( append @ A @ Ys2 @ Vs ) ) @ ( listrel1 @ A @ R ) ) ) ).
% append_listrel1I
thf(fact_3845_upto__split2,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_eq @ int @ I @ J )
=> ( ( ord_less_eq @ int @ J @ K )
=> ( ( upto @ I @ K )
= ( append @ int @ ( upto @ I @ J ) @ ( upto @ ( plus_plus @ int @ J @ ( one_one @ int ) ) @ K ) ) ) ) ) ).
% upto_split2
thf(fact_3846_upto__rec1,axiom,
! [I: int,J: int] :
( ( ord_less_eq @ int @ I @ J )
=> ( ( upto @ I @ J )
= ( cons @ int @ I @ ( upto @ ( plus_plus @ int @ I @ ( one_one @ int ) ) @ J ) ) ) ) ).
% upto_rec1
thf(fact_3847_upto_Osimps,axiom,
( upto
= ( ^ [I3: int,J2: int] : ( if @ ( list @ int ) @ ( ord_less_eq @ int @ I3 @ J2 ) @ ( cons @ int @ I3 @ ( upto @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) @ J2 ) ) @ ( nil @ int ) ) ) ) ).
% upto.simps
thf(fact_3848_upto_Oelims,axiom,
! [X: int,Xa: int,Y: list @ int] :
( ( ( upto @ X @ Xa )
= Y )
=> ( ( ( ord_less_eq @ int @ X @ Xa )
=> ( Y
= ( cons @ int @ X @ ( upto @ ( plus_plus @ int @ X @ ( one_one @ int ) ) @ Xa ) ) ) )
& ( ~ ( ord_less_eq @ int @ X @ Xa )
=> ( Y
= ( nil @ int ) ) ) ) ) ).
% upto.elims
thf(fact_3849_Cons__listrel1E2,axiom,
! [A: $tType,Xs: list @ A,Y: A,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ ( cons @ A @ Y @ Ys2 ) ) @ ( listrel1 @ A @ R ) )
=> ( ! [X2: A] :
( ( Xs
= ( cons @ A @ X2 @ Ys2 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R ) )
=> ~ ! [Zs2: list @ A] :
( ( Xs
= ( cons @ A @ Y @ Zs2 ) )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Zs2 @ Ys2 ) @ ( listrel1 @ A @ R ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_3850_Cons__listrel1E1,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ Ys2 ) @ ( listrel1 @ A @ R ) )
=> ( ! [Y4: A] :
( ( Ys2
= ( cons @ A @ Y4 @ Xs ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y4 ) @ R ) )
=> ~ ! [Zs2: list @ A] :
( ( Ys2
= ( cons @ A @ X @ Zs2 ) )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Zs2 ) @ ( listrel1 @ A @ R ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_3851_listrel1I1,axiom,
! [A: $tType,X: A,Y: A,R: set @ ( product_prod @ A @ A ),Xs: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Xs ) ) @ ( listrel1 @ A @ R ) ) ) ).
% listrel1I1
thf(fact_3852_greaterThanAtMost__upto,axiom,
( ( set_or3652927894154168847AtMost @ int )
= ( ^ [I3: int,J2: int] : ( set2 @ int @ ( upto @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) @ J2 ) ) ) ) ).
% greaterThanAtMost_upto
thf(fact_3853_listrel1I,axiom,
! [A: $tType,X: A,Y: A,R: set @ ( product_prod @ A @ A ),Xs: list @ A,Us: list @ A,Vs: list @ A,Ys2: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
=> ( ( Xs
= ( append @ A @ Us @ ( cons @ A @ X @ Vs ) ) )
=> ( ( Ys2
= ( append @ A @ Us @ ( cons @ A @ Y @ Vs ) ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( listrel1 @ A @ R ) ) ) ) ) ).
% listrel1I
thf(fact_3854_listrel1E,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( listrel1 @ A @ R ) )
=> ~ ! [X2: A,Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y4 ) @ R )
=> ! [Us3: list @ A,Vs3: list @ A] :
( ( Xs
= ( append @ A @ Us3 @ ( cons @ A @ X2 @ Vs3 ) ) )
=> ( Ys2
!= ( append @ A @ Us3 @ ( cons @ A @ Y4 @ Vs3 ) ) ) ) ) ) ).
% listrel1E
thf(fact_3855_upto__split3,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_eq @ int @ I @ J )
=> ( ( ord_less_eq @ int @ J @ K )
=> ( ( upto @ I @ K )
= ( append @ int @ ( upto @ I @ ( minus_minus @ int @ J @ ( one_one @ int ) ) ) @ ( cons @ int @ J @ ( upto @ ( plus_plus @ int @ J @ ( one_one @ int ) ) @ K ) ) ) ) ) ) ).
% upto_split3
thf(fact_3856_rtrancl__listrel1__ConsI2,axiom,
! [A: $tType,X: A,Y: A,R: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys2: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R ) ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys2 ) ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R ) ) ) ) ) ).
% rtrancl_listrel1_ConsI2
thf(fact_3857_greaterThanLessThan__upto,axiom,
( ( set_or5935395276787703475ssThan @ int )
= ( ^ [I3: int,J2: int] : ( set2 @ int @ ( upto @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) @ ( minus_minus @ int @ J2 @ ( one_one @ int ) ) ) ) ) ) ).
% greaterThanLessThan_upto
thf(fact_3858_snoc__listrel1__snoc__iff,axiom,
! [A: $tType,Xs: list @ A,X: A,Ys2: list @ A,Y: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) @ ( append @ A @ Ys2 @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) @ ( listrel1 @ A @ R ) )
= ( ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( listrel1 @ A @ R ) )
& ( X = Y ) )
| ( ( Xs = Ys2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_3859_listrel1p__def,axiom,
! [A: $tType] :
( ( listrel1p @ A )
= ( ^ [R3: A > A > $o,Xs3: list @ A,Ys: list @ A] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs3 @ Ys ) @ ( listrel1 @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R3 ) ) ) ) ) ) ).
% listrel1p_def
thf(fact_3860_sorted__wrt__map__rev__linord,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [L: list @ ( product_prod @ A @ B )] :
( ( sorted_wrt @ ( product_prod @ A @ B )
@ ^ [X3: product_prod @ A @ B,Y3: product_prod @ A @ B] : ( ord_less_eq @ A @ ( product_fst @ A @ B @ Y3 ) @ ( product_fst @ A @ B @ X3 ) )
@ L )
= ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ L ) ) ) ) ) ).
% sorted_wrt_map_rev_linord
thf(fact_3861_set__zip,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys2: list @ B] :
( ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys2 ) )
= ( collect @ ( product_prod @ A @ B )
@ ^ [Uu: product_prod @ A @ B] :
? [I3: nat] :
( ( Uu
= ( product_Pair @ A @ B @ ( nth @ A @ Xs @ I3 ) @ ( nth @ B @ Ys2 @ I3 ) ) )
& ( ord_less @ nat @ I3 @ ( ord_min @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys2 ) ) ) ) ) ) ).
% set_zip
thf(fact_3862_arg__min__list_Opelims,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [X: A > B,Xa: list @ A,Y: A] :
( ( ( arg_min_list @ A @ B @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X @ Xa ) )
=> ( ! [X2: A] :
( ( Xa
= ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ( ( Y = X2 )
=> ~ ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) ) )
=> ( ! [X2: A,Y4: A,Zs2: list @ A] :
( ( Xa
= ( cons @ A @ X2 @ ( cons @ A @ Y4 @ Zs2 ) ) )
=> ( ( Y
= ( if @ A @ ( ord_less_eq @ B @ ( X @ X2 ) @ ( 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 @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X @ ( cons @ A @ X2 @ ( cons @ A @ Y4 @ Zs2 ) ) ) ) ) )
=> ~ ( ( Xa
= ( nil @ A ) )
=> ( ( Y
= ( undefined @ A ) )
=> ~ ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X @ ( nil @ A ) ) ) ) ) ) ) ) ) ) ).
% arg_min_list.pelims
thf(fact_3863_zip__eq__zip__same__len,axiom,
! [A: $tType,B: $tType,A3: list @ A,B2: list @ B,A6: list @ A,B4: list @ B] :
( ( ( size_size @ ( list @ A ) @ A3 )
= ( size_size @ ( list @ B ) @ B2 ) )
=> ( ( ( size_size @ ( list @ A ) @ A6 )
= ( size_size @ ( list @ B ) @ B4 ) )
=> ( ( ( zip @ A @ B @ A3 @ B2 )
= ( zip @ A @ B @ A6 @ B4 ) )
= ( ( A3 = A6 )
& ( B2 = B4 ) ) ) ) ) ).
% zip_eq_zip_same_len
thf(fact_3864_sorted__wrt__rev__linord,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: list @ A] :
( ( sorted_wrt @ A
@ ^ [X3: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ X3 )
@ L )
= ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ L ) ) ) ) ).
% sorted_wrt_rev_linord
thf(fact_3865_zip__Cons__Cons,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ A,Y: B,Ys2: list @ B] :
( ( zip @ A @ B @ ( cons @ A @ X @ Xs ) @ ( cons @ B @ Y @ Ys2 ) )
= ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( zip @ A @ B @ Xs @ Ys2 ) ) ) ).
% zip_Cons_Cons
thf(fact_3866_map__of__rev__distinct,axiom,
! [B: $tType,A: $tType,M2: list @ ( product_prod @ A @ B )] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ M2 ) )
=> ( ( map_of @ A @ B @ ( rev @ ( product_prod @ A @ B ) @ M2 ) )
= ( map_of @ A @ B @ M2 ) ) ) ).
% map_of_rev_distinct
thf(fact_3867_zip__replicate,axiom,
! [A: $tType,B: $tType,I: nat,X: A,J: nat,Y: B] :
( ( zip @ A @ B @ ( replicate @ A @ I @ X ) @ ( replicate @ B @ J @ Y ) )
= ( replicate @ ( product_prod @ A @ B ) @ ( ord_min @ nat @ I @ J ) @ ( product_Pair @ A @ B @ X @ Y ) ) ) ).
% zip_replicate
thf(fact_3868_map__of__zip__is__None,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys2: list @ B,X: A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys2 ) @ X )
= ( none @ B ) )
= ( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) ) ) ) ) ).
% map_of_zip_is_None
thf(fact_3869_nth__zip,axiom,
! [A: $tType,B: $tType,I: nat,Xs: list @ A,Ys2: list @ B] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( nth @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys2 ) @ I )
= ( product_Pair @ A @ B @ ( nth @ A @ Xs @ I ) @ ( nth @ B @ Ys2 @ I ) ) ) ) ) ).
% nth_zip
thf(fact_3870_zip__assoc,axiom,
! [B: $tType,A: $tType,C: $tType,Xs: list @ A,Ys2: list @ B,Zs: list @ C] :
( ( zip @ A @ ( product_prod @ B @ C ) @ Xs @ ( zip @ B @ C @ Ys2 @ Zs ) )
= ( map @ ( product_prod @ ( product_prod @ A @ B ) @ C ) @ ( product_prod @ A @ ( product_prod @ B @ C ) )
@ ( product_case_prod @ ( product_prod @ A @ B ) @ C @ ( product_prod @ A @ ( product_prod @ B @ C ) )
@ ( product_case_prod @ A @ B @ ( C > ( product_prod @ A @ ( product_prod @ B @ C ) ) )
@ ^ [X3: A,Y3: B,Z5: C] : ( product_Pair @ A @ ( product_prod @ B @ C ) @ X3 @ ( product_Pair @ B @ C @ Y3 @ Z5 ) ) ) )
@ ( zip @ ( product_prod @ A @ B ) @ C @ ( zip @ A @ B @ Xs @ Ys2 ) @ Zs ) ) ) ).
% zip_assoc
thf(fact_3871_pair__list__split,axiom,
! [A: $tType,B: $tType,L: list @ ( product_prod @ A @ B )] :
~ ! [L12: list @ A,L22: list @ B] :
( ( L
= ( zip @ A @ B @ L12 @ L22 ) )
=> ( ( ( size_size @ ( list @ A ) @ L12 )
= ( size_size @ ( list @ B ) @ L22 ) )
=> ( ( size_size @ ( list @ ( product_prod @ A @ B ) ) @ L )
!= ( size_size @ ( list @ B ) @ L22 ) ) ) ) ).
% pair_list_split
thf(fact_3872_zip__inj,axiom,
! [A: $tType,B: $tType,A3: list @ A,B2: list @ B,A6: list @ A,B4: list @ B] :
( ( ( size_size @ ( list @ A ) @ A3 )
= ( size_size @ ( list @ B ) @ B2 ) )
=> ( ( ( size_size @ ( list @ A ) @ A6 )
= ( size_size @ ( list @ B ) @ B4 ) )
=> ( ( ( zip @ A @ B @ A3 @ B2 )
= ( zip @ A @ B @ A6 @ B4 ) )
=> ( ( A3 = A6 )
& ( B2 = B4 ) ) ) ) ) ).
% zip_inj
thf(fact_3873_zip__left__commute,axiom,
! [B: $tType,A: $tType,C: $tType,Xs: list @ A,Ys2: list @ B,Zs: list @ C] :
( ( zip @ A @ ( product_prod @ B @ C ) @ Xs @ ( zip @ B @ C @ Ys2 @ Zs ) )
= ( map @ ( product_prod @ B @ ( product_prod @ A @ C ) ) @ ( product_prod @ A @ ( product_prod @ B @ C ) )
@ ( product_case_prod @ B @ ( product_prod @ A @ C ) @ ( product_prod @ A @ ( product_prod @ B @ C ) )
@ ^ [Y3: B] :
( product_case_prod @ A @ C @ ( product_prod @ A @ ( product_prod @ B @ C ) )
@ ^ [X3: A,Z5: C] : ( product_Pair @ A @ ( product_prod @ B @ C ) @ X3 @ ( product_Pair @ B @ C @ Y3 @ Z5 ) ) ) )
@ ( zip @ B @ ( product_prod @ A @ C ) @ Ys2 @ ( zip @ A @ C @ Xs @ Zs ) ) ) ) ).
% zip_left_commute
thf(fact_3874_zip__same__conv__map,axiom,
! [A: $tType,Xs: list @ A] :
( ( zip @ A @ A @ Xs @ Xs )
= ( map @ A @ ( product_prod @ A @ A )
@ ^ [X3: A] : ( product_Pair @ A @ A @ X3 @ X3 )
@ Xs ) ) ).
% zip_same_conv_map
thf(fact_3875_set__zip__rightD,axiom,
! [A: $tType,B: $tType,X: A,Y: B,Xs: list @ A,Ys2: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys2 ) ) )
=> ( member @ B @ Y @ ( set2 @ B @ Ys2 ) ) ) ).
% set_zip_rightD
thf(fact_3876_set__zip__leftD,axiom,
! [B: $tType,A: $tType,X: A,Y: B,Xs: list @ A,Ys2: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys2 ) ) )
=> ( member @ A @ X @ ( set2 @ A @ Xs ) ) ) ).
% set_zip_leftD
thf(fact_3877_in__set__zipE,axiom,
! [A: $tType,B: $tType,X: A,Y: B,Xs: list @ A,Ys2: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys2 ) ) )
=> ~ ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ~ ( member @ B @ Y @ ( set2 @ B @ Ys2 ) ) ) ) ).
% in_set_zipE
thf(fact_3878_zip__same,axiom,
! [A: $tType,A3: A,B2: A,Xs: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( set2 @ ( product_prod @ A @ A ) @ ( zip @ A @ A @ Xs @ Xs ) ) )
= ( ( member @ A @ A3 @ ( set2 @ A @ Xs ) )
& ( A3 = B2 ) ) ) ).
% zip_same
thf(fact_3879_zip__eq__ConsE,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B,Xy2: product_prod @ A @ B,Xys: list @ ( product_prod @ A @ B )] :
( ( ( zip @ A @ B @ Xs @ Ys2 )
= ( cons @ ( product_prod @ A @ B ) @ Xy2 @ Xys ) )
=> ~ ! [X2: A,Xs6: list @ A] :
( ( Xs
= ( cons @ A @ X2 @ Xs6 ) )
=> ! [Y4: B,Ys4: list @ B] :
( ( Ys2
= ( cons @ B @ Y4 @ Ys4 ) )
=> ( ( Xy2
= ( product_Pair @ A @ B @ X2 @ Y4 ) )
=> ( Xys
!= ( zip @ A @ B @ Xs6 @ Ys4 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_3880_set__zip__cart,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B,L: list @ A,L4: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ X @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ L @ L4 ) ) )
=> ( member @ ( product_prod @ A @ B ) @ X
@ ( product_Sigma @ A @ B @ ( set2 @ A @ L )
@ ^ [Uu: A] : ( set2 @ B @ L4 ) ) ) ) ).
% set_zip_cart
thf(fact_3881_zip__commute,axiom,
! [B: $tType,A: $tType] :
( ( zip @ A @ B )
= ( ^ [Xs3: list @ A,Ys: list @ B] :
( map @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ B @ A @ ( product_prod @ A @ B )
@ ^ [X3: B,Y3: A] : ( product_Pair @ A @ B @ Y3 @ X3 ) )
@ ( zip @ B @ A @ Ys @ Xs3 ) ) ) ) ).
% zip_commute
thf(fact_3882_last__zip,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B] :
( ( Xs
!= ( nil @ A ) )
=> ( ( Ys2
!= ( nil @ B ) )
=> ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( last @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys2 ) )
= ( product_Pair @ A @ B @ ( last @ A @ Xs ) @ ( last @ B @ Ys2 ) ) ) ) ) ) ).
% last_zip
thf(fact_3883_in__set__impl__in__set__zip1,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B,X: A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ~ ! [Y4: B] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys2 ) ) ) ) ) ).
% in_set_impl_in_set_zip1
thf(fact_3884_in__set__impl__in__set__zip2,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B,Y: B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( member @ B @ Y @ ( set2 @ B @ Ys2 ) )
=> ~ ! [X2: A] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys2 ) ) ) ) ) ).
% in_set_impl_in_set_zip2
thf(fact_3885_map__of__zip__is__Some,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B,X: A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
= ( ? [Y3: B] :
( ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys2 ) @ X )
= ( some @ B @ Y3 ) ) ) ) ) ).
% map_of_zip_is_Some
thf(fact_3886_map__upds__fold__map__upd,axiom,
! [B: $tType,A: $tType] :
( ( map_upds @ A @ B )
= ( ^ [M: A > ( option @ B ),Ks2: list @ A,Vs2: list @ B] :
( foldl @ ( A > ( option @ B ) ) @ ( product_prod @ A @ B )
@ ^ [N5: A > ( option @ B )] :
( product_case_prod @ A @ B @ ( A > ( option @ B ) )
@ ^ [K4: A,V2: B] : ( fun_upd @ A @ ( option @ B ) @ N5 @ K4 @ ( some @ B @ V2 ) ) )
@ M
@ ( zip @ A @ B @ Ks2 @ Vs2 ) ) ) ) ).
% map_upds_fold_map_upd
thf(fact_3887_enumerate__eq__zip,axiom,
! [A: $tType] :
( ( enumerate @ A )
= ( ^ [N5: nat,Xs3: list @ A] : ( zip @ nat @ A @ ( upt @ N5 @ ( plus_plus @ nat @ N5 @ ( size_size @ ( list @ A ) @ Xs3 ) ) ) @ Xs3 ) ) ) ).
% enumerate_eq_zip
thf(fact_3888_map__zip__map,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F: ( product_prod @ B @ C ) > A,G: D > B,Xs: list @ D,Ys2: list @ C] :
( ( map @ ( product_prod @ B @ C ) @ A @ F @ ( zip @ B @ C @ ( map @ D @ B @ G @ Xs ) @ Ys2 ) )
= ( map @ ( product_prod @ D @ C ) @ A
@ ( product_case_prod @ D @ C @ A
@ ^ [X3: D,Y3: C] : ( F @ ( product_Pair @ B @ C @ ( G @ X3 ) @ Y3 ) ) )
@ ( zip @ D @ C @ Xs @ Ys2 ) ) ) ).
% map_zip_map
thf(fact_3889_map__zip__map2,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F: ( product_prod @ B @ C ) > A,Xs: list @ B,G: D > C,Ys2: list @ D] :
( ( map @ ( product_prod @ B @ C ) @ A @ F @ ( zip @ B @ C @ Xs @ ( map @ D @ C @ G @ Ys2 ) ) )
= ( map @ ( product_prod @ B @ D ) @ A
@ ( product_case_prod @ B @ D @ A
@ ^ [X3: B,Y3: D] : ( F @ ( product_Pair @ B @ C @ X3 @ ( G @ Y3 ) ) ) )
@ ( zip @ B @ D @ Xs @ Ys2 ) ) ) ).
% map_zip_map2
thf(fact_3890_foldl__snd__zip,axiom,
! [B: $tType,C: $tType,A: $tType,Ys2: list @ A,Xs: list @ B,F: C > A > C,B2: C] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ys2 ) @ ( size_size @ ( list @ B ) @ Xs ) )
=> ( ( foldl @ C @ ( product_prod @ B @ A )
@ ^ [B5: C] :
( product_case_prod @ B @ A @ C
@ ^ [X3: B] : ( F @ B5 ) )
@ B2
@ ( zip @ B @ A @ Xs @ Ys2 ) )
= ( foldl @ C @ A @ F @ B2 @ Ys2 ) ) ) ).
% foldl_snd_zip
thf(fact_3891_map__zip2,axiom,
! [A: $tType,B: $tType,K: A,L: list @ B] :
( ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K ) @ L )
= ( zip @ A @ B @ ( replicate @ A @ ( size_size @ ( list @ B ) @ L ) @ K ) @ L ) ) ).
% map_zip2
thf(fact_3892_map__zip1,axiom,
! [A: $tType,B: $tType,K: B,L: list @ A] :
( ( map @ A @ ( product_prod @ A @ B )
@ ^ [X3: A] : ( product_Pair @ A @ B @ X3 @ K )
@ L )
= ( zip @ A @ B @ L @ ( replicate @ B @ ( size_size @ ( list @ A ) @ L ) @ K ) ) ) ).
% map_zip1
thf(fact_3893_map__prod__fun__zip,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F: C > A,G: D > B,Xs: list @ C,Ys2: list @ D] :
( ( map @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ C @ D @ ( product_prod @ A @ B )
@ ^ [X3: C,Y3: D] : ( product_Pair @ A @ B @ ( F @ X3 ) @ ( G @ Y3 ) ) )
@ ( zip @ C @ D @ Xs @ Ys2 ) )
= ( zip @ A @ B @ ( map @ C @ A @ F @ Xs ) @ ( map @ D @ B @ G @ Ys2 ) ) ) ).
% map_prod_fun_zip
thf(fact_3894_zip__map2,axiom,
! [B: $tType,A: $tType,C: $tType,Xs: list @ A,F: C > B,Ys2: list @ C] :
( ( zip @ A @ B @ Xs @ ( map @ C @ B @ F @ Ys2 ) )
= ( map @ ( product_prod @ A @ C ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ C @ ( product_prod @ A @ B )
@ ^ [X3: A,Y3: C] : ( product_Pair @ A @ B @ X3 @ ( F @ Y3 ) ) )
@ ( zip @ A @ C @ Xs @ Ys2 ) ) ) ).
% zip_map2
thf(fact_3895_zip__map1,axiom,
! [A: $tType,C: $tType,B: $tType,F: C > A,Xs: list @ C,Ys2: list @ B] :
( ( zip @ A @ B @ ( map @ C @ A @ F @ Xs ) @ Ys2 )
= ( map @ ( product_prod @ C @ B ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ C @ B @ ( product_prod @ A @ B )
@ ^ [X3: C] : ( product_Pair @ A @ B @ ( F @ X3 ) ) )
@ ( zip @ C @ B @ Xs @ Ys2 ) ) ) ).
% zip_map1
thf(fact_3896_map__of__zip__map,axiom,
! [B: $tType,A: $tType,Xs: list @ A,F: A > B] :
( ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ ( map @ A @ B @ F @ Xs ) ) )
= ( ^ [X3: A] : ( if @ ( option @ B ) @ ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) @ ( some @ B @ ( F @ X3 ) ) @ ( none @ B ) ) ) ) ).
% map_of_zip_map
thf(fact_3897_map__of__zip__upd,axiom,
! [A: $tType,B: $tType,Ys2: list @ B,Xs: list @ A,Zs: list @ B,X: A,Y: B,Z2: B] :
( ( ( size_size @ ( list @ B ) @ Ys2 )
= ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( size_size @ ( list @ B ) @ Zs )
= ( size_size @ ( list @ A ) @ Xs ) )
=> ( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys2 ) ) @ X @ ( some @ B @ Y ) )
= ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Zs ) ) @ X @ ( some @ B @ Z2 ) ) )
=> ( ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys2 ) )
= ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Zs ) ) ) ) ) ) ) ).
% map_of_zip_upd
thf(fact_3898_map__of__zip__nth,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B,I: nat] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( distinct @ A @ Xs )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys2 ) @ ( nth @ A @ Xs @ I ) )
= ( some @ B @ ( nth @ B @ Ys2 @ I ) ) ) ) ) ) ).
% map_of_zip_nth
thf(fact_3899_foldr__snd__zip,axiom,
! [B: $tType,A: $tType,C: $tType,Ys2: list @ A,Xs: list @ B,F: A > C > C,B2: C] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ys2 ) @ ( size_size @ ( list @ B ) @ Xs ) )
=> ( ( foldr @ ( product_prod @ B @ A ) @ C
@ ( product_case_prod @ B @ A @ ( C > C )
@ ^ [X3: B] : F )
@ ( zip @ B @ A @ Xs @ Ys2 )
@ B2 )
= ( foldr @ A @ C @ F @ Ys2 @ B2 ) ) ) ).
% foldr_snd_zip
thf(fact_3900_nths__shift__lemma,axiom,
! [A: $tType,A4: set @ nat,Xs: list @ A,I: nat] :
( ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
@ ( filter @ ( product_prod @ A @ nat )
@ ^ [P6: product_prod @ A @ nat] : ( member @ nat @ ( product_snd @ A @ nat @ P6 ) @ A4 )
@ ( zip @ A @ nat @ Xs @ ( upt @ I @ ( plus_plus @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) )
= ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
@ ( filter @ ( product_prod @ A @ nat )
@ ^ [P6: product_prod @ A @ nat] : ( member @ nat @ ( plus_plus @ nat @ ( product_snd @ A @ nat @ P6 ) @ I ) @ A4 )
@ ( zip @ A @ nat @ Xs @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ).
% nths_shift_lemma
thf(fact_3901_remove__rev__alt__def,axiom,
! [A: $tType] :
( ( remove_rev @ A )
= ( ^ [X3: A,Xs3: list @ A] :
( filter @ A
@ ^ [Y3: A] : Y3 != X3
@ ( rev @ A @ Xs3 ) ) ) ) ).
% remove_rev_alt_def
thf(fact_3902_zip__Cons1,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ A,Ys2: list @ B] :
( ( zip @ A @ B @ ( cons @ A @ X @ Xs ) @ Ys2 )
= ( case_list @ ( list @ ( product_prod @ A @ B ) ) @ B @ ( nil @ ( product_prod @ A @ B ) )
@ ^ [Y3: B,Ys: list @ B] : ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y3 ) @ ( zip @ A @ B @ Xs @ Ys ) )
@ Ys2 ) ) ).
% zip_Cons1
thf(fact_3903_zip__Cons,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Y: B,Ys2: list @ B] :
( ( zip @ A @ B @ Xs @ ( cons @ B @ Y @ Ys2 ) )
= ( case_list @ ( list @ ( product_prod @ A @ B ) ) @ A @ ( nil @ ( product_prod @ A @ B ) )
@ ^ [Z5: A,Zs3: list @ A] : ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Z5 @ Y ) @ ( zip @ A @ B @ Zs3 @ Ys2 ) )
@ Xs ) ) ).
% zip_Cons
thf(fact_3904_filter__rev__alt,axiom,
! [A: $tType] :
( ( filter_rev @ A )
= ( ^ [P4: A > $o,L3: list @ A] : ( filter @ A @ P4 @ ( rev @ A @ L3 ) ) ) ) ).
% filter_rev_alt
thf(fact_3905_remove__rev__def,axiom,
! [A: $tType] :
( ( remove_rev @ A )
= ( ^ [X3: A] :
( filter_rev @ A
@ ( comp @ $o @ $o @ A @ (~)
@ ( ^ [Y5: A,Z4: A] : Y5 = Z4
@ X3 ) ) ) ) ) ).
% remove_rev_def
thf(fact_3906_min__list_Opelims,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: list @ A,Y: A] :
( ( ( min_list @ A @ X )
= Y )
=> ( ( accp @ ( list @ A ) @ ( min_list_rel @ A ) @ X )
=> ( ! [X2: A,Xs2: list @ A] :
( ( X
= ( cons @ A @ X2 @ Xs2 ) )
=> ( ( Y
= ( case_list @ A @ A @ X2
@ ^ [A7: A,List2: list @ A] : ( ord_min @ A @ X2 @ ( min_list @ A @ Xs2 ) )
@ Xs2 ) )
=> ~ ( accp @ ( list @ A ) @ ( min_list_rel @ A ) @ ( cons @ A @ X2 @ Xs2 ) ) ) )
=> ~ ( ( X
= ( nil @ A ) )
=> ( ( Y
= ( undefined @ A ) )
=> ~ ( accp @ ( list @ A ) @ ( min_list_rel @ A ) @ ( nil @ A ) ) ) ) ) ) ) ) ).
% min_list.pelims
thf(fact_3907_merge__list_Opsimps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: list @ A] :
( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L @ ( nil @ ( list @ A ) ) ) ) )
=> ( ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L @ ( nil @ ( list @ A ) ) ) )
= L ) ) ) ).
% merge_list.psimps(2)
thf(fact_3908_merge__list_Opsimps_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [La: list @ A,Acc22: list @ ( list @ A )] :
( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La @ Acc22 ) @ ( nil @ ( list @ A ) ) ) )
=> ( ( merge_list @ A @ ( cons @ ( list @ A ) @ La @ Acc22 ) @ ( nil @ ( list @ A ) ) )
= ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La @ Acc22 ) ) ) ) ) ).
% merge_list.psimps(3)
thf(fact_3909_merge__list_Opsimps_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [La: list @ A,Acc22: list @ ( list @ A ),L: list @ A] :
( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La @ Acc22 ) @ ( cons @ ( list @ A ) @ L @ ( nil @ ( list @ A ) ) ) ) )
=> ( ( merge_list @ A @ ( cons @ ( list @ A ) @ La @ Acc22 ) @ ( cons @ ( list @ A ) @ L @ ( nil @ ( list @ A ) ) ) )
= ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L @ ( cons @ ( list @ A ) @ La @ Acc22 ) ) ) ) ) ) ).
% merge_list.psimps(4)
thf(fact_3910_merge__list_Opsimps_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) ) )
=> ( ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) )
= ( nil @ A ) ) ) ) ).
% merge_list.psimps(1)
thf(fact_3911_merge__list_Opelims,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: list @ ( list @ A ),Xa: list @ ( list @ A ),Y: list @ A] :
( ( ( merge_list @ A @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ X @ Xa ) )
=> ( ( ( X
= ( nil @ ( list @ A ) ) )
=> ( ( Xa
= ( nil @ ( list @ A ) ) )
=> ( ( Y
= ( nil @ A ) )
=> ~ ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) ) ) ) ) )
=> ( ( ( X
= ( nil @ ( list @ A ) ) )
=> ! [L5: list @ A] :
( ( Xa
= ( cons @ ( list @ A ) @ L5 @ ( nil @ ( list @ A ) ) ) )
=> ( ( Y = L5 )
=> ~ ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L5 @ ( nil @ ( list @ A ) ) ) ) ) ) ) )
=> ( ! [La2: list @ A,Acc23: list @ ( list @ A )] :
( ( X
= ( cons @ ( list @ A ) @ La2 @ Acc23 ) )
=> ( ( Xa
= ( nil @ ( list @ A ) ) )
=> ( ( Y
= ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La2 @ Acc23 ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La2 @ Acc23 ) @ ( nil @ ( list @ A ) ) ) ) ) ) )
=> ( ! [La2: list @ A,Acc23: list @ ( list @ A )] :
( ( X
= ( cons @ ( list @ A ) @ La2 @ Acc23 ) )
=> ! [L5: list @ A] :
( ( Xa
= ( cons @ ( list @ A ) @ L5 @ ( nil @ ( list @ A ) ) ) )
=> ( ( Y
= ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L5 @ ( cons @ ( list @ A ) @ La2 @ Acc23 ) ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La2 @ Acc23 ) @ ( cons @ ( list @ A ) @ L5 @ ( nil @ ( list @ A ) ) ) ) ) ) ) )
=> ~ ! [L12: list @ A,L22: list @ A,Ls2: list @ ( list @ A )] :
( ( Xa
= ( cons @ ( list @ A ) @ L12 @ ( cons @ ( list @ A ) @ L22 @ Ls2 ) ) )
=> ( ( Y
= ( merge_list @ A @ ( cons @ ( list @ A ) @ ( merge @ A @ L12 @ L22 ) @ X ) @ Ls2 ) )
=> ~ ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ X @ ( cons @ ( list @ A ) @ L12 @ ( cons @ ( list @ A ) @ L22 @ Ls2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% merge_list.pelims
thf(fact_3912_merge__list_Opsimps_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Acc22: list @ ( list @ A ),L1: list @ A,L2: list @ A,Ls: list @ ( list @ A )] :
( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ Acc22 @ ( cons @ ( list @ A ) @ L1 @ ( cons @ ( list @ A ) @ L2 @ Ls ) ) ) )
=> ( ( merge_list @ A @ Acc22 @ ( cons @ ( list @ A ) @ L1 @ ( cons @ ( list @ A ) @ L2 @ Ls ) ) )
= ( merge_list @ A @ ( cons @ ( list @ A ) @ ( merge @ A @ L1 @ L2 ) @ Acc22 ) @ Ls ) ) ) ) ).
% merge_list.psimps(5)
thf(fact_3913_partition__filter__conv,axiom,
! [A: $tType] :
( ( partition @ A )
= ( ^ [F2: A > $o,Xs3: list @ A] : ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( filter @ A @ F2 @ Xs3 ) @ ( filter @ A @ ( comp @ $o @ $o @ A @ (~) @ F2 ) @ Xs3 ) ) ) ) ).
% partition_filter_conv
thf(fact_3914_partition__rev__filter__conv,axiom,
! [A: $tType,P: A > $o,Yes2: list @ A,No2: list @ A,Xs: list @ A] :
( ( partition_rev @ A @ P @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes2 @ No2 ) @ Xs )
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ ( rev @ A @ ( filter @ A @ P @ Xs ) ) @ Yes2 ) @ ( append @ A @ ( rev @ A @ ( filter @ A @ ( comp @ $o @ $o @ A @ (~) @ P ) @ Xs ) ) @ No2 ) ) ) ).
% partition_rev_filter_conv
thf(fact_3915_merge_Osimps_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L2: list @ A] :
( ( merge @ A @ ( nil @ A ) @ L2 )
= L2 ) ) ).
% merge.simps(1)
thf(fact_3916_merge_Osimps_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X1: A,X22: A,L1: list @ A,L2: list @ A] :
( ( ( ord_less @ A @ X1 @ X22 )
=> ( ( merge @ A @ ( cons @ A @ X1 @ L1 ) @ ( cons @ A @ X22 @ L2 ) )
= ( cons @ A @ X1 @ ( merge @ A @ L1 @ ( cons @ A @ X22 @ L2 ) ) ) ) )
& ( ~ ( ord_less @ A @ X1 @ X22 )
=> ( ( ( X1 = X22 )
=> ( ( merge @ A @ ( cons @ A @ X1 @ L1 ) @ ( cons @ A @ X22 @ L2 ) )
= ( cons @ A @ X1 @ ( merge @ A @ L1 @ L2 ) ) ) )
& ( ( X1 != X22 )
=> ( ( merge @ A @ ( cons @ A @ X1 @ L1 ) @ ( cons @ A @ X22 @ L2 ) )
= ( cons @ A @ X22 @ ( merge @ A @ ( cons @ A @ X1 @ L1 ) @ L2 ) ) ) ) ) ) ) ) ).
% merge.simps(3)
thf(fact_3917_merge_Osimps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [V: A,Va2: list @ A] :
( ( merge @ A @ ( cons @ A @ V @ Va2 ) @ ( nil @ A ) )
= ( cons @ A @ V @ Va2 ) ) ) ).
% merge.simps(2)
thf(fact_3918_partition__rev_Osimps_I2_J,axiom,
! [A: $tType,P: A > $o,Yes2: list @ A,No2: list @ A,X: A,Xs: list @ A] :
( ( partition_rev @ A @ P @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes2 @ No2 ) @ ( cons @ A @ X @ Xs ) )
= ( partition_rev @ A @ P @ ( if @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( P @ X ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Yes2 ) @ No2 ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes2 @ ( cons @ A @ X @ No2 ) ) ) @ Xs ) ) ).
% partition_rev.simps(2)
thf(fact_3919_partition__rev_Osimps_I1_J,axiom,
! [A: $tType,P: A > $o,Yes2: list @ A,No2: list @ A] :
( ( partition_rev @ A @ P @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes2 @ No2 ) @ ( nil @ A ) )
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes2 @ No2 ) ) ).
% partition_rev.simps(1)
thf(fact_3920_merge__list_Osimps_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Acc22: list @ ( list @ A ),L1: list @ A,L2: list @ A,Ls: list @ ( list @ A )] :
( ( merge_list @ A @ Acc22 @ ( cons @ ( list @ A ) @ L1 @ ( cons @ ( list @ A ) @ L2 @ Ls ) ) )
= ( merge_list @ A @ ( cons @ ( list @ A ) @ ( merge @ A @ L1 @ L2 ) @ Acc22 ) @ Ls ) ) ) ).
% merge_list.simps(5)
thf(fact_3921_partition_Osimps_I1_J,axiom,
! [A: $tType,P: A > $o] :
( ( partition @ A @ P @ ( nil @ A ) )
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) ) ).
% partition.simps(1)
thf(fact_3922_partition__P,axiom,
! [A: $tType,P: A > $o,Xs: list @ A,Yes2: list @ A,No2: list @ A] :
( ( ( partition @ A @ P @ Xs )
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes2 @ No2 ) )
=> ( ! [X7: A] :
( ( member @ A @ X7 @ ( set2 @ A @ Yes2 ) )
=> ( P @ X7 ) )
& ! [X7: A] :
( ( member @ A @ X7 @ ( set2 @ A @ No2 ) )
=> ~ ( P @ X7 ) ) ) ) ).
% partition_P
thf(fact_3923_merge_Oelims,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: list @ A,Xa: list @ A,Y: list @ A] :
( ( ( merge @ A @ X @ Xa )
= Y )
=> ( ( ( X
= ( nil @ A ) )
=> ( Y != Xa ) )
=> ( ! [V4: A,Va: list @ A] :
( ( X
= ( cons @ A @ V4 @ Va ) )
=> ( ( Xa
= ( nil @ A ) )
=> ( Y
!= ( cons @ A @ V4 @ Va ) ) ) )
=> ~ ! [X13: A,L12: list @ A] :
( ( X
= ( cons @ A @ X13 @ L12 ) )
=> ! [X23: A,L22: list @ A] :
( ( Xa
= ( cons @ A @ X23 @ L22 ) )
=> ~ ( ( ( ord_less @ A @ X13 @ X23 )
=> ( Y
= ( cons @ A @ X13 @ ( merge @ A @ L12 @ ( cons @ A @ X23 @ L22 ) ) ) ) )
& ( ~ ( ord_less @ A @ X13 @ X23 )
=> ( ( ( X13 = X23 )
=> ( Y
= ( cons @ A @ X13 @ ( merge @ A @ L12 @ L22 ) ) ) )
& ( ( X13 != X23 )
=> ( Y
= ( cons @ A @ X23 @ ( merge @ A @ ( cons @ A @ X13 @ L12 ) @ L22 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% merge.elims
thf(fact_3924_partition_Osimps_I2_J,axiom,
! [A: $tType,P: A > $o,X: A,Xs: list @ A] :
( ( partition @ A @ P @ ( cons @ A @ X @ Xs ) )
= ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ^ [Yes3: list @ A,No3: list @ A] : ( if @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( P @ X ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Yes3 ) @ No3 ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes3 @ ( cons @ A @ X @ No3 ) ) )
@ ( partition @ A @ P @ Xs ) ) ) ).
% partition.simps(2)
thf(fact_3925_partition__rev_Oelims,axiom,
! [A: $tType,X: A > $o,Xa: product_prod @ ( list @ A ) @ ( list @ A ),Xb: list @ A,Y: product_prod @ ( list @ A ) @ ( list @ A )] :
( ( ( partition_rev @ A @ X @ Xa @ Xb )
= Y )
=> ( ! [Yes: list @ A,No: list @ A] :
( ( Xa
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) )
=> ( ( Xb
= ( nil @ A ) )
=> ( Y
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) ) ) )
=> ~ ! [Yes: list @ A,No: list @ A] :
( ( Xa
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) )
=> ! [X2: A,Xs2: list @ A] :
( ( Xb
= ( cons @ A @ X2 @ Xs2 ) )
=> ( Y
!= ( partition_rev @ A @ X @ ( if @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( X @ X2 ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Yes ) @ No ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ ( cons @ A @ X2 @ No ) ) ) @ Xs2 ) ) ) ) ) ) ).
% partition_rev.elims
thf(fact_3926_partition__set,axiom,
! [A: $tType,P: A > $o,Xs: list @ A,Yes2: list @ A,No2: list @ A] :
( ( ( partition @ A @ P @ Xs )
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes2 @ No2 ) )
=> ( ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Yes2 ) @ ( set2 @ A @ No2 ) )
= ( set2 @ A @ Xs ) ) ) ).
% partition_set
thf(fact_3927_inv__image__partition,axiom,
! [A: $tType,Xs: list @ A,P: A > $o,Ys2: list @ A] :
( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( P @ X2 ) )
=> ( ! [Y4: A] :
( ( member @ A @ Y4 @ ( set2 @ A @ Ys2 ) )
=> ~ ( P @ Y4 ) )
=> ( ( vimage @ ( list @ A ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( partition @ A @ P ) @ ( insert2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( bot_bot @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ) )
= ( shuffles @ A @ Xs @ Ys2 ) ) ) ) ).
% inv_image_partition
thf(fact_3928_merge__list_Oelims,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: list @ ( list @ A ),Xa: list @ ( list @ A ),Y: list @ A] :
( ( ( merge_list @ A @ X @ Xa )
= Y )
=> ( ( ( X
= ( nil @ ( list @ A ) ) )
=> ( ( Xa
= ( nil @ ( list @ A ) ) )
=> ( Y
!= ( nil @ A ) ) ) )
=> ( ( ( X
= ( nil @ ( list @ A ) ) )
=> ! [L5: list @ A] :
( ( Xa
= ( cons @ ( list @ A ) @ L5 @ ( nil @ ( list @ A ) ) ) )
=> ( Y != L5 ) ) )
=> ( ! [La2: list @ A,Acc23: list @ ( list @ A )] :
( ( X
= ( cons @ ( list @ A ) @ La2 @ Acc23 ) )
=> ( ( Xa
= ( nil @ ( list @ A ) ) )
=> ( Y
!= ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La2 @ Acc23 ) ) ) ) )
=> ( ! [La2: list @ A,Acc23: list @ ( list @ A )] :
( ( X
= ( cons @ ( list @ A ) @ La2 @ Acc23 ) )
=> ! [L5: list @ A] :
( ( Xa
= ( cons @ ( list @ A ) @ L5 @ ( nil @ ( list @ A ) ) ) )
=> ( Y
!= ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L5 @ ( cons @ ( list @ A ) @ La2 @ Acc23 ) ) ) ) ) )
=> ~ ! [L12: list @ A,L22: list @ A,Ls2: list @ ( list @ A )] :
( ( Xa
= ( cons @ ( list @ A ) @ L12 @ ( cons @ ( list @ A ) @ L22 @ Ls2 ) ) )
=> ( Y
!= ( merge_list @ A @ ( cons @ ( list @ A ) @ ( merge @ A @ L12 @ L22 ) @ X ) @ Ls2 ) ) ) ) ) ) ) ) ) ).
% merge_list.elims
thf(fact_3929_merge__correct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L1: list @ A,L2: list @ A] :
( ( ( distinct @ A @ L1 )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L1 ) )
=> ( ( ( distinct @ A @ L2 )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L2 ) )
=> ( ( distinct @ A @ ( merge @ A @ L1 @ L2 ) )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( merge @ A @ L1 @ L2 ) )
& ( ( set2 @ A @ ( merge @ A @ L1 @ L2 ) )
= ( sup_sup @ ( set @ A ) @ ( set2 @ A @ L1 ) @ ( set2 @ A @ L2 ) ) ) ) ) ) ) ).
% merge_correct
thf(fact_3930_merge__list_Opinduct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A0: list @ ( list @ A ),A1: list @ ( list @ A ),P: ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) > $o] :
( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ A0 @ A1 ) )
=> ( ( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) ) )
=> ( P @ ( nil @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) ) )
=> ( ! [L5: list @ A] :
( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L5 @ ( nil @ ( list @ A ) ) ) ) )
=> ( P @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L5 @ ( nil @ ( list @ A ) ) ) ) )
=> ( ! [La2: list @ A,Acc23: list @ ( list @ A )] :
( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La2 @ Acc23 ) @ ( nil @ ( list @ A ) ) ) )
=> ( ( P @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La2 @ Acc23 ) )
=> ( P @ ( cons @ ( list @ A ) @ La2 @ Acc23 ) @ ( nil @ ( list @ A ) ) ) ) )
=> ( ! [La2: list @ A,Acc23: list @ ( list @ A ),L5: list @ A] :
( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La2 @ Acc23 ) @ ( cons @ ( list @ A ) @ L5 @ ( nil @ ( list @ A ) ) ) ) )
=> ( ( P @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L5 @ ( cons @ ( list @ A ) @ La2 @ Acc23 ) ) )
=> ( P @ ( cons @ ( list @ A ) @ La2 @ Acc23 ) @ ( cons @ ( list @ A ) @ L5 @ ( nil @ ( list @ A ) ) ) ) ) )
=> ( ! [Acc23: list @ ( list @ A ),L12: list @ A,L22: list @ A,Ls2: list @ ( list @ A )] :
( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ Acc23 @ ( cons @ ( list @ A ) @ L12 @ ( cons @ ( list @ A ) @ L22 @ Ls2 ) ) ) )
=> ( ( P @ ( cons @ ( list @ A ) @ ( merge @ A @ L12 @ L22 ) @ Acc23 ) @ Ls2 )
=> ( P @ Acc23 @ ( cons @ ( list @ A ) @ L12 @ ( cons @ ( list @ A ) @ L22 @ Ls2 ) ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ) ) ) ) ).
% merge_list.pinduct
thf(fact_3931_partition__rev_Opelims,axiom,
! [A: $tType,X: A > $o,Xa: product_prod @ ( list @ A ) @ ( list @ A ),Xb: list @ A,Y: product_prod @ ( list @ A ) @ ( list @ A )] :
( ( ( partition_rev @ A @ X @ Xa @ Xb )
= Y )
=> ( ( accp @ ( product_prod @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) ) @ ( partition_rev_rel @ A ) @ ( product_Pair @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ X @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ Xa @ Xb ) ) )
=> ( ! [Yes: list @ A,No: list @ A] :
( ( Xa
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) )
=> ( ( Xb
= ( nil @ A ) )
=> ( ( Y
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) )
=> ~ ( accp @ ( product_prod @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) ) @ ( partition_rev_rel @ A ) @ ( product_Pair @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ X @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) @ ( nil @ A ) ) ) ) ) ) )
=> ~ ! [Yes: list @ A,No: list @ A] :
( ( Xa
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) )
=> ! [X2: A,Xs2: list @ A] :
( ( Xb
= ( cons @ A @ X2 @ Xs2 ) )
=> ( ( Y
= ( partition_rev @ A @ X @ ( if @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( X @ X2 ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Yes ) @ No ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ ( cons @ A @ X2 @ No ) ) ) @ Xs2 ) )
=> ~ ( accp @ ( product_prod @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) ) @ ( partition_rev_rel @ A ) @ ( product_Pair @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ X @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) @ ( cons @ A @ X2 @ Xs2 ) ) ) ) ) ) ) ) ) ) ).
% partition_rev.pelims
thf(fact_3932_merge_Opelims,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: list @ A,Xa: list @ A,Y: list @ A] :
( ( ( merge @ A @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( merge_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Xa ) )
=> ( ( ( X
= ( nil @ A ) )
=> ( ( Y = Xa )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( merge_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xa ) ) ) )
=> ( ! [V4: A,Va: list @ A] :
( ( X
= ( cons @ A @ V4 @ Va ) )
=> ( ( Xa
= ( nil @ A ) )
=> ( ( Y
= ( cons @ A @ V4 @ Va ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( merge_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ V4 @ Va ) @ ( nil @ A ) ) ) ) ) )
=> ~ ! [X13: A,L12: list @ A] :
( ( X
= ( cons @ A @ X13 @ L12 ) )
=> ! [X23: A,L22: list @ A] :
( ( Xa
= ( cons @ A @ X23 @ L22 ) )
=> ( ( ( ( ord_less @ A @ X13 @ X23 )
=> ( Y
= ( cons @ A @ X13 @ ( merge @ A @ L12 @ ( cons @ A @ X23 @ L22 ) ) ) ) )
& ( ~ ( ord_less @ A @ X13 @ X23 )
=> ( ( ( X13 = X23 )
=> ( Y
= ( cons @ A @ X13 @ ( merge @ A @ L12 @ L22 ) ) ) )
& ( ( X13 != X23 )
=> ( Y
= ( cons @ A @ X23 @ ( merge @ A @ ( cons @ A @ X13 @ L12 ) @ L22 ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( merge_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X13 @ L12 ) @ ( cons @ A @ X23 @ L22 ) ) ) ) ) ) ) ) ) ) ) ).
% merge.pelims
thf(fact_3933_transpose_Opinduct,axiom,
! [A: $tType,A0: list @ ( list @ A ),P: ( list @ ( list @ A ) ) > $o] :
( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ A0 )
=> ( ( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( nil @ ( list @ A ) ) )
=> ( P @ ( nil @ ( list @ A ) ) ) )
=> ( ! [Xss2: list @ ( list @ A )] :
( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) )
=> ( ( P @ Xss2 )
=> ( P @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) ) ) )
=> ( ! [X2: A,Xs2: list @ A,Xss2: list @ ( list @ A )] :
( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ Xss2 ) )
=> ( ( P
@ ( cons @ ( list @ A ) @ Xs2
@ ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
@ ^ [H: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) )
@ Xss2 ) ) ) )
=> ( P @ ( cons @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ Xss2 ) ) ) )
=> ( P @ A0 ) ) ) ) ) ).
% transpose.pinduct
thf(fact_3934_quicksort__by__rel_Oelims,axiom,
! [A: $tType,X: A > A > $o,Xa: list @ A,Xb: list @ A,Y: list @ A] :
( ( ( quicksort_by_rel @ A @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xb
= ( nil @ A ) )
=> ( Y != Xa ) )
=> ~ ! [X2: A,Xs2: list @ A] :
( ( Xb
= ( cons @ A @ X2 @ Xs2 ) )
=> ( Y
!= ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ ( list @ A )
@ ^ [Xs_s: list @ A,Xs_b: list @ A] : ( quicksort_by_rel @ A @ X @ ( cons @ A @ X2 @ ( quicksort_by_rel @ A @ X @ Xa @ Xs_b ) ) @ Xs_s )
@ ( partition_rev @ A
@ ^ [Y3: A] : ( X @ Y3 @ X2 )
@ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) )
@ Xs2 ) ) ) ) ) ) ).
% quicksort_by_rel.elims
thf(fact_3935_set__quicksort__by__rel,axiom,
! [A: $tType,R5: A > A > $o,Sl2: list @ A,Xs: list @ A] :
( ( set2 @ A @ ( quicksort_by_rel @ A @ R5 @ Sl2 @ Xs ) )
= ( set2 @ A @ ( append @ A @ Xs @ Sl2 ) ) ) ).
% set_quicksort_by_rel
thf(fact_3936_quicksort__by__rel_Osimps_I1_J,axiom,
! [A: $tType,R5: A > A > $o,Sl2: list @ A] :
( ( quicksort_by_rel @ A @ R5 @ Sl2 @ ( nil @ A ) )
= Sl2 ) ).
% quicksort_by_rel.simps(1)
thf(fact_3937_quicksort__by__rel__remove__acc,axiom,
! [A: $tType] :
( ( quicksort_by_rel @ A )
= ( ^ [R6: A > A > $o,Sl3: list @ A,Xs3: list @ A] : ( append @ A @ ( quicksort_by_rel @ A @ R6 @ ( nil @ A ) @ Xs3 ) @ Sl3 ) ) ) ).
% quicksort_by_rel_remove_acc
thf(fact_3938_quicksort__by__rel__remove__acc__guared,axiom,
! [A: $tType,Sl2: list @ A,R5: A > A > $o,Xs: list @ A] :
( ( Sl2
!= ( nil @ A ) )
=> ( ( quicksort_by_rel @ A @ R5 @ Sl2 @ Xs )
= ( append @ A @ ( quicksort_by_rel @ A @ R5 @ ( nil @ A ) @ Xs ) @ Sl2 ) ) ) ).
% quicksort_by_rel_remove_acc_guared
thf(fact_3939_sorted__wrt__quicksort__by__rel,axiom,
! [X14: $tType,R5: X14 > X14 > $o,Xs: list @ X14] :
( ! [X2: X14,Y4: X14] :
( ( R5 @ X2 @ Y4 )
| ( R5 @ Y4 @ X2 ) )
=> ( ! [X2: X14,Y4: X14,Z3: X14] :
( ( R5 @ X2 @ Y4 )
=> ( ( R5 @ Y4 @ Z3 )
=> ( R5 @ X2 @ Z3 ) ) )
=> ( sorted_wrt @ X14 @ R5 @ ( quicksort_by_rel @ X14 @ R5 @ ( nil @ X14 ) @ Xs ) ) ) ) ).
% sorted_wrt_quicksort_by_rel
thf(fact_3940_sorted__quicksort__by__rel,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( quicksort_by_rel @ A @ ( ord_less_eq @ A ) @ ( nil @ A ) @ Xs ) ) ) ).
% sorted_quicksort_by_rel
thf(fact_3941_quicksort__by__rel_Osimps_I2_J,axiom,
! [A: $tType,R5: A > A > $o,Sl2: list @ A,X: A,Xs: list @ A] :
( ( quicksort_by_rel @ A @ R5 @ Sl2 @ ( cons @ A @ X @ Xs ) )
= ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ ( list @ A )
@ ^ [Xs_s: list @ A,Xs_b: list @ A] : ( quicksort_by_rel @ A @ R5 @ ( cons @ A @ X @ ( quicksort_by_rel @ A @ R5 @ Sl2 @ Xs_b ) ) @ Xs_s )
@ ( partition_rev @ A
@ ^ [Y3: A] : ( R5 @ Y3 @ X )
@ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) )
@ Xs ) ) ) ).
% quicksort_by_rel.simps(2)
thf(fact_3942_transpose_Opsimps_I3_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Xss: list @ ( list @ A )] :
( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ Xss ) )
=> ( ( transpose @ A @ ( cons @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ Xss ) )
= ( cons @ ( list @ A )
@ ( cons @ A @ X
@ ( concat @ A
@ ( map @ ( list @ A ) @ ( list @ A )
@ ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [H: A,T3: list @ A] : ( cons @ A @ H @ ( nil @ A ) ) )
@ Xss ) ) )
@ ( transpose @ A
@ ( cons @ ( list @ A ) @ Xs
@ ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
@ ^ [H: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) )
@ Xss ) ) ) ) ) ) ) ).
% transpose.psimps(3)
thf(fact_3943_transpose_Opelims,axiom,
! [A: $tType,X: list @ ( list @ A ),Y: list @ ( list @ A )] :
( ( ( transpose @ A @ X )
= Y )
=> ( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ X )
=> ( ( ( X
= ( nil @ ( list @ A ) ) )
=> ( ( Y
= ( nil @ ( list @ A ) ) )
=> ~ ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( nil @ ( list @ A ) ) ) ) )
=> ( ! [Xss2: list @ ( list @ A )] :
( ( X
= ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) )
=> ( ( Y
= ( transpose @ A @ Xss2 ) )
=> ~ ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) ) ) )
=> ~ ! [X2: A,Xs2: list @ A,Xss2: list @ ( list @ A )] :
( ( X
= ( cons @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ Xss2 ) )
=> ( ( Y
= ( cons @ ( list @ A )
@ ( cons @ A @ X2
@ ( concat @ A
@ ( map @ ( list @ A ) @ ( list @ A )
@ ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [H: A,T3: list @ A] : ( cons @ A @ H @ ( nil @ A ) ) )
@ Xss2 ) ) )
@ ( transpose @ A
@ ( cons @ ( list @ A ) @ Xs2
@ ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
@ ^ [H: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) )
@ Xss2 ) ) ) ) ) )
=> ~ ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ Xss2 ) ) ) ) ) ) ) ) ).
% transpose.pelims
thf(fact_3944_quicksort__by__rel_Opsimps_I2_J,axiom,
! [A: $tType,R5: A > A > $o,Sl2: list @ A,X: A,Xs: list @ A] :
( ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( quicksort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R5 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Sl2 @ ( cons @ A @ X @ Xs ) ) ) )
=> ( ( quicksort_by_rel @ A @ R5 @ Sl2 @ ( cons @ A @ X @ Xs ) )
= ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ ( list @ A )
@ ^ [Xs_s: list @ A,Xs_b: list @ A] : ( quicksort_by_rel @ A @ R5 @ ( cons @ A @ X @ ( quicksort_by_rel @ A @ R5 @ Sl2 @ Xs_b ) ) @ Xs_s )
@ ( partition_rev @ A
@ ^ [Y3: A] : ( R5 @ Y3 @ X )
@ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) )
@ Xs ) ) ) ) ).
% quicksort_by_rel.psimps(2)
thf(fact_3945_quicksort__by__rel_Opelims,axiom,
! [A: $tType,X: A > A > $o,Xa: list @ A,Xb: list @ A,Y: list @ A] :
( ( ( quicksort_by_rel @ A @ X @ Xa @ Xb )
= Y )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( quicksort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xa @ Xb ) ) )
=> ( ( ( Xb
= ( nil @ A ) )
=> ( ( Y = Xa )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( quicksort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xa @ ( nil @ A ) ) ) ) ) )
=> ~ ! [X2: A,Xs2: list @ A] :
( ( Xb
= ( cons @ A @ X2 @ Xs2 ) )
=> ( ( Y
= ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ ( list @ A )
@ ^ [Xs_s: list @ A,Xs_b: list @ A] : ( quicksort_by_rel @ A @ X @ ( cons @ A @ X2 @ ( quicksort_by_rel @ A @ X @ Xa @ Xs_b ) ) @ Xs_s )
@ ( partition_rev @ A
@ ^ [Y3: A] : ( X @ Y3 @ X2 )
@ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) )
@ Xs2 ) ) )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( quicksort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xa @ ( cons @ A @ X2 @ Xs2 ) ) ) ) ) ) ) ) ) ).
% quicksort_by_rel.pelims
thf(fact_3946_transpose_Opsimps_I1_J,axiom,
! [A: $tType] :
( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( nil @ ( list @ A ) ) )
=> ( ( transpose @ A @ ( nil @ ( list @ A ) ) )
= ( nil @ ( list @ A ) ) ) ) ).
% transpose.psimps(1)
thf(fact_3947_transpose_Opsimps_I2_J,axiom,
! [A: $tType,Xss: list @ ( list @ A )] :
( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
=> ( ( transpose @ A @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
= ( transpose @ A @ Xss ) ) ) ).
% transpose.psimps(2)
thf(fact_3948_quicksort__by__rel_Opsimps_I1_J,axiom,
! [A: $tType,R5: A > A > $o,Sl2: list @ A] :
( ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( quicksort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R5 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Sl2 @ ( nil @ A ) ) ) )
=> ( ( quicksort_by_rel @ A @ R5 @ Sl2 @ ( nil @ A ) )
= Sl2 ) ) ).
% quicksort_by_rel.psimps(1)
thf(fact_3949_quicksort__by__rel_Opinduct,axiom,
! [A: $tType,A0: A > A > $o,A1: list @ A,A22: list @ A,P: ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o] :
( ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( quicksort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ A0 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ A1 @ A22 ) ) )
=> ( ! [R8: A > A > $o,Sl: list @ A] :
( ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( quicksort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R8 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Sl @ ( nil @ A ) ) ) )
=> ( P @ R8 @ Sl @ ( nil @ A ) ) )
=> ( ! [R8: A > A > $o,Sl: list @ A,X2: A,Xs2: list @ A] :
( ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( quicksort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R8 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Sl @ ( cons @ A @ X2 @ Xs2 ) ) ) )
=> ( ! [Xa2: product_prod @ ( list @ A ) @ ( list @ A ),Xb2: list @ A,Y6: list @ A] :
( ( Xa2
= ( partition_rev @ A
@ ^ [Z5: A] : ( R8 @ Z5 @ X2 )
@ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) )
@ Xs2 ) )
=> ( ( ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xb2 @ Y6 )
= Xa2 )
=> ( P @ R8 @ Sl @ Y6 ) ) )
=> ( ! [Xa2: product_prod @ ( list @ A ) @ ( list @ A ),Xb2: list @ A,Y6: list @ A] :
( ( Xa2
= ( partition_rev @ A
@ ^ [Z5: A] : ( R8 @ Z5 @ X2 )
@ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) )
@ Xs2 ) )
=> ( ( ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xb2 @ Y6 )
= Xa2 )
=> ( P @ R8 @ ( cons @ A @ X2 @ ( quicksort_by_rel @ A @ R8 @ Sl @ Y6 ) ) @ Xb2 ) ) )
=> ( P @ R8 @ Sl @ ( cons @ A @ X2 @ Xs2 ) ) ) ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ) ).
% quicksort_by_rel.pinduct
thf(fact_3950_filter__rev__aux__alt,axiom,
! [A: $tType] :
( ( filter_rev_aux @ A )
= ( ^ [A7: list @ A,P4: A > $o,L3: list @ A] : ( append @ A @ ( filter @ A @ P4 @ ( rev @ A @ L3 ) ) @ A7 ) ) ) ).
% filter_rev_aux_alt
thf(fact_3951_lexord__take__index__conv,axiom,
! [A: $tType,X: list @ A,Y: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lexord @ A @ R ) )
= ( ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ X ) @ ( size_size @ ( list @ A ) @ Y ) )
& ( ( take @ A @ ( size_size @ ( list @ A ) @ X ) @ Y )
= X ) )
| ? [I3: nat] :
( ( ord_less @ nat @ I3 @ ( ord_min @ nat @ ( size_size @ ( list @ A ) @ X ) @ ( size_size @ ( list @ A ) @ Y ) ) )
& ( ( take @ A @ I3 @ X )
= ( take @ A @ I3 @ Y ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ X @ I3 ) @ ( nth @ A @ Y @ I3 ) ) @ R ) ) ) ) ).
% lexord_take_index_conv
thf(fact_3952_last__take__nth__conv,axiom,
! [A: $tType,N: nat,L: list @ A] :
( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( N
!= ( zero_zero @ nat ) )
=> ( ( last @ A @ ( take @ A @ N @ L ) )
= ( nth @ A @ L @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ).
% last_take_nth_conv
thf(fact_3953_revg__fun,axiom,
! [A: $tType] :
( ( revg @ A )
= ( ^ [A7: list @ A] : ( append @ A @ ( rev @ A @ A7 ) ) ) ) ).
% revg_fun
thf(fact_3954_take__upt,axiom,
! [I: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ M2 ) @ N )
=> ( ( take @ nat @ M2 @ ( upt @ I @ N ) )
= ( upt @ I @ ( plus_plus @ nat @ I @ M2 ) ) ) ) ).
% take_upt
thf(fact_3955_take__Cons__numeral,axiom,
! [A: $tType,V: num,X: A,Xs: list @ A] :
( ( take @ A @ ( numeral_numeral @ nat @ V ) @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X @ ( take @ A @ ( minus_minus @ nat @ ( numeral_numeral @ nat @ V ) @ ( one_one @ nat ) ) @ Xs ) ) ) ).
% take_Cons_numeral
thf(fact_3956_take__drop,axiom,
! [A: $tType,N: nat,M2: nat,Xs: list @ A] :
( ( take @ A @ N @ ( drop @ A @ M2 @ Xs ) )
= ( drop @ A @ M2 @ ( take @ A @ ( plus_plus @ nat @ N @ M2 ) @ Xs ) ) ) ).
% take_drop
thf(fact_3957_revg_Osimps_I2_J,axiom,
! [A: $tType,A3: A,As2: list @ A,B2: list @ A] :
( ( revg @ A @ ( cons @ A @ A3 @ As2 ) @ B2 )
= ( revg @ A @ As2 @ ( cons @ A @ A3 @ B2 ) ) ) ).
% revg.simps(2)
thf(fact_3958_revg_Osimps_I1_J,axiom,
! [A: $tType,B2: list @ A] :
( ( revg @ A @ ( nil @ A ) @ B2 )
= B2 ) ).
% revg.simps(1)
thf(fact_3959_filter__rev__aux_Osimps_I2_J,axiom,
! [A: $tType,P: A > $o,X: A,A3: list @ A,Xs: list @ A] :
( ( ( P @ X )
=> ( ( filter_rev_aux @ A @ A3 @ P @ ( cons @ A @ X @ Xs ) )
= ( filter_rev_aux @ A @ ( cons @ A @ X @ A3 ) @ P @ Xs ) ) )
& ( ~ ( P @ X )
=> ( ( filter_rev_aux @ A @ A3 @ P @ ( cons @ A @ X @ Xs ) )
= ( filter_rev_aux @ A @ A3 @ P @ Xs ) ) ) ) ).
% filter_rev_aux.simps(2)
thf(fact_3960_filter__rev__aux_Osimps_I1_J,axiom,
! [A: $tType,A3: list @ A,P: A > $o] :
( ( filter_rev_aux @ A @ A3 @ P @ ( nil @ A ) )
= A3 ) ).
% filter_rev_aux.simps(1)
thf(fact_3961_drop__take__drop__unsplit,axiom,
! [A: $tType,I: nat,J: nat,L: list @ A] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( append @ A @ ( drop @ A @ I @ ( take @ A @ J @ L ) ) @ ( drop @ A @ J @ L ) )
= ( drop @ A @ I @ L ) ) ) ).
% drop_take_drop_unsplit
thf(fact_3962_take__add,axiom,
! [A: $tType,I: nat,J: nat,Xs: list @ A] :
( ( take @ A @ ( plus_plus @ nat @ I @ J ) @ Xs )
= ( append @ A @ ( take @ A @ I @ Xs ) @ ( take @ A @ J @ ( drop @ A @ I @ Xs ) ) ) ) ).
% take_add
thf(fact_3963_slice__def,axiom,
! [A: $tType] :
( ( slice @ A )
= ( ^ [From2: nat,To2: nat,List2: list @ A] : ( take @ A @ ( minus_minus @ nat @ To2 @ From2 ) @ ( drop @ A @ From2 @ List2 ) ) ) ) ).
% slice_def
thf(fact_3964_filter__upt__take__conv,axiom,
! [A: $tType,P: A > $o,M2: nat,L: list @ A,N: nat] :
( ( filter @ nat
@ ^ [I3: nat] : ( P @ ( nth @ A @ ( take @ A @ M2 @ L ) @ I3 ) )
@ ( upt @ N @ M2 ) )
= ( filter @ nat
@ ^ [I3: nat] : ( P @ ( nth @ A @ L @ I3 ) )
@ ( upt @ N @ M2 ) ) ) ).
% filter_upt_take_conv
thf(fact_3965_zip__replicate1,axiom,
! [A: $tType,B: $tType,N: nat,X: A,Ys2: list @ B] :
( ( zip @ A @ B @ ( replicate @ A @ N @ X ) @ Ys2 )
= ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X ) @ ( take @ B @ N @ Ys2 ) ) ) ).
% zip_replicate1
thf(fact_3966_revg_Oelims,axiom,
! [A: $tType,X: list @ A,Xa: list @ A,Y: list @ A] :
( ( ( revg @ A @ X @ Xa )
= Y )
=> ( ( ( X
= ( nil @ A ) )
=> ( Y != Xa ) )
=> ~ ! [A5: A,As: list @ A] :
( ( X
= ( cons @ A @ A5 @ As ) )
=> ( Y
!= ( revg @ A @ As @ ( cons @ A @ A5 @ Xa ) ) ) ) ) ) ).
% revg.elims
thf(fact_3967_zip__replicate2,axiom,
! [B: $tType,A: $tType,Xs: list @ A,N: nat,Y: B] :
( ( zip @ A @ B @ Xs @ ( replicate @ B @ N @ Y ) )
= ( map @ A @ ( product_prod @ A @ B )
@ ^ [X3: A] : ( product_Pair @ A @ B @ X3 @ Y )
@ ( take @ A @ N @ Xs ) ) ) ).
% zip_replicate2
thf(fact_3968_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 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ Xs ) ) ) ) ) ).
% take_Cons'
thf(fact_3969_filter__nth__ex__nth,axiom,
! [A: $tType,N: nat,P: A > $o,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ ( filter @ A @ P @ Xs ) ) )
=> ? [M5: nat] :
( ( ord_less_eq @ nat @ N @ M5 )
& ( ord_less @ nat @ M5 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ( nth @ A @ ( filter @ A @ P @ Xs ) @ N )
= ( nth @ A @ Xs @ M5 ) )
& ( ( filter @ A @ P @ ( take @ A @ M5 @ Xs ) )
= ( take @ A @ N @ ( filter @ A @ P @ Xs ) ) ) ) ) ).
% filter_nth_ex_nth
thf(fact_3970_map__upd__upds__conv__if,axiom,
! [A: $tType,B: $tType,X: A,Ys2: list @ B,Xs: list @ A,F: A > ( option @ B ),Y: B] :
( ( ( member @ A @ X @ ( set2 @ A @ ( take @ A @ ( size_size @ ( list @ B ) @ Ys2 ) @ Xs ) ) )
=> ( ( map_upds @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F @ X @ ( some @ B @ Y ) ) @ Xs @ Ys2 )
= ( map_upds @ A @ B @ F @ Xs @ Ys2 ) ) )
& ( ~ ( member @ A @ X @ ( set2 @ A @ ( take @ A @ ( size_size @ ( list @ B ) @ Ys2 ) @ Xs ) ) )
=> ( ( map_upds @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F @ X @ ( some @ B @ Y ) ) @ Xs @ Ys2 )
= ( fun_upd @ A @ ( option @ B ) @ ( map_upds @ A @ B @ F @ Xs @ Ys2 ) @ X @ ( some @ B @ Y ) ) ) ) ) ).
% map_upd_upds_conv_if
thf(fact_3971_Union__take__drop__id,axiom,
! [A: $tType,N: nat,L: list @ ( set @ A )] :
( ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( set2 @ ( set @ A ) @ ( drop @ ( set @ A ) @ N @ L ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( set2 @ ( set @ A ) @ ( take @ ( set @ A ) @ N @ L ) ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( set2 @ ( set @ A ) @ L ) ) ) ).
% Union_take_drop_id
thf(fact_3972_lex__take__index,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( lex @ A @ R ) )
=> ~ ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Ys2 ) )
=> ( ( ( take @ A @ I2 @ Xs )
= ( take @ A @ I2 @ Ys2 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ I2 ) @ ( nth @ A @ Ys2 @ I2 ) ) @ R ) ) ) ) ) ).
% lex_take_index
thf(fact_3973_map__nth__upt__drop__take__conv,axiom,
! [A: $tType,N6: nat,L: list @ A,M3: nat] :
( ( ord_less_eq @ nat @ N6 @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( map @ nat @ A @ ( nth @ A @ L ) @ ( upt @ M3 @ N6 ) )
= ( drop @ A @ M3 @ ( take @ A @ N6 @ L ) ) ) ) ).
% map_nth_upt_drop_take_conv
thf(fact_3974_filter__rev__def,axiom,
! [A: $tType] :
( ( filter_rev @ A )
= ( filter_rev_aux @ A @ ( nil @ A ) ) ) ).
% filter_rev_def
thf(fact_3975_revg_Opelims,axiom,
! [A: $tType,X: list @ A,Xa: list @ A,Y: list @ A] :
( ( ( revg @ A @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( revg_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Xa ) )
=> ( ( ( X
= ( nil @ A ) )
=> ( ( Y = Xa )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( revg_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xa ) ) ) )
=> ~ ! [A5: A,As: list @ A] :
( ( X
= ( cons @ A @ A5 @ As ) )
=> ( ( Y
= ( revg @ A @ As @ ( cons @ A @ A5 @ Xa ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( revg_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ A5 @ As ) @ Xa ) ) ) ) ) ) ) ).
% revg.pelims
thf(fact_3976_take__butlast__conv,axiom,
! [A: $tType,L: list @ A] :
( ( take @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ L ) @ ( suc @ ( zero_zero @ nat ) ) ) @ L )
= ( butlast @ A @ L ) ) ).
% take_butlast_conv
thf(fact_3977_take__minus__one__conv__butlast,axiom,
! [A: $tType,N: nat,L: list @ A] :
( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( take @ A @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) @ L )
= ( butlast @ A @ ( take @ A @ N @ L ) ) ) ) ).
% take_minus_one_conv_butlast
thf(fact_3978_take__update__last,axiom,
! [A: $tType,N: nat,List: list @ A,X: A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ List ) )
=> ( ( list_update @ A @ ( take @ A @ ( suc @ N ) @ List ) @ N @ X )
= ( append @ A @ ( take @ A @ N @ List ) @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ) ).
% take_update_last
thf(fact_3979_take__update,axiom,
! [A: $tType,N: nat,L: list @ A,I: nat,X: A] :
( ( take @ A @ N @ ( list_update @ A @ L @ I @ X ) )
= ( list_update @ A @ ( take @ A @ N @ L ) @ I @ X ) ) ).
% take_update
thf(fact_3980_drop__upd__irrelevant,axiom,
! [A: $tType,M2: nat,N: nat,L: list @ A,X: A] :
( ( ord_less @ nat @ M2 @ N )
=> ( ( drop @ A @ N @ ( list_update @ A @ L @ M2 @ X ) )
= ( drop @ A @ N @ L ) ) ) ).
% drop_upd_irrelevant
thf(fact_3981_nth__update__invalid,axiom,
! [A: $tType,I: nat,L: list @ A,J: nat,X: A] :
( ~ ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( nth @ A @ ( list_update @ A @ L @ J @ X ) @ I )
= ( nth @ A @ L @ I ) ) ) ).
% nth_update_invalid
thf(fact_3982_length__butlast,axiom,
! [A: $tType,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( butlast @ A @ Xs ) )
= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) ) ).
% length_butlast
thf(fact_3983_butlast__upd__last__eq,axiom,
! [A: $tType,L: list @ A,X: A] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( list_update @ A @ ( butlast @ A @ L ) @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ L ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ X )
= ( append @ A @ ( take @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ L ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ L ) @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ) ).
% butlast_upd_last_eq
thf(fact_3984_butlast__update_H,axiom,
! [A: $tType,L: list @ A,I: nat,X: A] :
( ( list_update @ A @ ( butlast @ A @ L ) @ I @ X )
= ( butlast @ A @ ( list_update @ A @ L @ I @ X ) ) ) ).
% butlast_update'
thf(fact_3985_distinct__butlast__swap,axiom,
! [A: $tType,Pq: list @ A,I: nat] :
( ( distinct @ A @ Pq )
=> ( distinct @ A @ ( butlast @ A @ ( list_update @ A @ Pq @ I @ ( last @ A @ Pq ) ) ) ) ) ).
% distinct_butlast_swap
thf(fact_3986_zip__update,axiom,
! [A: $tType,B: $tType,Xs: list @ A,I: nat,X: A,Ys2: list @ B,Y: B] :
( ( zip @ A @ B @ ( list_update @ A @ Xs @ I @ X ) @ ( list_update @ B @ Ys2 @ I @ Y ) )
= ( list_update @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys2 ) @ I @ ( product_Pair @ A @ B @ X @ Y ) ) ) ).
% zip_update
thf(fact_3987_butlast__list__update,axiom,
! [A: $tType,K: nat,Xs: list @ A,X: A] :
( ( ( K
= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) )
=> ( ( butlast @ A @ ( list_update @ A @ Xs @ K @ X ) )
= ( butlast @ A @ Xs ) ) )
& ( ( K
!= ( minus_minus @ 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
thf(fact_3988_in__set__upd__eq__aux,axiom,
! [A: $tType,I: nat,L: list @ A,X: A,Y: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( member @ A @ X @ ( set2 @ A @ ( list_update @ A @ L @ I @ Y ) ) )
= ( ( X = Y )
| ! [Y3: A] : ( member @ A @ X @ ( set2 @ A @ ( list_update @ A @ L @ I @ Y3 ) ) ) ) ) ) ).
% in_set_upd_eq_aux
thf(fact_3989_in__set__upd__cases,axiom,
! [A: $tType,X: A,L: list @ A,I: nat,Y: A] :
( ( member @ A @ X @ ( set2 @ A @ ( list_update @ A @ L @ I @ Y ) ) )
=> ( ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ( X != Y ) )
=> ( member @ A @ X @ ( set2 @ A @ L ) ) ) ) ).
% in_set_upd_cases
thf(fact_3990_in__set__upd__eq,axiom,
! [A: $tType,I: nat,L: list @ A,X: A,Y: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( member @ A @ X @ ( set2 @ A @ ( list_update @ A @ L @ I @ Y ) ) )
= ( ( X = Y )
| ( ( member @ A @ X @ ( set2 @ A @ L ) )
& ! [Y3: A] : ( member @ A @ X @ ( set2 @ A @ ( list_update @ A @ L @ I @ Y3 ) ) ) ) ) ) ) ).
% in_set_upd_eq
thf(fact_3991_nth__list__update_H,axiom,
! [A: $tType,I: nat,J: nat,L: list @ A,X: A] :
( ( ( ( I = J )
& ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) ) )
=> ( ( nth @ A @ ( list_update @ A @ L @ I @ X ) @ J )
= X ) )
& ( ~ ( ( I = J )
& ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) ) )
=> ( ( nth @ A @ ( list_update @ A @ L @ I @ X ) @ J )
= ( nth @ A @ L @ J ) ) ) ) ).
% nth_list_update'
thf(fact_3992_butlast__subset,axiom,
! [A: $tType,Xs: list @ A,A4: set @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( butlast @ A @ Xs ) ) @ A4 ) ) ) ).
% butlast_subset
thf(fact_3993_butlast__eq__cons__conv,axiom,
! [A: $tType,L: list @ A,X: A,Xs: list @ A] :
( ( ( butlast @ A @ L )
= ( cons @ A @ X @ Xs ) )
= ( ? [Xl: A] :
( L
= ( cons @ A @ X @ ( append @ A @ Xs @ ( cons @ A @ Xl @ ( nil @ A ) ) ) ) ) ) ) ).
% butlast_eq_cons_conv
thf(fact_3994_butlast__eq__consE,axiom,
! [A: $tType,L: list @ A,X: A,Xs: list @ A] :
( ( ( butlast @ A @ L )
= ( cons @ A @ X @ Xs ) )
=> ~ ! [Xl2: A] :
( L
!= ( cons @ A @ X @ ( append @ A @ Xs @ ( cons @ A @ Xl2 @ ( nil @ A ) ) ) ) ) ) ).
% butlast_eq_consE
thf(fact_3995_butlast__upt,axiom,
! [M2: nat,N: nat] :
( ( butlast @ nat @ ( upt @ M2 @ N ) )
= ( upt @ M2 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ).
% butlast_upt
thf(fact_3996_map__upd__eq,axiom,
! [B: $tType,A: $tType,I: nat,L: list @ A,F: A > B,X: A] :
( ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( F @ ( nth @ A @ L @ I ) )
= ( F @ X ) ) )
=> ( ( map @ A @ B @ F @ ( list_update @ A @ L @ I @ X ) )
= ( map @ A @ B @ F @ L ) ) ) ).
% map_upd_eq
thf(fact_3997_snoc__eq__iff__butlast_H,axiom,
! [A: $tType,Ys2: list @ A,Xs: list @ A,X: A] :
( ( Ys2
= ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) )
= ( ( Ys2
!= ( nil @ A ) )
& ( ( butlast @ A @ Ys2 )
= Xs )
& ( ( last @ A @ Ys2 )
= X ) ) ) ).
% snoc_eq_iff_butlast'
thf(fact_3998_butlast__conv__take,axiom,
! [A: $tType] :
( ( butlast @ A )
= ( ^ [Xs3: list @ A] : ( take @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ ( one_one @ nat ) ) @ Xs3 ) ) ) ).
% butlast_conv_take
thf(fact_3999_insert__swap__set__eq,axiom,
! [A: $tType,I: nat,L: list @ A,X: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( insert2 @ A @ ( nth @ A @ L @ I ) @ ( set2 @ A @ ( list_update @ A @ L @ I @ X ) ) )
= ( insert2 @ A @ X @ ( set2 @ A @ L ) ) ) ) ).
% insert_swap_set_eq
thf(fact_4000_rev__update,axiom,
! [A: $tType,K: nat,Xs: list @ A,Y: A] :
( ( ord_less @ nat @ K @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( rev @ A @ ( list_update @ A @ Xs @ K @ Y ) )
= ( list_update @ A @ ( rev @ A @ Xs ) @ ( minus_minus @ nat @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ K ) @ ( one_one @ nat ) ) @ Y ) ) ) ).
% rev_update
thf(fact_4001_last__list__update,axiom,
! [A: $tType,Xs: list @ A,K: nat,X: A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( ( K
= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) )
=> ( ( last @ A @ ( list_update @ A @ Xs @ K @ X ) )
= X ) )
& ( ( K
!= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) )
=> ( ( last @ A @ ( list_update @ A @ Xs @ K @ X ) )
= ( last @ A @ Xs ) ) ) ) ) ).
% last_list_update
thf(fact_4002_butlast__take,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( butlast @ A @ ( take @ A @ N @ Xs ) )
= ( take @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ Xs ) ) ) ).
% butlast_take
thf(fact_4003_listrel1__iff__update,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( listrel1 @ A @ R ) )
= ( ? [Y3: A,N5: nat] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ N5 ) @ Y3 ) @ R )
& ( ord_less @ nat @ N5 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( Ys2
= ( list_update @ A @ Xs @ N5 @ Y3 ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_4004_map__distinct__upd__conv,axiom,
! [B: $tType,A: $tType,I: nat,L: list @ A,F: A > B,X: B] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( distinct @ A @ L )
=> ( ( list_update @ B @ ( map @ A @ B @ F @ L ) @ I @ X )
= ( map @ A @ B @ ( fun_upd @ A @ B @ F @ ( nth @ A @ L @ I ) @ X ) @ L ) ) ) ) ).
% map_distinct_upd_conv
thf(fact_4005_foldl__list__update,axiom,
! [B: $tType,A: $tType,N: nat,Xs: list @ A,F: B > A > B,A3: B,X: A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( foldl @ B @ A @ F @ A3 @ ( list_update @ A @ Xs @ N @ X ) )
= ( foldl @ B @ A @ F @ ( F @ ( foldl @ B @ A @ F @ A3 @ ( take @ A @ N @ Xs ) ) @ X ) @ ( drop @ A @ ( suc @ N ) @ Xs ) ) ) ) ).
% foldl_list_update
thf(fact_4006_case__natural__def,axiom,
! [T: $tType] :
( ( code_case_natural @ T )
= ( ^ [F12: T,F23: code_natural > T] :
( code_rec_natural @ T @ F12
@ ^ [X12: code_natural,X24: T] : ( F23 @ X12 ) ) ) ) ).
% case_natural_def
thf(fact_4007_nth__rotate,axiom,
! [A: $tType,N: nat,Xs: list @ A,M2: nat] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( rotate @ A @ M2 @ Xs ) @ N )
= ( nth @ A @ Xs @ ( modulo_modulo @ nat @ ( plus_plus @ nat @ M2 @ N ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ).
% nth_rotate
thf(fact_4008_list_Osize__gen_I2_J,axiom,
! [A: $tType,X: A > nat,X21: A,X222: list @ A] :
( ( size_list @ A @ X @ ( cons @ A @ X21 @ X222 ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( X @ X21 ) @ ( size_list @ A @ X @ X222 ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% list.size_gen(2)
thf(fact_4009_rotate1__length01,axiom,
! [A: $tType,Xs: list @ A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) )
=> ( ( rotate1 @ A @ Xs )
= Xs ) ) ).
% rotate1_length01
thf(fact_4010_natural_Osimps_I6_J,axiom,
! [T: $tType,F1: T,F22: code_natural > T > T] :
( ( code_rec_natural @ T @ F1 @ F22 @ ( zero_zero @ code_natural ) )
= F1 ) ).
% natural.simps(6)
thf(fact_4011_size__list__append,axiom,
! [A: $tType,F: A > nat,Xs: list @ A,Ys2: list @ A] :
( ( size_list @ A @ F @ ( append @ A @ Xs @ Ys2 ) )
= ( plus_plus @ nat @ ( size_list @ A @ F @ Xs ) @ ( size_list @ A @ F @ Ys2 ) ) ) ).
% size_list_append
thf(fact_4012_rotate__length01,axiom,
! [A: $tType,Xs: list @ A,N: nat] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) )
=> ( ( rotate @ A @ N @ Xs )
= Xs ) ) ).
% rotate_length01
thf(fact_4013_rotate__rotate,axiom,
! [A: $tType,M2: nat,N: nat,Xs: list @ A] :
( ( rotate @ A @ M2 @ ( rotate @ A @ N @ Xs ) )
= ( rotate @ A @ ( plus_plus @ nat @ M2 @ N ) @ Xs ) ) ).
% rotate_rotate
thf(fact_4014_rotate__add,axiom,
! [A: $tType,M2: nat,N: nat] :
( ( rotate @ A @ ( plus_plus @ nat @ M2 @ N ) )
= ( comp @ ( list @ A ) @ ( list @ A ) @ ( list @ A ) @ ( rotate @ A @ M2 ) @ ( rotate @ A @ N ) ) ) ).
% rotate_add
thf(fact_4015_rec__natural__def,axiom,
! [T: $tType] :
( ( code_rec_natural @ T )
= ( ^ [F12: T,F23: code_natural > T > T,X3: code_natural] : ( the @ T @ ( code_rec_set_natural @ T @ F12 @ F23 @ X3 ) ) ) ) ).
% rec_natural_def
thf(fact_4016_lenlex__append2,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A ),Us: list @ A,Xs: list @ A,Ys2: list @ A] :
( ( irrefl @ A @ R5 )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Us @ Xs ) @ ( append @ A @ Us @ Ys2 ) ) @ ( lenlex @ A @ R5 ) )
= ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( lenlex @ A @ R5 ) ) ) ) ).
% lenlex_append2
thf(fact_4017_zipf__zip,axiom,
! [A: $tType,B: $tType,L1: list @ A,L2: list @ B] :
( ( ( size_size @ ( list @ A ) @ L1 )
= ( size_size @ ( list @ B ) @ L2 ) )
=> ( ( zipf @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) @ L1 @ L2 )
= ( zip @ A @ B @ L1 @ L2 ) ) ) ).
% zipf_zip
thf(fact_4018_map__tailrec__rev_Opelims,axiom,
! [A: $tType,B: $tType,X: A > B,Xa: list @ A,Xb: list @ B,Y: list @ B] :
( ( ( map_tailrec_rev @ A @ B @ X @ Xa @ Xb )
= Y )
=> ( ( accp @ ( product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( map_tailrec_rev_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xa @ Xb ) ) )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( ( Y = Xb )
=> ~ ( accp @ ( product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( map_tailrec_rev_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ Xb ) ) ) ) )
=> ~ ! [A5: A,As: list @ A] :
( ( Xa
= ( cons @ A @ A5 @ As ) )
=> ( ( Y
= ( map_tailrec_rev @ A @ B @ X @ As @ ( cons @ B @ ( X @ A5 ) @ Xb ) ) )
=> ~ ( accp @ ( product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( map_tailrec_rev_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A5 @ As ) @ Xb ) ) ) ) ) ) ) ) ).
% map_tailrec_rev.pelims
thf(fact_4019_lexord__same__pref__if__irrefl,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys2: list @ A,Zs: list @ A] :
( ( irrefl @ A @ R )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys2 ) @ ( append @ A @ Xs @ Zs ) ) @ ( lexord @ A @ R ) )
= ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys2 @ Zs ) @ ( lexord @ A @ R ) ) ) ) ).
% lexord_same_pref_if_irrefl
thf(fact_4020_irreflI,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
( ! [A5: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ A5 ) @ R5 )
=> ( irrefl @ A @ R5 ) ) ).
% irreflI
thf(fact_4021_irrefl__def,axiom,
! [A: $tType] :
( ( irrefl @ A )
= ( ^ [R3: set @ ( product_prod @ A @ A )] :
! [A7: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ A7 ) @ R3 ) ) ) ).
% irrefl_def
thf(fact_4022_zipf_Osimps_I2_J,axiom,
! [A: $tType,C: $tType,B: $tType,F: A > B > C,A3: A,As2: list @ A,B2: B,Bs2: list @ B] :
( ( zipf @ A @ B @ C @ F @ ( cons @ A @ A3 @ As2 ) @ ( cons @ B @ B2 @ Bs2 ) )
= ( cons @ C @ ( F @ A3 @ B2 ) @ ( zipf @ A @ B @ C @ F @ As2 @ Bs2 ) ) ) ).
% zipf.simps(2)
thf(fact_4023_zipf_Osimps_I1_J,axiom,
! [B: $tType,A: $tType,C: $tType,F: A > B > C] :
( ( zipf @ A @ B @ C @ F @ ( nil @ A ) @ ( nil @ B ) )
= ( nil @ C ) ) ).
% zipf.simps(1)
thf(fact_4024_lexl__not__refl,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),X: list @ A] :
( ( irrefl @ A @ R )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ X ) @ ( lex @ A @ R ) ) ) ).
% lexl_not_refl
thf(fact_4025_zipf_Oelims,axiom,
! [B: $tType,A: $tType,C: $tType,X: A > B > C,Xa: list @ A,Xb: list @ B,Y: list @ C] :
( ( ( zipf @ A @ B @ C @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( ( Xb
= ( nil @ B ) )
=> ( Y
!= ( nil @ C ) ) ) )
=> ( ! [A5: A,As: list @ A] :
( ( Xa
= ( cons @ A @ A5 @ As ) )
=> ! [B3: B,Bs: list @ B] :
( ( Xb
= ( cons @ B @ B3 @ Bs ) )
=> ( Y
!= ( cons @ C @ ( X @ A5 @ B3 ) @ ( zipf @ A @ B @ C @ X @ As @ Bs ) ) ) ) )
=> ( ( ? [V4: A,Va: list @ A] :
( Xa
= ( cons @ A @ V4 @ Va ) )
=> ( ( Xb
= ( nil @ B ) )
=> ( Y
!= ( undefined @ ( list @ C ) ) ) ) )
=> ~ ( ( Xa
= ( nil @ A ) )
=> ( ? [V4: B,Va: list @ B] :
( Xb
= ( cons @ B @ V4 @ Va ) )
=> ( Y
!= ( undefined @ ( list @ C ) ) ) ) ) ) ) ) ) ).
% zipf.elims
thf(fact_4026_zipf_Opelims,axiom,
! [C: $tType,A: $tType,B: $tType,X: A > B > C,Xa: list @ A,Xb: list @ B,Y: list @ C] :
( ( ( zipf @ A @ B @ C @ X @ Xa @ Xb )
= Y )
=> ( ( accp @ ( product_prod @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( zipf_rel @ A @ B @ C ) @ ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xa @ Xb ) ) )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( ( Xb
= ( nil @ B ) )
=> ( ( Y
= ( nil @ C ) )
=> ~ ( accp @ ( product_prod @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( zipf_rel @ A @ B @ C ) @ ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( nil @ B ) ) ) ) ) ) )
=> ( ! [A5: A,As: list @ A] :
( ( Xa
= ( cons @ A @ A5 @ As ) )
=> ! [B3: B,Bs: list @ B] :
( ( Xb
= ( cons @ B @ B3 @ Bs ) )
=> ( ( Y
= ( cons @ C @ ( X @ A5 @ B3 ) @ ( zipf @ A @ B @ C @ X @ As @ Bs ) ) )
=> ~ ( accp @ ( product_prod @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( zipf_rel @ A @ B @ C ) @ ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A5 @ As ) @ ( cons @ B @ B3 @ Bs ) ) ) ) ) ) )
=> ( ! [V4: A,Va: list @ A] :
( ( Xa
= ( cons @ A @ V4 @ Va ) )
=> ( ( Xb
= ( nil @ B ) )
=> ( ( Y
= ( undefined @ ( list @ C ) ) )
=> ~ ( accp @ ( product_prod @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( zipf_rel @ A @ B @ C ) @ ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ V4 @ Va ) @ ( nil @ B ) ) ) ) ) ) )
=> ~ ( ( Xa
= ( nil @ A ) )
=> ! [V4: B,Va: list @ B] :
( ( Xb
= ( cons @ B @ V4 @ Va ) )
=> ( ( Y
= ( undefined @ ( list @ C ) ) )
=> ~ ( accp @ ( product_prod @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( zipf_rel @ A @ B @ C ) @ ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( cons @ B @ V4 @ Va ) ) ) ) ) ) ) ) ) ) ) ) ).
% zipf.pelims
thf(fact_4027_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 )
= ( 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)
thf(fact_4028_effect__makeE,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [N: nat,F: nat > A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: array @ A,N2: nat] :
( ( heap_Time_effect @ ( array @ A ) @ ( array_make @ A @ N @ F ) @ H3 @ H4 @ R @ N2 )
=> ~ ( ( R
= ( product_fst @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ ( map @ nat @ A @ F @ ( upt @ ( zero_zero @ nat ) @ N ) ) @ H3 ) ) )
=> ( ( H4
= ( product_snd @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ ( map @ nat @ A @ F @ ( upt @ ( zero_zero @ nat ) @ N ) ) @ H3 ) ) )
=> ( ( ( array_get @ A @ H4 @ R )
= ( map @ nat @ A @ F @ ( upt @ ( zero_zero @ nat ) @ N ) ) )
=> ( ( array_present @ A @ H4 @ R )
=> ( ~ ( array_present @ A @ H3 @ R )
=> ( ( plus_plus @ nat @ N @ ( one_one @ nat ) )
!= N2 ) ) ) ) ) ) ) ) ).
% effect_makeE
thf(fact_4029_listrel__iff__nth,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B,R: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys2 ) @ ( listrel @ A @ B @ R ) )
= ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys2 ) )
& ! [N5: nat] :
( ( ord_less @ nat @ N5 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ ( nth @ A @ Xs @ N5 ) @ ( nth @ B @ Ys2 @ N5 ) ) @ R ) ) ) ) ).
% listrel_iff_nth
thf(fact_4030_listrel__rtrancl__refl,axiom,
! [A: $tType,Xs: list @ A,R: set @ ( product_prod @ A @ A )] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Xs ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R ) ) ) ).
% listrel_rtrancl_refl
thf(fact_4031_Array__Time_Oget__alloc,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [Xs: list @ A,H3: heap_ext @ product_unit,Ys2: list @ A] :
( ( array_get @ A @ ( product_snd @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Xs @ H3 ) ) @ ( product_fst @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Ys2 @ H3 ) ) )
= Xs ) ) ).
% Array_Time.get_alloc
thf(fact_4032_Array__Time_Opresent__alloc,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [Xs: list @ A,H3: heap_ext @ product_unit] : ( array_present @ A @ ( product_snd @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Xs @ H3 ) ) @ ( product_fst @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Xs @ H3 ) ) ) ) ).
% Array_Time.present_alloc
thf(fact_4033_Array__Time_Onot__present__alloc,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [H3: heap_ext @ product_unit,Xs: list @ A] :
~ ( array_present @ A @ H3 @ ( product_fst @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Xs @ H3 ) ) ) ) ).
% Array_Time.not_present_alloc
thf(fact_4034_listrel_ONil,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B )] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( nil @ B ) ) @ ( listrel @ A @ B @ R ) ) ).
% listrel.Nil
thf(fact_4035_listrel__Nil1,axiom,
! [A: $tType,B: $tType,Xs: list @ B,R: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ Xs ) @ ( listrel @ A @ B @ R ) )
=> ( Xs
= ( nil @ B ) ) ) ).
% listrel_Nil1
thf(fact_4036_listrel__Nil2,axiom,
! [B: $tType,A: $tType,Xs: list @ A,R: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ ( nil @ B ) ) @ ( listrel @ A @ B @ R ) )
=> ( Xs
= ( nil @ A ) ) ) ).
% listrel_Nil2
thf(fact_4037_listrel__eq__len,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B,R: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys2 ) @ ( listrel @ A @ B @ R ) )
=> ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys2 ) ) ) ).
% listrel_eq_len
thf(fact_4038_listrel__rtrancl__trans,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A ),Zs: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R ) ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys2 @ Zs ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R ) ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Zs ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R ) ) ) ) ) ).
% listrel_rtrancl_trans
thf(fact_4039_listrel_OCons,axiom,
! [B: $tType,A: $tType,X: A,Y: B,R: set @ ( product_prod @ A @ B ),Xs: list @ A,Ys2: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ R )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys2 ) @ ( listrel @ A @ B @ R ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ X @ Xs ) @ ( cons @ B @ Y @ Ys2 ) ) @ ( listrel @ A @ B @ R ) ) ) ) ).
% listrel.Cons
thf(fact_4040_listrel__Cons1,axiom,
! [B: $tType,A: $tType,Y: A,Ys2: list @ A,Xs: list @ B,R: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ Y @ Ys2 ) @ Xs ) @ ( listrel @ A @ B @ R ) )
=> ~ ! [Y4: B,Ys3: list @ B] :
( ( Xs
= ( cons @ B @ Y4 @ Ys3 ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y @ Y4 ) @ R )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Ys2 @ Ys3 ) @ ( listrel @ A @ B @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_4041_listrel__Cons2,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Y: B,Ys2: list @ B,R: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ ( cons @ B @ Y @ Ys2 ) ) @ ( listrel @ A @ B @ R ) )
=> ~ ! [X2: A,Xs2: list @ A] :
( ( Xs
= ( cons @ A @ X2 @ Xs2 ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ R )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs2 @ Ys2 ) @ ( listrel @ A @ B @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_4042_listrel__reflcl__if__listrel1,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( listrel1 @ A @ R ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R ) ) ) ) ).
% listrel_reflcl_if_listrel1
thf(fact_4043_rtrancl__listrel1__if__listrel,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( listrel @ A @ A @ R ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R ) ) ) ) ).
% rtrancl_listrel1_if_listrel
thf(fact_4044_effect__of__listE,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [Xs: list @ A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: array @ A,N2: nat] :
( ( heap_Time_effect @ ( array @ A ) @ ( array_of_list @ A @ Xs ) @ H3 @ H4 @ R @ N2 )
=> ~ ( ( R
= ( product_fst @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Xs @ H3 ) ) )
=> ( ( H4
= ( product_snd @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Xs @ H3 ) ) )
=> ( ( ( array_get @ A @ H4 @ R )
= Xs )
=> ( ( array_present @ A @ H4 @ R )
=> ( ~ ( array_present @ A @ H3 @ R )
=> ( N2
!= ( plus_plus @ nat @ ( one_one @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ) ) ) ) ).
% effect_of_listE
thf(fact_4045_effect__newE,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [N: nat,X: A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: array @ A,N2: nat] :
( ( heap_Time_effect @ ( array @ A ) @ ( array_new @ A @ N @ X ) @ H3 @ H4 @ R @ N2 )
=> ~ ( ( R
= ( product_fst @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ ( replicate @ A @ N @ X ) @ H3 ) ) )
=> ( ( H4
= ( product_snd @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ ( replicate @ A @ N @ X ) @ H3 ) ) )
=> ( ( ( array_get @ A @ H4 @ R )
= ( replicate @ A @ N @ X ) )
=> ( ( array_present @ A @ H4 @ R )
=> ( ~ ( array_present @ A @ H3 @ R )
=> ( ( plus_plus @ nat @ N @ ( one_one @ nat ) )
!= N2 ) ) ) ) ) ) ) ) ).
% effect_newE
thf(fact_4046_listrel_Ocases,axiom,
! [B: $tType,A: $tType,A1: list @ A,A22: list @ B,R: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ A1 @ A22 ) @ ( listrel @ A @ B @ R ) )
=> ( ( ( A1
= ( nil @ A ) )
=> ( A22
!= ( nil @ B ) ) )
=> ~ ! [X2: A,Y4: B,Xs2: list @ A] :
( ( A1
= ( cons @ A @ X2 @ Xs2 ) )
=> ! [Ys3: list @ B] :
( ( A22
= ( cons @ B @ Y4 @ Ys3 ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y4 ) @ R )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs2 @ Ys3 ) @ ( listrel @ A @ B @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_4047_listrel_Osimps,axiom,
! [B: $tType,A: $tType,A1: list @ A,A22: list @ B,R: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ A1 @ A22 ) @ ( listrel @ A @ B @ R ) )
= ( ( ( A1
= ( nil @ A ) )
& ( A22
= ( nil @ B ) ) )
| ? [X3: A,Y3: B,Xs3: list @ A,Ys: list @ B] :
( ( A1
= ( cons @ A @ X3 @ Xs3 ) )
& ( A22
= ( cons @ B @ Y3 @ Ys ) )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs3 @ Ys ) @ ( listrel @ A @ B @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_4048_listrel__iff__zip,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys2: list @ B,R: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys2 ) @ ( listrel @ A @ B @ R ) )
= ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys2 ) )
& ! [X3: product_prod @ A @ B] :
( ( member @ ( product_prod @ A @ B ) @ X3 @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys2 ) ) )
=> ( product_case_prod @ A @ B @ $o
@ ^ [Y3: A,Z5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y3 @ Z5 ) @ R )
@ X3 ) ) ) ) ).
% listrel_iff_zip
thf(fact_4049_present__alloc__noteq,axiom,
! [A: $tType,B: $tType] :
( ( ( heap @ B )
& ( heap @ A ) )
=> ! [H3: heap_ext @ product_unit,A3: array @ A,Xs: list @ B] :
( ( array_present @ A @ H3 @ A3 )
=> ( array_noteq @ A @ B @ A3 @ ( product_fst @ ( array @ B ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ B @ Xs @ H3 ) ) ) ) ) ).
% present_alloc_noteq
thf(fact_4050_listrel__def,axiom,
! [B: $tType,A: $tType] :
( ( listrel @ A @ B )
= ( ^ [R3: set @ ( product_prod @ A @ B )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ B ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ B ) @ $o
@ ( listrelp @ A @ B
@ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R3 ) ) ) ) ) ) ).
% listrel_def
thf(fact_4051_set,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [Ls: list @ A,H3: heap_ext @ product_unit,New_ls: list @ A] :
( ( array_set @ A @ ( product_fst @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Ls @ H3 ) ) @ New_ls @ ( product_snd @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Ls @ H3 ) ) )
= ( product_snd @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ New_ls @ H3 ) ) ) ) ).
% set
thf(fact_4052_Array__Time_Oset__same,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R: array @ A,X: list @ A,Y: list @ A,H3: heap_ext @ product_unit] :
( ( array_set @ A @ R @ X @ ( array_set @ A @ R @ Y @ H3 ) )
= ( array_set @ A @ R @ X @ H3 ) ) ) ).
% Array_Time.set_same
thf(fact_4053_Array__Time_Ounequal,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [A3: array @ A,A6: array @ A] :
( ( A3 != A6 )
= ( array_noteq @ A @ A @ A3 @ A6 ) ) ) ).
% Array_Time.unequal
thf(fact_4054_Array__Time_Oget__set__eq,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R: array @ A,X: list @ A,H3: heap_ext @ product_unit] :
( ( array_get @ A @ ( array_set @ A @ R @ X @ H3 ) @ R )
= X ) ) ).
% Array_Time.get_set_eq
thf(fact_4055_Array__Time_Oget__set__neq,axiom,
! [B: $tType,A: $tType] :
( ( ( heap @ A )
& ( heap @ B ) )
=> ! [R: array @ A,S: array @ B,X: list @ B,H3: heap_ext @ product_unit] :
( ( array_noteq @ A @ B @ R @ S )
=> ( ( array_get @ A @ ( array_set @ B @ S @ X @ H3 ) @ R )
= ( array_get @ A @ H3 @ R ) ) ) ) ).
% Array_Time.get_set_neq
thf(fact_4056_Array__Time_Oset__set__swap,axiom,
! [B: $tType,A: $tType] :
( ( ( heap @ A )
& ( heap @ B ) )
=> ! [R: array @ A,R4: array @ B,X: list @ A,X6: list @ B,H3: heap_ext @ product_unit] :
( ( array_noteq @ A @ B @ R @ R4 )
=> ( ( array_set @ A @ R @ X @ ( array_set @ B @ R4 @ X6 @ H3 ) )
= ( array_set @ B @ R4 @ X6 @ ( array_set @ A @ R @ X @ H3 ) ) ) ) ) ).
% Array_Time.set_set_swap
thf(fact_4057_Array__Time_Onoteq__irrefl,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R: array @ A] :
~ ( array_noteq @ A @ A @ R @ R ) ) ).
% Array_Time.noteq_irrefl
thf(fact_4058_Array__Time_Onoteq__sym,axiom,
! [B: $tType,A: $tType] :
( ( ( heap @ A )
& ( heap @ B ) )
=> ! [A3: array @ A,B2: array @ B] :
( ( array_noteq @ A @ B @ A3 @ B2 )
=> ( array_noteq @ B @ A @ B2 @ A3 ) ) ) ).
% Array_Time.noteq_sym
thf(fact_4059_Field__not__elem,axiom,
! [A: $tType,V: A,R5: set @ ( product_prod @ A @ A )] :
( ~ ( member @ A @ V @ ( field2 @ A @ R5 ) )
=> ! [X7: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X7 @ R5 )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y3: A,Z5: A] :
( ( Y3 != V )
& ( Z5 != V ) )
@ X7 ) ) ) ).
% Field_not_elem
thf(fact_4060_refl__on__def_H,axiom,
! [A: $tType] :
( ( refl_on @ A )
= ( ^ [A9: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ! [X3: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X3 @ R3 )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y3: A,Z5: A] :
( ( member @ A @ Y3 @ A9 )
& ( member @ A @ Z5 @ A9 ) )
@ X3 ) )
& ! [X3: A] :
( ( member @ A @ X3 @ A9 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ R3 ) ) ) ) ) ).
% refl_on_def'
thf(fact_4061_listrelp__listrel__eq,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B )] :
( ( listrelp @ A @ B
@ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R ) )
= ( ^ [X3: list @ A,Y3: list @ B] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ X3 @ Y3 ) @ ( listrel @ A @ B @ R ) ) ) ) ).
% listrelp_listrel_eq
thf(fact_4062_Under__def,axiom,
! [A: $tType] :
( ( order_Under @ A )
= ( ^ [R3: set @ ( product_prod @ A @ A ),A9: set @ A] :
( collect @ A
@ ^ [B5: A] :
( ( member @ A @ B5 @ ( field2 @ A @ R3 ) )
& ! [X3: A] :
( ( member @ A @ X3 @ A9 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ X3 ) @ R3 ) ) ) ) ) ) ).
% Under_def
thf(fact_4063_UnderS__def,axiom,
! [A: $tType] :
( ( order_UnderS @ A )
= ( ^ [R3: set @ ( product_prod @ A @ A ),A9: set @ A] :
( collect @ A
@ ^ [B5: A] :
( ( member @ A @ B5 @ ( field2 @ A @ R3 ) )
& ! [X3: A] :
( ( member @ A @ X3 @ A9 )
=> ( ( B5 != X3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ X3 ) @ R3 ) ) ) ) ) ) ) ).
% UnderS_def
thf(fact_4064_sorted__wrt_Opelims_I2_J,axiom,
! [A: $tType,X: A > A > $o,Xa: list @ A] :
( ( sorted_wrt @ A @ X @ Xa )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ Xa ) )
=> ( ( ( Xa
= ( nil @ A ) )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( nil @ A ) ) ) )
=> ~ ! [X2: A,Ys3: list @ A] :
( ( Xa
= ( cons @ A @ X2 @ Ys3 ) )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( cons @ A @ X2 @ Ys3 ) ) )
=> ~ ( ! [Xa2: A] :
( ( member @ A @ Xa2 @ ( set2 @ A @ Ys3 ) )
=> ( X @ X2 @ Xa2 ) )
& ( sorted_wrt @ A @ X @ Ys3 ) ) ) ) ) ) ) ).
% sorted_wrt.pelims(2)
thf(fact_4065_sorted__wrt_Opelims_I1_J,axiom,
! [A: $tType,X: A > A > $o,Xa: list @ A,Y: $o] :
( ( ( sorted_wrt @ A @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ Xa ) )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( Y
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( nil @ A ) ) ) ) )
=> ~ ! [X2: A,Ys3: list @ A] :
( ( Xa
= ( cons @ A @ X2 @ Ys3 ) )
=> ( ( Y
= ( ! [Y3: A] :
( ( member @ A @ Y3 @ ( set2 @ A @ Ys3 ) )
=> ( X @ X2 @ Y3 ) )
& ( sorted_wrt @ A @ X @ Ys3 ) ) )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( cons @ A @ X2 @ Ys3 ) ) ) ) ) ) ) ) ).
% sorted_wrt.pelims(1)
thf(fact_4066_sorted__wrt_Opelims_I3_J,axiom,
! [A: $tType,X: A > A > $o,Xa: list @ A] :
( ~ ( sorted_wrt @ A @ X @ Xa )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ Xa ) )
=> ~ ! [X2: A,Ys3: list @ A] :
( ( Xa
= ( cons @ A @ X2 @ Ys3 ) )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( cons @ A @ X2 @ Ys3 ) ) )
=> ( ! [Xa3: A] :
( ( member @ A @ Xa3 @ ( set2 @ A @ Ys3 ) )
=> ( X @ X2 @ Xa3 ) )
& ( sorted_wrt @ A @ X @ Ys3 ) ) ) ) ) ) ).
% sorted_wrt.pelims(3)
thf(fact_4067_Above__def,axiom,
! [A: $tType] :
( ( order_Above @ A )
= ( ^ [R3: set @ ( product_prod @ A @ A ),A9: set @ A] :
( collect @ A
@ ^ [B5: A] :
( ( member @ A @ B5 @ ( field2 @ A @ R3 ) )
& ! [X3: A] :
( ( member @ A @ X3 @ A9 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ B5 ) @ R3 ) ) ) ) ) ) ).
% Above_def
thf(fact_4068_execute__nth_I1_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,H3: heap_ext @ product_unit,A3: array @ A] :
( ( ord_less @ nat @ I @ ( array_length @ A @ H3 @ A3 ) )
=> ( ( heap_Time_execute @ A @ ( array_nth @ A @ A3 @ I ) @ H3 )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( nth @ A @ ( array_get @ A @ H3 @ A3 ) @ I ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% execute_nth(1)
thf(fact_4069_Array__Time_Olength__alloc,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [Xs: list @ A,H3: heap_ext @ product_unit,Ys2: list @ A] :
( ( array_length @ A @ ( product_snd @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Xs @ H3 ) ) @ ( product_fst @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Ys2 @ H3 ) ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% Array_Time.length_alloc
thf(fact_4070_Array__Time_Onth__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_nth @ A )
= ( ^ [A7: array @ A,I3: nat] :
( heap_Time_guard @ A
@ ^ [H: heap_ext @ product_unit] : ( ord_less @ nat @ I3 @ ( array_length @ A @ H @ A7 ) )
@ ^ [H: heap_ext @ product_unit] : ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( nth @ A @ ( array_get @ A @ H @ A7 ) @ I3 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H @ ( one_one @ nat ) ) ) ) ) ) ) ).
% Array_Time.nth_def
thf(fact_4071_success__nthI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,H3: heap_ext @ product_unit,A3: array @ A] :
( ( ord_less @ nat @ I @ ( array_length @ A @ H3 @ A3 ) )
=> ( heap_Time_success @ A @ ( array_nth @ A @ A3 @ I ) @ H3 ) ) ) ).
% success_nthI
thf(fact_4072_fails__array__nth,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [P5: array @ A,I: nat,H3: heap_ext @ product_unit] :
( ( time_fails @ A @ ( array_nth @ A @ P5 @ I ) @ H3 )
= ( ~ ( ord_less @ nat @ I @ ( array_length @ A @ H3 @ P5 ) ) ) ) ) ).
% fails_array_nth
thf(fact_4073_execute__nth_I2_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [H3: heap_ext @ product_unit,A3: array @ A,I: nat] :
( ( ord_less_eq @ nat @ ( array_length @ A @ H3 @ A3 ) @ I )
=> ( ( heap_Time_execute @ A @ ( array_nth @ A @ A3 @ I ) @ H3 )
= ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ).
% execute_nth(2)
thf(fact_4074_length__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_length @ A )
= ( ^ [H: heap_ext @ product_unit,A7: array @ A] : ( size_size @ ( list @ A ) @ ( array_get @ A @ H @ A7 ) ) ) ) ) ).
% length_def
thf(fact_4075_effect__nthE,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [A3: array @ A,I: nat,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A,N: nat] :
( ( heap_Time_effect @ A @ ( array_nth @ A @ A3 @ I ) @ H3 @ H4 @ R @ N )
=> ~ ( ( ord_less @ nat @ I @ ( array_length @ A @ H3 @ A3 ) )
=> ( ( R
= ( nth @ A @ ( array_get @ A @ H3 @ A3 ) @ I ) )
=> ( ( H4 = H3 )
=> ( N
!= ( one_one @ nat ) ) ) ) ) ) ) ).
% effect_nthE
thf(fact_4076_effect__nthI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,H3: heap_ext @ product_unit,A3: array @ A,H4: heap_ext @ product_unit,R: A,N: nat] :
( ( ord_less @ nat @ I @ ( array_length @ A @ H3 @ A3 ) )
=> ( ( H4 = H3 )
=> ( ( R
= ( nth @ A @ ( array_get @ A @ H3 @ A3 ) @ I ) )
=> ( ( N
= ( one_one @ nat ) )
=> ( heap_Time_effect @ A @ ( array_nth @ A @ A3 @ I ) @ H3 @ H4 @ R @ N ) ) ) ) ) ) ).
% effect_nthI
thf(fact_4077_nth_H__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_nth2 @ A )
= ( ^ [A7: array @ A] : ( comp @ nat @ ( heap_Time_Heap @ A ) @ code_integer @ ( array_nth @ A @ A7 ) @ code_nat_of_integer ) ) ) ) ).
% nth'_def
thf(fact_4078_freeze__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_freeze @ A )
= ( ^ [A7: array @ A] :
( heap_Time_heap @ ( list @ A )
@ ^ [H: heap_ext @ product_unit] : ( product_Pair @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( array_get @ A @ H @ A7 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( array_length @ A @ H @ A7 ) ) ) ) ) ) ) ) ).
% freeze_def
thf(fact_4079_execute__map__entry_I2_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [H3: heap_ext @ product_unit,A3: array @ A,I: nat,F: A > A] :
( ( ord_less_eq @ nat @ ( array_length @ A @ H3 @ A3 ) @ I )
=> ( ( heap_Time_execute @ ( array @ A ) @ ( array_map_entry @ A @ I @ F @ A3 ) @ H3 )
= ( none @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ).
% execute_map_entry(2)
thf(fact_4080_execute__freeze,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [A3: array @ A,H3: heap_ext @ product_unit] :
( ( heap_Time_execute @ ( list @ A ) @ ( array_freeze @ A @ A3 ) @ H3 )
= ( some @ ( 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 @ H3 @ A3 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( array_length @ A @ H3 @ A3 ) ) ) ) ) ) ) ).
% execute_freeze
thf(fact_4081_success__freezeI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [A3: array @ A,H3: heap_ext @ product_unit] : ( heap_Time_success @ ( list @ A ) @ ( array_freeze @ A @ A3 ) @ H3 ) ) ).
% success_freezeI
thf(fact_4082_success__map__entryI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,H3: heap_ext @ product_unit,A3: array @ A,F: A > A] :
( ( ord_less @ nat @ I @ ( array_length @ A @ H3 @ A3 ) )
=> ( heap_Time_success @ ( array @ A ) @ ( array_map_entry @ A @ I @ F @ A3 ) @ H3 ) ) ) ).
% success_map_entryI
thf(fact_4083_effect__freezeE,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [A3: array @ A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: list @ A,N: nat] :
( ( heap_Time_effect @ ( list @ A ) @ ( array_freeze @ A @ A3 ) @ H3 @ H4 @ R @ N )
=> ~ ( ( H4 = H3 )
=> ( ( R
= ( array_get @ A @ H3 @ A3 ) )
=> ( N
!= ( plus_plus @ nat @ ( array_length @ A @ H3 @ A3 ) @ ( one_one @ nat ) ) ) ) ) ) ) ).
% effect_freezeE
thf(fact_4084_effect__freezeI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [H4: heap_ext @ product_unit,H3: heap_ext @ product_unit,R: list @ A,A3: array @ A,N: nat] :
( ( H4 = H3 )
=> ( ( R
= ( array_get @ A @ H3 @ A3 ) )
=> ( ( N
= ( array_length @ A @ H3 @ A3 ) )
=> ( heap_Time_effect @ ( list @ A ) @ ( array_freeze @ A @ A3 ) @ H3 @ H4 @ R @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ) ).
% effect_freezeI
thf(fact_4085_execute__map__entry_I1_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,H3: heap_ext @ product_unit,A3: array @ A,F: A > A] :
( ( ord_less @ nat @ I @ ( array_length @ A @ H3 @ A3 ) )
=> ( ( heap_Time_execute @ ( array @ A ) @ ( array_map_entry @ A @ I @ F @ A3 ) @ H3 )
= ( some @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ A3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ ( array_update @ A @ A3 @ I @ ( F @ ( nth @ A @ ( array_get @ A @ H3 @ A3 ) @ I ) ) @ H3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% execute_map_entry(1)
thf(fact_4086_map__entry__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_map_entry @ A )
= ( ^ [I3: nat,F2: A > A,A7: array @ A] :
( heap_Time_guard @ ( array @ A )
@ ^ [H: heap_ext @ product_unit] : ( ord_less @ nat @ I3 @ ( array_length @ A @ H @ A7 ) )
@ ^ [H: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ A7 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ ( array_update @ A @ A7 @ I3 @ ( F2 @ ( nth @ A @ ( array_get @ A @ H @ A7 ) @ I3 ) ) @ H ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% map_entry_def
thf(fact_4087_effect__map__entryI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,H3: heap_ext @ product_unit,A3: array @ A,H4: heap_ext @ product_unit,F: A > A,R: array @ A,N: nat] :
( ( ord_less @ nat @ I @ ( array_length @ A @ H3 @ A3 ) )
=> ( ( H4
= ( array_update @ A @ A3 @ I @ ( F @ ( nth @ A @ ( array_get @ A @ H3 @ A3 ) @ I ) ) @ H3 ) )
=> ( ( R = A3 )
=> ( ( N
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( heap_Time_effect @ ( array @ A ) @ ( array_map_entry @ A @ I @ F @ A3 ) @ H3 @ H4 @ R @ N ) ) ) ) ) ) ).
% effect_map_entryI
thf(fact_4088_effect__map__entryE,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,F: A > A,A3: array @ A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: array @ A,N: nat] :
( ( heap_Time_effect @ ( array @ A ) @ ( array_map_entry @ A @ I @ F @ A3 ) @ H3 @ H4 @ R @ N )
=> ~ ( ( R = A3 )
=> ( ( H4
= ( array_update @ A @ A3 @ I @ ( F @ ( nth @ A @ ( array_get @ A @ H3 @ A3 ) @ I ) ) @ H3 ) )
=> ( ( ord_less @ nat @ I @ ( array_length @ A @ H3 @ A3 ) )
=> ( N
!= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% effect_map_entryE
thf(fact_4089_length__update,axiom,
! [B: $tType,A: $tType] :
( ( ( heap @ A )
& ( heap @ B ) )
=> ! [B2: array @ B,I: nat,V: B,H3: heap_ext @ product_unit] :
( ( array_length @ A @ ( array_update @ B @ B2 @ I @ V @ H3 ) )
= ( array_length @ A @ H3 ) ) ) ).
% length_update
thf(fact_4090_Array__Time_Opresent__update,axiom,
! [B: $tType,A: $tType] :
( ( ( heap @ A )
& ( heap @ B ) )
=> ! [B2: array @ B,I: nat,V: B,H3: heap_ext @ product_unit] :
( ( array_present @ A @ ( array_update @ B @ B2 @ I @ V @ H3 ) )
= ( array_present @ A @ H3 ) ) ) ).
% Array_Time.present_update
thf(fact_4091_get__update__elem__neqIndex,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,J: nat,A3: array @ A,V: A,H3: heap_ext @ product_unit] :
( ( I != J )
=> ( ( nth @ A @ ( array_get @ A @ ( array_update @ A @ A3 @ J @ V @ H3 ) @ A3 ) @ I )
= ( nth @ A @ ( array_get @ A @ H3 @ A3 ) @ I ) ) ) ) ).
% get_update_elem_neqIndex
thf(fact_4092_get__update__eq,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [A3: array @ A,I: nat,V: A,H3: heap_ext @ product_unit] :
( ( array_get @ A @ ( array_update @ A @ A3 @ I @ V @ H3 ) @ A3 )
= ( list_update @ A @ ( array_get @ A @ H3 @ A3 ) @ I @ V ) ) ) ).
% get_update_eq
thf(fact_4093_nth__update__neq,axiom,
! [B: $tType,A: $tType] :
( ( ( heap @ A )
& ( heap @ B ) )
=> ! [A3: array @ A,B2: array @ B,J: nat,V: B,H3: heap_ext @ product_unit,I: nat] :
( ( array_noteq @ A @ B @ A3 @ B2 )
=> ( ( nth @ A @ ( array_get @ A @ ( array_update @ B @ B2 @ J @ V @ H3 ) @ A3 ) @ I )
= ( nth @ A @ ( array_get @ A @ H3 @ A3 ) @ I ) ) ) ) ).
% nth_update_neq
thf(fact_4094_update__swap__neqIndex,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,I7: nat,A3: array @ A,V: A,V7: A,H3: heap_ext @ product_unit] :
( ( I != I7 )
=> ( ( array_update @ A @ A3 @ I @ V @ ( array_update @ A @ A3 @ I7 @ V7 @ H3 ) )
= ( array_update @ A @ A3 @ I7 @ V7 @ ( array_update @ A @ A3 @ I @ V @ H3 ) ) ) ) ) ).
% update_swap_neqIndex
thf(fact_4095_update__swap__neq,axiom,
! [B: $tType,A: $tType] :
( ( ( heap @ A )
& ( heap @ B ) )
=> ! [A3: array @ A,A6: array @ B,I: nat,V: A,I7: nat,V7: B,H3: heap_ext @ product_unit] :
( ( array_noteq @ A @ B @ A3 @ A6 )
=> ( ( array_update @ A @ A3 @ I @ V @ ( array_update @ B @ A6 @ I7 @ V7 @ H3 ) )
= ( array_update @ B @ A6 @ I7 @ V7 @ ( array_update @ A @ A3 @ I @ V @ H3 ) ) ) ) ) ).
% update_swap_neq
thf(fact_4096_Array__Time_Oupdate__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_update @ A )
= ( ^ [A7: array @ A,I3: nat,X3: A,H: heap_ext @ product_unit] : ( array_set @ A @ A7 @ ( list_update @ A @ ( array_get @ A @ H @ A7 ) @ I3 @ X3 ) @ H ) ) ) ) ).
% Array_Time.update_def
thf(fact_4097_execute__swap_I1_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,H3: heap_ext @ product_unit,A3: array @ A,X: A] :
( ( ord_less @ nat @ I @ ( array_length @ A @ H3 @ A3 ) )
=> ( ( heap_Time_execute @ A @ ( array_swap @ A @ I @ X @ A3 ) @ H3 )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( nth @ A @ ( array_get @ A @ H3 @ A3 ) @ I ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ ( array_update @ A @ A3 @ I @ X @ H3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% execute_swap(1)
thf(fact_4098_swap__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_swap @ A )
= ( ^ [I3: nat,X3: A,A7: array @ A] :
( heap_Time_guard @ A
@ ^ [H: heap_ext @ product_unit] : ( ord_less @ nat @ I3 @ ( array_length @ A @ H @ A7 ) )
@ ^ [H: heap_ext @ product_unit] : ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( nth @ A @ ( array_get @ A @ H @ A7 ) @ I3 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ ( array_update @ A @ A7 @ I3 @ X3 @ H ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% swap_def
thf(fact_4099_effect__swapI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,H3: heap_ext @ product_unit,A3: array @ A,H4: heap_ext @ product_unit,X: A,R: A,N: nat] :
( ( ord_less @ nat @ I @ ( array_length @ A @ H3 @ A3 ) )
=> ( ( H4
= ( array_update @ A @ A3 @ I @ X @ H3 ) )
=> ( ( R
= ( nth @ A @ ( array_get @ A @ H3 @ A3 ) @ I ) )
=> ( ( N
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( heap_Time_effect @ A @ ( array_swap @ A @ I @ X @ A3 ) @ H3 @ H4 @ R @ N ) ) ) ) ) ) ).
% effect_swapI
thf(fact_4100_effect__swapE,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,X: A,A3: array @ A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A,N: nat] :
( ( heap_Time_effect @ A @ ( array_swap @ A @ I @ X @ A3 ) @ H3 @ H4 @ R @ N )
=> ~ ( ( R
= ( nth @ A @ ( array_get @ A @ H3 @ A3 ) @ I ) )
=> ( ( H4
= ( array_update @ A @ A3 @ I @ X @ H3 ) )
=> ( ( ord_less @ nat @ I @ ( array_length @ A @ H3 @ A3 ) )
=> ( N
!= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% effect_swapE
thf(fact_4101_success__swapI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,H3: heap_ext @ product_unit,A3: array @ A,X: A] :
( ( ord_less @ nat @ I @ ( array_length @ A @ H3 @ A3 ) )
=> ( heap_Time_success @ A @ ( array_swap @ A @ I @ X @ A3 ) @ H3 ) ) ) ).
% success_swapI
thf(fact_4102_execute__swap_I2_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [H3: heap_ext @ product_unit,A3: array @ A,I: nat,X: A] :
( ( ord_less_eq @ nat @ ( array_length @ A @ H3 @ A3 ) @ I )
=> ( ( heap_Time_execute @ A @ ( array_swap @ A @ I @ X @ A3 ) @ H3 )
= ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ).
% execute_swap(2)
thf(fact_4103_execute__upd_I1_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,H3: heap_ext @ product_unit,A3: array @ A,X: A] :
( ( ord_less @ nat @ I @ ( array_length @ A @ H3 @ A3 ) )
=> ( ( heap_Time_execute @ ( array @ A ) @ ( array_upd @ A @ I @ X @ A3 ) @ H3 )
= ( some @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ A3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ ( array_update @ A @ A3 @ I @ X @ H3 ) @ ( one_one @ nat ) ) ) ) ) ) ) ).
% execute_upd(1)
thf(fact_4104_upd__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_upd @ A )
= ( ^ [I3: nat,X3: A,A7: array @ A] :
( heap_Time_guard @ ( array @ A )
@ ^ [H: heap_ext @ product_unit] : ( ord_less @ nat @ I3 @ ( array_length @ A @ H @ A7 ) )
@ ^ [H: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ A7 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ ( array_update @ A @ A7 @ I3 @ X3 @ H ) @ ( one_one @ nat ) ) ) ) ) ) ) ).
% upd_def
thf(fact_4105_execute__upd_I2_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [H3: heap_ext @ product_unit,A3: array @ A,I: nat,X: A] :
( ( ord_less_eq @ nat @ ( array_length @ A @ H3 @ A3 ) @ I )
=> ( ( heap_Time_execute @ ( array @ A ) @ ( array_upd @ A @ I @ X @ A3 ) @ H3 )
= ( none @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ).
% execute_upd(2)
thf(fact_4106_effect__updI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,H3: heap_ext @ product_unit,A3: array @ A,H4: heap_ext @ product_unit,V: A,N: nat] :
( ( ord_less @ nat @ I @ ( array_length @ A @ H3 @ A3 ) )
=> ( ( H4
= ( array_update @ A @ A3 @ I @ V @ H3 ) )
=> ( ( N
= ( one_one @ nat ) )
=> ( heap_Time_effect @ ( array @ A ) @ ( array_upd @ A @ I @ V @ A3 ) @ H3 @ H4 @ A3 @ N ) ) ) ) ) ).
% effect_updI
thf(fact_4107_upd__ureturn,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,X: A,A3: array @ A] :
( ( heap_Time_bind @ ( array @ A ) @ ( array @ A ) @ ( array_upd @ A @ I @ X @ A3 )
@ ^ [Uu: array @ A] : ( heap_Time_ureturn @ ( array @ A ) @ A3 ) )
= ( array_upd @ A @ I @ X @ A3 ) ) ) ).
% upd_ureturn
thf(fact_4108_success__updI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,H3: heap_ext @ product_unit,A3: array @ A,X: A] :
( ( ord_less @ nat @ I @ ( array_length @ A @ H3 @ A3 ) )
=> ( heap_Time_success @ ( array @ A ) @ ( array_upd @ A @ I @ X @ A3 ) @ H3 ) ) ) ).
% success_updI
thf(fact_4109_fails__array__upd,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,X: A,P5: array @ A,H3: heap_ext @ product_unit] :
( ( time_fails @ ( array @ A ) @ ( array_upd @ A @ I @ X @ P5 ) @ H3 )
= ( ~ ( ord_less @ nat @ I @ ( array_length @ A @ H3 @ P5 ) ) ) ) ) ).
% fails_array_upd
thf(fact_4110_effect__updE,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,V: A,A3: array @ A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: array @ A,N: nat] :
( ( heap_Time_effect @ ( array @ A ) @ ( array_upd @ A @ I @ V @ A3 ) @ H3 @ H4 @ R @ N )
=> ~ ( ( R = A3 )
=> ( ( H4
= ( array_update @ A @ A3 @ I @ V @ H3 ) )
=> ( ( ord_less @ nat @ I @ ( array_length @ A @ H3 @ A3 ) )
=> ( N
!= ( one_one @ nat ) ) ) ) ) ) ) ).
% effect_updE
thf(fact_4111_upd_H__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_upd2 @ A )
= ( ^ [A7: array @ A,I3: code_integer,X3: A] :
( heap_Time_bind @ ( array @ A ) @ product_unit @ ( array_upd @ A @ ( code_nat_of_integer @ I3 ) @ X3 @ A7 )
@ ^ [Uu: array @ A] : ( heap_Time_ureturn @ product_unit @ product_Unity ) ) ) ) ) ).
% upd'_def
thf(fact_4112_execute__len,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [A3: array @ A,H3: heap_ext @ product_unit] :
( ( heap_Time_execute @ nat @ ( array_len @ A @ A3 ) @ H3 )
= ( some @ ( product_prod @ nat @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ nat @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( array_length @ A @ H3 @ A3 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ ( one_one @ nat ) ) ) ) ) ) ).
% execute_len
thf(fact_4113_in__measures_I2_J,axiom,
! [A: $tType,X: A,Y: A,F: A > nat,Fs: list @ ( A > nat )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measures @ A @ ( cons @ ( A > nat ) @ F @ Fs ) ) )
= ( ( ord_less @ nat @ ( F @ X ) @ ( F @ Y ) )
| ( ( ( F @ X )
= ( F @ Y ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measures @ A @ Fs ) ) ) ) ) ).
% in_measures(2)
thf(fact_4114_mset__mergesort__by__rel__split,axiom,
! [A: $tType,Xs1: list @ A,Xs22: list @ A,Xs: list @ A] :
( ( plus_plus @ ( multiset @ A ) @ ( mset @ A @ ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) @ Xs ) ) ) @ ( mset @ A @ ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) @ Xs ) ) ) )
= ( plus_plus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ ( mset @ A @ Xs ) @ ( mset @ A @ Xs1 ) ) @ ( mset @ A @ Xs22 ) ) ) ).
% mset_mergesort_by_rel_split
thf(fact_4115_union__eq__empty,axiom,
! [A: $tType,M3: multiset @ A,N6: multiset @ A] :
( ( ( plus_plus @ ( multiset @ A ) @ M3 @ N6 )
= ( zero_zero @ ( multiset @ A ) ) )
= ( ( M3
= ( zero_zero @ ( multiset @ A ) ) )
& ( N6
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% union_eq_empty
thf(fact_4116_empty__eq__union,axiom,
! [A: $tType,M3: multiset @ A,N6: multiset @ A] :
( ( ( zero_zero @ ( multiset @ A ) )
= ( plus_plus @ ( multiset @ A ) @ M3 @ N6 ) )
= ( ( M3
= ( zero_zero @ ( multiset @ A ) ) )
& ( N6
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% empty_eq_union
thf(fact_4117_subset__mset_Ozero__eq__add__iff__both__eq__0,axiom,
! [A: $tType,X: multiset @ A,Y: multiset @ A] :
( ( ( zero_zero @ ( multiset @ A ) )
= ( plus_plus @ ( multiset @ A ) @ X @ Y ) )
= ( ( X
= ( zero_zero @ ( multiset @ A ) ) )
& ( Y
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% subset_mset.zero_eq_add_iff_both_eq_0
thf(fact_4118_subset__mset_Oadd__eq__0__iff__both__eq__0,axiom,
! [A: $tType,X: multiset @ A,Y: multiset @ A] :
( ( ( plus_plus @ ( multiset @ A ) @ X @ Y )
= ( zero_zero @ ( multiset @ A ) ) )
= ( ( X
= ( zero_zero @ ( multiset @ A ) ) )
& ( Y
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% subset_mset.add_eq_0_iff_both_eq_0
thf(fact_4119_diff__diff__add__mset,axiom,
! [A: $tType,M3: multiset @ A,N6: multiset @ A,P: multiset @ A] :
( ( minus_minus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ M3 @ N6 ) @ P )
= ( minus_minus @ ( multiset @ A ) @ M3 @ ( plus_plus @ ( multiset @ A ) @ N6 @ P ) ) ) ).
% diff_diff_add_mset
thf(fact_4120_mergesort__by__rel__permutes,axiom,
! [A: $tType,R5: A > A > $o,Xs: list @ A] :
( ( mset @ A @ ( mergesort_by_rel @ A @ R5 @ Xs ) )
= ( mset @ A @ Xs ) ) ).
% mergesort_by_rel_permutes
thf(fact_4121_in__measures_I1_J,axiom,
! [A: $tType,X: A,Y: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measures @ A @ ( nil @ ( A > nat ) ) ) ) ).
% in_measures(1)
thf(fact_4122_mset__append,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( mset @ A @ ( append @ A @ Xs @ Ys2 ) )
= ( plus_plus @ ( multiset @ A ) @ ( mset @ A @ Xs ) @ ( mset @ A @ Ys2 ) ) ) ).
% mset_append
thf(fact_4123_quicksort__by__rel__permutes,axiom,
! [A: $tType,R5: A > A > $o,Sl2: list @ A,Xs: list @ A] :
( ( mset @ A @ ( quicksort_by_rel @ A @ R5 @ Sl2 @ Xs ) )
= ( mset @ A @ ( append @ A @ Xs @ Sl2 ) ) ) ).
% quicksort_by_rel_permutes
thf(fact_4124_mset__mergesort__by__rel__merge,axiom,
! [A: $tType,R5: A > A > $o,Xs: list @ A,Ys2: list @ A] :
( ( mset @ A @ ( merges9089515139780605204_merge @ A @ R5 @ Xs @ Ys2 ) )
= ( plus_plus @ ( multiset @ A ) @ ( mset @ A @ Xs ) @ ( mset @ A @ Ys2 ) ) ) ).
% mset_mergesort_by_rel_merge
thf(fact_4125_success__lenI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [A3: array @ A,H3: heap_ext @ product_unit] : ( heap_Time_success @ nat @ ( array_len @ A @ A3 ) @ H3 ) ) ).
% success_lenI
thf(fact_4126_fails__array__len,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [P5: array @ A,H3: heap_ext @ product_unit] :
~ ( time_fails @ nat @ ( array_len @ A @ P5 ) @ H3 ) ) ).
% fails_array_len
thf(fact_4127_mset__distrib,axiom,
! [A: $tType,A4: multiset @ A,B6: multiset @ A,M3: multiset @ A,N6: multiset @ A] :
( ( ( plus_plus @ ( multiset @ A ) @ A4 @ B6 )
= ( plus_plus @ ( multiset @ A ) @ M3 @ N6 ) )
=> ~ ! [Am: multiset @ A,An: multiset @ A] :
( ( A4
= ( plus_plus @ ( multiset @ A ) @ Am @ An ) )
=> ! [Bm: multiset @ A,Bn: multiset @ A] :
( ( B6
= ( plus_plus @ ( multiset @ A ) @ Bm @ Bn ) )
=> ( ( M3
= ( plus_plus @ ( multiset @ A ) @ Am @ Bm ) )
=> ( N6
!= ( plus_plus @ ( multiset @ A ) @ An @ Bn ) ) ) ) ) ) ).
% mset_distrib
thf(fact_4128_mset__shuffles,axiom,
! [A: $tType,Zs: list @ A,Xs: list @ A,Ys2: list @ A] :
( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys2 ) )
=> ( ( mset @ A @ Zs )
= ( plus_plus @ ( multiset @ A ) @ ( mset @ A @ Xs ) @ ( mset @ A @ Ys2 ) ) ) ) ).
% mset_shuffles
thf(fact_4129_union__less__mono,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A4: multiset @ A,C5: multiset @ A,B6: multiset @ A,D4: multiset @ A] :
( ( ord_less @ ( multiset @ A ) @ A4 @ C5 )
=> ( ( ord_less @ ( multiset @ A ) @ B6 @ D4 )
=> ( ord_less @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) @ ( plus_plus @ ( multiset @ A ) @ C5 @ D4 ) ) ) ) ) ).
% union_less_mono
thf(fact_4130_union__le__mono2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B6: multiset @ A,D4: multiset @ A,C5: multiset @ A] :
( ( ord_less @ ( multiset @ A ) @ B6 @ D4 )
=> ( ord_less @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ C5 @ B6 ) @ ( plus_plus @ ( multiset @ A ) @ C5 @ D4 ) ) ) ) ).
% union_le_mono2
thf(fact_4131_union__le__mono1,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B6: multiset @ A,D4: multiset @ A,C5: multiset @ A] :
( ( ord_less @ ( multiset @ A ) @ B6 @ D4 )
=> ( ord_less @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ B6 @ C5 ) @ ( plus_plus @ ( multiset @ A ) @ D4 @ C5 ) ) ) ) ).
% union_le_mono1
thf(fact_4132_multi__union__self__other__eq,axiom,
! [A: $tType,A4: multiset @ A,X10: multiset @ A,Y9: multiset @ A] :
( ( ( plus_plus @ ( multiset @ A ) @ A4 @ X10 )
= ( plus_plus @ ( multiset @ A ) @ A4 @ Y9 ) )
=> ( X10 = Y9 ) ) ).
% multi_union_self_other_eq
thf(fact_4133_union__right__cancel,axiom,
! [A: $tType,M3: multiset @ A,K2: multiset @ A,N6: multiset @ A] :
( ( ( plus_plus @ ( multiset @ A ) @ M3 @ K2 )
= ( plus_plus @ ( multiset @ A ) @ N6 @ K2 ) )
= ( M3 = N6 ) ) ).
% union_right_cancel
thf(fact_4134_union__left__cancel,axiom,
! [A: $tType,K2: multiset @ A,M3: multiset @ A,N6: multiset @ A] :
( ( ( plus_plus @ ( multiset @ A ) @ K2 @ M3 )
= ( plus_plus @ ( multiset @ A ) @ K2 @ N6 ) )
= ( M3 = N6 ) ) ).
% union_left_cancel
thf(fact_4135_union__commute,axiom,
! [A: $tType] :
( ( plus_plus @ ( multiset @ A ) )
= ( ^ [M8: multiset @ A,N9: multiset @ A] : ( plus_plus @ ( multiset @ A ) @ N9 @ M8 ) ) ) ).
% union_commute
thf(fact_4136_union__lcomm,axiom,
! [A: $tType,M3: multiset @ A,N6: multiset @ A,K2: multiset @ A] :
( ( plus_plus @ ( multiset @ A ) @ M3 @ ( plus_plus @ ( multiset @ A ) @ N6 @ K2 ) )
= ( plus_plus @ ( multiset @ A ) @ N6 @ ( plus_plus @ ( multiset @ A ) @ M3 @ K2 ) ) ) ).
% union_lcomm
thf(fact_4137_union__assoc,axiom,
! [A: $tType,M3: multiset @ A,N6: multiset @ A,K2: multiset @ A] :
( ( plus_plus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ M3 @ N6 ) @ K2 )
= ( plus_plus @ ( multiset @ A ) @ M3 @ ( plus_plus @ ( multiset @ A ) @ N6 @ K2 ) ) ) ).
% union_assoc
thf(fact_4138_empty__neutral_I1_J,axiom,
! [A: $tType,X: multiset @ A] :
( ( plus_plus @ ( multiset @ A ) @ ( zero_zero @ ( multiset @ A ) ) @ X )
= X ) ).
% empty_neutral(1)
thf(fact_4139_empty__neutral_I2_J,axiom,
! [A: $tType,X: multiset @ A] :
( ( plus_plus @ ( multiset @ A ) @ X @ ( zero_zero @ ( multiset @ A ) ) )
= X ) ).
% empty_neutral(2)
thf(fact_4140_diff__union__cancelR,axiom,
! [A: $tType,M3: multiset @ A,N6: multiset @ A] :
( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ M3 @ N6 ) @ N6 )
= M3 ) ).
% diff_union_cancelR
thf(fact_4141_diff__union__cancelL,axiom,
! [A: $tType,N6: multiset @ A,M3: multiset @ A] :
( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ N6 @ M3 ) @ N6 )
= M3 ) ).
% diff_union_cancelL
thf(fact_4142_Multiset_Odiff__add,axiom,
! [A: $tType,M3: multiset @ A,N6: multiset @ A,Q: multiset @ A] :
( ( minus_minus @ ( multiset @ A ) @ M3 @ ( plus_plus @ ( multiset @ A ) @ N6 @ Q ) )
= ( minus_minus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ M3 @ N6 ) @ Q ) ) ).
% Multiset.diff_add
thf(fact_4143_union__diff__assoc,axiom,
! [A: $tType,C5: multiset @ A,B6: multiset @ A,A4: multiset @ A] :
( ( ( minus_minus @ ( multiset @ A ) @ C5 @ B6 )
= ( zero_zero @ ( multiset @ A ) ) )
=> ( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) @ C5 )
= ( plus_plus @ ( multiset @ A ) @ A4 @ ( minus_minus @ ( multiset @ A ) @ B6 @ C5 ) ) ) ) ).
% union_diff_assoc
thf(fact_4144_comp__fun__commute__plus__mset,axiom,
! [A: $tType] : ( finite6289374366891150609ommute @ ( multiset @ A ) @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) ) ).
% comp_fun_commute_plus_mset
thf(fact_4145_len__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_len @ A )
= ( ^ [A7: array @ A] :
( heap_Time_tap @ nat
@ ^ [H: heap_ext @ product_unit] : ( array_length @ A @ H @ A7 ) ) ) ) ) ).
% len_def
thf(fact_4146_measures__less,axiom,
! [A: $tType,F: A > nat,X: A,Y: A,Fs: list @ ( A > nat )] :
( ( ord_less @ nat @ ( F @ X ) @ ( F @ Y ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measures @ A @ ( cons @ ( A > nat ) @ F @ Fs ) ) ) ) ).
% measures_less
thf(fact_4147_measures__lesseq,axiom,
! [A: $tType,F: A > nat,X: A,Y: A,Fs: list @ ( A > nat )] :
( ( ord_less_eq @ nat @ ( F @ X ) @ ( F @ Y ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measures @ A @ Fs ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measures @ A @ ( cons @ ( A > nat ) @ F @ Fs ) ) ) ) ) ).
% measures_lesseq
thf(fact_4148_effect__lengthE,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [A3: array @ A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: nat,N: nat] :
( ( heap_Time_effect @ nat @ ( array_len @ A @ A3 ) @ H3 @ H4 @ R @ N )
=> ~ ( ( R
= ( array_length @ A @ H4 @ A3 ) )
=> ( ( H4 = H3 )
=> ( N
!= ( one_one @ nat ) ) ) ) ) ) ).
% effect_lengthE
thf(fact_4149_effect__lengthI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [H4: heap_ext @ product_unit,H3: heap_ext @ product_unit,R: nat,A3: array @ A,N: nat] :
( ( H4 = H3 )
=> ( ( R
= ( array_length @ A @ H3 @ A3 ) )
=> ( ( N
= ( one_one @ nat ) )
=> ( heap_Time_effect @ nat @ ( array_len @ A @ A3 ) @ H3 @ H4 @ R @ N ) ) ) ) ) ).
% effect_lengthI
thf(fact_4150_len_H__def,axiom,
! [A: $tType,B: $tType] :
( ( ( heap @ B )
& ( semiring_1 @ A ) )
=> ( ( array_len2 @ B @ A )
= ( ^ [A7: array @ B] :
( heap_Time_bind @ nat @ A @ ( array_len @ B @ A7 )
@ ^ [N5: nat] : ( heap_Time_ureturn @ A @ ( semiring_1_of_nat @ A @ N5 ) ) ) ) ) ) ).
% len'_def
thf(fact_4151_mset__zip__take__Cons__drop__twice,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B,J: nat,X: A,Y: B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( ord_less_eq @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( mset @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ ( append @ A @ ( take @ A @ J @ Xs ) @ ( cons @ A @ X @ ( drop @ A @ J @ Xs ) ) ) @ ( append @ B @ ( take @ B @ J @ Ys2 ) @ ( cons @ B @ Y @ ( drop @ B @ J @ Ys2 ) ) ) ) )
= ( add_mset @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( mset @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys2 ) ) ) ) ) ) ).
% mset_zip_take_Cons_drop_twice
thf(fact_4152_image__mset__map__of,axiom,
! [B: $tType,A: $tType,Xs: list @ ( product_prod @ A @ B )] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) )
=> ( ( image_mset @ A @ B
@ ^ [I3: A] : ( the2 @ B @ ( map_of @ A @ B @ Xs @ I3 ) )
@ ( mset @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) ) )
= ( mset @ B @ ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Xs ) ) ) ) ).
% image_mset_map_of
thf(fact_4153_mset__set__Union,axiom,
! [A: $tType,A4: set @ A,B6: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ A @ B6 )
=> ( ( ( inf_inf @ ( set @ A ) @ A4 @ B6 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( mset_set @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B6 ) )
= ( plus_plus @ ( multiset @ A ) @ ( mset_set @ A @ A4 ) @ ( mset_set @ A @ B6 ) ) ) ) ) ) ).
% mset_set_Union
thf(fact_4154_union__mset__add__mset__right,axiom,
! [A: $tType,A4: multiset @ A,A3: A,B6: multiset @ A] :
( ( plus_plus @ ( multiset @ A ) @ A4 @ ( add_mset @ A @ A3 @ B6 ) )
= ( add_mset @ A @ A3 @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) ) ) ).
% union_mset_add_mset_right
thf(fact_4155_union__mset__add__mset__left,axiom,
! [A: $tType,A3: A,A4: multiset @ A,B6: multiset @ A] :
( ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ A3 @ A4 ) @ B6 )
= ( add_mset @ A @ A3 @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) ) ) ).
% union_mset_add_mset_left
thf(fact_4156_image__mset__union,axiom,
! [A: $tType,B: $tType,F: B > A,M3: multiset @ B,N6: multiset @ B] :
( ( image_mset @ B @ A @ F @ ( plus_plus @ ( multiset @ B ) @ M3 @ N6 ) )
= ( plus_plus @ ( multiset @ A ) @ ( image_mset @ B @ A @ F @ M3 ) @ ( image_mset @ B @ A @ F @ N6 ) ) ) ).
% image_mset_union
thf(fact_4157_mset__map__split__orig,axiom,
! [B: $tType,A: $tType,F: B > A,P: multiset @ B,M13: multiset @ A,M24: multiset @ A] :
( ( ( image_mset @ B @ A @ F @ P )
= ( plus_plus @ ( multiset @ A ) @ M13 @ M24 ) )
=> ~ ! [P1: multiset @ B,P22: multiset @ B] :
( ( P
= ( plus_plus @ ( multiset @ B ) @ P1 @ P22 ) )
=> ( ( ( image_mset @ B @ A @ F @ P1 )
= M13 )
=> ( ( image_mset @ B @ A @ F @ P22 )
!= M24 ) ) ) ) ).
% mset_map_split_orig
thf(fact_4158_add__mset__add__single,axiom,
! [A: $tType] :
( ( add_mset @ A )
= ( ^ [A7: A,A9: multiset @ A] : ( plus_plus @ ( multiset @ A ) @ A9 @ ( add_mset @ A @ A7 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ).
% add_mset_add_single
thf(fact_4159_union__is__single,axiom,
! [A: $tType,M3: multiset @ A,N6: multiset @ A,A3: A] :
( ( ( plus_plus @ ( multiset @ A ) @ M3 @ N6 )
= ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) )
= ( ( ( M3
= ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) )
& ( N6
= ( zero_zero @ ( multiset @ A ) ) ) )
| ( ( M3
= ( zero_zero @ ( multiset @ A ) ) )
& ( N6
= ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ).
% union_is_single
thf(fact_4160_single__is__union,axiom,
! [A: $tType,A3: A,M3: multiset @ A,N6: multiset @ A] :
( ( ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) )
= ( plus_plus @ ( multiset @ A ) @ M3 @ N6 ) )
= ( ( ( ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) )
= M3 )
& ( N6
= ( zero_zero @ ( multiset @ A ) ) ) )
| ( ( M3
= ( zero_zero @ ( multiset @ A ) ) )
& ( ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) )
= N6 ) ) ) ) ).
% single_is_union
thf(fact_4161_add__mset__replicate__mset__safe,axiom,
! [A: $tType,B: $tType,M3: multiset @ B,A3: B] :
( ( nO_MATCH @ ( multiset @ A ) @ ( multiset @ B ) @ ( zero_zero @ ( multiset @ A ) ) @ M3 )
=> ( ( add_mset @ B @ A3 @ M3 )
= ( plus_plus @ ( multiset @ B ) @ ( add_mset @ B @ A3 @ ( zero_zero @ ( multiset @ B ) ) ) @ M3 ) ) ) ).
% add_mset_replicate_mset_safe
thf(fact_4162_mset__single__cases,axiom,
! [A: $tType,S: A,C2: multiset @ A,R4: A,C6: multiset @ A] :
( ( ( add_mset @ A @ S @ C2 )
= ( add_mset @ A @ R4 @ C6 ) )
=> ( ( ( S = R4 )
=> ( C2 != C6 ) )
=> ~ ( ( C6
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ C6 @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
=> ( ( C2
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ R4 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ C2 @ ( add_mset @ A @ R4 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
=> ( ( minus_minus @ ( multiset @ A ) @ C2 @ ( add_mset @ A @ R4 @ ( zero_zero @ ( multiset @ A ) ) ) )
!= ( minus_minus @ ( multiset @ A ) @ C6 @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ) ) ).
% mset_single_cases
thf(fact_4163_mset__single__cases_H,axiom,
! [A: $tType,S: A,C2: multiset @ A,R4: A,C6: multiset @ A] :
( ( ( add_mset @ A @ S @ C2 )
= ( add_mset @ A @ R4 @ C6 ) )
=> ( ( ( S = R4 )
=> ( C2 != C6 ) )
=> ~ ! [Cc: multiset @ A] :
( ( C6
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) @ Cc ) )
=> ( ( C2
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ R4 @ ( zero_zero @ ( multiset @ A ) ) ) @ Cc ) )
=> ( ( ( minus_minus @ ( multiset @ A ) @ C6 @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) )
= Cc )
=> ( ( minus_minus @ ( multiset @ A ) @ C2 @ ( add_mset @ A @ R4 @ ( zero_zero @ ( multiset @ A ) ) ) )
!= Cc ) ) ) ) ) ) ).
% mset_single_cases'
thf(fact_4164_mset__single__cases2,axiom,
! [A: $tType,S: A,C2: multiset @ A,R4: A,C6: multiset @ A] :
( ( ( add_mset @ A @ S @ C2 )
= ( add_mset @ A @ R4 @ C6 ) )
=> ( ( ( S = R4 )
=> ( C2 != C6 ) )
=> ~ ( ( C6
= ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ C6 @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) ) @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) ) )
=> ( ( C2
= ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ C2 @ ( add_mset @ A @ R4 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ ( add_mset @ A @ R4 @ ( zero_zero @ ( multiset @ A ) ) ) ) )
=> ( ( minus_minus @ ( multiset @ A ) @ C2 @ ( add_mset @ A @ R4 @ ( zero_zero @ ( multiset @ A ) ) ) )
!= ( minus_minus @ ( multiset @ A ) @ C6 @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ) ) ).
% mset_single_cases2
thf(fact_4165_mset__single__cases2_H,axiom,
! [A: $tType,S: A,C2: multiset @ A,R4: A,C6: multiset @ A] :
( ( ( add_mset @ A @ S @ C2 )
= ( add_mset @ A @ R4 @ C6 ) )
=> ( ( ( S = R4 )
=> ( C2 != C6 ) )
=> ~ ! [Cc: multiset @ A] :
( ( C6
= ( plus_plus @ ( multiset @ A ) @ Cc @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) ) )
=> ( ( C2
= ( plus_plus @ ( multiset @ A ) @ Cc @ ( add_mset @ A @ R4 @ ( zero_zero @ ( multiset @ A ) ) ) ) )
=> ( ( ( minus_minus @ ( multiset @ A ) @ C6 @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) )
= Cc )
=> ( ( minus_minus @ ( multiset @ A ) @ C2 @ ( add_mset @ A @ R4 @ ( zero_zero @ ( multiset @ A ) ) ) )
!= Cc ) ) ) ) ) ) ).
% mset_single_cases2'
thf(fact_4166_mset__unplusm__dist__cases,axiom,
! [A: $tType,S: A,A4: multiset @ A,B6: multiset @ A,C5: multiset @ A] :
( ( ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) @ A4 )
= ( plus_plus @ ( multiset @ A ) @ B6 @ C5 ) )
=> ( ( ( B6
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ B6 @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
=> ( A4
!= ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ B6 @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) ) @ C5 ) ) )
=> ~ ( ( C5
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ C5 @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
=> ( A4
!= ( plus_plus @ ( multiset @ A ) @ B6 @ ( minus_minus @ ( multiset @ A ) @ C5 @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ) ) ).
% mset_unplusm_dist_cases
thf(fact_4167_mset__unplusm__dist__cases2,axiom,
! [A: $tType,B6: multiset @ A,C5: multiset @ A,S: A,A4: multiset @ A] :
( ( ( plus_plus @ ( multiset @ A ) @ B6 @ C5 )
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) @ A4 ) )
=> ( ( ( B6
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ B6 @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
=> ( A4
!= ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ B6 @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) ) @ C5 ) ) )
=> ~ ( ( C5
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ C5 @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
=> ( A4
!= ( plus_plus @ ( multiset @ A ) @ B6 @ ( minus_minus @ ( multiset @ A ) @ C5 @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ) ) ).
% mset_unplusm_dist_cases2
thf(fact_4168_image__mset_Ocompositionality,axiom,
! [B: $tType,C: $tType,A: $tType,F: B > C,G: A > B,Multiset: multiset @ A] :
( ( image_mset @ B @ C @ F @ ( image_mset @ A @ B @ G @ Multiset ) )
= ( image_mset @ A @ C @ ( comp @ B @ C @ A @ F @ G ) @ Multiset ) ) ).
% image_mset.compositionality
thf(fact_4169_multiset_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,G: B > C,F: A > B,V: multiset @ A] :
( ( image_mset @ B @ C @ G @ ( image_mset @ A @ B @ F @ V ) )
= ( image_mset @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ V ) ) ).
% multiset.map_comp
thf(fact_4170_image__mset_Ocomp,axiom,
! [C: $tType,B: $tType,A: $tType,F: B > C,G: A > B] :
( ( comp @ ( multiset @ B ) @ ( multiset @ C ) @ ( multiset @ A ) @ ( image_mset @ B @ C @ F ) @ ( image_mset @ A @ B @ G ) )
= ( image_mset @ A @ C @ ( comp @ B @ C @ A @ F @ G ) ) ) ).
% image_mset.comp
thf(fact_4171_mset__map__id,axiom,
! [B: $tType,A: $tType,F: B > A,G: A > B,X10: multiset @ A] :
( ! [X2: A] :
( ( F @ ( G @ X2 ) )
= X2 )
=> ( ( image_mset @ B @ A @ F @ ( image_mset @ A @ B @ G @ X10 ) )
= X10 ) ) ).
% mset_map_id
thf(fact_4172_mset__union__2__elem,axiom,
! [A: $tType,A3: A,B2: A,C2: A,M3: multiset @ A] :
( ( ( add_mset @ A @ A3 @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) )
= ( add_mset @ A @ C2 @ M3 ) )
=> ( ( ( ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) )
= M3 )
& ( B2 = C2 ) )
| ( ( A3 = C2 )
& ( ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) )
= M3 ) ) ) ) ).
% mset_union_2_elem
thf(fact_4173_wcount__add__mset,axiom,
! [A: $tType,F: A > nat,X: A,M3: multiset @ A,A3: A] :
( ( wcount @ A @ F @ ( add_mset @ A @ X @ M3 ) @ A3 )
= ( plus_plus @ nat @ ( if @ nat @ ( X = A3 ) @ ( suc @ ( F @ A3 ) ) @ ( zero_zero @ nat ) ) @ ( wcount @ A @ F @ M3 @ A3 ) ) ) ).
% wcount_add_mset
thf(fact_4174_subset__mset_Osum__list__update,axiom,
! [A: $tType,K: nat,Xs: list @ ( multiset @ A ),X: multiset @ A] :
( ( ord_less @ nat @ K @ ( size_size @ ( list @ ( multiset @ A ) ) @ Xs ) )
=> ( ( groups4543113879258116180m_list @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ ( list_update @ ( multiset @ A ) @ Xs @ K @ X ) )
= ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ ( groups4543113879258116180m_list @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ Xs ) @ X ) @ ( nth @ ( multiset @ A ) @ Xs @ K ) ) ) ) ).
% subset_mset.sum_list_update
thf(fact_4175_size__multiset__add__mset,axiom,
! [A: $tType,F: A > nat,A3: A,M3: multiset @ A] :
( ( size_multiset @ A @ F @ ( add_mset @ A @ A3 @ M3 ) )
= ( plus_plus @ nat @ ( suc @ ( F @ A3 ) ) @ ( size_multiset @ A @ F @ M3 ) ) ) ).
% size_multiset_add_mset
thf(fact_4176_subset__mset_Osum__eq__0__iff,axiom,
! [A: $tType,B: $tType,F6: set @ B,F: B > ( multiset @ A )] :
( ( finite_finite2 @ B @ F6 )
=> ( ( ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F @ F6 )
= ( zero_zero @ ( multiset @ A ) ) )
= ( ! [X3: B] :
( ( member @ B @ X3 @ F6 )
=> ( ( F @ X3 )
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ).
% subset_mset.sum_eq_0_iff
thf(fact_4177_size__multiset__union,axiom,
! [A: $tType,F: A > nat,M3: multiset @ A,N6: multiset @ A] :
( ( size_multiset @ A @ F @ ( plus_plus @ ( multiset @ A ) @ M3 @ N6 ) )
= ( plus_plus @ nat @ ( size_multiset @ A @ F @ M3 ) @ ( size_multiset @ A @ F @ N6 ) ) ) ).
% size_multiset_union
thf(fact_4178_subset__mset_Osum__list__eq__0__iff,axiom,
! [A: $tType,Ns: list @ ( multiset @ A )] :
( ( ( groups4543113879258116180m_list @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ Ns )
= ( zero_zero @ ( multiset @ A ) ) )
= ( ! [X3: multiset @ A] :
( ( member @ ( multiset @ A ) @ X3 @ ( set2 @ ( multiset @ A ) @ Ns ) )
=> ( X3
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ).
% subset_mset.sum_list_eq_0_iff
thf(fact_4179_wcount__union,axiom,
! [A: $tType,F: A > nat,M3: multiset @ A,N6: multiset @ A,A3: A] :
( ( wcount @ A @ F @ ( plus_plus @ ( multiset @ A ) @ M3 @ N6 ) @ A3 )
= ( plus_plus @ nat @ ( wcount @ A @ F @ M3 @ A3 ) @ ( wcount @ A @ F @ N6 @ A3 ) ) ) ).
% wcount_union
thf(fact_4180_subset__mset_Oelem__le__sum__list,axiom,
! [A: $tType,K: nat,Ns: list @ ( multiset @ A )] :
( ( ord_less @ nat @ K @ ( size_size @ ( list @ ( multiset @ A ) ) @ Ns ) )
=> ( subseteq_mset @ A @ ( nth @ ( multiset @ A ) @ Ns @ K ) @ ( groups4543113879258116180m_list @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ Ns ) ) ) ).
% subset_mset.elem_le_sum_list
thf(fact_4181_subset__mset_Osum__mono2,axiom,
! [A: $tType,B: $tType,B6: set @ B,A4: set @ B,F: B > ( multiset @ A )] :
( ( finite_finite2 @ B @ B6 )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ B6 )
=> ( ! [B3: B] :
( ( member @ B @ B3 @ ( minus_minus @ ( set @ B ) @ B6 @ A4 ) )
=> ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( F @ B3 ) ) )
=> ( subseteq_mset @ A @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F @ A4 ) @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F @ B6 ) ) ) ) ) ).
% subset_mset.sum_mono2
thf(fact_4182_next_Osimps,axiom,
! [V: code_natural,W: code_natural] :
( ( next @ ( product_Pair @ code_natural @ code_natural @ V @ W ) )
= ( product_Pair @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( plus_plus @ code_natural @ ( minus_shift @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( minus_shift @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_times @ code_natural @ ( modulo_modulo @ code_natural @ V @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_times @ code_natural @ ( divide_divide @ code_natural @ V @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( plus_plus @ code_natural @ ( minus_shift @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_times @ code_natural @ ( modulo_modulo @ code_natural @ W @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_times @ code_natural @ ( divide_divide @ code_natural @ W @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( one_one @ code_natural ) ) ) @ ( one_one @ code_natural ) ) @ ( product_Pair @ code_natural @ code_natural @ ( minus_shift @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_times @ code_natural @ ( modulo_modulo @ code_natural @ V @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_times @ code_natural @ ( divide_divide @ code_natural @ V @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( minus_shift @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_times @ code_natural @ ( modulo_modulo @ code_natural @ W @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_times @ code_natural @ ( divide_divide @ code_natural @ W @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% next.simps
thf(fact_4183_find__dropWhile,axiom,
! [A: $tType] :
( ( find @ A )
= ( ^ [P4: A > $o,Xs3: list @ A] :
( case_list @ ( option @ A ) @ A @ ( none @ A )
@ ^ [X3: A,Xa4: list @ A] : ( some @ A @ X3 )
@ ( dropWhile @ A @ ( comp @ $o @ $o @ A @ (~) @ P4 ) @ Xs3 ) ) ) ) ).
% find_dropWhile
thf(fact_4184_mset__subset__eq__mono__add__right__cancel,axiom,
! [A: $tType,A4: multiset @ A,C5: multiset @ A,B6: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A4 @ C5 ) @ ( plus_plus @ ( multiset @ A ) @ B6 @ C5 ) )
= ( subseteq_mset @ A @ A4 @ B6 ) ) ).
% mset_subset_eq_mono_add_right_cancel
thf(fact_4185_mset__subset__eq__mono__add__left__cancel,axiom,
! [A: $tType,C5: multiset @ A,A4: multiset @ A,B6: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ C5 @ A4 ) @ ( plus_plus @ ( multiset @ A ) @ C5 @ B6 ) )
= ( subseteq_mset @ A @ A4 @ B6 ) ) ).
% mset_subset_eq_mono_add_left_cancel
thf(fact_4186_subset__mset_Oadd__le__cancel__right,axiom,
! [A: $tType,A3: multiset @ A,C2: multiset @ A,B2: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ C2 ) @ ( plus_plus @ ( multiset @ A ) @ B2 @ C2 ) )
= ( subseteq_mset @ A @ A3 @ B2 ) ) ).
% subset_mset.add_le_cancel_right
thf(fact_4187_subset__mset_Oadd__le__cancel__left,axiom,
! [A: $tType,C2: multiset @ A,A3: multiset @ A,B2: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ C2 @ A3 ) @ ( plus_plus @ ( multiset @ A ) @ C2 @ B2 ) )
= ( subseteq_mset @ A @ A3 @ B2 ) ) ).
% subset_mset.add_le_cancel_left
thf(fact_4188_subset__mset_Ole__add__same__cancel2,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
( ( subseteq_mset @ A @ A3 @ ( plus_plus @ ( multiset @ A ) @ B2 @ A3 ) )
= ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ B2 ) ) ).
% subset_mset.le_add_same_cancel2
thf(fact_4189_subset__mset_Ole__add__same__cancel1,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
( ( subseteq_mset @ A @ A3 @ ( plus_plus @ ( multiset @ A ) @ A3 @ B2 ) )
= ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ B2 ) ) ).
% subset_mset.le_add_same_cancel1
thf(fact_4190_subset__mset_Oadd__le__same__cancel2,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ B2 ) @ B2 )
= ( subseteq_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ).
% subset_mset.add_le_same_cancel2
thf(fact_4191_subset__mset_Oadd__le__same__cancel1,axiom,
! [A: $tType,B2: multiset @ A,A3: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ B2 @ A3 ) @ B2 )
= ( subseteq_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ).
% subset_mset.add_le_same_cancel1
thf(fact_4192_mset__subset__eq__multiset__union__diff__commute,axiom,
! [A: $tType,B6: multiset @ A,A4: multiset @ A,C5: multiset @ A] :
( ( subseteq_mset @ A @ B6 @ A4 )
=> ( ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ A4 @ B6 ) @ C5 )
= ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ A4 @ C5 ) @ B6 ) ) ) ).
% mset_subset_eq_multiset_union_diff_commute
thf(fact_4193_subset__mset_Oadd__diff__assoc2,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
( ( subseteq_mset @ A @ A3 @ B2 )
=> ( ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ B2 @ A3 ) @ C2 )
= ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ B2 @ C2 ) @ A3 ) ) ) ).
% subset_mset.add_diff_assoc2
thf(fact_4194_subset__mset_Oadd__diff__assoc,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
( ( subseteq_mset @ A @ A3 @ B2 )
=> ( ( plus_plus @ ( multiset @ A ) @ C2 @ ( minus_minus @ ( multiset @ A ) @ B2 @ A3 ) )
= ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ C2 @ B2 ) @ A3 ) ) ) ).
% subset_mset.add_diff_assoc
thf(fact_4195_subset__mset_Odiff__add,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
( ( subseteq_mset @ A @ A3 @ B2 )
=> ( ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ B2 @ A3 ) @ A3 )
= B2 ) ) ).
% subset_mset.diff_add
thf(fact_4196_mset__le__add__mset__decr__left1,axiom,
! [A: $tType,C2: A,A3: multiset @ A,B2: multiset @ A] :
( ( subseteq_mset @ A @ ( add_mset @ A @ C2 @ A3 ) @ B2 )
=> ( subseteq_mset @ A @ A3 @ B2 ) ) ).
% mset_le_add_mset_decr_left1
thf(fact_4197_mset__le__incr__right2,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
( ( subseteq_mset @ A @ A3 @ B2 )
=> ( subseteq_mset @ A @ A3 @ ( plus_plus @ ( multiset @ A ) @ C2 @ B2 ) ) ) ).
% mset_le_incr_right2
thf(fact_4198_mset__le__incr__right1,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
( ( subseteq_mset @ A @ A3 @ B2 )
=> ( subseteq_mset @ A @ A3 @ ( plus_plus @ ( multiset @ A ) @ B2 @ C2 ) ) ) ).
% mset_le_incr_right1
thf(fact_4199_mset__le__decr__left2,axiom,
! [A: $tType,C2: multiset @ A,A3: multiset @ A,B2: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ C2 @ A3 ) @ B2 )
=> ( subseteq_mset @ A @ A3 @ B2 ) ) ).
% mset_le_decr_left2
thf(fact_4200_mset__le__decr__left1,axiom,
! [A: $tType,A3: multiset @ A,C2: multiset @ A,B2: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ C2 ) @ B2 )
=> ( subseteq_mset @ A @ A3 @ B2 ) ) ).
% mset_le_decr_left1
thf(fact_4201_mset__union__subset,axiom,
! [A: $tType,A4: multiset @ A,B6: multiset @ A,C5: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) @ C5 )
=> ( ( subseteq_mset @ A @ A4 @ C5 )
& ( subseteq_mset @ A @ B6 @ C5 ) ) ) ).
% mset_union_subset
thf(fact_4202_mset__le__distrib,axiom,
! [A: $tType,X10: multiset @ A,A4: multiset @ A,B6: multiset @ A] :
( ( subseteq_mset @ A @ X10 @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) )
=> ~ ! [Xa5: multiset @ A,Xb4: multiset @ A] :
( ( X10
= ( plus_plus @ ( multiset @ A ) @ Xa5 @ Xb4 ) )
=> ( ( subseteq_mset @ A @ Xa5 @ A4 )
=> ~ ( subseteq_mset @ A @ Xb4 @ B6 ) ) ) ) ).
% mset_le_distrib
thf(fact_4203_mset__le__addE,axiom,
! [A: $tType,Xs: multiset @ A,Ys2: multiset @ A] :
( ( subseteq_mset @ A @ Xs @ Ys2 )
=> ~ ! [Zs2: multiset @ A] :
( Ys2
!= ( plus_plus @ ( multiset @ A ) @ Xs @ Zs2 ) ) ) ).
% mset_le_addE
thf(fact_4204_mset__subset__eq__exists__conv,axiom,
! [A: $tType] :
( ( subseteq_mset @ A )
= ( ^ [A9: multiset @ A,B9: multiset @ A] :
? [C7: multiset @ A] :
( B9
= ( plus_plus @ ( multiset @ A ) @ A9 @ C7 ) ) ) ) ).
% mset_subset_eq_exists_conv
thf(fact_4205_mset__subset__eq__add__right,axiom,
! [A: $tType,B6: multiset @ A,A4: multiset @ A] : ( subseteq_mset @ A @ B6 @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) ) ).
% mset_subset_eq_add_right
thf(fact_4206_mset__subset__eq__mono__add,axiom,
! [A: $tType,A4: multiset @ A,B6: multiset @ A,C5: multiset @ A,D4: multiset @ A] :
( ( subseteq_mset @ A @ A4 @ B6 )
=> ( ( subseteq_mset @ A @ C5 @ D4 )
=> ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A4 @ C5 ) @ ( plus_plus @ ( multiset @ A ) @ B6 @ D4 ) ) ) ) ).
% mset_subset_eq_mono_add
thf(fact_4207_mset__subset__eq__add__left,axiom,
! [A: $tType,A4: multiset @ A,B6: multiset @ A] : ( subseteq_mset @ A @ A4 @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) ) ).
% mset_subset_eq_add_left
thf(fact_4208_subset__mset_Oadd__le__imp__le__right,axiom,
! [A: $tType,A3: multiset @ A,C2: multiset @ A,B2: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ C2 ) @ ( plus_plus @ ( multiset @ A ) @ B2 @ C2 ) )
=> ( subseteq_mset @ A @ A3 @ B2 ) ) ).
% subset_mset.add_le_imp_le_right
thf(fact_4209_subset__mset_Oadd__le__imp__le__left,axiom,
! [A: $tType,C2: multiset @ A,A3: multiset @ A,B2: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ C2 @ A3 ) @ ( plus_plus @ ( multiset @ A ) @ C2 @ B2 ) )
=> ( subseteq_mset @ A @ A3 @ B2 ) ) ).
% subset_mset.add_le_imp_le_left
thf(fact_4210_subset__mset_Oadd__right__mono,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
( ( subseteq_mset @ A @ A3 @ B2 )
=> ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ C2 ) @ ( plus_plus @ ( multiset @ A ) @ B2 @ C2 ) ) ) ).
% subset_mset.add_right_mono
thf(fact_4211_subset__mset_Oadd__left__mono,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
( ( subseteq_mset @ A @ A3 @ B2 )
=> ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ C2 @ A3 ) @ ( plus_plus @ ( multiset @ A ) @ C2 @ B2 ) ) ) ).
% subset_mset.add_left_mono
thf(fact_4212_subset__mset_Ole__iff__add,axiom,
! [A: $tType] :
( ( subseteq_mset @ A )
= ( ^ [A7: multiset @ A,B5: multiset @ A] :
? [C3: multiset @ A] :
( B5
= ( plus_plus @ ( multiset @ A ) @ A7 @ C3 ) ) ) ) ).
% subset_mset.le_iff_add
thf(fact_4213_subset__mset_Oless__eqE,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
( ( subseteq_mset @ A @ A3 @ B2 )
=> ~ ! [C4: multiset @ A] :
( B2
!= ( plus_plus @ ( multiset @ A ) @ A3 @ C4 ) ) ) ).
% subset_mset.less_eqE
thf(fact_4214_subset__mset_Oadd__mono,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A,D3: multiset @ A] :
( ( subseteq_mset @ A @ A3 @ B2 )
=> ( ( subseteq_mset @ A @ C2 @ D3 )
=> ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ C2 ) @ ( plus_plus @ ( multiset @ A ) @ B2 @ D3 ) ) ) ) ).
% subset_mset.add_mono
thf(fact_4215_full__exhaustive__natural_H_Ocases,axiom,
! [X: product_prod @ ( ( product_prod @ code_natural @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_natural @ code_natural )] :
~ ! [F4: ( product_prod @ code_natural @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D2: code_natural,I2: code_natural] :
( X
!= ( product_Pair @ ( ( product_prod @ code_natural @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_natural @ code_natural ) @ F4 @ ( product_Pair @ code_natural @ code_natural @ D2 @ I2 ) ) ) ).
% full_exhaustive_natural'.cases
thf(fact_4216_full__exhaustive__fun_H_Ocases,axiom,
! [A: $tType,B: $tType] :
( ( ( quickc3360725361186068524ustive @ B )
& ( cl_HOL_Oequal @ A )
& ( quickc3360725361186068524ustive @ A ) )
=> ! [X: product_prod @ ( ( product_prod @ ( A > B ) @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_natural @ code_natural )] :
~ ! [F4: ( product_prod @ ( A > B ) @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),I2: code_natural,D2: code_natural] :
( X
!= ( product_Pair @ ( ( product_prod @ ( A > B ) @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_natural @ code_natural ) @ F4 @ ( product_Pair @ code_natural @ code_natural @ I2 @ D2 ) ) ) ) ).
% full_exhaustive_fun'.cases
thf(fact_4217_exhaustive__natural_H_Ocases,axiom,
! [X: product_prod @ ( code_natural > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_natural @ code_natural )] :
~ ! [F4: code_natural > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D2: code_natural,I2: code_natural] :
( X
!= ( product_Pair @ ( code_natural > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_natural @ code_natural ) @ F4 @ ( product_Pair @ code_natural @ code_natural @ D2 @ I2 ) ) ) ).
% exhaustive_natural'.cases
thf(fact_4218_exhaustive__fun_H_Ocases,axiom,
! [A: $tType,B: $tType] :
( ( ( quickc658316121487927005ustive @ B )
& ( cl_HOL_Oequal @ A )
& ( quickc658316121487927005ustive @ A ) )
=> ! [X: product_prod @ ( ( A > B ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_natural @ code_natural )] :
~ ! [F4: ( A > B ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),I2: code_natural,D2: code_natural] :
( X
!= ( product_Pair @ ( ( A > B ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_natural @ code_natural ) @ F4 @ ( product_Pair @ code_natural @ code_natural @ I2 @ D2 ) ) ) ) ).
% exhaustive_fun'.cases
thf(fact_4219_Lazy__Sequence_Oiterate__upto_Ocases,axiom,
! [A: $tType,X: product_prod @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural )] :
~ ! [F4: code_natural > A,N3: code_natural,M5: code_natural] :
( X
!= ( product_Pair @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) @ F4 @ ( product_Pair @ code_natural @ code_natural @ N3 @ M5 ) ) ) ).
% Lazy_Sequence.iterate_upto.cases
thf(fact_4220_iter_Ocases,axiom,
! [A: $tType,X: product_prod @ ( ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( product_prod @ A @ ( product_unit > code_term ) ) @ ( product_prod @ code_natural @ code_natural ) ) ) @ ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) )] :
~ ! [Random: ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( product_prod @ A @ ( product_unit > code_term ) ) @ ( product_prod @ code_natural @ code_natural ) ),Nrandom: code_natural,Seed: product_prod @ code_natural @ code_natural] :
( X
!= ( product_Pair @ ( ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( product_prod @ A @ ( product_unit > code_term ) ) @ ( product_prod @ code_natural @ code_natural ) ) ) @ ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) ) @ Random @ ( product_Pair @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ Nrandom @ Seed ) ) ) ).
% iter.cases
thf(fact_4221_log_Ocases,axiom,
! [X: product_prod @ code_natural @ code_natural] :
~ ! [B3: code_natural,I2: code_natural] :
( X
!= ( product_Pair @ code_natural @ code_natural @ B3 @ I2 ) ) ).
% log.cases
thf(fact_4222_mset__le__subtract,axiom,
! [A: $tType,A4: multiset @ A,B6: multiset @ A,C5: multiset @ A] :
( ( subseteq_mset @ A @ A4 @ B6 )
=> ( subseteq_mset @ A @ ( minus_minus @ ( multiset @ A ) @ A4 @ C5 ) @ ( minus_minus @ ( multiset @ A ) @ B6 @ C5 ) ) ) ).
% mset_le_subtract
thf(fact_4223_mset__le__add__mset__decr__left2,axiom,
! [A: $tType,C2: A,A3: multiset @ A,B2: multiset @ A] :
( ( subseteq_mset @ A @ ( add_mset @ A @ C2 @ A3 ) @ B2 )
=> ( subseteq_mset @ A @ ( add_mset @ A @ C2 @ ( zero_zero @ ( multiset @ A ) ) ) @ B2 ) ) ).
% mset_le_add_mset_decr_left2
thf(fact_4224_mset__le__single__cases,axiom,
! [A: $tType,M3: multiset @ A,A3: A] :
( ( subseteq_mset @ A @ M3 @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) )
=> ( ( M3
!= ( zero_zero @ ( multiset @ A ) ) )
=> ( M3
= ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ).
% mset_le_single_cases
thf(fact_4225_mset__le__add__mset,axiom,
! [A: $tType,X: A,B6: multiset @ A,C5: multiset @ A] :
( ( subseteq_mset @ A @ ( add_mset @ A @ X @ B6 ) @ C5 )
=> ( ( subseteq_mset @ A @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) @ C5 )
& ( subseteq_mset @ A @ B6 @ C5 ) ) ) ).
% mset_le_add_mset
thf(fact_4226_subset__mset_Oadd__nonpos__eq__0__iff,axiom,
! [A: $tType,X: multiset @ A,Y: multiset @ A] :
( ( subseteq_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) )
=> ( ( subseteq_mset @ A @ Y @ ( zero_zero @ ( multiset @ A ) ) )
=> ( ( ( plus_plus @ ( multiset @ A ) @ X @ Y )
= ( zero_zero @ ( multiset @ A ) ) )
= ( ( X
= ( zero_zero @ ( multiset @ A ) ) )
& ( Y
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ).
% subset_mset.add_nonpos_eq_0_iff
thf(fact_4227_subset__mset_Oadd__nonneg__eq__0__iff,axiom,
! [A: $tType,X: multiset @ A,Y: multiset @ A] :
( ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ X )
=> ( ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ Y )
=> ( ( ( plus_plus @ ( multiset @ A ) @ X @ Y )
= ( zero_zero @ ( multiset @ A ) ) )
= ( ( X
= ( zero_zero @ ( multiset @ A ) ) )
& ( Y
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ).
% subset_mset.add_nonneg_eq_0_iff
thf(fact_4228_subset__mset_Oadd__nonpos__nonpos,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
( ( subseteq_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) )
=> ( ( subseteq_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) )
=> ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ B2 ) @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% subset_mset.add_nonpos_nonpos
thf(fact_4229_subset__mset_Oadd__nonneg__nonneg,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
( ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ A3 )
=> ( ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ B2 )
=> ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( plus_plus @ ( multiset @ A ) @ A3 @ B2 ) ) ) ) ).
% subset_mset.add_nonneg_nonneg
thf(fact_4230_subset__mset_Oadd__increasing2,axiom,
! [A: $tType,C2: multiset @ A,B2: multiset @ A,A3: multiset @ A] :
( ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ C2 )
=> ( ( subseteq_mset @ A @ B2 @ A3 )
=> ( subseteq_mset @ A @ B2 @ ( plus_plus @ ( multiset @ A ) @ A3 @ C2 ) ) ) ) ).
% subset_mset.add_increasing2
thf(fact_4231_subset__mset_Oadd__decreasing2,axiom,
! [A: $tType,C2: multiset @ A,A3: multiset @ A,B2: multiset @ A] :
( ( subseteq_mset @ A @ C2 @ ( zero_zero @ ( multiset @ A ) ) )
=> ( ( subseteq_mset @ A @ A3 @ B2 )
=> ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ C2 ) @ B2 ) ) ) ).
% subset_mset.add_decreasing2
thf(fact_4232_subset__mset_Oadd__increasing,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
( ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ A3 )
=> ( ( subseteq_mset @ A @ B2 @ C2 )
=> ( subseteq_mset @ A @ B2 @ ( plus_plus @ ( multiset @ A ) @ A3 @ C2 ) ) ) ) ).
% subset_mset.add_increasing
thf(fact_4233_subset__mset_Oadd__decreasing,axiom,
! [A: $tType,A3: multiset @ A,C2: multiset @ A,B2: multiset @ A] :
( ( subseteq_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) )
=> ( ( subseteq_mset @ A @ C2 @ B2 )
=> ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ C2 ) @ B2 ) ) ) ).
% subset_mset.add_decreasing
thf(fact_4234_mset__le__subtract__left,axiom,
! [A: $tType,A4: multiset @ A,B6: multiset @ A,X10: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) @ X10 )
=> ( ( subseteq_mset @ A @ B6 @ ( minus_minus @ ( multiset @ A ) @ X10 @ A4 ) )
& ( subseteq_mset @ A @ A4 @ X10 ) ) ) ).
% mset_le_subtract_left
thf(fact_4235_mset__le__subtract__right,axiom,
! [A: $tType,A4: multiset @ A,B6: multiset @ A,X10: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) @ X10 )
=> ( ( subseteq_mset @ A @ A4 @ ( minus_minus @ ( multiset @ A ) @ X10 @ B6 ) )
& ( subseteq_mset @ A @ B6 @ X10 ) ) ) ).
% mset_le_subtract_right
thf(fact_4236_multiset__diff__union__assoc,axiom,
! [A: $tType,C5: multiset @ A,B6: multiset @ A,A4: multiset @ A] :
( ( subseteq_mset @ A @ C5 @ B6 )
=> ( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) @ C5 )
= ( plus_plus @ ( multiset @ A ) @ A4 @ ( minus_minus @ ( multiset @ A ) @ B6 @ C5 ) ) ) ) ).
% multiset_diff_union_assoc
thf(fact_4237_subset__eq__diff__conv,axiom,
! [A: $tType,A4: multiset @ A,C5: multiset @ A,B6: multiset @ A] :
( ( subseteq_mset @ A @ ( minus_minus @ ( multiset @ A ) @ A4 @ C5 ) @ B6 )
= ( subseteq_mset @ A @ A4 @ ( plus_plus @ ( multiset @ A ) @ B6 @ C5 ) ) ) ).
% subset_eq_diff_conv
thf(fact_4238_subset__mset_Ole__imp__diff__is__add,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
( ( subseteq_mset @ A @ A3 @ B2 )
=> ( ( subseteq_mset @ A @ A3 @ B2 )
=> ( ( ( minus_minus @ ( multiset @ A ) @ B2 @ A3 )
= C2 )
= ( B2
= ( plus_plus @ ( multiset @ A ) @ C2 @ A3 ) ) ) ) ) ).
% subset_mset.le_imp_diff_is_add
thf(fact_4239_subset__mset_Oadd__diff__inverse,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
( ( subseteq_mset @ A @ A3 @ B2 )
=> ( ( plus_plus @ ( multiset @ A ) @ A3 @ ( minus_minus @ ( multiset @ A ) @ B2 @ A3 ) )
= B2 ) ) ).
% subset_mset.add_diff_inverse
thf(fact_4240_subset__mset_Odiff__diff__right,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
( ( subseteq_mset @ A @ A3 @ B2 )
=> ( ( minus_minus @ ( multiset @ A ) @ C2 @ ( minus_minus @ ( multiset @ A ) @ B2 @ A3 ) )
= ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ C2 @ A3 ) @ B2 ) ) ) ).
% subset_mset.diff_diff_right
thf(fact_4241_subset__mset_Odiff__add__assoc2,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
( ( subseteq_mset @ A @ A3 @ B2 )
=> ( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ B2 @ C2 ) @ A3 )
= ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ B2 @ A3 ) @ C2 ) ) ) ).
% subset_mset.diff_add_assoc2
thf(fact_4242_subset__mset_Odiff__add__assoc,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
( ( subseteq_mset @ A @ A3 @ B2 )
=> ( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ C2 @ B2 ) @ A3 )
= ( plus_plus @ ( multiset @ A ) @ C2 @ ( minus_minus @ ( multiset @ A ) @ B2 @ A3 ) ) ) ) ).
% subset_mset.diff_add_assoc
thf(fact_4243_subset__mset_Ole__diff__conv2,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
( ( subseteq_mset @ A @ A3 @ B2 )
=> ( ( subseteq_mset @ A @ C2 @ ( minus_minus @ ( multiset @ A ) @ B2 @ A3 ) )
= ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ C2 @ A3 ) @ B2 ) ) ) ).
% subset_mset.le_diff_conv2
thf(fact_4244_subset__mset_Ole__add__diff,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
( ( subseteq_mset @ A @ A3 @ B2 )
=> ( subseteq_mset @ A @ C2 @ ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ B2 @ C2 ) @ A3 ) ) ) ).
% subset_mset.le_add_diff
thf(fact_4245_mset__map__split__orig__le,axiom,
! [B: $tType,A: $tType,F: B > A,P: multiset @ B,M13: multiset @ A,M24: multiset @ A] :
( ( subseteq_mset @ A @ ( image_mset @ B @ A @ F @ P ) @ ( plus_plus @ ( multiset @ A ) @ M13 @ M24 ) )
=> ~ ! [P1: multiset @ B,P22: multiset @ B] :
( ( P
= ( plus_plus @ ( multiset @ B ) @ P1 @ P22 ) )
=> ( ( subseteq_mset @ A @ ( image_mset @ B @ A @ F @ P1 ) @ M13 )
=> ~ ( subseteq_mset @ A @ ( image_mset @ B @ A @ F @ P22 ) @ M24 ) ) ) ) ).
% mset_map_split_orig_le
thf(fact_4246_mset__le__subtract__add__mset__right,axiom,
! [A: $tType,X: A,B6: multiset @ A,X10: multiset @ A] :
( ( subseteq_mset @ A @ ( add_mset @ A @ X @ B6 ) @ X10 )
=> ( ( subseteq_mset @ A @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ X10 @ B6 ) )
& ( subseteq_mset @ A @ B6 @ X10 ) ) ) ).
% mset_le_subtract_add_mset_right
thf(fact_4247_mset__le__subtract__add__mset__left,axiom,
! [A: $tType,X: A,B6: multiset @ A,X10: multiset @ A] :
( ( subseteq_mset @ A @ ( add_mset @ A @ X @ B6 ) @ X10 )
=> ( ( subseteq_mset @ A @ B6 @ ( minus_minus @ ( multiset @ A ) @ X10 @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) ) )
& ( subseteq_mset @ A @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) @ X10 ) ) ) ).
% mset_le_subtract_add_mset_left
thf(fact_4248_subset__mset_Osum__nonneg,axiom,
! [A: $tType,B: $tType,A4: set @ B,F: B > ( multiset @ A )] :
( ! [X2: B] :
( ( member @ B @ X2 @ A4 )
=> ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( F @ X2 ) ) )
=> ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F @ A4 ) ) ) ).
% subset_mset.sum_nonneg
thf(fact_4249_subset__mset_Osum__nonpos,axiom,
! [B: $tType,A: $tType,A4: set @ B,F: B > ( multiset @ A )] :
( ! [X2: B] :
( ( member @ B @ X2 @ A4 )
=> ( subseteq_mset @ A @ ( F @ X2 ) @ ( zero_zero @ ( multiset @ A ) ) ) )
=> ( subseteq_mset @ A @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F @ A4 ) @ ( zero_zero @ ( multiset @ A ) ) ) ) ).
% subset_mset.sum_nonpos
thf(fact_4250_subset__mset_Osum__list__nonneg,axiom,
! [A: $tType,Xs: list @ ( multiset @ A )] :
( ! [X2: multiset @ A] :
( ( member @ ( multiset @ A ) @ X2 @ ( set2 @ ( multiset @ A ) @ Xs ) )
=> ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ X2 ) )
=> ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( groups4543113879258116180m_list @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ Xs ) ) ) ).
% subset_mset.sum_list_nonneg
thf(fact_4251_subset__mset_Osum__list__nonpos,axiom,
! [A: $tType,Xs: list @ ( multiset @ A )] :
( ! [X2: multiset @ A] :
( ( member @ ( multiset @ A ) @ X2 @ ( set2 @ ( multiset @ A ) @ Xs ) )
=> ( subseteq_mset @ A @ X2 @ ( zero_zero @ ( multiset @ A ) ) ) )
=> ( subseteq_mset @ A @ ( groups4543113879258116180m_list @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ Xs ) @ ( zero_zero @ ( multiset @ A ) ) ) ) ).
% subset_mset.sum_list_nonpos
thf(fact_4252_subset__mset_Omember__le__sum__list,axiom,
! [A: $tType,X: multiset @ A,Xs: list @ ( multiset @ A )] :
( ( member @ ( multiset @ A ) @ X @ ( set2 @ ( multiset @ A ) @ Xs ) )
=> ( subseteq_mset @ A @ X @ ( groups4543113879258116180m_list @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ Xs ) ) ) ).
% subset_mset.member_le_sum_list
thf(fact_4253_subset__mset_Osum__list__nonneg__eq__0__iff,axiom,
! [A: $tType,Xs: list @ ( multiset @ A )] :
( ! [X2: multiset @ A] :
( ( member @ ( multiset @ A ) @ X2 @ ( set2 @ ( multiset @ A ) @ Xs ) )
=> ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ X2 ) )
=> ( ( ( groups4543113879258116180m_list @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ Xs )
= ( zero_zero @ ( multiset @ A ) ) )
= ( ! [X3: multiset @ A] :
( ( member @ ( multiset @ A ) @ X3 @ ( set2 @ ( multiset @ A ) @ Xs ) )
=> ( X3
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ).
% subset_mset.sum_list_nonneg_eq_0_iff
thf(fact_4254_subset__mset_Osum__mono,axiom,
! [A: $tType,B: $tType,K2: set @ B,F: B > ( multiset @ A ),G: B > ( multiset @ A )] :
( ! [I2: B] :
( ( member @ B @ I2 @ K2 )
=> ( subseteq_mset @ A @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( subseteq_mset @ A @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F @ K2 ) @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ G @ K2 ) ) ) ).
% subset_mset.sum_mono
thf(fact_4255_subset__mset_Osum__le__included,axiom,
! [B: $tType,A: $tType,C: $tType,S: set @ B,T2: set @ C,G: C > ( multiset @ A ),I: C > B,F: B > ( multiset @ A )] :
( ( finite_finite2 @ B @ S )
=> ( ( finite_finite2 @ C @ T2 )
=> ( ! [X2: C] :
( ( member @ C @ X2 @ T2 )
=> ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( G @ X2 ) ) )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ S )
=> ? [Xa2: C] :
( ( member @ C @ Xa2 @ T2 )
& ( ( I @ Xa2 )
= X2 )
& ( subseteq_mset @ A @ ( F @ X2 ) @ ( G @ Xa2 ) ) ) )
=> ( subseteq_mset @ A @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F @ S ) @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ C @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ G @ T2 ) ) ) ) ) ) ).
% subset_mset.sum_le_included
thf(fact_4256_subset__mset_Osum__nonneg__eq__0__iff,axiom,
! [A: $tType,B: $tType,A4: set @ B,F: B > ( multiset @ A )] :
( ( finite_finite2 @ B @ A4 )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ A4 )
=> ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( F @ X2 ) ) )
=> ( ( ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F @ A4 )
= ( zero_zero @ ( multiset @ A ) ) )
= ( ! [X3: B] :
( ( member @ B @ X3 @ A4 )
=> ( ( F @ X3 )
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ) ).
% subset_mset.sum_nonneg_eq_0_iff
thf(fact_4257_subset__mset_Osum__nonneg__0,axiom,
! [B: $tType,A: $tType,S: set @ B,F: B > ( multiset @ A ),I: B] :
( ( finite_finite2 @ B @ S )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ S )
=> ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( F @ I2 ) ) )
=> ( ( ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F @ S )
= ( zero_zero @ ( multiset @ A ) ) )
=> ( ( member @ B @ I @ S )
=> ( ( F @ I )
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ).
% subset_mset.sum_nonneg_0
thf(fact_4258_subset__mset_Osum__nonneg__leq__bound,axiom,
! [B: $tType,A: $tType,S: set @ B,F: B > ( multiset @ A ),B6: multiset @ A,I: B] :
( ( finite_finite2 @ B @ S )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ S )
=> ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( F @ I2 ) ) )
=> ( ( ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F @ S )
= B6 )
=> ( ( member @ B @ I @ S )
=> ( subseteq_mset @ A @ ( F @ I ) @ B6 ) ) ) ) ) ).
% subset_mset.sum_nonneg_leq_bound
thf(fact_4259_minus__shift__def,axiom,
( minus_shift
= ( ^ [R3: code_natural,K4: code_natural,L3: code_natural] : ( if @ code_natural @ ( ord_less @ code_natural @ K4 @ L3 ) @ ( minus_minus @ code_natural @ ( plus_plus @ code_natural @ R3 @ K4 ) @ L3 ) @ ( minus_minus @ code_natural @ K4 @ L3 ) ) ) ) ).
% minus_shift_def
thf(fact_4260_split__seed__def,axiom,
( split_seed
= ( ^ [S4: product_prod @ code_natural @ code_natural] :
( product_case_prod @ code_natural @ code_natural @ ( product_prod @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ code_natural @ code_natural ) )
@ ^ [V2: code_natural,W2: code_natural] :
( product_case_prod @ code_natural @ code_natural @ ( product_prod @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ code_natural @ code_natural ) )
@ ^ [V5: code_natural,W3: code_natural] : ( product_Pair @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ code_natural @ code_natural ) @ ( product_Pair @ code_natural @ code_natural @ ( inc_shift @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ V2 ) @ W3 ) @ ( product_Pair @ code_natural @ code_natural @ V5 @ ( inc_shift @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ W2 ) ) )
@ ( product_snd @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( next @ S4 ) ) )
@ S4 ) ) ) ).
% split_seed_def
thf(fact_4261_log_Osimps,axiom,
( log
= ( ^ [B5: code_natural,I3: code_natural] :
( if @ code_natural
@ ( ( ord_less_eq @ code_natural @ B5 @ ( one_one @ code_natural ) )
| ( ord_less @ code_natural @ I3 @ B5 ) )
@ ( one_one @ code_natural )
@ ( plus_plus @ code_natural @ ( one_one @ code_natural ) @ ( log @ B5 @ ( divide_divide @ code_natural @ I3 @ B5 ) ) ) ) ) ) ).
% log.simps
thf(fact_4262_log_Oelims,axiom,
! [X: code_natural,Xa: code_natural,Y: code_natural] :
( ( ( log @ X @ Xa )
= Y )
=> ( ( ( ( ord_less_eq @ code_natural @ X @ ( one_one @ code_natural ) )
| ( ord_less @ code_natural @ Xa @ X ) )
=> ( Y
= ( one_one @ code_natural ) ) )
& ( ~ ( ( ord_less_eq @ code_natural @ X @ ( one_one @ code_natural ) )
| ( ord_less @ code_natural @ Xa @ X ) )
=> ( Y
= ( plus_plus @ code_natural @ ( one_one @ code_natural ) @ ( log @ X @ ( divide_divide @ code_natural @ Xa @ X ) ) ) ) ) ) ) ).
% log.elims
thf(fact_4263_extract__def,axiom,
! [A: $tType] :
( ( extract @ A )
= ( ^ [P4: A > $o,Xs3: list @ A] :
( case_list @ ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) @ A @ ( none @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) )
@ ^ [Y3: A,Ys: list @ A] : ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( takeWhile @ A @ ( comp @ $o @ $o @ A @ (~) @ P4 ) @ Xs3 ) @ ( product_Pair @ A @ ( list @ A ) @ Y3 @ Ys ) ) )
@ ( dropWhile @ A @ ( comp @ $o @ $o @ A @ (~) @ P4 ) @ Xs3 ) ) ) ) ).
% extract_def
thf(fact_4264_drop__takeWhile,axiom,
! [A: $tType,I: nat,P: A > $o,L: list @ A] :
( ( ord_less_eq @ nat @ I @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ L ) ) )
=> ( ( drop @ A @ I @ ( takeWhile @ A @ P @ L ) )
= ( takeWhile @ A @ P @ ( drop @ A @ I @ L ) ) ) ) ).
% drop_takeWhile
thf(fact_4265_eq__len__takeWhile__conv,axiom,
! [A: $tType,I: nat,P: A > $o,L: list @ A] :
( ( I
= ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ L ) ) )
= ( ( ord_less_eq @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
& ! [J2: nat] :
( ( ord_less @ nat @ J2 @ I )
=> ( P @ ( nth @ A @ L @ J2 ) ) )
& ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ~ ( P @ ( nth @ A @ L @ I ) ) ) ) ) ).
% eq_len_takeWhile_conv
thf(fact_4266_less__length__takeWhile__conv,axiom,
! [A: $tType,I: nat,P: A > $o,L: list @ A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ L ) ) )
= ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
& ! [J2: nat] :
( ( ord_less_eq @ nat @ J2 @ I )
=> ( P @ ( nth @ A @ L @ J2 ) ) ) ) ) ).
% less_length_takeWhile_conv
thf(fact_4267_length__dropWhile__takeWhile,axiom,
! [A: $tType,X: nat,P: A > $o,Xs: list @ A] :
( ( ord_less @ nat @ X @ ( size_size @ ( list @ A ) @ ( dropWhile @ A @ P @ Xs ) ) )
=> ( ord_less @ nat @ ( plus_plus @ nat @ X @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% length_dropWhile_takeWhile
thf(fact_4268_dropWhile__nth,axiom,
! [A: $tType,J: nat,P: A > $o,Xs: list @ A] :
( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ ( dropWhile @ A @ P @ Xs ) ) )
=> ( ( nth @ A @ ( dropWhile @ A @ P @ Xs ) @ J )
= ( nth @ A @ Xs @ ( plus_plus @ nat @ J @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) ) ) ) ) ).
% dropWhile_nth
thf(fact_4269_inc__shift__def,axiom,
( inc_shift
= ( ^ [V2: code_natural,K4: code_natural] : ( if @ code_natural @ ( V2 = K4 ) @ ( one_one @ code_natural ) @ ( plus_plus @ code_natural @ K4 @ ( one_one @ code_natural ) ) ) ) ) ).
% inc_shift_def
thf(fact_4270_log_Opelims,axiom,
! [X: code_natural,Xa: code_natural,Y: code_natural] :
( ( ( log @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ code_natural @ code_natural ) @ log_rel @ ( product_Pair @ code_natural @ code_natural @ X @ Xa ) )
=> ~ ( ( ( ( ( ord_less_eq @ code_natural @ X @ ( one_one @ code_natural ) )
| ( ord_less @ code_natural @ Xa @ X ) )
=> ( Y
= ( one_one @ code_natural ) ) )
& ( ~ ( ( ord_less_eq @ code_natural @ X @ ( one_one @ code_natural ) )
| ( ord_less @ code_natural @ Xa @ X ) )
=> ( Y
= ( plus_plus @ code_natural @ ( one_one @ code_natural ) @ ( log @ X @ ( divide_divide @ code_natural @ Xa @ X ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ code_natural @ code_natural ) @ log_rel @ ( product_Pair @ code_natural @ code_natural @ X @ Xa ) ) ) ) ) ).
% log.pelims
thf(fact_4271_Random_Orange__def,axiom,
( range
= ( ^ [K4: code_natural] :
( product_scomp @ ( product_prod @ code_natural @ code_natural ) @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) )
@ ( iterate @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( log @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ K4 )
@ ^ [L3: code_natural] :
( product_scomp @ ( product_prod @ code_natural @ code_natural ) @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) ) @ next
@ ^ [V2: code_natural] : ( product_Pair @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( plus_plus @ code_natural @ V2 @ ( times_times @ code_natural @ L3 @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( one_one @ code_natural ) )
@ ^ [V2: code_natural] : ( product_Pair @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( modulo_modulo @ code_natural @ V2 @ K4 ) ) ) ) ) ).
% Random.range_def
thf(fact_4272_Predicate_Oiterate__upto_Opinduct,axiom,
! [A: $tType,A0: code_natural > A,A1: code_natural,A22: code_natural,P: ( code_natural > A ) > code_natural > code_natural > $o] :
( ( accp @ ( product_prod @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) ) @ ( iterate_upto_rel @ A ) @ ( product_Pair @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) @ A0 @ ( product_Pair @ code_natural @ code_natural @ A1 @ A22 ) ) )
=> ( ! [F4: code_natural > A,N3: code_natural,M5: code_natural] :
( ( accp @ ( product_prod @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) ) @ ( iterate_upto_rel @ A ) @ ( product_Pair @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) @ F4 @ ( product_Pair @ code_natural @ code_natural @ N3 @ M5 ) ) )
=> ( ! [X7: product_unit] :
( ~ ( ord_less @ code_natural @ M5 @ N3 )
=> ( P @ F4 @ ( plus_plus @ code_natural @ N3 @ ( one_one @ code_natural ) ) @ M5 ) )
=> ( P @ F4 @ N3 @ M5 ) ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% Predicate.iterate_upto.pinduct
thf(fact_4273_mset__size2elem,axiom,
! [A: $tType,P: multiset @ A,Q3: A,Q4: A] :
( ( ord_less_eq @ nat @ ( size_size @ ( multiset @ A ) @ P ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ Q3 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( add_mset @ A @ Q4 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ P )
=> ( P
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ Q3 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( add_mset @ A @ Q4 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ).
% mset_size2elem
thf(fact_4274_scomp__apply,axiom,
! [A: $tType,D: $tType,C: $tType,B: $tType] :
( ( product_scomp @ B @ C @ D @ A )
= ( ^ [F2: B > ( product_prod @ C @ D ),G2: C > D > A,X3: B] : ( product_case_prod @ C @ D @ A @ G2 @ ( F2 @ X3 ) ) ) ) ).
% scomp_apply
thf(fact_4275_size__union,axiom,
! [A: $tType,M3: multiset @ A,N6: multiset @ A] :
( ( size_size @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ M3 @ N6 ) )
= ( plus_plus @ nat @ ( size_size @ ( multiset @ A ) @ M3 ) @ ( size_size @ ( multiset @ A ) @ N6 ) ) ) ).
% size_union
thf(fact_4276_scomp__scomp,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F5: $tType,E: $tType,F: A > ( product_prod @ E @ F5 ),G: E > F5 > ( product_prod @ C @ D ),H3: C > D > B] :
( ( product_scomp @ A @ C @ D @ B @ ( product_scomp @ A @ E @ F5 @ ( product_prod @ C @ D ) @ F @ G ) @ H3 )
= ( product_scomp @ A @ E @ F5 @ B @ F
@ ^ [X3: E] : ( product_scomp @ F5 @ C @ D @ B @ ( G @ X3 ) @ H3 ) ) ) ).
% scomp_scomp
thf(fact_4277_scomp__Pair,axiom,
! [C: $tType,B: $tType,A: $tType,X: A > ( product_prod @ B @ C )] :
( ( product_scomp @ A @ B @ C @ ( product_prod @ B @ C ) @ X @ ( product_Pair @ B @ C ) )
= X ) ).
% scomp_Pair
thf(fact_4278_Pair__scomp,axiom,
! [A: $tType,B: $tType,C: $tType,X: C,F: C > A > B] :
( ( product_scomp @ A @ C @ A @ B @ ( product_Pair @ C @ A @ X ) @ F )
= ( F @ X ) ) ).
% Pair_scomp
thf(fact_4279_scomp__def,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType] :
( ( product_scomp @ A @ B @ C @ D )
= ( ^ [F2: A > ( product_prod @ B @ C ),G2: B > C > D,X3: A] : ( product_case_prod @ B @ C @ D @ G2 @ ( F2 @ X3 ) ) ) ) ).
% scomp_def
thf(fact_4280_size__1__singleton__mset,axiom,
! [A: $tType,M3: multiset @ A] :
( ( ( size_size @ ( multiset @ A ) @ M3 )
= ( one_one @ nat ) )
=> ? [A5: A] :
( M3
= ( add_mset @ A @ A5 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% size_1_singleton_mset
thf(fact_4281_size__single,axiom,
! [A: $tType,B2: A] :
( ( size_size @ ( multiset @ A ) @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) )
= ( one_one @ nat ) ) ).
% size_single
thf(fact_4282_iterate_Oelims,axiom,
! [A: $tType,B: $tType,X: code_natural,Xa: B > A > ( product_prod @ B @ A ),Xb: B,Y: A > ( product_prod @ B @ A )] :
( ( ( iterate @ B @ A @ X @ Xa @ Xb )
= Y )
=> ( ( ( X
= ( zero_zero @ code_natural ) )
=> ( Y
= ( product_Pair @ B @ A @ Xb ) ) )
& ( ( X
!= ( zero_zero @ code_natural ) )
=> ( Y
= ( product_scomp @ A @ B @ A @ ( product_prod @ B @ A ) @ ( Xa @ Xb ) @ ( iterate @ B @ A @ ( minus_minus @ code_natural @ X @ ( one_one @ code_natural ) ) @ Xa ) ) ) ) ) ) ).
% iterate.elims
thf(fact_4283_iterate_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( iterate @ B @ A )
= ( ^ [K4: code_natural,F2: B > A > ( product_prod @ B @ A ),X3: B] :
( if @ ( A > ( product_prod @ B @ A ) )
@ ( K4
= ( zero_zero @ code_natural ) )
@ ( product_Pair @ B @ A @ X3 )
@ ( product_scomp @ A @ B @ A @ ( product_prod @ B @ A ) @ ( F2 @ X3 ) @ ( iterate @ B @ A @ ( minus_minus @ code_natural @ K4 @ ( one_one @ code_natural ) ) @ F2 ) ) ) ) ) ).
% iterate.simps
thf(fact_4284_size__mset__SucE,axiom,
! [A: $tType,A4: multiset @ A,N: nat] :
( ( ( size_size @ ( multiset @ A ) @ A4 )
= ( suc @ N ) )
=> ~ ! [A5: A,B10: multiset @ A] :
( ( A4
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ A5 @ ( zero_zero @ ( multiset @ A ) ) ) @ B10 ) )
=> ( ( size_size @ ( multiset @ A ) @ B10 )
!= N ) ) ) ).
% size_mset_SucE
thf(fact_4285_scomp__unfold,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType] :
( ( product_scomp @ A @ B @ C @ D )
= ( ^ [F2: A > ( product_prod @ B @ C ),G2: B > C > D,X3: A] : ( G2 @ ( product_fst @ B @ C @ ( F2 @ X3 ) ) @ ( product_snd @ B @ C @ ( F2 @ X3 ) ) ) ) ) ).
% scomp_unfold
thf(fact_4286_mset__size__le1__cases,axiom,
! [A: $tType,M3: multiset @ A] :
( ( ord_less_eq @ nat @ ( size_size @ ( multiset @ A ) @ M3 ) @ ( suc @ ( zero_zero @ nat ) ) )
=> ( ( M3
!= ( zero_zero @ ( multiset @ A ) ) )
=> ~ ! [M5: A] :
( M3
!= ( add_mset @ A @ M5 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ).
% mset_size_le1_cases
thf(fact_4287_iterate_Opelims,axiom,
! [A: $tType,B: $tType,X: code_natural,Xa: B > A > ( product_prod @ B @ A ),Xb: B,Y: A > ( product_prod @ B @ A )] :
( ( ( iterate @ B @ A @ X @ Xa @ Xb )
= Y )
=> ( ( accp @ ( product_prod @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B ) ) @ ( iterate_rel @ B @ A ) @ ( product_Pair @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B ) @ X @ ( product_Pair @ ( B > A > ( product_prod @ B @ A ) ) @ B @ Xa @ Xb ) ) )
=> ~ ( ( ( ( X
= ( zero_zero @ code_natural ) )
=> ( Y
= ( product_Pair @ B @ A @ Xb ) ) )
& ( ( X
!= ( zero_zero @ code_natural ) )
=> ( Y
= ( product_scomp @ A @ B @ A @ ( product_prod @ B @ A ) @ ( Xa @ Xb ) @ ( iterate @ B @ A @ ( minus_minus @ code_natural @ X @ ( one_one @ code_natural ) ) @ Xa ) ) ) ) )
=> ~ ( accp @ ( product_prod @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B ) ) @ ( iterate_rel @ B @ A ) @ ( product_Pair @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B ) @ X @ ( product_Pair @ ( B > A > ( product_prod @ B @ A ) ) @ B @ Xa @ Xb ) ) ) ) ) ) ).
% iterate.pelims
thf(fact_4288_size__Diff__singleton__if,axiom,
! [A: $tType,X: A,A4: multiset @ A] :
( ( ( member @ A @ X @ ( set_mset @ A @ A4 ) )
=> ( ( size_size @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ A4 @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) ) )
= ( minus_minus @ nat @ ( size_size @ ( multiset @ A ) @ A4 ) @ ( one_one @ nat ) ) ) )
& ( ~ ( member @ A @ X @ ( set_mset @ A @ A4 ) )
=> ( ( size_size @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ A4 @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) ) )
= ( size_size @ ( multiset @ A ) @ A4 ) ) ) ) ).
% size_Diff_singleton_if
thf(fact_4289_size__Diff__singleton,axiom,
! [A: $tType,X: A,M3: multiset @ A] :
( ( member @ A @ X @ ( set_mset @ A @ M3 ) )
=> ( ( size_size @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ M3 @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) ) )
= ( minus_minus @ nat @ ( size_size @ ( multiset @ A ) @ M3 ) @ ( one_one @ nat ) ) ) ) ).
% size_Diff_singleton
thf(fact_4290_size__diff__se,axiom,
! [A: $tType,T2: A,S2: multiset @ A] :
( ( member @ A @ T2 @ ( set_mset @ A @ S2 ) )
=> ( ( size_size @ ( multiset @ A ) @ S2 )
= ( plus_plus @ nat @ ( size_size @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ S2 @ ( add_mset @ A @ T2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) @ ( one_one @ nat ) ) ) ) ).
% size_diff_se
thf(fact_4291_set__mset__union,axiom,
! [A: $tType,M3: multiset @ A,N6: multiset @ A] :
( ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ M3 @ N6 ) )
= ( sup_sup @ ( set @ A ) @ ( set_mset @ A @ M3 ) @ ( set_mset @ A @ N6 ) ) ) ).
% set_mset_union
thf(fact_4292_mset__diff__cancel1elem,axiom,
! [A: $tType,A3: A,B6: multiset @ A] :
( ~ ( member @ A @ A3 @ ( set_mset @ A @ B6 ) )
=> ( ( minus_minus @ ( multiset @ A ) @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) @ B6 )
= ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% mset_diff_cancel1elem
thf(fact_4293_ex__Melem__conv,axiom,
! [A: $tType,A4: multiset @ A] :
( ( ? [X3: A] : ( member @ A @ X3 @ ( set_mset @ A @ A4 ) ) )
= ( A4
!= ( zero_zero @ ( multiset @ A ) ) ) ) ).
% ex_Melem_conv
thf(fact_4294_union__iff,axiom,
! [A: $tType,A3: A,A4: multiset @ A,B6: multiset @ A] :
( ( member @ A @ A3 @ ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) ) )
= ( ( member @ A @ A3 @ ( set_mset @ A @ A4 ) )
| ( member @ A @ A3 @ ( set_mset @ A @ B6 ) ) ) ) ).
% union_iff
thf(fact_4295_mset__un__cases,axiom,
! [A: $tType,A3: A,A4: multiset @ A,B6: multiset @ A] :
( ( member @ A @ A3 @ ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) ) )
=> ( ~ ( member @ A @ A3 @ ( set_mset @ A @ A4 ) )
=> ( member @ A @ A3 @ ( set_mset @ A @ B6 ) ) ) ) ).
% mset_un_cases
thf(fact_4296_mset__left__cancel__union,axiom,
! [A: $tType,A3: A,A4: multiset @ A,B6: multiset @ A] :
( ( member @ A @ A3 @ ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) ) )
=> ( ~ ( member @ A @ A3 @ ( set_mset @ A @ A4 ) )
=> ( member @ A @ A3 @ ( set_mset @ A @ B6 ) ) ) ) ).
% mset_left_cancel_union
thf(fact_4297_mset__right__cancel__union,axiom,
! [A: $tType,A3: A,A4: multiset @ A,B6: multiset @ A] :
( ( member @ A @ A3 @ ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) ) )
=> ( ~ ( member @ A @ A3 @ ( set_mset @ A @ B6 ) )
=> ( member @ A @ A3 @ ( set_mset @ A @ A4 ) ) ) ) ).
% mset_right_cancel_union
thf(fact_4298_iterate_Ocases,axiom,
! [A: $tType,B: $tType,X: product_prod @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B )] :
~ ! [K5: code_natural,F4: B > A > ( product_prod @ B @ A ),X2: B] :
( X
!= ( product_Pair @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B ) @ K5 @ ( product_Pair @ ( B > A > ( product_prod @ B @ A ) ) @ B @ F4 @ X2 ) ) ) ).
% iterate.cases
thf(fact_4299_image__mset__cong__pair,axiom,
! [C: $tType,B: $tType,A: $tType,M3: multiset @ ( product_prod @ A @ B ),F: A > B > C,G: A > B > C] :
( ! [X2: A,Y4: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y4 ) @ ( set_mset @ ( product_prod @ A @ B ) @ M3 ) )
=> ( ( F @ X2 @ Y4 )
= ( G @ X2 @ Y4 ) ) )
=> ( ( image_mset @ ( product_prod @ A @ B ) @ C @ ( product_case_prod @ A @ B @ C @ F ) @ M3 )
= ( image_mset @ ( product_prod @ A @ B ) @ C @ ( product_case_prod @ A @ B @ C @ G ) @ M3 ) ) ) ).
% image_mset_cong_pair
thf(fact_4300_multi__member__skip,axiom,
! [A: $tType,X: A,XS: multiset @ A,Y: A] :
( ( member @ A @ X @ ( set_mset @ A @ XS ) )
=> ( member @ A @ X @ ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ Y @ ( zero_zero @ ( multiset @ A ) ) ) @ XS ) ) ) ) ).
% multi_member_skip
thf(fact_4301_multi__member__this,axiom,
! [A: $tType,X: A,XS: multiset @ A] : ( member @ A @ X @ ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) @ XS ) ) ) ).
% multi_member_this
thf(fact_4302_mset__right__cancel__elem,axiom,
! [A: $tType,A3: A,A4: multiset @ A,B2: A] :
( ( member @ A @ A3 @ ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A4 @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
=> ( ( A3 != B2 )
=> ( member @ A @ A3 @ ( set_mset @ A @ A4 ) ) ) ) ).
% mset_right_cancel_elem
thf(fact_4303_mset__left__cancel__elem,axiom,
! [A: $tType,A3: A,B2: A,A4: multiset @ A] :
( ( member @ A @ A3 @ ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) @ A4 ) ) )
=> ( ( A3 != B2 )
=> ( member @ A @ A3 @ ( set_mset @ A @ A4 ) ) ) ) ).
% mset_left_cancel_elem
thf(fact_4304_mset__le__mono__add__single,axiom,
! [A: $tType,A3: A,Ys2: multiset @ A,B2: A,Ws: multiset @ A] :
( ( member @ A @ A3 @ ( set_mset @ A @ Ys2 ) )
=> ( ( member @ A @ B2 @ ( set_mset @ A @ Ws ) )
=> ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ ( plus_plus @ ( multiset @ A ) @ Ys2 @ Ws ) ) ) ) ).
% mset_le_mono_add_single
thf(fact_4305_mset__union__subset__s,axiom,
! [A: $tType,A3: A,B6: multiset @ A,C5: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) @ B6 ) @ C5 )
=> ( ( member @ A @ A3 @ ( set_mset @ A @ C5 ) )
& ( subseteq_mset @ A @ B6 @ C5 ) ) ) ).
% mset_union_subset_s
thf(fact_4306_mset__2dist2__cases,axiom,
! [A: $tType,A3: A,B2: A,A4: multiset @ A,B6: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) )
=> ( ~ ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ A4 )
=> ( ~ ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ B6 )
=> ( ( ( member @ A @ A3 @ ( set_mset @ A @ A4 ) )
=> ~ ( member @ A @ B2 @ ( set_mset @ A @ B6 ) ) )
=> ~ ( ( member @ A @ A3 @ ( set_mset @ A @ B6 ) )
=> ~ ( member @ A @ B2 @ ( set_mset @ A @ A4 ) ) ) ) ) ) ) ).
% mset_2dist2_cases
thf(fact_4307_insert__DiffM2,axiom,
! [A: $tType,X: A,M3: multiset @ A] :
( ( member @ A @ X @ ( set_mset @ A @ M3 ) )
=> ( ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ M3 @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) ) @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) )
= M3 ) ) ).
% insert_DiffM2
thf(fact_4308_diff__union__single__conv,axiom,
! [A: $tType,A3: A,J4: multiset @ A,I4: multiset @ A] :
( ( member @ A @ A3 @ ( set_mset @ A @ J4 ) )
=> ( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ I4 @ J4 ) @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) )
= ( plus_plus @ ( multiset @ A ) @ I4 @ ( minus_minus @ ( multiset @ A ) @ J4 @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ).
% diff_union_single_conv
thf(fact_4309_mset__un__single__un__cases,axiom,
! [A: $tType,A3: A,A4: multiset @ A,B6: multiset @ A,C5: multiset @ A] :
( ( ( add_mset @ A @ A3 @ A4 )
= ( plus_plus @ ( multiset @ A ) @ B6 @ C5 ) )
=> ( ( ( member @ A @ A3 @ ( set_mset @ A @ B6 ) )
=> ( A4
!= ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ B6 @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ C5 ) ) )
=> ~ ( ( member @ A @ A3 @ ( set_mset @ A @ C5 ) )
=> ( A4
!= ( plus_plus @ ( multiset @ A ) @ B6 @ ( minus_minus @ ( multiset @ A ) @ C5 @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ) ) ).
% mset_un_single_un_cases
thf(fact_4310_diff__union__single__conv2,axiom,
! [A: $tType,A3: A,J4: multiset @ A,I4: multiset @ A] :
( ( member @ A @ A3 @ ( set_mset @ A @ J4 ) )
=> ( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ J4 @ I4 ) @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) )
= ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ J4 @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ I4 ) ) ) ).
% diff_union_single_conv2
thf(fact_4311_mset__union__diff__comm,axiom,
! [A: $tType,T2: A,S2: multiset @ A,T6: multiset @ A] :
( ( member @ A @ T2 @ ( set_mset @ A @ S2 ) )
=> ( ( plus_plus @ ( multiset @ A ) @ T6 @ ( minus_minus @ ( multiset @ A ) @ S2 @ ( add_mset @ A @ T2 @ ( zero_zero @ ( multiset @ A ) ) ) ) )
= ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ T6 @ S2 ) @ ( add_mset @ A @ T2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ).
% mset_union_diff_comm
thf(fact_4312_mset__contains__eq,axiom,
! [A: $tType,M2: A,M3: multiset @ A] :
( ( member @ A @ M2 @ ( set_mset @ A @ M3 ) )
= ( ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ M2 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ M3 @ ( add_mset @ A @ M2 @ ( zero_zero @ ( multiset @ A ) ) ) ) )
= M3 ) ) ).
% mset_contains_eq
thf(fact_4313_mset__size1elem,axiom,
! [A: $tType,P: multiset @ A,Q3: A] :
( ( ord_less_eq @ nat @ ( size_size @ ( multiset @ A ) @ P ) @ ( one_one @ nat ) )
=> ( ( member @ A @ Q3 @ ( set_mset @ A @ P ) )
=> ( P
= ( add_mset @ A @ Q3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ).
% mset_size1elem
thf(fact_4314_subset__mset_Osum__mset__mono,axiom,
! [A: $tType,B: $tType,K2: multiset @ B,F: B > ( multiset @ A ),G: B > ( multiset @ A )] :
( ! [I2: B] :
( ( member @ B @ I2 @ ( set_mset @ B @ K2 ) )
=> ( subseteq_mset @ A @ ( F @ I2 ) @ ( G @ I2 ) ) )
=> ( subseteq_mset @ A @ ( comm_monoid_sum_mset @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ ( image_mset @ B @ ( multiset @ A ) @ F @ K2 ) ) @ ( comm_monoid_sum_mset @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ ( image_mset @ B @ ( multiset @ A ) @ G @ K2 ) ) ) ) ).
% subset_mset.sum_mset_mono
thf(fact_4315_subset__mset_Osum__mset__0__iff,axiom,
! [A: $tType,M3: multiset @ ( multiset @ A )] :
( ( ( comm_monoid_sum_mset @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ M3 )
= ( zero_zero @ ( multiset @ A ) ) )
= ( ! [X3: multiset @ A] :
( ( member @ ( multiset @ A ) @ X3 @ ( set_mset @ ( multiset @ A ) @ M3 ) )
=> ( X3
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ).
% subset_mset.sum_mset_0_iff
thf(fact_4316_wmsI,axiom,
! [A4: multiset @ ( product_prod @ nat @ nat ),B6: multiset @ ( product_prod @ nat @ nat ),Z9: multiset @ ( product_prod @ nat @ nat )] :
( ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set_mset @ ( product_prod @ nat @ nat ) @ A4 ) @ ( set_mset @ ( product_prod @ nat @ nat ) @ B6 ) ) @ fun_max_strict )
| ( ( A4
= ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) )
& ( B6
= ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) ) )
=> ( member @ ( product_prod @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ Z9 @ A4 ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ Z9 @ B6 ) ) @ ms_weak ) ) ).
% wmsI
thf(fact_4317_smsI,axiom,
! [A4: multiset @ ( product_prod @ nat @ nat ),B6: multiset @ ( product_prod @ nat @ nat ),Z9: multiset @ ( product_prod @ nat @ nat )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set_mset @ ( product_prod @ nat @ nat ) @ A4 ) @ ( set_mset @ ( product_prod @ nat @ nat ) @ B6 ) ) @ fun_max_strict )
=> ( member @ ( product_prod @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ Z9 @ A4 ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ Z9 @ B6 ) ) @ ms_strict ) ) ).
% smsI
thf(fact_4318_ms__reduction__pair,axiom,
fun_reduction_pair @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) @ ( set @ ( product_prod @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) @ ms_strict @ ms_weak ) ).
% ms_reduction_pair
thf(fact_4319_ms__strictI,axiom,
! [Z9: multiset @ ( product_prod @ nat @ nat ),Z10: multiset @ ( product_prod @ nat @ nat ),A4: multiset @ ( product_prod @ nat @ nat ),B6: multiset @ ( product_prod @ nat @ nat )] :
( ( pw_leq @ Z9 @ Z10 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set_mset @ ( product_prod @ nat @ nat ) @ A4 ) @ ( set_mset @ ( product_prod @ nat @ nat ) @ B6 ) ) @ fun_max_strict )
=> ( member @ ( product_prod @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ Z9 @ A4 ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ Z10 @ B6 ) ) @ ms_strict ) ) ) ).
% ms_strictI
thf(fact_4320_ms__weakI1,axiom,
! [Z9: multiset @ ( product_prod @ nat @ nat ),Z10: multiset @ ( product_prod @ nat @ nat ),A4: multiset @ ( product_prod @ nat @ nat ),B6: multiset @ ( product_prod @ nat @ nat )] :
( ( pw_leq @ Z9 @ Z10 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set_mset @ ( product_prod @ nat @ nat ) @ A4 ) @ ( set_mset @ ( product_prod @ nat @ nat ) @ B6 ) ) @ fun_max_strict )
=> ( member @ ( product_prod @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ Z9 @ A4 ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ Z10 @ B6 ) ) @ ms_weak ) ) ) ).
% ms_weakI1
thf(fact_4321_pw__leq__split,axiom,
! [X10: multiset @ ( product_prod @ nat @ nat ),Y9: multiset @ ( product_prod @ nat @ nat )] :
( ( pw_leq @ X10 @ Y9 )
=> ? [A15: multiset @ ( product_prod @ nat @ nat ),B10: multiset @ ( product_prod @ nat @ nat ),Z11: multiset @ ( product_prod @ nat @ nat )] :
( ( X10
= ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ A15 @ Z11 ) )
& ( Y9
= ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ B10 @ Z11 ) )
& ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set_mset @ ( product_prod @ nat @ nat ) @ A15 ) @ ( set_mset @ ( product_prod @ nat @ nat ) @ B10 ) ) @ fun_max_strict )
| ( ( B10
= ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) )
& ( A15
= ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) ) ) ) ) ).
% pw_leq_split
thf(fact_4322_ms__weakI2,axiom,
! [Z9: multiset @ ( product_prod @ nat @ nat ),Z10: multiset @ ( product_prod @ nat @ nat )] :
( ( pw_leq @ Z9 @ Z10 )
=> ( member @ ( product_prod @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ Z9 @ ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ Z10 @ ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) ) @ ms_weak ) ) ).
% ms_weakI2
thf(fact_4323_pw__leq__lstep,axiom,
! [X: product_prod @ nat @ nat,Y: product_prod @ nat @ nat] :
( ( member @ ( 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 )
=> ( pw_leq @ ( add_mset @ ( product_prod @ nat @ nat ) @ X @ ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) @ ( add_mset @ ( product_prod @ nat @ nat ) @ Y @ ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) ) ) ).
% pw_leq_lstep
thf(fact_4324_pw__leq__step,axiom,
! [X: product_prod @ nat @ nat,Y: product_prod @ nat @ nat,X10: multiset @ ( product_prod @ nat @ nat ),Y9: multiset @ ( product_prod @ nat @ nat )] :
( ( member @ ( 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 )
=> ( ( pw_leq @ X10 @ Y9 )
=> ( pw_leq @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( add_mset @ ( product_prod @ nat @ nat ) @ X @ ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) @ X10 ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( add_mset @ ( product_prod @ nat @ nat ) @ Y @ ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) @ Y9 ) ) ) ) ).
% pw_leq_step
thf(fact_4325_pw__leq_Osimps,axiom,
( pw_leq
= ( ^ [A12: multiset @ ( product_prod @ nat @ nat ),A23: multiset @ ( product_prod @ nat @ nat )] :
( ( ( A12
= ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) )
& ( A23
= ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) )
| ? [X3: product_prod @ nat @ nat,Y3: product_prod @ nat @ nat,X8: multiset @ ( product_prod @ nat @ nat ),Y10: multiset @ ( product_prod @ nat @ nat )] :
( ( A12
= ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( add_mset @ ( product_prod @ nat @ nat ) @ X3 @ ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) @ X8 ) )
& ( A23
= ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( add_mset @ ( product_prod @ nat @ nat ) @ Y3 @ ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) @ Y10 ) )
& ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X3 @ Y3 ) @ fun_pair_leq )
& ( pw_leq @ X8 @ Y10 ) ) ) ) ) ).
% pw_leq.simps
thf(fact_4326_pw__leq_Ocases,axiom,
! [A1: multiset @ ( product_prod @ nat @ nat ),A22: multiset @ ( product_prod @ nat @ nat )] :
( ( pw_leq @ A1 @ A22 )
=> ( ( ( A1
= ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) )
=> ( A22
!= ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) )
=> ~ ! [X2: product_prod @ nat @ nat,Y4: product_prod @ nat @ nat,X15: multiset @ ( product_prod @ nat @ nat )] :
( ( A1
= ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( add_mset @ ( product_prod @ nat @ nat ) @ X2 @ ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) @ X15 ) )
=> ! [Y11: multiset @ ( product_prod @ nat @ nat )] :
( ( A22
= ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( add_mset @ ( product_prod @ nat @ nat ) @ Y4 @ ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) @ Y11 ) )
=> ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X2 @ Y4 ) @ fun_pair_leq )
=> ~ ( pw_leq @ X15 @ Y11 ) ) ) ) ) ) ).
% pw_leq.cases
thf(fact_4327_mult1E,axiom,
! [A: $tType,N6: multiset @ A,M3: multiset @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ N6 @ M3 ) @ ( mult1 @ A @ R ) )
=> ~ ! [A5: A,M0: multiset @ A] :
( ( M3
= ( add_mset @ A @ A5 @ M0 ) )
=> ! [K10: multiset @ A] :
( ( N6
= ( plus_plus @ ( multiset @ A ) @ M0 @ K10 ) )
=> ~ ! [B8: A] :
( ( member @ A @ B8 @ ( set_mset @ A @ K10 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B8 @ A5 ) @ R ) ) ) ) ) ).
% mult1E
thf(fact_4328_mult1I,axiom,
! [A: $tType,M3: multiset @ A,A3: A,M02: multiset @ A,N6: multiset @ A,K2: multiset @ A,R: set @ ( product_prod @ A @ A )] :
( ( M3
= ( add_mset @ A @ A3 @ M02 ) )
=> ( ( N6
= ( plus_plus @ ( multiset @ A ) @ M02 @ K2 ) )
=> ( ! [B3: A] :
( ( member @ A @ B3 @ ( set_mset @ A @ K2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A3 ) @ R ) )
=> ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ N6 @ M3 ) @ ( mult1 @ A @ R ) ) ) ) ) ).
% mult1I
thf(fact_4329_less__add,axiom,
! [A: $tType,N6: multiset @ A,A3: A,M02: multiset @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ N6 @ ( add_mset @ A @ A3 @ M02 ) ) @ ( mult1 @ A @ R ) )
=> ( ? [M9: multiset @ A] :
( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ M9 @ M02 ) @ ( mult1 @ A @ R ) )
& ( N6
= ( add_mset @ A @ A3 @ M9 ) ) )
| ? [K10: multiset @ A] :
( ! [B8: A] :
( ( member @ A @ B8 @ ( set_mset @ A @ K10 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B8 @ A3 ) @ R ) )
& ( N6
= ( plus_plus @ ( multiset @ A ) @ M02 @ K10 ) ) ) ) ) ).
% less_add
thf(fact_4330_one__step__implies__mult,axiom,
! [A: $tType,J4: multiset @ A,K2: multiset @ A,R: set @ ( product_prod @ A @ A ),I4: multiset @ A] :
( ( J4
!= ( zero_zero @ ( multiset @ A ) ) )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ ( set_mset @ A @ K2 ) )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ ( set_mset @ A @ J4 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Xa2 ) @ R ) ) )
=> ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ I4 @ K2 ) @ ( plus_plus @ ( multiset @ A ) @ I4 @ J4 ) ) @ ( mult @ A @ R ) ) ) ) ).
% one_step_implies_mult
thf(fact_4331_not__less__empty,axiom,
! [A: $tType,M3: multiset @ A,R: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ M3 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( mult1 @ A @ R ) ) ).
% not_less_empty
thf(fact_4332_mult1__union,axiom,
! [A: $tType,B6: multiset @ A,D4: multiset @ A,R: set @ ( product_prod @ A @ A ),C5: multiset @ A] :
( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ B6 @ D4 ) @ ( mult1 @ A @ R ) )
=> ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ C5 @ B6 ) @ ( plus_plus @ ( multiset @ A ) @ C5 @ D4 ) ) @ ( mult1 @ A @ R ) ) ) ).
% mult1_union
thf(fact_4333_mult1__lessE,axiom,
! [A: $tType,N6: multiset @ A,M3: multiset @ A,R: A > A > $o] :
( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ N6 @ M3 ) @ ( mult1 @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R ) ) ) )
=> ( ( asymp @ A @ R )
=> ~ ! [A5: A,M0: multiset @ A] :
( ( M3
= ( add_mset @ A @ A5 @ M0 ) )
=> ! [K10: multiset @ A] :
( ( N6
= ( plus_plus @ ( multiset @ A ) @ M0 @ K10 ) )
=> ( ~ ( member @ A @ A5 @ ( set_mset @ A @ K10 ) )
=> ~ ! [B8: A] :
( ( member @ A @ B8 @ ( set_mset @ A @ K10 ) )
=> ( R @ B8 @ A5 ) ) ) ) ) ) ) ).
% mult1_lessE
thf(fact_4334_in__mset__fold__plus__iff,axiom,
! [A: $tType,X: A,M3: multiset @ A,NN: multiset @ ( multiset @ A )] :
( ( member @ A @ X @ ( set_mset @ A @ ( fold_mset @ ( multiset @ A ) @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ M3 @ NN ) ) )
= ( ( member @ A @ X @ ( set_mset @ A @ M3 ) )
| ? [N9: multiset @ A] :
( ( member @ ( multiset @ A ) @ N9 @ ( set_mset @ ( multiset @ A ) @ NN ) )
& ( member @ A @ X @ ( set_mset @ A @ N9 ) ) ) ) ) ).
% in_mset_fold_plus_iff
thf(fact_4335_mult__implies__one__step,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),M3: multiset @ A,N6: multiset @ A] :
( ( trans @ A @ R )
=> ( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ M3 @ N6 ) @ ( mult @ A @ R ) )
=> ? [I8: multiset @ A,J5: multiset @ A] :
( ( N6
= ( plus_plus @ ( multiset @ A ) @ I8 @ J5 ) )
& ? [K10: multiset @ A] :
( ( M3
= ( plus_plus @ ( multiset @ A ) @ I8 @ K10 ) )
& ( J5
!= ( zero_zero @ ( multiset @ A ) ) )
& ! [X7: A] :
( ( member @ A @ X7 @ ( set_mset @ A @ K10 ) )
=> ? [Xa3: A] :
( ( member @ A @ Xa3 @ ( set_mset @ A @ J5 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X7 @ Xa3 ) @ R ) ) ) ) ) ) ) ).
% mult_implies_one_step
thf(fact_4336_mult1__def,axiom,
! [A: $tType] :
( ( mult1 @ A )
= ( ^ [R3: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) )
@ ( product_case_prod @ ( multiset @ A ) @ ( multiset @ A ) @ $o
@ ^ [N9: multiset @ A,M8: multiset @ A] :
? [A7: A,M03: multiset @ A,K3: multiset @ A] :
( ( M8
= ( add_mset @ A @ A7 @ M03 ) )
& ( N9
= ( plus_plus @ ( multiset @ A ) @ M03 @ K3 ) )
& ! [B5: A] :
( ( member @ A @ B5 @ ( set_mset @ A @ K3 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ A7 ) @ R3 ) ) ) ) ) ) ) ).
% mult1_def
thf(fact_4337_trans__def,axiom,
! [A: $tType] :
( ( trans @ A )
= ( ^ [R3: set @ ( product_prod @ A @ A )] :
! [X3: A,Y3: A,Z5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z5 ) @ R3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z5 ) @ R3 ) ) ) ) ) ).
% trans_def
thf(fact_4338_transI,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ! [X2: A,Y4: A,Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y4 ) @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z3 ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Z3 ) @ R ) ) )
=> ( trans @ A @ R ) ) ).
% transI
thf(fact_4339_transE,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),X: A,Y: A,Z2: A] :
( ( trans @ A @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ R ) ) ) ) ).
% transE
thf(fact_4340_transD,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),X: A,Y: A,Z2: A] :
( ( trans @ A @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ R ) ) ) ) ).
% transD
thf(fact_4341_lexord__trans,axiom,
! [A: $tType,X: list @ A,Y: list @ A,R: set @ ( product_prod @ A @ A ),Z2: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lexord @ A @ R ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Y @ Z2 ) @ ( lexord @ A @ R ) )
=> ( ( trans @ A @ R )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Z2 ) @ ( lexord @ A @ R ) ) ) ) ) ).
% lexord_trans
thf(fact_4342_lenlex__trans,axiom,
! [A: $tType,X: list @ A,Y: list @ A,R: set @ ( product_prod @ A @ A ),Z2: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lenlex @ A @ R ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Y @ Z2 ) @ ( lenlex @ A @ R ) )
=> ( ( trans @ A @ R )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Z2 ) @ ( lenlex @ A @ R ) ) ) ) ) ).
% lenlex_trans
thf(fact_4343_union__fold__mset__add__mset,axiom,
! [A: $tType] :
( ( plus_plus @ ( multiset @ A ) )
= ( fold_mset @ A @ ( multiset @ A ) @ ( add_mset @ A ) ) ) ).
% union_fold_mset_add_mset
thf(fact_4344_trans__singleton,axiom,
! [A: $tType,A3: A] : ( trans @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% trans_singleton
thf(fact_4345_trans__rtrancl__eq__reflcl,axiom,
! [A: $tType,A4: set @ ( product_prod @ A @ A )] :
( ( trans @ A @ A4 )
=> ( ( transitive_rtrancl @ A @ A4 )
= ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ A4 @ ( id2 @ A ) ) ) ) ).
% trans_rtrancl_eq_reflcl
thf(fact_4346_comp__fun__commute_Ofold__mset__union,axiom,
! [B: $tType,A: $tType,F: A > B > B,S: B,M3: multiset @ A,N6: multiset @ A] :
( ( finite6289374366891150609ommute @ A @ B @ F )
=> ( ( fold_mset @ A @ B @ F @ S @ ( plus_plus @ ( multiset @ A ) @ M3 @ N6 ) )
= ( fold_mset @ A @ B @ F @ ( fold_mset @ A @ B @ F @ S @ M3 ) @ N6 ) ) ) ).
% comp_fun_commute.fold_mset_union
thf(fact_4347_trans__join,axiom,
! [A: $tType] :
( ( trans @ A )
= ( ^ [R3: set @ ( product_prod @ A @ A )] :
! [X3: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X3 @ R3 )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y3: A,Y13: A] :
! [Z5: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ Z5 @ R3 )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y23: A,Aa2: A] :
( ( Y13 = Y23 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Aa2 ) @ R3 ) )
@ Z5 ) )
@ X3 ) ) ) ) ).
% trans_join
thf(fact_4348_Image__absorb__rtrancl,axiom,
! [A: $tType,A4: set @ ( product_prod @ A @ A ),B6: set @ A,C5: set @ A] :
( ( trans @ A @ A4 )
=> ( ( refl_on @ A @ B6 @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ C5 @ B6 )
=> ( ( image @ A @ A @ ( transitive_rtrancl @ A @ A4 ) @ C5 )
= ( image @ A @ A @ A4 @ C5 ) ) ) ) ) ).
% Image_absorb_rtrancl
thf(fact_4349_wf__finite__segments,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( irrefl @ A @ R )
=> ( ( trans @ A @ R )
=> ( ! [X2: A] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X2 ) @ R ) ) )
=> ( wf @ A @ R ) ) ) ) ).
% wf_finite_segments
thf(fact_4350_mult__cancel__add__mset,axiom,
! [A: $tType,S: set @ ( product_prod @ A @ A ),Uu2: A,X10: multiset @ A,Y9: multiset @ A] :
( ( trans @ A @ S )
=> ( ( irrefl @ A @ S )
=> ( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ ( add_mset @ A @ Uu2 @ X10 ) @ ( add_mset @ A @ Uu2 @ Y9 ) ) @ ( mult @ A @ S ) )
= ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ X10 @ Y9 ) @ ( mult @ A @ S ) ) ) ) ) ).
% mult_cancel_add_mset
thf(fact_4351_mult__cancel,axiom,
! [A: $tType,S: set @ ( product_prod @ A @ A ),X10: multiset @ A,Z9: multiset @ A,Y9: multiset @ A] :
( ( trans @ A @ S )
=> ( ( irrefl @ A @ S )
=> ( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ X10 @ Z9 ) @ ( plus_plus @ ( multiset @ A ) @ Y9 @ Z9 ) ) @ ( mult @ A @ S ) )
= ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ X10 @ Y9 ) @ ( mult @ A @ S ) ) ) ) ) ).
% mult_cancel
thf(fact_4352_mult__cancel__max,axiom,
! [A: $tType,S: set @ ( product_prod @ A @ A ),X10: multiset @ A,Y9: multiset @ A] :
( ( trans @ A @ S )
=> ( ( irrefl @ A @ S )
=> ( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ X10 @ Y9 ) @ ( mult @ A @ S ) )
= ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ X10 @ Y9 ) @ ( minus_minus @ ( multiset @ A ) @ Y9 @ X10 ) ) @ ( mult @ A @ S ) ) ) ) ) ).
% mult_cancel_max
thf(fact_4353_multeqp__code__iff__reflcl__mult,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A ),P: A > A > $o,N6: multiset @ A,M3: multiset @ A] :
( ( irrefl @ A @ R5 )
=> ( ( trans @ A @ R5 )
=> ( ! [X2: A,Y4: A] :
( ( P @ X2 @ Y4 )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y4 ) @ R5 ) )
=> ( ( multeqp_code @ A @ P @ N6 @ M3 )
= ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ N6 @ M3 ) @ ( sup_sup @ ( set @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) ) @ ( mult @ A @ R5 ) @ ( id2 @ ( multiset @ A ) ) ) ) ) ) ) ) ).
% multeqp_code_iff_reflcl_mult
thf(fact_4354_multp__code__iff__mult,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A ),P: A > A > $o,N6: multiset @ A,M3: multiset @ A] :
( ( irrefl @ A @ R5 )
=> ( ( trans @ A @ R5 )
=> ( ! [X2: A,Y4: A] :
( ( P @ X2 @ Y4 )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y4 ) @ R5 ) )
=> ( ( multp_code @ A @ P @ N6 @ M3 )
= ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ N6 @ M3 ) @ ( mult @ A @ R5 ) ) ) ) ) ) ).
% multp_code_iff_mult
thf(fact_4355_max__ext__eq,axiom,
! [A: $tType] :
( ( max_ext @ A )
= ( ^ [R6: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
@ ^ [X8: set @ A,Y10: set @ A] :
( ( finite_finite2 @ A @ X8 )
& ( finite_finite2 @ A @ Y10 )
& ( Y10
!= ( bot_bot @ ( set @ A ) ) )
& ! [X3: A] :
( ( member @ A @ X3 @ X8 )
=> ? [Y3: A] :
( ( member @ A @ Y3 @ Y10 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R6 ) ) ) ) ) ) ) ) ).
% max_ext_eq
thf(fact_4356_init__seg__of__def,axiom,
! [A: $tType] :
( ( init_seg_of @ A )
= ( collect @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_case_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ $o
@ ^ [R3: set @ ( product_prod @ A @ A ),S4: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ S4 )
& ! [A7: A,B5: A,C3: A] :
( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B5 ) @ S4 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ C3 ) @ R3 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B5 ) @ R3 ) ) ) ) ) ) ).
% init_seg_of_def
thf(fact_4357_refl__on__init__seg__of,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ R ) @ ( init_seg_of @ A ) ) ).
% refl_on_init_seg_of
thf(fact_4358_antisym__init__seg__of,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ S ) @ ( init_seg_of @ A ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ S @ R ) @ ( init_seg_of @ A ) )
=> ( R = S ) ) ) ).
% antisym_init_seg_of
thf(fact_4359_trans__init__seg__of,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A ),T2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ S ) @ ( init_seg_of @ A ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ S @ T2 ) @ ( init_seg_of @ A ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ T2 ) @ ( init_seg_of @ A ) ) ) ) ).
% trans_init_seg_of
thf(fact_4360_Image__def,axiom,
! [B: $tType,A: $tType] :
( ( image @ A @ B )
= ( ^ [R3: set @ ( product_prod @ A @ B ),S4: set @ A] :
( collect @ B
@ ^ [Y3: B] :
? [X3: A] :
( ( member @ A @ X3 @ S4 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R3 ) ) ) ) ) ).
% Image_def
thf(fact_4361_initial__segment__of__Diff,axiom,
! [A: $tType,P5: set @ ( product_prod @ A @ A ),Q3: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P5 @ Q3 ) @ ( init_seg_of @ A ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ P5 @ S ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ Q3 @ S ) ) @ ( init_seg_of @ A ) ) ) ).
% initial_segment_of_Diff
thf(fact_4362_min__ext__def,axiom,
! [A: $tType] :
( ( min_ext @ A )
= ( ^ [R3: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ^ [Uu: product_prod @ ( set @ A ) @ ( set @ A )] :
? [X8: set @ A,Y10: set @ A] :
( ( Uu
= ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X8 @ Y10 ) )
& ( X8
!= ( bot_bot @ ( set @ A ) ) )
& ! [X3: A] :
( ( member @ A @ X3 @ Y10 )
=> ? [Y3: A] :
( ( member @ A @ Y3 @ X8 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R3 ) ) ) ) ) ) ) ).
% min_ext_def
thf(fact_4363_map__project__def,axiom,
! [B: $tType,A: $tType] :
( ( map_project @ A @ B )
= ( ^ [F2: A > ( option @ B ),A9: set @ A] :
( collect @ B
@ ^ [B5: B] :
? [X3: A] :
( ( member @ A @ X3 @ A9 )
& ( ( F2 @ X3 )
= ( some @ B @ B5 ) ) ) ) ) ) ).
% map_project_def
thf(fact_4364_Chains__init__seg__of__Union,axiom,
! [A: $tType,R5: set @ ( set @ ( product_prod @ A @ A ) ),R: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( set @ ( product_prod @ A @ A ) ) ) @ R5 @ ( chains @ ( set @ ( product_prod @ A @ A ) ) @ ( init_seg_of @ A ) ) )
=> ( ( member @ ( set @ ( product_prod @ A @ A ) ) @ R @ R5 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) ) @ R5 ) ) @ ( init_seg_of @ A ) ) ) ) ).
% Chains_init_seg_of_Union
thf(fact_4365_mult__cancel__max0,axiom,
! [A: $tType,S: set @ ( product_prod @ A @ A ),X10: multiset @ A,Y9: multiset @ A] :
( ( trans @ A @ S )
=> ( ( irrefl @ A @ S )
=> ( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ X10 @ Y9 ) @ ( mult @ A @ S ) )
= ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ X10 @ ( inter_mset @ A @ X10 @ Y9 ) ) @ ( minus_minus @ ( multiset @ A ) @ Y9 @ ( inter_mset @ A @ X10 @ Y9 ) ) ) @ ( mult @ A @ S ) ) ) ) ) ).
% mult_cancel_max0
thf(fact_4366_sum__mset_Oremove,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [X: A,A4: multiset @ A] :
( ( member @ A @ X @ ( set_mset @ A @ A4 ) )
=> ( ( comm_m7189776963980413722m_mset @ A @ A4 )
= ( plus_plus @ A @ X @ ( comm_m7189776963980413722m_mset @ A @ ( minus_minus @ ( multiset @ A ) @ A4 @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ) ) ).
% sum_mset.remove
thf(fact_4367_sum__mset_Oadd__mset,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [X: A,N6: multiset @ A] :
( ( comm_m7189776963980413722m_mset @ A @ ( add_mset @ A @ X @ N6 ) )
= ( plus_plus @ A @ X @ ( comm_m7189776963980413722m_mset @ A @ N6 ) ) ) ) ).
% sum_mset.add_mset
thf(fact_4368_sum__mset_Ounion,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M3: multiset @ A,N6: multiset @ A] :
( ( comm_m7189776963980413722m_mset @ A @ ( plus_plus @ ( multiset @ A ) @ M3 @ N6 ) )
= ( plus_plus @ A @ ( comm_m7189776963980413722m_mset @ A @ M3 ) @ ( comm_m7189776963980413722m_mset @ A @ N6 ) ) ) ) ).
% sum_mset.union
thf(fact_4369_sum__mset_Oinsert,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: B > A,X: B,A4: multiset @ B] :
( ( comm_m7189776963980413722m_mset @ A @ ( image_mset @ B @ A @ G @ ( add_mset @ B @ X @ A4 ) ) )
= ( plus_plus @ A @ ( G @ X ) @ ( comm_m7189776963980413722m_mset @ A @ ( image_mset @ B @ A @ G @ A4 ) ) ) ) ) ).
% sum_mset.insert
thf(fact_4370_inter__union__distrib__right,axiom,
! [A: $tType,C5: multiset @ A,A4: multiset @ A,B6: multiset @ A] :
( ( plus_plus @ ( multiset @ A ) @ C5 @ ( inter_mset @ A @ A4 @ B6 ) )
= ( inter_mset @ A @ ( plus_plus @ ( multiset @ A ) @ C5 @ A4 ) @ ( plus_plus @ ( multiset @ A ) @ C5 @ B6 ) ) ) ).
% inter_union_distrib_right
thf(fact_4371_inter__union__distrib__left,axiom,
! [A: $tType,A4: multiset @ A,B6: multiset @ A,C5: multiset @ A] :
( ( plus_plus @ ( multiset @ A ) @ ( inter_mset @ A @ A4 @ B6 ) @ C5 )
= ( inter_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A4 @ C5 ) @ ( plus_plus @ ( multiset @ A ) @ B6 @ C5 ) ) ) ).
% inter_union_distrib_left
thf(fact_4372_inter__subset__eq__union,axiom,
! [A: $tType,A4: multiset @ A,B6: multiset @ A] : ( subseteq_mset @ A @ ( inter_mset @ A @ A4 @ B6 ) @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) ) ).
% inter_subset_eq_union
thf(fact_4373_inter__mset__empty__distrib__left,axiom,
! [A: $tType,A4: multiset @ A,B6: multiset @ A,C5: multiset @ A] :
( ( ( inter_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) @ C5 )
= ( zero_zero @ ( multiset @ A ) ) )
= ( ( ( inter_mset @ A @ A4 @ C5 )
= ( zero_zero @ ( multiset @ A ) ) )
& ( ( inter_mset @ A @ B6 @ C5 )
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% inter_mset_empty_distrib_left
thf(fact_4374_inter__mset__empty__distrib__right,axiom,
! [A: $tType,A4: multiset @ A,B6: multiset @ A,C5: multiset @ A] :
( ( ( inter_mset @ A @ A4 @ ( plus_plus @ ( multiset @ A ) @ B6 @ C5 ) )
= ( zero_zero @ ( multiset @ A ) ) )
= ( ( ( inter_mset @ A @ A4 @ B6 )
= ( zero_zero @ ( multiset @ A ) ) )
& ( ( inter_mset @ A @ A4 @ C5 )
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% inter_mset_empty_distrib_right
thf(fact_4375_sum__mset_Odistrib,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: B > A,H3: B > A,A4: multiset @ B] :
( ( comm_m7189776963980413722m_mset @ A
@ ( image_mset @ B @ A
@ ^ [X3: B] : ( plus_plus @ A @ ( G @ X3 ) @ ( H3 @ X3 ) )
@ A4 ) )
= ( plus_plus @ A @ ( comm_m7189776963980413722m_mset @ A @ ( image_mset @ B @ A @ G @ A4 ) ) @ ( comm_m7189776963980413722m_mset @ A @ ( image_mset @ B @ A @ H3 @ A4 ) ) ) ) ) ).
% sum_mset.distrib
thf(fact_4376_size__eq__sum__mset,axiom,
! [A: $tType] :
( ( size_size @ ( multiset @ A ) )
= ( ^ [M8: multiset @ A] :
( comm_m7189776963980413722m_mset @ nat
@ ( image_mset @ A @ nat
@ ^ [A7: A] : ( one_one @ nat )
@ M8 ) ) ) ) ).
% size_eq_sum_mset
thf(fact_4377_multiset__union__diff__commute,axiom,
! [A: $tType,B6: multiset @ A,C5: multiset @ A,A4: multiset @ A] :
( ( ( inter_mset @ A @ B6 @ C5 )
= ( zero_zero @ ( multiset @ A ) ) )
=> ( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) @ C5 )
= ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ A4 @ C5 ) @ B6 ) ) ) ).
% multiset_union_diff_commute
thf(fact_4378_sum__mset_Oeq__fold,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ( ( comm_m7189776963980413722m_mset @ A )
= ( fold_mset @ A @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ) ).
% sum_mset.eq_fold
thf(fact_4379_Chains__def,axiom,
! [A: $tType] :
( ( chains @ A )
= ( ^ [R3: set @ ( product_prod @ A @ A )] :
( collect @ ( set @ A )
@ ^ [C7: set @ A] :
! [X3: A] :
( ( member @ A @ X3 @ C7 )
=> ! [Y3: A] :
( ( member @ A @ Y3 @ C7 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R3 ) ) ) ) ) ) ) ).
% Chains_def
thf(fact_4380_Chains__subset_H,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( refl_on @ A @ ( top_top @ ( set @ A ) ) @ R )
=> ( ord_less_eq @ ( set @ ( set @ A ) )
@ ( collect @ ( set @ A )
@ ( pred_chain @ A @ ( top_top @ ( set @ A ) )
@ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R ) ) )
@ ( chains @ A @ R ) ) ) ).
% Chains_subset'
thf(fact_4381_Chains__subset,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ord_less_eq @ ( set @ ( set @ A ) ) @ ( chains @ A @ R )
@ ( collect @ ( set @ A )
@ ( pred_chain @ A @ ( top_top @ ( set @ A ) )
@ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R ) ) ) ) ).
% Chains_subset
thf(fact_4382_Chains__alt__def,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( refl_on @ A @ ( top_top @ ( set @ A ) ) @ R )
=> ( ( chains @ A @ R )
= ( collect @ ( set @ A )
@ ( pred_chain @ A @ ( top_top @ ( set @ A ) )
@ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R ) ) ) ) ) ).
% Chains_alt_def
thf(fact_4383_iter_H_Ocases,axiom,
! [A: $tType] :
( ( quickcheck_random @ A )
=> ! [X: product_prod @ ( itself @ A ) @ ( product_prod @ code_natural @ ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) ) )] :
~ ! [T4: itself @ A,Nrandom: code_natural,Sz: code_natural,Seed: product_prod @ code_natural @ code_natural] :
( X
!= ( product_Pair @ ( itself @ A ) @ ( product_prod @ code_natural @ ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) ) ) @ T4 @ ( product_Pair @ code_natural @ ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) ) @ Nrandom @ ( product_Pair @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ Sz @ Seed ) ) ) ) ) ).
% iter'.cases
thf(fact_4384_multp__def,axiom,
! [A: $tType] :
( ( multp @ A )
= ( ^ [R3: A > A > $o,M8: multiset @ A,N9: multiset @ A] : ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ M8 @ N9 ) @ ( mult @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R3 ) ) ) ) ) ) ).
% multp_def
thf(fact_4385_AboveS__def,axiom,
! [A: $tType] :
( ( order_AboveS @ A )
= ( ^ [R3: set @ ( product_prod @ A @ A ),A9: set @ A] :
( collect @ A
@ ^ [B5: A] :
( ( member @ A @ B5 @ ( field2 @ A @ R3 ) )
& ! [X3: A] :
( ( member @ A @ X3 @ A9 )
=> ( ( B5 != X3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ B5 ) @ R3 ) ) ) ) ) ) ) ).
% AboveS_def
thf(fact_4386_lexordp__conv__lexord,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_lexordp @ A )
= ( ^ [Xs3: list @ A,Ys: list @ A] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs3 @ Ys ) @ ( lexord @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ ( ord_less @ A ) ) ) ) ) ) ) ) ).
% lexordp_conv_lexord
thf(fact_4387_select__weight__cons__zero,axiom,
! [A: $tType,X: A,Xs: list @ ( product_prod @ code_natural @ A )] :
( ( select_weight @ A @ ( cons @ ( product_prod @ code_natural @ A ) @ ( product_Pair @ code_natural @ A @ ( zero_zero @ code_natural ) @ X ) @ Xs ) )
= ( select_weight @ A @ Xs ) ) ).
% select_weight_cons_zero
thf(fact_4388_one__step__implies__multp,axiom,
! [A: $tType,J4: multiset @ A,K2: multiset @ A,R: A > A > $o,I4: multiset @ A] :
( ( J4
!= ( zero_zero @ ( multiset @ A ) ) )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ ( set_mset @ A @ K2 ) )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ ( set_mset @ A @ J4 ) )
& ( R @ X2 @ Xa2 ) ) )
=> ( multp @ A @ R @ ( plus_plus @ ( multiset @ A ) @ I4 @ K2 ) @ ( plus_plus @ ( multiset @ A ) @ I4 @ J4 ) ) ) ) ).
% one_step_implies_multp
thf(fact_4389_select__weight__select,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( select_weight @ A @ ( map @ A @ ( product_prod @ code_natural @ A ) @ ( product_Pair @ code_natural @ A @ ( one_one @ code_natural ) ) @ Xs ) )
= ( select @ A @ Xs ) ) ) ).
% select_weight_select
thf(fact_4390_wo__rel_Osuc__greater,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),B6: set @ A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ ( field2 @ A @ R ) )
=> ( ( ( order_AboveS @ A @ R @ B6 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ A @ B2 @ B6 )
=> ( ( ( bNF_Wellorder_wo_suc @ A @ R @ B6 )
!= B2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ ( bNF_Wellorder_wo_suc @ A @ R @ B6 ) ) @ R ) ) ) ) ) ) ).
% wo_rel.suc_greater
thf(fact_4391_wo__rel_Oequals__suc__AboveS,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),B6: set @ A,A3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ ( field2 @ A @ R ) )
=> ( ( member @ A @ A3 @ ( order_AboveS @ A @ R @ B6 ) )
=> ( ! [A16: A] :
( ( member @ A @ A16 @ ( order_AboveS @ A @ R @ B6 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A16 ) @ R ) )
=> ( A3
= ( bNF_Wellorder_wo_suc @ A @ R @ B6 ) ) ) ) ) ) ).
% wo_rel.equals_suc_AboveS
thf(fact_4392_wo__rel_Osuc__least__AboveS,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B6: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R )
=> ( ( member @ A @ A3 @ ( order_AboveS @ A @ R @ B6 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( bNF_Wellorder_wo_suc @ A @ R @ B6 ) @ A3 ) @ R ) ) ) ).
% wo_rel.suc_least_AboveS
thf(fact_4393_wo__rel_Osuc__ofilter__in,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A4: set @ A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R )
=> ( ( order_ofilter @ A @ R @ A4 )
=> ( ( ( order_AboveS @ A @ R @ A4 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ ( bNF_Wellorder_wo_suc @ A @ R @ A4 ) ) @ R )
=> ( ( B2
!= ( bNF_Wellorder_wo_suc @ A @ R @ A4 ) )
=> ( member @ A @ B2 @ A4 ) ) ) ) ) ) ).
% wo_rel.suc_ofilter_in
thf(fact_4394_slice__head,axiom,
! [A: $tType,From: nat,To: nat,Xs: list @ A] :
( ( ord_less @ nat @ From @ To )
=> ( ( ord_less_eq @ nat @ To @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( hd @ A @ ( slice @ A @ From @ To @ Xs ) )
= ( nth @ A @ Xs @ From ) ) ) ) ).
% slice_head
thf(fact_4395_gen__length__def,axiom,
! [A: $tType] :
( ( gen_length @ A )
= ( ^ [N5: nat,Xs3: list @ A] : ( plus_plus @ nat @ N5 @ ( size_size @ ( list @ A ) @ Xs3 ) ) ) ) ).
% gen_length_def
thf(fact_4396_subset__mset_Osum__pos2,axiom,
! [A: $tType,B: $tType,I4: set @ B,I: B,F: B > ( multiset @ A )] :
( ( finite_finite2 @ B @ I4 )
=> ( ( member @ B @ I @ I4 )
=> ( ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( F @ I ) )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ I4 )
=> ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( F @ I2 ) ) )
=> ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F @ I4 ) ) ) ) ) ) ).
% subset_mset.sum_pos2
thf(fact_4397_subset__mset_Oadd__less__cancel__right,axiom,
! [A: $tType,A3: multiset @ A,C2: multiset @ A,B2: multiset @ A] :
( ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ C2 ) @ ( plus_plus @ ( multiset @ A ) @ B2 @ C2 ) )
= ( subset_mset @ A @ A3 @ B2 ) ) ).
% subset_mset.add_less_cancel_right
thf(fact_4398_subset__mset_Oadd__less__cancel__left,axiom,
! [A: $tType,C2: multiset @ A,A3: multiset @ A,B2: multiset @ A] :
( ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ C2 @ A3 ) @ ( plus_plus @ ( multiset @ A ) @ C2 @ B2 ) )
= ( subset_mset @ A @ A3 @ B2 ) ) ).
% subset_mset.add_less_cancel_left
thf(fact_4399_subset__mset_Oadd__less__same__cancel1,axiom,
! [A: $tType,B2: multiset @ A,A3: multiset @ A] :
( ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ B2 @ A3 ) @ B2 )
= ( subset_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ).
% subset_mset.add_less_same_cancel1
thf(fact_4400_subset__mset_Oadd__less__same__cancel2,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
( ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ B2 ) @ B2 )
= ( subset_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ).
% subset_mset.add_less_same_cancel2
thf(fact_4401_subset__mset_Oless__add__same__cancel1,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
( ( subset_mset @ A @ A3 @ ( plus_plus @ ( multiset @ A ) @ A3 @ B2 ) )
= ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ B2 ) ) ).
% subset_mset.less_add_same_cancel1
thf(fact_4402_subset__mset_Oless__add__same__cancel2,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
( ( subset_mset @ A @ A3 @ ( plus_plus @ ( multiset @ A ) @ B2 @ A3 ) )
= ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ B2 ) ) ).
% subset_mset.less_add_same_cancel2
thf(fact_4403_subset__mset_Oadd__less__imp__less__right,axiom,
! [A: $tType,A3: multiset @ A,C2: multiset @ A,B2: multiset @ A] :
( ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ C2 ) @ ( plus_plus @ ( multiset @ A ) @ B2 @ C2 ) )
=> ( subset_mset @ A @ A3 @ B2 ) ) ).
% subset_mset.add_less_imp_less_right
thf(fact_4404_subset__mset_Oadd__less__imp__less__left,axiom,
! [A: $tType,C2: multiset @ A,A3: multiset @ A,B2: multiset @ A] :
( ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ C2 @ A3 ) @ ( plus_plus @ ( multiset @ A ) @ C2 @ B2 ) )
=> ( subset_mset @ A @ A3 @ B2 ) ) ).
% subset_mset.add_less_imp_less_left
thf(fact_4405_subset__mset_Oadd__strict__right__mono,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
( ( subset_mset @ A @ A3 @ B2 )
=> ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ C2 ) @ ( plus_plus @ ( multiset @ A ) @ B2 @ C2 ) ) ) ).
% subset_mset.add_strict_right_mono
thf(fact_4406_subset__mset_Oadd__strict__left__mono,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
( ( subset_mset @ A @ A3 @ B2 )
=> ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ C2 @ A3 ) @ ( plus_plus @ ( multiset @ A ) @ C2 @ B2 ) ) ) ).
% subset_mset.add_strict_left_mono
thf(fact_4407_subset__mset_Oadd__strict__mono,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A,D3: multiset @ A] :
( ( subset_mset @ A @ A3 @ B2 )
=> ( ( subset_mset @ A @ C2 @ D3 )
=> ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ C2 ) @ ( plus_plus @ ( multiset @ A ) @ B2 @ D3 ) ) ) ) ).
% subset_mset.add_strict_mono
thf(fact_4408_hd__zip,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B] :
( ( Xs
!= ( nil @ A ) )
=> ( ( Ys2
!= ( nil @ B ) )
=> ( ( hd @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys2 ) )
= ( product_Pair @ A @ B @ ( hd @ A @ Xs ) @ ( hd @ B @ Ys2 ) ) ) ) ) ).
% hd_zip
thf(fact_4409_subset__mset_Oadd__le__less__mono,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A,D3: multiset @ A] :
( ( subseteq_mset @ A @ A3 @ B2 )
=> ( ( subset_mset @ A @ C2 @ D3 )
=> ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ C2 ) @ ( plus_plus @ ( multiset @ A ) @ B2 @ D3 ) ) ) ) ).
% subset_mset.add_le_less_mono
thf(fact_4410_subset__mset_Oadd__less__le__mono,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A,D3: multiset @ A] :
( ( subset_mset @ A @ A3 @ B2 )
=> ( ( subseteq_mset @ A @ C2 @ D3 )
=> ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ C2 ) @ ( plus_plus @ ( multiset @ A ) @ B2 @ D3 ) ) ) ) ).
% subset_mset.add_less_le_mono
thf(fact_4411_subset__mset_OlessE,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
( ( subset_mset @ A @ A3 @ B2 )
=> ~ ! [C4: multiset @ A] :
( ( B2
= ( plus_plus @ ( multiset @ A ) @ A3 @ C4 ) )
=> ( C4
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% subset_mset.lessE
thf(fact_4412_subset__mset_Oadd__neg__neg,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
( ( subset_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) )
=> ( ( subset_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) )
=> ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ B2 ) @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% subset_mset.add_neg_neg
thf(fact_4413_subset__mset_Oadd__pos__pos,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
( ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ A3 )
=> ( ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ B2 )
=> ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( plus_plus @ ( multiset @ A ) @ A3 @ B2 ) ) ) ) ).
% subset_mset.add_pos_pos
thf(fact_4414_subset__mset_Opos__add__strict,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
( ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ A3 )
=> ( ( subset_mset @ A @ B2 @ C2 )
=> ( subset_mset @ A @ B2 @ ( plus_plus @ ( multiset @ A ) @ A3 @ C2 ) ) ) ) ).
% subset_mset.pos_add_strict
thf(fact_4415_subset__implies__mult,axiom,
! [A: $tType,A4: multiset @ A,B6: multiset @ A,R: set @ ( product_prod @ A @ A )] :
( ( subset_mset @ A @ A4 @ B6 )
=> ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ A4 @ B6 ) @ ( mult @ A @ R ) ) ) ).
% subset_implies_mult
thf(fact_4416_subset__mset_Oadd__strict__increasing2,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
( ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ A3 )
=> ( ( subset_mset @ A @ B2 @ C2 )
=> ( subset_mset @ A @ B2 @ ( plus_plus @ ( multiset @ A ) @ A3 @ C2 ) ) ) ) ).
% subset_mset.add_strict_increasing2
thf(fact_4417_subset__mset_Oadd__strict__increasing,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
( ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ A3 )
=> ( ( subseteq_mset @ A @ B2 @ C2 )
=> ( subset_mset @ A @ B2 @ ( plus_plus @ ( multiset @ A ) @ A3 @ C2 ) ) ) ) ).
% subset_mset.add_strict_increasing
thf(fact_4418_subset__mset_Oadd__pos__nonneg,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
( ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ A3 )
=> ( ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ B2 )
=> ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( plus_plus @ ( multiset @ A ) @ A3 @ B2 ) ) ) ) ).
% subset_mset.add_pos_nonneg
thf(fact_4419_subset__mset_Oadd__nonpos__neg,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
( ( subseteq_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) )
=> ( ( subset_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) )
=> ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ B2 ) @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% subset_mset.add_nonpos_neg
thf(fact_4420_subset__mset_Oadd__nonneg__pos,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
( ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ A3 )
=> ( ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ B2 )
=> ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( plus_plus @ ( multiset @ A ) @ A3 @ B2 ) ) ) ) ).
% subset_mset.add_nonneg_pos
thf(fact_4421_subset__mset_Oadd__neg__nonpos,axiom,
! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
( ( subset_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) )
=> ( ( subseteq_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) )
=> ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ B2 ) @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% subset_mset.add_neg_nonpos
thf(fact_4422_sorted__hd__min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ! [X7: A] :
( ( member @ A @ X7 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ ( hd @ A @ Xs ) @ X7 ) ) ) ) ) ).
% sorted_hd_min
thf(fact_4423_sorted__hd__last,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L )
=> ( ( L
!= ( nil @ A ) )
=> ( ord_less_eq @ A @ ( hd @ A @ L ) @ ( last @ A @ L ) ) ) ) ) ).
% sorted_hd_last
thf(fact_4424_hd__last__singletonI,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( ( hd @ A @ Xs )
= ( last @ A @ Xs ) )
=> ( ( distinct @ A @ Xs )
=> ( Xs
= ( cons @ A @ ( hd @ A @ Xs ) @ ( nil @ A ) ) ) ) ) ) ).
% hd_last_singletonI
thf(fact_4425_hd__butlast,axiom,
! [A: $tType,Xs: list @ A] :
( ( ord_less @ nat @ ( one_one @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( hd @ A @ ( butlast @ A @ Xs ) )
= ( hd @ A @ Xs ) ) ) ).
% hd_butlast
thf(fact_4426_subset__mset_Osum__pos,axiom,
! [A: $tType,B: $tType,I4: set @ B,F: B > ( multiset @ A )] :
( ( finite_finite2 @ B @ I4 )
=> ( ( I4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I2: B] :
( ( member @ B @ I2 @ I4 )
=> ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( F @ I2 ) ) )
=> ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F @ I4 ) ) ) ) ) ).
% subset_mset.sum_pos
thf(fact_4427_subset__mset_Osum__strict__mono,axiom,
! [A: $tType,B: $tType,A4: set @ B,F: B > ( multiset @ A ),G: B > ( multiset @ A )] :
( ( finite_finite2 @ B @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ A4 )
=> ( subset_mset @ A @ ( F @ X2 ) @ ( G @ X2 ) ) )
=> ( subset_mset @ A @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F @ A4 ) @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ G @ A4 ) ) ) ) ) ).
% subset_mset.sum_strict_mono
thf(fact_4428_ofilter__ordLess,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A4: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( order_ofilter @ A @ R @ A4 )
=> ( ( ord_less @ ( set @ A ) @ A4 @ ( field2 @ A @ R ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu: A] : A4 ) )
@ R )
@ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ) ).
% ofilter_ordLess
thf(fact_4429_ofilter__subset__ordLess,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A4: set @ A,B6: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( order_ofilter @ A @ R @ A4 )
=> ( ( order_ofilter @ A @ R @ B6 )
=> ( ( ord_less @ ( set @ A ) @ A4 @ B6 )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu: A] : A4 ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ B6
@ ^ [Uu: A] : B6 ) ) )
@ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ) ) ).
% ofilter_subset_ordLess
thf(fact_4430_ofilter__subset__ordLeq,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A4: set @ A,B6: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( order_ofilter @ A @ R @ A4 )
=> ( ( order_ofilter @ A @ R @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B6 )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu: A] : A4 ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ B6
@ ^ [Uu: A] : B6 ) ) )
@ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ) ) ) ).
% ofilter_subset_ordLeq
thf(fact_4431_Nitpick_Osize__list__simp_I1_J,axiom,
! [A: $tType] :
( ( size_list @ A )
= ( ^ [F2: A > nat,Xs3: list @ A] :
( if @ nat
@ ( Xs3
= ( nil @ A ) )
@ ( zero_zero @ nat )
@ ( suc @ ( plus_plus @ nat @ ( F2 @ ( hd @ A @ Xs3 ) ) @ ( size_list @ A @ F2 @ ( tl @ A @ Xs3 ) ) ) ) ) ) ) ).
% Nitpick.size_list_simp(1)
thf(fact_4432_length__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( tl @ A @ Xs ) )
= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) ) ).
% length_tl
thf(fact_4433_in__hd__or__tl__conv,axiom,
! [A: $tType,L: list @ A,X: A] :
( ( L
!= ( nil @ A ) )
=> ( ( ( X
= ( hd @ A @ L ) )
| ( member @ A @ X @ ( set2 @ A @ ( tl @ A @ L ) ) ) )
= ( member @ A @ X @ ( set2 @ A @ L ) ) ) ) ).
% in_hd_or_tl_conv
thf(fact_4434_tl__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( tl @ A @ ( replicate @ A @ N @ X ) )
= ( replicate @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ X ) ) ).
% tl_replicate
thf(fact_4435_rev__split__conv,axiom,
! [A: $tType,L: list @ A] :
( ( L
!= ( nil @ A ) )
=> ( ( append @ A @ ( rev @ A @ ( tl @ A @ L ) ) @ ( cons @ A @ ( hd @ A @ L ) @ ( nil @ A ) ) )
= ( rev @ A @ L ) ) ) ).
% rev_split_conv
thf(fact_4436_ordLess__irreflexive,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ R ) @ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ).
% ordLess_irreflexive
thf(fact_4437_ordLess__transitive,axiom,
! [B: $tType,C: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),R9: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R4 @ R9 ) @ ( bNF_We4044943003108391690rdLess @ B @ C ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R @ R9 ) @ ( bNF_We4044943003108391690rdLess @ A @ C ) ) ) ) ).
% ordLess_transitive
thf(fact_4438_ordLeq__transitive,axiom,
! [B: $tType,C: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),R9: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R4 @ R9 ) @ ( bNF_Wellorder_ordLeq @ B @ C ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R @ R9 ) @ ( bNF_Wellorder_ordLeq @ A @ C ) ) ) ) ).
% ordLeq_transitive
thf(fact_4439_not__ordLess__ordLeq,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
=> ~ ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ).
% not_ordLess_ordLeq
thf(fact_4440_ordLess__imp__ordLeq,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ).
% ordLess_imp_ordLeq
thf(fact_4441_ordLeq__ordLess__trans,axiom,
! [B: $tType,C: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),R9: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R4 @ R9 ) @ ( bNF_We4044943003108391690rdLess @ B @ C ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R @ R9 ) @ ( bNF_We4044943003108391690rdLess @ A @ C ) ) ) ) ).
% ordLeq_ordLess_trans
thf(fact_4442_ordLess__ordLeq__trans,axiom,
! [B: $tType,C: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),R9: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R4 @ R9 ) @ ( bNF_Wellorder_ordLeq @ B @ C ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R @ R9 ) @ ( bNF_We4044943003108391690rdLess @ A @ C ) ) ) ) ).
% ordLess_ordLeq_trans
thf(fact_4443_not__ordLess__iff__ordLeq,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( order_well_order_on @ B @ ( field2 @ B @ R4 ) @ R4 )
=> ( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ R ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ) ) ).
% not_ordLess_iff_ordLeq
thf(fact_4444_not__ordLeq__iff__ordLess,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( order_well_order_on @ B @ ( field2 @ B @ R4 ) @ R4 )
=> ( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) ) ) ) ) ).
% not_ordLeq_iff_ordLess
thf(fact_4445_ordLess__or__ordLeq,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( order_well_order_on @ B @ ( field2 @ B @ R4 ) @ R4 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
| ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ) ) ).
% ordLess_or_ordLeq
thf(fact_4446_in__set__tlD,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ ( tl @ A @ Xs ) ) )
=> ( member @ A @ X @ ( set2 @ A @ Xs ) ) ) ).
% in_set_tlD
thf(fact_4447_tl__obtain__elem,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( ( tl @ A @ Xs )
= ( nil @ A ) )
=> ~ ! [E3: A] :
( Xs
!= ( cons @ A @ E3 @ ( nil @ A ) ) ) ) ) ).
% tl_obtain_elem
thf(fact_4448_not__hd__in__tl,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( X
!= ( hd @ A @ Xs ) )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( member @ A @ X @ ( set2 @ A @ ( tl @ A @ Xs ) ) ) ) ) ).
% not_hd_in_tl
thf(fact_4449_tl__last,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( tl @ A @ Xs )
!= ( nil @ A ) )
=> ( ( last @ A @ Xs )
= ( last @ A @ ( tl @ A @ Xs ) ) ) ) ).
% tl_last
thf(fact_4450_rev__butlast__is__tl__rev,axiom,
! [A: $tType,L: list @ A] :
( ( rev @ A @ ( butlast @ A @ L ) )
= ( tl @ A @ ( rev @ A @ L ) ) ) ).
% rev_butlast_is_tl_rev
thf(fact_4451_tl__subset,axiom,
! [A: $tType,Xs: list @ A,A4: set @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( tl @ A @ Xs ) ) @ A4 ) ) ) ).
% tl_subset
thf(fact_4452_Misc_Onth__tl,axiom,
! [A: $tType,Xs: list @ A,N: nat] :
( ( Xs
!= ( nil @ A ) )
=> ( ( nth @ A @ ( tl @ A @ Xs ) @ N )
= ( nth @ A @ Xs @ ( suc @ N ) ) ) ) ).
% Misc.nth_tl
thf(fact_4453_list__take__induct__tl2,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys2: list @ B,P: B > A > $o] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ! [N3: nat] :
( ( ord_less @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ B @ Ys2 @ N3 ) @ ( nth @ A @ Xs @ N3 ) ) )
=> ! [N10: nat] :
( ( ord_less @ nat @ N10 @ ( size_size @ ( list @ A ) @ ( tl @ A @ Xs ) ) )
=> ( P @ ( nth @ B @ ( tl @ B @ Ys2 ) @ N10 ) @ ( nth @ A @ ( tl @ A @ Xs ) @ N10 ) ) ) ) ) ).
% list_take_induct_tl2
thf(fact_4454_exists__minim__Well__order,axiom,
! [A: $tType,R5: set @ ( set @ ( product_prod @ A @ A ) )] :
( ( R5
!= ( bot_bot @ ( set @ ( set @ ( product_prod @ A @ A ) ) ) ) )
=> ( ! [X2: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ X2 @ R5 )
=> ( order_well_order_on @ A @ ( field2 @ A @ X2 ) @ X2 ) )
=> ? [X2: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ X2 @ R5 )
& ! [Xa2: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ Xa2 @ R5 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ X2 @ Xa2 ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ) ) ) ).
% exists_minim_Well_order
thf(fact_4455_ordLeq__total,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( order_well_order_on @ B @ ( field2 @ B @ R4 ) @ R4 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
| ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ) ) ).
% ordLeq_total
thf(fact_4456_ordLeq__reflexive,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ R ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ).
% ordLeq_reflexive
thf(fact_4457_ordLeq__Well__order__simp,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
& ( order_well_order_on @ B @ ( field2 @ B @ R4 ) @ R4 ) ) ) ).
% ordLeq_Well_order_simp
thf(fact_4458_tl__take,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( tl @ A @ ( take @ A @ N @ Xs ) )
= ( take @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( tl @ A @ Xs ) ) ) ).
% tl_take
thf(fact_4459_distinct__hd__tl,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( distinct @ A @ Xs )
=> ( ( X
= ( hd @ A @ Xs ) )
=> ~ ( member @ A @ X @ ( set2 @ A @ ( tl @ A @ Xs ) ) ) ) ) ).
% distinct_hd_tl
thf(fact_4460_butlast__rev__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( butlast @ A @ ( rev @ A @ Xs ) )
= ( rev @ A @ ( tl @ A @ Xs ) ) ) ) ).
% butlast_rev_tl
thf(fact_4461_finite__ordLess__infinite,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( order_well_order_on @ B @ ( field2 @ B @ R4 ) @ R4 )
=> ( ( finite_finite2 @ A @ ( field2 @ A @ R ) )
=> ( ~ ( finite_finite2 @ B @ ( field2 @ B @ R4 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) ) ) ) ) ) ).
% finite_ordLess_infinite
thf(fact_4462_ordLeq__iff__ordLess__Restr,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( order_well_order_on @ B @ ( field2 @ B @ R4 ) @ R4 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ ( field2 @ A @ R ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ ( order_underS @ A @ R @ X3 )
@ ^ [Uu: A] : ( order_underS @ A @ R @ X3 ) ) )
@ R4 )
@ ( bNF_We4044943003108391690rdLess @ A @ B ) ) ) ) ) ) ) ).
% ordLeq_iff_ordLess_Restr
thf(fact_4463_underS__Restr__ordLess,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( ( field2 @ A @ R )
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ ( order_underS @ A @ R @ A3 )
@ ^ [Uu: A] : ( order_underS @ A @ R @ A3 ) ) )
@ R )
@ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ).
% underS_Restr_ordLess
thf(fact_4464_horner__sum__append,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ! [F: B > A,A3: A,Xs: list @ B,Ys2: list @ B] :
( ( groups4207007520872428315er_sum @ B @ A @ F @ A3 @ ( append @ B @ Xs @ Ys2 ) )
= ( plus_plus @ A @ ( groups4207007520872428315er_sum @ B @ A @ F @ A3 @ Xs ) @ ( times_times @ A @ ( power_power @ A @ A3 @ ( size_size @ ( list @ B ) @ Xs ) ) @ ( groups4207007520872428315er_sum @ B @ A @ F @ A3 @ Ys2 ) ) ) ) ) ).
% horner_sum_append
thf(fact_4465_Code__Numeral_Osub__code_I4_J,axiom,
! [N: num] :
( ( code_sub @ one2 @ ( bit0 @ N ) )
= ( code_Neg @ ( bitM @ N ) ) ) ).
% Code_Numeral.sub_code(4)
thf(fact_4466_horner__sum__simps_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_0 @ A )
=> ! [F: B > A,A3: A,X: B,Xs: list @ B] :
( ( groups4207007520872428315er_sum @ B @ A @ F @ A3 @ ( cons @ B @ X @ Xs ) )
= ( plus_plus @ A @ ( F @ X ) @ ( times_times @ A @ A3 @ ( groups4207007520872428315er_sum @ B @ A @ F @ A3 @ Xs ) ) ) ) ) ).
% horner_sum_simps(2)
thf(fact_4467_underS__E,axiom,
! [A: $tType,I: A,R5: set @ ( product_prod @ A @ A ),J: A] :
( ( member @ A @ I @ ( order_underS @ A @ R5 @ J ) )
=> ( ( I != J )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J ) @ R5 ) ) ) ).
% underS_E
thf(fact_4468_underS__I,axiom,
! [A: $tType,I: A,J: A,R5: set @ ( product_prod @ A @ A )] :
( ( I != J )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J ) @ R5 )
=> ( member @ A @ I @ ( order_underS @ A @ R5 @ J ) ) ) ) ).
% underS_I
thf(fact_4469_underS__def,axiom,
! [A: $tType] :
( ( order_underS @ A )
= ( ^ [R3: set @ ( product_prod @ A @ A ),A7: A] :
( collect @ A
@ ^ [B5: A] :
( ( B5 != A7 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ A7 ) @ R3 ) ) ) ) ) ).
% underS_def
thf(fact_4470_Code__Numeral_Odup__code_I3_J,axiom,
! [N: num] :
( ( code_dup @ ( code_Neg @ N ) )
= ( code_Neg @ ( bit0 @ N ) ) ) ).
% Code_Numeral.dup_code(3)
thf(fact_4471_less__integer__code_I3_J,axiom,
! [L: num] :
~ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ ( code_Neg @ L ) ) ).
% less_integer_code(3)
thf(fact_4472_less__integer__code_I7_J,axiom,
! [K: num] : ( ord_less @ code_integer @ ( code_Neg @ K ) @ ( zero_zero @ code_integer ) ) ).
% less_integer_code(7)
thf(fact_4473_plus__integer__code_I6_J,axiom,
! [M2: num,N: num] :
( ( plus_plus @ code_integer @ ( code_Neg @ M2 ) @ ( code_Neg @ N ) )
= ( code_Neg @ ( plus_plus @ num @ M2 @ N ) ) ) ).
% plus_integer_code(6)
thf(fact_4474_less__eq__integer__code_I7_J,axiom,
! [K: num] : ( ord_less_eq @ code_integer @ ( code_Neg @ K ) @ ( zero_zero @ code_integer ) ) ).
% less_eq_integer_code(7)
thf(fact_4475_less__eq__integer__code_I3_J,axiom,
! [L: num] :
~ ( ord_less_eq @ code_integer @ ( zero_zero @ code_integer ) @ ( code_Neg @ L ) ) ).
% less_eq_integer_code(3)
thf(fact_4476_less__integer__code_I9_J,axiom,
! [K: num,L: num] :
( ( ord_less @ code_integer @ ( code_Neg @ K ) @ ( code_Neg @ L ) )
= ( ord_less @ num @ L @ K ) ) ).
% less_integer_code(9)
thf(fact_4477_less__eq__integer__code_I9_J,axiom,
! [K: num,L: num] :
( ( ord_less_eq @ code_integer @ ( code_Neg @ K ) @ ( code_Neg @ L ) )
= ( ord_less_eq @ num @ L @ K ) ) ).
% less_eq_integer_code(9)
thf(fact_4478_minus__integer__code_I6_J,axiom,
! [M2: num,N: num] :
( ( minus_minus @ code_integer @ ( code_Neg @ M2 ) @ ( code_Neg @ N ) )
= ( code_sub @ N @ M2 ) ) ).
% minus_integer_code(6)
thf(fact_4479_divmod__abs__code_I2_J,axiom,
! [K: num,L: num] :
( ( code_divmod_abs @ ( code_Neg @ K ) @ ( code_Neg @ L ) )
= ( unique8689654367752047608divmod @ code_integer @ K @ L ) ) ).
% divmod_abs_code(2)
thf(fact_4480_horner__sum__foldr,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_0 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F2: B > A,A7: A,Xs3: list @ B] :
( foldr @ B @ A
@ ^ [X3: B,B5: A] : ( plus_plus @ A @ ( F2 @ X3 ) @ ( times_times @ A @ A7 @ B5 ) )
@ Xs3
@ ( zero_zero @ A ) ) ) ) ) ).
% horner_sum_foldr
thf(fact_4481_underS__incl__iff,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R ) )
=> ( ( member @ A @ B2 @ ( field2 @ A @ R ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( order_underS @ A @ R @ A3 ) @ ( order_underS @ A @ R @ B2 ) )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R ) ) ) ) ) ).
% underS_incl_iff
thf(fact_4482_Code__Numeral_Osub__code_I5_J,axiom,
! [N: num] :
( ( code_sub @ one2 @ ( bit1 @ N ) )
= ( code_Neg @ ( bit0 @ N ) ) ) ).
% Code_Numeral.sub_code(5)
thf(fact_4483_ordLess__iff__ordIso__Restr,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( order_well_order_on @ B @ ( field2 @ B @ R4 ) @ R4 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ R ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) )
= ( ? [X3: A] :
( ( member @ A @ X3 @ ( field2 @ A @ R ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R4
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ ( order_underS @ A @ R @ X3 )
@ ^ [Uu: A] : ( order_underS @ A @ R @ X3 ) ) ) )
@ ( bNF_Wellorder_ordIso @ B @ A ) ) ) ) ) ) ) ).
% ordLess_iff_ordIso_Restr
thf(fact_4484_sum__list__update,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [K: nat,Xs: list @ A,X: A] :
( ( ord_less @ nat @ K @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( groups8242544230860333062m_list @ A @ ( list_update @ A @ Xs @ K @ X ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( groups8242544230860333062m_list @ A @ Xs ) @ X ) @ ( nth @ A @ Xs @ K ) ) ) ) ) ).
% sum_list_update
thf(fact_4485_length__product__lists,axiom,
! [B: $tType,Xss: list @ ( list @ B )] :
( ( 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
thf(fact_4486_horner__sum__transfer,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType] :
( ( ( comm_semiring_0 @ B )
& ( comm_semiring_0 @ A ) )
=> ! [A4: A > B > $o,B6: C > D > $o] :
( ( A4 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A4 @ ( bNF_rel_fun @ A @ B @ A @ B @ A4 @ A4 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A4 @ ( bNF_rel_fun @ A @ B @ A @ B @ A4 @ A4 ) @ ( times_times @ A ) @ ( times_times @ B ) )
=> ( bNF_rel_fun @ ( C > A ) @ ( D > B ) @ ( A > ( list @ C ) > A ) @ ( B > ( list @ D ) > B ) @ ( bNF_rel_fun @ C @ D @ A @ B @ B6 @ A4 ) @ ( bNF_rel_fun @ A @ B @ ( ( list @ C ) > A ) @ ( ( list @ D ) > B ) @ A4 @ ( bNF_rel_fun @ ( list @ C ) @ ( list @ D ) @ A @ B @ ( list_all2 @ C @ D @ B6 ) @ A4 ) ) @ ( groups4207007520872428315er_sum @ C @ A ) @ ( groups4207007520872428315er_sum @ D @ B ) ) ) ) ) ) ).
% horner_sum_transfer
thf(fact_4487_sum__list_OCons,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [X: A,Xs: list @ A] :
( ( groups8242544230860333062m_list @ A @ ( cons @ A @ X @ Xs ) )
= ( plus_plus @ A @ X @ ( groups8242544230860333062m_list @ A @ Xs ) ) ) ) ).
% sum_list.Cons
thf(fact_4488_sum__list__append,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [Xs: list @ A,Ys2: list @ A] :
( ( groups8242544230860333062m_list @ A @ ( append @ A @ Xs @ Ys2 ) )
= ( plus_plus @ A @ ( groups8242544230860333062m_list @ A @ Xs ) @ ( groups8242544230860333062m_list @ A @ Ys2 ) ) ) ) ).
% sum_list_append
thf(fact_4489_ordIso__transitive,axiom,
! [B: $tType,C: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),R9: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R4 @ R9 ) @ ( bNF_Wellorder_ordIso @ B @ C ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R @ R9 ) @ ( bNF_Wellorder_ordIso @ A @ C ) ) ) ) ).
% ordIso_transitive
thf(fact_4490_ordIso__symmetric,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ R ) @ ( bNF_Wellorder_ordIso @ B @ A ) ) ) ).
% ordIso_symmetric
thf(fact_4491_sum__list__transfer,axiom,
! [A: $tType,B: $tType] :
( ( ( monoid_add @ B )
& ( monoid_add @ A ) )
=> ! [A4: A > B > $o] :
( ( A4 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A4 @ ( bNF_rel_fun @ A @ B @ A @ B @ A4 @ A4 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
=> ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ A @ B @ ( list_all2 @ A @ B @ A4 ) @ A4 @ ( groups8242544230860333062m_list @ A ) @ ( groups8242544230860333062m_list @ B ) ) ) ) ) ).
% sum_list_transfer
thf(fact_4492_Misc_Olist__all2__induct,axiom,
! [A: $tType,B: $tType,P: A > B > $o,L: list @ A,L4: list @ B,Q: ( list @ A ) > ( list @ B ) > $o] :
( ( list_all2 @ A @ B @ P @ L @ L4 )
=> ( ( Q @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X2: A,X5: B,Ls2: list @ A,Ls3: list @ B] :
( ( P @ X2 @ X5 )
=> ( ( list_all2 @ A @ B @ P @ Ls2 @ Ls3 )
=> ( ( Q @ Ls2 @ Ls3 )
=> ( Q @ ( cons @ A @ X2 @ Ls2 ) @ ( cons @ B @ X5 @ Ls3 ) ) ) ) )
=> ( Q @ L @ L4 ) ) ) ) ).
% Misc.list_all2_induct
thf(fact_4493_ordIso__iff__ordLeq,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
= ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ) ).
% ordIso_iff_ordLeq
thf(fact_4494_ordIso__imp__ordLeq,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ).
% ordIso_imp_ordLeq
thf(fact_4495_ordIso__ordLeq__trans,axiom,
! [B: $tType,C: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),R9: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R4 @ R9 ) @ ( bNF_Wellorder_ordLeq @ B @ C ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R @ R9 ) @ ( bNF_Wellorder_ordLeq @ A @ C ) ) ) ) ).
% ordIso_ordLeq_trans
thf(fact_4496_ordLeq__ordIso__trans,axiom,
! [B: $tType,C: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),R9: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R4 @ R9 ) @ ( bNF_Wellorder_ordIso @ B @ C ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R @ R9 ) @ ( bNF_Wellorder_ordLeq @ A @ C ) ) ) ) ).
% ordLeq_ordIso_trans
thf(fact_4497_not__ordLess__ordIso,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
=> ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ).
% not_ordLess_ordIso
thf(fact_4498_ordIso__ordLess__trans,axiom,
! [B: $tType,C: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),R9: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R4 @ R9 ) @ ( bNF_We4044943003108391690rdLess @ B @ C ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R @ R9 ) @ ( bNF_We4044943003108391690rdLess @ A @ C ) ) ) ) ).
% ordIso_ordLess_trans
thf(fact_4499_ordLess__ordIso__trans,axiom,
! [B: $tType,C: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),R9: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R4 @ R9 ) @ ( bNF_Wellorder_ordIso @ B @ C ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R @ R9 ) @ ( bNF_We4044943003108391690rdLess @ A @ C ) ) ) ) ).
% ordLess_ordIso_trans
thf(fact_4500_sum__list__addf,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [F: B > A,G: B > A,Xs: list @ B] :
( ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X3: B] : ( plus_plus @ A @ ( F @ X3 ) @ ( G @ X3 ) )
@ Xs ) )
= ( plus_plus @ A @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F @ Xs ) ) @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ G @ Xs ) ) ) ) ) ).
% sum_list_addf
thf(fact_4501_finite__well__order__on__ordIso,axiom,
! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ A @ A4 )
=> ( ( order_well_order_on @ A @ A4 @ R )
=> ( ( order_well_order_on @ A @ A4 @ R4 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ) ) ).
% finite_well_order_on_ordIso
thf(fact_4502_ordIso__reflexive,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ R ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ).
% ordIso_reflexive
thf(fact_4503_uminus__sum__list__map,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [F: B > A,Xs: list @ B] :
( ( uminus_uminus @ A @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F @ Xs ) ) )
= ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ ( comp @ A @ A @ B @ ( uminus_uminus @ A ) @ F ) @ Xs ) ) ) ) ).
% uminus_sum_list_map
thf(fact_4504_ordLeq__iff__ordLess__or__ordIso,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
= ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
| ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ) ).
% ordLeq_iff_ordLess_or_ordIso
thf(fact_4505_sum__list_Oeq__foldr,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ( ( groups8242544230860333062m_list @ A )
= ( ^ [Xs3: list @ A] : ( foldr @ A @ A @ ( plus_plus @ A ) @ Xs3 @ ( zero_zero @ A ) ) ) ) ) ).
% sum_list.eq_foldr
thf(fact_4506_internalize__ordLeq,axiom,
! [A: $tType,B: $tType,R4: set @ ( product_prod @ A @ A ),R: set @ ( product_prod @ B @ B )] :
( ( member @ ( 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_Wellorder_ordLeq @ A @ B ) )
= ( ? [P6: set @ ( product_prod @ B @ B )] :
( ( ord_less_eq @ ( set @ B ) @ ( field2 @ B @ P6 ) @ ( field2 @ B @ R ) )
& ( member @ ( 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 @ P6 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P6 @ R ) @ ( bNF_Wellorder_ordLeq @ B @ B ) ) ) ) ) ).
% internalize_ordLeq
thf(fact_4507_internalize__ordLess,axiom,
! [A: $tType,B: $tType,R4: set @ ( product_prod @ A @ A ),R: set @ ( product_prod @ B @ B )] :
( ( member @ ( 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 ) )
= ( ? [P6: set @ ( product_prod @ B @ B )] :
( ( ord_less @ ( set @ B ) @ ( field2 @ B @ P6 ) @ ( field2 @ B @ R ) )
& ( member @ ( 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 @ P6 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P6 @ R ) @ ( bNF_We4044943003108391690rdLess @ B @ B ) ) ) ) ) ).
% internalize_ordLess
thf(fact_4508_card__length__sum__list__rec,axiom,
! [M2: nat,N6: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ M2 )
=> ( ( finite_card @ ( list @ nat )
@ ( collect @ ( list @ nat )
@ ^ [L3: list @ nat] :
( ( ( size_size @ ( list @ nat ) @ L3 )
= M2 )
& ( ( groups8242544230860333062m_list @ nat @ L3 )
= N6 ) ) ) )
= ( plus_plus @ nat
@ ( finite_card @ ( list @ nat )
@ ( collect @ ( list @ nat )
@ ^ [L3: list @ nat] :
( ( ( size_size @ ( list @ nat ) @ L3 )
= ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) )
& ( ( groups8242544230860333062m_list @ nat @ L3 )
= N6 ) ) ) )
@ ( finite_card @ ( list @ nat )
@ ( collect @ ( list @ nat )
@ ^ [L3: list @ nat] :
( ( ( size_size @ ( list @ nat ) @ L3 )
= M2 )
& ( ( plus_plus @ nat @ ( groups8242544230860333062m_list @ nat @ L3 ) @ ( one_one @ nat ) )
= N6 ) ) ) ) ) ) ) ).
% card_length_sum_list_rec
thf(fact_4509_size__list__conv__sum__list,axiom,
! [B: $tType] :
( ( size_list @ B )
= ( ^ [F2: B > nat,Xs3: list @ B] : ( plus_plus @ nat @ ( groups8242544230860333062m_list @ nat @ ( map @ B @ nat @ F2 @ Xs3 ) ) @ ( size_size @ ( list @ B ) @ Xs3 ) ) ) ) ).
% size_list_conv_sum_list
thf(fact_4510_card__length__sum__list,axiom,
! [M2: nat,N6: nat] :
( ( finite_card @ ( list @ nat )
@ ( collect @ ( list @ nat )
@ ^ [L3: list @ nat] :
( ( ( size_size @ ( list @ nat ) @ L3 )
= M2 )
& ( ( groups8242544230860333062m_list @ nat @ L3 )
= N6 ) ) ) )
= ( binomial @ ( minus_minus @ nat @ ( plus_plus @ nat @ N6 @ M2 ) @ ( one_one @ nat ) ) @ N6 ) ) ).
% card_length_sum_list
thf(fact_4511_sum__list__Suc,axiom,
! [A: $tType,F: A > nat,Xs: list @ A] :
( ( groups8242544230860333062m_list @ nat
@ ( map @ A @ nat
@ ^ [X3: A] : ( suc @ ( F @ X3 ) )
@ Xs ) )
= ( plus_plus @ nat @ ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% sum_list_Suc
thf(fact_4512_select__weight__def,axiom,
! [A: $tType] :
( ( select_weight @ A )
= ( ^ [Xs3: list @ ( product_prod @ code_natural @ A )] :
( product_scomp @ ( product_prod @ code_natural @ code_natural ) @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ A @ ( product_prod @ code_natural @ code_natural ) ) @ ( range @ ( groups8242544230860333062m_list @ code_natural @ ( map @ ( product_prod @ code_natural @ A ) @ code_natural @ ( product_fst @ code_natural @ A ) @ Xs3 ) ) )
@ ^ [K4: code_natural] : ( product_Pair @ A @ ( product_prod @ code_natural @ code_natural ) @ ( pick @ A @ Xs3 @ K4 ) ) ) ) ) ).
% select_weight_def
thf(fact_4513_prod__list__transfer,axiom,
! [A: $tType,B: $tType] :
( ( ( monoid_mult @ B )
& ( monoid_mult @ A ) )
=> ! [A4: A > B > $o] :
( ( A4 @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A4 @ ( bNF_rel_fun @ A @ B @ A @ B @ A4 @ A4 ) @ ( times_times @ A ) @ ( times_times @ B ) )
=> ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ A @ B @ ( list_all2 @ A @ B @ A4 ) @ A4 @ ( groups5270119922927024881d_list @ A ) @ ( groups5270119922927024881d_list @ B ) ) ) ) ) ).
% prod_list_transfer
thf(fact_4514_dir__image__ordIso,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),F: A > B] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( inj_on @ A @ B @ F @ ( field2 @ A @ R ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ ( bNF_We2720479622203943262_image @ A @ B @ R @ F ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ) ).
% dir_image_ordIso
thf(fact_4515_underS__incr,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( trans @ A @ R )
=> ( ( antisym @ A @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
=> ( ord_less_eq @ ( set @ A ) @ ( order_underS @ A @ R @ A3 ) @ ( order_underS @ A @ R @ B2 ) ) ) ) ) ).
% underS_incr
thf(fact_4516_prod__list_ONil,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ( ( groups5270119922927024881d_list @ A @ ( nil @ A ) )
= ( one_one @ A ) ) ) ).
% prod_list.Nil
thf(fact_4517_antisymD,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( antisym @ A @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ R )
=> ( A3 = B2 ) ) ) ) ).
% antisymD
thf(fact_4518_antisymI,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ! [X2: A,Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y4 ) @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X2 ) @ R )
=> ( X2 = Y4 ) ) )
=> ( antisym @ A @ R ) ) ).
% antisymI
thf(fact_4519_antisym__def,axiom,
! [A: $tType] :
( ( antisym @ A )
= ( ^ [R3: set @ ( product_prod @ A @ A )] :
! [X3: A,Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R3 )
=> ( X3 = Y3 ) ) ) ) ) ).
% antisym_def
thf(fact_4520_dir__image__def,axiom,
! [A2: $tType,A: $tType] :
( ( bNF_We2720479622203943262_image @ A @ A2 )
= ( ^ [R3: set @ ( product_prod @ A @ A ),F2: A > A2] :
( collect @ ( product_prod @ A2 @ A2 )
@ ^ [Uu: product_prod @ A2 @ A2] :
? [A7: A,B5: A] :
( ( Uu
= ( product_Pair @ A2 @ A2 @ ( F2 @ A7 ) @ ( F2 @ B5 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B5 ) @ R3 ) ) ) ) ) ).
% dir_image_def
thf(fact_4521_prod__list_Oeq__foldr,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ( ( groups5270119922927024881d_list @ A )
= ( ^ [Xs3: list @ A] : ( foldr @ A @ A @ ( times_times @ A ) @ Xs3 @ ( one_one @ A ) ) ) ) ) ).
% prod_list.eq_foldr
thf(fact_4522_pick__same,axiom,
! [A: $tType,L: nat,Xs: list @ A] :
( ( ord_less @ nat @ L @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( pick @ A @ ( map @ A @ ( product_prod @ code_natural @ A ) @ ( product_Pair @ code_natural @ A @ ( one_one @ code_natural ) ) @ Xs ) @ ( code_natural_of_nat @ L ) )
= ( nth @ A @ Xs @ L ) ) ) ).
% pick_same
thf(fact_4523_flip__pred,axiom,
! [A: $tType,B: $tType,A4: set @ ( product_prod @ A @ B ),R5: B > A > $o] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A4 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( conversep @ B @ A @ R5 ) ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) )
@ ( image2 @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A )
@ ( product_case_prod @ A @ B @ ( product_prod @ B @ A )
@ ^ [X3: A,Y3: B] : ( product_Pair @ B @ A @ Y3 @ X3 ) )
@ A4 )
@ ( collect @ ( product_prod @ B @ A ) @ ( product_case_prod @ B @ A @ $o @ R5 ) ) ) ) ).
% flip_pred
thf(fact_4524_prod__set__simps_I1_J,axiom,
! [B: $tType,A: $tType,X: A,Y: B] :
( ( basic_fsts @ A @ B @ ( product_Pair @ A @ B @ X @ Y ) )
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% prod_set_simps(1)
thf(fact_4525_prod__set__simps_I2_J,axiom,
! [A: $tType,B: $tType,X: A,Y: B] :
( ( basic_snds @ A @ B @ ( product_Pair @ A @ B @ X @ Y ) )
= ( insert2 @ B @ Y @ ( bot_bot @ ( set @ B ) ) ) ) ).
% prod_set_simps(2)
thf(fact_4526_natural__of__nat__of__nat,axiom,
( code_natural_of_nat
= ( semiring_1_of_nat @ code_natural ) ) ).
% natural_of_nat_of_nat
thf(fact_4527_division__segment__natural_Oabs__eq,axiom,
! [X: nat] :
( ( euclid7384307370059645450egment @ code_natural @ ( code_natural_of_nat @ X ) )
= ( code_natural_of_nat @ ( euclid7384307370059645450egment @ nat @ X ) ) ) ).
% division_segment_natural.abs_eq
thf(fact_4528_set__bit__natural_Oabs__eq,axiom,
! [Xa: nat,X: nat] :
( ( bit_se5668285175392031749et_bit @ code_natural @ Xa @ ( code_natural_of_nat @ X ) )
= ( code_natural_of_nat @ ( bit_se5668285175392031749et_bit @ nat @ Xa @ X ) ) ) ).
% set_bit_natural.abs_eq
thf(fact_4529_flip__bit__natural_Oabs__eq,axiom,
! [Xa: nat,X: nat] :
( ( bit_se8732182000553998342ip_bit @ code_natural @ Xa @ ( code_natural_of_nat @ X ) )
= ( code_natural_of_nat @ ( bit_se8732182000553998342ip_bit @ nat @ Xa @ X ) ) ) ).
% flip_bit_natural.abs_eq
thf(fact_4530_unset__bit__natural_Oabs__eq,axiom,
! [Xa: nat,X: nat] :
( ( bit_se2638667681897837118et_bit @ code_natural @ Xa @ ( code_natural_of_nat @ X ) )
= ( code_natural_of_nat @ ( bit_se2638667681897837118et_bit @ nat @ Xa @ X ) ) ) ).
% unset_bit_natural.abs_eq
thf(fact_4531_euclidean__size__natural_Oabs__eq,axiom,
! [X: nat] :
( ( euclid6346220572633701492n_size @ code_natural @ ( code_natural_of_nat @ X ) )
= ( euclid6346220572633701492n_size @ nat @ X ) ) ).
% euclidean_size_natural.abs_eq
thf(fact_4532_bit__natural_Oabs__eq,axiom,
! [X: nat] :
( ( bit_se5641148757651400278ts_bit @ code_natural @ ( code_natural_of_nat @ X ) )
= ( bit_se5641148757651400278ts_bit @ nat @ X ) ) ).
% bit_natural.abs_eq
thf(fact_4533_natural__of__nat__cases,axiom,
! [X: code_natural] :
~ ! [Y4: nat] :
( ( X
= ( code_natural_of_nat @ Y4 ) )
=> ~ ( member @ nat @ Y4 @ ( top_top @ ( set @ nat ) ) ) ) ).
% natural_of_nat_cases
thf(fact_4534_natural__of__nat__induct,axiom,
! [P: code_natural > $o,X: code_natural] :
( ! [Y4: nat] :
( ( member @ nat @ Y4 @ ( top_top @ ( set @ nat ) ) )
=> ( P @ ( code_natural_of_nat @ Y4 ) ) )
=> ( P @ X ) ) ).
% natural_of_nat_induct
thf(fact_4535_natural__of__nat__inject,axiom,
! [X: nat,Y: nat] :
( ( member @ nat @ X @ ( top_top @ ( set @ nat ) ) )
=> ( ( member @ nat @ Y @ ( top_top @ ( set @ nat ) ) )
=> ( ( ( code_natural_of_nat @ X )
= ( code_natural_of_nat @ Y ) )
= ( X = Y ) ) ) ) ).
% natural_of_nat_inject
thf(fact_4536_take__bit__natural_Oabs__eq,axiom,
! [Xa: nat,X: nat] :
( ( bit_se2584673776208193580ke_bit @ code_natural @ Xa @ ( code_natural_of_nat @ X ) )
= ( code_natural_of_nat @ ( bit_se2584673776208193580ke_bit @ nat @ Xa @ X ) ) ) ).
% take_bit_natural.abs_eq
thf(fact_4537_and__natural_Oabs__eq,axiom,
! [Xa: nat,X: nat] :
( ( bit_se5824344872417868541ns_and @ code_natural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X ) )
= ( code_natural_of_nat @ ( bit_se5824344872417868541ns_and @ nat @ Xa @ X ) ) ) ).
% and_natural.abs_eq
thf(fact_4538_xor__natural_Oabs__eq,axiom,
! [Xa: nat,X: nat] :
( ( bit_se5824344971392196577ns_xor @ code_natural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X ) )
= ( code_natural_of_nat @ ( bit_se5824344971392196577ns_xor @ nat @ Xa @ X ) ) ) ).
% xor_natural.abs_eq
thf(fact_4539_or__natural_Oabs__eq,axiom,
! [Xa: nat,X: nat] :
( ( bit_se1065995026697491101ons_or @ code_natural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X ) )
= ( code_natural_of_nat @ ( bit_se1065995026697491101ons_or @ nat @ Xa @ X ) ) ) ).
% or_natural.abs_eq
thf(fact_4540_mask__natural_Oabs__eq,axiom,
( ( bit_se2239418461657761734s_mask @ code_natural )
= ( ^ [X3: nat] : ( code_natural_of_nat @ ( bit_se2239418461657761734s_mask @ nat @ X3 ) ) ) ) ).
% mask_natural.abs_eq
thf(fact_4541_push__bit__natural_Oabs__eq,axiom,
! [Xa: nat,X: nat] :
( ( bit_se4730199178511100633sh_bit @ code_natural @ Xa @ ( code_natural_of_nat @ X ) )
= ( code_natural_of_nat @ ( bit_se4730199178511100633sh_bit @ nat @ Xa @ X ) ) ) ).
% push_bit_natural.abs_eq
thf(fact_4542_drop__bit__natural_Oabs__eq,axiom,
! [Xa: nat,X: nat] :
( ( bit_se4197421643247451524op_bit @ code_natural @ Xa @ ( code_natural_of_nat @ X ) )
= ( code_natural_of_nat @ ( bit_se4197421643247451524op_bit @ nat @ Xa @ X ) ) ) ).
% drop_bit_natural.abs_eq
thf(fact_4543_zero__natural__def,axiom,
( ( zero_zero @ code_natural )
= ( code_natural_of_nat @ ( zero_zero @ nat ) ) ) ).
% zero_natural_def
thf(fact_4544_one__natural__def,axiom,
( ( one_one @ code_natural )
= ( code_natural_of_nat @ ( one_one @ nat ) ) ) ).
% one_natural_def
thf(fact_4545_less__natural_Oabs__eq,axiom,
! [Xa: nat,X: nat] :
( ( ord_less @ code_natural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X ) )
= ( ord_less @ nat @ Xa @ X ) ) ).
% less_natural.abs_eq
thf(fact_4546_plus__natural_Oabs__eq,axiom,
! [Xa: nat,X: nat] :
( ( plus_plus @ code_natural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X ) )
= ( code_natural_of_nat @ ( plus_plus @ nat @ Xa @ X ) ) ) ).
% plus_natural.abs_eq
thf(fact_4547_minus__natural_Oabs__eq,axiom,
! [Xa: nat,X: nat] :
( ( minus_minus @ code_natural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X ) )
= ( code_natural_of_nat @ ( minus_minus @ nat @ Xa @ X ) ) ) ).
% minus_natural.abs_eq
thf(fact_4548_less__eq__natural_Oabs__eq,axiom,
! [Xa: nat,X: nat] :
( ( ord_less_eq @ code_natural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X ) )
= ( ord_less_eq @ nat @ Xa @ X ) ) ).
% less_eq_natural.abs_eq
thf(fact_4549_times__natural_Oabs__eq,axiom,
! [Xa: nat,X: nat] :
( ( times_times @ code_natural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X ) )
= ( code_natural_of_nat @ ( times_times @ nat @ Xa @ X ) ) ) ).
% times_natural.abs_eq
thf(fact_4550_divide__natural_Oabs__eq,axiom,
! [Xa: nat,X: nat] :
( ( divide_divide @ code_natural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X ) )
= ( code_natural_of_nat @ ( divide_divide @ nat @ Xa @ X ) ) ) ).
% divide_natural.abs_eq
thf(fact_4551_mask__natural__def,axiom,
( ( bit_se2239418461657761734s_mask @ code_natural )
= ( map_fun @ nat @ nat @ nat @ code_natural @ ( id @ nat ) @ code_natural_of_nat @ ( bit_se2239418461657761734s_mask @ nat ) ) ) ).
% mask_natural_def
thf(fact_4552_modulo__natural_Oabs__eq,axiom,
! [Xa: nat,X: nat] :
( ( modulo_modulo @ code_natural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X ) )
= ( code_natural_of_nat @ ( modulo_modulo @ nat @ Xa @ X ) ) ) ).
% modulo_natural.abs_eq
thf(fact_4553_select__def,axiom,
! [A: $tType] :
( ( select @ A )
= ( ^ [Xs3: list @ A] :
( product_scomp @ ( product_prod @ code_natural @ code_natural ) @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ A @ ( product_prod @ code_natural @ code_natural ) ) @ ( range @ ( code_natural_of_nat @ ( size_size @ ( list @ A ) @ Xs3 ) ) )
@ ^ [K4: code_natural] : ( product_Pair @ A @ ( product_prod @ code_natural @ code_natural ) @ ( nth @ A @ Xs3 @ ( code_nat_of_natural @ K4 ) ) ) ) ) ) ).
% select_def
thf(fact_4554_Nat_Oabs__eq,axiom,
! [X: int] :
( ( code_Nat @ ( code_integer_of_int @ X ) )
= ( code_natural_of_nat @ ( nat2 @ X ) ) ) ).
% Nat.abs_eq
thf(fact_4555_Nat__def,axiom,
( code_Nat
= ( map_fun @ code_integer @ int @ nat @ code_natural @ code_int_of_integer @ code_natural_of_nat @ nat2 ) ) ).
% Nat_def
thf(fact_4556_natural__of__integer_Oabs__eq,axiom,
! [X: int] :
( ( code_n4118661773612635043nteger @ ( code_integer_of_int @ X ) )
= ( code_natural_of_nat @ ( nat2 @ X ) ) ) ).
% natural_of_integer.abs_eq
thf(fact_4557_natural__of__nat__of__natural__inverse,axiom,
! [N: code_natural] :
( ( code_natural_of_nat @ ( code_nat_of_natural @ N ) )
= N ) ).
% natural_of_nat_of_natural_inverse
thf(fact_4558_nat__of__natural__of__nat__inverse,axiom,
! [N: nat] :
( ( code_nat_of_natural @ ( code_natural_of_nat @ N ) )
= N ) ).
% nat_of_natural_of_nat_inverse
thf(fact_4559_nat__of__natural__inverse,axiom,
! [X: code_natural] :
( ( code_natural_of_nat @ ( code_nat_of_natural @ X ) )
= X ) ).
% nat_of_natural_inverse
thf(fact_4560_nat__of__natural__min,axiom,
! [K: code_natural,L: code_natural] :
( ( code_nat_of_natural @ ( ord_min @ code_natural @ K @ L ) )
= ( ord_min @ nat @ ( code_nat_of_natural @ K ) @ ( code_nat_of_natural @ L ) ) ) ).
% nat_of_natural_min
thf(fact_4561_nat__of__natural__max,axiom,
! [K: code_natural,L: code_natural] :
( ( code_nat_of_natural @ ( ord_max @ code_natural @ K @ L ) )
= ( ord_max @ nat @ ( code_nat_of_natural @ K ) @ ( code_nat_of_natural @ L ) ) ) ).
% nat_of_natural_max
thf(fact_4562_of__nat__of__natural,axiom,
! [N: code_natural] :
( ( semiring_1_of_nat @ code_natural @ ( code_nat_of_natural @ N ) )
= N ) ).
% of_nat_of_natural
thf(fact_4563_nat__of__natural__of__nat,axiom,
! [N: nat] :
( ( code_nat_of_natural @ ( semiring_1_of_nat @ code_natural @ N ) )
= N ) ).
% nat_of_natural_of_nat
thf(fact_4564_zero__natural_Orep__eq,axiom,
( ( code_nat_of_natural @ ( zero_zero @ code_natural ) )
= ( zero_zero @ nat ) ) ).
% zero_natural.rep_eq
thf(fact_4565_one__natural_Orep__eq,axiom,
( ( code_nat_of_natural @ ( one_one @ code_natural ) )
= ( one_one @ nat ) ) ).
% one_natural.rep_eq
thf(fact_4566_plus__natural_Orep__eq,axiom,
! [X: code_natural,Xa: code_natural] :
( ( code_nat_of_natural @ ( plus_plus @ code_natural @ X @ Xa ) )
= ( plus_plus @ nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa ) ) ) ).
% plus_natural.rep_eq
thf(fact_4567_minus__natural_Orep__eq,axiom,
! [X: code_natural,Xa: code_natural] :
( ( code_nat_of_natural @ ( minus_minus @ code_natural @ X @ Xa ) )
= ( minus_minus @ nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa ) ) ) ).
% minus_natural.rep_eq
thf(fact_4568_nat__of__natural__numeral,axiom,
! [K: num] :
( ( code_nat_of_natural @ ( numeral_numeral @ code_natural @ K ) )
= ( numeral_numeral @ nat @ K ) ) ).
% nat_of_natural_numeral
thf(fact_4569_times__natural_Orep__eq,axiom,
! [X: code_natural,Xa: code_natural] :
( ( code_nat_of_natural @ ( times_times @ code_natural @ X @ Xa ) )
= ( times_times @ nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa ) ) ) ).
% times_natural.rep_eq
thf(fact_4570_divide__natural_Orep__eq,axiom,
! [X: code_natural,Xa: code_natural] :
( ( code_nat_of_natural @ ( divide_divide @ code_natural @ X @ Xa ) )
= ( divide_divide @ nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa ) ) ) ).
% divide_natural.rep_eq
thf(fact_4571_modulo__natural_Orep__eq,axiom,
! [X: code_natural,Xa: code_natural] :
( ( code_nat_of_natural @ ( modulo_modulo @ code_natural @ X @ Xa ) )
= ( modulo_modulo @ nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa ) ) ) ).
% modulo_natural.rep_eq
thf(fact_4572_division__segment__natural_Orep__eq,axiom,
! [X: code_natural] :
( ( code_nat_of_natural @ ( euclid7384307370059645450egment @ code_natural @ X ) )
= ( euclid7384307370059645450egment @ nat @ ( code_nat_of_natural @ X ) ) ) ).
% division_segment_natural.rep_eq
thf(fact_4573_euclidean__size__natural_Orep__eq,axiom,
( ( euclid6346220572633701492n_size @ code_natural )
= ( ^ [X3: code_natural] : ( euclid6346220572633701492n_size @ nat @ ( code_nat_of_natural @ X3 ) ) ) ) ).
% euclidean_size_natural.rep_eq
thf(fact_4574_unset__bit__natural_Orep__eq,axiom,
! [X: nat,Xa: code_natural] :
( ( code_nat_of_natural @ ( bit_se2638667681897837118et_bit @ code_natural @ X @ Xa ) )
= ( bit_se2638667681897837118et_bit @ nat @ X @ ( code_nat_of_natural @ Xa ) ) ) ).
% unset_bit_natural.rep_eq
thf(fact_4575_flip__bit__natural_Orep__eq,axiom,
! [X: nat,Xa: code_natural] :
( ( code_nat_of_natural @ ( bit_se8732182000553998342ip_bit @ code_natural @ X @ Xa ) )
= ( bit_se8732182000553998342ip_bit @ nat @ X @ ( code_nat_of_natural @ Xa ) ) ) ).
% flip_bit_natural.rep_eq
thf(fact_4576_set__bit__natural_Orep__eq,axiom,
! [X: nat,Xa: code_natural] :
( ( code_nat_of_natural @ ( bit_se5668285175392031749et_bit @ code_natural @ X @ Xa ) )
= ( bit_se5668285175392031749et_bit @ nat @ X @ ( code_nat_of_natural @ Xa ) ) ) ).
% set_bit_natural.rep_eq
thf(fact_4577_Nat_Orep__eq,axiom,
! [X: code_integer] :
( ( code_nat_of_natural @ ( code_Nat @ X ) )
= ( nat2 @ ( code_int_of_integer @ X ) ) ) ).
% Nat.rep_eq
thf(fact_4578_push__bit__natural_Orep__eq,axiom,
! [X: nat,Xa: code_natural] :
( ( code_nat_of_natural @ ( bit_se4730199178511100633sh_bit @ code_natural @ X @ Xa ) )
= ( bit_se4730199178511100633sh_bit @ nat @ X @ ( code_nat_of_natural @ Xa ) ) ) ).
% push_bit_natural.rep_eq
thf(fact_4579_mask__natural_Orep__eq,axiom,
! [X: nat] :
( ( code_nat_of_natural @ ( bit_se2239418461657761734s_mask @ code_natural @ X ) )
= ( bit_se2239418461657761734s_mask @ nat @ X ) ) ).
% mask_natural.rep_eq
thf(fact_4580_and__natural_Orep__eq,axiom,
! [X: code_natural,Xa: code_natural] :
( ( code_nat_of_natural @ ( bit_se5824344872417868541ns_and @ code_natural @ X @ Xa ) )
= ( bit_se5824344872417868541ns_and @ nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa ) ) ) ).
% and_natural.rep_eq
thf(fact_4581_bit__natural_Orep__eq,axiom,
( ( bit_se5641148757651400278ts_bit @ code_natural )
= ( ^ [X3: code_natural] : ( bit_se5641148757651400278ts_bit @ nat @ ( code_nat_of_natural @ X3 ) ) ) ) ).
% bit_natural.rep_eq
thf(fact_4582_nat__of__natural__inject,axiom,
! [X: code_natural,Y: code_natural] :
( ( ( code_nat_of_natural @ X )
= ( code_nat_of_natural @ Y ) )
= ( X = Y ) ) ).
% nat_of_natural_inject
thf(fact_4583_natural__eqI,axiom,
! [M2: code_natural,N: code_natural] :
( ( ( code_nat_of_natural @ M2 )
= ( code_nat_of_natural @ N ) )
=> ( M2 = N ) ) ).
% natural_eqI
thf(fact_4584_natural__eq__iff,axiom,
( ( ^ [Y5: code_natural,Z4: code_natural] : Y5 = Z4 )
= ( ^ [M: code_natural,N5: code_natural] :
( ( code_nat_of_natural @ M )
= ( code_nat_of_natural @ N5 ) ) ) ) ).
% natural_eq_iff
thf(fact_4585_natural__cases,axiom,
! [M2: code_natural] :
~ ! [N3: nat] :
( M2
!= ( semiring_1_of_nat @ code_natural @ N3 ) ) ).
% natural_cases
thf(fact_4586_nat__of__natural,axiom,
! [X: code_natural] : ( member @ nat @ ( code_nat_of_natural @ X ) @ ( top_top @ ( set @ nat ) ) ) ).
% nat_of_natural
thf(fact_4587_nat__of__natural__cases,axiom,
! [Y: nat] :
( ( member @ nat @ Y @ ( top_top @ ( set @ nat ) ) )
=> ~ ! [X2: code_natural] :
( Y
!= ( code_nat_of_natural @ X2 ) ) ) ).
% nat_of_natural_cases
thf(fact_4588_nat__of__natural__induct,axiom,
! [Y: nat,P: nat > $o] :
( ( member @ nat @ Y @ ( top_top @ ( set @ nat ) ) )
=> ( ! [X2: code_natural] : ( P @ ( code_nat_of_natural @ X2 ) )
=> ( P @ Y ) ) ) ).
% nat_of_natural_induct
thf(fact_4589_take__bit__natural_Orep__eq,axiom,
! [X: nat,Xa: code_natural] :
( ( code_nat_of_natural @ ( bit_se2584673776208193580ke_bit @ code_natural @ X @ Xa ) )
= ( bit_se2584673776208193580ke_bit @ nat @ X @ ( code_nat_of_natural @ Xa ) ) ) ).
% take_bit_natural.rep_eq
thf(fact_4590_or__natural_Orep__eq,axiom,
! [X: code_natural,Xa: code_natural] :
( ( code_nat_of_natural @ ( bit_se1065995026697491101ons_or @ code_natural @ X @ Xa ) )
= ( bit_se1065995026697491101ons_or @ nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa ) ) ) ).
% or_natural.rep_eq
thf(fact_4591_xor__natural_Orep__eq,axiom,
! [X: code_natural,Xa: code_natural] :
( ( code_nat_of_natural @ ( bit_se5824344971392196577ns_xor @ code_natural @ X @ Xa ) )
= ( bit_se5824344971392196577ns_xor @ nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa ) ) ) ).
% xor_natural.rep_eq
thf(fact_4592_drop__bit__natural_Orep__eq,axiom,
! [X: nat,Xa: code_natural] :
( ( code_nat_of_natural @ ( bit_se4197421643247451524op_bit @ code_natural @ X @ Xa ) )
= ( bit_se4197421643247451524op_bit @ nat @ X @ ( code_nat_of_natural @ Xa ) ) ) ).
% drop_bit_natural.rep_eq
thf(fact_4593_natural__of__integer_Orep__eq,axiom,
! [X: code_integer] :
( ( code_nat_of_natural @ ( code_n4118661773612635043nteger @ X ) )
= ( nat2 @ ( code_int_of_integer @ X ) ) ) ).
% natural_of_integer.rep_eq
thf(fact_4594_less__natural_Orep__eq,axiom,
( ( ord_less @ code_natural )
= ( ^ [X3: code_natural,Xa4: code_natural] : ( ord_less @ nat @ ( code_nat_of_natural @ X3 ) @ ( code_nat_of_natural @ Xa4 ) ) ) ) ).
% less_natural.rep_eq
thf(fact_4595_natural__of__nat__inverse,axiom,
! [Y: nat] :
( ( member @ nat @ Y @ ( top_top @ ( set @ nat ) ) )
=> ( ( code_nat_of_natural @ ( code_natural_of_nat @ Y ) )
= Y ) ) ).
% natural_of_nat_inverse
thf(fact_4596_type__definition__natural,axiom,
type_definition @ code_natural @ nat @ code_nat_of_natural @ code_natural_of_nat @ ( top_top @ ( set @ nat ) ) ).
% type_definition_natural
thf(fact_4597_or__natural__def,axiom,
( ( bit_se1065995026697491101ons_or @ code_natural )
= ( map_fun @ code_natural @ nat @ ( nat > nat ) @ ( 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
thf(fact_4598_xor__natural__def,axiom,
( ( bit_se5824344971392196577ns_xor @ code_natural )
= ( map_fun @ code_natural @ nat @ ( nat > nat ) @ ( 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
thf(fact_4599_and__natural__def,axiom,
( ( bit_se5824344872417868541ns_and @ code_natural )
= ( map_fun @ code_natural @ nat @ ( nat > nat ) @ ( 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
thf(fact_4600_less__eq__natural_Orep__eq,axiom,
( ( ord_less_eq @ code_natural )
= ( ^ [X3: code_natural,Xa4: code_natural] : ( ord_less_eq @ nat @ ( code_nat_of_natural @ X3 ) @ ( code_nat_of_natural @ Xa4 ) ) ) ) ).
% less_eq_natural.rep_eq
thf(fact_4601_division__segment__natural__def,axiom,
( ( euclid7384307370059645450egment @ code_natural )
= ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat @ ( euclid7384307370059645450egment @ nat ) ) ) ).
% division_segment_natural_def
thf(fact_4602_euclidean__size__natural__def,axiom,
( ( euclid6346220572633701492n_size @ code_natural )
= ( map_fun @ code_natural @ nat @ nat @ nat @ code_nat_of_natural @ ( id @ nat ) @ ( euclid6346220572633701492n_size @ nat ) ) ) ).
% euclidean_size_natural_def
thf(fact_4603_plus__natural__def,axiom,
( ( plus_plus @ code_natural )
= ( map_fun @ code_natural @ nat @ ( nat > nat ) @ ( 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
thf(fact_4604_minus__natural__def,axiom,
( ( minus_minus @ code_natural )
= ( map_fun @ code_natural @ nat @ ( nat > nat ) @ ( 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
thf(fact_4605_bit__natural__def,axiom,
( ( bit_se5641148757651400278ts_bit @ code_natural )
= ( map_fun @ code_natural @ nat @ ( nat > $o ) @ ( nat > $o ) @ code_nat_of_natural @ ( id @ ( nat > $o ) ) @ ( bit_se5641148757651400278ts_bit @ nat ) ) ) ).
% bit_natural_def
thf(fact_4606_less__natural__def,axiom,
( ( ord_less @ code_natural )
= ( map_fun @ code_natural @ nat @ ( nat > $o ) @ ( code_natural > $o ) @ code_nat_of_natural @ ( map_fun @ code_natural @ nat @ $o @ $o @ code_nat_of_natural @ ( id @ $o ) ) @ ( ord_less @ nat ) ) ) ).
% less_natural_def
thf(fact_4607_times__natural__def,axiom,
( ( times_times @ code_natural )
= ( map_fun @ code_natural @ nat @ ( nat > nat ) @ ( 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
thf(fact_4608_divide__natural__def,axiom,
( ( divide_divide @ code_natural )
= ( map_fun @ code_natural @ nat @ ( nat > nat ) @ ( 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
thf(fact_4609_take__bit__natural__def,axiom,
( ( bit_se2584673776208193580ke_bit @ code_natural )
= ( map_fun @ nat @ nat @ ( nat > nat ) @ ( 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
thf(fact_4610_push__bit__natural__def,axiom,
( ( bit_se4730199178511100633sh_bit @ code_natural )
= ( map_fun @ nat @ nat @ ( nat > nat ) @ ( 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
thf(fact_4611_less__eq__natural__def,axiom,
( ( ord_less_eq @ code_natural )
= ( map_fun @ code_natural @ nat @ ( nat > $o ) @ ( code_natural > $o ) @ code_nat_of_natural @ ( map_fun @ code_natural @ nat @ $o @ $o @ code_nat_of_natural @ ( id @ $o ) ) @ ( ord_less_eq @ nat ) ) ) ).
% less_eq_natural_def
thf(fact_4612_modulo__natural__def,axiom,
( ( modulo_modulo @ code_natural )
= ( map_fun @ code_natural @ nat @ ( nat > nat ) @ ( 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
thf(fact_4613_drop__bit__natural__def,axiom,
( ( bit_se4197421643247451524op_bit @ code_natural )
= ( map_fun @ nat @ nat @ ( nat > nat ) @ ( 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
thf(fact_4614_unset__bit__natural__def,axiom,
( ( bit_se2638667681897837118et_bit @ code_natural )
= ( map_fun @ nat @ nat @ ( nat > nat ) @ ( 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
thf(fact_4615_flip__bit__natural__def,axiom,
( ( bit_se8732182000553998342ip_bit @ code_natural )
= ( map_fun @ nat @ nat @ ( nat > nat ) @ ( 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
thf(fact_4616_set__bit__natural__def,axiom,
( ( bit_se5668285175392031749et_bit @ code_natural )
= ( map_fun @ nat @ nat @ ( nat > nat ) @ ( 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
thf(fact_4617_natural__decr,axiom,
! [N: code_natural] :
( ( N
!= ( zero_zero @ code_natural ) )
=> ( ord_less @ nat @ ( minus_minus @ nat @ ( code_nat_of_natural @ N ) @ ( suc @ ( zero_zero @ nat ) ) ) @ ( code_nat_of_natural @ N ) ) ) ).
% natural_decr
thf(fact_4618_natural__of__integer__def,axiom,
( code_n4118661773612635043nteger
= ( map_fun @ code_integer @ int @ nat @ code_natural @ code_int_of_integer @ code_natural_of_nat @ nat2 ) ) ).
% natural_of_integer_def
thf(fact_4619_integer__of__natural__of__integer,axiom,
! [K: code_integer] :
( ( code_i5400310926305786745atural @ ( code_n4118661773612635043nteger @ K ) )
= ( ord_max @ code_integer @ ( zero_zero @ code_integer ) @ K ) ) ).
% integer_of_natural_of_integer
thf(fact_4620_integer__of__natural_Orep__eq,axiom,
! [X: code_natural] :
( ( code_int_of_integer @ ( code_i5400310926305786745atural @ X ) )
= ( semiring_1_of_nat @ int @ ( code_nat_of_natural @ X ) ) ) ).
% integer_of_natural.rep_eq
thf(fact_4621_int__of__integer__of__natural,axiom,
! [N: code_natural] :
( ( code_int_of_integer @ ( code_i5400310926305786745atural @ N ) )
= ( semiring_1_of_nat @ int @ ( code_nat_of_natural @ N ) ) ) ).
% int_of_integer_of_natural
thf(fact_4622_integer__of__natural__of__nat,axiom,
! [N: nat] :
( ( code_i5400310926305786745atural @ ( semiring_1_of_nat @ code_natural @ N ) )
= ( semiring_1_of_nat @ code_integer @ N ) ) ).
% integer_of_natural_of_nat
thf(fact_4623_natural__of__integer__of__natural,axiom,
! [N: code_natural] :
( ( code_n4118661773612635043nteger @ ( code_i5400310926305786745atural @ N ) )
= N ) ).
% natural_of_integer_of_natural
thf(fact_4624_integer__of__natural__def,axiom,
( code_i5400310926305786745atural
= ( 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
thf(fact_4625_integer__of__natural_Oabs__eq,axiom,
! [X: nat] :
( ( code_i5400310926305786745atural @ ( code_natural_of_nat @ X ) )
= ( code_integer_of_int @ ( semiring_1_of_nat @ int @ X ) ) ) ).
% integer_of_natural.abs_eq
thf(fact_4626_Code__Numeral_Osize__nat__def,axiom,
code_size_nat = code_nat_of_natural ).
% Code_Numeral.size_nat_def
thf(fact_4627_Code__Numeral_OSuc__def,axiom,
( code_Suc
= ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat @ suc ) ) ).
% Code_Numeral.Suc_def
thf(fact_4628_sum__mset_Ounion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: multiset @ B,B6: multiset @ B,G: B > A] :
( ( ( inter_mset @ B @ A4 @ B6 )
= ( zero_zero @ ( multiset @ B ) ) )
=> ( ( comm_m7189776963980413722m_mset @ A @ ( image_mset @ B @ A @ G @ ( union_mset @ B @ A4 @ B6 ) ) )
= ( plus_plus @ A @ ( comm_m7189776963980413722m_mset @ A @ ( image_mset @ B @ A @ G @ A4 ) ) @ ( comm_m7189776963980413722m_mset @ A @ ( image_mset @ B @ A @ G @ B6 ) ) ) ) ) ) ).
% sum_mset.union_disjoint
thf(fact_4629_natural_Oinject,axiom,
! [Natural: code_natural,Natural2: code_natural] :
( ( ( code_Suc @ Natural )
= ( code_Suc @ Natural2 ) )
= ( Natural = Natural2 ) ) ).
% natural.inject
thf(fact_4630_natural_Osimps_I7_J,axiom,
! [T: $tType,F1: T,F22: code_natural > T > T,Natural: code_natural] :
( ( code_rec_natural @ T @ F1 @ F22 @ ( code_Suc @ Natural ) )
= ( F22 @ Natural @ ( code_rec_natural @ T @ F1 @ F22 @ Natural ) ) ) ).
% natural.simps(7)
thf(fact_4631_natural_Osimps_I5_J,axiom,
! [T: $tType,F1: T,F22: code_natural > T,Natural: code_natural] :
( ( code_case_natural @ T @ F1 @ F22 @ ( code_Suc @ Natural ) )
= ( F22 @ Natural ) ) ).
% natural.simps(5)
thf(fact_4632_Suc_Orep__eq,axiom,
! [X: code_natural] :
( ( code_nat_of_natural @ ( code_Suc @ X ) )
= ( suc @ ( code_nat_of_natural @ X ) ) ) ).
% Suc.rep_eq
thf(fact_4633_union__sup__distrib__right,axiom,
! [A: $tType,C5: multiset @ A,A4: multiset @ A,B6: multiset @ A] :
( ( plus_plus @ ( multiset @ A ) @ C5 @ ( union_mset @ A @ A4 @ B6 ) )
= ( union_mset @ A @ ( plus_plus @ ( multiset @ A ) @ C5 @ A4 ) @ ( plus_plus @ ( multiset @ A ) @ C5 @ B6 ) ) ) ).
% union_sup_distrib_right
thf(fact_4634_sup__union__distrib__left,axiom,
! [A: $tType,A4: multiset @ A,B6: multiset @ A,C5: multiset @ A] :
( ( plus_plus @ ( multiset @ A ) @ ( union_mset @ A @ A4 @ B6 ) @ C5 )
= ( union_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A4 @ C5 ) @ ( plus_plus @ ( multiset @ A ) @ B6 @ C5 ) ) ) ).
% sup_union_distrib_left
thf(fact_4635_natural_Oexhaust,axiom,
! [Y: code_natural] :
( ( Y
!= ( zero_zero @ code_natural ) )
=> ~ ! [Natural3: code_natural] :
( Y
!= ( code_Suc @ Natural3 ) ) ) ).
% natural.exhaust
thf(fact_4636_natural_Odistinct_I1_J,axiom,
! [Natural2: code_natural] :
( ( zero_zero @ code_natural )
!= ( code_Suc @ Natural2 ) ) ).
% natural.distinct(1)
thf(fact_4637_natural_Odistinct_I2_J,axiom,
! [Natural4: code_natural] :
( ( code_Suc @ Natural4 )
!= ( zero_zero @ code_natural ) ) ).
% natural.distinct(2)
thf(fact_4638_union__mset__def,axiom,
! [A: $tType] :
( ( union_mset @ A )
= ( ^ [A9: multiset @ A,B9: multiset @ A] : ( plus_plus @ ( multiset @ A ) @ A9 @ ( minus_minus @ ( multiset @ A ) @ B9 @ A9 ) ) ) ) ).
% union_mset_def
thf(fact_4639_union__diff__inter__eq__sup,axiom,
! [A: $tType,A4: multiset @ A,B6: multiset @ A] :
( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) @ ( inter_mset @ A @ A4 @ B6 ) )
= ( union_mset @ A @ A4 @ B6 ) ) ).
% union_diff_inter_eq_sup
thf(fact_4640_union__diff__sup__eq__inter,axiom,
! [A: $tType,A4: multiset @ A,B6: multiset @ A] :
( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) @ ( union_mset @ A @ A4 @ B6 ) )
= ( inter_mset @ A @ A4 @ B6 ) ) ).
% union_diff_sup_eq_inter
thf(fact_4641_size__Un__Int,axiom,
! [A: $tType,A4: multiset @ A,B6: multiset @ A] :
( ( plus_plus @ nat @ ( size_size @ ( multiset @ A ) @ A4 ) @ ( size_size @ ( multiset @ A ) @ B6 ) )
= ( plus_plus @ nat @ ( size_size @ ( multiset @ A ) @ ( union_mset @ A @ A4 @ B6 ) ) @ ( size_size @ ( multiset @ A ) @ ( inter_mset @ A @ A4 @ B6 ) ) ) ) ).
% size_Un_Int
thf(fact_4642_Suc_Oabs__eq,axiom,
! [X: nat] :
( ( code_Suc @ ( code_natural_of_nat @ X ) )
= ( code_natural_of_nat @ ( suc @ X ) ) ) ).
% Suc.abs_eq
thf(fact_4643_Suc__natural__minus__one,axiom,
! [N: code_natural] :
( ( minus_minus @ code_natural @ ( code_Suc @ N ) @ ( one_one @ code_natural ) )
= N ) ).
% Suc_natural_minus_one
thf(fact_4644_size__Un__disjoint,axiom,
! [A: $tType,A4: multiset @ A,B6: multiset @ A] :
( ( ( inter_mset @ A @ A4 @ B6 )
= ( zero_zero @ ( multiset @ A ) ) )
=> ( ( size_size @ ( multiset @ A ) @ ( union_mset @ A @ A4 @ B6 ) )
= ( plus_plus @ nat @ ( size_size @ ( multiset @ A ) @ A4 ) @ ( size_size @ ( multiset @ A ) @ B6 ) ) ) ) ).
% size_Un_disjoint
thf(fact_4645_random__aux__set_Ocases,axiom,
! [X: product_prod @ code_natural @ code_natural] :
( ! [J3: code_natural] :
( X
!= ( product_Pair @ code_natural @ code_natural @ ( zero_zero @ code_natural ) @ J3 ) )
=> ~ ! [I2: code_natural,J3: code_natural] :
( X
!= ( product_Pair @ code_natural @ code_natural @ ( code_Suc @ I2 ) @ J3 ) ) ) ).
% random_aux_set.cases
thf(fact_4646_Eps__Opt__def,axiom,
! [A: $tType] :
( ( eps_Opt @ A )
= ( ^ [P4: A > $o] :
( if @ ( option @ A )
@ ? [X8: A] : ( P4 @ X8 )
@ ( some @ A @ ( fChoice @ A @ P4 ) )
@ ( none @ A ) ) ) ) ).
% Eps_Opt_def
thf(fact_4647_acyclic__insert,axiom,
! [A: $tType,Y: A,X: A,R: set @ ( product_prod @ A @ A )] :
( ( transitive_acyclic @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ R ) )
= ( ( transitive_acyclic @ A @ R )
& ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ).
% acyclic_insert
thf(fact_4648_acyclic__empty,axiom,
! [A: $tType] : ( transitive_acyclic @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% acyclic_empty
thf(fact_4649_Eps__case__prod__eq,axiom,
! [A: $tType,B: $tType,X: A,Y: B] :
( ( fChoice @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X9: A,Y8: B] :
( ( X = X9 )
& ( Y = Y8 ) ) ) )
= ( product_Pair @ A @ B @ X @ Y ) ) ).
% Eps_case_prod_eq
thf(fact_4650_some__insert__self,axiom,
! [A: $tType,S2: set @ A] :
( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( insert2 @ A
@ ( fChoice @ A
@ ^ [X3: A] : ( member @ A @ X3 @ S2 ) )
@ S2 )
= S2 ) ) ).
% some_insert_self
thf(fact_4651_some__elem,axiom,
! [A: $tType,S2: set @ A] :
( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A
@ ( fChoice @ A
@ ^ [X3: A] : ( member @ A @ X3 @ S2 ) )
@ S2 ) ) ).
% some_elem
thf(fact_4652_cyclic__subset,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ A @ A )] :
( ~ ( transitive_acyclic @ A @ R5 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R5 @ S2 )
=> ~ ( transitive_acyclic @ A @ S2 ) ) ) ).
% cyclic_subset
thf(fact_4653_acyclic__union_I1_J,axiom,
! [A: $tType,A4: set @ ( product_prod @ A @ A ),B6: set @ ( product_prod @ A @ A )] :
( ( transitive_acyclic @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ A4 @ B6 ) )
=> ( transitive_acyclic @ A @ A4 ) ) ).
% acyclic_union(1)
thf(fact_4654_acyclic__union_I2_J,axiom,
! [A: $tType,A4: set @ ( product_prod @ A @ A ),B6: set @ ( product_prod @ A @ A )] :
( ( transitive_acyclic @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ A4 @ B6 ) )
=> ( transitive_acyclic @ A @ B6 ) ) ).
% acyclic_union(2)
thf(fact_4655_inv__on__def,axiom,
! [B: $tType,A: $tType] :
( ( inv_on @ A @ B )
= ( ^ [F2: A > B,A9: set @ A,X3: B] :
( fChoice @ A
@ ^ [Y3: A] :
( ( member @ A @ Y3 @ A9 )
& ( ( F2 @ Y3 )
= X3 ) ) ) ) ) ).
% inv_on_def
thf(fact_4656_split__paired__Eps,axiom,
! [B: $tType,A: $tType] :
( ( fChoice @ ( product_prod @ A @ B ) )
= ( ^ [P4: ( product_prod @ A @ B ) > $o] :
( fChoice @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A7: A,B5: B] : ( P4 @ ( product_Pair @ A @ B @ A7 @ B5 ) ) ) ) ) ) ).
% split_paired_Eps
thf(fact_4657_acyclicI__order,axiom,
! [A: $tType,B: $tType] :
( ( preorder @ A )
=> ! [R: set @ ( product_prod @ B @ B ),F: B > A] :
( ! [A5: B,B3: B] :
( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A5 @ B3 ) @ R )
=> ( ord_less @ A @ ( F @ B3 ) @ ( F @ A5 ) ) )
=> ( transitive_acyclic @ B @ R ) ) ) ).
% acyclicI_order
thf(fact_4658_subset__mset_OacyclicI__order,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ B @ B ),F: B > ( multiset @ A )] :
( ! [A5: B,B3: B] :
( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A5 @ B3 ) @ R )
=> ( subset_mset @ A @ ( F @ B3 ) @ ( F @ A5 ) ) )
=> ( transitive_acyclic @ B @ R ) ) ).
% subset_mset.acyclicI_order
thf(fact_4659_acyclicI,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ! [X2: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ ( transitive_trancl @ A @ R ) )
=> ( transitive_acyclic @ A @ R ) ) ).
% acyclicI
thf(fact_4660_acyclic__def,axiom,
! [A: $tType] :
( ( transitive_acyclic @ A )
= ( ^ [R3: set @ ( product_prod @ A @ A )] :
! [X3: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ ( transitive_trancl @ A @ R3 ) ) ) ) ).
% acyclic_def
thf(fact_4661_cyclicE,axiom,
! [A: $tType,G: set @ ( product_prod @ A @ A )] :
( ~ ( transitive_acyclic @ A @ G )
=> ~ ! [X2: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ ( transitive_trancl @ A @ G ) ) ) ).
% cyclicE
thf(fact_4662_some__theI,axiom,
! [A: $tType,B: $tType,P: A > B > $o] :
( ? [A17: A,X_1: B] : ( P @ A17 @ X_1 )
=> ( ! [B13: B,B24: B] :
( ? [A5: A] : ( P @ A5 @ B13 )
=> ( ? [A5: A] : ( P @ A5 @ B24 )
=> ( B13 = B24 ) ) )
=> ( P
@ ( fChoice @ A
@ ^ [A7: A] :
? [X8: B] : ( P @ A7 @ X8 ) )
@ ( the @ B
@ ^ [B5: B] :
? [A7: A] : ( P @ A7 @ B5 ) ) ) ) ) ).
% some_theI
thf(fact_4663_acyclic__insert__cyclic,axiom,
! [A: $tType,G: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( transitive_acyclic @ A @ G )
=> ( ~ ( transitive_acyclic @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ G ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ ( transitive_rtrancl @ A @ G ) ) ) ) ).
% acyclic_insert_cyclic
thf(fact_4664_fun__of__rel__def,axiom,
! [A: $tType,B: $tType] :
( ( fun_of_rel @ B @ A )
= ( ^ [R6: set @ ( product_prod @ B @ A ),X3: B] :
( fChoice @ A
@ ^ [Y3: A] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 @ Y3 ) @ R6 ) ) ) ) ).
% fun_of_rel_def
thf(fact_4665_irrefl__tranclI,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),X: A] :
( ( ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ ( converse @ A @ A @ R ) @ ( transitive_rtrancl @ A @ R ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ ( transitive_trancl @ A @ R ) ) ) ).
% irrefl_tranclI
thf(fact_4666_quicksort_Opelims,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: list @ A,Y: list @ A] :
( ( ( linorder_quicksort @ A @ X )
= Y )
=> ( ( accp @ ( list @ A ) @ ( linord6200660962353139674rt_rel @ A ) @ X )
=> ( ( ( X
= ( nil @ A ) )
=> ( ( Y
= ( nil @ A ) )
=> ~ ( accp @ ( list @ A ) @ ( linord6200660962353139674rt_rel @ A ) @ ( nil @ A ) ) ) )
=> ~ ! [X2: A,Xs2: list @ A] :
( ( X
= ( cons @ A @ X2 @ Xs2 ) )
=> ( ( Y
= ( append @ A
@ ( linorder_quicksort @ A
@ ( filter @ A
@ ^ [Y3: A] :
~ ( ord_less_eq @ A @ X2 @ Y3 )
@ Xs2 ) )
@ ( append @ A @ ( cons @ A @ X2 @ ( nil @ A ) ) @ ( linorder_quicksort @ A @ ( filter @ A @ ( ord_less_eq @ A @ X2 ) @ Xs2 ) ) ) ) )
=> ~ ( accp @ ( list @ A ) @ ( linord6200660962353139674rt_rel @ A ) @ ( cons @ A @ X2 @ Xs2 ) ) ) ) ) ) ) ) ).
% quicksort.pelims
thf(fact_4667_converse__iff,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,R: set @ ( product_prod @ B @ A )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( converse @ B @ A @ R ) )
= ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ B2 @ A3 ) @ R ) ) ).
% converse_iff
thf(fact_4668_pair__set__inverse,axiom,
! [B: $tType,A: $tType,P: B > A > $o] :
( ( converse @ B @ A @ ( collect @ ( product_prod @ B @ A ) @ ( product_case_prod @ B @ A @ $o @ P ) ) )
= ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [B5: A,A7: B] : ( P @ A7 @ B5 ) ) ) ) ).
% pair_set_inverse
thf(fact_4669_below__Id__inv,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( converse @ A @ A @ R5 ) @ ( id2 @ A ) )
= ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R5 @ ( id2 @ A ) ) ) ).
% below_Id_inv
thf(fact_4670_converseI,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,R: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R )
=> ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ B2 @ A3 ) @ ( converse @ A @ B @ R ) ) ) ).
% converseI
thf(fact_4671_converseE,axiom,
! [A: $tType,B: $tType,Yx: product_prod @ A @ B,R: set @ ( product_prod @ B @ A )] :
( ( member @ ( product_prod @ A @ B ) @ Yx @ ( converse @ B @ A @ R ) )
=> ~ ! [X2: B,Y4: A] :
( ( Yx
= ( product_Pair @ A @ B @ Y4 @ X2 ) )
=> ~ ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X2 @ Y4 ) @ R ) ) ) ).
% converseE
thf(fact_4672_converseD,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,R: set @ ( product_prod @ B @ A )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( converse @ B @ A @ R ) )
=> ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ B2 @ A3 ) @ R ) ) ).
% converseD
thf(fact_4673_converse_Osimps,axiom,
! [B: $tType,A: $tType,A1: B,A22: A,R: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A1 @ A22 ) @ ( converse @ A @ B @ R ) )
= ( ? [A7: A,B5: B] :
( ( A1 = B5 )
& ( A22 = A7 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A7 @ B5 ) @ R ) ) ) ) ).
% converse.simps
thf(fact_4674_converse_Ocases,axiom,
! [B: $tType,A: $tType,A1: B,A22: A,R: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A1 @ A22 ) @ ( converse @ A @ B @ R ) )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A22 @ A1 ) @ R ) ) ).
% converse.cases
thf(fact_4675_in__listrel1__converse,axiom,
! [A: $tType,X: list @ A,Y: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( listrel1 @ A @ ( converse @ A @ A @ R ) ) )
= ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( converse @ ( list @ A ) @ ( list @ A ) @ ( listrel1 @ A @ R ) ) ) ) ).
% in_listrel1_converse
thf(fact_4676_rtrancl__converseI,axiom,
! [A: $tType,Y: A,X: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ ( transitive_rtrancl @ A @ R ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ ( converse @ A @ A @ R ) ) ) ) ).
% rtrancl_converseI
thf(fact_4677_rtrancl__converseD,axiom,
! [A: $tType,X: A,Y: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ ( converse @ A @ A @ R ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ ( transitive_rtrancl @ A @ R ) ) ) ).
% rtrancl_converseD
thf(fact_4678_trancl__converseI,axiom,
! [A: $tType,X: A,Y: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( converse @ A @ A @ ( transitive_trancl @ A @ R ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ ( converse @ A @ A @ R ) ) ) ) ).
% trancl_converseI
thf(fact_4679_trancl__converseD,axiom,
! [A: $tType,X: A,Y: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ ( converse @ A @ A @ R ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( converse @ A @ A @ ( transitive_trancl @ A @ R ) ) ) ) ).
% trancl_converseD
thf(fact_4680_finite__wf__eq__wf__converse,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R5 )
=> ( ( wf @ A @ ( converse @ A @ A @ R5 ) )
= ( wf @ A @ R5 ) ) ) ).
% finite_wf_eq_wf_converse
thf(fact_4681_converse__unfold,axiom,
! [A: $tType,B: $tType] :
( ( converse @ B @ A )
= ( ^ [R3: set @ ( product_prod @ B @ A )] :
( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [Y3: A,X3: B] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 @ Y3 ) @ R3 ) ) ) ) ) ).
% converse_unfold
thf(fact_4682_conversep__converse__eq,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B )] :
( ( conversep @ A @ B
@ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R ) )
= ( ^ [X3: B,Y3: A] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 @ Y3 ) @ ( converse @ A @ B @ R ) ) ) ) ).
% conversep_converse_eq
thf(fact_4683_converse__def,axiom,
! [B: $tType,A: $tType] :
( ( converse @ A @ B )
= ( ^ [R3: set @ ( product_prod @ A @ B )] :
( collect @ ( product_prod @ B @ A )
@ ( product_case_prod @ B @ A @ $o
@ ( conversep @ A @ B
@ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R3 ) ) ) ) ) ) ).
% converse_def
thf(fact_4684_is__singleton__altdef,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A9: set @ A] :
( ( finite_card @ A @ A9 )
= ( one_one @ nat ) ) ) ) ).
% is_singleton_altdef
thf(fact_4685_Code__Numeral_Osub__code_I2_J,axiom,
! [M2: num] :
( ( code_sub @ ( bit0 @ M2 ) @ one2 )
= ( code_Pos @ ( bitM @ M2 ) ) ) ).
% Code_Numeral.sub_code(2)
thf(fact_4686_su__rel__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( su_rel_fun @ A @ B )
= ( ^ [F7: set @ ( product_prod @ A @ B ),F2: A > B] :
( ! [A9: A,B9: B,B11: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A9 @ B9 ) @ F7 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A9 @ B11 ) @ F7 )
=> ( B9 = B11 ) ) )
& ! [A9: A,P4: $o] :
( ! [B9: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A9 @ B9 ) @ F7 )
=> P4 )
=> P4 )
& ! [A9: A] :
( ( F2 @ A9 )
= ( the @ B
@ ^ [B9: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A9 @ B9 ) @ F7 ) ) ) ) ) ) ).
% su_rel_fun_def
thf(fact_4687_Code__Numeral_OPos__def,axiom,
( code_Pos
= ( numeral_numeral @ code_integer ) ) ).
% Code_Numeral.Pos_def
thf(fact_4688_su__rel__fun_Osurjective,axiom,
! [B: $tType,A: $tType,F6: set @ ( product_prod @ A @ B ),F: A > B,A4: A] :
( ( su_rel_fun @ A @ B @ F6 @ F )
=> ~ ! [B10: B] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B10 ) @ F6 ) ) ).
% su_rel_fun.surjective
thf(fact_4689_su__rel__fun_Ounique,axiom,
! [A: $tType,B: $tType,F6: set @ ( product_prod @ A @ B ),F: A > B,A4: A,B6: B,B14: B] :
( ( su_rel_fun @ A @ B @ F6 @ F )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B6 ) @ F6 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B14 ) @ F6 )
=> ( B6 = B14 ) ) ) ) ).
% su_rel_fun.unique
thf(fact_4690_su__rel__fun_Orepr2,axiom,
! [B: $tType,A: $tType,F6: set @ ( product_prod @ A @ B ),F: A > B,A4: A,B6: B] :
( ( su_rel_fun @ A @ B @ F6 @ F )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B6 ) @ F6 )
=> ( B6
= ( F @ A4 ) ) ) ) ).
% su_rel_fun.repr2
thf(fact_4691_su__rel__fun_Orepr1,axiom,
! [B: $tType,A: $tType,F6: set @ ( product_prod @ A @ B ),F: A > B,A4: A] :
( ( su_rel_fun @ A @ B @ F6 @ F )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ ( F @ A4 ) ) @ F6 ) ) ).
% su_rel_fun.repr1
thf(fact_4692_su__rel__fun_Orepr,axiom,
! [B: $tType,A: $tType,F6: set @ ( product_prod @ A @ B ),F: A > B,A4: A,B6: B] :
( ( su_rel_fun @ A @ B @ F6 @ F )
=> ( ( ( F @ A4 )
= B6 )
= ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B6 ) @ F6 ) ) ) ).
% su_rel_fun.repr
thf(fact_4693_Pos__fold_I2_J,axiom,
! [K: num] :
( ( numeral_numeral @ code_integer @ ( bit0 @ K ) )
= ( code_Pos @ ( bit0 @ K ) ) ) ).
% Pos_fold(2)
thf(fact_4694_Pos__fold_I1_J,axiom,
( ( numeral_numeral @ code_integer @ one2 )
= ( code_Pos @ one2 ) ) ).
% Pos_fold(1)
thf(fact_4695_Pos__fold_I3_J,axiom,
! [K: num] :
( ( numeral_numeral @ code_integer @ ( bit1 @ K ) )
= ( code_Pos @ ( bit1 @ K ) ) ) ).
% Pos_fold(3)
thf(fact_4696_one__integer__code,axiom,
( ( one_one @ code_integer )
= ( code_Pos @ one2 ) ) ).
% one_integer_code
thf(fact_4697_Code__Numeral_Odup__code_I2_J,axiom,
! [N: num] :
( ( code_dup @ ( code_Pos @ N ) )
= ( code_Pos @ ( bit0 @ N ) ) ) ).
% Code_Numeral.dup_code(2)
thf(fact_4698_less__integer__code_I4_J,axiom,
! [K: num] :
~ ( ord_less @ code_integer @ ( code_Pos @ K ) @ ( zero_zero @ code_integer ) ) ).
% less_integer_code(4)
thf(fact_4699_less__integer__code_I2_J,axiom,
! [L: num] : ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ ( code_Pos @ L ) ) ).
% less_integer_code(2)
thf(fact_4700_plus__integer__code_I3_J,axiom,
! [M2: num,N: num] :
( ( plus_plus @ code_integer @ ( code_Pos @ M2 ) @ ( code_Pos @ N ) )
= ( code_Pos @ ( plus_plus @ num @ M2 @ N ) ) ) ).
% plus_integer_code(3)
thf(fact_4701_less__eq__integer__code_I2_J,axiom,
! [L: num] : ( ord_less_eq @ code_integer @ ( zero_zero @ code_integer ) @ ( code_Pos @ L ) ) ).
% less_eq_integer_code(2)
thf(fact_4702_less__eq__integer__code_I4_J,axiom,
! [K: num] :
~ ( ord_less_eq @ code_integer @ ( code_Pos @ K ) @ ( zero_zero @ code_integer ) ) ).
% less_eq_integer_code(4)
thf(fact_4703_less__integer__code_I8_J,axiom,
! [K: num,L: num] : ( ord_less @ code_integer @ ( code_Neg @ K ) @ ( code_Pos @ L ) ) ).
% less_integer_code(8)
thf(fact_4704_less__integer__code_I6_J,axiom,
! [K: num,L: num] :
~ ( ord_less @ code_integer @ ( code_Pos @ K ) @ ( code_Neg @ L ) ) ).
% less_integer_code(6)
thf(fact_4705_uminus__integer__code_I3_J,axiom,
! [M2: num] :
( ( uminus_uminus @ code_integer @ ( code_Neg @ M2 ) )
= ( code_Pos @ M2 ) ) ).
% uminus_integer_code(3)
thf(fact_4706_uminus__integer__code_I2_J,axiom,
! [M2: num] :
( ( uminus_uminus @ code_integer @ ( code_Pos @ M2 ) )
= ( code_Neg @ M2 ) ) ).
% uminus_integer_code(2)
thf(fact_4707_Code__Numeral_ONeg__def,axiom,
( code_Neg
= ( ^ [N5: num] : ( uminus_uminus @ code_integer @ ( code_Pos @ N5 ) ) ) ) ).
% Code_Numeral.Neg_def
thf(fact_4708_less__eq__integer__code_I8_J,axiom,
! [K: num,L: num] : ( ord_less_eq @ code_integer @ ( code_Neg @ K ) @ ( code_Pos @ L ) ) ).
% less_eq_integer_code(8)
thf(fact_4709_less__eq__integer__code_I6_J,axiom,
! [K: num,L: num] :
~ ( ord_less_eq @ code_integer @ ( code_Pos @ K ) @ ( code_Neg @ L ) ) ).
% less_eq_integer_code(6)
thf(fact_4710_times__integer__code_I3_J,axiom,
! [M2: num,N: num] :
( ( times_times @ code_integer @ ( code_Pos @ M2 ) @ ( code_Pos @ N ) )
= ( code_Pos @ ( times_times @ num @ M2 @ N ) ) ) ).
% times_integer_code(3)
thf(fact_4711_less__integer__code_I5_J,axiom,
! [K: num,L: num] :
( ( ord_less @ code_integer @ ( code_Pos @ K ) @ ( code_Pos @ L ) )
= ( ord_less @ num @ K @ L ) ) ).
% less_integer_code(5)
thf(fact_4712_less__eq__integer__code_I5_J,axiom,
! [K: num,L: num] :
( ( ord_less_eq @ code_integer @ ( code_Pos @ K ) @ ( code_Pos @ L ) )
= ( ord_less_eq @ num @ K @ L ) ) ).
% less_eq_integer_code(5)
thf(fact_4713_minus__integer__code_I3_J,axiom,
! [M2: num,N: num] :
( ( minus_minus @ code_integer @ ( code_Pos @ M2 ) @ ( code_Pos @ N ) )
= ( code_sub @ M2 @ N ) ) ).
% minus_integer_code(3)
thf(fact_4714_divmod__abs__code_I1_J,axiom,
! [K: num,L: num] :
( ( code_divmod_abs @ ( code_Pos @ K ) @ ( code_Pos @ L ) )
= ( unique8689654367752047608divmod @ code_integer @ K @ L ) ) ).
% divmod_abs_code(1)
thf(fact_4715_minus__integer__code_I4_J,axiom,
! [M2: num,N: num] :
( ( minus_minus @ code_integer @ ( code_Pos @ M2 ) @ ( code_Neg @ N ) )
= ( code_Pos @ ( plus_plus @ num @ M2 @ N ) ) ) ).
% minus_integer_code(4)
thf(fact_4716_minus__integer__code_I5_J,axiom,
! [M2: num,N: num] :
( ( minus_minus @ code_integer @ ( code_Neg @ M2 ) @ ( code_Pos @ N ) )
= ( code_Neg @ ( plus_plus @ num @ M2 @ N ) ) ) ).
% minus_integer_code(5)
thf(fact_4717_times__integer__code_I4_J,axiom,
! [M2: num,N: num] :
( ( times_times @ code_integer @ ( code_Pos @ M2 ) @ ( code_Neg @ N ) )
= ( code_Neg @ ( times_times @ num @ M2 @ N ) ) ) ).
% times_integer_code(4)
thf(fact_4718_times__integer__code_I5_J,axiom,
! [M2: num,N: num] :
( ( times_times @ code_integer @ ( code_Neg @ M2 ) @ ( code_Pos @ N ) )
= ( code_Neg @ ( times_times @ num @ M2 @ N ) ) ) ).
% times_integer_code(5)
thf(fact_4719_times__integer__code_I6_J,axiom,
! [M2: num,N: num] :
( ( times_times @ code_integer @ ( code_Neg @ M2 ) @ ( code_Neg @ N ) )
= ( code_Pos @ ( times_times @ num @ M2 @ N ) ) ) ).
% times_integer_code(6)
thf(fact_4720_plus__integer__code_I4_J,axiom,
! [M2: num,N: num] :
( ( plus_plus @ code_integer @ ( code_Pos @ M2 ) @ ( code_Neg @ N ) )
= ( code_sub @ M2 @ N ) ) ).
% plus_integer_code(4)
thf(fact_4721_plus__integer__code_I5_J,axiom,
! [M2: num,N: num] :
( ( plus_plus @ code_integer @ ( code_Neg @ M2 ) @ ( code_Pos @ N ) )
= ( code_sub @ N @ M2 ) ) ).
% plus_integer_code(5)
thf(fact_4722_divmod__abs__code_I3_J,axiom,
! [K: num,L: num] :
( ( code_divmod_abs @ ( code_Neg @ K ) @ ( code_Pos @ L ) )
= ( unique8689654367752047608divmod @ code_integer @ K @ L ) ) ).
% divmod_abs_code(3)
thf(fact_4723_divmod__abs__code_I4_J,axiom,
! [K: num,L: num] :
( ( code_divmod_abs @ ( code_Pos @ K ) @ ( code_Neg @ L ) )
= ( unique8689654367752047608divmod @ code_integer @ K @ L ) ) ).
% divmod_abs_code(4)
thf(fact_4724_Code__Numeral_Osub__code_I3_J,axiom,
! [M2: num] :
( ( code_sub @ ( bit1 @ M2 ) @ one2 )
= ( code_Pos @ ( bit0 @ M2 ) ) ) ).
% Code_Numeral.sub_code(3)
thf(fact_4725_su__rel__fun_Of__def,axiom,
! [A: $tType,B: $tType,F6: set @ ( product_prod @ A @ B ),F: A > B,A4: A] :
( ( su_rel_fun @ A @ B @ F6 @ F )
=> ( ( F @ A4 )
= ( the @ B
@ ^ [B9: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B9 ) @ F6 ) ) ) ) ).
% su_rel_fun.f_def
thf(fact_4726_su__rel__fun_Ointro,axiom,
! [B: $tType,A: $tType,F6: set @ ( product_prod @ A @ B ),F: A > B] :
( ! [A15: A,B10: B,B15: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A15 @ B10 ) @ F6 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A15 @ B15 ) @ F6 )
=> ( B10 = B15 ) ) )
=> ( ! [A15: A,P7: $o] :
( ! [B16: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A15 @ B16 ) @ F6 )
=> P7 )
=> P7 )
=> ( ! [A15: A] :
( ( F @ A15 )
= ( the @ B
@ ^ [B9: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A15 @ B9 ) @ F6 ) ) )
=> ( su_rel_fun @ A @ B @ F6 @ F ) ) ) ) ).
% su_rel_fun.intro
thf(fact_4727_aboveS__def,axiom,
! [A: $tType] :
( ( order_aboveS @ A )
= ( ^ [R3: set @ ( product_prod @ A @ A ),A7: A] :
( collect @ A
@ ^ [B5: A] :
( ( B5 != A7 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B5 ) @ R3 ) ) ) ) ) ).
% aboveS_def
thf(fact_4728_relInvImage__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Gr7122648621184425601vImage @ A @ B )
= ( ^ [A9: set @ A,R6: set @ ( product_prod @ B @ B ),F2: A > B] :
( collect @ ( product_prod @ A @ A )
@ ^ [Uu: product_prod @ A @ A] :
? [A12: A,A23: A] :
( ( Uu
= ( product_Pair @ A @ A @ A12 @ A23 ) )
& ( member @ A @ A12 @ A9 )
& ( member @ A @ A23 @ A9 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F2 @ A12 ) @ ( F2 @ A23 ) ) @ R6 ) ) ) ) ) ).
% relInvImage_def
thf(fact_4729_prod__list__def,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ( ( groups5270119922927024881d_list @ A )
= ( groups_monoid_F @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ) ).
% prod_list_def
thf(fact_4730_sum__list__def,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ( ( groups8242544230860333062m_list @ A )
= ( groups_monoid_F @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ) ).
% sum_list_def
thf(fact_4731_sum__list__map__remove1,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [X: B,Xs: list @ B,F: B > A] :
( ( member @ B @ X @ ( set2 @ B @ Xs ) )
=> ( ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F @ Xs ) )
= ( plus_plus @ A @ ( F @ X ) @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F @ ( remove1 @ B @ X @ Xs ) ) ) ) ) ) ) ).
% sum_list_map_remove1
thf(fact_4732_normalize__stable,axiom,
! [Q3: int,P5: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ Q3 )
=> ( ( algebr8660921524188924756oprime @ int @ P5 @ Q3 )
=> ( ( normalize @ ( product_Pair @ int @ int @ P5 @ Q3 ) )
= ( product_Pair @ int @ int @ P5 @ Q3 ) ) ) ) ).
% normalize_stable
thf(fact_4733_distinct__count__atmost__1,axiom,
! [A: $tType] :
( ( distinct @ A )
= ( ^ [X3: list @ A] :
! [A7: A] :
( ( ( member @ A @ A7 @ ( set2 @ A @ X3 ) )
=> ( ( count @ A @ ( mset @ A @ X3 ) @ A7 )
= ( one_one @ nat ) ) )
& ( ~ ( member @ A @ A7 @ ( set2 @ A @ X3 ) )
=> ( ( count @ A @ ( mset @ A @ X3 ) @ A7 )
= ( zero_zero @ nat ) ) ) ) ) ) ).
% distinct_count_atmost_1
thf(fact_4734_coprime__self,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A] :
( ( algebr8660921524188924756oprime @ A @ A3 @ A3 )
= ( dvd_dvd @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% coprime_self
thf(fact_4735_mset__empty__count,axiom,
! [A: $tType,M3: multiset @ A] :
( ( ! [P6: A] :
( ( count @ A @ M3 @ P6 )
= ( zero_zero @ nat ) ) )
= ( M3
= ( zero_zero @ ( multiset @ A ) ) ) ) ).
% mset_empty_count
thf(fact_4736_coprime__imp__gcd__eq__1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ A3 @ B2 )
=> ( ( gcd_gcd @ A @ A3 @ B2 )
= ( one_one @ A ) ) ) ) ).
% coprime_imp_gcd_eq_1
thf(fact_4737_count__union,axiom,
! [A: $tType,M3: multiset @ A,N6: multiset @ A,A3: A] :
( ( count @ A @ ( plus_plus @ ( multiset @ A ) @ M3 @ N6 ) @ A3 )
= ( plus_plus @ nat @ ( count @ A @ M3 @ A3 ) @ ( count @ A @ N6 @ A3 ) ) ) ).
% count_union
thf(fact_4738_coprime__0__right__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A] :
( ( algebr8660921524188924756oprime @ A @ A3 @ ( zero_zero @ A ) )
= ( dvd_dvd @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% coprime_0_right_iff
thf(fact_4739_coprime__0__left__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A] :
( ( algebr8660921524188924756oprime @ A @ ( zero_zero @ A ) @ A3 )
= ( dvd_dvd @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% coprime_0_left_iff
thf(fact_4740_coprime__mult__self__right__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
& ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ) ).
% coprime_mult_self_right_iff
thf(fact_4741_coprime__mult__self__left__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
& ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ) ).
% coprime_mult_self_left_iff
thf(fact_4742_is__unit__gcd,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ ( gcd_gcd @ A @ A3 @ B2 ) @ ( one_one @ A ) )
= ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% is_unit_gcd
thf(fact_4743_count__greater__eq__one__iff,axiom,
! [A: $tType,M3: multiset @ A,X: A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ ( count @ A @ M3 @ X ) )
= ( member @ A @ X @ ( set_mset @ A @ M3 ) ) ) ).
% count_greater_eq_one_iff
thf(fact_4744_count__mset__set_I1_J,axiom,
! [A: $tType,A4: set @ A,X: A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( count @ A @ ( mset_set @ A @ A4 ) @ X )
= ( one_one @ nat ) ) ) ) ).
% count_mset_set(1)
thf(fact_4745_coprime__1__left,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A] : ( algebr8660921524188924756oprime @ A @ ( one_one @ A ) @ A3 ) ) ).
% coprime_1_left
thf(fact_4746_coprime__1__right,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A] : ( algebr8660921524188924756oprime @ A @ A3 @ ( one_one @ A ) ) ) ).
% coprime_1_right
thf(fact_4747_coprime__add__one__left,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A] : ( algebr8660921524188924756oprime @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ A3 ) ) ).
% coprime_add_one_left
thf(fact_4748_coprime__add__one__right,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A] : ( algebr8660921524188924756oprime @ A @ A3 @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% coprime_add_one_right
thf(fact_4749_coprime__doff__one__right,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [A3: A] : ( algebr8660921524188924756oprime @ A @ A3 @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% coprime_doff_one_right
thf(fact_4750_coprime__diff__one__left,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [A3: A] : ( algebr8660921524188924756oprime @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ A3 ) ) ).
% coprime_diff_one_left
thf(fact_4751_is__unit__right__imp__coprime,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% is_unit_right_imp_coprime
thf(fact_4752_is__unit__left__imp__coprime,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% is_unit_left_imp_coprime
thf(fact_4753_coprime__common__divisor,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( algebr8660921524188924756oprime @ A @ A3 @ B2 )
=> ( ( dvd_dvd @ A @ C2 @ A3 )
=> ( ( dvd_dvd @ A @ C2 @ B2 )
=> ( dvd_dvd @ A @ C2 @ ( one_one @ A ) ) ) ) ) ) ).
% coprime_common_divisor
thf(fact_4754_coprime__absorb__right,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [Y: A,X: A] :
( ( dvd_dvd @ A @ Y @ X )
=> ( ( algebr8660921524188924756oprime @ A @ X @ Y )
= ( dvd_dvd @ A @ Y @ ( one_one @ A ) ) ) ) ) ).
% coprime_absorb_right
thf(fact_4755_coprime__imp__coprime,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,D3: A,A3: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ C2 @ D3 )
=> ( ! [E3: A] :
( ~ ( dvd_dvd @ A @ E3 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ E3 @ A3 )
=> ( ( dvd_dvd @ A @ E3 @ B2 )
=> ( dvd_dvd @ A @ E3 @ C2 ) ) ) )
=> ( ! [E3: A] :
( ~ ( dvd_dvd @ A @ E3 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ E3 @ A3 )
=> ( ( dvd_dvd @ A @ E3 @ B2 )
=> ( dvd_dvd @ A @ E3 @ D3 ) ) ) )
=> ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ) ) ).
% coprime_imp_coprime
thf(fact_4756_coprime__absorb__left,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [X: A,Y: A] :
( ( dvd_dvd @ A @ X @ Y )
=> ( ( algebr8660921524188924756oprime @ A @ X @ Y )
= ( dvd_dvd @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% coprime_absorb_left
thf(fact_4757_not__coprimeI,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( dvd_dvd @ A @ C2 @ A3 )
=> ( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ~ ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
=> ~ ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ) ) ).
% not_coprimeI
thf(fact_4758_not__coprimeE,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ~ ( algebr8660921524188924756oprime @ A @ A3 @ B2 )
=> ~ ! [C4: A] :
( ( dvd_dvd @ A @ C4 @ A3 )
=> ( ( dvd_dvd @ A @ C4 @ B2 )
=> ( dvd_dvd @ A @ C4 @ ( one_one @ A ) ) ) ) ) ) ).
% not_coprimeE
thf(fact_4759_coprime__def,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ( ( algebr8660921524188924756oprime @ A )
= ( ^ [A7: A,B5: A] :
! [C3: A] :
( ( dvd_dvd @ A @ C3 @ A7 )
=> ( ( dvd_dvd @ A @ C3 @ B5 )
=> ( dvd_dvd @ A @ C3 @ ( one_one @ A ) ) ) ) ) ) ) ).
% coprime_def
thf(fact_4760_coprimeI,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ! [C4: A] :
( ( dvd_dvd @ A @ C4 @ A3 )
=> ( ( dvd_dvd @ A @ C4 @ B2 )
=> ( dvd_dvd @ A @ C4 @ ( one_one @ A ) ) ) )
=> ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% coprimeI
thf(fact_4761_plus__multiset_Orep__eq,axiom,
! [A: $tType,X: multiset @ A,Xa: multiset @ A] :
( ( count @ A @ ( plus_plus @ ( multiset @ A ) @ X @ Xa ) )
= ( ^ [A7: A] : ( plus_plus @ nat @ ( count @ A @ X @ A7 ) @ ( count @ A @ Xa @ A7 ) ) ) ) ).
% plus_multiset.rep_eq
thf(fact_4762_coprime__iff__gcd__eq__1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( algebr8660921524188924756oprime @ A )
= ( ^ [A7: A,B5: A] :
( ( gcd_gcd @ A @ A7 @ B5 )
= ( one_one @ A ) ) ) ) ) ).
% coprime_iff_gcd_eq_1
thf(fact_4763_gcd__eq__1__imp__coprime,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( ( gcd_gcd @ A @ A3 @ B2 )
= ( one_one @ A ) )
=> ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% gcd_eq_1_imp_coprime
thf(fact_4764_quotient__of__coprime,axiom,
! [R: rat,P5: int,Q3: int] :
( ( ( quotient_of @ R )
= ( product_Pair @ int @ int @ P5 @ Q3 ) )
=> ( algebr8660921524188924756oprime @ int @ P5 @ Q3 ) ) ).
% quotient_of_coprime
thf(fact_4765_normalize__coprime,axiom,
! [R: product_prod @ int @ int,P5: int,Q3: int] :
( ( ( normalize @ R )
= ( product_Pair @ int @ int @ P5 @ Q3 ) )
=> ( algebr8660921524188924756oprime @ int @ P5 @ Q3 ) ) ).
% normalize_coprime
thf(fact_4766_invertible__coprime,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
= ( one_one @ A ) )
=> ( algebr8660921524188924756oprime @ A @ A3 @ C2 ) ) ) ).
% invertible_coprime
thf(fact_4767_count__ne__remove,axiom,
! [A: $tType,X: A,T2: A,S2: multiset @ A] :
( ( X != T2 )
=> ( ( count @ A @ S2 @ X )
= ( count @ A @ ( minus_minus @ ( multiset @ A ) @ S2 @ ( add_mset @ A @ T2 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ X ) ) ) ).
% count_ne_remove
thf(fact_4768_count__in__diffI,axiom,
! [A: $tType,N6: multiset @ A,X: A,M3: multiset @ A] :
( ! [N3: nat] :
( ( count @ A @ N6 @ X )
!= ( plus_plus @ nat @ N3 @ ( count @ A @ M3 @ X ) ) )
=> ( member @ A @ X @ ( set_mset @ A @ ( minus_minus @ ( multiset @ A ) @ M3 @ N6 ) ) ) ) ).
% count_in_diffI
thf(fact_4769_count__single,axiom,
! [A: $tType,B2: A,A3: A] :
( ( ( B2 = A3 )
=> ( ( count @ A @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) @ A3 )
= ( one_one @ nat ) ) )
& ( ( B2 != A3 )
=> ( ( count @ A @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) @ A3 )
= ( zero_zero @ nat ) ) ) ) ).
% count_single
thf(fact_4770_in__diff__countE,axiom,
! [A: $tType,X: A,M3: multiset @ A,N6: multiset @ A] :
( ( member @ A @ X @ ( set_mset @ A @ ( minus_minus @ ( multiset @ A ) @ M3 @ N6 ) ) )
=> ~ ! [N3: nat] :
( ( count @ A @ M3 @ X )
!= ( plus_plus @ nat @ ( suc @ N3 ) @ ( count @ A @ N6 @ X ) ) ) ) ).
% in_diff_countE
thf(fact_4771_remove1__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( remove1 @ A @ ( hd @ A @ Xs ) @ Xs )
= ( tl @ A @ Xs ) ) ) ).
% remove1_tl
thf(fact_4772_count__mset__set__finite__iff,axiom,
! [A: $tType,S2: set @ A,A3: A] :
( ( finite_finite2 @ A @ S2 )
=> ( ( ( member @ A @ A3 @ S2 )
=> ( ( count @ A @ ( mset_set @ A @ S2 ) @ A3 )
= ( one_one @ nat ) ) )
& ( ~ ( member @ A @ A3 @ S2 )
=> ( ( count @ A @ ( mset_set @ A @ S2 ) @ A3 )
= ( zero_zero @ nat ) ) ) ) ) ).
% count_mset_set_finite_iff
thf(fact_4773_count__mset__set_H,axiom,
! [A: $tType,A4: set @ A,X: A] :
( ( ( ( finite_finite2 @ A @ A4 )
& ( member @ A @ X @ A4 ) )
=> ( ( count @ A @ ( mset_set @ A @ A4 ) @ X )
= ( one_one @ nat ) ) )
& ( ~ ( ( finite_finite2 @ A @ A4 )
& ( member @ A @ X @ A4 ) )
=> ( ( count @ A @ ( mset_set @ A @ A4 ) @ X )
= ( zero_zero @ nat ) ) ) ) ).
% count_mset_set'
thf(fact_4774_length__remove1,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( size_size @ ( list @ A ) @ ( remove1 @ A @ X @ Xs ) )
= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) ) )
& ( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( size_size @ ( list @ A ) @ ( remove1 @ A @ X @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% length_remove1
thf(fact_4775_prod__mset__delta_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [Y: B,C2: A,A4: multiset @ B] :
( ( comm_m9189036328036947845d_mset @ A
@ ( image_mset @ B @ A
@ ^ [X3: B] : ( if @ A @ ( Y = X3 ) @ C2 @ ( one_one @ A ) )
@ A4 ) )
= ( power_power @ A @ C2 @ ( count @ B @ A4 @ Y ) ) ) ) ).
% prod_mset_delta'
thf(fact_4776_prod__mset__delta,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [Y: B,C2: A,A4: multiset @ B] :
( ( comm_m9189036328036947845d_mset @ A
@ ( image_mset @ B @ A
@ ^ [X3: B] : ( if @ A @ ( X3 = Y ) @ C2 @ ( one_one @ A ) )
@ A4 ) )
= ( power_power @ A @ C2 @ ( count @ B @ A4 @ Y ) ) ) ) ).
% prod_mset_delta
thf(fact_4777_prod_OatLeastLessThan__reindex,axiom,
! [B: $tType,A: $tType] :
( ( ( comm_monoid_mult @ A )
& ( ord @ B ) )
=> ! [H3: nat > B,M2: nat,N: nat,G: B > A] :
( ( bij_betw @ nat @ B @ H3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) @ ( set_or7035219750837199246ssThan @ B @ ( H3 @ M2 ) @ ( H3 @ N ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( set_or7035219750837199246ssThan @ B @ ( H3 @ M2 ) @ ( H3 @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ B @ A @ nat @ G @ H3 ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ) ).
% prod.atLeastLessThan_reindex
thf(fact_4778_bij__betw__id,axiom,
! [A: $tType,A4: set @ A] : ( bij_betw @ A @ A @ ( id @ A ) @ A4 @ A4 ) ).
% bij_betw_id
thf(fact_4779_bij__betw__add,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A3: A,A4: set @ A,B6: set @ A] :
( ( bij_betw @ A @ A @ ( plus_plus @ A @ A3 ) @ A4 @ B6 )
= ( ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ A4 )
= B6 ) ) ) ).
% bij_betw_add
thf(fact_4780_prod__mset__empty,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ( ( comm_m9189036328036947845d_mset @ A @ ( zero_zero @ ( multiset @ A ) ) )
= ( one_one @ A ) ) ) ).
% prod_mset_empty
thf(fact_4781_prod__mset_Ounion,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M3: multiset @ A,N6: multiset @ A] :
( ( comm_m9189036328036947845d_mset @ A @ ( plus_plus @ ( multiset @ A ) @ M3 @ N6 ) )
= ( times_times @ A @ ( comm_m9189036328036947845d_mset @ A @ M3 ) @ ( comm_m9189036328036947845d_mset @ A @ N6 ) ) ) ) ).
% prod_mset.union
thf(fact_4782_prod__mset__Un,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: multiset @ A,B6: multiset @ A] :
( ( comm_m9189036328036947845d_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) )
= ( times_times @ A @ ( comm_m9189036328036947845d_mset @ A @ A4 ) @ ( comm_m9189036328036947845d_mset @ A @ B6 ) ) ) ) ).
% prod_mset_Un
thf(fact_4783_bij__betw__of__nat,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N6: set @ nat,A4: set @ A] :
( ( bij_betw @ nat @ A @ ( semiring_1_of_nat @ A ) @ N6 @ A4 )
= ( ( image2 @ nat @ A @ ( semiring_1_of_nat @ A ) @ N6 )
= A4 ) ) ) ).
% bij_betw_of_nat
thf(fact_4784_prod__mset_Oneutral__const,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: multiset @ B] :
( ( comm_m9189036328036947845d_mset @ A
@ ( image_mset @ B @ A
@ ^ [Uu: B] : ( one_one @ A )
@ A4 ) )
= ( one_one @ A ) ) ) ).
% prod_mset.neutral_const
thf(fact_4785_bij__betw__imp__surj__on,axiom,
! [A: $tType,B: $tType,F: A > B,A4: set @ A,B6: set @ B] :
( ( bij_betw @ A @ B @ F @ A4 @ B6 )
=> ( ( image2 @ A @ B @ F @ A4 )
= B6 ) ) ).
% bij_betw_imp_surj_on
thf(fact_4786_f__the__inv__into__f__bij__betw,axiom,
! [A: $tType,B: $tType,F: A > B,A4: set @ A,B6: set @ B,X: B] :
( ( bij_betw @ A @ B @ F @ A4 @ B6 )
=> ( ( ( bij_betw @ A @ B @ F @ A4 @ B6 )
=> ( member @ B @ X @ B6 ) )
=> ( ( F @ ( the_inv_into @ A @ B @ A4 @ F @ X ) )
= X ) ) ) ).
% f_the_inv_into_f_bij_betw
thf(fact_4787_bij__betw__the__inv__into,axiom,
! [B: $tType,A: $tType,F: A > B,A4: set @ A,B6: set @ B] :
( ( bij_betw @ A @ B @ F @ A4 @ B6 )
=> ( bij_betw @ B @ A @ ( the_inv_into @ A @ B @ A4 @ F ) @ B6 @ A4 ) ) ).
% bij_betw_the_inv_into
thf(fact_4788_bij__betw__id__iff,axiom,
! [A: $tType,A4: set @ A,B6: set @ A] :
( ( bij_betw @ A @ A @ ( id @ A ) @ A4 @ B6 )
= ( A4 = B6 ) ) ).
% bij_betw_id_iff
thf(fact_4789_bij__betwE,axiom,
! [A: $tType,B: $tType,F: A > B,A4: set @ A,B6: set @ B] :
( ( bij_betw @ A @ B @ F @ A4 @ B6 )
=> ! [X7: A] :
( ( member @ A @ X7 @ A4 )
=> ( member @ B @ ( F @ X7 ) @ B6 ) ) ) ).
% bij_betwE
thf(fact_4790_bij__betw__inv,axiom,
! [B: $tType,A: $tType,F: A > B,A4: set @ A,B6: set @ B] :
( ( bij_betw @ A @ B @ F @ A4 @ B6 )
=> ? [G4: B > A] : ( bij_betw @ B @ A @ G4 @ B6 @ A4 ) ) ).
% bij_betw_inv
thf(fact_4791_bij__betw__cong,axiom,
! [A: $tType,B: $tType,A4: set @ A,F: A > B,G: A > B,A18: set @ B] :
( ! [A5: A] :
( ( member @ A @ A5 @ A4 )
=> ( ( F @ A5 )
= ( G @ A5 ) ) )
=> ( ( bij_betw @ A @ B @ F @ A4 @ A18 )
= ( bij_betw @ A @ B @ G @ A4 @ A18 ) ) ) ).
% bij_betw_cong
thf(fact_4792_bij__betw__apply,axiom,
! [A: $tType,B: $tType,F: A > B,A4: set @ A,B6: set @ B,A3: A] :
( ( bij_betw @ A @ B @ F @ A4 @ B6 )
=> ( ( member @ A @ A3 @ A4 )
=> ( member @ B @ ( F @ A3 ) @ B6 ) ) ) ).
% bij_betw_apply
thf(fact_4793_bij__betw__iff__bijections,axiom,
! [B: $tType,A: $tType] :
( ( bij_betw @ A @ B )
= ( ^ [F2: A > B,A9: set @ A,B9: set @ B] :
? [G2: B > A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A9 )
=> ( ( member @ B @ ( F2 @ X3 ) @ B9 )
& ( ( G2 @ ( F2 @ X3 ) )
= X3 ) ) )
& ! [X3: B] :
( ( member @ B @ X3 @ B9 )
=> ( ( member @ A @ ( G2 @ X3 ) @ A9 )
& ( ( F2 @ ( G2 @ X3 ) )
= X3 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_4794_involuntory__imp__bij,axiom,
! [A: $tType,F: A > A] :
( ! [X2: A] :
( ( F @ ( F @ X2 ) )
= X2 )
=> ( bij_betw @ A @ A @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) ) ) ).
% involuntory_imp_bij
thf(fact_4795_bij__pointE,axiom,
! [B: $tType,A: $tType,F: A > B,Y: B] :
( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ~ ! [X2: A] :
( ( Y
= ( F @ X2 ) )
=> ~ ! [X16: A] :
( ( Y
= ( F @ X16 ) )
=> ( X16 = X2 ) ) ) ) ).
% bij_pointE
thf(fact_4796_bij__iff,axiom,
! [A: $tType,B: $tType,F: A > B] :
( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
= ( ! [X3: B] :
? [Y3: A] :
( ( ( F @ Y3 )
= X3 )
& ! [Z5: A] :
( ( ( F @ Z5 )
= X3 )
=> ( Z5 = Y3 ) ) ) ) ) ).
% bij_iff
thf(fact_4797_bij__betw__imp__inj__on,axiom,
! [B: $tType,A: $tType,F: A > B,A4: set @ A,B6: set @ B] :
( ( bij_betw @ A @ B @ F @ A4 @ B6 )
=> ( inj_on @ A @ B @ F @ A4 ) ) ).
% bij_betw_imp_inj_on
thf(fact_4798_bij__betw__trans,axiom,
! [A: $tType,B: $tType,C: $tType,F: A > B,A4: set @ A,B6: set @ B,G: B > C,C5: set @ C] :
( ( bij_betw @ A @ B @ F @ A4 @ B6 )
=> ( ( bij_betw @ B @ C @ G @ B6 @ C5 )
=> ( bij_betw @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ A4 @ C5 ) ) ) ).
% bij_betw_trans
thf(fact_4799_bij__betw__comp__iff,axiom,
! [A: $tType,B: $tType,C: $tType,F: A > B,A4: set @ A,A18: set @ B,F3: B > C,A19: set @ C] :
( ( bij_betw @ A @ B @ F @ A4 @ A18 )
=> ( ( bij_betw @ B @ C @ F3 @ A18 @ A19 )
= ( bij_betw @ A @ C @ ( comp @ B @ C @ A @ F3 @ F ) @ A4 @ A19 ) ) ) ).
% bij_betw_comp_iff
thf(fact_4800_bij__betw__empty2,axiom,
! [B: $tType,A: $tType,F: A > B,A4: set @ A] :
( ( bij_betw @ A @ B @ F @ A4 @ ( bot_bot @ ( set @ B ) ) )
=> ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% bij_betw_empty2
thf(fact_4801_bij__betw__empty1,axiom,
! [A: $tType,B: $tType,F: A > B,A4: set @ B] :
( ( bij_betw @ A @ B @ F @ ( bot_bot @ ( set @ A ) ) @ A4 )
=> ( A4
= ( bot_bot @ ( set @ B ) ) ) ) ).
% bij_betw_empty1
thf(fact_4802_bij__betw__byWitness,axiom,
! [A: $tType,B: $tType,A4: set @ A,F3: B > A,F: A > B,A18: set @ B] :
( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( ( F3 @ ( F @ X2 ) )
= X2 ) )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ A18 )
=> ( ( F @ ( F3 @ X2 ) )
= X2 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F @ A4 ) @ A18 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F3 @ A18 ) @ A4 )
=> ( bij_betw @ A @ B @ F @ A4 @ A18 ) ) ) ) ) ).
% bij_betw_byWitness
thf(fact_4803_bij__betw__subset,axiom,
! [A: $tType,B: $tType,F: A > B,A4: set @ A,A18: set @ B,B6: set @ A,B14: set @ B] :
( ( bij_betw @ A @ B @ F @ A4 @ A18 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ A4 )
=> ( ( ( image2 @ A @ B @ F @ B6 )
= B14 )
=> ( bij_betw @ A @ B @ F @ B6 @ B14 ) ) ) ) ).
% bij_betw_subset
thf(fact_4804_bij__betw__imp__surj,axiom,
! [A: $tType,B: $tType,F: A > B,A4: set @ A] :
( ( bij_betw @ A @ B @ F @ A4 @ ( top_top @ ( set @ B ) ) )
=> ( ( image2 @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ B ) ) ) ) ).
% bij_betw_imp_surj
thf(fact_4805_bij__is__surj,axiom,
! [A: $tType,B: $tType,F: A > B] :
( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( image2 @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ B ) ) ) ) ).
% bij_is_surj
thf(fact_4806_bij__uminus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ( bij_betw @ A @ A @ ( uminus_uminus @ A ) @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) ) ) ).
% bij_uminus
thf(fact_4807_bij__comp,axiom,
! [A: $tType,B: $tType,C: $tType,F: A > B,G: B > C] :
( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( bij_betw @ B @ C @ G @ ( top_top @ ( set @ B ) ) @ ( top_top @ ( set @ C ) ) )
=> ( bij_betw @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ C ) ) ) ) ) ).
% bij_comp
thf(fact_4808_inj__on__imp__bij__betw,axiom,
! [B: $tType,A: $tType,F: A > B,A4: set @ A] :
( ( inj_on @ A @ B @ F @ A4 )
=> ( bij_betw @ A @ B @ F @ A4 @ ( image2 @ A @ B @ F @ A4 ) ) ) ).
% inj_on_imp_bij_betw
thf(fact_4809_bij__betw__imageI,axiom,
! [A: $tType,B: $tType,F: A > B,A4: set @ A,B6: set @ B] :
( ( inj_on @ A @ B @ F @ A4 )
=> ( ( ( image2 @ A @ B @ F @ A4 )
= B6 )
=> ( bij_betw @ A @ B @ F @ A4 @ B6 ) ) ) ).
% bij_betw_imageI
thf(fact_4810_bij__betw__def,axiom,
! [B: $tType,A: $tType] :
( ( bij_betw @ A @ B )
= ( ^ [F2: A > B,A9: set @ A,B9: set @ B] :
( ( inj_on @ A @ B @ F2 @ A9 )
& ( ( image2 @ A @ B @ F2 @ A9 )
= B9 ) ) ) ) ).
% bij_betw_def
thf(fact_4811_bij__betw__iff__card,axiom,
! [A: $tType,B: $tType,A4: set @ A,B6: set @ B] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ B @ B6 )
=> ( ( ? [F2: A > B] : ( bij_betw @ A @ B @ F2 @ A4 @ B6 ) )
= ( ( finite_card @ A @ A4 )
= ( finite_card @ B @ B6 ) ) ) ) ) ).
% bij_betw_iff_card
thf(fact_4812_finite__same__card__bij,axiom,
! [A: $tType,B: $tType,A4: set @ A,B6: set @ B] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ B @ B6 )
=> ( ( ( finite_card @ A @ A4 )
= ( finite_card @ B @ B6 ) )
=> ? [H2: A > B] : ( bij_betw @ A @ B @ H2 @ A4 @ B6 ) ) ) ) ).
% finite_same_card_bij
thf(fact_4813_bij__is__inj,axiom,
! [B: $tType,A: $tType,F: A > B] :
( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) ) ) ).
% bij_is_inj
thf(fact_4814_bij__id,axiom,
! [A: $tType] : ( bij_betw @ A @ A @ ( id @ A ) @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) ) ).
% bij_id
thf(fact_4815_coprime__common__divisor__nat,axiom,
! [A3: nat,B2: nat,X: nat] :
( ( algebr8660921524188924756oprime @ nat @ A3 @ B2 )
=> ( ( dvd_dvd @ nat @ X @ A3 )
=> ( ( dvd_dvd @ nat @ X @ B2 )
=> ( X
= ( one_one @ nat ) ) ) ) ) ).
% coprime_common_divisor_nat
thf(fact_4816_prod__mset_Oneutral,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: multiset @ A] :
( ! [X2: A] :
( ( member @ A @ X2 @ ( set_mset @ A @ A4 ) )
=> ( X2
= ( one_one @ A ) ) )
=> ( ( comm_m9189036328036947845d_mset @ A @ A4 )
= ( one_one @ A ) ) ) ) ).
% prod_mset.neutral
thf(fact_4817_bij__betw__comp__iff2,axiom,
! [C: $tType,A: $tType,B: $tType,F3: A > B,A18: set @ A,A19: set @ B,F: C > A,A4: set @ C] :
( ( bij_betw @ A @ B @ F3 @ A18 @ A19 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ C @ A @ F @ A4 ) @ A18 )
=> ( ( bij_betw @ C @ A @ F @ A4 @ A18 )
= ( bij_betw @ C @ B @ ( comp @ A @ B @ C @ F3 @ F ) @ A4 @ A19 ) ) ) ) ).
% bij_betw_comp_iff2
thf(fact_4818_notIn__Un__bij__betw,axiom,
! [A: $tType,B: $tType,B2: A,A4: set @ A,F: A > B,A18: set @ B] :
( ~ ( member @ A @ B2 @ A4 )
=> ( ~ ( member @ B @ ( F @ B2 ) @ A18 )
=> ( ( bij_betw @ A @ B @ F @ A4 @ A18 )
=> ( bij_betw @ A @ B @ F @ ( sup_sup @ ( set @ A ) @ A4 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( sup_sup @ ( set @ B ) @ A18 @ ( insert2 @ B @ ( F @ B2 ) @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ).
% notIn_Un_bij_betw
thf(fact_4819_notIn__Un__bij__betw3,axiom,
! [A: $tType,B: $tType,B2: A,A4: set @ A,F: A > B,A18: set @ B] :
( ~ ( member @ A @ B2 @ A4 )
=> ( ~ ( member @ B @ ( F @ B2 ) @ A18 )
=> ( ( bij_betw @ A @ B @ F @ A4 @ A18 )
= ( bij_betw @ A @ B @ F @ ( sup_sup @ ( set @ A ) @ A4 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( sup_sup @ ( set @ B ) @ A18 @ ( insert2 @ B @ ( F @ B2 ) @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ).
% notIn_Un_bij_betw3
thf(fact_4820_bijI,axiom,
! [A: $tType,B: $tType,F: A > B] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( ( image2 @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ B ) ) )
=> ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) ) ) ) ).
% bijI
thf(fact_4821_bij__def,axiom,
! [A: $tType,B: $tType,F: A > B] :
( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
= ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
& ( ( image2 @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ B ) ) ) ) ) ).
% bij_def
thf(fact_4822_bij__betw__partition,axiom,
! [A: $tType,B: $tType,F: A > B,A4: set @ A,C5: set @ A,B6: set @ B,D4: set @ B] :
( ( bij_betw @ A @ B @ F @ ( sup_sup @ ( set @ A ) @ A4 @ C5 ) @ ( sup_sup @ ( set @ B ) @ B6 @ D4 ) )
=> ( ( bij_betw @ A @ B @ F @ C5 @ D4 )
=> ( ( ( inf_inf @ ( set @ A ) @ A4 @ C5 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( inf_inf @ ( set @ B ) @ B6 @ D4 )
= ( bot_bot @ ( set @ B ) ) )
=> ( bij_betw @ A @ B @ F @ A4 @ B6 ) ) ) ) ) ).
% bij_betw_partition
thf(fact_4823_bij__betw__combine,axiom,
! [A: $tType,B: $tType,F: A > B,A4: set @ A,B6: set @ B,C5: set @ A,D4: set @ B] :
( ( bij_betw @ A @ B @ F @ A4 @ B6 )
=> ( ( bij_betw @ A @ B @ F @ C5 @ D4 )
=> ( ( ( inf_inf @ ( set @ B ) @ B6 @ D4 )
= ( bot_bot @ ( set @ B ) ) )
=> ( bij_betw @ A @ B @ F @ ( sup_sup @ ( set @ A ) @ A4 @ C5 ) @ ( sup_sup @ ( set @ B ) @ B6 @ D4 ) ) ) ) ) ).
% bij_betw_combine
thf(fact_4824_o__bij,axiom,
! [A: $tType,B: $tType,G: B > A,F: A > B] :
( ( ( comp @ B @ A @ A @ G @ F )
= ( id @ A ) )
=> ( ( ( comp @ A @ B @ B @ F @ G )
= ( id @ B ) )
=> ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) ) ) ) ).
% o_bij
thf(fact_4825_bij__image__Compl__eq,axiom,
! [B: $tType,A: $tType,F: A > B,A4: set @ A] :
( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( image2 @ A @ B @ F @ ( uminus_uminus @ ( set @ A ) @ A4 ) )
= ( uminus_uminus @ ( set @ B ) @ ( image2 @ A @ B @ F @ A4 ) ) ) ) ).
% bij_image_Compl_eq
thf(fact_4826_Well__order__iso__copy,axiom,
! [B: $tType,A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),F: A > B,A18: set @ B] :
( ( order_well_order_on @ A @ A4 @ R )
=> ( ( bij_betw @ A @ B @ F @ A4 @ A18 )
=> ? [R10: set @ ( product_prod @ B @ B )] :
( ( order_well_order_on @ B @ A18 @ R10 )
& ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R10 ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ) ) ).
% Well_order_iso_copy
thf(fact_4827_bij__betw__disjoint__Un,axiom,
! [A: $tType,B: $tType,F: A > B,A4: set @ A,C5: set @ B,G: A > B,B6: set @ A,D4: set @ B] :
( ( bij_betw @ A @ B @ F @ A4 @ C5 )
=> ( ( bij_betw @ A @ B @ G @ B6 @ D4 )
=> ( ( ( inf_inf @ ( set @ A ) @ A4 @ B6 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( inf_inf @ ( set @ B ) @ C5 @ D4 )
= ( bot_bot @ ( set @ B ) ) )
=> ( bij_betw @ A @ B
@ ^ [X3: A] : ( if @ B @ ( member @ A @ X3 @ A4 ) @ ( F @ X3 ) @ ( G @ X3 ) )
@ ( sup_sup @ ( set @ A ) @ A4 @ B6 )
@ ( sup_sup @ ( set @ B ) @ C5 @ D4 ) ) ) ) ) ) ).
% bij_betw_disjoint_Un
thf(fact_4828_is__unit__prod__mset__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A4: multiset @ A] :
( ( dvd_dvd @ A @ ( comm_m9189036328036947845d_mset @ A @ A4 ) @ ( one_one @ A ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ ( set_mset @ A @ A4 ) )
=> ( dvd_dvd @ A @ X3 @ ( one_one @ A ) ) ) ) ) ) ).
% is_unit_prod_mset_iff
thf(fact_4829_prod__mset_Oeq__fold,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ( ( comm_m9189036328036947845d_mset @ A )
= ( fold_mset @ A @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ) ).
% prod_mset.eq_fold
thf(fact_4830_vimage__subset__eq,axiom,
! [B: $tType,A: $tType,F: A > B,B6: set @ B,A4: set @ A] :
( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( vimage @ A @ B @ F @ B6 ) @ A4 )
= ( ord_less_eq @ ( set @ B ) @ B6 @ ( image2 @ A @ B @ F @ A4 ) ) ) ) ).
% vimage_subset_eq
thf(fact_4831_ex__bij__betw__nat__finite,axiom,
! [A: $tType,M3: set @ A] :
( ( finite_finite2 @ A @ M3 )
=> ? [H2: nat > A] : ( bij_betw @ nat @ A @ H2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ M3 ) ) @ M3 ) ) ).
% ex_bij_betw_nat_finite
thf(fact_4832_ex__bij__betw__nat__finite__1,axiom,
! [A: $tType,M3: set @ A] :
( ( finite_finite2 @ A @ M3 )
=> ? [H2: nat > A] : ( bij_betw @ nat @ A @ H2 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ ( finite_card @ A @ M3 ) ) @ M3 ) ) ).
% ex_bij_betw_nat_finite_1
thf(fact_4833_coprime__diff__one__right__nat,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( algebr8660921524188924756oprime @ nat @ N @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ).
% coprime_diff_one_right_nat
thf(fact_4834_coprime__diff__one__left__nat,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( algebr8660921524188924756oprime @ nat @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ N ) ) ).
% coprime_diff_one_left_nat
thf(fact_4835_prod_Oreindex__bij__betw__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S7: set @ B,T5: set @ C,H3: B > C,S2: set @ B,T6: set @ C,G: C > A] :
( ( finite_finite2 @ B @ S7 )
=> ( ( finite_finite2 @ C @ T5 )
=> ( ( bij_betw @ B @ C @ H3 @ ( minus_minus @ ( set @ B ) @ S2 @ S7 ) @ ( minus_minus @ ( set @ C ) @ T6 @ T5 ) )
=> ( ! [A5: B] :
( ( member @ B @ A5 @ S7 )
=> ( ( G @ ( H3 @ A5 ) )
= ( one_one @ A ) ) )
=> ( ! [B3: C] :
( ( member @ C @ B3 @ T5 )
=> ( ( G @ B3 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X3: B] : ( G @ ( H3 @ X3 ) )
@ S2 )
= ( groups7121269368397514597t_prod @ C @ A @ G @ T6 ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_betw_not_neutral
thf(fact_4836_sum_OatLeastAtMost__reindex,axiom,
! [B: $tType,A: $tType] :
( ( ( comm_monoid_add @ A )
& ( ord @ B ) )
=> ! [H3: nat > B,M2: nat,N: nat,G: B > A] :
( ( bij_betw @ nat @ B @ H3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) @ ( set_or1337092689740270186AtMost @ B @ ( H3 @ M2 ) @ ( H3 @ N ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( set_or1337092689740270186AtMost @ B @ ( H3 @ M2 ) @ ( H3 @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ B @ A @ nat @ G @ H3 ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ) ).
% sum.atLeastAtMost_reindex
thf(fact_4837_sum_OatLeastLessThan__reindex,axiom,
! [B: $tType,A: $tType] :
( ( ( comm_monoid_add @ A )
& ( ord @ B ) )
=> ! [H3: nat > B,M2: nat,N: nat,G: B > A] :
( ( bij_betw @ nat @ B @ H3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) @ ( set_or7035219750837199246ssThan @ B @ ( H3 @ M2 ) @ ( H3 @ N ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( set_or7035219750837199246ssThan @ B @ ( H3 @ M2 ) @ ( H3 @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ B @ A @ nat @ G @ H3 ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ) ).
% sum.atLeastLessThan_reindex
thf(fact_4838_prod_OatLeastAtMost__reindex,axiom,
! [B: $tType,A: $tType] :
( ( ( comm_monoid_mult @ A )
& ( ord @ B ) )
=> ! [H3: nat > B,M2: nat,N: nat,G: B > A] :
( ( bij_betw @ nat @ B @ H3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) @ ( set_or1337092689740270186AtMost @ B @ ( H3 @ M2 ) @ ( H3 @ N ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( set_or1337092689740270186AtMost @ B @ ( H3 @ M2 ) @ ( H3 @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ B @ A @ nat @ G @ H3 ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ) ).
% prod.atLeastAtMost_reindex
thf(fact_4839_antisymp__antisym__eq,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( antisymp @ A
@ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R ) )
= ( antisym @ A @ R ) ) ).
% antisymp_antisym_eq
thf(fact_4840_convol__expand__snd_H,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > ( product_prod @ B @ C ),G: A > B,H3: A > C] :
( ( ( comp @ ( product_prod @ B @ C ) @ B @ A @ ( product_fst @ B @ C ) @ F )
= G )
=> ( ( H3
= ( comp @ ( product_prod @ B @ C ) @ C @ A @ ( product_snd @ B @ C ) @ F ) )
= ( ( bNF_convol @ A @ B @ C @ G @ H3 )
= F ) ) ) ).
% convol_expand_snd'
thf(fact_4841_convol__expand__snd,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > ( product_prod @ B @ C ),G: A > B] :
( ( ( comp @ ( product_prod @ B @ C ) @ B @ A @ ( product_fst @ B @ C ) @ F )
= G )
=> ( ( bNF_convol @ A @ B @ C @ G @ ( comp @ ( product_prod @ B @ C ) @ C @ A @ ( product_snd @ B @ C ) @ F ) )
= F ) ) ).
% convol_expand_snd
thf(fact_4842_convol__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( bNF_convol @ A @ B @ C )
= ( ^ [F2: A > B,G2: A > C,A7: A] : ( product_Pair @ B @ C @ ( F2 @ A7 ) @ ( G2 @ A7 ) ) ) ) ).
% convol_def
thf(fact_4843_bij__swap,axiom,
! [A: $tType,B: $tType] : ( bij_betw @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A ) @ ( product_swap @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) ) ).
% bij_swap
thf(fact_4844_fst__convol,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > B,G: A > C] :
( ( comp @ ( product_prod @ B @ C ) @ B @ A @ ( product_fst @ B @ C ) @ ( bNF_convol @ A @ B @ C @ F @ G ) )
= F ) ).
% fst_convol
thf(fact_4845_snd__convol,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > C,G: A > B] :
( ( comp @ ( product_prod @ C @ B ) @ B @ A @ ( product_snd @ C @ B ) @ ( bNF_convol @ A @ C @ B @ F @ G ) )
= G ) ).
% snd_convol
thf(fact_4846_convol__o,axiom,
! [B: $tType,C: $tType,D: $tType,A: $tType,F: D > B,G: D > C,H3: A > D] :
( ( comp @ D @ ( product_prod @ B @ C ) @ A @ ( bNF_convol @ D @ B @ C @ F @ G ) @ H3 )
= ( bNF_convol @ A @ B @ C @ ( comp @ D @ B @ A @ F @ H3 ) @ ( comp @ D @ C @ A @ G @ H3 ) ) ) ).
% convol_o
thf(fact_4847_ex__bij__betw__finite__nat,axiom,
! [A: $tType,M3: set @ A] :
( ( finite_finite2 @ A @ M3 )
=> ? [H2: A > nat] : ( bij_betw @ A @ nat @ H2 @ M3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ M3 ) ) ) ) ).
% ex_bij_betw_finite_nat
thf(fact_4848_map__prod__o__convol__id,axiom,
! [B: $tType,A: $tType,C: $tType,F: C > A,G: C > B,X: C] :
( ( comp @ ( product_prod @ C @ B ) @ ( product_prod @ A @ B ) @ C @ ( product_map_prod @ C @ A @ B @ B @ F @ ( id @ B ) ) @ ( bNF_convol @ C @ C @ B @ ( id @ C ) @ G ) @ X )
= ( bNF_convol @ C @ A @ B @ ( comp @ A @ A @ C @ ( id @ A ) @ F ) @ G @ X ) ) ).
% map_prod_o_convol_id
thf(fact_4849_Heap__lub__empty,axiom,
! [A: $tType] :
( ( heap_Time_Heap_lub @ A @ ( bot_bot @ ( set @ ( heap_Time_Heap @ A ) ) ) )
= ( heap_Time_Heap2 @ A
@ ^ [X3: heap_ext @ product_unit] : ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ).
% Heap_lub_empty
thf(fact_4850_and_Osemilattice__neutr__axioms,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( semilattice_neutr @ A @ ( bit_se5824344872417868541ns_and @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% and.semilattice_neutr_axioms
thf(fact_4851_map__prod__ident,axiom,
! [B: $tType,A: $tType] :
( ( product_map_prod @ A @ A @ B @ B
@ ^ [X3: A] : X3
@ ^ [Y3: B] : Y3 )
= ( ^ [Z5: product_prod @ A @ B] : Z5 ) ) ).
% map_prod_ident
thf(fact_4852_map__prod__simp,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F: C > A,G: D > B,A3: C,B2: D] :
( ( product_map_prod @ C @ A @ D @ B @ F @ G @ ( product_Pair @ C @ D @ A3 @ B2 ) )
= ( product_Pair @ A @ B @ ( F @ A3 ) @ ( G @ B2 ) ) ) ).
% map_prod_simp
thf(fact_4853_fst__map__prod,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F: C > A,G: D > B,X: product_prod @ C @ D] :
( ( product_fst @ A @ B @ ( product_map_prod @ C @ A @ D @ B @ F @ G @ X ) )
= ( F @ ( product_fst @ C @ D @ X ) ) ) ).
% fst_map_prod
thf(fact_4854_snd__map__prod,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F: C > B,G: D > A,X: product_prod @ C @ D] :
( ( product_snd @ B @ A @ ( product_map_prod @ C @ B @ D @ A @ F @ G @ X ) )
= ( G @ ( product_snd @ C @ D @ X ) ) ) ).
% snd_map_prod
thf(fact_4855_map__prod__imageI,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,A3: A,B2: B,R5: set @ ( product_prod @ A @ B ),F: A > C,G: B > D] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R5 )
=> ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ ( F @ A3 ) @ ( G @ B2 ) ) @ ( image2 @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F @ G ) @ R5 ) ) ) ).
% map_prod_imageI
thf(fact_4856_fst__comp__map__prod,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,F: A > C,G: B > D] :
( ( comp @ ( product_prod @ C @ D ) @ C @ ( product_prod @ A @ B ) @ ( product_fst @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F @ G ) )
= ( comp @ A @ C @ ( product_prod @ A @ B ) @ F @ ( product_fst @ A @ B ) ) ) ).
% fst_comp_map_prod
thf(fact_4857_snd__comp__map__prod,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,F: A > D,G: B > C] :
( ( comp @ ( product_prod @ D @ C ) @ C @ ( product_prod @ A @ B ) @ ( product_snd @ D @ C ) @ ( product_map_prod @ A @ D @ B @ C @ F @ G ) )
= ( comp @ B @ C @ ( product_prod @ A @ B ) @ G @ ( product_snd @ A @ B ) ) ) ).
% snd_comp_map_prod
thf(fact_4858_prod_Omap__comp,axiom,
! [D: $tType,F5: $tType,E: $tType,C: $tType,B: $tType,A: $tType,G1: C > E,G22: D > F5,F1: A > C,F22: B > D,V: product_prod @ A @ B] :
( ( product_map_prod @ C @ E @ D @ F5 @ G1 @ G22 @ ( product_map_prod @ A @ C @ B @ D @ F1 @ F22 @ V ) )
= ( product_map_prod @ A @ E @ B @ F5 @ ( comp @ C @ E @ A @ G1 @ F1 ) @ ( comp @ D @ F5 @ B @ G22 @ F22 ) @ V ) ) ).
% prod.map_comp
thf(fact_4859_map__prod_Ocompositionality,axiom,
! [D: $tType,F5: $tType,E: $tType,C: $tType,B: $tType,A: $tType,F: C > E,G: D > F5,H3: A > C,I: B > D,Prod: product_prod @ A @ B] :
( ( product_map_prod @ C @ E @ D @ F5 @ F @ G @ ( product_map_prod @ A @ C @ B @ D @ H3 @ I @ Prod ) )
= ( product_map_prod @ A @ E @ B @ F5 @ ( comp @ C @ E @ A @ F @ H3 ) @ ( comp @ D @ F5 @ B @ G @ I ) @ Prod ) ) ).
% map_prod.compositionality
thf(fact_4860_map__prod_Oidentity,axiom,
! [B: $tType,A: $tType] :
( ( product_map_prod @ A @ A @ B @ B
@ ^ [X3: A] : X3
@ ^ [X3: B] : X3 )
= ( id @ ( product_prod @ A @ B ) ) ) ).
% map_prod.identity
thf(fact_4861_prod__fun__imageE,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,C2: product_prod @ A @ B,F: C > A,G: D > B,R5: set @ ( product_prod @ C @ D )] :
( ( member @ ( product_prod @ A @ B ) @ C2 @ ( image2 @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ A @ D @ B @ F @ G ) @ R5 ) )
=> ~ ! [X2: C,Y4: D] :
( ( C2
= ( product_Pair @ A @ B @ ( F @ X2 ) @ ( G @ Y4 ) ) )
=> ~ ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ X2 @ Y4 ) @ R5 ) ) ) ).
% prod_fun_imageE
thf(fact_4862_apfst__def,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( product_apfst @ A @ C @ B )
= ( ^ [F2: A > C] : ( product_map_prod @ A @ C @ B @ B @ F2 @ ( id @ B ) ) ) ) ).
% apfst_def
thf(fact_4863_apsnd__def,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( product_apsnd @ B @ C @ A )
= ( product_map_prod @ A @ A @ B @ C @ ( id @ A ) ) ) ).
% apsnd_def
thf(fact_4864_map__prod__def,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType] :
( ( product_map_prod @ A @ C @ B @ D )
= ( ^ [F2: A > C,G2: B > D] :
( product_case_prod @ A @ B @ ( product_prod @ C @ D )
@ ^ [X3: A,Y3: B] : ( product_Pair @ C @ D @ ( F2 @ X3 ) @ ( G2 @ Y3 ) ) ) ) ) ).
% map_prod_def
thf(fact_4865_map__prod__compose,axiom,
! [D: $tType,C: $tType,A: $tType,E: $tType,F5: $tType,B: $tType,F1: E > C,F22: A > E,G1: F5 > D,G22: B > F5] :
( ( product_map_prod @ A @ C @ B @ D @ ( comp @ E @ C @ A @ F1 @ F22 ) @ ( comp @ F5 @ D @ B @ G1 @ G22 ) )
= ( comp @ ( product_prod @ E @ F5 ) @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ E @ C @ F5 @ D @ F1 @ G1 ) @ ( product_map_prod @ A @ E @ B @ F5 @ F22 @ G22 ) ) ) ).
% map_prod_compose
thf(fact_4866_map__prod_Ocomp,axiom,
! [A: $tType,C: $tType,E: $tType,F5: $tType,D: $tType,B: $tType,F: C > E,G: D > F5,H3: A > C,I: B > D] :
( ( comp @ ( product_prod @ C @ D ) @ ( product_prod @ E @ F5 ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ E @ D @ F5 @ F @ G ) @ ( product_map_prod @ A @ C @ B @ D @ H3 @ I ) )
= ( product_map_prod @ A @ E @ B @ F5 @ ( comp @ C @ E @ A @ F @ H3 ) @ ( comp @ D @ F5 @ B @ G @ I ) ) ) ).
% map_prod.comp
thf(fact_4867_map__prod__surj__on,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType,F: B > A,A4: set @ B,A18: set @ A,G: D > C,B6: set @ D,B14: set @ C] :
( ( ( image2 @ B @ A @ F @ A4 )
= A18 )
=> ( ( ( image2 @ D @ C @ G @ B6 )
= B14 )
=> ( ( image2 @ ( product_prod @ B @ D ) @ ( product_prod @ A @ C ) @ ( product_map_prod @ B @ A @ D @ C @ F @ G )
@ ( product_Sigma @ B @ D @ A4
@ ^ [Uu: B] : B6 ) )
= ( product_Sigma @ A @ C @ A18
@ ^ [Uu: A] : B14 ) ) ) ) ).
% map_prod_surj_on
thf(fact_4868_map__prod__o__convol,axiom,
! [D: $tType,B: $tType,C: $tType,E: $tType,A: $tType,H1: D > B,H22: E > C,F: A > D,G: A > E] :
( ( comp @ ( product_prod @ D @ E ) @ ( product_prod @ B @ C ) @ A @ ( product_map_prod @ D @ B @ E @ C @ H1 @ H22 ) @ ( bNF_convol @ A @ D @ E @ F @ G ) )
= ( bNF_convol @ A @ B @ C @ ( comp @ D @ B @ A @ H1 @ F ) @ ( comp @ E @ C @ A @ H22 @ G ) ) ) ).
% map_prod_o_convol
thf(fact_4869_map__prod__inj__on,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,F: A > B,A4: set @ A,G: C > D,B6: set @ C] :
( ( inj_on @ A @ B @ F @ A4 )
=> ( ( inj_on @ C @ D @ G @ B6 )
=> ( inj_on @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( product_map_prod @ A @ B @ C @ D @ F @ G )
@ ( product_Sigma @ A @ C @ A4
@ ^ [Uu: A] : B6 ) ) ) ) ).
% map_prod_inj_on
thf(fact_4870_map__prod__surj,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F: A > B,G: C > D] :
( ( ( image2 @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ B ) ) )
=> ( ( ( image2 @ C @ D @ G @ ( top_top @ ( set @ C ) ) )
= ( top_top @ ( set @ D ) ) )
=> ( ( image2 @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( product_map_prod @ A @ B @ C @ D @ F @ G ) @ ( top_top @ ( set @ ( product_prod @ A @ C ) ) ) )
= ( top_top @ ( set @ ( product_prod @ B @ D ) ) ) ) ) ) ).
% map_prod_surj
thf(fact_4871_Zorns__po__lemma,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( order_7125193373082350890der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ! [C8: set @ A] :
( ( member @ ( set @ A ) @ C8 @ ( chains @ A @ R ) )
=> ? [X7: A] :
( ( member @ A @ X7 @ ( field2 @ A @ R ) )
& ! [Xa3: A] :
( ( member @ A @ Xa3 @ C8 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Xa3 @ X7 ) @ R ) ) ) )
=> ? [X2: A] :
( ( member @ A @ X2 @ ( field2 @ A @ R ) )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ ( field2 @ A @ R ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Xa2 ) @ R )
=> ( Xa2 = X2 ) ) ) ) ) ) ).
% Zorns_po_lemma
thf(fact_4872_assert__def,axiom,
! [A: $tType] :
( ( heap_Time_assert @ A )
= ( ^ [P4: A > $o,X3: A] : ( if @ ( heap_Time_Heap @ A ) @ ( P4 @ X3 ) @ ( heap_Time_return @ A @ X3 ) @ ( heap_Time_raise @ A @ ( cons @ char @ ( char2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ $false ) @ ( cons @ char @ ( char2 @ $true @ $true @ $false @ $false @ $true @ $true @ $true @ $false ) @ ( cons @ char @ ( char2 @ $true @ $true @ $false @ $false @ $true @ $true @ $true @ $false ) @ ( cons @ char @ ( char2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ $false ) @ ( cons @ char @ ( char2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ $false ) @ ( cons @ char @ ( char2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ $false ) @ ( nil @ char ) ) ) ) ) ) ) ) ) ) ) ).
% assert_def
thf(fact_4873_power__int__add__1,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M2: int] :
( ( ( X
!= ( zero_zero @ A ) )
| ( M2
!= ( uminus_uminus @ int @ ( one_one @ int ) ) ) )
=> ( ( power_int @ A @ X @ ( plus_plus @ int @ M2 @ ( one_one @ int ) ) )
= ( times_times @ A @ ( power_int @ A @ X @ M2 ) @ X ) ) ) ) ).
% power_int_add_1
thf(fact_4874_power__int__1__left,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [N: int] :
( ( power_int @ A @ ( one_one @ A ) @ N )
= ( one_one @ A ) ) ) ).
% power_int_1_left
thf(fact_4875_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
thf(fact_4876_power__int__minus__one__mult__self,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [M2: int] :
( ( times_times @ A @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M2 ) @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M2 ) )
= ( one_one @ A ) ) ) ).
% power_int_minus_one_mult_self
thf(fact_4877_power__int__minus__one__mult__self_H,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [M2: int,B2: A] :
( ( times_times @ A @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M2 ) @ ( times_times @ A @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M2 ) @ B2 ) )
= B2 ) ) ).
% power_int_minus_one_mult_self'
thf(fact_4878_power__int__add__numeral,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M2: num,N: num] :
( ( times_times @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ M2 ) ) @ ( power_int @ A @ X @ ( numeral_numeral @ int @ N ) ) )
= ( power_int @ A @ X @ ( numeral_numeral @ int @ ( plus_plus @ num @ M2 @ N ) ) ) ) ) ).
% power_int_add_numeral
thf(fact_4879_power__int__add__numeral2,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M2: num,N: num,B2: A] :
( ( times_times @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ M2 ) ) @ ( times_times @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ N ) ) @ B2 ) )
= ( times_times @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ ( plus_plus @ num @ M2 @ N ) ) ) @ B2 ) ) ) ).
% power_int_add_numeral2
thf(fact_4880_power__int__one__over,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,N: int] :
( ( power_int @ A @ ( divide_divide @ A @ ( one_one @ A ) @ X ) @ N )
= ( divide_divide @ A @ ( one_one @ A ) @ ( power_int @ A @ X @ N ) ) ) ) ).
% power_int_one_over
thf(fact_4881_power__int__0__left__If,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [M2: int] :
( ( ( M2
= ( zero_zero @ int ) )
=> ( ( power_int @ A @ ( zero_zero @ A ) @ M2 )
= ( one_one @ A ) ) )
& ( ( M2
!= ( zero_zero @ int ) )
=> ( ( power_int @ A @ ( zero_zero @ A ) @ M2 )
= ( zero_zero @ A ) ) ) ) ) ).
% power_int_0_left_If
thf(fact_4882_power__int__increasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N: int,N6: int,A3: A] :
( ( ord_less_eq @ int @ N @ N6 )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( power_int @ A @ A3 @ N ) @ ( power_int @ A @ A3 @ N6 ) ) ) ) ) ).
% power_int_increasing
thf(fact_4883_power__int__strict__increasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N: int,N6: int,A3: A] :
( ( ord_less @ int @ N @ N6 )
=> ( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less @ A @ ( power_int @ A @ A3 @ N ) @ ( power_int @ A @ A3 @ N6 ) ) ) ) ) ).
% power_int_strict_increasing
thf(fact_4884_power__int__minus__one__minus,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [N: int] :
( ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ int @ N ) )
= ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) ) ) ).
% power_int_minus_one_minus
thf(fact_4885_power__int__minus__one__diff__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: int,B2: int] :
( ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( minus_minus @ int @ A3 @ B2 ) )
= ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( minus_minus @ int @ B2 @ A3 ) ) ) ) ).
% power_int_minus_one_diff_commute
thf(fact_4886_power__int__strict__decreasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N: int,N6: int,A3: A] :
( ( ord_less @ int @ N @ N6 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( power_int @ A @ A3 @ N6 ) @ ( power_int @ A @ A3 @ N ) ) ) ) ) ) ).
% power_int_strict_decreasing
thf(fact_4887_one__le__power__int,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,N: int] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( power_int @ A @ X @ N ) ) ) ) ) ).
% one_le_power_int
thf(fact_4888_one__less__power__int,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,N: int] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less @ A @ ( one_one @ A ) @ ( power_int @ A @ A3 @ N ) ) ) ) ) ).
% one_less_power_int
thf(fact_4889_power__int__add,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M2: int,N: int] :
( ( ( X
!= ( zero_zero @ A ) )
| ( ( plus_plus @ int @ M2 @ N )
!= ( zero_zero @ int ) ) )
=> ( ( power_int @ A @ X @ ( plus_plus @ int @ M2 @ N ) )
= ( times_times @ A @ ( power_int @ A @ X @ M2 ) @ ( power_int @ A @ X @ N ) ) ) ) ) ).
% power_int_add
thf(fact_4890_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 @ ( uminus_uminus @ B @ ( one_one @ B ) ) @ X )
=> ( ( power_int @ A @ ( uminus_uminus @ A @ A3 ) @ N )
= ( times_times @ A @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( power_int @ A @ A3 @ N ) ) ) ) ) ).
% power_int_minus_left_distrib
thf(fact_4891_power__int__decreasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N: int,N6: int,A3: A] :
( ( ord_less_eq @ int @ N @ N6 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
=> ( ( ( A3
!= ( zero_zero @ A ) )
| ( N6
!= ( zero_zero @ int ) )
| ( N
= ( zero_zero @ int ) ) )
=> ( ord_less_eq @ A @ ( power_int @ A @ A3 @ N6 ) @ ( power_int @ A @ A3 @ N ) ) ) ) ) ) ) ).
% power_int_decreasing
thf(fact_4892_power__int__le__one,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,N: int] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ( ord_less_eq @ A @ X @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_int @ A @ X @ N ) @ ( one_one @ A ) ) ) ) ) ) ).
% power_int_le_one
thf(fact_4893_power__int__le__imp__le__exp,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,M2: int,N: int] :
( ( ord_less @ A @ ( one_one @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( power_int @ A @ X @ M2 ) @ ( power_int @ A @ X @ N ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less_eq @ int @ M2 @ N ) ) ) ) ) ).
% power_int_le_imp_le_exp
thf(fact_4894_power__int__le__imp__less__exp,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,M2: int,N: int] :
( ( ord_less @ A @ ( one_one @ A ) @ X )
=> ( ( ord_less @ A @ ( power_int @ A @ X @ M2 ) @ ( power_int @ A @ X @ N ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less @ int @ M2 @ N ) ) ) ) ) ).
% power_int_le_imp_less_exp
thf(fact_4895_power__int__add__1_H,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M2: int] :
( ( ( X
!= ( zero_zero @ A ) )
| ( M2
!= ( uminus_uminus @ int @ ( one_one @ int ) ) ) )
=> ( ( power_int @ A @ X @ ( plus_plus @ int @ M2 @ ( one_one @ int ) ) )
= ( times_times @ A @ X @ ( power_int @ A @ X @ M2 ) ) ) ) ) ).
% power_int_add_1'
thf(fact_4896_integer__of__char__code,axiom,
! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] :
( ( integer_of_char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ B72 ) )
= ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( zero_neq_one_of_bool @ code_integer @ B72 ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B62 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B52 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B42 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B32 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B22 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B1 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B0 ) ) ) ).
% integer_of_char_code
thf(fact_4897_char__of__def,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ( ( unique5772411509450598832har_of @ A )
= ( ^ [N5: A] :
( char2
@ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N5 )
@ ( bit_se5641148757651400278ts_bit @ A @ N5 @ ( one_one @ nat ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N5 @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N5 @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ one2 ) ) ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N5 @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ).
% char_of_def
thf(fact_4898_numeral__unfold__funpow,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( numeral_numeral @ A )
= ( ^ [K4: num] : ( compow @ ( A > A ) @ ( numeral_numeral @ nat @ K4 ) @ ( plus_plus @ A @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ) ) ).
% numeral_unfold_funpow
thf(fact_4899_relpowp__1,axiom,
! [A: $tType,P: A > A > $o] :
( ( compow @ ( A > A > $o ) @ ( one_one @ nat ) @ P )
= P ) ).
% relpowp_1
thf(fact_4900_Suc__funpow,axiom,
! [N: nat] :
( ( compow @ ( nat > nat ) @ N @ suc )
= ( plus_plus @ nat @ N ) ) ).
% Suc_funpow
thf(fact_4901_comp__funpow,axiom,
! [B: $tType,A: $tType,N: nat,F: A > A] :
( ( compow @ ( ( B > A ) > B > A ) @ N @ ( comp @ A @ A @ B @ F ) )
= ( comp @ A @ A @ B @ ( compow @ ( A > A ) @ N @ F ) ) ) ).
% comp_funpow
thf(fact_4902_funpow_Osimps_I2_J,axiom,
! [A: $tType,N: nat,F: A > A] :
( ( compow @ ( A > A ) @ ( suc @ N ) @ F )
= ( comp @ A @ A @ A @ F @ ( compow @ ( A > A ) @ N @ F ) ) ) ).
% funpow.simps(2)
thf(fact_4903_funpow__Suc__right,axiom,
! [A: $tType,N: nat,F: A > A] :
( ( compow @ ( A > A ) @ ( suc @ N ) @ F )
= ( comp @ A @ A @ A @ ( compow @ ( A > A ) @ N @ F ) @ F ) ) ).
% funpow_Suc_right
thf(fact_4904_funpow__add,axiom,
! [A: $tType,M2: nat,N: nat,F: A > A] :
( ( compow @ ( A > A ) @ ( plus_plus @ nat @ M2 @ N ) @ F )
= ( comp @ A @ A @ A @ ( compow @ ( A > A ) @ M2 @ F ) @ ( compow @ ( A > A ) @ N @ F ) ) ) ).
% funpow_add
thf(fact_4905_of__nat__def,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A )
= ( ^ [N5: nat] : ( compow @ ( A > A ) @ N5 @ ( plus_plus @ A @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ) ) ).
% of_nat_def
thf(fact_4906_numeral__add__unfold__funpow,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [K: num,A3: A] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ K ) @ A3 )
= ( compow @ ( A > A ) @ ( numeral_numeral @ nat @ K ) @ ( plus_plus @ A @ ( one_one @ A ) ) @ A3 ) ) ) ).
% numeral_add_unfold_funpow
thf(fact_4907_inj__on__funpow__least,axiom,
! [A: $tType,N: nat,F: A > A,S: A] :
( ( ( compow @ ( A > A ) @ N @ F @ S )
= S )
=> ( ! [M5: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M5 )
=> ( ( ord_less @ nat @ M5 @ N )
=> ( ( compow @ ( A > A ) @ M5 @ F @ S )
!= S ) ) )
=> ( inj_on @ nat @ A
@ ^ [K4: nat] : ( compow @ ( A > A ) @ K4 @ F @ S )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% inj_on_funpow_least
thf(fact_4908_card__Un__disjnt,axiom,
! [A: $tType,A4: set @ A,B6: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ A @ B6 )
=> ( ( disjnt @ A @ A4 @ B6 )
=> ( ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B6 ) )
= ( plus_plus @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ A @ B6 ) ) ) ) ) ) ).
% card_Un_disjnt
thf(fact_4909_plus__multiset__def,axiom,
! [A: $tType] :
( ( plus_plus @ ( multiset @ A ) )
= ( map_fun @ ( multiset @ A ) @ ( A > nat ) @ ( ( A > nat ) > A > nat ) @ ( ( multiset @ A ) > ( multiset @ A ) ) @ ( count @ A ) @ ( map_fun @ ( multiset @ A ) @ ( A > nat ) @ ( A > nat ) @ ( multiset @ A ) @ ( count @ A ) @ ( abs_multiset @ A ) )
@ ^ [M8: A > nat,N9: A > nat,A7: A] : ( plus_plus @ nat @ ( M8 @ A7 ) @ ( N9 @ A7 ) ) ) ) ).
% plus_multiset_def
thf(fact_4910_mset__subset__add__iff2,axiom,
! [A: $tType,I: nat,J: nat,U: multiset @ A,M2: multiset @ A,N: multiset @ A] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ I @ U ) @ M2 ) @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ J @ U ) @ N ) )
= ( subset_mset @ A @ M2 @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% mset_subset_add_iff2
thf(fact_4911_repeat__mset__distrib2,axiom,
! [A: $tType,N: nat,A4: multiset @ A,B6: multiset @ A] :
( ( repeat_mset @ A @ N @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) )
= ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ N @ A4 ) @ ( repeat_mset @ A @ N @ B6 ) ) ) ).
% repeat_mset_distrib2
thf(fact_4912_repeat__mset__Suc,axiom,
! [A: $tType,N: nat,M3: multiset @ A] :
( ( repeat_mset @ A @ ( suc @ N ) @ M3 )
= ( plus_plus @ ( multiset @ A ) @ M3 @ ( repeat_mset @ A @ N @ M3 ) ) ) ).
% repeat_mset_Suc
thf(fact_4913_disjnt__Times1__iff,axiom,
! [A: $tType,B: $tType,C5: set @ A,A4: set @ B,B6: set @ B] :
( ( disjnt @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ C5
@ ^ [Uu: A] : A4 )
@ ( product_Sigma @ A @ B @ C5
@ ^ [Uu: A] : B6 ) )
= ( ( C5
= ( bot_bot @ ( set @ A ) ) )
| ( disjnt @ B @ A4 @ B6 ) ) ) ).
% disjnt_Times1_iff
thf(fact_4914_disjnt__Times2__iff,axiom,
! [B: $tType,A: $tType,A4: set @ A,C5: set @ B,B6: set @ A] :
( ( disjnt @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : C5 )
@ ( product_Sigma @ A @ B @ B6
@ ^ [Uu: A] : C5 ) )
= ( ( C5
= ( bot_bot @ ( set @ B ) ) )
| ( disjnt @ A @ A4 @ B6 ) ) ) ).
% disjnt_Times2_iff
thf(fact_4915_disjnt__Sigma__iff,axiom,
! [B: $tType,A: $tType,A4: set @ A,C5: A > ( set @ B ),B6: set @ A] :
( ( disjnt @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ A4 @ C5 ) @ ( product_Sigma @ A @ B @ B6 @ C5 ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ ( inf_inf @ ( set @ A ) @ A4 @ B6 ) )
=> ( ( C5 @ X3 )
= ( bot_bot @ ( set @ B ) ) ) )
| ( disjnt @ A @ A4 @ B6 ) ) ) ).
% disjnt_Sigma_iff
thf(fact_4916_left__add__mult__distrib__mset,axiom,
! [A: $tType,I: nat,U: multiset @ A,J: nat,K: multiset @ A] :
( ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ I @ U ) @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ J @ U ) @ K ) )
= ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ ( plus_plus @ nat @ I @ J ) @ U ) @ K ) ) ).
% left_add_mult_distrib_mset
thf(fact_4917_repeat__mset__distrib,axiom,
! [A: $tType,M2: nat,N: nat,A4: multiset @ A] :
( ( repeat_mset @ A @ ( plus_plus @ nat @ M2 @ N ) @ A4 )
= ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ M2 @ A4 ) @ ( repeat_mset @ A @ N @ A4 ) ) ) ).
% repeat_mset_distrib
thf(fact_4918_mset__subseteq__add__iff1,axiom,
! [A: $tType,J: nat,I: nat,U: multiset @ A,M2: multiset @ A,N: multiset @ A] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ I @ U ) @ M2 ) @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ J @ U ) @ N ) )
= ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ ( minus_minus @ nat @ I @ J ) @ U ) @ M2 ) @ N ) ) ) ).
% mset_subseteq_add_iff1
thf(fact_4919_mset__subseteq__add__iff2,axiom,
! [A: $tType,I: nat,J: nat,U: multiset @ A,M2: multiset @ A,N: multiset @ A] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ I @ U ) @ M2 ) @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ J @ U ) @ N ) )
= ( subseteq_mset @ A @ M2 @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% mset_subseteq_add_iff2
thf(fact_4920_mset__subset__add__iff1,axiom,
! [A: $tType,J: nat,I: nat,U: multiset @ A,M2: multiset @ A,N: multiset @ A] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ I @ U ) @ M2 ) @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ J @ U ) @ N ) )
= ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ ( minus_minus @ nat @ I @ J ) @ U ) @ M2 ) @ N ) ) ) ).
% mset_subset_add_iff1
thf(fact_4921_plus__multiset_Oabs__eq,axiom,
! [A: $tType,Xa: A > nat,X: A > nat] :
( ( bNF_eq_onp @ ( A > nat )
@ ^ [F2: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X3 ) ) ) )
@ Xa
@ Xa )
=> ( ( bNF_eq_onp @ ( A > nat )
@ ^ [F2: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X3 ) ) ) )
@ X
@ X )
=> ( ( plus_plus @ ( multiset @ A ) @ ( abs_multiset @ A @ Xa ) @ ( abs_multiset @ A @ X ) )
= ( abs_multiset @ A
@ ^ [A7: A] : ( plus_plus @ nat @ ( Xa @ A7 ) @ ( X @ A7 ) ) ) ) ) ) ).
% plus_multiset.abs_eq
thf(fact_4922_repeat__mset__distrib__add__mset,axiom,
! [A: $tType,N: nat,A3: A,A4: multiset @ A] :
( ( repeat_mset @ A @ N @ ( add_mset @ A @ A3 @ A4 ) )
= ( plus_plus @ ( multiset @ A ) @ ( replicate_mset @ A @ N @ A3 ) @ ( repeat_mset @ A @ N @ A4 ) ) ) ).
% repeat_mset_distrib_add_mset
thf(fact_4923_o__prs_I2_J,axiom,
! [F5: $tType,E: $tType,C: $tType,D: $tType,A: $tType,B: $tType,H8: $tType,G3: $tType,R12: A > A > $o,Abs1: A > B,Rep1: B > A,R23: C > C > $o,Abs22: C > D,Rep22: D > C,R32: E > E > $o,Abs32: E > F5,Rep32: F5 > E] :
( ( quotient3 @ A @ B @ R12 @ Abs1 @ Rep1 )
=> ( ( quotient3 @ C @ D @ R23 @ Abs22 @ Rep22 )
=> ( ( quotient3 @ E @ F5 @ R32 @ Abs32 @ Rep32 )
=> ( ( map_fun @ ( G3 > H8 ) @ ( G3 > H8 ) @ ( ( A > G3 ) > A > H8 ) @ ( ( B > G3 ) > B > H8 ) @ ( id @ ( G3 > H8 ) ) @ ( map_fun @ ( B > G3 ) @ ( A > G3 ) @ ( A > H8 ) @ ( B > H8 ) @ ( map_fun @ A @ B @ G3 @ G3 @ Abs1 @ ( id @ G3 ) ) @ ( map_fun @ B @ A @ H8 @ H8 @ Rep1 @ ( id @ H8 ) ) ) @ ( comp @ G3 @ H8 @ A ) )
= ( comp @ G3 @ H8 @ B ) ) ) ) ) ).
% o_prs(2)
thf(fact_4924_abs__o__rep,axiom,
! [A: $tType,B: $tType,R5: A > A > $o,Abs: A > B,Rep: B > A] :
( ( quotient3 @ A @ B @ R5 @ Abs @ Rep )
=> ( ( comp @ A @ B @ B @ Abs @ Rep )
= ( id @ B ) ) ) ).
% abs_o_rep
thf(fact_4925_o__prs_I1_J,axiom,
! [C: $tType,E: $tType,A: $tType,B: $tType,F5: $tType,D: $tType,R12: A > A > $o,Abs1: A > B,Rep1: B > A,R23: C > C > $o,Abs22: C > D,Rep22: D > C,R32: E > E > $o,Abs32: E > F5,Rep32: F5 > E] :
( ( quotient3 @ A @ B @ R12 @ Abs1 @ Rep1 )
=> ( ( quotient3 @ C @ D @ R23 @ Abs22 @ Rep22 )
=> ( ( quotient3 @ E @ F5 @ R32 @ Abs32 @ Rep32 )
=> ( ( map_fun @ ( D > F5 ) @ ( C > E ) @ ( ( A > C ) > A > E ) @ ( ( B > D ) > B > F5 ) @ ( map_fun @ C @ D @ F5 @ E @ Abs22 @ Rep32 ) @ ( map_fun @ ( B > D ) @ ( A > C ) @ ( A > E ) @ ( B > F5 ) @ ( map_fun @ A @ B @ D @ C @ Abs1 @ Rep22 ) @ ( map_fun @ B @ A @ E @ F5 @ Rep1 @ Abs32 ) ) @ ( comp @ C @ E @ A ) )
= ( comp @ D @ F5 @ B ) ) ) ) ) ).
% o_prs(1)
thf(fact_4926_plus__multiset_Orsp,axiom,
! [A: $tType] :
( bNF_rel_fun @ ( A > nat ) @ ( A > nat ) @ ( ( A > nat ) > A > nat ) @ ( ( A > nat ) > A > nat )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F2: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X3 ) ) ) ) )
@ ( bNF_rel_fun @ ( A > nat ) @ ( A > nat ) @ ( A > nat ) @ ( A > nat )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F2: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X3 ) ) ) ) )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F2: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X3 ) ) ) ) ) )
@ ^ [M8: A > nat,N9: A > nat,A7: A] : ( plus_plus @ nat @ ( M8 @ A7 ) @ ( N9 @ A7 ) )
@ ^ [M8: A > nat,N9: A > nat,A7: A] : ( plus_plus @ nat @ ( M8 @ A7 ) @ ( N9 @ A7 ) ) ) ).
% plus_multiset.rsp
thf(fact_4927_or__not__num__neg_Opelims,axiom,
! [X: num,Xa: num,Y: num] :
( ( ( bit_or_not_num_neg @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ X @ Xa ) )
=> ( ( ( X = one2 )
=> ( ( Xa = one2 )
=> ( ( Y = one2 )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ one2 @ one2 ) ) ) ) )
=> ( ( ( X = one2 )
=> ! [M5: num] :
( ( Xa
= ( bit0 @ M5 ) )
=> ( ( Y
= ( bit1 @ M5 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ one2 @ ( bit0 @ M5 ) ) ) ) ) )
=> ( ( ( X = one2 )
=> ! [M5: num] :
( ( Xa
= ( bit1 @ M5 ) )
=> ( ( Y
= ( bit1 @ M5 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ one2 @ ( bit1 @ M5 ) ) ) ) ) )
=> ( ! [N3: num] :
( ( X
= ( bit0 @ N3 ) )
=> ( ( Xa = one2 )
=> ( ( Y
= ( bit0 @ one2 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit0 @ N3 ) @ one2 ) ) ) ) )
=> ( ! [N3: num] :
( ( X
= ( bit0 @ N3 ) )
=> ! [M5: num] :
( ( Xa
= ( bit0 @ M5 ) )
=> ( ( Y
= ( bitM @ ( bit_or_not_num_neg @ N3 @ M5 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit0 @ N3 ) @ ( bit0 @ M5 ) ) ) ) ) )
=> ( ! [N3: num] :
( ( X
= ( bit0 @ N3 ) )
=> ! [M5: num] :
( ( Xa
= ( bit1 @ M5 ) )
=> ( ( Y
= ( bit0 @ ( bit_or_not_num_neg @ N3 @ M5 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit0 @ N3 ) @ ( bit1 @ M5 ) ) ) ) ) )
=> ( ! [N3: num] :
( ( X
= ( bit1 @ N3 ) )
=> ( ( Xa = one2 )
=> ( ( Y = one2 )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit1 @ N3 ) @ one2 ) ) ) ) )
=> ( ! [N3: num] :
( ( X
= ( bit1 @ N3 ) )
=> ! [M5: num] :
( ( Xa
= ( bit0 @ M5 ) )
=> ( ( Y
= ( bitM @ ( bit_or_not_num_neg @ N3 @ M5 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit1 @ N3 ) @ ( bit0 @ M5 ) ) ) ) ) )
=> ~ ! [N3: num] :
( ( X
= ( bit1 @ N3 ) )
=> ! [M5: num] :
( ( Xa
= ( bit1 @ M5 ) )
=> ( ( Y
= ( bitM @ ( bit_or_not_num_neg @ N3 @ M5 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit1 @ N3 ) @ ( bit1 @ M5 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% or_not_num_neg.pelims
thf(fact_4928_cr__natural__def,axiom,
( code_cr_natural
= ( ^ [X3: nat,Y3: code_natural] :
( X3
= ( code_nat_of_natural @ Y3 ) ) ) ) ).
% cr_natural_def
thf(fact_4929_inv__o__cancel,axiom,
! [B: $tType,A: $tType,F: A > B] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( comp @ B @ A @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F ) @ F )
= ( id @ A ) ) ) ).
% inv_o_cancel
thf(fact_4930_o__inv__o__cancel,axiom,
! [B: $tType,C: $tType,A: $tType,F: A > B,G: A > C] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( comp @ B @ C @ A @ ( comp @ A @ C @ B @ G @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F ) ) @ F )
= G ) ) ).
% o_inv_o_cancel
thf(fact_4931_inj__map__inv__f,axiom,
! [B: $tType,A: $tType,F: A > B,L: list @ A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( map @ B @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F ) @ ( map @ A @ B @ F @ L ) )
= L ) ) ).
% inj_map_inv_f
thf(fact_4932_inv__into__comp,axiom,
! [A: $tType,C: $tType,B: $tType,F: A > B,G: C > A,A4: set @ C,X: B] :
( ( inj_on @ A @ B @ F @ ( image2 @ C @ A @ G @ A4 ) )
=> ( ( inj_on @ C @ A @ G @ A4 )
=> ( ( member @ B @ X @ ( image2 @ A @ B @ F @ ( image2 @ C @ A @ G @ A4 ) ) )
=> ( ( hilbert_inv_into @ C @ B @ A4 @ ( comp @ A @ B @ C @ F @ G ) @ X )
= ( comp @ A @ C @ B @ ( hilbert_inv_into @ C @ A @ A4 @ G ) @ ( hilbert_inv_into @ A @ B @ ( image2 @ C @ A @ G @ A4 ) @ F ) @ X ) ) ) ) ) ).
% inv_into_comp
thf(fact_4933_inv__unique__comp,axiom,
! [B: $tType,A: $tType,F: B > A,G: A > B] :
( ( ( comp @ B @ A @ A @ F @ G )
= ( id @ A ) )
=> ( ( ( comp @ A @ B @ B @ G @ F )
= ( id @ B ) )
=> ( ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F )
= G ) ) ) ).
% inv_unique_comp
thf(fact_4934_o__inv__distrib,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > B,G: 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 ) ) @ ( comp @ A @ B @ C @ F @ G ) )
= ( 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
thf(fact_4935_surj__iff,axiom,
! [B: $tType,A: $tType,F: B > A] :
( ( ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
= ( ( comp @ B @ A @ A @ F @ ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F ) )
= ( id @ A ) ) ) ).
% surj_iff
thf(fact_4936_inj__iff,axiom,
! [B: $tType,A: $tType,F: A > B] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
= ( ( comp @ B @ A @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F ) @ F )
= ( id @ A ) ) ) ).
% inj_iff
thf(fact_4937_fn__o__inv__fn__is__id,axiom,
! [A: $tType,F: A > A,N: nat] :
( ( bij_betw @ A @ A @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) )
=> ( ( comp @ A @ A @ A @ ( compow @ ( A > A ) @ N @ F ) @ ( compow @ ( A > A ) @ N @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F ) ) )
= ( ^ [X3: A] : X3 ) ) ) ).
% fn_o_inv_fn_is_id
thf(fact_4938_inv__fn__o__fn__is__id,axiom,
! [A: $tType,F: A > A,N: nat] :
( ( bij_betw @ A @ A @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) )
=> ( ( comp @ A @ A @ A @ ( compow @ ( A > A ) @ N @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F ) ) @ ( compow @ ( A > A ) @ N @ F ) )
= ( ^ [X3: A] : X3 ) ) ) ).
% inv_fn_o_fn_is_id
thf(fact_4939_bijection_Oinv__comp__right,axiom,
! [A: $tType,F: A > A] :
( ( hilbert_bijection @ A @ F )
=> ( ( comp @ A @ A @ A @ F @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F ) )
= ( id @ A ) ) ) ).
% bijection.inv_comp_right
thf(fact_4940_bijection_Oinv__comp__left,axiom,
! [A: $tType,F: A > A] :
( ( hilbert_bijection @ A @ F )
=> ( ( comp @ A @ A @ A @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F ) @ F )
= ( id @ A ) ) ) ).
% bijection.inv_comp_left
thf(fact_4941_finite__reachable__restrictedI,axiom,
! [A: $tType,Q: set @ A,I4: set @ A,E4: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ A @ Q )
=> ( ( ord_less_eq @ ( set @ A ) @ I4 @ Q )
=> ( ( ord_less_eq @ ( set @ A ) @ ( range2 @ A @ A @ E4 ) @ Q )
=> ( finite_finite2 @ A @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E4 ) @ I4 ) ) ) ) ) ).
% finite_reachable_restrictedI
thf(fact_4942_Range__insert,axiom,
! [A: $tType,B: $tType,A3: B,B2: A,R: set @ ( product_prod @ B @ A )] :
( ( range2 @ B @ A @ ( insert2 @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A3 @ B2 ) @ R ) )
= ( insert2 @ A @ B2 @ ( range2 @ B @ A @ R ) ) ) ).
% Range_insert
thf(fact_4943_Range__iff,axiom,
! [A: $tType,B: $tType,A3: A,R: set @ ( product_prod @ B @ A )] :
( ( member @ A @ A3 @ ( range2 @ B @ A @ R ) )
= ( ? [Y3: B] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ Y3 @ A3 ) @ R ) ) ) ).
% Range_iff
thf(fact_4944_RangeE,axiom,
! [A: $tType,B: $tType,B2: A,R: set @ ( product_prod @ B @ A )] :
( ( member @ A @ B2 @ ( range2 @ B @ A @ R ) )
=> ~ ! [A5: B] :
~ ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A5 @ B2 ) @ R ) ) ).
% RangeE
thf(fact_4945_Range_Ointros,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,R: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R )
=> ( member @ B @ B2 @ ( range2 @ A @ B @ R ) ) ) ).
% Range.intros
thf(fact_4946_Range_Osimps,axiom,
! [B: $tType,A: $tType,A3: B,R: set @ ( product_prod @ A @ B )] :
( ( member @ B @ A3 @ ( range2 @ A @ B @ R ) )
= ( ? [A7: A,B5: B] :
( ( A3 = B5 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A7 @ B5 ) @ R ) ) ) ) ).
% Range.simps
thf(fact_4947_Range_Ocases,axiom,
! [B: $tType,A: $tType,A3: B,R: set @ ( product_prod @ A @ B )] :
( ( member @ B @ A3 @ ( range2 @ A @ B @ R ) )
=> ~ ! [A5: A] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ A3 ) @ R ) ) ).
% Range.cases
thf(fact_4948_trancl__Image__in__Range,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A ),V6: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ ( transitive_trancl @ A @ R5 ) @ V6 ) @ ( range2 @ A @ A @ R5 ) ) ).
% trancl_Image_in_Range
thf(fact_4949_Range__rel__restrict,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A ),A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( range2 @ A @ A @ ( rel_restrict @ A @ R5 @ A4 ) ) @ ( minus_minus @ ( set @ A ) @ ( range2 @ A @ A @ R5 ) @ A4 ) ) ).
% Range_rel_restrict
thf(fact_4950_dom__ran__disj__comp,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
( ( ( inf_inf @ ( set @ A ) @ ( domain @ A @ A @ R5 ) @ ( range2 @ A @ A @ R5 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( relcomp @ A @ A @ A @ R5 @ R5 )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% dom_ran_disj_comp
thf(fact_4951_wf__max,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ ( converse @ A @ A @ R5 ) )
=> ( ( R5
!= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ~ ! [M5: A] :
~ ( member @ A @ M5 @ ( minus_minus @ ( set @ A ) @ ( range2 @ A @ A @ R5 ) @ ( domain @ A @ A @ R5 ) ) ) ) ) ).
% wf_max
thf(fact_4952_Rangep__Range__eq,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B )] :
( ( rangep @ A @ B
@ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R ) )
= ( ^ [X3: B] : ( member @ B @ X3 @ ( range2 @ A @ B @ R ) ) ) ) ).
% Rangep_Range_eq
thf(fact_4953_Domain__insert,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,R: set @ ( product_prod @ A @ B )] :
( ( domain @ A @ B @ ( insert2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R ) )
= ( insert2 @ A @ A3 @ ( domain @ A @ B @ R ) ) ) ).
% Domain_insert
thf(fact_4954_Domain_Ocases,axiom,
! [B: $tType,A: $tType,A3: A,R: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A3 @ ( domain @ A @ B @ R ) )
=> ~ ! [B3: B] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ R ) ) ).
% Domain.cases
thf(fact_4955_Domain_Osimps,axiom,
! [B: $tType,A: $tType,A3: A,R: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A3 @ ( domain @ A @ B @ R ) )
= ( ? [A7: A,B5: B] :
( ( A3 = A7 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A7 @ B5 ) @ R ) ) ) ) ).
% Domain.simps
thf(fact_4956_Domain_ODomainI,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,R: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R )
=> ( member @ A @ A3 @ ( domain @ A @ B @ R ) ) ) ).
% Domain.DomainI
thf(fact_4957_DomainE,axiom,
! [B: $tType,A: $tType,A3: A,R: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A3 @ ( domain @ A @ B @ R ) )
=> ~ ! [B3: B] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ R ) ) ).
% DomainE
thf(fact_4958_Domain__iff,axiom,
! [A: $tType,B: $tType,A3: A,R: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A3 @ ( domain @ A @ B @ R ) )
= ( ? [Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ Y3 ) @ R ) ) ) ).
% Domain_iff
thf(fact_4959_Not__Domain__rtrancl,axiom,
! [A: $tType,X: A,R5: set @ ( product_prod @ A @ A ),Y: A] :
( ~ ( member @ A @ X @ ( domain @ A @ A @ R5 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R5 ) )
= ( X = Y ) ) ) ).
% Not_Domain_rtrancl
thf(fact_4960_Domain__unfold,axiom,
! [B: $tType,A: $tType] :
( ( domain @ A @ B )
= ( ^ [R3: set @ ( product_prod @ A @ B )] :
( collect @ A
@ ^ [X3: A] :
? [Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R3 ) ) ) ) ).
% Domain_unfold
thf(fact_4961_for__in__RI,axiom,
! [B: $tType,A: $tType,X: A,R5: set @ ( product_prod @ A @ B )] :
( ( member @ A @ X @ ( domain @ A @ B @ R5 ) )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ ( fun_of_rel @ A @ B @ R5 @ X ) ) @ R5 ) ) ).
% for_in_RI
thf(fact_4962_Domain__rel__restrict,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A ),A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( domain @ A @ A @ ( rel_restrict @ A @ R5 @ A4 ) ) @ ( minus_minus @ ( set @ A ) @ ( domain @ A @ A @ R5 ) @ A4 ) ) ).
% Domain_rel_restrict
thf(fact_4963_wf__no__path,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
( ( ( inf_inf @ ( set @ A ) @ ( domain @ A @ A @ R5 ) @ ( range2 @ A @ A @ R5 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( wf @ A @ R5 ) ) ).
% wf_no_path
thf(fact_4964_wf__min,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ R5 )
=> ( ( R5
!= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ~ ! [M5: A] :
~ ( member @ A @ M5 @ ( minus_minus @ ( set @ A ) @ ( domain @ A @ A @ R5 ) @ ( range2 @ A @ A @ R5 ) ) ) ) ) ).
% wf_min
thf(fact_4965_Range__def,axiom,
! [B: $tType,A: $tType] :
( ( range2 @ A @ B )
= ( ^ [R3: set @ ( product_prod @ A @ B )] :
( collect @ B
@ ( rangep @ A @ B
@ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R3 ) ) ) ) ) ).
% Range_def
thf(fact_4966_and_Omonoid__axioms,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( monoid @ A @ ( bit_se5824344872417868541ns_and @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% and.monoid_axioms
thf(fact_4967_cr__integer__def,axiom,
( code_cr_integer
= ( ^ [X3: int,Y3: code_integer] :
( X3
= ( code_int_of_integer @ Y3 ) ) ) ) ).
% cr_integer_def
thf(fact_4968_wo__rel_OisMinim__def,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A4: set @ A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R )
=> ( ( bNF_We4791949203932849705sMinim @ A @ R @ A4 @ B2 )
= ( ( member @ A @ B2 @ A4 )
& ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ X3 ) @ R ) ) ) ) ) ).
% wo_rel.isMinim_def
thf(fact_4969_monoid_Oleft__neutral,axiom,
! [A: $tType,F: A > A > A,Z2: A,A3: A] :
( ( monoid @ A @ F @ Z2 )
=> ( ( F @ Z2 @ A3 )
= A3 ) ) ).
% monoid.left_neutral
thf(fact_4970_monoid_Oright__neutral,axiom,
! [A: $tType,F: A > A > A,Z2: A,A3: A] :
( ( monoid @ A @ F @ Z2 )
=> ( ( F @ A3 @ Z2 )
= A3 ) ) ).
% monoid.right_neutral
thf(fact_4971_add_Omonoid__axioms,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ( monoid @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ).
% add.monoid_axioms
thf(fact_4972_mult_Omonoid__axioms,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ( monoid @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ).
% mult.monoid_axioms
thf(fact_4973_integer_Opcr__cr__eq,axiom,
code_pcr_integer = code_cr_integer ).
% integer.pcr_cr_eq
thf(fact_4974_multp__implies__one__step,axiom,
! [A: $tType,R: A > A > $o,M3: multiset @ A,N6: multiset @ A] :
( ( transp @ A @ R )
=> ( ( multp @ A @ R @ M3 @ N6 )
=> ? [I8: multiset @ A,J5: multiset @ A] :
( ( N6
= ( plus_plus @ ( multiset @ A ) @ I8 @ J5 ) )
& ? [K10: multiset @ A] :
( ( M3
= ( plus_plus @ ( multiset @ A ) @ I8 @ K10 ) )
& ( J5
!= ( zero_zero @ ( multiset @ A ) ) )
& ! [X7: A] :
( ( member @ A @ X7 @ ( set_mset @ A @ K10 ) )
=> ? [Xa3: A] :
( ( member @ A @ Xa3 @ ( set_mset @ A @ J5 ) )
& ( R @ X7 @ Xa3 ) ) ) ) ) ) ) ).
% multp_implies_one_step
thf(fact_4975_relpow__fun__conv,axiom,
! [A: $tType,A3: A,B2: A,N: nat,R5: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R5 ) )
= ( ? [F2: nat > A] :
( ( ( F2 @ ( zero_zero @ nat ) )
= A3 )
& ( ( F2 @ N )
= B2 )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F2 @ I3 ) @ ( F2 @ ( suc @ I3 ) ) ) @ R5 ) ) ) ) ) ).
% relpow_fun_conv
thf(fact_4976_irreflp__irrefl__eq,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
( ( irreflp @ A
@ ^ [A7: A,B5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B5 ) @ R5 ) )
= ( irrefl @ A @ R5 ) ) ).
% irreflp_irrefl_eq
thf(fact_4977_relpow__1,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
( ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( one_one @ nat ) @ R5 )
= R5 ) ).
% relpow_1
thf(fact_4978_multp__cancel,axiom,
! [A: $tType,R: A > A > $o,X10: multiset @ A,Z9: multiset @ A,Y9: multiset @ A] :
( ( transp @ A @ R )
=> ( ( irreflp @ A @ R )
=> ( ( multp @ A @ R @ ( plus_plus @ ( multiset @ A ) @ X10 @ Z9 ) @ ( plus_plus @ ( multiset @ A ) @ Y9 @ Z9 ) )
= ( multp @ A @ R @ X10 @ Y9 ) ) ) ) ).
% multp_cancel
thf(fact_4979_relpow__Suc__D2_H,axiom,
! [A: $tType,N: nat,R5: set @ ( product_prod @ A @ A ),X7: A,Y6: A,Z6: A] :
( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X7 @ Y6 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R5 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ Z6 ) @ R5 ) )
=> ? [W4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X7 @ W4 ) @ R5 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ W4 @ Z6 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R5 ) ) ) ) ).
% relpow_Suc_D2'
thf(fact_4980_relpow__0__E,axiom,
! [A: $tType,X: A,Y: A,R5: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( zero_zero @ nat ) @ R5 ) )
=> ( X = Y ) ) ).
% relpow_0_E
thf(fact_4981_relpow__0__I,axiom,
! [A: $tType,X: A,R5: set @ ( product_prod @ A @ A )] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( zero_zero @ nat ) @ R5 ) ) ).
% relpow_0_I
thf(fact_4982_relpow__Suc__E,axiom,
! [A: $tType,X: A,Z2: A,N: nat,R5: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R5 ) )
=> ~ ! [Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y4 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R5 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z2 ) @ R5 ) ) ) ).
% relpow_Suc_E
thf(fact_4983_relpow__Suc__I,axiom,
! [A: $tType,X: A,Y: A,N: nat,R5: set @ ( product_prod @ A @ A ),Z2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R5 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ R5 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R5 ) ) ) ) ).
% relpow_Suc_I
thf(fact_4984_relpow__Suc__D2,axiom,
! [A: $tType,X: A,Z2: A,N: nat,R5: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R5 ) )
=> ? [Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y4 ) @ R5 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R5 ) ) ) ) ).
% relpow_Suc_D2
thf(fact_4985_relpow__Suc__E2,axiom,
! [A: $tType,X: A,Z2: A,N: nat,R5: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R5 ) )
=> ~ ! [Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y4 ) @ R5 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R5 ) ) ) ) ).
% relpow_Suc_E2
thf(fact_4986_relpow__Suc__I2,axiom,
! [A: $tType,X: A,Y: A,R5: set @ ( product_prod @ A @ A ),Z2: A,N: nat] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R5 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R5 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R5 ) ) ) ) ).
% relpow_Suc_I2
thf(fact_4987_relpow__add,axiom,
! [A: $tType,M2: nat,N: nat,R5: set @ ( product_prod @ A @ A )] :
( ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( plus_plus @ nat @ M2 @ N ) @ R5 )
= ( relcomp @ A @ A @ A @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ M2 @ R5 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R5 ) ) ) ).
% relpow_add
thf(fact_4988_transp__trans__eq,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( transp @ A
@ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R ) )
= ( trans @ A @ R ) ) ).
% transp_trans_eq
thf(fact_4989_relpowp__relpow__eq,axiom,
! [A: $tType,N: nat,R5: set @ ( product_prod @ A @ A )] :
( ( compow @ ( A > A > $o ) @ N
@ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R5 ) )
= ( ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R5 ) ) ) ) ).
% relpowp_relpow_eq
thf(fact_4990_relpow__E2,axiom,
! [A: $tType,X: A,Z2: A,N: nat,R5: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R5 ) )
=> ( ( ( N
= ( zero_zero @ nat ) )
=> ( X != Z2 ) )
=> ~ ! [Y4: A,M5: nat] :
( ( N
= ( suc @ M5 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y4 ) @ R5 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ M5 @ R5 ) ) ) ) ) ) ).
% relpow_E2
thf(fact_4991_relpow__E,axiom,
! [A: $tType,X: A,Z2: A,N: nat,R5: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R5 ) )
=> ( ( ( N
= ( zero_zero @ nat ) )
=> ( X != Z2 ) )
=> ~ ! [Y4: A,M5: nat] :
( ( N
= ( suc @ M5 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y4 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ M5 @ R5 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z2 ) @ R5 ) ) ) ) ) ).
% relpow_E
thf(fact_4992_nths__drop,axiom,
! [A: $tType,N: nat,Xs: list @ A,I4: set @ nat] :
( ( nths @ A @ ( drop @ A @ N @ Xs ) @ I4 )
= ( nths @ A @ Xs @ ( image2 @ nat @ nat @ ( plus_plus @ nat @ N ) @ I4 ) ) ) ).
% nths_drop
thf(fact_4993_nths__append,axiom,
! [A: $tType,L: list @ A,L4: list @ A,A4: set @ nat] :
( ( nths @ A @ ( append @ A @ L @ L4 ) @ A4 )
= ( append @ A @ ( nths @ A @ L @ A4 )
@ ( nths @ A @ L4
@ ( collect @ nat
@ ^ [J2: nat] : ( member @ nat @ ( plus_plus @ nat @ J2 @ ( size_size @ ( list @ A ) @ L ) ) @ A4 ) ) ) ) ) ).
% nths_append
thf(fact_4994_asymp__asym__eq,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
( ( asymp @ A
@ ^ [A7: A,B5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B5 ) @ R5 ) )
= ( asym @ A @ R5 ) ) ).
% asymp_asym_eq
thf(fact_4995_asym_Ocases,axiom,
! [A: $tType,A3: set @ ( product_prod @ A @ A )] :
( ( asym @ A @ A3 )
=> ! [A17: A,B8: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A17 @ B8 ) @ A3 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B8 @ A17 ) @ A3 ) ) ) ).
% asym.cases
thf(fact_4996_asym_Osimps,axiom,
! [A: $tType] :
( ( asym @ A )
= ( ^ [A7: set @ ( product_prod @ A @ A )] :
? [R6: set @ ( product_prod @ A @ A )] :
( ( A7 = R6 )
& ! [X3: A,Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R6 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R6 ) ) ) ) ) ).
% asym.simps
thf(fact_4997_asym_Ointros,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
( ! [A5: A,B3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B3 ) @ R5 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A5 ) @ R5 ) )
=> ( asym @ A @ R5 ) ) ).
% asym.intros
thf(fact_4998_asymD,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( asym @ A @ R5 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R5 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ R5 ) ) ) ).
% asymD
thf(fact_4999_asym__iff,axiom,
! [A: $tType] :
( ( asym @ A )
= ( ^ [R6: set @ ( product_prod @ A @ A )] :
! [X3: A,Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R6 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R6 ) ) ) ) ).
% asym_iff
thf(fact_5000_lexord__asymmetric,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A ),A3: list @ A,B2: list @ A] :
( ( asym @ A @ R5 )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ A3 @ B2 ) @ ( lexord @ A @ R5 ) )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ B2 @ A3 ) @ ( lexord @ A @ R5 ) ) ) ) ).
% lexord_asymmetric
thf(fact_5001_ordLess__iff,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
= ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
& ( order_well_order_on @ B @ ( field2 @ B @ R4 ) @ R4 )
& ~ ? [X8: B > A] : ( bNF_Wellorder_embed @ B @ A @ R4 @ R @ X8 ) ) ) ).
% ordLess_iff
thf(fact_5002_option_Oin__rel,axiom,
! [B: $tType,A: $tType] :
( ( rel_option @ A @ B )
= ( ^ [R6: A > B > $o,A7: option @ A,B5: option @ B] :
? [Z5: option @ ( product_prod @ A @ B )] :
( ( member @ ( option @ ( product_prod @ A @ B ) ) @ Z5
@ ( collect @ ( option @ ( product_prod @ A @ B ) )
@ ^ [X3: option @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set_option @ ( product_prod @ A @ B ) @ X3 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) ) ) )
& ( ( map_option @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Z5 )
= A7 )
& ( ( map_option @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Z5 )
= B5 ) ) ) ) ).
% option.in_rel
thf(fact_5003_bit__double__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ A3 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
& ( N
!= ( zero_zero @ nat ) )
& ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N ) ) ) ) ).
% bit_double_iff
thf(fact_5004_rel__option__None2,axiom,
! [B: $tType,A: $tType,P: A > B > $o,X: option @ A] :
( ( rel_option @ A @ B @ P @ X @ ( none @ B ) )
= ( X
= ( none @ A ) ) ) ).
% rel_option_None2
thf(fact_5005_rel__option__None1,axiom,
! [A: $tType,B: $tType,P: A > B > $o,X: option @ B] :
( ( rel_option @ A @ B @ P @ ( none @ A ) @ X )
= ( X
= ( none @ B ) ) ) ).
% rel_option_None1
thf(fact_5006_bit__minus__1__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N )
= ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N ) ) ) ).
% bit_minus_1_iff
thf(fact_5007_rel__option__iff,axiom,
! [B: $tType,A: $tType] :
( ( rel_option @ A @ B )
= ( ^ [R6: A > B > $o,X3: option @ A,Y3: option @ B] :
( product_case_prod @ ( option @ A ) @ ( option @ B ) @ $o
@ ^ [A7: option @ A,B5: option @ B] :
( case_option @ $o @ A
@ ( case_option @ $o @ B @ $true
@ ^ [C3: B] : $false
@ B5 )
@ ^ [Z5: A] : ( case_option @ $o @ B @ $false @ ( R6 @ Z5 ) @ B5 )
@ A7 )
@ ( product_Pair @ ( option @ A ) @ ( option @ B ) @ X3 @ Y3 ) ) ) ) ).
% rel_option_iff
thf(fact_5008_option_Orel__mono,axiom,
! [B: $tType,A: $tType,R5: A > B > $o,Ra2: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ R5 @ Ra2 )
=> ( ord_less_eq @ ( ( option @ A ) > ( option @ B ) > $o ) @ ( rel_option @ A @ B @ R5 ) @ ( rel_option @ A @ B @ Ra2 ) ) ) ).
% option.rel_mono
thf(fact_5009_option_Orel__inject_I2_J,axiom,
! [A: $tType,B: $tType,R5: A > B > $o,X22: A,Y2: B] :
( ( rel_option @ A @ B @ R5 @ ( some @ A @ X22 ) @ ( some @ B @ Y2 ) )
= ( R5 @ X22 @ Y2 ) ) ).
% option.rel_inject(2)
thf(fact_5010_option_Orel__intros_I2_J,axiom,
! [A: $tType,B: $tType,R5: A > B > $o,X22: A,Y2: B] :
( ( R5 @ X22 @ Y2 )
=> ( rel_option @ A @ B @ R5 @ ( some @ A @ X22 ) @ ( some @ B @ Y2 ) ) ) ).
% option.rel_intros(2)
thf(fact_5011_option__rel__Some1,axiom,
! [A: $tType,B: $tType,A4: A > B > $o,X: A,Y: option @ B] :
( ( rel_option @ A @ B @ A4 @ ( some @ A @ X ) @ Y )
= ( ? [Y8: B] :
( ( Y
= ( some @ B @ Y8 ) )
& ( A4 @ X @ Y8 ) ) ) ) ).
% option_rel_Some1
thf(fact_5012_option__rel__Some2,axiom,
! [B: $tType,A: $tType,A4: A > B > $o,X: option @ A,Y: B] :
( ( rel_option @ A @ B @ A4 @ X @ ( some @ B @ Y ) )
= ( ? [X9: A] :
( ( X
= ( some @ A @ X9 ) )
& ( A4 @ X9 @ Y ) ) ) ) ).
% option_rel_Some2
thf(fact_5013_option_Octr__transfer_I1_J,axiom,
! [A: $tType,B: $tType,R5: A > B > $o] : ( rel_option @ A @ B @ R5 @ ( none @ A ) @ ( none @ B ) ) ).
% option.ctr_transfer(1)
thf(fact_5014_rel__option__inf,axiom,
! [B: $tType,A: $tType,A4: A > B > $o,B6: A > B > $o] :
( ( inf_inf @ ( ( option @ A ) > ( option @ B ) > $o ) @ ( rel_option @ A @ B @ A4 ) @ ( rel_option @ A @ B @ B6 ) )
= ( rel_option @ A @ B @ ( inf_inf @ ( A > B > $o ) @ A4 @ B6 ) ) ) ).
% rel_option_inf
thf(fact_5015_option_Orel__refl,axiom,
! [B: $tType,Ra2: B > B > $o,X: option @ B] :
( ! [X2: B] : ( Ra2 @ X2 @ X2 )
=> ( rel_option @ B @ B @ Ra2 @ X @ X ) ) ).
% option.rel_refl
thf(fact_5016_option_Orel__eq,axiom,
! [A: $tType] :
( ( rel_option @ A @ A
@ ^ [Y5: A,Z4: A] : Y5 = Z4 )
= ( ^ [Y5: option @ A,Z4: option @ A] : Y5 = Z4 ) ) ).
% option.rel_eq
thf(fact_5017_option_Orel__cong,axiom,
! [A: $tType,B: $tType,X: option @ A,Ya: option @ A,Y: option @ B,Xa: option @ B,R5: A > B > $o,Ra2: A > B > $o] :
( ( X = Ya )
=> ( ( Y = Xa )
=> ( ! [Z3: A,Yb: B] :
( ( member @ A @ Z3 @ ( set_option @ A @ Ya ) )
=> ( ( member @ B @ Yb @ ( set_option @ B @ Xa ) )
=> ( ( R5 @ Z3 @ Yb )
= ( Ra2 @ Z3 @ Yb ) ) ) )
=> ( ( rel_option @ A @ B @ R5 @ X @ Y )
= ( rel_option @ A @ B @ Ra2 @ Ya @ Xa ) ) ) ) ) ).
% option.rel_cong
thf(fact_5018_option_Orel__mono__strong,axiom,
! [A: $tType,B: $tType,R5: A > B > $o,X: option @ A,Y: option @ B,Ra2: A > B > $o] :
( ( rel_option @ A @ B @ R5 @ X @ Y )
=> ( ! [Z3: A,Yb: B] :
( ( member @ A @ Z3 @ ( set_option @ A @ X ) )
=> ( ( member @ B @ Yb @ ( set_option @ B @ Y ) )
=> ( ( R5 @ Z3 @ Yb )
=> ( Ra2 @ Z3 @ Yb ) ) ) )
=> ( rel_option @ A @ B @ Ra2 @ X @ Y ) ) ) ).
% option.rel_mono_strong
thf(fact_5019_option_Orel__refl__strong,axiom,
! [A: $tType,X: option @ A,Ra2: A > A > $o] :
( ! [Z3: A] :
( ( member @ A @ Z3 @ ( set_option @ A @ X ) )
=> ( Ra2 @ Z3 @ Z3 ) )
=> ( rel_option @ A @ A @ Ra2 @ X @ X ) ) ).
% option.rel_refl_strong
thf(fact_5020_rel__option__reflI,axiom,
! [A: $tType,Y: option @ A,P: A > A > $o] :
( ! [X2: A] :
( ( member @ A @ X2 @ ( set_option @ A @ Y ) )
=> ( P @ X2 @ X2 ) )
=> ( rel_option @ A @ A @ P @ Y @ Y ) ) ).
% rel_option_reflI
thf(fact_5021_option_Orel__map_I1_J,axiom,
! [A: $tType,C: $tType,B: $tType,Sb: C > B > $o,I: A > C,X: option @ A,Y: option @ B] :
( ( rel_option @ C @ B @ Sb @ ( map_option @ A @ C @ I @ X ) @ Y )
= ( rel_option @ A @ B
@ ^ [X3: A] : ( Sb @ ( I @ X3 ) )
@ X
@ Y ) ) ).
% option.rel_map(1)
thf(fact_5022_option_Orel__map_I2_J,axiom,
! [A: $tType,C: $tType,B: $tType,Sa: A > C > $o,X: option @ A,G: B > C,Y: option @ B] :
( ( rel_option @ A @ C @ Sa @ X @ ( map_option @ B @ C @ G @ Y ) )
= ( rel_option @ A @ B
@ ^ [X3: A,Y3: B] : ( Sa @ X3 @ ( G @ Y3 ) )
@ X
@ Y ) ) ).
% option.rel_map(2)
thf(fact_5023_option_Orel__flip,axiom,
! [A: $tType,B: $tType,R5: A > B > $o,A3: option @ B,B2: option @ A] :
( ( rel_option @ B @ A @ ( conversep @ A @ B @ R5 ) @ A3 @ B2 )
= ( rel_option @ A @ B @ R5 @ B2 @ A3 ) ) ).
% option.rel_flip
thf(fact_5024_option_Orel__conversep,axiom,
! [B: $tType,A: $tType,R5: A > B > $o] :
( ( rel_option @ B @ A @ ( conversep @ A @ B @ R5 ) )
= ( conversep @ ( option @ A ) @ ( option @ B ) @ ( rel_option @ A @ B @ R5 ) ) ) ).
% option.rel_conversep
thf(fact_5025_option_Octr__transfer_I2_J,axiom,
! [A: $tType,B: $tType,R5: A > B > $o] : ( bNF_rel_fun @ A @ B @ ( option @ A ) @ ( option @ B ) @ R5 @ ( rel_option @ A @ B @ R5 ) @ ( some @ A ) @ ( some @ B ) ) ).
% option.ctr_transfer(2)
thf(fact_5026_option_Odisc__transfer_I2_J,axiom,
! [A: $tType,B: $tType,R5: A > B > $o] :
( bNF_rel_fun @ ( option @ A ) @ ( option @ B ) @ $o @ $o @ ( rel_option @ A @ B @ R5 )
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4
@ ^ [Option3: option @ A] :
( Option3
!= ( none @ A ) )
@ ^ [Option3: option @ B] :
( Option3
!= ( none @ B ) ) ) ).
% option.disc_transfer(2)
thf(fact_5027_option_Odisc__transfer_I1_J,axiom,
! [A: $tType,B: $tType,R5: A > B > $o] :
( bNF_rel_fun @ ( option @ A ) @ ( option @ B ) @ $o @ $o @ ( rel_option @ A @ B @ R5 )
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4
@ ^ [Option3: option @ A] :
( Option3
= ( none @ A ) )
@ ^ [Option3: option @ B] :
( Option3
= ( none @ B ) ) ) ).
% option.disc_transfer(1)
thf(fact_5028_option__bind__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A4: A > B > $o,B6: C > D > $o] : ( bNF_rel_fun @ ( option @ A ) @ ( option @ B ) @ ( ( A > ( option @ C ) ) > ( option @ C ) ) @ ( ( B > ( option @ D ) ) > ( option @ D ) ) @ ( rel_option @ A @ B @ A4 ) @ ( bNF_rel_fun @ ( A > ( option @ C ) ) @ ( B > ( option @ D ) ) @ ( option @ C ) @ ( option @ D ) @ ( bNF_rel_fun @ A @ B @ ( option @ C ) @ ( option @ D ) @ A4 @ ( rel_option @ C @ D @ B6 ) ) @ ( rel_option @ C @ D @ B6 ) ) @ ( bind @ A @ C ) @ ( bind @ B @ D ) ) ).
% option_bind_transfer
thf(fact_5029_option_Orel__transp,axiom,
! [A: $tType,R5: A > A > $o] :
( ( transp @ A @ R5 )
=> ( transp @ ( option @ A ) @ ( rel_option @ A @ A @ R5 ) ) ) ).
% option.rel_transp
thf(fact_5030_option_Orel__distinct_I2_J,axiom,
! [A: $tType,B: $tType,R5: A > B > $o,Y2: A] :
~ ( rel_option @ A @ B @ R5 @ ( some @ A @ Y2 ) @ ( none @ B ) ) ).
% option.rel_distinct(2)
thf(fact_5031_option_Orel__distinct_I1_J,axiom,
! [A: $tType,B: $tType,R5: A > B > $o,Y2: B] :
~ ( rel_option @ A @ B @ R5 @ ( none @ A ) @ ( some @ B @ Y2 ) ) ).
% option.rel_distinct(1)
thf(fact_5032_option_Orel__cases,axiom,
! [A: $tType,B: $tType,R5: A > B > $o,A3: option @ A,B2: option @ B] :
( ( rel_option @ A @ B @ R5 @ A3 @ B2 )
=> ( ( ( A3
= ( none @ A ) )
=> ( B2
!= ( none @ B ) ) )
=> ~ ! [X2: A] :
( ( A3
= ( some @ A @ X2 ) )
=> ! [Y4: B] :
( ( B2
= ( some @ B @ Y4 ) )
=> ~ ( R5 @ X2 @ Y4 ) ) ) ) ) ).
% option.rel_cases
thf(fact_5033_option_Orel__induct,axiom,
! [A: $tType,B: $tType,R5: A > B > $o,X: option @ A,Y: option @ B,Q: ( option @ A ) > ( option @ B ) > $o] :
( ( rel_option @ A @ B @ R5 @ X @ Y )
=> ( ( Q @ ( none @ A ) @ ( none @ B ) )
=> ( ! [A24: A,B24: B] :
( ( R5 @ A24 @ B24 )
=> ( Q @ ( some @ A @ A24 ) @ ( some @ B @ B24 ) ) )
=> ( Q @ X @ Y ) ) ) ) ).
% option.rel_induct
thf(fact_5034_option_Omap__transfer,axiom,
! [A: $tType,B: $tType,F5: $tType,E: $tType,Rb2: A > E > $o,Sd: B > F5 > $o] : ( bNF_rel_fun @ ( A > B ) @ ( E > F5 ) @ ( ( option @ A ) > ( option @ B ) ) @ ( ( option @ E ) > ( option @ F5 ) ) @ ( bNF_rel_fun @ A @ E @ B @ F5 @ Rb2 @ Sd ) @ ( bNF_rel_fun @ ( option @ A ) @ ( option @ E ) @ ( option @ B ) @ ( option @ F5 ) @ ( rel_option @ A @ E @ Rb2 ) @ ( rel_option @ B @ F5 @ Sd ) ) @ ( map_option @ A @ B ) @ ( map_option @ E @ F5 ) ) ).
% option.map_transfer
thf(fact_5035_option_Ocase__transfer,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,S2: C > D > $o,R5: A > B > $o] : ( bNF_rel_fun @ C @ D @ ( ( A > C ) > ( option @ A ) > C ) @ ( ( B > D ) > ( option @ B ) > D ) @ S2 @ ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ ( ( option @ A ) > C ) @ ( ( option @ B ) > D ) @ ( bNF_rel_fun @ A @ B @ C @ D @ R5 @ S2 ) @ ( bNF_rel_fun @ ( option @ A ) @ ( option @ B ) @ C @ D @ ( rel_option @ A @ B @ R5 ) @ S2 ) ) @ ( case_option @ C @ A ) @ ( case_option @ D @ B ) ) ).
% option.case_transfer
thf(fact_5036_option_Orel__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,Sa: A > C > $o,Sc: B > D > $o] :
( bNF_rel_fun @ ( A > B > $o ) @ ( C > D > $o ) @ ( ( option @ A ) > ( option @ B ) > $o ) @ ( ( option @ C ) > ( option @ D ) > $o )
@ ( bNF_rel_fun @ A @ C @ ( B > $o ) @ ( D > $o ) @ Sa
@ ( bNF_rel_fun @ B @ D @ $o @ $o @ Sc
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 ) )
@ ( bNF_rel_fun @ ( option @ A ) @ ( option @ C ) @ ( ( option @ B ) > $o ) @ ( ( option @ D ) > $o ) @ ( rel_option @ A @ C @ Sa )
@ ( bNF_rel_fun @ ( option @ B ) @ ( option @ D ) @ $o @ $o @ ( rel_option @ B @ D @ Sc )
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 ) )
@ ( rel_option @ A @ B )
@ ( rel_option @ C @ D ) ) ).
% option.rel_transfer
thf(fact_5037_bit__minus__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( uminus_uminus @ A @ A3 ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ~ ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ N ) ) ) ) ).
% bit_minus_iff
thf(fact_5038_option_Orel__sel,axiom,
! [B: $tType,A: $tType] :
( ( rel_option @ A @ B )
= ( ^ [R6: A > B > $o,A7: option @ A,B5: option @ B] :
( ( ( A7
= ( none @ A ) )
= ( B5
= ( none @ B ) ) )
& ( ( A7
!= ( none @ A ) )
=> ( ( B5
!= ( none @ B ) )
=> ( R6 @ ( the2 @ A @ A7 ) @ ( the2 @ B @ B5 ) ) ) ) ) ) ) ).
% option.rel_sel
thf(fact_5039_ordLess__not__embed,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
=> ~ ? [X_1: B > A] : ( bNF_Wellorder_embed @ B @ A @ R4 @ R @ X_1 ) ) ).
% ordLess_not_embed
thf(fact_5040_option_Orec__transfer,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,S2: C > D > $o,R5: A > B > $o] : ( bNF_rel_fun @ C @ D @ ( ( A > C ) > ( option @ A ) > C ) @ ( ( B > D ) > ( option @ B ) > D ) @ S2 @ ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ ( ( option @ A ) > C ) @ ( ( option @ B ) > D ) @ ( bNF_rel_fun @ A @ B @ C @ D @ R5 @ S2 ) @ ( bNF_rel_fun @ ( option @ A ) @ ( option @ B ) @ C @ D @ ( rel_option @ A @ B @ R5 ) @ S2 ) ) @ ( rec_option @ C @ A ) @ ( rec_option @ D @ B ) ) ).
% option.rec_transfer
thf(fact_5041_bit__2__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ ( one_one @ nat ) )
& ( N
= ( one_one @ nat ) ) ) ) ) ).
% bit_2_iff
thf(fact_5042_BNF__Wellorder__Constructions_OordLess__Field,axiom,
! [A: $tType,B: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),F: A > B] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R1 @ R22 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
=> ( ( bNF_Wellorder_embed @ A @ B @ R1 @ R22 @ F )
=> ( ( image2 @ A @ B @ F @ ( field2 @ A @ R1 ) )
!= ( field2 @ B @ R22 ) ) ) ) ).
% BNF_Wellorder_Constructions.ordLess_Field
thf(fact_5043_bit__mask__sub__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M2: nat,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) @ ( one_one @ A ) ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ( ord_less @ nat @ N @ M2 ) ) ) ) ).
% bit_mask_sub_iff
thf(fact_5044_embed__ordLess__ofilterIncl,axiom,
! [B: $tType,A: $tType,C: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),R33: set @ ( product_prod @ C @ C ),F13: A > C,F232: B > C] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R1 @ R22 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R22 @ R33 ) @ ( bNF_We4044943003108391690rdLess @ B @ C ) )
=> ( ( bNF_Wellorder_embed @ A @ C @ R1 @ R33 @ F13 )
=> ( ( bNF_Wellorder_embed @ B @ C @ R22 @ R33 @ F232 )
=> ( member @ ( product_prod @ ( set @ C ) @ ( set @ C ) ) @ ( product_Pair @ ( set @ C ) @ ( set @ C ) @ ( image2 @ A @ C @ F13 @ ( field2 @ A @ R1 ) ) @ ( image2 @ B @ C @ F232 @ ( field2 @ B @ R22 ) ) ) @ ( bNF_We413866401316099525erIncl @ C @ R33 ) ) ) ) ) ) ).
% embed_ordLess_ofilterIncl
thf(fact_5045_comp__embed,axiom,
! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),F: A > B,R9: set @ ( product_prod @ C @ C ),F3: B > C] :
( ( bNF_Wellorder_embed @ A @ B @ R @ R4 @ F )
=> ( ( bNF_Wellorder_embed @ B @ C @ R4 @ R9 @ F3 )
=> ( bNF_Wellorder_embed @ A @ C @ R @ R9 @ ( comp @ B @ C @ A @ F3 @ F ) ) ) ) ).
% comp_embed
thf(fact_5046_override__on__insert,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B,X: A,X10: set @ A] :
( ( override_on @ A @ B @ F @ G @ ( insert2 @ A @ X @ X10 ) )
= ( fun_upd @ A @ B @ ( override_on @ A @ B @ F @ G @ X10 ) @ X @ ( G @ X ) ) ) ).
% override_on_insert
thf(fact_5047_override__on__apply__in,axiom,
! [B: $tType,A: $tType,A3: A,A4: set @ A,F: A > B,G: A > B] :
( ( member @ A @ A3 @ A4 )
=> ( ( override_on @ A @ B @ F @ G @ A4 @ A3 )
= ( G @ A3 ) ) ) ).
% override_on_apply_in
thf(fact_5048_override__on__apply__notin,axiom,
! [B: $tType,A: $tType,A3: A,A4: set @ A,F: A > B,G: A > B] :
( ~ ( member @ A @ A3 @ A4 )
=> ( ( override_on @ A @ B @ F @ G @ A4 @ A3 )
= ( F @ A3 ) ) ) ).
% override_on_apply_notin
thf(fact_5049_override__on__emptyset,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ( override_on @ A @ B @ F @ G @ ( bot_bot @ ( set @ A ) ) )
= F ) ).
% override_on_emptyset
thf(fact_5050_override__on__def,axiom,
! [B: $tType,A: $tType] :
( ( override_on @ A @ B )
= ( ^ [F2: A > B,G2: A > B,A9: set @ A,A7: A] : ( if @ B @ ( member @ A @ A7 @ A9 ) @ ( G2 @ A7 ) @ ( F2 @ A7 ) ) ) ) ).
% override_on_def
thf(fact_5051_override__on__insert_H,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B,X: A,X10: set @ A] :
( ( override_on @ A @ B @ F @ G @ ( insert2 @ A @ X @ X10 ) )
= ( override_on @ A @ B @ ( fun_upd @ A @ B @ F @ X @ ( G @ X ) ) @ G @ X10 ) ) ).
% override_on_insert'
thf(fact_5052_embedS__comp__embed,axiom,
! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),F: A > B,R9: set @ ( product_prod @ C @ C ),F3: B > C] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( order_well_order_on @ B @ ( field2 @ B @ R4 ) @ R4 )
=> ( ( bNF_Wellorder_embedS @ A @ B @ R @ R4 @ F )
=> ( ( bNF_Wellorder_embed @ B @ C @ R4 @ R9 @ F3 )
=> ( bNF_Wellorder_embedS @ A @ C @ R @ R9 @ ( comp @ B @ C @ A @ F3 @ F ) ) ) ) ) ) ).
% embedS_comp_embed
thf(fact_5053_embed__comp__embedS,axiom,
! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),F: A > B,R9: set @ ( product_prod @ C @ C ),F3: B > C] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( order_well_order_on @ B @ ( field2 @ B @ R4 ) @ R4 )
=> ( ( bNF_Wellorder_embed @ A @ B @ R @ R4 @ F )
=> ( ( bNF_Wellorder_embedS @ B @ C @ R4 @ R9 @ F3 )
=> ( bNF_Wellorder_embedS @ A @ C @ R @ R9 @ ( comp @ B @ C @ A @ F3 @ F ) ) ) ) ) ) ).
% embed_comp_embedS
thf(fact_5054_pred__option__parametric,axiom,
! [A: $tType,B: $tType,A4: A > B > $o] :
( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( option @ A ) > $o ) @ ( ( option @ B ) > $o )
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A4
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 )
@ ( bNF_rel_fun @ ( option @ A ) @ ( option @ B ) @ $o @ $o @ ( rel_option @ A @ B @ A4 )
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 )
@ ( pred_option @ A )
@ ( pred_option @ B ) ) ).
% pred_option_parametric
thf(fact_5055_option_Opred__inject_I2_J,axiom,
! [A: $tType,P: A > $o,A3: A] :
( ( pred_option @ A @ P @ ( some @ A @ A3 ) )
= ( P @ A3 ) ) ).
% option.pred_inject(2)
thf(fact_5056_option_Opred__mono,axiom,
! [A: $tType,P: A > $o,Pa: A > $o] :
( ( ord_less_eq @ ( A > $o ) @ P @ Pa )
=> ( ord_less_eq @ ( ( option @ A ) > $o ) @ ( pred_option @ A @ P ) @ ( pred_option @ A @ Pa ) ) ) ).
% option.pred_mono
thf(fact_5057_option_Opred__inject_I1_J,axiom,
! [A: $tType,P: A > $o] : ( pred_option @ A @ P @ ( none @ A ) ) ).
% option.pred_inject(1)
thf(fact_5058_option_Opred__True,axiom,
! [A: $tType] :
( ( pred_option @ A
@ ^ [Uu: A] : $true )
= ( ^ [Uu: option @ A] : $true ) ) ).
% option.pred_True
thf(fact_5059_option_Opred__mono__strong,axiom,
! [A: $tType,P: A > $o,X: option @ A,Pa: A > $o] :
( ( pred_option @ A @ P @ X )
=> ( ! [Z3: A] :
( ( member @ A @ Z3 @ ( set_option @ A @ X ) )
=> ( ( P @ Z3 )
=> ( Pa @ Z3 ) ) )
=> ( pred_option @ A @ Pa @ X ) ) ) ).
% option.pred_mono_strong
thf(fact_5060_option_Opred__cong,axiom,
! [A: $tType,X: option @ A,Ya: option @ A,P: A > $o,Pa: A > $o] :
( ( X = Ya )
=> ( ! [Z3: A] :
( ( member @ A @ Z3 @ ( set_option @ A @ Ya ) )
=> ( ( P @ Z3 )
= ( Pa @ Z3 ) ) )
=> ( ( pred_option @ A @ P @ X )
= ( pred_option @ A @ Pa @ Ya ) ) ) ) ).
% option.pred_cong
thf(fact_5061_option_Omap__cong__pred,axiom,
! [B: $tType,A: $tType,X: option @ A,Ya: option @ A,F: A > B,G: A > B] :
( ( X = Ya )
=> ( ( pred_option @ A
@ ^ [Z5: A] :
( ( F @ Z5 )
= ( G @ Z5 ) )
@ Ya )
=> ( ( map_option @ A @ B @ F @ X )
= ( map_option @ A @ B @ G @ Ya ) ) ) ) ).
% option.map_cong_pred
thf(fact_5062_option_Opred__set,axiom,
! [A: $tType] :
( ( pred_option @ A )
= ( ^ [P4: A > $o,X3: option @ A] :
! [Y3: A] :
( ( member @ A @ Y3 @ ( set_option @ A @ X3 ) )
=> ( P4 @ Y3 ) ) ) ) ).
% option.pred_set
thf(fact_5063_option_Opred__rel,axiom,
! [A: $tType] :
( ( pred_option @ A )
= ( ^ [P4: A > $o,X3: option @ A] : ( rel_option @ A @ A @ ( bNF_eq_onp @ A @ P4 ) @ X3 @ X3 ) ) ) ).
% option.pred_rel
thf(fact_5064_option_Orel__eq__onp,axiom,
! [A: $tType,P: A > $o] :
( ( rel_option @ A @ A @ ( bNF_eq_onp @ A @ P ) )
= ( bNF_eq_onp @ ( option @ A ) @ ( pred_option @ A @ P ) ) ) ).
% option.rel_eq_onp
thf(fact_5065_option_Opred__map,axiom,
! [B: $tType,A: $tType,Q: B > $o,F: A > B,X: option @ A] :
( ( pred_option @ B @ Q @ ( map_option @ A @ B @ F @ X ) )
= ( pred_option @ A @ ( comp @ B @ $o @ A @ Q @ F ) @ X ) ) ).
% option.pred_map
thf(fact_5066_option_Opred__transfer,axiom,
! [A: $tType,B: $tType,R5: A > B > $o] :
( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( option @ A ) > $o ) @ ( ( option @ B ) > $o )
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ R5
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 )
@ ( bNF_rel_fun @ ( option @ A ) @ ( option @ B ) @ $o @ $o @ ( rel_option @ A @ B @ R5 )
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 )
@ ( pred_option @ A )
@ ( pred_option @ B ) ) ).
% option.pred_transfer
thf(fact_5067_iso__comp__embedS,axiom,
! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),F: A > B,R9: set @ ( product_prod @ C @ C ),F3: B > C] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( order_well_order_on @ B @ ( field2 @ B @ R4 ) @ R4 )
=> ( ( bNF_Wellorder_iso @ A @ B @ R @ R4 @ F )
=> ( ( bNF_Wellorder_embedS @ B @ C @ R4 @ R9 @ F3 )
=> ( bNF_Wellorder_embedS @ A @ C @ R @ R9 @ ( comp @ B @ C @ A @ F3 @ F ) ) ) ) ) ) ).
% iso_comp_embedS
thf(fact_5068_cut__def,axiom,
! [B: $tType,A: $tType] :
( ( cut @ A @ B )
= ( ^ [F2: A > B,R6: set @ ( product_prod @ A @ A ),X3: A,Y3: A] : ( if @ B @ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R6 ) @ ( F2 @ Y3 ) @ ( undefined @ B ) ) ) ) ).
% cut_def
thf(fact_5069_above__def,axiom,
! [A: $tType] :
( ( order_above @ A )
= ( ^ [R3: set @ ( product_prod @ A @ A ),A7: A] :
( collect @ A
@ ^ [B5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B5 ) @ R3 ) ) ) ) ).
% above_def
thf(fact_5070_comp__iso,axiom,
! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),F: A > B,R9: set @ ( product_prod @ C @ C ),F3: B > C] :
( ( bNF_Wellorder_iso @ A @ B @ R @ R4 @ F )
=> ( ( bNF_Wellorder_iso @ B @ C @ R4 @ R9 @ F3 )
=> ( bNF_Wellorder_iso @ A @ C @ R @ R9 @ ( comp @ B @ C @ A @ F3 @ F ) ) ) ) ).
% comp_iso
thf(fact_5071_iso__forward,axiom,
! [A: $tType,B: $tType,X: A,Y: A,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),F: A > B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
=> ( ( bNF_Wellorder_iso @ A @ B @ R @ R4 @ F )
=> ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F @ X ) @ ( F @ Y ) ) @ R4 ) ) ) ).
% iso_forward
thf(fact_5072_cut__apply,axiom,
! [B: $tType,A: $tType,X: A,A3: A,R5: set @ ( product_prod @ A @ A ),F: A > B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ A3 ) @ R5 )
=> ( ( cut @ A @ B @ F @ R5 @ A3 @ X )
= ( F @ X ) ) ) ).
% cut_apply
thf(fact_5073_cuts__eq,axiom,
! [B: $tType,A: $tType,F: A > B,R5: set @ ( product_prod @ A @ A ),X: A,G: A > B] :
( ( ( cut @ A @ B @ F @ R5 @ X )
= ( cut @ A @ B @ G @ R5 @ X ) )
= ( ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X ) @ R5 )
=> ( ( F @ Y3 )
= ( G @ Y3 ) ) ) ) ) ).
% cuts_eq
thf(fact_5074_iso__comp__embed,axiom,
! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),F: A > B,R9: set @ ( product_prod @ C @ C ),F3: B > C] :
( ( bNF_Wellorder_iso @ A @ B @ R @ R4 @ F )
=> ( ( bNF_Wellorder_embed @ B @ C @ R4 @ R9 @ F3 )
=> ( bNF_Wellorder_embed @ A @ C @ R @ R9 @ ( comp @ B @ C @ A @ F3 @ F ) ) ) ) ).
% iso_comp_embed
thf(fact_5075_embed__comp__iso,axiom,
! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),F: A > B,R9: set @ ( product_prod @ C @ C ),F3: B > C] :
( ( bNF_Wellorder_embed @ A @ B @ R @ R4 @ F )
=> ( ( bNF_Wellorder_iso @ B @ C @ R4 @ R9 @ F3 )
=> ( bNF_Wellorder_embed @ A @ C @ R @ R9 @ ( comp @ B @ C @ A @ F3 @ F ) ) ) ) ).
% embed_comp_iso
thf(fact_5076_embedS__comp__iso,axiom,
! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),F: A > B,R9: set @ ( product_prod @ C @ C ),F3: B > C] :
( ( bNF_Wellorder_embedS @ A @ B @ R @ R4 @ F )
=> ( ( bNF_Wellorder_iso @ B @ C @ R4 @ R9 @ F3 )
=> ( bNF_Wellorder_embedS @ A @ C @ R @ R9 @ ( comp @ B @ C @ A @ F3 @ F ) ) ) ) ).
% embedS_comp_iso
thf(fact_5077_iso__iff2,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Wellorder_iso @ A @ B )
= ( ^ [R3: set @ ( product_prod @ A @ A ),R11: set @ ( product_prod @ B @ B ),F2: A > B] :
( ( bij_betw @ A @ B @ F2 @ ( field2 @ A @ R3 ) @ ( field2 @ B @ R11 ) )
& ! [X3: A] :
( ( member @ A @ X3 @ ( field2 @ A @ R3 ) )
=> ! [Y3: A] :
( ( member @ A @ Y3 @ ( field2 @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 )
= ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) @ R11 ) ) ) ) ) ) ) ).
% iso_iff2
thf(fact_5078_iso__backward,axiom,
! [A: $tType,B: $tType,X: A,Y: A,R4: set @ ( product_prod @ A @ A ),R: set @ ( product_prod @ B @ B ),F: B > A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R4 )
=> ( ( bNF_Wellorder_iso @ B @ A @ R @ R4 @ F )
=> ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( hilbert_inv_into @ B @ A @ ( field2 @ B @ R ) @ F @ X ) @ ( hilbert_inv_into @ B @ A @ ( field2 @ B @ R ) @ F @ Y ) ) @ R ) ) ) ).
% iso_backward
thf(fact_5079_lists__of__len__fin1,axiom,
! [A: $tType,P: set @ A,N: nat] :
( ( finite_finite2 @ A @ P )
=> ( finite_finite2 @ ( list @ A )
@ ( inf_inf @ ( set @ ( list @ A ) ) @ ( lists @ A @ P )
@ ( collect @ ( list @ A )
@ ^ [L3: list @ A] :
( ( size_size @ ( list @ A ) @ L3 )
= N ) ) ) ) ) ).
% lists_of_len_fin1
thf(fact_5080_lists__of__len__fin2,axiom,
! [A: $tType,P: set @ A,N: nat] :
( ( finite_finite2 @ A @ P )
=> ( finite_finite2 @ ( list @ A )
@ ( inf_inf @ ( set @ ( list @ A ) ) @ ( lists @ A @ P )
@ ( collect @ ( list @ A )
@ ^ [L3: list @ A] :
( N
= ( size_size @ ( list @ A ) @ L3 ) ) ) ) ) ) ).
% lists_of_len_fin2
thf(fact_5081_Random__Pred_Ounion__def,axiom,
! [A: $tType] :
( ( random_union @ A )
= ( ^ [R13: ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ),R24: ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ),S4: product_prod @ code_natural @ code_natural] :
( product_case_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) )
@ ^ [P12: pred @ A,S8: product_prod @ code_natural @ code_natural] :
( product_case_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) )
@ ^ [P23: pred @ A] : ( product_Pair @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) @ ( sup_sup @ ( pred @ A ) @ P12 @ P23 ) )
@ ( R24 @ S8 ) )
@ ( R13 @ S4 ) ) ) ) ).
% Random_Pred.union_def
thf(fact_5082_lists__image__witness,axiom,
! [A: $tType,B: $tType,X: list @ A,F: B > A,Q: set @ B] :
( ( member @ ( list @ A ) @ X @ ( lists @ A @ ( image2 @ B @ A @ F @ Q ) ) )
=> ~ ! [Xo2: list @ B] :
( ( member @ ( list @ B ) @ Xo2 @ ( lists @ B @ Q ) )
=> ( X
!= ( map @ B @ A @ F @ Xo2 ) ) ) ) ).
% lists_image_witness
thf(fact_5083_Random__Pred_Oempty__def,axiom,
! [A: $tType] :
( ( random_empty @ A )
= ( product_Pair @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) @ ( bot_bot @ ( pred @ A ) ) ) ) ).
% Random_Pred.empty_def
thf(fact_5084_Random__Pred_Obind__def,axiom,
! [B: $tType,A: $tType] :
( ( random_bind @ A @ B )
= ( ^ [R6: ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ),F2: A > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ B ) @ ( product_prod @ code_natural @ code_natural ) ),S4: product_prod @ code_natural @ code_natural] :
( product_case_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ ( pred @ B ) @ ( product_prod @ code_natural @ code_natural ) )
@ ^ [P4: pred @ A,S8: product_prod @ code_natural @ code_natural] :
( product_case_prod @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ ( pred @ B ) @ ( product_prod @ code_natural @ code_natural ) )
@ ^ [S12: product_prod @ code_natural @ code_natural] :
( product_Pair @ ( pred @ B ) @ ( product_prod @ code_natural @ code_natural )
@ ( bind2 @ A @ B @ P4
@ ^ [A7: A] : ( product_fst @ ( pred @ B ) @ ( product_prod @ code_natural @ code_natural ) @ ( F2 @ A7 @ S12 ) ) ) )
@ ( split_seed @ S8 ) )
@ ( R6 @ S4 ) ) ) ) ).
% Random_Pred.bind_def
thf(fact_5085_card__Plus__conv__if,axiom,
! [B: $tType,A: $tType,A4: set @ A,B6: set @ B] :
( ( ( ( finite_finite2 @ A @ A4 )
& ( finite_finite2 @ B @ B6 ) )
=> ( ( finite_card @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A4 @ B6 ) )
= ( plus_plus @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ B @ B6 ) ) ) )
& ( ~ ( ( finite_finite2 @ A @ A4 )
& ( finite_finite2 @ B @ B6 ) )
=> ( ( finite_card @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A4 @ B6 ) )
= ( zero_zero @ nat ) ) ) ) ).
% card_Plus_conv_if
thf(fact_5086_card__Plus,axiom,
! [A: $tType,B: $tType,A4: set @ A,B6: set @ B] :
( ( finite_finite2 @ A @ A4 )
=> ( ( finite_finite2 @ B @ B6 )
=> ( ( finite_card @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A4 @ B6 ) )
= ( plus_plus @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ B @ B6 ) ) ) ) ) ).
% card_Plus
thf(fact_5087_option_Orel__compp__Grp,axiom,
! [B: $tType,A: $tType] :
( ( rel_option @ A @ B )
= ( ^ [R6: A > B > $o] :
( relcompp @ ( option @ A ) @ ( option @ ( product_prod @ A @ B ) ) @ ( option @ B )
@ ( conversep @ ( option @ ( product_prod @ A @ B ) ) @ ( option @ A )
@ ( bNF_Grp @ ( option @ ( product_prod @ A @ B ) ) @ ( option @ A )
@ ( collect @ ( option @ ( product_prod @ A @ B ) )
@ ^ [X3: option @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set_option @ ( product_prod @ A @ B ) @ X3 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) ) )
@ ( map_option @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) ) ) )
@ ( bNF_Grp @ ( option @ ( product_prod @ A @ B ) ) @ ( option @ B )
@ ( collect @ ( option @ ( product_prod @ A @ B ) )
@ ^ [X3: option @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set_option @ ( product_prod @ A @ B ) @ X3 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) ) )
@ ( map_option @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) ) ) ) ) ) ).
% option.rel_compp_Grp
thf(fact_5088_mult__one__div__unit__factor,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: A] :
( ( times_times @ A @ A3 @ ( divide_divide @ A @ ( one_one @ A ) @ ( unit_f5069060285200089521factor @ A @ B2 ) ) )
= ( divide_divide @ A @ A3 @ ( unit_f5069060285200089521factor @ A @ B2 ) ) ) ) ).
% mult_one_div_unit_factor
thf(fact_5089_unit__factor__mult__unit__left,axiom,
! [A: $tType] :
( ( semido2269285787275462019factor @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( times_times @ A @ A3 @ ( unit_f5069060285200089521factor @ A @ B2 ) ) ) ) ) ).
% unit_factor_mult_unit_left
thf(fact_5090_unit__factor__simps_I2_J,axiom,
! [N: nat] :
( ( unit_f5069060285200089521factor @ nat @ ( suc @ N ) )
= ( one_one @ nat ) ) ).
% unit_factor_simps(2)
thf(fact_5091_unit__factor__1,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ( ( unit_f5069060285200089521factor @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% unit_factor_1
thf(fact_5092_inv__unit__factor__eq__0__iff,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( ( divide_divide @ A @ ( one_one @ A ) @ ( unit_f5069060285200089521factor @ A @ A3 ) )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% inv_unit_factor_eq_0_iff
thf(fact_5093_unit__factor__mult__unit__right,axiom,
! [A: $tType] :
( ( semido2269285787275462019factor @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( times_times @ A @ B2 @ A3 ) )
= ( times_times @ A @ ( unit_f5069060285200089521factor @ A @ B2 ) @ A3 ) ) ) ) ).
% unit_factor_mult_unit_right
thf(fact_5094_option_Orel__compp,axiom,
! [A: $tType,C: $tType,B: $tType,R5: A > B > $o,S2: B > C > $o] :
( ( rel_option @ A @ C @ ( relcompp @ A @ B @ C @ R5 @ S2 ) )
= ( relcompp @ ( option @ A ) @ ( option @ B ) @ ( option @ C ) @ ( rel_option @ A @ B @ R5 ) @ ( rel_option @ B @ C @ S2 ) ) ) ).
% option.rel_compp
thf(fact_5095_OOO__quotient3,axiom,
! [A: $tType,B: $tType,C: $tType,R12: A > A > $o,Abs1: A > B,Rep1: B > A,R23: B > B > $o,Abs22: B > C,Rep22: C > B,R25: A > A > $o] :
( ( quotient3 @ A @ B @ R12 @ Abs1 @ Rep1 )
=> ( ( quotient3 @ B @ C @ R23 @ Abs22 @ Rep22 )
=> ( ! [X2: A,Y4: A] :
( ( R25 @ X2 @ Y4 )
=> ( ( R12 @ X2 @ X2 )
=> ( ( R12 @ Y4 @ Y4 )
=> ( R23 @ ( Abs1 @ X2 ) @ ( Abs1 @ Y4 ) ) ) ) )
=> ( ! [X2: B,Y4: B] :
( ( R23 @ X2 @ Y4 )
=> ( R25 @ ( Rep1 @ X2 ) @ ( Rep1 @ Y4 ) ) )
=> ( quotient3 @ A @ C @ ( relcompp @ A @ A @ A @ R12 @ ( relcompp @ A @ A @ A @ R25 @ R12 ) ) @ ( comp @ B @ C @ A @ Abs22 @ Abs1 ) @ ( comp @ B @ A @ C @ Rep1 @ Rep22 ) ) ) ) ) ) ).
% OOO_quotient3
thf(fact_5096_OOO__eq__quotient3,axiom,
! [A: $tType,B: $tType,C: $tType,R12: A > A > $o,Abs1: A > B,Rep1: B > A,Abs22: B > C,Rep22: C > B] :
( ( quotient3 @ A @ B @ R12 @ Abs1 @ Rep1 )
=> ( ( quotient3 @ B @ C
@ ^ [Y5: B,Z4: B] : Y5 = Z4
@ Abs22
@ Rep22 )
=> ( quotient3 @ A @ C
@ ( relcompp @ A @ A @ A @ R12
@ ( relcompp @ A @ A @ A
@ ^ [Y5: A,Z4: A] : Y5 = Z4
@ R12 ) )
@ ( comp @ B @ C @ A @ Abs22 @ Abs1 )
@ ( comp @ B @ A @ C @ Rep1 @ Rep22 ) ) ) ) ).
% OOO_eq_quotient3
thf(fact_5097_relpowp__add,axiom,
! [A: $tType,M2: nat,N: nat,P: A > A > $o] :
( ( compow @ ( A > A > $o ) @ ( plus_plus @ nat @ M2 @ N ) @ P )
= ( relcompp @ A @ A @ A @ ( compow @ ( A > A > $o ) @ M2 @ P ) @ ( compow @ ( A > A > $o ) @ N @ P ) ) ) ).
% relpowp_add
thf(fact_5098_relcompp__relcomp__eq,axiom,
! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ B @ C )] :
( ( relcompp @ A @ B @ C
@ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R )
@ ^ [X3: B,Y3: C] : ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ X3 @ Y3 ) @ S ) )
= ( ^ [X3: A,Y3: C] : ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X3 @ Y3 ) @ ( relcomp @ A @ B @ C @ R @ S ) ) ) ) ).
% relcompp_relcomp_eq
thf(fact_5099_is__unit__unit__factor,axiom,
! [A: $tType] :
( ( semido2269285787275462019factor @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ A3 )
= A3 ) ) ) ).
% is_unit_unit_factor
thf(fact_5100_unit__factor__nat__def,axiom,
( ( unit_f5069060285200089521factor @ nat )
= ( ^ [N5: nat] :
( if @ nat
@ ( N5
= ( zero_zero @ nat ) )
@ ( zero_zero @ nat )
@ ( one_one @ nat ) ) ) ) ).
% unit_factor_nat_def
thf(fact_5101_option_Orel__Grp,axiom,
! [B: $tType,A: $tType,A4: set @ A,F: A > B] :
( ( rel_option @ A @ B @ ( bNF_Grp @ A @ B @ A4 @ F ) )
= ( bNF_Grp @ ( option @ A ) @ ( option @ B )
@ ( collect @ ( option @ A )
@ ^ [X3: option @ A] : ( ord_less_eq @ ( set @ A ) @ ( set_option @ A @ X3 ) @ A4 ) )
@ ( map_option @ A @ B @ F ) ) ) ).
% option.rel_Grp
thf(fact_5102_fun_Orel__Grp,axiom,
! [D: $tType,B: $tType,A: $tType,A4: set @ A,F: A > B] :
( ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y5: D,Z4: D] : Y5 = Z4
@ ( bNF_Grp @ A @ B @ A4 @ F ) )
= ( bNF_Grp @ ( D > A ) @ ( D > B )
@ ( collect @ ( D > A )
@ ^ [X3: D > A] : ( ord_less_eq @ ( set @ A ) @ ( image2 @ D @ A @ X3 @ ( top_top @ ( set @ D ) ) ) @ A4 ) )
@ ( comp @ A @ B @ D @ F ) ) ) ).
% fun.rel_Grp
thf(fact_5103_relcomp__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( relcomp @ A @ B @ C )
= ( ^ [R3: set @ ( product_prod @ A @ B ),S4: set @ ( product_prod @ B @ C )] :
( collect @ ( product_prod @ A @ C )
@ ( product_case_prod @ A @ C @ $o
@ ( relcompp @ A @ B @ C
@ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R3 )
@ ^ [X3: B,Y3: C] : ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ X3 @ Y3 ) @ S4 ) ) ) ) ) ) ).
% relcomp_def
thf(fact_5104_unit__factor__is__unit,axiom,
! [A: $tType] :
( ( semido2269285787275462019factor @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( dvd_dvd @ A @ ( unit_f5069060285200089521factor @ A @ A3 ) @ ( one_one @ A ) ) ) ) ).
% unit_factor_is_unit
thf(fact_5105_unit__factor__gcd,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( ( ( A3
= ( zero_zero @ A ) )
& ( B2
= ( zero_zero @ A ) ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_gcd @ A @ A3 @ B2 ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ( A3
= ( zero_zero @ A ) )
& ( B2
= ( zero_zero @ A ) ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_gcd @ A @ A3 @ B2 ) )
= ( one_one @ A ) ) ) ) ) ).
% unit_factor_gcd
thf(fact_5106_type__copy__vimage2p__Grp__Abs,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Rep: A > B,Abs: B > A,G: D > C,P: C > $o,H3: C > A] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ( ( bNF_vimage2p @ D @ C @ B @ A @ $o @ G @ Abs @ ( bNF_Grp @ C @ A @ ( collect @ C @ P ) @ H3 ) )
= ( bNF_Grp @ D @ B
@ ( collect @ D
@ ^ [X3: D] : ( P @ ( G @ X3 ) ) )
@ ( comp @ C @ B @ D @ ( comp @ A @ B @ C @ Rep @ H3 ) @ G ) ) ) ) ).
% type_copy_vimage2p_Grp_Abs
thf(fact_5107_type__copy__vimage2p__Grp__Rep,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,Rep: A > B,Abs: B > A,F: C > D,P: D > $o,H3: D > B] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ( ( bNF_vimage2p @ C @ D @ A @ B @ $o @ F @ Rep @ ( bNF_Grp @ D @ B @ ( collect @ D @ P ) @ H3 ) )
= ( bNF_Grp @ C @ A
@ ( collect @ C
@ ^ [X3: C] : ( P @ ( F @ X3 ) ) )
@ ( comp @ D @ A @ C @ ( comp @ B @ A @ D @ Abs @ H3 ) @ F ) ) ) ) ).
% type_copy_vimage2p_Grp_Rep
thf(fact_5108_Nat_Otransfer,axiom,
bNF_rel_fun @ int @ code_integer @ nat @ code_natural @ code_pcr_integer @ code_pcr_natural @ nat2 @ code_Nat ).
% Nat.transfer
thf(fact_5109_pcr__natural__def,axiom,
( code_pcr_natural
= ( relcompp @ nat @ nat @ code_natural
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ code_cr_natural ) ) ).
% pcr_natural_def
thf(fact_5110_natural_Orep__transfer,axiom,
( bNF_rel_fun @ nat @ code_natural @ nat @ nat @ code_pcr_natural
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ^ [X3: nat] : X3
@ code_nat_of_natural ) ).
% natural.rep_transfer
thf(fact_5111_natural_Oid__abs__transfer,axiom,
( bNF_rel_fun @ nat @ nat @ nat @ code_natural
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ code_pcr_natural
@ ^ [X3: nat] : X3
@ code_natural_of_nat ) ).
% natural.id_abs_transfer
thf(fact_5112_natural_Opcr__cr__eq,axiom,
code_pcr_natural = code_cr_natural ).
% natural.pcr_cr_eq
thf(fact_5113_zero__natural_Otransfer,axiom,
code_pcr_natural @ ( zero_zero @ nat ) @ ( zero_zero @ code_natural ) ).
% zero_natural.transfer
thf(fact_5114_one__natural_Otransfer,axiom,
code_pcr_natural @ ( one_one @ nat ) @ ( one_one @ code_natural ) ).
% one_natural.transfer
thf(fact_5115_less__natural_Otransfer,axiom,
( bNF_rel_fun @ nat @ code_natural @ ( nat > $o ) @ ( code_natural > $o ) @ code_pcr_natural
@ ( bNF_rel_fun @ nat @ code_natural @ $o @ $o @ code_pcr_natural
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 )
@ ( ord_less @ nat )
@ ( ord_less @ code_natural ) ) ).
% less_natural.transfer
thf(fact_5116_vimage2p__comp,axiom,
! [E: $tType,D: $tType,F5: $tType,A: $tType,C: $tType,B: $tType,G3: $tType,F1: F5 > A,F22: D > F5,G1: G3 > B,G22: E > G3] :
( ( bNF_vimage2p @ D @ A @ E @ B @ C @ ( comp @ F5 @ A @ D @ F1 @ F22 ) @ ( comp @ G3 @ B @ E @ G1 @ G22 ) )
= ( comp @ ( F5 > G3 > C ) @ ( D > E > C ) @ ( A > B > C ) @ ( bNF_vimage2p @ D @ F5 @ E @ G3 @ C @ F22 @ G22 ) @ ( bNF_vimage2p @ F5 @ A @ G3 @ B @ C @ F1 @ G1 ) ) ) ).
% vimage2p_comp
thf(fact_5117_plus__natural_Otransfer,axiom,
bNF_rel_fun @ nat @ code_natural @ ( nat > nat ) @ ( 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
thf(fact_5118_Suc_Otransfer,axiom,
bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural @ suc @ code_Suc ).
% Suc.transfer
thf(fact_5119_minus__natural_Otransfer,axiom,
bNF_rel_fun @ nat @ code_natural @ ( nat > nat ) @ ( 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
thf(fact_5120_pcr__integer__def,axiom,
( code_pcr_integer
= ( relcompp @ int @ int @ code_integer
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ code_cr_integer ) ) ).
% pcr_integer_def
thf(fact_5121_less__eq__natural_Otransfer,axiom,
( bNF_rel_fun @ nat @ code_natural @ ( nat > $o ) @ ( code_natural > $o ) @ code_pcr_natural
@ ( bNF_rel_fun @ nat @ code_natural @ $o @ $o @ code_pcr_natural
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 )
@ ( ord_less_eq @ nat )
@ ( ord_less_eq @ code_natural ) ) ).
% less_eq_natural.transfer
thf(fact_5122_times__natural_Otransfer,axiom,
bNF_rel_fun @ nat @ code_natural @ ( nat > nat ) @ ( 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
thf(fact_5123_divide__natural_Otransfer,axiom,
bNF_rel_fun @ nat @ code_natural @ ( nat > nat ) @ ( 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
thf(fact_5124_modulo__natural_Otransfer,axiom,
bNF_rel_fun @ nat @ code_natural @ ( nat > nat ) @ ( 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
thf(fact_5125_take__bit__natural_Otransfer,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( code_natural > code_natural )
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ( 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
thf(fact_5126_xor__natural_Otransfer,axiom,
bNF_rel_fun @ nat @ code_natural @ ( nat > nat ) @ ( 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
thf(fact_5127_or__natural_Otransfer,axiom,
bNF_rel_fun @ nat @ code_natural @ ( nat > nat ) @ ( 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
thf(fact_5128_and__natural_Otransfer,axiom,
bNF_rel_fun @ nat @ code_natural @ ( nat > nat ) @ ( 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
thf(fact_5129_bit__natural_Otransfer,axiom,
( bNF_rel_fun @ nat @ code_natural @ ( nat > $o ) @ ( nat > $o ) @ code_pcr_natural
@ ^ [Y5: nat > $o,Z4: nat > $o] : Y5 = Z4
@ ( bit_se5641148757651400278ts_bit @ nat )
@ ( bit_se5641148757651400278ts_bit @ code_natural ) ) ).
% bit_natural.transfer
thf(fact_5130_push__bit__natural_Otransfer,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( code_natural > code_natural )
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ( 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
thf(fact_5131_mask__natural_Otransfer,axiom,
( bNF_rel_fun @ nat @ nat @ nat @ code_natural
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ code_pcr_natural
@ ( bit_se2239418461657761734s_mask @ nat )
@ ( bit_se2239418461657761734s_mask @ code_natural ) ) ).
% mask_natural.transfer
thf(fact_5132_drop__bit__natural_Otransfer,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( code_natural > code_natural )
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ( 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
thf(fact_5133_division__segment__natural_Otransfer,axiom,
bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural @ ( euclid7384307370059645450egment @ nat ) @ ( euclid7384307370059645450egment @ code_natural ) ).
% division_segment_natural.transfer
thf(fact_5134_flip__bit__natural_Otransfer,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( code_natural > code_natural )
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ( 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
thf(fact_5135_set__bit__natural_Otransfer,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( code_natural > code_natural )
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ( 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
thf(fact_5136_unset__bit__natural_Otransfer,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( code_natural > code_natural )
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ( 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
thf(fact_5137_euclidean__size__natural_Otransfer,axiom,
( bNF_rel_fun @ nat @ code_natural @ nat @ nat @ code_pcr_natural
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ ( euclid6346220572633701492n_size @ nat )
@ ( euclid6346220572633701492n_size @ code_natural ) ) ).
% euclidean_size_natural.transfer
thf(fact_5138_integer__of__natural_Otransfer,axiom,
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
thf(fact_5139_natural__of__integer_Otransfer,axiom,
bNF_rel_fun @ int @ code_integer @ nat @ code_natural @ code_pcr_integer @ code_pcr_natural @ nat2 @ code_n4118661773612635043nteger ).
% natural_of_integer.transfer
thf(fact_5140_fun_Orel__compp__Grp,axiom,
! [D: $tType,B: $tType,A: $tType,R5: A > B > $o] :
( ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y5: D,Z4: D] : Y5 = Z4
@ R5 )
= ( relcompp @ ( D > A ) @ ( D > ( product_prod @ A @ B ) ) @ ( D > B )
@ ( conversep @ ( D > ( product_prod @ A @ B ) ) @ ( D > A )
@ ( bNF_Grp @ ( D > ( product_prod @ A @ B ) ) @ ( D > A )
@ ( collect @ ( D > ( product_prod @ A @ B ) )
@ ^ [X3: D > ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ D @ ( product_prod @ A @ B ) @ X3 @ ( top_top @ ( set @ D ) ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R5 ) ) ) )
@ ( comp @ ( product_prod @ A @ B ) @ A @ D @ ( product_fst @ A @ B ) ) ) )
@ ( bNF_Grp @ ( D > ( product_prod @ A @ B ) ) @ ( D > B )
@ ( collect @ ( D > ( product_prod @ A @ B ) )
@ ^ [X3: D > ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ D @ ( product_prod @ A @ B ) @ X3 @ ( top_top @ ( set @ D ) ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R5 ) ) ) )
@ ( comp @ ( product_prod @ A @ B ) @ B @ D @ ( product_snd @ A @ B ) ) ) ) ) ).
% fun.rel_compp_Grp
thf(fact_5141_plus__num__def,axiom,
( ( plus_plus @ num )
= ( ^ [M: num,N5: num] : ( num_of_nat @ ( plus_plus @ nat @ ( nat_of_num @ M ) @ ( nat_of_num @ N5 ) ) ) ) ) ).
% plus_num_def
thf(fact_5142_Lcm__no__units,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ( ( gcd_Lcm @ A )
= ( ^ [A9: set @ A] :
( gcd_Lcm @ A
@ ( minus_minus @ ( set @ A ) @ A9
@ ( collect @ A
@ ^ [A7: A] : ( dvd_dvd @ A @ A7 @ ( one_one @ A ) ) ) ) ) ) ) ) ).
% Lcm_no_units
thf(fact_5143_remdups__adj__altdef,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( ( remdups_adj @ A @ Xs )
= Ys2 )
= ( ? [F2: nat > nat] :
( ( order_mono @ nat @ nat @ F2 )
& ( ( image2 @ nat @ nat @ F2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) )
= ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Ys2 ) ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ Xs @ I3 )
= ( nth @ A @ Ys2 @ ( F2 @ I3 ) ) ) )
& ! [I3: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ I3 @ ( one_one @ nat ) ) @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( nth @ A @ Xs @ I3 )
= ( nth @ A @ Xs @ ( plus_plus @ nat @ I3 @ ( one_one @ nat ) ) ) )
= ( ( F2 @ I3 )
= ( F2 @ ( plus_plus @ nat @ I3 @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ).
% remdups_adj_altdef
thf(fact_5144_Lcm__empty,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ( ( gcd_Lcm @ A @ ( bot_bot @ ( set @ A ) ) )
= ( one_one @ A ) ) ) ).
% Lcm_empty
thf(fact_5145_Lcm__1__iff,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A4: set @ A] :
( ( ( gcd_Lcm @ A @ A4 )
= ( one_one @ A ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( dvd_dvd @ A @ X3 @ ( one_one @ A ) ) ) ) ) ) ).
% Lcm_1_iff
thf(fact_5146_mono__add,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A3: A] : ( order_mono @ A @ A @ ( plus_plus @ A @ A3 ) ) ) ).
% mono_add
thf(fact_5147_Lcm__nat__empty,axiom,
( ( gcd_Lcm @ nat @ ( bot_bot @ ( set @ nat ) ) )
= ( one_one @ nat ) ) ).
% Lcm_nat_empty
thf(fact_5148_nat__of__num_Osimps_I2_J,axiom,
! [X: num] :
( ( nat_of_num @ ( bit0 @ X ) )
= ( plus_plus @ nat @ ( nat_of_num @ X ) @ ( nat_of_num @ X ) ) ) ).
% nat_of_num.simps(2)
thf(fact_5149_nat__of__num__code_I1_J,axiom,
( ( nat_of_num @ one2 )
= ( one_one @ nat ) ) ).
% nat_of_num_code(1)
thf(fact_5150_nat__of__num__add,axiom,
! [X: num,Y: num] :
( ( nat_of_num @ ( plus_plus @ num @ X @ Y ) )
= ( plus_plus @ nat @ ( nat_of_num @ X ) @ ( nat_of_num @ Y ) ) ) ).
% nat_of_num_add
thf(fact_5151_nat__of__num__code_I2_J,axiom,
! [N: num] :
( ( nat_of_num @ ( bit0 @ N ) )
= ( plus_plus @ nat @ ( nat_of_num @ N ) @ ( nat_of_num @ N ) ) ) ).
% nat_of_num_code(2)
thf(fact_5152_mono__image__least,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [F: A > B,M2: A,N: A,M7: B,N2: B] :
( ( order_mono @ A @ B @ F )
=> ( ( ( image2 @ A @ B @ F @ ( set_or7035219750837199246ssThan @ A @ M2 @ N ) )
= ( set_or7035219750837199246ssThan @ B @ M7 @ N2 ) )
=> ( ( ord_less @ A @ M2 @ N )
=> ( ( F @ M2 )
= M7 ) ) ) ) ) ).
% mono_image_least
thf(fact_5153_nat__of__num_Osimps_I3_J,axiom,
! [X: num] :
( ( nat_of_num @ ( bit1 @ X ) )
= ( suc @ ( plus_plus @ nat @ ( nat_of_num @ X ) @ ( nat_of_num @ X ) ) ) ) ).
% nat_of_num.simps(3)
thf(fact_5154_unit__factor__Lcm,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A4: set @ A] :
( ( ( ( gcd_Lcm @ A @ A4 )
= ( zero_zero @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_Lcm @ A @ A4 ) )
= ( zero_zero @ A ) ) )
& ( ( ( gcd_Lcm @ A @ A4 )
!= ( zero_zero @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_Lcm @ A @ A4 ) )
= ( one_one @ A ) ) ) ) ) ).
% unit_factor_Lcm
thf(fact_5155_nat__of__num__code_I3_J,axiom,
! [N: num] :
( ( nat_of_num @ ( bit1 @ N ) )
= ( suc @ ( plus_plus @ nat @ ( nat_of_num @ N ) @ ( nat_of_num @ N ) ) ) ) ).
% nat_of_num_code(3)
thf(fact_5156_remdups__adj_Opelims,axiom,
! [A: $tType,X: list @ A,Y: list @ A] :
( ( ( remdups_adj @ A @ X )
= Y )
=> ( ( accp @ ( list @ A ) @ ( remdups_adj_rel @ A ) @ X )
=> ( ( ( X
= ( nil @ A ) )
=> ( ( Y
= ( nil @ A ) )
=> ~ ( accp @ ( list @ A ) @ ( remdups_adj_rel @ A ) @ ( nil @ A ) ) ) )
=> ( ! [X2: A] :
( ( X
= ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ( ( Y
= ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ~ ( accp @ ( list @ A ) @ ( remdups_adj_rel @ A ) @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) )
=> ~ ! [X2: A,Y4: A,Xs2: list @ A] :
( ( X
= ( cons @ A @ X2 @ ( cons @ A @ Y4 @ Xs2 ) ) )
=> ( ( ( ( X2 = Y4 )
=> ( Y
= ( remdups_adj @ A @ ( cons @ A @ X2 @ Xs2 ) ) ) )
& ( ( X2 != Y4 )
=> ( Y
= ( cons @ A @ X2 @ ( remdups_adj @ A @ ( cons @ A @ Y4 @ Xs2 ) ) ) ) ) )
=> ~ ( accp @ ( list @ A ) @ ( remdups_adj_rel @ A ) @ ( cons @ A @ X2 @ ( cons @ A @ Y4 @ Xs2 ) ) ) ) ) ) ) ) ) ).
% remdups_adj.pelims
thf(fact_5157_relChain__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( bNF_Ca3754400796208372196lChain @ A @ B )
= ( ^ [R3: set @ ( product_prod @ A @ A ),As3: A > B] :
! [I3: A,J2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I3 @ J2 ) @ R3 )
=> ( ord_less_eq @ B @ ( As3 @ I3 ) @ ( As3 @ J2 ) ) ) ) ) ) ).
% relChain_def
thf(fact_5158_unit__factor__Gcd,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A4: set @ A] :
( ( ( ( gcd_Gcd @ A @ A4 )
= ( zero_zero @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_Gcd @ A @ A4 ) )
= ( zero_zero @ A ) ) )
& ( ( ( gcd_Gcd @ A @ A4 )
!= ( zero_zero @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_Gcd @ A @ A4 ) )
= ( one_one @ A ) ) ) ) ) ).
% unit_factor_Gcd
thf(fact_5159_Gcd__UNIV,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ( ( gcd_Gcd @ A @ ( top_top @ ( set @ A ) ) )
= ( one_one @ A ) ) ) ).
% Gcd_UNIV
thf(fact_5160_Gcd__1,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A4: set @ A] :
( ( member @ A @ ( one_one @ A ) @ A4 )
=> ( ( gcd_Gcd @ A @ A4 )
= ( one_one @ A ) ) ) ) ).
% Gcd_1
thf(fact_5161_Gcd__nat__eq__one,axiom,
! [N6: set @ nat] :
( ( member @ nat @ ( one_one @ nat ) @ N6 )
=> ( ( gcd_Gcd @ nat @ N6 )
= ( one_one @ nat ) ) ) ).
% Gcd_nat_eq_one
thf(fact_5162_Gcd__eq__1__I,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A3: A,A4: set @ A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( member @ A @ A3 @ A4 )
=> ( ( gcd_Gcd @ A @ A4 )
= ( one_one @ A ) ) ) ) ) ).
% Gcd_eq_1_I
thf(fact_5163_cofinal__def,axiom,
! [A: $tType] :
( ( bNF_Ca7293521722713021262ofinal @ A )
= ( ^ [A9: set @ A,R3: set @ ( product_prod @ A @ A )] :
! [X3: A] :
( ( member @ A @ X3 @ ( field2 @ A @ R3 ) )
=> ? [Y3: A] :
( ( member @ A @ Y3 @ A9 )
& ( X3 != Y3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 ) ) ) ) ) ).
% cofinal_def
thf(fact_5164_card__of__UNION__ordLeq__infinite,axiom,
! [B: $tType,A: $tType,C: $tType,B6: set @ A,I4: set @ B,A4: B > ( set @ C )] :
( ~ ( finite_finite2 @ A @ B6 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ I4 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ I4 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( A4 @ X2 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ A4 @ I4 ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) ) ) ) ).
% card_of_UNION_ordLeq_infinite
thf(fact_5165_card__of__ordLess,axiom,
! [A: $tType,B: $tType,A4: set @ A,B6: set @ B] :
( ( ~ ? [F2: A > B] :
( ( inj_on @ A @ B @ F2 @ A4 )
& ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ A4 ) @ B6 ) ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) ) ) ).
% card_of_ordLess
thf(fact_5166_card__of__Times3,axiom,
! [A: $tType,A4: set @ A] :
( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ A )
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu: A] : A4 ) ) )
@ ( bNF_Wellorder_ordLeq @ A @ ( product_prod @ A @ A ) ) ) ).
% card_of_Times3
thf(fact_5167_card__of__Sigma__mono1,axiom,
! [C: $tType,B: $tType,A: $tType,I4: set @ A,A4: A > ( set @ B ),B6: A > ( set @ C )] :
( ! [X2: A] :
( ( member @ A @ X2 @ I4 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( A4 @ X2 ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( B6 @ X2 ) ) ) @ ( bNF_Wellorder_ordLeq @ B @ C ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ I4 @ A4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ C ) @ ( product_Sigma @ A @ C @ I4 @ B6 ) ) ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) ) ) ).
% card_of_Sigma_mono1
thf(fact_5168_card__of__Times__mono1,axiom,
! [B: $tType,C: $tType,A: $tType,A4: set @ A,B6: set @ B,C5: set @ C] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ C )
@ ( product_Sigma @ A @ C @ A4
@ ^ [Uu: A] : C5 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ C )
@ ( product_Sigma @ B @ C @ B6
@ ^ [Uu: B] : C5 ) ) )
@ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ C ) @ ( product_prod @ B @ C ) ) ) ) ).
% card_of_Times_mono1
thf(fact_5169_card__of__Times__mono2,axiom,
! [B: $tType,A: $tType,C: $tType,A4: set @ A,B6: set @ B,C5: set @ C] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ C @ A ) @ ( product_prod @ C @ A ) ) ) @ ( set @ ( product_prod @ ( product_prod @ C @ B ) @ ( product_prod @ C @ B ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ C @ A ) @ ( product_prod @ C @ A ) ) ) @ ( set @ ( product_prod @ ( product_prod @ C @ B ) @ ( product_prod @ C @ B ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ C @ A )
@ ( product_Sigma @ C @ A @ C5
@ ^ [Uu: C] : A4 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ C @ B )
@ ( product_Sigma @ C @ B @ C5
@ ^ [Uu: C] : B6 ) ) )
@ ( bNF_Wellorder_ordLeq @ ( product_prod @ C @ A ) @ ( product_prod @ C @ B ) ) ) ) ).
% card_of_Times_mono2
thf(fact_5170_infinite__iff__card__of__nat,axiom,
! [A: $tType,A4: set @ A] :
( ( ~ ( finite_finite2 @ A @ A4 ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ nat @ ( top_top @ ( set @ nat ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ nat @ A ) ) ) ).
% infinite_iff_card_of_nat
thf(fact_5171_card__of__UNION__Sigma,axiom,
! [B: $tType,A: $tType,A4: B > ( set @ A ),I4: set @ B] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A4 @ I4 ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A ) @ ( product_Sigma @ B @ A @ I4 @ A4 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( product_prod @ B @ A ) ) ) ).
% card_of_UNION_Sigma
thf(fact_5172_card__of__Times__commute,axiom,
! [B: $tType,A: $tType,A4: set @ A,B6: set @ B] :
( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : B6 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
@ ( product_Sigma @ B @ A @ B6
@ ^ [Uu: B] : A4 ) ) )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A ) ) ) ).
% card_of_Times_commute
thf(fact_5173_card__of__Times__same__infinite,axiom,
! [A: $tType,A4: set @ A] :
( ~ ( finite_finite2 @ A @ A4 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ A )
@ ( product_Sigma @ A @ A @ A4
@ ^ [Uu: A] : A4 ) )
@ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ A ) @ A ) ) ) ).
% card_of_Times_same_infinite
thf(fact_5174_card__of__Times1,axiom,
! [A: $tType,B: $tType,A4: set @ A,B6: set @ B] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
@ ( product_Sigma @ B @ A @ B6
@ ^ [Uu: B] : A4 ) ) )
@ ( bNF_Wellorder_ordLeq @ B @ ( product_prod @ B @ A ) ) ) ) ).
% card_of_Times1
thf(fact_5175_card__of__Times2,axiom,
! [A: $tType,B: $tType,A4: set @ A,B6: set @ B] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : B6 ) ) )
@ ( bNF_Wellorder_ordLeq @ B @ ( product_prod @ A @ B ) ) ) ) ).
% card_of_Times2
thf(fact_5176_card__of__bool,axiom,
! [A: $tType,A1: A,A22: A] :
( ( A1 != A22 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ $o @ ( top_top @ ( set @ $o ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( insert2 @ A @ A1 @ ( insert2 @ A @ A22 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) @ ( bNF_Wellorder_ordIso @ $o @ A ) ) ) ).
% card_of_bool
thf(fact_5177_card__of__Plus__empty1,axiom,
! [B: $tType,A: $tType,A4: set @ A] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A4 @ ( bot_bot @ ( set @ B ) ) ) ) ) @ ( bNF_Wellorder_ordIso @ A @ ( sum_sum @ A @ B ) ) ) ).
% card_of_Plus_empty1
thf(fact_5178_card__of__Plus__empty2,axiom,
! [B: $tType,A: $tType,A4: set @ A] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ A ) @ ( sum_Plus @ B @ A @ ( bot_bot @ ( set @ B ) ) @ A4 ) ) ) @ ( bNF_Wellorder_ordIso @ A @ ( sum_sum @ B @ A ) ) ) ).
% card_of_Plus_empty2
thf(fact_5179_card__of__Plus__Times__bool,axiom,
! [A: $tType,A4: set @ A] :
( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ A ) @ ( sum_sum @ A @ A ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ $o ) @ ( product_prod @ A @ $o ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ A ) @ ( sum_sum @ A @ A ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ $o ) @ ( product_prod @ A @ $o ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ A ) @ ( sum_Plus @ A @ A @ A4 @ A4 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ $o )
@ ( product_Sigma @ A @ $o @ A4
@ ^ [Uu: A] : ( top_top @ ( set @ $o ) ) ) ) )
@ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ A ) @ ( product_prod @ A @ $o ) ) ) ).
% card_of_Plus_Times_bool
thf(fact_5180_card__of__Plus__commute,axiom,
! [B: $tType,A: $tType,A4: set @ A,B6: set @ B] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A4 @ B6 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ A ) @ ( sum_Plus @ B @ A @ B6 @ A4 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ ( sum_sum @ B @ A ) ) ) ).
% card_of_Plus_commute
thf(fact_5181_card__of__Plus__cong2,axiom,
! [B: $tType,A: $tType,C: $tType,A4: set @ A,B6: set @ B,C5: set @ C] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ A ) @ ( sum_Plus @ C @ A @ C5 @ A4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ B ) @ ( sum_Plus @ C @ B @ C5 @ B6 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ B ) ) ) ) ).
% card_of_Plus_cong2
thf(fact_5182_card__of__Plus__cong1,axiom,
! [B: $tType,C: $tType,A: $tType,A4: set @ A,B6: set @ B,C5: set @ C] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ C ) @ ( sum_Plus @ A @ C @ A4 @ C5 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ C ) @ ( sum_Plus @ B @ C @ B6 @ C5 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ C ) ) ) ) ).
% card_of_Plus_cong1
thf(fact_5183_card__of__Plus__assoc,axiom,
! [C: $tType,B: $tType,A: $tType,A4: set @ A,B6: set @ B,C5: set @ C] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) @ ( sum_Plus @ ( sum_sum @ A @ B ) @ C @ ( sum_Plus @ A @ B @ A4 @ B6 ) @ C5 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) @ ( sum_Plus @ A @ ( sum_sum @ B @ C ) @ A4 @ ( sum_Plus @ B @ C @ B6 @ C5 ) ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) ) ).
% card_of_Plus_assoc
thf(fact_5184_card__of__Plus__cong,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,A4: set @ A,B6: set @ B,C5: set @ C,D4: set @ D] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ C5 ) @ ( bNF_Ca6860139660246222851ard_of @ D @ D4 ) ) @ ( bNF_Wellorder_ordIso @ C @ D ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ C ) @ ( sum_Plus @ A @ C @ A4 @ C5 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ D ) @ ( sum_Plus @ B @ D @ B6 @ D4 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) ).
% card_of_Plus_cong
thf(fact_5185_card__of__Times__Plus__distrib,axiom,
! [C: $tType,B: $tType,A: $tType,A4: set @ A,B6: set @ B,C5: set @ C] :
( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ ( sum_sum @ B @ C ) )
@ ( product_Sigma @ A @ ( sum_sum @ B @ C ) @ A4
@ ^ [Uu: A] : ( sum_Plus @ B @ C @ B6 @ C5 ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) )
@ ( sum_Plus @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : B6 )
@ ( product_Sigma @ A @ C @ A4
@ ^ [Uu: A] : C5 ) ) ) )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) ) ) ).
% card_of_Times_Plus_distrib
thf(fact_5186_card__of__Plus__mono2,axiom,
! [B: $tType,A: $tType,C: $tType,A4: set @ A,B6: set @ B,C5: set @ C] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ A ) @ ( sum_Plus @ C @ A @ C5 @ A4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ B ) @ ( sum_Plus @ C @ B @ C5 @ B6 ) ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ B ) ) ) ) ).
% card_of_Plus_mono2
thf(fact_5187_card__of__Plus__mono1,axiom,
! [B: $tType,C: $tType,A: $tType,A4: set @ A,B6: set @ B,C5: set @ C] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ C ) @ ( sum_Plus @ A @ C @ A4 @ C5 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ C ) @ ( sum_Plus @ B @ C @ B6 @ C5 ) ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ C ) ) ) ) ).
% card_of_Plus_mono1
thf(fact_5188_card__of__Plus__mono,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,A4: set @ A,B6: set @ B,C5: set @ C,D4: set @ D] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ C5 ) @ ( bNF_Ca6860139660246222851ard_of @ D @ D4 ) ) @ ( bNF_Wellorder_ordLeq @ C @ D ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ C ) @ ( sum_Plus @ A @ C @ A4 @ C5 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ D ) @ ( sum_Plus @ B @ D @ B6 @ D4 ) ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) ).
% card_of_Plus_mono
thf(fact_5189_card__of__Plus2,axiom,
! [B: $tType,A: $tType,B6: set @ A,A4: set @ B] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B6 ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ A ) @ ( sum_Plus @ B @ A @ A4 @ B6 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ B @ A ) ) ) ).
% card_of_Plus2
thf(fact_5190_card__of__Plus1,axiom,
! [B: $tType,A: $tType,A4: set @ A,B6: set @ B] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A4 @ B6 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ A @ B ) ) ) ).
% card_of_Plus1
thf(fact_5191_card__of__Un__Plus__ordLeq,axiom,
! [A: $tType,A4: set @ A,B6: set @ A] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ A ) @ ( sum_sum @ A @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ A ) @ ( sum_sum @ A @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B6 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ A ) @ ( sum_Plus @ A @ A @ A4 @ B6 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ A @ A ) ) ) ).
% card_of_Un_Plus_ordLeq
thf(fact_5192_card__of__Plus__infinite,axiom,
! [A: $tType,B: $tType,A4: set @ A,B6: set @ B] :
( ~ ( finite_finite2 @ A @ A4 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A4 @ B6 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ A ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ A ) @ ( sum_Plus @ B @ A @ B6 @ A4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ B @ A ) @ A ) ) ) ) ) ).
% card_of_Plus_infinite
thf(fact_5193_card__of__Plus__infinite1,axiom,
! [B: $tType,A: $tType,A4: set @ A,B6: set @ B] :
( ~ ( finite_finite2 @ A @ A4 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A4 @ B6 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ A ) ) ) ) ).
% card_of_Plus_infinite1
thf(fact_5194_card__of__Plus__infinite2,axiom,
! [A: $tType,B: $tType,A4: set @ A,B6: set @ B] :
( ~ ( finite_finite2 @ A @ A4 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ A ) @ ( sum_Plus @ B @ A @ B6 @ A4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ B @ A ) @ A ) ) ) ) ).
% card_of_Plus_infinite2
thf(fact_5195_card__of__Plus__ordLess__infinite,axiom,
! [A: $tType,C: $tType,B: $tType,C5: set @ A,A4: set @ B,B6: set @ C] :
( ~ ( finite_finite2 @ A @ C5 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ C5 ) ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ B6 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ C5 ) ) @ ( bNF_We4044943003108391690rdLess @ C @ A ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ C ) @ ( sum_Plus @ B @ C @ A4 @ B6 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ C5 ) ) @ ( bNF_We4044943003108391690rdLess @ ( sum_sum @ B @ C ) @ A ) ) ) ) ) ).
% card_of_Plus_ordLess_infinite
thf(fact_5196_card__of__Sigma__ordLeq__infinite,axiom,
! [A: $tType,C: $tType,B: $tType,B6: set @ A,I4: set @ B,A4: B > ( set @ C )] :
( ~ ( finite_finite2 @ A @ B6 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ I4 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ I4 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( A4 @ X2 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ C ) @ ( product_Sigma @ B @ C @ I4 @ A4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ B @ C ) @ A ) ) ) ) ) ).
% card_of_Sigma_ordLeq_infinite
thf(fact_5197_card__of__refl,axiom,
! [A: $tType,A4: set @ A] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ).
% card_of_refl
thf(fact_5198_card__of__Plus__Times__aux,axiom,
! [B: $tType,A: $tType,A1: A,A22: A,A4: set @ A,B6: set @ B] :
( ( ( A1 != A22 )
& ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ A1 @ ( insert2 @ A @ A22 @ ( bot_bot @ ( set @ A ) ) ) ) @ A4 ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A4 @ B6 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : B6 ) ) )
@ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ) ).
% card_of_Plus_Times_aux
thf(fact_5199_card__of__Times__infinite__simps_I4_J,axiom,
! [B: $tType,A: $tType,A4: set @ A,B6: set @ B] :
( ~ ( finite_finite2 @ A @ A4 )
=> ( ( B6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
@ ( product_Sigma @ B @ A @ B6
@ ^ [Uu: B] : A4 ) ) )
@ ( bNF_Wellorder_ordIso @ A @ ( product_prod @ B @ A ) ) ) ) ) ) ).
% card_of_Times_infinite_simps(4)
thf(fact_5200_card__of__Times__infinite__simps_I3_J,axiom,
! [A: $tType,B: $tType,A4: set @ A,B6: set @ B] :
( ~ ( finite_finite2 @ A @ A4 )
=> ( ( B6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
@ ( product_Sigma @ B @ A @ B6
@ ^ [Uu: B] : A4 ) )
@ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ B @ A ) @ A ) ) ) ) ) ).
% card_of_Times_infinite_simps(3)
thf(fact_5201_card__of__Times__infinite__simps_I2_J,axiom,
! [B: $tType,A: $tType,A4: set @ A,B6: set @ B] :
( ~ ( finite_finite2 @ A @ A4 )
=> ( ( B6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : B6 ) ) )
@ ( bNF_Wellorder_ordIso @ A @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% card_of_Times_infinite_simps(2)
thf(fact_5202_card__of__Times__infinite__simps_I1_J,axiom,
! [B: $tType,A: $tType,A4: set @ A,B6: set @ B] :
( ~ ( finite_finite2 @ A @ A4 )
=> ( ( B6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : B6 ) )
@ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ B ) @ A ) ) ) ) ) ).
% card_of_Times_infinite_simps(1)
thf(fact_5203_card__of__Times__infinite,axiom,
! [A: $tType,B: $tType,A4: set @ A,B6: set @ B] :
( ~ ( finite_finite2 @ A @ A4 )
=> ( ( B6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : B6 ) )
@ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ B ) @ A ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
@ ( product_Sigma @ B @ A @ B6
@ ^ [Uu: B] : A4 ) )
@ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ B @ A ) @ A ) ) ) ) ) ) ).
% card_of_Times_infinite
thf(fact_5204_card__of__Plus__Times,axiom,
! [B: $tType,A: $tType,A1: A,A22: A,A4: set @ A,B1: B,B22: B,B6: set @ B] :
( ( ( A1 != A22 )
& ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ A1 @ ( insert2 @ A @ A22 @ ( bot_bot @ ( set @ A ) ) ) ) @ A4 ) )
=> ( ( ( B1 != B22 )
& ( ord_less_eq @ ( set @ B ) @ ( insert2 @ B @ B1 @ ( insert2 @ B @ B22 @ ( bot_bot @ ( set @ B ) ) ) ) @ B6 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A4 @ B6 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : B6 ) ) )
@ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ) ).
% card_of_Plus_Times
thf(fact_5205_card__of__image,axiom,
! [B: $tType,A: $tType,F: B > A,A4: set @ B] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( image2 @ B @ A @ F @ A4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ).
% card_of_image
thf(fact_5206_card__of__empty,axiom,
! [B: $tType,A: $tType,A4: set @ B] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( bot_bot @ ( set @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ).
% card_of_empty
thf(fact_5207_card__of__empty3,axiom,
! [B: $tType,A: $tType,A4: set @ A] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( bot_bot @ ( set @ B ) ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% card_of_empty3
thf(fact_5208_card__of__ordLeq__finite,axiom,
! [B: $tType,A: $tType,A4: set @ A,B6: set @ B] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( finite_finite2 @ B @ B6 )
=> ( finite_finite2 @ A @ A4 ) ) ) ).
% card_of_ordLeq_finite
thf(fact_5209_card__of__ordLeq__infinite,axiom,
! [A: $tType,B: $tType,A4: set @ A,B6: set @ B] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ~ ( finite_finite2 @ A @ A4 )
=> ~ ( finite_finite2 @ B @ B6 ) ) ) ).
% card_of_ordLeq_infinite
thf(fact_5210_card__of__mono1,axiom,
! [A: $tType,A4: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B6 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ).
% card_of_mono1
thf(fact_5211_card__of__empty__ordIso,axiom,
! [B: $tType,A: $tType] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( bot_bot @ ( set @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( bot_bot @ ( set @ B ) ) ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ).
% card_of_empty_ordIso
thf(fact_5212_card__of__empty2,axiom,
! [B: $tType,A: $tType,A4: set @ A] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( bot_bot @ ( set @ B ) ) ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% card_of_empty2
thf(fact_5213_card__of__ordIso__finite,axiom,
! [A: $tType,B: $tType,A4: set @ A,B6: set @ B] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( finite_finite2 @ A @ A4 )
= ( finite_finite2 @ B @ B6 ) ) ) ).
% card_of_ordIso_finite
thf(fact_5214_card__of__mono2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( field2 @ A @ R ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( field2 @ B @ R4 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ).
% card_of_mono2
thf(fact_5215_card__of__cong,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( field2 @ A @ R ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( field2 @ B @ R4 ) ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ).
% card_of_cong
thf(fact_5216_card__of__ordLeqI,axiom,
! [B: $tType,A: $tType,F: A > B,A4: set @ A,B6: set @ B] :
( ( inj_on @ A @ B @ F @ A4 )
=> ( ! [A5: A] :
( ( member @ A @ A5 @ A4 )
=> ( member @ B @ ( F @ A5 ) @ B6 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ) ).
% card_of_ordLeqI
thf(fact_5217_ex__bij__betw,axiom,
! [B: $tType,A: $tType,A4: set @ A,R: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ R ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ? [F4: B > A,B10: set @ B] : ( bij_betw @ B @ A @ F4 @ B10 @ A4 ) ) ).
% ex_bij_betw
thf(fact_5218_card__of__least,axiom,
! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A )] :
( ( order_well_order_on @ A @ A4 @ R )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ R ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ).
% card_of_least
thf(fact_5219_BNF__Cardinal__Order__Relation_OordLess__Field,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( field2 @ A @ R ) ) @ R4 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) ) ) ).
% BNF_Cardinal_Order_Relation.ordLess_Field
thf(fact_5220_card__of__ordIsoI,axiom,
! [B: $tType,A: $tType,F: A > B,A4: set @ A,B6: set @ B] :
( ( bij_betw @ A @ B @ F @ A4 @ B6 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ).
% card_of_ordIsoI
thf(fact_5221_card__of__ordIso,axiom,
! [B: $tType,A: $tType,A4: set @ A,B6: set @ B] :
( ( ? [F2: A > B] : ( bij_betw @ A @ B @ F2 @ A4 @ B6 ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ).
% card_of_ordIso
thf(fact_5222_type__copy__set__bd,axiom,
! [A: $tType,D: $tType,C: $tType,B: $tType,S2: A > ( set @ B ),Bd: set @ ( product_prod @ C @ C ),Rep: D > A,X: D] :
( ! [Y4: A] : ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( S2 @ Y4 ) ) @ Bd ) @ ( bNF_Wellorder_ordLeq @ B @ C ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( comp @ A @ ( set @ B ) @ D @ S2 @ Rep @ X ) ) @ Bd ) @ ( bNF_Wellorder_ordLeq @ B @ C ) ) ) ).
% type_copy_set_bd
thf(fact_5223_card__of__ordLeq2,axiom,
! [B: $tType,A: $tType,A4: set @ A,B6: set @ B] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ? [G2: B > A] :
( ( image2 @ B @ A @ G2 @ B6 )
= A4 ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ) ).
% card_of_ordLeq2
thf(fact_5224_surj__imp__ordLeq,axiom,
! [B: $tType,A: $tType,B6: set @ A,F: B > A,A4: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B6 @ ( image2 @ B @ A @ F @ A4 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B6 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ).
% surj_imp_ordLeq
thf(fact_5225_card__of__singl__ordLeq,axiom,
! [A: $tType,B: $tType,A4: set @ A,B2: B] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( insert2 @ B @ B2 @ ( bot_bot @ ( set @ B ) ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ).
% card_of_singl_ordLeq
thf(fact_5226_card__of__ordLess2,axiom,
! [A: $tType,B: $tType,B6: set @ A,A4: set @ B] :
( ( B6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ~ ? [F2: B > A] :
( ( image2 @ B @ A @ F2 @ A4 )
= B6 ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B6 ) ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) ) ) ) ).
% card_of_ordLess2
thf(fact_5227_internalize__card__of__ordLeq2,axiom,
! [A: $tType,B: $tType,A4: set @ A,C5: set @ B] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ C5 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
= ( ? [B9: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ B9 @ C5 )
& ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B9 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B9 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ C5 ) ) @ ( bNF_Wellorder_ordLeq @ B @ B ) ) ) ) ) ).
% internalize_card_of_ordLeq2
thf(fact_5228_card__of__Field__ordLess,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( field2 @ A @ R ) ) @ R ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ).
% card_of_Field_ordLess
thf(fact_5229_ordLeq__Plus__mono,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),P5: set @ ( product_prod @ C @ C ),P8: set @ ( product_prod @ D @ D )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ P5 @ P8 ) @ ( bNF_Wellorder_ordLeq @ C @ D ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ C ) @ ( sum_Plus @ A @ C @ ( field2 @ A @ R ) @ ( field2 @ C @ P5 ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ D ) @ ( sum_Plus @ B @ D @ ( field2 @ B @ R4 ) @ ( field2 @ D @ P8 ) ) ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) ).
% ordLeq_Plus_mono
thf(fact_5230_ordLeq__Plus__mono1,axiom,
! [B: $tType,C: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),C5: set @ C] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ C ) @ ( sum_Plus @ A @ C @ ( field2 @ A @ R ) @ C5 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ C ) @ ( sum_Plus @ B @ C @ ( field2 @ B @ R4 ) @ C5 ) ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ C ) ) ) ) ).
% ordLeq_Plus_mono1
thf(fact_5231_ordLeq__Plus__mono2,axiom,
! [B: $tType,A: $tType,C: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),A4: set @ C] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ A ) @ ( sum_Plus @ C @ A @ A4 @ ( field2 @ A @ R ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ B ) @ ( sum_Plus @ C @ B @ A4 @ ( field2 @ B @ R4 ) ) ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ B ) ) ) ) ).
% ordLeq_Plus_mono2
thf(fact_5232_ordIso__Plus__cong,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),P5: set @ ( product_prod @ C @ C ),P8: set @ ( product_prod @ D @ D )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ P5 @ P8 ) @ ( bNF_Wellorder_ordIso @ C @ D ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ C ) @ ( sum_Plus @ A @ C @ ( field2 @ A @ R ) @ ( field2 @ C @ P5 ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ D ) @ ( sum_Plus @ B @ D @ ( field2 @ B @ R4 ) @ ( field2 @ D @ P8 ) ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) ).
% ordIso_Plus_cong
thf(fact_5233_ordIso__Plus__cong1,axiom,
! [B: $tType,C: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),C5: set @ C] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ C ) @ ( sum_Plus @ A @ C @ ( field2 @ A @ R ) @ C5 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ C ) @ ( sum_Plus @ B @ C @ ( field2 @ B @ R4 ) @ C5 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ C ) ) ) ) ).
% ordIso_Plus_cong1
thf(fact_5234_ordIso__Plus__cong2,axiom,
! [B: $tType,A: $tType,C: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),A4: set @ C] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ A ) @ ( sum_Plus @ C @ A @ A4 @ ( field2 @ A @ R ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ B ) @ ( sum_Plus @ C @ B @ A4 @ ( field2 @ B @ R4 ) ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ B ) ) ) ) ).
% ordIso_Plus_cong2
thf(fact_5235_ordLeq__Times__mono1,axiom,
! [B: $tType,C: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),C5: set @ C] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ C )
@ ( product_Sigma @ A @ C @ ( field2 @ A @ R )
@ ^ [Uu: A] : C5 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ C )
@ ( product_Sigma @ B @ C @ ( field2 @ B @ R4 )
@ ^ [Uu: B] : C5 ) ) )
@ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ C ) @ ( product_prod @ B @ C ) ) ) ) ).
% ordLeq_Times_mono1
thf(fact_5236_ordLeq__Times__mono2,axiom,
! [B: $tType,A: $tType,C: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),A4: set @ C] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ C @ A ) @ ( product_prod @ C @ A ) ) ) @ ( set @ ( product_prod @ ( product_prod @ C @ B ) @ ( product_prod @ C @ B ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ C @ A ) @ ( product_prod @ C @ A ) ) ) @ ( set @ ( product_prod @ ( product_prod @ C @ B ) @ ( product_prod @ C @ B ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ C @ A )
@ ( product_Sigma @ C @ A @ A4
@ ^ [Uu: C] : ( field2 @ A @ R ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ C @ B )
@ ( product_Sigma @ C @ B @ A4
@ ^ [Uu: C] : ( field2 @ B @ R4 ) ) ) )
@ ( bNF_Wellorder_ordLeq @ ( product_prod @ C @ A ) @ ( product_prod @ C @ B ) ) ) ) ).
% ordLeq_Times_mono2
thf(fact_5237_card__of__ordLeq,axiom,
! [B: $tType,A: $tType,A4: set @ A,B6: set @ B] :
( ( ? [F2: A > B] :
( ( inj_on @ A @ B @ F2 @ A4 )
& ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ A4 ) @ B6 ) ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ).
% card_of_ordLeq
thf(fact_5238_internalize__card__of__ordLeq,axiom,
! [A: $tType,B: $tType,A4: set @ A,R: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ R ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
= ( ? [B9: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ B9 @ ( field2 @ B @ R ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B9 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B9 ) @ R ) @ ( bNF_Wellorder_ordLeq @ B @ B ) ) ) ) ) ).
% internalize_card_of_ordLeq
thf(fact_5239_regularCard__def,axiom,
! [A: $tType] :
( ( bNF_Ca7133664381575040944arCard @ A )
= ( ^ [R3: set @ ( product_prod @ A @ A )] :
! [K3: set @ A] :
( ( ( ord_less_eq @ ( set @ A ) @ K3 @ ( field2 @ A @ R3 ) )
& ( bNF_Ca7293521722713021262ofinal @ A @ K3 @ R3 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ K3 ) @ R3 ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ) ) ).
% regularCard_def
thf(fact_5240_card__of__ordIso__subst,axiom,
! [A: $tType,A4: set @ A,B6: set @ A] :
( ( A4 = B6 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B6 ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ).
% card_of_ordIso_subst
thf(fact_5241_comp__set__bd__Union__o__collect,axiom,
! [C: $tType,B: $tType,A: $tType,X: C,X10: set @ ( C > ( set @ ( set @ A ) ) ),Hbd: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) )
@ ( bNF_Ca6860139660246222851ard_of @ A
@ ( complete_Sup_Sup @ ( set @ A )
@ ( complete_Sup_Sup @ ( set @ ( set @ A ) )
@ ( image2 @ ( C > ( set @ ( set @ A ) ) ) @ ( set @ ( set @ A ) )
@ ^ [F2: C > ( set @ ( set @ A ) )] : ( F2 @ X )
@ X10 ) ) ) )
@ Hbd )
@ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( comp @ ( set @ ( set @ A ) ) @ ( set @ A ) @ C @ ( complete_Sup_Sup @ ( set @ A ) ) @ ( bNF_collect @ C @ ( set @ A ) @ X10 ) @ X ) ) @ Hbd ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ).
% comp_set_bd_Union_o_collect
thf(fact_5242_card__of__Plus__ordLess__infinite__Field,axiom,
! [A: $tType,C: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),A4: set @ B,B6: set @ C] :
( ~ ( finite_finite2 @ A @ ( field2 @ A @ R ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A4 ) @ R ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ B6 ) @ R ) @ ( bNF_We4044943003108391690rdLess @ C @ A ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ C ) @ ( sum_Plus @ B @ C @ A4 @ B6 ) ) @ R ) @ ( bNF_We4044943003108391690rdLess @ ( sum_sum @ B @ C ) @ A ) ) ) ) ) ) ).
% card_of_Plus_ordLess_infinite_Field
thf(fact_5243_card__of__Times__ordLeq__infinite__Field,axiom,
! [A: $tType,C: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),A4: set @ B,B6: set @ C] :
( ~ ( finite_finite2 @ A @ ( field2 @ A @ R ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A4 ) @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ B6 ) @ R ) @ ( bNF_Wellorder_ordLeq @ C @ A ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ C )
@ ( product_Sigma @ B @ C @ A4
@ ^ [Uu: B] : B6 ) )
@ R )
@ ( bNF_Wellorder_ordLeq @ ( product_prod @ B @ C ) @ A ) ) ) ) ) ) ).
% card_of_Times_ordLeq_infinite_Field
thf(fact_5244_card__of__Sigma__ordLeq__infinite__Field,axiom,
! [A: $tType,C: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),I4: set @ B,A4: B > ( set @ C )] :
( ~ ( finite_finite2 @ A @ ( field2 @ A @ R ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ I4 ) @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ I4 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( A4 @ X2 ) ) @ R ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ C ) @ ( product_Sigma @ B @ C @ I4 @ A4 ) ) @ R ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ B @ C ) @ A ) ) ) ) ) ) ).
% card_of_Sigma_ordLeq_infinite_Field
thf(fact_5245_Card__order__trans,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),X: A,Y: A,Z2: A] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( X != Y )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
=> ( ( Y != Z2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ R )
=> ( ( X != Z2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ R ) ) ) ) ) ) ) ).
% Card_order_trans
thf(fact_5246_card__order__on__ordIso,axiom,
! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ A4 @ R )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ A4 @ R4 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ) ).
% card_order_on_ordIso
thf(fact_5247_infinite__Card__order__limit,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ~ ( finite_finite2 @ A @ ( field2 @ A @ R ) )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R ) )
=> ? [X2: A] :
( ( member @ A @ X2 @ ( field2 @ A @ R ) )
& ( A3 != X2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ X2 ) @ R ) ) ) ) ) ).
% infinite_Card_order_limit
thf(fact_5248_ordLeq__refl,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ R ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ).
% ordLeq_refl
thf(fact_5249_Card__order__ordIso,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ R ) @ ( bNF_Wellorder_ordIso @ B @ A ) )
=> ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R4 ) @ R4 ) ) ) ).
% Card_order_ordIso
thf(fact_5250_Card__order__ordIso2,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R4 ) @ R4 ) ) ) ).
% Card_order_ordIso2
thf(fact_5251_ordIso__refl,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ R ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ).
% ordIso_refl
thf(fact_5252_card__of__unique,axiom,
! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ A4 @ R )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ).
% card_of_unique
thf(fact_5253_card__order__on__def,axiom,
! [A: $tType] :
( ( bNF_Ca8970107618336181345der_on @ A )
= ( ^ [A9: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( order_well_order_on @ A @ A9 @ R3 )
& ! [R11: set @ ( product_prod @ A @ A )] :
( ( order_well_order_on @ A @ A9 @ R11 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ R11 ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ) ) ) ).
% card_order_on_def
thf(fact_5254_Card__order__iff__ordLeq__card__of,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( bNF_Ca6860139660246222851ard_of @ A @ ( field2 @ A @ R ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ).
% Card_order_iff_ordLeq_card_of
thf(fact_5255_ordIso__card__of__imp__Card__order,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),A4: set @ B] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ ( bNF_Ca6860139660246222851ard_of @ B @ A4 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) ) ).
% ordIso_card_of_imp_Card_order
thf(fact_5256_Card__order__iff__ordIso__card__of,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( bNF_Ca6860139660246222851ard_of @ A @ ( field2 @ A @ R ) ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ).
% Card_order_iff_ordIso_card_of
thf(fact_5257_card__of__Field__ordIso,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( field2 @ A @ R ) ) @ R ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ).
% card_of_Field_ordIso
thf(fact_5258_dir__image,axiom,
! [B: $tType,A: $tType,F: A > B,R: set @ ( product_prod @ A @ A )] :
( ! [X2: A,Y4: A] :
( ( ( F @ X2 )
= ( F @ Y4 ) )
= ( X2 = Y4 ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ ( bNF_We2720479622203943262_image @ A @ B @ R @ F ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ) ).
% dir_image
thf(fact_5259_exists__minim__Card__order,axiom,
! [A: $tType,R5: set @ ( set @ ( product_prod @ A @ A ) )] :
( ( R5
!= ( bot_bot @ ( set @ ( set @ ( product_prod @ A @ A ) ) ) ) )
=> ( ! [X2: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ X2 @ R5 )
=> ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ X2 ) @ X2 ) )
=> ? [X2: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ X2 @ R5 )
& ! [Xa2: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ Xa2 @ R5 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ X2 @ Xa2 ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ) ) ) ).
% exists_minim_Card_order
thf(fact_5260_Card__order__empty,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( bot_bot @ ( set @ B ) ) ) @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ).
% Card_order_empty
thf(fact_5261_card__of__ordIso__finite__Field,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),A4: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ ( bNF_Ca6860139660246222851ard_of @ B @ A4 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( finite_finite2 @ A @ ( field2 @ A @ R ) )
= ( finite_finite2 @ B @ A4 ) ) ) ) ).
% card_of_ordIso_finite_Field
thf(fact_5262_card__of__underS,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( order_underS @ A @ R @ A3 ) ) @ R ) @ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ).
% card_of_underS
thf(fact_5263_Card__order__Plus1,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),B6: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ R @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ ( field2 @ A @ R ) @ B6 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ A @ B ) ) ) ) ).
% Card_order_Plus1
thf(fact_5264_Card__order__Plus2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),A4: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ R @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ A ) @ ( sum_Plus @ B @ A @ A4 @ ( field2 @ A @ R ) ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ B @ A ) ) ) ) ).
% Card_order_Plus2
thf(fact_5265_Card__order__singl__ordLeq,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),B2: B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( ( field2 @ A @ R )
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( insert2 @ B @ B2 @ ( bot_bot @ ( set @ B ) ) ) ) @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ) ).
% Card_order_singl_ordLeq
thf(fact_5266_card__of__Un__ordLeq__infinite__Field,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),A4: set @ B,B6: set @ B] :
( ~ ( finite_finite2 @ A @ ( field2 @ A @ R ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A4 ) @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( sup_sup @ ( set @ B ) @ A4 @ B6 ) ) @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ) ) ) ).
% card_of_Un_ordLeq_infinite_Field
thf(fact_5267_card__of__empty1,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A )] :
( ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
| ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( bot_bot @ ( set @ B ) ) ) @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ).
% card_of_empty1
thf(fact_5268_Card__order__Times1,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),B6: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( B6
!= ( bot_bot @ ( set @ B ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ R
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ ( field2 @ A @ R )
@ ^ [Uu: A] : B6 ) ) )
@ ( bNF_Wellorder_ordLeq @ A @ ( product_prod @ A @ B ) ) ) ) ) ).
% Card_order_Times1
thf(fact_5269_Card__order__Times2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),A4: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ R
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
@ ( product_Sigma @ B @ A @ A4
@ ^ [Uu: B] : ( field2 @ A @ R ) ) ) )
@ ( bNF_Wellorder_ordLeq @ A @ ( product_prod @ B @ A ) ) ) ) ) ).
% Card_order_Times2
thf(fact_5270_Card__order__Times__same__infinite,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ~ ( finite_finite2 @ A @ ( field2 @ A @ R ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ A )
@ ( product_Sigma @ A @ A @ ( field2 @ A @ R )
@ ^ [Uu: A] : ( field2 @ A @ R ) ) )
@ R )
@ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ A ) @ A ) ) ) ) ).
% Card_order_Times_same_infinite
thf(fact_5271_card__of__UNION__ordLeq__infinite__Field,axiom,
! [B: $tType,A: $tType,C: $tType,R: set @ ( product_prod @ A @ A ),I4: set @ B,A4: B > ( set @ C )] :
( ~ ( finite_finite2 @ A @ ( field2 @ A @ R ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ I4 ) @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ I4 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( A4 @ X2 ) ) @ R ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ A4 @ I4 ) ) ) @ R ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) ) ) ) ) ).
% card_of_UNION_ordLeq_infinite_Field
thf(fact_5272_Card__order__Plus__infinite,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),P5: set @ ( product_prod @ B @ B )] :
( ~ ( finite_finite2 @ A @ ( field2 @ A @ R ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P5 @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ ( field2 @ A @ R ) @ ( field2 @ B @ P5 ) ) ) @ R ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ A ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ A ) @ ( sum_Plus @ B @ A @ ( field2 @ B @ P5 ) @ ( field2 @ A @ R ) ) ) @ R ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ B @ A ) @ A ) ) ) ) ) ) ).
% Card_order_Plus_infinite
thf(fact_5273_Card__order__iff__Restr__underS,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
= ( ! [X3: A] :
( ( member @ A @ X3 @ ( field2 @ A @ R ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ ( order_underS @ A @ R @ X3 )
@ ^ [Uu: A] : ( order_underS @ A @ R @ X3 ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ A @ ( field2 @ A @ R ) ) )
@ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ) ) ).
% Card_order_iff_Restr_underS
thf(fact_5274_regularCard__UNION,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),As4: A > ( set @ B ),B6: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( bNF_Ca7133664381575040944arCard @ A @ R )
=> ( ( bNF_Ca3754400796208372196lChain @ A @ ( set @ B ) @ R @ As4 )
=> ( ( ord_less_eq @ ( set @ B ) @ B6 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ As4 @ ( field2 @ A @ R ) ) ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) @ R ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) )
=> ? [X2: A] :
( ( member @ A @ X2 @ ( field2 @ A @ R ) )
& ( ord_less_eq @ ( set @ B ) @ B6 @ ( As4 @ X2 ) ) ) ) ) ) ) ) ).
% regularCard_UNION
thf(fact_5275_card__of__Plus__ordLeq__infinite__Field,axiom,
! [A: $tType,C: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),A4: set @ B,B6: set @ C] :
( ~ ( finite_finite2 @ A @ ( field2 @ A @ R ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A4 ) @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ B6 ) @ R ) @ ( bNF_Wellorder_ordLeq @ C @ A ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ C ) @ ( sum_Plus @ B @ C @ A4 @ B6 ) ) @ R ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ B @ C ) @ A ) ) ) ) ) ) ).
% card_of_Plus_ordLeq_infinite_Field
thf(fact_5276_Card__order__Times__infinite,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),P5: set @ ( product_prod @ B @ B )] :
( ~ ( finite_finite2 @ A @ ( field2 @ A @ R ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( ( field2 @ B @ P5 )
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P5 @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ ( field2 @ A @ R )
@ ^ [Uu: A] : ( field2 @ B @ P5 ) ) )
@ R )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ B ) @ A ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
@ ( product_Sigma @ B @ A @ ( field2 @ B @ P5 )
@ ^ [Uu: B] : ( field2 @ A @ R ) ) )
@ R )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ B @ A ) @ A ) ) ) ) ) ) ) ).
% Card_order_Times_infinite
thf(fact_5277_ex__toCard__pred,axiom,
! [B: $tType,A: $tType,A4: set @ A,R: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ R ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R ) @ R )
=> ? [X_12: A > B] : ( bNF_Gr1419584066657907630d_pred @ A @ B @ A4 @ R @ X_12 ) ) ) ).
% ex_toCard_pred
thf(fact_5278_cardSuc__UNION,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),As4: ( set @ A ) > ( set @ B ),B6: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ~ ( finite_finite2 @ A @ ( field2 @ A @ R ) )
=> ( ( bNF_Ca3754400796208372196lChain @ ( set @ A ) @ ( set @ B ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R ) @ As4 )
=> ( ( ord_less_eq @ ( set @ B ) @ B6 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ ( set @ A ) @ ( set @ B ) @ As4 @ ( field2 @ ( set @ A ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R ) ) ) ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ? [X2: set @ A] :
( ( member @ ( set @ A ) @ X2 @ ( field2 @ ( set @ A ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R ) ) )
& ( ord_less_eq @ ( set @ B ) @ B6 @ ( As4 @ X2 ) ) ) ) ) ) ) ) ).
% cardSuc_UNION
thf(fact_5279_toCard__inj,axiom,
! [B: $tType,A: $tType,A4: set @ A,R: set @ ( product_prod @ B @ B ),X: A,Y: A] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ R ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R ) @ R )
=> ( ( member @ A @ X @ A4 )
=> ( ( member @ A @ Y @ A4 )
=> ( ( ( bNF_Greatest_toCard @ A @ B @ A4 @ R @ X )
= ( bNF_Greatest_toCard @ A @ B @ A4 @ R @ Y ) )
= ( X = Y ) ) ) ) ) ) ).
% toCard_inj
thf(fact_5280_cardSuc__greater,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ R @ ( bNF_Ca8387033319878233205ardSuc @ A @ R ) ) @ ( bNF_We4044943003108391690rdLess @ A @ ( set @ A ) ) ) ) ).
% cardSuc_greater
thf(fact_5281_cardSuc__ordLeq,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ R @ ( bNF_Ca8387033319878233205ardSuc @ A @ R ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( set @ A ) ) ) ) ).
% cardSuc_ordLeq
thf(fact_5282_toCard__pred__toCard,axiom,
! [A: $tType,B: $tType,A4: set @ A,R: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ R ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R ) @ R )
=> ( bNF_Gr1419584066657907630d_pred @ A @ B @ A4 @ R @ ( bNF_Greatest_toCard @ A @ B @ A4 @ R ) ) ) ) ).
% toCard_pred_toCard
thf(fact_5283_cardSuc__least__aux,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ ( set @ A ) @ ( set @ A ) )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( bNF_Ca8970107618336181345der_on @ ( set @ A ) @ ( field2 @ ( set @ A ) @ R4 ) @ R4 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ R @ R4 ) @ ( bNF_We4044943003108391690rdLess @ A @ ( set @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R ) @ R4 ) @ ( bNF_Wellorder_ordLeq @ ( set @ A ) @ ( set @ A ) ) ) ) ) ) ).
% cardSuc_least_aux
thf(fact_5284_cardSuc__ordLeq__ordLess,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R4 ) @ R4 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ R4 @ ( bNF_Ca8387033319878233205ardSuc @ A @ R ) ) @ ( bNF_We4044943003108391690rdLess @ B @ ( set @ A ) ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ) ) ).
% cardSuc_ordLeq_ordLess
thf(fact_5285_cardSuc__ordLess__ordLeq,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R4 ) @ R4 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R ) @ R4 ) @ ( bNF_Wellorder_ordLeq @ ( set @ A ) @ B ) ) ) ) ) ).
% cardSuc_ordLess_ordLeq
thf(fact_5286_cardSuc__least,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R4 ) @ R4 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R ) @ R4 ) @ ( bNF_Wellorder_ordLeq @ ( set @ A ) @ B ) ) ) ) ) ).
% cardSuc_least
thf(fact_5287_cardSuc__mono__ordLeq,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R4 ) @ R4 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( set @ B ) @ ( set @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( set @ B ) @ ( set @ B ) ) ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R ) @ ( bNF_Ca8387033319878233205ardSuc @ B @ R4 ) ) @ ( bNF_Wellorder_ordLeq @ ( set @ A ) @ ( set @ B ) ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ) ) ).
% cardSuc_mono_ordLeq
thf(fact_5288_cardSuc__invar__ordIso,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R4 ) @ R4 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( set @ B ) @ ( set @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( set @ B ) @ ( set @ B ) ) ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R ) @ ( bNF_Ca8387033319878233205ardSuc @ B @ R4 ) ) @ ( bNF_Wellorder_ordIso @ ( set @ A ) @ ( set @ B ) ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ) ) ).
% cardSuc_invar_ordIso
thf(fact_5289_fromCard__toCard,axiom,
! [B: $tType,A: $tType,A4: set @ A,R: set @ ( product_prod @ B @ B ),B2: A] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ R ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R ) @ R )
=> ( ( member @ A @ B2 @ A4 )
=> ( ( bNF_Gr5436034075474128252omCard @ A @ B @ A4 @ R @ ( bNF_Greatest_toCard @ A @ B @ A4 @ R @ B2 ) )
= B2 ) ) ) ) ).
% fromCard_toCard
thf(fact_5290_cardSuc__UNION__Cinfinite,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),As4: ( set @ A ) > ( set @ B ),B6: set @ B] :
( ( ( bNF_Ca4139267488887388095finite @ A @ R )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
=> ( ( bNF_Ca3754400796208372196lChain @ ( set @ A ) @ ( set @ B ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R ) @ As4 )
=> ( ( ord_less_eq @ ( set @ B ) @ B6 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ ( set @ A ) @ ( set @ B ) @ As4 @ ( field2 @ ( set @ A ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R ) ) ) ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ? [X2: set @ A] :
( ( member @ ( set @ A ) @ X2 @ ( field2 @ ( set @ A ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R ) ) )
& ( ord_less_eq @ ( set @ B ) @ B6 @ ( As4 @ X2 ) ) ) ) ) ) ) ).
% cardSuc_UNION_Cinfinite
thf(fact_5291_isCardSuc__def,axiom,
! [A: $tType] :
( ( bNF_Ca6246979054910435723ardSuc @ A )
= ( ^ [R3: set @ ( product_prod @ A @ A ),R11: set @ ( product_prod @ ( set @ A ) @ ( set @ A ) )] :
( ( bNF_Ca8970107618336181345der_on @ ( set @ A ) @ ( field2 @ ( set @ A ) @ R11 ) @ R11 )
& ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ R3 @ R11 ) @ ( bNF_We4044943003108391690rdLess @ A @ ( set @ A ) ) )
& ! [R14: set @ ( product_prod @ ( set @ A ) @ ( set @ A ) )] :
( ( ( bNF_Ca8970107618336181345der_on @ ( set @ A ) @ ( field2 @ ( set @ A ) @ R14 ) @ R14 )
& ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ R3 @ R14 ) @ ( bNF_We4044943003108391690rdLess @ A @ ( set @ A ) ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ R11 @ R14 ) @ ( bNF_Wellorder_ordLeq @ ( set @ A ) @ ( set @ A ) ) ) ) ) ) ) ).
% isCardSuc_def
thf(fact_5292_cinfinite__mono,axiom,
! [A: $tType,B: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R1 @ R22 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca4139267488887388095finite @ A @ R1 )
=> ( bNF_Ca4139267488887388095finite @ B @ R22 ) ) ) ).
% cinfinite_mono
thf(fact_5293_Cinfinite__limit2,axiom,
! [A: $tType,X1: A,R: set @ ( product_prod @ A @ A ),X22: A] :
( ( member @ A @ X1 @ ( field2 @ A @ R ) )
=> ( ( member @ A @ X22 @ ( field2 @ A @ R ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ A @ R )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
=> ? [X2: A] :
( ( member @ A @ X2 @ ( field2 @ A @ R ) )
& ( X1 != X2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X1 @ X2 ) @ R )
& ( X22 != X2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X22 @ X2 ) @ R ) ) ) ) ) ).
% Cinfinite_limit2
thf(fact_5294_Cinfinite__limit,axiom,
! [A: $tType,X: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ A @ X @ ( field2 @ A @ R ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ A @ R )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
=> ? [X2: A] :
( ( member @ A @ X2 @ ( field2 @ A @ R ) )
& ( X != X2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X2 ) @ R ) ) ) ) ).
% Cinfinite_limit
thf(fact_5295_Cinfinite__cong,axiom,
! [A: $tType,B: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R1 @ R22 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ A @ R1 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R1 ) @ R1 ) )
=> ( ( bNF_Ca4139267488887388095finite @ B @ R22 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R22 ) @ R22 ) ) ) ) ).
% Cinfinite_cong
thf(fact_5296_Cinfinite__limit__finite,axiom,
! [A: $tType,X10: set @ A,R: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ A @ X10 )
=> ( ( ord_less_eq @ ( set @ A ) @ X10 @ ( field2 @ A @ R ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ A @ R )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
=> ? [X2: A] :
( ( member @ A @ X2 @ ( field2 @ A @ R ) )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ X10 )
=> ( ( Xa2 != X2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Xa2 @ X2 ) @ R ) ) ) ) ) ) ) ).
% Cinfinite_limit_finite
thf(fact_5297_Un__Cinfinite__bound,axiom,
! [B: $tType,A: $tType,A4: set @ A,R: set @ ( product_prod @ B @ B ),B6: set @ A] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ R ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B6 ) @ R ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R ) @ R ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B6 ) ) @ R ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ) ) ).
% Un_Cinfinite_bound
thf(fact_5298_UNION__Cinfinite__bound,axiom,
! [A: $tType,B: $tType,C: $tType,I4: set @ A,R: set @ ( product_prod @ B @ B ),A4: A > ( set @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ I4 ) @ R ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ I4 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( A4 @ X2 ) ) @ R ) @ ( bNF_Wellorder_ordLeq @ C @ B ) ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R ) @ R ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ A @ ( set @ C ) @ A4 @ I4 ) ) ) @ R ) @ ( bNF_Wellorder_ordLeq @ C @ B ) ) ) ) ) ).
% UNION_Cinfinite_bound
thf(fact_5299_Cfinite__ordLess__Cinfinite,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ B @ B )] :
( ( ( bNF_Cardinal_cfinite @ A @ R )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ S )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ S ) @ S ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ S ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) ) ) ) ).
% Cfinite_ordLess_Cinfinite
thf(fact_5300_comp__single__set__bd,axiom,
! [B: $tType,D: $tType,A: $tType,E: $tType,C: $tType,Fbd: set @ ( product_prod @ A @ A ),Fset: B > ( set @ C ),Gset: D > ( set @ B ),Gbd: set @ ( product_prod @ E @ E ),X: D] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ Fbd ) @ Fbd )
=> ( ! [X2: B] : ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( Fset @ X2 ) ) @ Fbd ) @ ( bNF_Wellorder_ordLeq @ C @ A ) )
=> ( ! [X2: D] : ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ E @ E ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ E @ E ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( Gset @ X2 ) ) @ Gbd ) @ ( bNF_Wellorder_ordLeq @ B @ E ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ ( product_prod @ E @ A ) @ ( product_prod @ E @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ ( product_prod @ E @ A ) @ ( product_prod @ E @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ Fset @ ( Gset @ X ) ) ) ) @ ( bNF_Cardinal_cprod @ E @ A @ Gbd @ Fbd ) ) @ ( bNF_Wellorder_ordLeq @ C @ ( product_prod @ E @ A ) ) ) ) ) ) ).
% comp_single_set_bd
thf(fact_5301_cprod__dup,axiom,
! [A: $tType,C: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),P5: set @ ( product_prod @ B @ B ),P8: set @ ( product_prod @ C @ C )] :
( ( bNF_Ca4139267488887388095finite @ A @ R )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) @ ( bNF_Cardinal_cprod @ B @ C @ P5 @ P8 ) @ ( bNF_Cardinal_cprod @ A @ A @ R @ R ) ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ B @ C ) @ ( product_prod @ A @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Cardinal_cprod @ B @ C @ P5 @ P8 ) @ R ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ B @ C ) @ A ) ) ) ) ) ).
% cprod_dup
thf(fact_5302_cprod__com,axiom,
! [B: $tType,A: $tType,P13: set @ ( product_prod @ A @ A ),P24: set @ ( product_prod @ B @ B )] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( bNF_Cardinal_cprod @ A @ B @ P13 @ P24 ) @ ( bNF_Cardinal_cprod @ B @ A @ P24 @ P13 ) ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A ) ) ) ).
% cprod_com
thf(fact_5303_Cfinite__cprod__Cinfinite,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ B @ B )] :
( ( ( bNF_Cardinal_cfinite @ A @ R )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ S )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ S ) @ S ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Cardinal_cprod @ A @ B @ R @ S ) @ S ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ B ) @ B ) ) ) ) ).
% Cfinite_cprod_Cinfinite
thf(fact_5304_cprod__mono,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,P13: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),P24: set @ ( product_prod @ C @ C ),R22: set @ ( product_prod @ D @ D )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P13 @ R1 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ P24 @ R22 ) @ ( bNF_Wellorder_ordLeq @ C @ D ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ D ) @ ( product_prod @ B @ D ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ D ) @ ( product_prod @ B @ D ) ) ) @ ( bNF_Cardinal_cprod @ A @ C @ P13 @ P24 ) @ ( bNF_Cardinal_cprod @ B @ D @ R1 @ R22 ) ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) ) ) ) ).
% cprod_mono
thf(fact_5305_cprod__mono1,axiom,
! [B: $tType,C: $tType,A: $tType,P13: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),Q3: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P13 @ R1 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( bNF_Cardinal_cprod @ A @ C @ P13 @ Q3 ) @ ( bNF_Cardinal_cprod @ B @ C @ R1 @ Q3 ) ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ C ) @ ( product_prod @ B @ C ) ) ) ) ).
% cprod_mono1
thf(fact_5306_cprod__mono2,axiom,
! [B: $tType,A: $tType,C: $tType,P24: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),Q3: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P24 @ R22 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ C @ A ) @ ( product_prod @ C @ A ) ) ) @ ( set @ ( product_prod @ ( product_prod @ C @ B ) @ ( product_prod @ C @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ C @ A ) @ ( product_prod @ C @ A ) ) ) @ ( set @ ( product_prod @ ( product_prod @ C @ B ) @ ( product_prod @ C @ B ) ) ) @ ( bNF_Cardinal_cprod @ C @ A @ Q3 @ P24 ) @ ( bNF_Cardinal_cprod @ C @ B @ Q3 @ R22 ) ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ C @ A ) @ ( product_prod @ C @ B ) ) ) ) ).
% cprod_mono2
thf(fact_5307_cprod__cong,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,P13: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),P24: set @ ( product_prod @ C @ C ),R22: set @ ( product_prod @ D @ D )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P13 @ R1 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ P24 @ R22 ) @ ( bNF_Wellorder_ordIso @ C @ D ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ D ) @ ( product_prod @ B @ D ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ D ) @ ( product_prod @ B @ D ) ) ) @ ( bNF_Cardinal_cprod @ A @ C @ P13 @ P24 ) @ ( bNF_Cardinal_cprod @ B @ D @ R1 @ R22 ) ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) ) ) ) ).
% cprod_cong
thf(fact_5308_cprod__cong1,axiom,
! [B: $tType,C: $tType,A: $tType,P13: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),P24: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P13 @ R1 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( bNF_Cardinal_cprod @ A @ C @ P13 @ P24 ) @ ( bNF_Cardinal_cprod @ B @ C @ R1 @ P24 ) ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ C ) @ ( product_prod @ B @ C ) ) ) ) ).
% cprod_cong1
thf(fact_5309_cprod__cong2,axiom,
! [B: $tType,A: $tType,C: $tType,P24: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),Q3: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P24 @ R22 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ C @ A ) @ ( product_prod @ C @ A ) ) ) @ ( set @ ( product_prod @ ( product_prod @ C @ B ) @ ( product_prod @ C @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ C @ A ) @ ( product_prod @ C @ A ) ) ) @ ( set @ ( product_prod @ ( product_prod @ C @ B ) @ ( product_prod @ C @ B ) ) ) @ ( bNF_Cardinal_cprod @ C @ A @ Q3 @ P24 ) @ ( bNF_Cardinal_cprod @ C @ B @ Q3 @ R22 ) ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ C @ A ) @ ( product_prod @ C @ B ) ) ) ) ).
% cprod_cong2
thf(fact_5310_cprod__infinite,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( ( bNF_Ca4139267488887388095finite @ A @ R )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Cardinal_cprod @ A @ A @ R @ R ) @ R ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ A ) @ A ) ) ) ).
% cprod_infinite
thf(fact_5311_cprod__cinfinite__bound,axiom,
! [B: $tType,C: $tType,A: $tType,P5: set @ ( product_prod @ A @ A ),R: set @ ( product_prod @ B @ B ),Q3: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P5 @ R ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) @ Q3 @ R ) @ ( bNF_Wellorder_ordLeq @ C @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ P5 ) @ P5 )
=> ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q3 ) @ Q3 )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R ) @ R ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Cardinal_cprod @ A @ C @ P5 @ Q3 ) @ R ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ C ) @ B ) ) ) ) ) ) ) ).
% cprod_cinfinite_bound
thf(fact_5312_card__of__Csum__Times_H,axiom,
! [A: $tType,C: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),I4: set @ B,A4: B > ( set @ C )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ! [X2: B] :
( ( member @ B @ X2 @ I4 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( A4 @ X2 ) ) @ R ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) )
@ ( bNF_Cardinal_Csum @ B @ C @ ( bNF_Ca6860139660246222851ard_of @ B @ I4 )
@ ^ [I3: B] : ( bNF_Ca6860139660246222851ard_of @ C @ ( A4 @ I3 ) ) )
@ ( bNF_Cardinal_cprod @ B @ A @ ( bNF_Ca6860139660246222851ard_of @ B @ I4 ) @ R ) )
@ ( bNF_Wellorder_ordLeq @ ( product_prod @ B @ C ) @ ( product_prod @ B @ A ) ) ) ) ) ).
% card_of_Csum_Times'
thf(fact_5313_card__of__Csum__Times,axiom,
! [C: $tType,B: $tType,A: $tType,I4: set @ A,A4: A > ( set @ B ),B6: set @ C] :
( ! [X2: A] :
( ( member @ A @ X2 @ I4 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( A4 @ X2 ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ B @ C ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) )
@ ( bNF_Cardinal_Csum @ A @ B @ ( bNF_Ca6860139660246222851ard_of @ A @ I4 )
@ ^ [I3: A] : ( bNF_Ca6860139660246222851ard_of @ B @ ( A4 @ I3 ) ) )
@ ( bNF_Cardinal_cprod @ A @ C @ ( bNF_Ca6860139660246222851ard_of @ A @ I4 ) @ ( bNF_Ca6860139660246222851ard_of @ C @ B6 ) ) )
@ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) ) ) ).
% card_of_Csum_Times
thf(fact_5314_cprod__infinite1_H,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),P5: set @ ( product_prod @ B @ B )] :
( ( ( bNF_Ca4139267488887388095finite @ A @ R )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
=> ( ( ~ ( member @ ( 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 @ ( bNF_Cardinal_czero @ B ) ) @ ( bNF_Wellorder_ordIso @ B @ B ) )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ P5 ) @ P5 ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P5 @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Cardinal_cprod @ A @ B @ R @ P5 ) @ R ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ B ) @ A ) ) ) ) ) ).
% cprod_infinite1'
thf(fact_5315_czero__ordIso,axiom,
! [B: $tType,A: $tType] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Cardinal_czero @ A ) @ ( bNF_Cardinal_czero @ B ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ).
% czero_ordIso
thf(fact_5316_cinfinite__not__czero,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca4139267488887388095finite @ B @ R )
=> ~ ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ B @ A ) ) ) ).
% cinfinite_not_czero
thf(fact_5317_czeroE,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ ( bNF_Cardinal_czero @ B ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( field2 @ A @ R )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% czeroE
thf(fact_5318_card__of__ordIso__czero__iff__empty,axiom,
! [B: $tType,A: $tType,A4: set @ A] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Cardinal_czero @ B ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
= ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% card_of_ordIso_czero_iff_empty
thf(fact_5319_czeroI,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( ( field2 @ A @ R )
= ( bot_bot @ ( set @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ ( bNF_Cardinal_czero @ B ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ) ).
% czeroI
thf(fact_5320_Cnotzero__imp__not__empty,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
=> ( ( field2 @ A @ R )
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% Cnotzero_imp_not_empty
thf(fact_5321_Cnotzero__mono,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),Q3: set @ ( product_prod @ B @ B )] :
( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ Q3 ) @ Q3 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ Q3 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ Q3 @ ( bNF_Cardinal_czero @ B ) ) @ ( bNF_Wellorder_ordIso @ B @ B ) )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ Q3 ) @ Q3 ) ) ) ) ) ).
% Cnotzero_mono
thf(fact_5322_Cinfinite__Cnotzero,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( ( bNF_Ca4139267488887388095finite @ A @ R )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
=> ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) ) ) ).
% Cinfinite_Cnotzero
thf(fact_5323_Cnotzero__UNIV,axiom,
! [A: $tType] :
( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( top_top @ ( set @ A ) ) ) @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ ( bNF_Ca6860139660246222851ard_of @ A @ ( top_top @ ( set @ A ) ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% Cnotzero_UNIV
thf(fact_5324_cinfinite__cprod2,axiom,
! [A: $tType,B: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B )] :
( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R1 @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R1 ) @ R1 ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R22 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R22 ) @ R22 ) )
=> ( bNF_Ca4139267488887388095finite @ ( product_prod @ A @ B ) @ ( bNF_Cardinal_cprod @ A @ B @ R1 @ R22 ) ) ) ) ).
% cinfinite_cprod2
thf(fact_5325_Cinfinite__cprod2,axiom,
! [A: $tType,B: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B )] :
( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R1 @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R1 ) @ R1 ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R22 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R22 ) @ R22 ) )
=> ( ( bNF_Ca4139267488887388095finite @ ( product_prod @ A @ B ) @ ( bNF_Cardinal_cprod @ A @ B @ R1 @ R22 ) )
& ( bNF_Ca8970107618336181345der_on @ ( product_prod @ A @ B ) @ ( field2 @ ( product_prod @ A @ B ) @ ( bNF_Cardinal_cprod @ A @ B @ R1 @ R22 ) ) @ ( bNF_Cardinal_cprod @ A @ B @ R1 @ R22 ) ) ) ) ) ).
% Cinfinite_cprod2
thf(fact_5326_ordLeq__cprod2,axiom,
! [A: $tType,B: $tType,P13: set @ ( product_prod @ A @ A ),P24: set @ ( product_prod @ B @ B )] :
( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P13 @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ P13 ) @ P13 ) )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ P24 ) @ P24 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ P24 @ ( bNF_Cardinal_cprod @ A @ B @ P13 @ P24 ) ) @ ( bNF_Wellorder_ordLeq @ B @ ( product_prod @ A @ B ) ) ) ) ) ).
% ordLeq_cprod2
thf(fact_5327_cone__ordLeq__Cnotzero,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ product_unit @ product_unit ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ product_unit @ product_unit ) ) @ ( set @ ( product_prod @ A @ A ) ) @ bNF_Cardinal_cone @ R ) @ ( bNF_Wellorder_ordLeq @ product_unit @ A ) ) ) ).
% cone_ordLeq_Cnotzero
thf(fact_5328_csum__dup,axiom,
! [A: $tType,C: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),P5: set @ ( product_prod @ B @ B ),P8: set @ ( product_prod @ C @ C )] :
( ( bNF_Ca4139267488887388095finite @ A @ R )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ A ) @ ( sum_sum @ A @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ A ) @ ( sum_sum @ A @ A ) ) ) @ ( bNF_Cardinal_csum @ B @ C @ P5 @ P8 ) @ ( bNF_Cardinal_csum @ A @ A @ R @ R ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ B @ C ) @ ( sum_sum @ A @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Cardinal_csum @ B @ C @ P5 @ P8 ) @ R ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ B @ C ) @ A ) ) ) ) ) ).
% csum_dup
thf(fact_5329_csum__Cnotzero1,axiom,
! [A: $tType,B: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B )] :
( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R1 @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R1 ) @ R1 ) )
=> ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( bNF_Cardinal_csum @ A @ B @ R1 @ R22 ) @ ( bNF_Cardinal_czero @ ( sum_sum @ A @ B ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) )
& ( bNF_Ca8970107618336181345der_on @ ( sum_sum @ A @ B ) @ ( field2 @ ( sum_sum @ A @ B ) @ ( bNF_Cardinal_csum @ A @ B @ R1 @ R22 ) ) @ ( bNF_Cardinal_csum @ A @ B @ R1 @ R22 ) ) ) ) ).
% csum_Cnotzero1
thf(fact_5330_csum__csum,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),R33: set @ ( product_prod @ C @ C ),R42: set @ ( product_prod @ D @ D )] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ ( sum_sum @ A @ B ) @ ( sum_sum @ C @ D ) ) @ ( sum_sum @ ( sum_sum @ A @ B ) @ ( sum_sum @ C @ D ) ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) @ ( sum_sum @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ ( sum_sum @ A @ B ) @ ( sum_sum @ C @ D ) ) @ ( sum_sum @ ( sum_sum @ A @ B ) @ ( sum_sum @ C @ D ) ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) @ ( sum_sum @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) @ ( bNF_Cardinal_csum @ ( sum_sum @ A @ B ) @ ( sum_sum @ C @ D ) @ ( bNF_Cardinal_csum @ A @ B @ R1 @ R22 ) @ ( bNF_Cardinal_csum @ C @ D @ R33 @ R42 ) ) @ ( bNF_Cardinal_csum @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) @ ( bNF_Cardinal_csum @ A @ C @ R1 @ R33 ) @ ( bNF_Cardinal_csum @ B @ D @ R22 @ R42 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ ( sum_sum @ A @ B ) @ ( sum_sum @ C @ D ) ) @ ( sum_sum @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ).
% csum_csum
thf(fact_5331_csum__assoc,axiom,
! [C: $tType,B: $tType,A: $tType,P13: set @ ( product_prod @ A @ A ),P24: set @ ( product_prod @ B @ B ),P32: set @ ( product_prod @ C @ C )] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) ) @ ( bNF_Cardinal_csum @ ( sum_sum @ A @ B ) @ C @ ( bNF_Cardinal_csum @ A @ B @ P13 @ P24 ) @ P32 ) @ ( bNF_Cardinal_csum @ A @ ( sum_sum @ B @ C ) @ P13 @ ( bNF_Cardinal_csum @ B @ C @ P24 @ P32 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) ) ).
% csum_assoc
thf(fact_5332_cprod__csum__distrib1,axiom,
! [C: $tType,B: $tType,A: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),R33: set @ ( product_prod @ C @ C )] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) ) ) @ ( bNF_Cardinal_csum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) @ ( bNF_Cardinal_cprod @ A @ B @ R1 @ R22 ) @ ( bNF_Cardinal_cprod @ A @ C @ R1 @ R33 ) ) @ ( bNF_Cardinal_cprod @ A @ ( sum_sum @ B @ C ) @ R1 @ ( bNF_Cardinal_csum @ B @ C @ R22 @ R33 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) ) ) ).
% cprod_csum_distrib1
thf(fact_5333_cone__not__czero,axiom,
! [A: $tType] :
~ ( member @ ( product_prod @ ( set @ ( product_prod @ product_unit @ product_unit ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ product_unit @ product_unit ) ) @ ( set @ ( product_prod @ A @ A ) ) @ bNF_Cardinal_cone @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ product_unit @ A ) ) ).
% cone_not_czero
thf(fact_5334_csum__com,axiom,
! [B: $tType,A: $tType,P13: set @ ( product_prod @ A @ A ),P24: set @ ( product_prod @ B @ B )] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( bNF_Cardinal_csum @ A @ B @ P13 @ P24 ) @ ( bNF_Cardinal_csum @ B @ A @ P24 @ P13 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ ( sum_sum @ B @ A ) ) ) ).
% csum_com
thf(fact_5335_Un__csum,axiom,
! [A: $tType,A4: set @ A,B6: set @ A] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ A ) @ ( sum_sum @ A @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ A ) @ ( sum_sum @ A @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B6 ) ) @ ( bNF_Cardinal_csum @ A @ A @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B6 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ A @ A ) ) ) ).
% Un_csum
thf(fact_5336_ordLeq__csum1,axiom,
! [B: $tType,A: $tType,P13: set @ ( product_prod @ A @ A ),P24: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ P13 ) @ P13 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ P13 @ ( bNF_Cardinal_csum @ A @ B @ P13 @ P24 ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ A @ B ) ) ) ) ).
% ordLeq_csum1
thf(fact_5337_ordLeq__csum2,axiom,
! [B: $tType,A: $tType,P24: set @ ( product_prod @ A @ A ),P13: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ P24 ) @ P24 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ P24 @ ( bNF_Cardinal_csum @ B @ A @ P13 @ P24 ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ B @ A ) ) ) ) ).
% ordLeq_csum2
thf(fact_5338_csum__cong,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,P13: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),P24: set @ ( product_prod @ C @ C ),R22: set @ ( product_prod @ D @ D )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P13 @ R1 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ P24 @ R22 ) @ ( bNF_Wellorder_ordIso @ C @ D ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) @ ( bNF_Cardinal_csum @ A @ C @ P13 @ P24 ) @ ( bNF_Cardinal_csum @ B @ D @ R1 @ R22 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) ).
% csum_cong
thf(fact_5339_csum__cong1,axiom,
! [B: $tType,C: $tType,A: $tType,P13: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),Q3: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P13 @ R1 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( bNF_Cardinal_csum @ A @ C @ P13 @ Q3 ) @ ( bNF_Cardinal_csum @ B @ C @ R1 @ Q3 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ C ) ) ) ) ).
% csum_cong1
thf(fact_5340_csum__cong2,axiom,
! [B: $tType,A: $tType,C: $tType,P24: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),Q3: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P24 @ R22 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) @ ( bNF_Cardinal_csum @ C @ A @ Q3 @ P24 ) @ ( bNF_Cardinal_csum @ C @ B @ Q3 @ R22 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ B ) ) ) ) ).
% csum_cong2
thf(fact_5341_csum__mono,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,P13: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),P24: set @ ( product_prod @ C @ C ),R22: set @ ( product_prod @ D @ D )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P13 @ R1 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ P24 @ R22 ) @ ( bNF_Wellorder_ordLeq @ C @ D ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) @ ( bNF_Cardinal_csum @ A @ C @ P13 @ P24 ) @ ( bNF_Cardinal_csum @ B @ D @ R1 @ R22 ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) ).
% csum_mono
thf(fact_5342_csum__mono1,axiom,
! [B: $tType,C: $tType,A: $tType,P13: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),Q3: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P13 @ R1 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( bNF_Cardinal_csum @ A @ C @ P13 @ Q3 ) @ ( bNF_Cardinal_csum @ B @ C @ R1 @ Q3 ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ C ) ) ) ) ).
% csum_mono1
thf(fact_5343_csum__mono2,axiom,
! [B: $tType,A: $tType,C: $tType,P24: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),Q3: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P24 @ R22 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) @ ( bNF_Cardinal_csum @ C @ A @ Q3 @ P24 ) @ ( bNF_Cardinal_csum @ C @ B @ Q3 @ R22 ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ B ) ) ) ) ).
% csum_mono2
thf(fact_5344_Plus__csum,axiom,
! [B: $tType,A: $tType,A4: set @ A,B6: set @ B] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A4 @ B6 ) ) @ ( bNF_Cardinal_csum @ A @ B @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) ).
% Plus_csum
thf(fact_5345_csum__absorb2_H,axiom,
! [A: $tType,B: $tType,R22: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R22 ) @ R22 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R1 @ R22 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R1 )
| ( bNF_Ca4139267488887388095finite @ A @ R22 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Cardinal_csum @ B @ A @ R1 @ R22 ) @ R22 ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ B @ A ) @ A ) ) ) ) ) ).
% csum_absorb2'
thf(fact_5346_csum__absorb1_H,axiom,
! [B: $tType,A: $tType,R22: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R22 ) @ R22 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R1 @ R22 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R1 )
| ( bNF_Ca4139267488887388095finite @ A @ R22 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Cardinal_csum @ A @ B @ R22 @ R1 ) @ R22 ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ A ) ) ) ) ) ).
% csum_absorb1'
thf(fact_5347_csum__absorb1,axiom,
! [B: $tType,A: $tType,R22: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B )] :
( ( ( bNF_Ca4139267488887388095finite @ A @ R22 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R22 ) @ R22 ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R1 @ R22 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Cardinal_csum @ A @ B @ R22 @ R1 ) @ R22 ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ A ) ) ) ) ).
% csum_absorb1
thf(fact_5348_csum__cinfinite__bound,axiom,
! [B: $tType,C: $tType,A: $tType,P5: set @ ( product_prod @ A @ A ),R: set @ ( product_prod @ B @ B ),Q3: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P5 @ R ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) @ Q3 @ R ) @ ( bNF_Wellorder_ordLeq @ C @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ P5 ) @ P5 )
=> ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q3 ) @ Q3 )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R ) @ R ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Cardinal_csum @ A @ C @ P5 @ Q3 ) @ R ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ B ) ) ) ) ) ) ) ).
% csum_cinfinite_bound
thf(fact_5349_natLeq__ordLeq__cinfinite,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( ( bNF_Ca4139267488887388095finite @ A @ R )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ A @ A ) ) @ bNF_Ca8665028551170535155natLeq @ R ) @ ( bNF_Wellorder_ordLeq @ nat @ A ) ) ) ).
% natLeq_ordLeq_cinfinite
thf(fact_5350_cexp__mono2__Cnotzero,axiom,
! [B: $tType,C: $tType,A: $tType,P24: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),Q3: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P24 @ R22 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q3 ) @ Q3 )
=> ( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P24 @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ P24 ) @ P24 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( A > C ) @ ( A > C ) ) ) @ ( set @ ( product_prod @ ( B > C ) @ ( B > C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( A > C ) @ ( A > C ) ) ) @ ( set @ ( product_prod @ ( B > C ) @ ( B > C ) ) ) @ ( bNF_Cardinal_cexp @ C @ A @ Q3 @ P24 ) @ ( bNF_Cardinal_cexp @ C @ B @ Q3 @ R22 ) ) @ ( bNF_Wellorder_ordLeq @ ( A > C ) @ ( B > C ) ) ) ) ) ) ).
% cexp_mono2_Cnotzero
thf(fact_5351_cexp__mono2,axiom,
! [D: $tType,E: $tType,B: $tType,C: $tType,A: $tType,P24: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),Q3: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P24 @ R22 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q3 ) @ Q3 )
=> ( ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ D @ D ) ) @ P24 @ ( bNF_Cardinal_czero @ D ) ) @ ( bNF_Wellorder_ordIso @ A @ D ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ E @ E ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ E @ E ) ) @ R22 @ ( bNF_Cardinal_czero @ E ) ) @ ( bNF_Wellorder_ordIso @ B @ E ) ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ P24 ) @ P24 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( A > C ) @ ( A > C ) ) ) @ ( set @ ( product_prod @ ( B > C ) @ ( B > C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( A > C ) @ ( A > C ) ) ) @ ( set @ ( product_prod @ ( B > C ) @ ( B > C ) ) ) @ ( bNF_Cardinal_cexp @ C @ A @ Q3 @ P24 ) @ ( bNF_Cardinal_cexp @ C @ B @ Q3 @ R22 ) ) @ ( bNF_Wellorder_ordLeq @ ( A > C ) @ ( B > C ) ) ) ) ) ) ) ).
% cexp_mono2
thf(fact_5352_cprod__cexp,axiom,
! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ B @ B ),S: set @ ( product_prod @ C @ C ),T2: set @ ( product_prod @ A @ A )] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( A > ( product_prod @ B @ C ) ) @ ( A > ( product_prod @ B @ C ) ) ) ) @ ( set @ ( product_prod @ ( product_prod @ ( A > B ) @ ( A > C ) ) @ ( product_prod @ ( A > B ) @ ( A > C ) ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( A > ( product_prod @ B @ C ) ) @ ( A > ( product_prod @ B @ C ) ) ) ) @ ( set @ ( product_prod @ ( product_prod @ ( A > B ) @ ( A > C ) ) @ ( product_prod @ ( A > B ) @ ( A > C ) ) ) ) @ ( bNF_Cardinal_cexp @ ( product_prod @ B @ C ) @ A @ ( bNF_Cardinal_cprod @ B @ C @ R @ S ) @ T2 ) @ ( bNF_Cardinal_cprod @ ( A > B ) @ ( A > C ) @ ( bNF_Cardinal_cexp @ B @ A @ R @ T2 ) @ ( bNF_Cardinal_cexp @ C @ A @ S @ T2 ) ) ) @ ( bNF_Wellorder_ordIso @ ( A > ( product_prod @ B @ C ) ) @ ( product_prod @ ( A > B ) @ ( A > C ) ) ) ) ).
% cprod_cexp
thf(fact_5353_cexp__cprod,axiom,
! [A: $tType,C: $tType,B: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ C @ C ),R33: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R1 ) @ R1 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( B > C > A ) @ ( B > C > A ) ) ) @ ( set @ ( product_prod @ ( ( product_prod @ C @ B ) > A ) @ ( ( product_prod @ C @ B ) > A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( B > C > A ) @ ( B > C > A ) ) ) @ ( set @ ( product_prod @ ( ( product_prod @ C @ B ) > A ) @ ( ( product_prod @ C @ B ) > A ) ) ) @ ( bNF_Cardinal_cexp @ ( C > A ) @ B @ ( bNF_Cardinal_cexp @ A @ C @ R1 @ R22 ) @ R33 ) @ ( bNF_Cardinal_cexp @ A @ ( product_prod @ C @ B ) @ R1 @ ( bNF_Cardinal_cprod @ C @ B @ R22 @ R33 ) ) ) @ ( bNF_Wellorder_ordIso @ ( B > C > A ) @ ( ( product_prod @ C @ B ) > A ) ) ) ) ).
% cexp_cprod
thf(fact_5354_cprod__cexp__csum__cexp__Cinfinite,axiom,
! [C: $tType,B: $tType,A: $tType,T2: set @ ( product_prod @ A @ A ),R: set @ ( product_prod @ B @ B ),S: set @ ( product_prod @ C @ C )] :
( ( ( bNF_Ca4139267488887388095finite @ A @ T2 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ T2 ) @ T2 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( A > ( product_prod @ B @ C ) ) @ ( A > ( product_prod @ B @ C ) ) ) ) @ ( set @ ( product_prod @ ( A > ( sum_sum @ B @ C ) ) @ ( A > ( sum_sum @ B @ C ) ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( A > ( product_prod @ B @ C ) ) @ ( A > ( product_prod @ B @ C ) ) ) ) @ ( set @ ( product_prod @ ( A > ( sum_sum @ B @ C ) ) @ ( A > ( sum_sum @ B @ C ) ) ) ) @ ( bNF_Cardinal_cexp @ ( product_prod @ B @ C ) @ A @ ( bNF_Cardinal_cprod @ B @ C @ R @ S ) @ T2 ) @ ( bNF_Cardinal_cexp @ ( sum_sum @ B @ C ) @ A @ ( bNF_Cardinal_csum @ B @ C @ R @ S ) @ T2 ) ) @ ( bNF_Wellorder_ordLeq @ ( A > ( product_prod @ B @ C ) ) @ ( A > ( sum_sum @ B @ C ) ) ) ) ) ).
% cprod_cexp_csum_cexp_Cinfinite
thf(fact_5355_card__of__Field__natLeq,axiom,
member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( bNF_Ca6860139660246222851ard_of @ nat @ ( field2 @ nat @ bNF_Ca8665028551170535155natLeq ) ) @ bNF_Ca8665028551170535155natLeq ) @ ( bNF_Wellorder_ordIso @ nat @ nat ) ).
% card_of_Field_natLeq
thf(fact_5356_cexp__cone,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_unit > A ) @ ( product_unit > A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_unit > A ) @ ( product_unit > A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Cardinal_cexp @ A @ product_unit @ R @ bNF_Cardinal_cone ) @ R ) @ ( bNF_Wellorder_ordIso @ ( product_unit > A ) @ A ) ) ) ).
% cexp_cone
thf(fact_5357_finite__iff__ordLess__natLeq,axiom,
! [A: $tType] :
( ( finite_finite2 @ A )
= ( ^ [A9: set @ A] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A9 ) @ bNF_Ca8665028551170535155natLeq ) @ ( bNF_We4044943003108391690rdLess @ A @ nat ) ) ) ) ).
% finite_iff_ordLess_natLeq
thf(fact_5358_cexp__cprod__ordLeq,axiom,
! [A: $tType,B: $tType,C: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),R33: set @ ( product_prod @ C @ C )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R1 ) @ R1 )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R22 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R22 ) @ R22 ) )
=> ( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R33 @ ( bNF_Cardinal_czero @ C ) ) @ ( bNF_Wellorder_ordIso @ C @ C ) )
& ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ R33 ) @ R33 ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R33 @ R22 ) @ ( bNF_Wellorder_ordLeq @ C @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( C > B > A ) @ ( C > B > A ) ) ) @ ( set @ ( product_prod @ ( B > A ) @ ( B > A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( C > B > A ) @ ( C > B > A ) ) ) @ ( set @ ( product_prod @ ( B > A ) @ ( B > A ) ) ) @ ( bNF_Cardinal_cexp @ ( B > A ) @ C @ ( bNF_Cardinal_cexp @ A @ B @ R1 @ R22 ) @ R33 ) @ ( bNF_Cardinal_cexp @ A @ B @ R1 @ R22 ) ) @ ( bNF_Wellorder_ordIso @ ( C > B > A ) @ ( B > A ) ) ) ) ) ) ) ).
% cexp_cprod_ordLeq
thf(fact_5359_cexp__mono_H,axiom,
! [B: $tType,D: $tType,A: $tType,C: $tType,P13: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),P24: set @ ( product_prod @ C @ C ),R22: set @ ( product_prod @ D @ D )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P13 @ R1 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ P24 @ R22 ) @ ( bNF_Wellorder_ordLeq @ C @ D ) )
=> ( ( ( ( field2 @ C @ P24 )
= ( bot_bot @ ( set @ C ) ) )
=> ( ( field2 @ D @ R22 )
= ( bot_bot @ ( set @ D ) ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( D > B ) @ ( D > B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( D > B ) @ ( D > B ) ) ) @ ( bNF_Cardinal_cexp @ A @ C @ P13 @ P24 ) @ ( bNF_Cardinal_cexp @ B @ D @ R1 @ R22 ) ) @ ( bNF_Wellorder_ordLeq @ ( C > A ) @ ( D > B ) ) ) ) ) ) ).
% cexp_mono'
thf(fact_5360_cexp__mono1,axiom,
! [B: $tType,A: $tType,C: $tType,P13: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),Q3: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P13 @ R1 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q3 ) @ Q3 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( C > B ) @ ( C > B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( C > B ) @ ( C > B ) ) ) @ ( bNF_Cardinal_cexp @ A @ C @ P13 @ Q3 ) @ ( bNF_Cardinal_cexp @ B @ C @ R1 @ Q3 ) ) @ ( bNF_Wellorder_ordLeq @ ( C > A ) @ ( C > B ) ) ) ) ) ).
% cexp_mono1
thf(fact_5361_cexp__cong2,axiom,
! [B: $tType,C: $tType,A: $tType,P24: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),Q3: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P24 @ R22 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q3 ) @ Q3 )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ P24 ) @ P24 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( A > C ) @ ( A > C ) ) ) @ ( set @ ( product_prod @ ( B > C ) @ ( B > C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( A > C ) @ ( A > C ) ) ) @ ( set @ ( product_prod @ ( B > C ) @ ( B > C ) ) ) @ ( bNF_Cardinal_cexp @ C @ A @ Q3 @ P24 ) @ ( bNF_Cardinal_cexp @ C @ B @ Q3 @ R22 ) ) @ ( bNF_Wellorder_ordIso @ ( A > C ) @ ( B > C ) ) ) ) ) ) ).
% cexp_cong2
thf(fact_5362_cexp__cong1,axiom,
! [B: $tType,A: $tType,C: $tType,P13: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),Q3: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P13 @ R1 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q3 ) @ Q3 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( C > B ) @ ( C > B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( C > B ) @ ( C > B ) ) ) @ ( bNF_Cardinal_cexp @ A @ C @ P13 @ Q3 ) @ ( bNF_Cardinal_cexp @ B @ C @ R1 @ Q3 ) ) @ ( bNF_Wellorder_ordIso @ ( C > A ) @ ( C > B ) ) ) ) ) ).
% cexp_cong1
thf(fact_5363_cexp__cong,axiom,
! [B: $tType,D: $tType,A: $tType,C: $tType,P13: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),P24: set @ ( product_prod @ C @ C ),R22: set @ ( product_prod @ D @ D )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P13 @ R1 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ P24 @ R22 ) @ ( bNF_Wellorder_ordIso @ C @ D ) )
=> ( ( bNF_Ca8970107618336181345der_on @ D @ ( field2 @ D @ R22 ) @ R22 )
=> ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ P24 ) @ P24 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( D > B ) @ ( D > B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( D > B ) @ ( D > B ) ) ) @ ( bNF_Cardinal_cexp @ A @ C @ P13 @ P24 ) @ ( bNF_Cardinal_cexp @ B @ D @ R1 @ R22 ) ) @ ( bNF_Wellorder_ordIso @ ( C > A ) @ ( D > B ) ) ) ) ) ) ) ).
% cexp_cong
thf(fact_5364_card__of__nat,axiom,
member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( bNF_Ca6860139660246222851ard_of @ nat @ ( top_top @ ( set @ nat ) ) ) @ bNF_Ca8665028551170535155natLeq ) @ ( bNF_Wellorder_ordIso @ nat @ nat ) ).
% card_of_nat
thf(fact_5365_cexp__mono2_H,axiom,
! [B: $tType,C: $tType,A: $tType,P24: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),Q3: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P24 @ R22 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q3 ) @ Q3 )
=> ( ( ( ( field2 @ A @ P24 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( field2 @ B @ R22 )
= ( bot_bot @ ( set @ B ) ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( A > C ) @ ( A > C ) ) ) @ ( set @ ( product_prod @ ( B > C ) @ ( B > C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( A > C ) @ ( A > C ) ) ) @ ( set @ ( product_prod @ ( B > C ) @ ( B > C ) ) ) @ ( bNF_Cardinal_cexp @ C @ A @ Q3 @ P24 ) @ ( bNF_Cardinal_cexp @ C @ B @ Q3 @ R22 ) ) @ ( bNF_Wellorder_ordLeq @ ( A > C ) @ ( B > C ) ) ) ) ) ) ).
% cexp_mono2'
thf(fact_5366_ordLeq__cexp1,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),Q3: set @ ( product_prod @ B @ B )] :
( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ Q3 ) @ Q3 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) @ Q3 @ ( bNF_Cardinal_cexp @ B @ A @ Q3 @ R ) ) @ ( bNF_Wellorder_ordLeq @ B @ ( A > B ) ) ) ) ) ).
% ordLeq_cexp1
thf(fact_5367_infinite__iff__natLeq__ordLeq,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite2 @ A @ A4 )
!= ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ A @ A ) ) @ bNF_Ca8665028551170535155natLeq @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ nat @ A ) ) ) ).
% infinite_iff_natLeq_ordLeq
thf(fact_5368_cexp__mono,axiom,
! [E: $tType,F5: $tType,B: $tType,D: $tType,A: $tType,C: $tType,P13: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),P24: set @ ( product_prod @ C @ C ),R22: set @ ( product_prod @ D @ D )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P13 @ R1 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ P24 @ R22 ) @ ( bNF_Wellorder_ordLeq @ C @ D ) )
=> ( ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ E @ E ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ E @ E ) ) @ P24 @ ( bNF_Cardinal_czero @ E ) ) @ ( bNF_Wellorder_ordIso @ C @ E ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ D @ D ) ) @ ( set @ ( product_prod @ F5 @ F5 ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ D @ D ) ) @ ( set @ ( product_prod @ F5 @ F5 ) ) @ R22 @ ( bNF_Cardinal_czero @ F5 ) ) @ ( bNF_Wellorder_ordIso @ D @ F5 ) ) )
=> ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ P24 ) @ P24 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( D > B ) @ ( D > B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( D > B ) @ ( D > B ) ) ) @ ( bNF_Cardinal_cexp @ A @ C @ P13 @ P24 ) @ ( bNF_Cardinal_cexp @ B @ D @ R1 @ R22 ) ) @ ( bNF_Wellorder_ordLeq @ ( C > A ) @ ( D > B ) ) ) ) ) ) ) ).
% cexp_mono
thf(fact_5369_ordLeq__cexp2,axiom,
! [A: $tType,B: $tType,Q3: set @ ( product_prod @ A @ A ),R: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) @ bNF_Cardinal_ctwo @ Q3 ) @ ( bNF_Wellorder_ordLeq @ $o @ A ) )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R ) @ R )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( B > A ) @ ( B > A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( B > A ) @ ( B > A ) ) ) @ R @ ( bNF_Cardinal_cexp @ A @ B @ Q3 @ R ) ) @ ( bNF_Wellorder_ordLeq @ B @ ( B > A ) ) ) ) ) ).
% ordLeq_cexp2
thf(fact_5370_Cfinite__cexp__Cinfinite,axiom,
! [A: $tType,B: $tType,S: set @ ( product_prod @ A @ A ),T2: set @ ( product_prod @ B @ B )] :
( ( ( bNF_Cardinal_cfinite @ A @ S )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ S ) @ S ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ T2 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ T2 ) @ T2 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( B > A ) @ ( B > A ) ) ) @ ( set @ ( product_prod @ ( B > $o ) @ ( B > $o ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( B > A ) @ ( B > A ) ) ) @ ( set @ ( product_prod @ ( B > $o ) @ ( B > $o ) ) ) @ ( bNF_Cardinal_cexp @ A @ B @ S @ T2 ) @ ( bNF_Cardinal_cexp @ $o @ B @ bNF_Cardinal_ctwo @ T2 ) ) @ ( bNF_Wellorder_ordLeq @ ( B > A ) @ ( B > $o ) ) ) ) ) ).
% Cfinite_cexp_Cinfinite
thf(fact_5371_cinfinite__cexp,axiom,
! [A: $tType,B: $tType,Q3: set @ ( product_prod @ A @ A ),R: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) @ bNF_Cardinal_ctwo @ Q3 ) @ ( bNF_Wellorder_ordLeq @ $o @ A ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R ) @ R ) )
=> ( bNF_Ca4139267488887388095finite @ ( B > A ) @ ( bNF_Cardinal_cexp @ A @ B @ Q3 @ R ) ) ) ) ).
% cinfinite_cexp
thf(fact_5372_ctwo__Cnotzero,axiom,
( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ $o @ $o ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ $o @ $o ) ) @ bNF_Cardinal_ctwo @ ( bNF_Cardinal_czero @ $o ) ) @ ( bNF_Wellorder_ordIso @ $o @ $o ) )
& ( bNF_Ca8970107618336181345der_on @ $o @ ( field2 @ $o @ bNF_Cardinal_ctwo ) @ bNF_Cardinal_ctwo ) ) ).
% ctwo_Cnotzero
thf(fact_5373_ctwo__ordLess__natLeq,axiom,
member @ ( product_prod @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ bNF_Cardinal_ctwo @ bNF_Ca8665028551170535155natLeq ) @ ( bNF_We4044943003108391690rdLess @ $o @ nat ) ).
% ctwo_ordLess_natLeq
thf(fact_5374_ordLess__ctwo__cexp,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( A > $o ) @ ( A > $o ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( A > $o ) @ ( A > $o ) ) ) @ R @ ( bNF_Cardinal_cexp @ $o @ A @ bNF_Cardinal_ctwo @ R ) ) @ ( bNF_We4044943003108391690rdLess @ A @ ( A > $o ) ) ) ) ).
% ordLess_ctwo_cexp
thf(fact_5375_cprod__csum__cexp,axiom,
! [B: $tType,A: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B )] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( $o > ( sum_sum @ A @ B ) ) @ ( $o > ( sum_sum @ A @ B ) ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( $o > ( sum_sum @ A @ B ) ) @ ( $o > ( sum_sum @ A @ B ) ) ) ) @ ( bNF_Cardinal_cprod @ A @ B @ R1 @ R22 ) @ ( bNF_Cardinal_cexp @ ( sum_sum @ A @ B ) @ $o @ ( bNF_Cardinal_csum @ A @ B @ R1 @ R22 ) @ bNF_Cardinal_ctwo ) ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ B ) @ ( $o > ( sum_sum @ A @ B ) ) ) ) ).
% cprod_csum_cexp
thf(fact_5376_ctwo__not__czero,axiom,
! [A: $tType] :
~ ( member @ ( product_prod @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) @ bNF_Cardinal_ctwo @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ $o @ A ) ) ).
% ctwo_not_czero
thf(fact_5377_ctwo__ordLeq__Cinfinite,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( ( bNF_Ca4139267488887388095finite @ A @ R )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) @ bNF_Cardinal_ctwo @ R ) @ ( bNF_Wellorder_ordLeq @ $o @ A ) ) ) ).
% ctwo_ordLeq_Cinfinite
thf(fact_5378_ctwo__ordLess__Cinfinite,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( ( bNF_Ca4139267488887388095finite @ A @ R )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) @ bNF_Cardinal_ctwo @ R ) @ ( bNF_We4044943003108391690rdLess @ $o @ A ) ) ) ).
% ctwo_ordLess_Cinfinite
thf(fact_5379_csum__Cfinite__cexp__Cinfinite,axiom,
! [B: $tType,A: $tType,C: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ B @ B ),T2: set @ ( product_prod @ C @ C )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( ( bNF_Cardinal_cfinite @ B @ S )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ S ) @ S ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ C @ T2 )
& ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ T2 ) @ T2 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( C > ( sum_sum @ A @ B ) ) @ ( C > ( sum_sum @ A @ B ) ) ) ) @ ( set @ ( product_prod @ ( C > ( sum_sum @ A @ $o ) ) @ ( C > ( sum_sum @ A @ $o ) ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( C > ( sum_sum @ A @ B ) ) @ ( C > ( sum_sum @ A @ B ) ) ) ) @ ( set @ ( product_prod @ ( C > ( sum_sum @ A @ $o ) ) @ ( C > ( sum_sum @ A @ $o ) ) ) ) @ ( bNF_Cardinal_cexp @ ( sum_sum @ A @ B ) @ C @ ( bNF_Cardinal_csum @ A @ B @ R @ S ) @ T2 ) @ ( bNF_Cardinal_cexp @ ( sum_sum @ A @ $o ) @ C @ ( bNF_Cardinal_csum @ A @ $o @ R @ bNF_Cardinal_ctwo ) @ T2 ) ) @ ( bNF_Wellorder_ordLeq @ ( C > ( sum_sum @ A @ B ) ) @ ( C > ( sum_sum @ A @ $o ) ) ) ) ) ) ) ).
% csum_Cfinite_cexp_Cinfinite
thf(fact_5380_Cinfinite__cexp,axiom,
! [A: $tType,B: $tType,Q3: set @ ( product_prod @ A @ A ),R: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) @ bNF_Cardinal_ctwo @ Q3 ) @ ( bNF_Wellorder_ordLeq @ $o @ A ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R ) @ R ) )
=> ( ( bNF_Ca4139267488887388095finite @ ( B > A ) @ ( bNF_Cardinal_cexp @ A @ B @ Q3 @ R ) )
& ( bNF_Ca8970107618336181345der_on @ ( B > A ) @ ( field2 @ ( B > A ) @ ( bNF_Cardinal_cexp @ A @ B @ Q3 @ R ) ) @ ( bNF_Cardinal_cexp @ A @ B @ Q3 @ R ) ) ) ) ) ).
% Cinfinite_cexp
thf(fact_5381_card__of__Func__UNIV,axiom,
! [B: $tType,A: $tType,B6: set @ B] :
( member @ ( product_prod @ ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) @ ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) @ ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( A > B ) @ ( bNF_Wellorder_Func @ A @ B @ ( top_top @ ( set @ A ) ) @ B6 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( A > B )
@ ( collect @ ( A > B )
@ ^ [F2: A > B] : ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) ) @ B6 ) ) ) )
@ ( bNF_Wellorder_ordIso @ ( A > B ) @ ( A > B ) ) ) ).
% card_of_Func_UNIV
thf(fact_5382_Card__order__Pow,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ R @ ( bNF_Ca6860139660246222851ard_of @ ( set @ A ) @ ( pow @ A @ ( field2 @ A @ R ) ) ) ) @ ( bNF_We4044943003108391690rdLess @ A @ ( set @ A ) ) ) ) ).
% Card_order_Pow
thf(fact_5383_Func__Times__Range,axiom,
! [C: $tType,B: $tType,A: $tType,A4: set @ A,B6: set @ B,C5: set @ C] :
( member @ ( product_prod @ ( set @ ( product_prod @ ( A > ( product_prod @ B @ C ) ) @ ( A > ( product_prod @ B @ C ) ) ) ) @ ( set @ ( product_prod @ ( product_prod @ ( A > B ) @ ( A > C ) ) @ ( product_prod @ ( A > B ) @ ( A > C ) ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( A > ( product_prod @ B @ C ) ) @ ( A > ( product_prod @ B @ C ) ) ) ) @ ( set @ ( product_prod @ ( product_prod @ ( A > B ) @ ( A > C ) ) @ ( product_prod @ ( A > B ) @ ( A > C ) ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( A > ( product_prod @ B @ C ) )
@ ( bNF_Wellorder_Func @ A @ ( product_prod @ B @ C ) @ A4
@ ( product_Sigma @ B @ C @ B6
@ ^ [Uu: B] : C5 ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ ( A > B ) @ ( A > C ) )
@ ( product_Sigma @ ( A > B ) @ ( A > C ) @ ( bNF_Wellorder_Func @ A @ B @ A4 @ B6 )
@ ^ [Uu: A > B] : ( bNF_Wellorder_Func @ A @ C @ A4 @ C5 ) ) ) )
@ ( bNF_Wellorder_ordIso @ ( A > ( product_prod @ B @ C ) ) @ ( product_prod @ ( A > B ) @ ( A > C ) ) ) ) ).
% Func_Times_Range
thf(fact_5384_card__of__Func__Times,axiom,
! [C: $tType,B: $tType,A: $tType,A4: set @ A,B6: set @ B,C5: set @ C] :
( member @ ( product_prod @ ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > C ) @ ( ( product_prod @ A @ B ) > C ) ) ) @ ( set @ ( product_prod @ ( A > B > C ) @ ( A > B > C ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > C ) @ ( ( product_prod @ A @ B ) > C ) ) ) @ ( set @ ( product_prod @ ( A > B > C ) @ ( A > B > C ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( ( product_prod @ A @ B ) > C )
@ ( bNF_Wellorder_Func @ ( product_prod @ A @ B ) @ C
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu: A] : B6 )
@ C5 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( A > B > C ) @ ( bNF_Wellorder_Func @ A @ ( B > C ) @ A4 @ ( bNF_Wellorder_Func @ B @ C @ B6 @ C5 ) ) ) )
@ ( bNF_Wellorder_ordIso @ ( ( product_prod @ A @ B ) > C ) @ ( A > B > C ) ) ) ).
% card_of_Func_Times
thf(fact_5385_card__of__Pow__Func,axiom,
! [A: $tType,A4: set @ A] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( A > $o ) @ ( A > $o ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( A > $o ) @ ( A > $o ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( set @ A ) @ ( pow @ A @ A4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( A > $o ) @ ( bNF_Wellorder_Func @ A @ $o @ A4 @ ( top_top @ ( set @ $o ) ) ) ) ) @ ( bNF_Wellorder_ordIso @ ( set @ A ) @ ( A > $o ) ) ) ).
% card_of_Pow_Func
thf(fact_5386_Cantors__paradox,axiom,
! [A: $tType,A4: set @ A] :
~ ? [F8: A > ( set @ A )] :
( ( image2 @ A @ ( set @ A ) @ F8 @ A4 )
= ( pow @ A @ A4 ) ) ).
% Cantors_paradox
thf(fact_5387_inj__on__image__Pow,axiom,
! [B: $tType,A: $tType,F: A > B,A4: set @ A] :
( ( inj_on @ A @ B @ F @ A4 )
=> ( inj_on @ ( set @ A ) @ ( set @ B ) @ ( image2 @ A @ B @ F ) @ ( pow @ A @ A4 ) ) ) ).
% inj_on_image_Pow
thf(fact_5388_card__of__Pow,axiom,
! [A: $tType,A4: set @ A] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ ( set @ A ) @ ( pow @ A @ A4 ) ) ) @ ( bNF_We4044943003108391690rdLess @ A @ ( set @ A ) ) ) ).
% card_of_Pow
thf(fact_5389_lfp__induct2,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,F: ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ A @ B ) ),P: A > B > $o] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( complete_lattice_lfp @ ( set @ ( product_prod @ A @ B ) ) @ F ) )
=> ( ( order_mono @ ( set @ ( product_prod @ A @ B ) ) @ ( set @ ( product_prod @ A @ B ) ) @ F )
=> ( ! [A5: A,B3: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B3 ) @ ( F @ ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ ( complete_lattice_lfp @ ( set @ ( product_prod @ A @ B ) ) @ F ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P ) ) ) ) )
=> ( P @ A5 @ B3 ) )
=> ( P @ A3 @ B2 ) ) ) ) ).
% lfp_induct2
thf(fact_5390_sort__quicksort__by__rel,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( linorder_sort_key @ A @ A
@ ^ [X3: A] : X3 )
= ( quicksort_by_rel @ A @ ( ord_less_eq @ A ) @ ( nil @ A ) ) ) ) ).
% sort_quicksort_by_rel
thf(fact_5391_Gr__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Gr @ A @ B )
= ( ^ [A9: set @ A,F2: A > B] :
( collect @ ( product_prod @ A @ B )
@ ^ [Uu: product_prod @ A @ B] :
? [A7: A] :
( ( Uu
= ( product_Pair @ A @ B @ A7 @ ( F2 @ A7 ) ) )
& ( member @ A @ A7 @ A9 ) ) ) ) ) ).
% Gr_def
thf(fact_5392_GrD2,axiom,
! [A: $tType,B: $tType,X: A,Fx: B,A4: set @ A,F: A > B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Fx ) @ ( bNF_Gr @ A @ B @ A4 @ F ) )
=> ( ( F @ X )
= Fx ) ) ).
% GrD2
thf(fact_5393_GrD1,axiom,
! [B: $tType,A: $tType,X: A,Fx: B,A4: set @ A,F: A > B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Fx ) @ ( bNF_Gr @ A @ B @ A4 @ F ) )
=> ( member @ A @ X @ A4 ) ) ).
% GrD1
thf(fact_5394_sort__mergesort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( linorder_sort_key @ A @ A
@ ^ [X3: A] : X3 )
= ( mergesort @ A ) ) ) ).
% sort_mergesort
thf(fact_5395_sort__mergesort__by__rel,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( linorder_sort_key @ A @ A
@ ^ [X3: A] : X3 )
= ( mergesort_by_rel @ A @ ( ord_less_eq @ A ) ) ) ) ).
% sort_mergesort_by_rel
thf(fact_5396_remdup__sort__mergesort__remdups,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( comp @ ( list @ A ) @ ( list @ A ) @ ( list @ A ) @ ( remdups @ A )
@ ( linorder_sort_key @ A @ A
@ ^ [X3: A] : X3 ) )
= ( mergesort_remdups @ A ) ) ) ).
% remdup_sort_mergesort_remdups
thf(fact_5397_prod_H__def,axiom,
! [C: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ( ( groups1962203154675924110t_prod @ C @ A )
= ( groups_comm_monoid_G @ A @ C @ ( times_times @ A ) @ ( one_one @ A ) ) ) ) ).
% prod'_def
thf(fact_5398_prod_Ocomm__monoid__list__set__axioms,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ( groups4802862169904069756st_set @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ).
% prod.comm_monoid_list_set_axioms
thf(fact_5399_length__remdups__card,axiom,
! [A: $tType,L: list @ A] :
( ( size_size @ ( list @ A ) @ ( remdups @ A @ L ) )
= ( finite_card @ A @ ( set2 @ A @ L ) ) ) ).
% length_remdups_card
thf(fact_5400_sum_H__def,axiom,
! [C: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ( ( groups1027152243600224163dd_sum @ C @ A )
= ( groups_comm_monoid_G @ A @ C @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ) ).
% sum'_def
thf(fact_5401_sum_Ocomm__monoid__list__set__axioms,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ( groups4802862169904069756st_set @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ).
% sum.comm_monoid_list_set_axioms
thf(fact_5402_and_Ocomm__monoid__axioms,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( comm_monoid @ A @ ( bit_se5824344872417868541ns_and @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% and.comm_monoid_axioms
thf(fact_5403_bijective__alt,axiom,
! [B: $tType,A: $tType] :
( ( bijective @ A @ B )
= ( ^ [R6: set @ ( product_prod @ A @ B )] :
( ( single_valued @ A @ B @ R6 )
& ( single_valued @ B @ A @ ( converse @ A @ B @ R6 ) ) ) ) ) ).
% bijective_alt
thf(fact_5404_Heap_Osize__gen__o__map,axiom,
! [B: $tType,A: $tType,F: B > nat,G: A > B] :
( ( comp @ ( heap_Time_Heap @ B ) @ nat @ ( heap_Time_Heap @ A ) @ ( heap_Time_size_Heap @ B @ F ) @ ( heap_Time_map_Heap @ A @ B @ G ) )
= ( heap_Time_size_Heap @ A @ ( comp @ B @ nat @ A @ F @ G ) ) ) ).
% Heap.size_gen_o_map
thf(fact_5405_single__valued__inter1,axiom,
! [B: $tType,A: $tType,R5: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
( ( single_valued @ A @ B @ R5 )
=> ( single_valued @ A @ B @ ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ R5 @ S2 ) ) ) ).
% single_valued_inter1
thf(fact_5406_single__valued__inter2,axiom,
! [B: $tType,A: $tType,R5: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
( ( single_valued @ A @ B @ R5 )
=> ( single_valued @ A @ B @ ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ S2 @ R5 ) ) ) ).
% single_valued_inter2
thf(fact_5407_comm__monoid_Ocomm__neutral,axiom,
! [A: $tType,F: A > A > A,Z2: A,A3: A] :
( ( comm_monoid @ A @ F @ Z2 )
=> ( ( F @ A3 @ Z2 )
= A3 ) ) ).
% comm_monoid.comm_neutral
thf(fact_5408_Heap_Omap__ident,axiom,
! [A: $tType,T2: heap_Time_Heap @ A] :
( ( heap_Time_map_Heap @ A @ A
@ ^ [X3: A] : X3
@ T2 )
= T2 ) ).
% Heap.map_ident
thf(fact_5409_Heap_Omap__id0,axiom,
! [A: $tType] :
( ( heap_Time_map_Heap @ A @ A @ ( id @ A ) )
= ( id @ ( heap_Time_Heap @ A ) ) ) ).
% Heap.map_id0
thf(fact_5410_Heap_Omap__id,axiom,
! [A: $tType,T2: heap_Time_Heap @ A] :
( ( heap_Time_map_Heap @ A @ A @ ( id @ A ) @ T2 )
= T2 ) ).
% Heap.map_id
thf(fact_5411_Heap_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,G: B > C,F: A > B,V: heap_Time_Heap @ A] :
( ( heap_Time_map_Heap @ B @ C @ G @ ( heap_Time_map_Heap @ A @ B @ F @ V ) )
= ( heap_Time_map_Heap @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ V ) ) ).
% Heap.map_comp
thf(fact_5412_Heap_Omap,axiom,
! [B: $tType,A: $tType,F: A > B,X: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
( ( heap_Time_map_Heap @ A @ B @ F @ ( heap_Time_Heap2 @ A @ X ) )
= ( heap_Time_Heap2 @ B @ ( comp @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( heap_ext @ product_unit ) @ ( map_option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_map_prod @ A @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ F @ ( id @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ X ) ) ) ).
% Heap.map
thf(fact_5413_single__valuedD,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B ),X: A,Y: B,Z2: B] :
( ( single_valued @ A @ B @ R )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ R )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Z2 ) @ R )
=> ( Y = Z2 ) ) ) ) ).
% single_valuedD
thf(fact_5414_single__valuedI,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B )] :
( ! [X2: A,Y4: B,Z3: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y4 ) @ R )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Z3 ) @ R )
=> ( Y4 = Z3 ) ) )
=> ( single_valued @ A @ B @ R ) ) ).
% single_valuedI
thf(fact_5415_single__valued__def,axiom,
! [B: $tType,A: $tType] :
( ( single_valued @ A @ B )
= ( ^ [R3: set @ ( product_prod @ A @ B )] :
! [X3: A,Y3: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R3 )
=> ! [Z5: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Z5 ) @ R3 )
=> ( Y3 = Z5 ) ) ) ) ) ).
% single_valued_def
thf(fact_5416_add_Ocomm__monoid__axioms,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ( comm_monoid @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ).
% add.comm_monoid_axioms
thf(fact_5417_mult_Ocomm__monoid__axioms,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ( comm_monoid @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ).
% mult.comm_monoid_axioms
thf(fact_5418_Heap_Oinj__map,axiom,
! [B: $tType,A: $tType,F: A > B] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( inj_on @ ( 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
thf(fact_5419_single__valued__confluent,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),X: A,Y: A,Z2: A] :
( ( single_valued @ A @ A @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ ( transitive_rtrancl @ A @ R ) )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z2 @ Y ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ) ) ).
% single_valued_confluent
thf(fact_5420_single__valued__below__Id,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R5 @ ( id2 @ A ) )
=> ( single_valued @ A @ A @ R5 ) ) ).
% single_valued_below_Id
thf(fact_5421_Heap_Orec__o__map,axiom,
! [B: $tType,C: $tType,A: $tType,G: ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > C,F: A > B] :
( ( comp @ ( heap_Time_Heap @ B ) @ C @ ( heap_Time_Heap @ A ) @ ( heap_Time_rec_Heap @ B @ C @ G ) @ ( heap_Time_map_Heap @ A @ B @ F ) )
= ( heap_Time_rec_Heap @ A @ C
@ ^ [X3: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] : ( G @ ( comp @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( heap_ext @ product_unit ) @ ( map_option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_map_prod @ A @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ F @ ( id @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ X3 ) ) ) ) ).
% Heap.rec_o_map
thf(fact_5422_single__valuedp__single__valued__eq,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B )] :
( ( single_valuedp @ A @ B
@ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R ) )
= ( single_valued @ A @ B @ R ) ) ).
% single_valuedp_single_valued_eq
thf(fact_5423_disjnt__equiv__class,axiom,
! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( equiv_equiv @ A @ A4 @ R )
=> ( ( disjnt @ A @ ( image @ A @ A @ R @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R ) ) ) ) ).
% disjnt_equiv_class
thf(fact_5424_equiv__class__eq__iff,axiom,
! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( equiv_equiv @ A @ A4 @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
= ( ( ( image @ A @ A @ R @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( member @ A @ X @ A4 )
& ( member @ A @ Y @ A4 ) ) ) ) ).
% equiv_class_eq_iff
thf(fact_5425_eq__equiv__class__iff,axiom,
! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( equiv_equiv @ A @ A4 @ R )
=> ( ( member @ A @ X @ A4 )
=> ( ( member @ A @ Y @ A4 )
=> ( ( ( image @ A @ A @ R @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R ) ) ) ) ) ).
% eq_equiv_class_iff
thf(fact_5426_equiv__class__eq,axiom,
! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( equiv_equiv @ A @ A4 @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
=> ( ( image @ A @ A @ R @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% equiv_class_eq
thf(fact_5427_eq__equiv__class,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B2: A,A4: set @ A] :
( ( ( image @ A @ A @ R @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( equiv_equiv @ A @ A4 @ R )
=> ( ( member @ A @ B2 @ A4 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R ) ) ) ) ).
% eq_equiv_class
thf(fact_5428_Heap_Orec,axiom,
! [C: $tType,A: $tType,F: ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > C,X: ( 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 ) )
= ( F @ X ) ) ).
% Heap.rec
thf(fact_5429_subset__equiv__class,axiom,
! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),B2: A,A3: A] :
( ( equiv_equiv @ A @ A4 @ R )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( member @ A @ B2 @ A4 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R ) ) ) ) ).
% subset_equiv_class
thf(fact_5430_equiv__class__subset,axiom,
! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( equiv_equiv @ A @ A4 @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
=> ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% equiv_class_subset
thf(fact_5431_equiv__class__nondisjoint,axiom,
! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),X: A,A3: A,B2: A] :
( ( equiv_equiv @ A @ A4 @ R )
=> ( ( member @ A @ X @ ( inf_inf @ ( set @ A ) @ ( image @ A @ A @ R @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R ) ) ) ).
% equiv_class_nondisjoint
thf(fact_5432_proj__iff,axiom,
! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( equiv_equiv @ A @ A4 @ R )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) @ A4 )
=> ( ( ( equiv_proj @ A @ A @ R @ X )
= ( equiv_proj @ A @ A @ R @ Y ) )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R ) ) ) ) ).
% proj_iff
thf(fact_5433_in__quotient__imp__in__rel,axiom,
! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),X10: set @ A,X: A,Y: A] :
( ( equiv_equiv @ A @ A4 @ R )
=> ( ( member @ ( set @ A ) @ X10 @ ( equiv_quotient @ A @ A4 @ R ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) @ X10 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R ) ) ) ) ).
% in_quotient_imp_in_rel
thf(fact_5434_congruent2__commuteI,axiom,
! [B: $tType,A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),F: A > A > B] :
( ( equiv_equiv @ A @ A4 @ R )
=> ( ! [Y4: A,Z3: A] :
( ( member @ A @ Y4 @ A4 )
=> ( ( member @ A @ Z3 @ A4 )
=> ( ( F @ Y4 @ Z3 )
= ( F @ Z3 @ Y4 ) ) ) )
=> ( ! [Y4: A,Z3: A,W4: A] :
( ( member @ A @ W4 @ A4 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z3 ) @ R )
=> ( ( F @ W4 @ Y4 )
= ( F @ W4 @ Z3 ) ) ) )
=> ( equiv_congruent2 @ A @ A @ B @ R @ R @ F ) ) ) ) ).
% congruent2_commuteI
thf(fact_5435_quotient__eqI,axiom,
! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),X10: set @ A,Y9: set @ A,X: A,Y: A] :
( ( equiv_equiv @ A @ A4 @ R )
=> ( ( member @ ( set @ A ) @ X10 @ ( equiv_quotient @ A @ A4 @ R ) )
=> ( ( member @ ( set @ A ) @ Y9 @ ( equiv_quotient @ A @ A4 @ R ) )
=> ( ( member @ A @ X @ X10 )
=> ( ( member @ A @ Y @ Y9 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
=> ( X10 = Y9 ) ) ) ) ) ) ) ).
% quotient_eqI
thf(fact_5436_quotient__eq__iff,axiom,
! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),X10: set @ A,Y9: set @ A,X: A,Y: A] :
( ( equiv_equiv @ A @ A4 @ R )
=> ( ( member @ ( set @ A ) @ X10 @ ( equiv_quotient @ A @ A4 @ R ) )
=> ( ( member @ ( set @ A ) @ Y9 @ ( equiv_quotient @ A @ A4 @ R ) )
=> ( ( member @ A @ X @ X10 )
=> ( ( member @ A @ Y @ Y9 )
=> ( ( X10 = Y9 )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R ) ) ) ) ) ) ) ).
% quotient_eq_iff
thf(fact_5437_in__quotient__imp__closed,axiom,
! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),X10: set @ A,X: A,Y: A] :
( ( equiv_equiv @ A @ A4 @ R )
=> ( ( member @ ( set @ A ) @ X10 @ ( equiv_quotient @ A @ A4 @ R ) )
=> ( ( member @ A @ X @ X10 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
=> ( member @ A @ Y @ X10 ) ) ) ) ) ).
% in_quotient_imp_closed
thf(fact_5438_congruent2D,axiom,
! [A: $tType,C: $tType,B: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),F: A > B > C,Y1: A,Z1: A,Y2: B,Z22: B] :
( ( equiv_congruent2 @ A @ B @ C @ R1 @ R22 @ F )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y1 @ Z1 ) @ R1 )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y2 @ Z22 ) @ R22 )
=> ( ( F @ Y1 @ Y2 )
= ( F @ Z1 @ Z22 ) ) ) ) ) ).
% congruent2D
thf(fact_5439_congruent2I_H,axiom,
! [C: $tType,B: $tType,A: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),F: A > B > C] :
( ! [Y12: A,Z12: A,Y22: B,Z23: B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y12 @ Z12 ) @ R1 )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y22 @ Z23 ) @ R22 )
=> ( ( F @ Y12 @ Y22 )
= ( F @ Z12 @ Z23 ) ) ) )
=> ( equiv_congruent2 @ A @ B @ C @ R1 @ R22 @ F ) ) ).
% congruent2I'
thf(fact_5440_eq__equiv__class__iff2,axiom,
! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( equiv_equiv @ A @ A4 @ R )
=> ( ( member @ A @ X @ A4 )
=> ( ( member @ A @ Y @ A4 )
=> ( ( ( equiv_quotient @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) @ R )
= ( equiv_quotient @ A @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) @ R ) )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R ) ) ) ) ) ).
% eq_equiv_class_iff2
thf(fact_5441_congruent2I,axiom,
! [C: $tType,B: $tType,A: $tType,A110: set @ A,R1: set @ ( product_prod @ A @ A ),A26: set @ B,R22: set @ ( product_prod @ B @ B ),F: A > B > C] :
( ( equiv_equiv @ A @ A110 @ R1 )
=> ( ( equiv_equiv @ B @ A26 @ R22 )
=> ( ! [Y4: A,Z3: A,W4: B] :
( ( member @ B @ W4 @ A26 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z3 ) @ R1 )
=> ( ( F @ Y4 @ W4 )
= ( F @ Z3 @ W4 ) ) ) )
=> ( ! [Y4: B,Z3: B,W4: A] :
( ( member @ A @ W4 @ A110 )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y4 @ Z3 ) @ R22 )
=> ( ( F @ W4 @ Y4 )
= ( F @ W4 @ Z3 ) ) ) )
=> ( equiv_congruent2 @ A @ B @ C @ R1 @ R22 @ F ) ) ) ) ) ).
% congruent2I
thf(fact_5442_UN__equiv__class__inject,axiom,
! [B: $tType,A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),F: A > ( set @ B ),X10: set @ A,Y9: set @ A] :
( ( equiv_equiv @ A @ A4 @ R )
=> ( ( equiv_congruent @ A @ ( set @ B ) @ R @ F )
=> ( ( ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F @ X10 ) )
= ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F @ Y9 ) ) )
=> ( ( member @ ( set @ A ) @ X10 @ ( equiv_quotient @ A @ A4 @ R ) )
=> ( ( member @ ( set @ A ) @ Y9 @ ( equiv_quotient @ A @ A4 @ R ) )
=> ( ! [X2: A,Y4: A] :
( ( member @ A @ X2 @ A4 )
=> ( ( member @ A @ Y4 @ A4 )
=> ( ( ( F @ X2 )
= ( F @ Y4 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y4 ) @ R ) ) ) )
=> ( X10 = Y9 ) ) ) ) ) ) ) ).
% UN_equiv_class_inject
thf(fact_5443_equivp__equiv,axiom,
! [A: $tType,A4: set @ ( product_prod @ A @ A )] :
( ( equiv_equiv @ A @ ( top_top @ ( set @ A ) ) @ A4 )
= ( equiv_equivp @ A
@ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ A4 ) ) ) ).
% equivp_equiv
thf(fact_5444_prod__mset__def,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ( ( comm_m9189036328036947845d_mset @ A )
= ( comm_monoid_F @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ) ).
% prod_mset_def
thf(fact_5445_congruentI,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),F: A > B] :
( ! [Y4: A,Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z3 ) @ R )
=> ( ( F @ Y4 )
= ( F @ Z3 ) ) )
=> ( equiv_congruent @ A @ B @ R @ F ) ) ).
% congruentI
thf(fact_5446_congruentD,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),F: A > B,Y: A,Z2: A] :
( ( equiv_congruent @ A @ B @ R @ F )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ R )
=> ( ( F @ Y )
= ( F @ Z2 ) ) ) ) ).
% congruentD
thf(fact_5447_sum__mset__def,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ( ( comm_m7189776963980413722m_mset @ A )
= ( comm_monoid_F @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ) ).
% sum_mset_def
thf(fact_5448_or__num_Opelims,axiom,
! [X: num,Xa: num,Y: num] :
( ( ( bit_un6697907153464112080or_num @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ num @ num ) @ bit_un4773296044027857193um_rel @ ( product_Pair @ num @ num @ X @ Xa ) )
=> ( ( ( X = one2 )
=> ( ( Xa = one2 )
=> ( ( Y = one2 )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4773296044027857193um_rel @ ( product_Pair @ num @ num @ one2 @ one2 ) ) ) ) )
=> ( ( ( X = one2 )
=> ! [N3: num] :
( ( Xa
= ( bit0 @ N3 ) )
=> ( ( Y
= ( bit1 @ N3 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4773296044027857193um_rel @ ( product_Pair @ num @ num @ one2 @ ( bit0 @ N3 ) ) ) ) ) )
=> ( ( ( X = one2 )
=> ! [N3: num] :
( ( Xa
= ( bit1 @ N3 ) )
=> ( ( Y
= ( bit1 @ N3 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4773296044027857193um_rel @ ( product_Pair @ num @ num @ one2 @ ( bit1 @ N3 ) ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ( ( Xa = one2 )
=> ( ( Y
= ( bit1 @ M5 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4773296044027857193um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M5 ) @ one2 ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ! [N3: num] :
( ( Xa
= ( bit0 @ N3 ) )
=> ( ( Y
= ( bit0 @ ( bit_un6697907153464112080or_num @ M5 @ N3 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4773296044027857193um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M5 ) @ ( bit0 @ N3 ) ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit0 @ M5 ) )
=> ! [N3: num] :
( ( Xa
= ( bit1 @ N3 ) )
=> ( ( Y
= ( bit1 @ ( bit_un6697907153464112080or_num @ M5 @ N3 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4773296044027857193um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M5 ) @ ( bit1 @ N3 ) ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ( ( Xa = one2 )
=> ( ( Y
= ( bit1 @ M5 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4773296044027857193um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M5 ) @ one2 ) ) ) ) )
=> ( ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ! [N3: num] :
( ( Xa
= ( bit0 @ N3 ) )
=> ( ( Y
= ( bit1 @ ( bit_un6697907153464112080or_num @ M5 @ N3 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4773296044027857193um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M5 ) @ ( bit0 @ N3 ) ) ) ) ) )
=> ~ ! [M5: num] :
( ( X
= ( bit1 @ M5 ) )
=> ! [N3: num] :
( ( Xa
= ( bit1 @ N3 ) )
=> ( ( Y
= ( bit1 @ ( bit_un6697907153464112080or_num @ M5 @ N3 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4773296044027857193um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M5 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% or_num.pelims
thf(fact_5449_Heap_Oset,axiom,
! [A: $tType,X: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
( ( heap_Time_set_Heap @ A @ ( heap_Time_Heap2 @ A @ X ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( set @ A )
@ ^ [Uu3: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] : ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( set @ A ) @ ( basic_fsts @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( set_option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ Uu3 ) ) )
@ ( image2 @ ( heap_ext @ product_unit ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ X @ ( top_top @ ( set @ ( heap_ext @ product_unit ) ) ) ) ) ) ) ).
% Heap.set
thf(fact_5450_under__incr,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( trans @ A @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
=> ( ord_less_eq @ ( set @ A ) @ ( order_under @ A @ R @ A3 ) @ ( order_under @ A @ R @ B2 ) ) ) ) ).
% under_incr
thf(fact_5451_Heap_Oset__intros,axiom,
! [A: $tType,Y: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ),X: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ),Ya: product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ),Yb2: A] :
( ( member @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ Y @ ( image2 @ ( heap_ext @ product_unit ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ X @ ( top_top @ ( set @ ( heap_ext @ product_unit ) ) ) ) )
=> ( ( member @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ Ya @ ( set_option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ Y ) )
=> ( ( member @ A @ Yb2 @ ( basic_fsts @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ Ya ) )
=> ( member @ A @ Yb2 @ ( heap_Time_set_Heap @ A @ ( heap_Time_Heap2 @ A @ X ) ) ) ) ) ) ).
% Heap.set_intros
thf(fact_5452_Heap_Oset__cases,axiom,
! [A: $tType,E2: A,A3: heap_Time_Heap @ A] :
( ( member @ A @ E2 @ ( heap_Time_set_Heap @ A @ A3 ) )
=> ~ ! [Z3: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
( ( A3
= ( heap_Time_Heap2 @ A @ Z3 ) )
=> ! [Xa3: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
( ( member @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ Xa3 @ ( image2 @ ( heap_ext @ product_unit ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ Z3 @ ( top_top @ ( set @ ( heap_ext @ product_unit ) ) ) ) )
=> ! [Xb3: product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat )] :
( ( member @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ Xb3 @ ( set_option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ Xa3 ) )
=> ~ ( member @ A @ E2 @ ( basic_fsts @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ Xb3 ) ) ) ) ) ) ).
% Heap.set_cases
thf(fact_5453_under__def,axiom,
! [A: $tType] :
( ( order_under @ A )
= ( ^ [R3: set @ ( product_prod @ A @ A ),A7: A] :
( collect @ A
@ ^ [B5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ A7 ) @ R3 ) ) ) ) ).
% under_def
thf(fact_5454_Heap_Oinj__map__strong,axiom,
! [B: $tType,A: $tType,X: heap_Time_Heap @ A,Xa: heap_Time_Heap @ A,F: A > B,Fa: A > B] :
( ! [Z3: A,Za: A] :
( ( member @ A @ Z3 @ ( heap_Time_set_Heap @ A @ X ) )
=> ( ( member @ A @ Za @ ( heap_Time_set_Heap @ A @ Xa ) )
=> ( ( ( F @ Z3 )
= ( Fa @ Za ) )
=> ( Z3 = Za ) ) ) )
=> ( ( ( heap_Time_map_Heap @ A @ B @ F @ X )
= ( heap_Time_map_Heap @ A @ B @ Fa @ Xa ) )
=> ( X = Xa ) ) ) ).
% Heap.inj_map_strong
thf(fact_5455_Heap_Omap__cong0,axiom,
! [B: $tType,A: $tType,X: heap_Time_Heap @ A,F: A > B,G: A > B] :
( ! [Z3: A] :
( ( member @ A @ Z3 @ ( heap_Time_set_Heap @ A @ X ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( heap_Time_map_Heap @ A @ B @ F @ X )
= ( heap_Time_map_Heap @ A @ B @ G @ X ) ) ) ).
% Heap.map_cong0
thf(fact_5456_Heap_Omap__cong,axiom,
! [B: $tType,A: $tType,X: heap_Time_Heap @ A,Ya: heap_Time_Heap @ A,F: A > B,G: A > B] :
( ( X = Ya )
=> ( ! [Z3: A] :
( ( member @ A @ Z3 @ ( heap_Time_set_Heap @ A @ Ya ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( heap_Time_map_Heap @ A @ B @ F @ X )
= ( heap_Time_map_Heap @ A @ B @ G @ Ya ) ) ) ) ).
% Heap.map_cong
thf(fact_5457_Heap_Oset__map,axiom,
! [B: $tType,A: $tType,F: A > B,V: heap_Time_Heap @ A] :
( ( heap_Time_set_Heap @ B @ ( heap_Time_map_Heap @ A @ B @ F @ V ) )
= ( image2 @ A @ B @ F @ ( heap_Time_set_Heap @ A @ V ) ) ) ).
% Heap.set_map
thf(fact_5458_Heap_Orel__Grp,axiom,
! [B: $tType,A: $tType,A4: set @ A,F: A > B] :
( ( heap_Time_rel_Heap @ A @ B @ ( bNF_Grp @ A @ B @ A4 @ F ) )
= ( bNF_Grp @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ B )
@ ( collect @ ( heap_Time_Heap @ A )
@ ^ [X3: heap_Time_Heap @ A] : ( ord_less_eq @ ( set @ A ) @ ( heap_Time_set_Heap @ A @ X3 ) @ A4 ) )
@ ( heap_Time_map_Heap @ A @ B @ F ) ) ) ).
% Heap.rel_Grp
thf(fact_5459_Heap_Orel__compp__Grp,axiom,
! [B: $tType,A: $tType] :
( ( heap_Time_rel_Heap @ A @ B )
= ( ^ [R6: A > B > $o] :
( relcompp @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ ( product_prod @ A @ B ) ) @ ( heap_Time_Heap @ B )
@ ( conversep @ ( heap_Time_Heap @ ( product_prod @ A @ B ) ) @ ( heap_Time_Heap @ A )
@ ( bNF_Grp @ ( heap_Time_Heap @ ( product_prod @ A @ B ) ) @ ( heap_Time_Heap @ A )
@ ( collect @ ( heap_Time_Heap @ ( product_prod @ A @ B ) )
@ ^ [X3: heap_Time_Heap @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( heap_Time_set_Heap @ ( product_prod @ A @ B ) @ X3 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) ) )
@ ( heap_Time_map_Heap @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) ) ) )
@ ( bNF_Grp @ ( heap_Time_Heap @ ( product_prod @ A @ B ) ) @ ( heap_Time_Heap @ B )
@ ( collect @ ( heap_Time_Heap @ ( product_prod @ A @ B ) )
@ ^ [X3: heap_Time_Heap @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( heap_Time_set_Heap @ ( product_prod @ A @ B ) @ X3 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) ) )
@ ( heap_Time_map_Heap @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) ) ) ) ) ) ).
% Heap.rel_compp_Grp
thf(fact_5460_Heap_Oin__rel,axiom,
! [B: $tType,A: $tType] :
( ( heap_Time_rel_Heap @ A @ B )
= ( ^ [R6: A > B > $o,A7: heap_Time_Heap @ A,B5: heap_Time_Heap @ B] :
? [Z5: heap_Time_Heap @ ( product_prod @ A @ B )] :
( ( member @ ( heap_Time_Heap @ ( product_prod @ A @ B ) ) @ Z5
@ ( collect @ ( heap_Time_Heap @ ( product_prod @ A @ B ) )
@ ^ [X3: heap_Time_Heap @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( heap_Time_set_Heap @ ( product_prod @ A @ B ) @ X3 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) ) ) )
& ( ( heap_Time_map_Heap @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Z5 )
= A7 )
& ( ( heap_Time_map_Heap @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Z5 )
= B5 ) ) ) ) ).
% Heap.in_rel
thf(fact_5461_Heap_Orel__cong,axiom,
! [A: $tType,B: $tType,X: heap_Time_Heap @ A,Ya: heap_Time_Heap @ A,Y: heap_Time_Heap @ B,Xa: heap_Time_Heap @ B,R5: A > B > $o,Ra2: A > B > $o] :
( ( X = Ya )
=> ( ( Y = Xa )
=> ( ! [Z3: A,Yb: B] :
( ( member @ A @ Z3 @ ( heap_Time_set_Heap @ A @ Ya ) )
=> ( ( member @ B @ Yb @ ( heap_Time_set_Heap @ B @ Xa ) )
=> ( ( R5 @ Z3 @ Yb )
= ( Ra2 @ Z3 @ Yb ) ) ) )
=> ( ( heap_Time_rel_Heap @ A @ B @ R5 @ X @ Y )
= ( heap_Time_rel_Heap @ A @ B @ Ra2 @ Ya @ Xa ) ) ) ) ) ).
% Heap.rel_cong
thf(fact_5462_Heap_Orel__mono__strong,axiom,
! [A: $tType,B: $tType,R5: A > B > $o,X: heap_Time_Heap @ A,Y: heap_Time_Heap @ B,Ra2: A > B > $o] :
( ( heap_Time_rel_Heap @ A @ B @ R5 @ X @ Y )
=> ( ! [Z3: A,Yb: B] :
( ( member @ A @ Z3 @ ( heap_Time_set_Heap @ A @ X ) )
=> ( ( member @ B @ Yb @ ( heap_Time_set_Heap @ B @ Y ) )
=> ( ( R5 @ Z3 @ Yb )
=> ( Ra2 @ Z3 @ Yb ) ) ) )
=> ( heap_Time_rel_Heap @ A @ B @ Ra2 @ X @ Y ) ) ) ).
% Heap.rel_mono_strong
thf(fact_5463_Heap_Orel__refl__strong,axiom,
! [A: $tType,X: heap_Time_Heap @ A,Ra2: A > A > $o] :
( ! [Z3: A] :
( ( member @ A @ Z3 @ ( heap_Time_set_Heap @ A @ X ) )
=> ( Ra2 @ Z3 @ Z3 ) )
=> ( heap_Time_rel_Heap @ A @ A @ Ra2 @ X @ X ) ) ).
% Heap.rel_refl_strong
thf(fact_5464_Heap_Orel__map_I2_J,axiom,
! [A: $tType,C: $tType,B: $tType,Sa: A > C > $o,X: heap_Time_Heap @ A,G: B > C,Y: heap_Time_Heap @ B] :
( ( heap_Time_rel_Heap @ A @ C @ Sa @ X @ ( heap_Time_map_Heap @ B @ C @ G @ Y ) )
= ( heap_Time_rel_Heap @ A @ B
@ ^ [X3: A,Y3: B] : ( Sa @ X3 @ ( G @ Y3 ) )
@ X
@ Y ) ) ).
% Heap.rel_map(2)
thf(fact_5465_Heap_Orel__map_I1_J,axiom,
! [A: $tType,C: $tType,B: $tType,Sb: C > B > $o,I: A > C,X: heap_Time_Heap @ A,Y: heap_Time_Heap @ B] :
( ( heap_Time_rel_Heap @ C @ B @ Sb @ ( heap_Time_map_Heap @ A @ C @ I @ X ) @ Y )
= ( heap_Time_rel_Heap @ A @ B
@ ^ [X3: A] : ( Sb @ ( I @ X3 ) )
@ X
@ Y ) ) ).
% Heap.rel_map(1)
thf(fact_5466_Heap_Orel__mono,axiom,
! [B: $tType,A: $tType,R5: A > B > $o,Ra2: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ R5 @ Ra2 )
=> ( ord_less_eq @ ( ( heap_Time_Heap @ A ) > ( heap_Time_Heap @ B ) > $o ) @ ( heap_Time_rel_Heap @ A @ B @ R5 ) @ ( heap_Time_rel_Heap @ A @ B @ Ra2 ) ) ) ).
% Heap.rel_mono
thf(fact_5467_Heap_Orel__transp,axiom,
! [A: $tType,R5: A > A > $o] :
( ( transp @ A @ R5 )
=> ( transp @ ( heap_Time_Heap @ A ) @ ( heap_Time_rel_Heap @ A @ A @ R5 ) ) ) ).
% Heap.rel_transp
thf(fact_5468_Heap_Orel__refl,axiom,
! [B: $tType,Ra2: B > B > $o,X: heap_Time_Heap @ B] :
( ! [X2: B] : ( Ra2 @ X2 @ X2 )
=> ( heap_Time_rel_Heap @ B @ B @ Ra2 @ X @ X ) ) ).
% Heap.rel_refl
thf(fact_5469_Heap_Orel__eq,axiom,
! [A: $tType] :
( ( heap_Time_rel_Heap @ A @ A
@ ^ [Y5: A,Z4: A] : Y5 = Z4 )
= ( ^ [Y5: heap_Time_Heap @ A,Z4: heap_Time_Heap @ A] : Y5 = Z4 ) ) ).
% Heap.rel_eq
thf(fact_5470_Heap_Orel__conversep,axiom,
! [B: $tType,A: $tType,R5: A > B > $o] :
( ( heap_Time_rel_Heap @ B @ A @ ( conversep @ A @ B @ R5 ) )
= ( conversep @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ B ) @ ( heap_Time_rel_Heap @ A @ B @ R5 ) ) ) ).
% Heap.rel_conversep
thf(fact_5471_Heap_Orel__flip,axiom,
! [A: $tType,B: $tType,R5: A > B > $o,A3: heap_Time_Heap @ B,B2: heap_Time_Heap @ A] :
( ( heap_Time_rel_Heap @ B @ A @ ( conversep @ A @ B @ R5 ) @ A3 @ B2 )
= ( heap_Time_rel_Heap @ A @ B @ R5 @ B2 @ A3 ) ) ).
% Heap.rel_flip
thf(fact_5472_Heap_Orel__compp,axiom,
! [A: $tType,C: $tType,B: $tType,R5: A > B > $o,S2: B > C > $o] :
( ( heap_Time_rel_Heap @ A @ C @ ( relcompp @ A @ B @ C @ R5 @ S2 ) )
= ( relcompp @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ B ) @ ( heap_Time_Heap @ C ) @ ( heap_Time_rel_Heap @ A @ B @ R5 ) @ ( heap_Time_rel_Heap @ B @ C @ S2 ) ) ) ).
% Heap.rel_compp
thf(fact_5473_Heap_Orel__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,Sa: A > C > $o,Sc: B > D > $o] :
( bNF_rel_fun @ ( A > B > $o ) @ ( C > D > $o ) @ ( ( heap_Time_Heap @ A ) > ( heap_Time_Heap @ B ) > $o ) @ ( ( heap_Time_Heap @ C ) > ( heap_Time_Heap @ D ) > $o )
@ ( bNF_rel_fun @ A @ C @ ( B > $o ) @ ( D > $o ) @ Sa
@ ( bNF_rel_fun @ B @ D @ $o @ $o @ Sc
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 ) )
@ ( bNF_rel_fun @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ C ) @ ( ( heap_Time_Heap @ B ) > $o ) @ ( ( heap_Time_Heap @ D ) > $o ) @ ( heap_Time_rel_Heap @ A @ C @ Sa )
@ ( bNF_rel_fun @ ( heap_Time_Heap @ B ) @ ( heap_Time_Heap @ D ) @ $o @ $o @ ( heap_Time_rel_Heap @ B @ D @ Sc )
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 ) )
@ ( heap_Time_rel_Heap @ A @ B )
@ ( heap_Time_rel_Heap @ C @ D ) ) ).
% Heap.rel_transfer
thf(fact_5474_Heap_Omap__transfer,axiom,
! [A: $tType,B: $tType,F5: $tType,E: $tType,Rb2: A > E > $o,Sd: B > F5 > $o] : ( bNF_rel_fun @ ( A > B ) @ ( E > F5 ) @ ( ( heap_Time_Heap @ A ) > ( heap_Time_Heap @ B ) ) @ ( ( heap_Time_Heap @ E ) > ( heap_Time_Heap @ F5 ) ) @ ( bNF_rel_fun @ A @ E @ B @ F5 @ Rb2 @ Sd ) @ ( bNF_rel_fun @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ E ) @ ( heap_Time_Heap @ B ) @ ( heap_Time_Heap @ F5 ) @ ( heap_Time_rel_Heap @ A @ E @ Rb2 ) @ ( heap_Time_rel_Heap @ B @ F5 @ Sd ) ) @ ( heap_Time_map_Heap @ A @ B ) @ ( heap_Time_map_Heap @ E @ F5 ) ) ).
% Heap.map_transfer
thf(fact_5475_those_Osimps_I2_J,axiom,
! [A: $tType,X: option @ A,Xs: list @ ( option @ A )] :
( ( those @ A @ ( cons @ ( option @ A ) @ X @ Xs ) )
= ( case_option @ ( option @ ( list @ A ) ) @ A @ ( none @ ( list @ A ) )
@ ^ [Y3: A] : ( map_option @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y3 ) @ ( those @ A @ Xs ) )
@ X ) ) ).
% those.simps(2)
thf(fact_5476_Quotient__natural,axiom,
( quotient @ nat @ code_natural
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4
@ code_natural_of_nat
@ code_nat_of_natural
@ code_cr_natural ) ).
% Quotient_natural
thf(fact_5477_Quotient__integer,axiom,
( quotient @ int @ code_integer
@ ^ [Y5: int,Z4: int] : Y5 = Z4
@ code_integer_of_int
@ code_int_of_integer
@ code_cr_integer ) ).
% Quotient_integer
thf(fact_5478_Heap_OQuotient,axiom,
! [B: $tType,A: $tType,R5: A > A > $o,Abs: A > B,Rep: B > A,T6: A > B > $o] :
( ( quotient @ A @ B @ R5 @ Abs @ Rep @ T6 )
=> ( quotient @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ B ) @ ( heap_Time_rel_Heap @ A @ A @ R5 ) @ ( heap_Time_map_Heap @ A @ B @ Abs ) @ ( heap_Time_map_Heap @ B @ A @ Rep ) @ ( heap_Time_rel_Heap @ A @ B @ T6 ) ) ) ).
% Heap.Quotient
thf(fact_5479_option_OQuotient,axiom,
! [B: $tType,A: $tType,R5: A > A > $o,Abs: A > B,Rep: B > A,T6: A > B > $o] :
( ( quotient @ A @ B @ R5 @ Abs @ Rep @ T6 )
=> ( quotient @ ( option @ A ) @ ( option @ B ) @ ( rel_option @ A @ A @ R5 ) @ ( map_option @ A @ B @ Abs ) @ ( map_option @ B @ A @ Rep ) @ ( rel_option @ A @ B @ T6 ) ) ) ).
% option.Quotient
thf(fact_5480_Quotient__compose,axiom,
! [A: $tType,C: $tType,B: $tType,R12: A > A > $o,Abs1: A > B,Rep1: B > A,T1: A > B > $o,R23: B > B > $o,Abs22: B > C,Rep22: C > B,T22: B > C > $o] :
( ( quotient @ A @ B @ R12 @ Abs1 @ Rep1 @ T1 )
=> ( ( quotient @ B @ C @ R23 @ Abs22 @ Rep22 @ T22 )
=> ( quotient @ A @ C @ ( relcompp @ A @ B @ A @ T1 @ ( relcompp @ B @ B @ A @ R23 @ ( conversep @ A @ B @ T1 ) ) ) @ ( comp @ B @ C @ A @ Abs22 @ Abs1 ) @ ( comp @ B @ A @ C @ Rep1 @ Rep22 ) @ ( relcompp @ A @ B @ C @ T1 @ T22 ) ) ) ) ).
% Quotient_compose
thf(fact_5481_Heap_Opred__transfer,axiom,
! [A: $tType,B: $tType,R5: A > B > $o] :
( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( heap_Time_Heap @ A ) > $o ) @ ( ( heap_Time_Heap @ B ) > $o )
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ R5
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 )
@ ( bNF_rel_fun @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ B ) @ $o @ $o @ ( heap_Time_rel_Heap @ A @ B @ R5 )
@ ^ [Y5: $o,Z4: $o] : Y5 = Z4 )
@ ( heap_Time_pred_Heap @ A )
@ ( heap_Time_pred_Heap @ B ) ) ).
% Heap.pred_transfer
thf(fact_5482_Heap_Orec__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,R5: A > B > $o,S2: C > D > $o] :
( bNF_rel_fun @ ( ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > C ) @ ( ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > D ) @ ( ( heap_Time_Heap @ A ) > C ) @ ( ( heap_Time_Heap @ B ) > D )
@ ( bNF_rel_fun @ ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ C @ D
@ ( bNF_rel_fun @ ( heap_ext @ product_unit ) @ ( heap_ext @ product_unit ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
@ ^ [Y5: heap_ext @ product_unit,Z4: heap_ext @ product_unit] : Y5 = Z4
@ ( rel_option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
@ ( basic_rel_prod @ A @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R5
@ ^ [Y5: product_prod @ ( heap_ext @ product_unit ) @ nat,Z4: product_prod @ ( heap_ext @ product_unit ) @ nat] : Y5 = Z4 ) ) )
@ S2 )
@ ( bNF_rel_fun @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ B ) @ C @ D @ ( heap_Time_rel_Heap @ A @ B @ R5 ) @ S2 )
@ ( heap_Time_rec_Heap @ A @ C )
@ ( heap_Time_rec_Heap @ B @ D ) ) ).
% Heap.rec_transfer
thf(fact_5483_rel__prod__inject,axiom,
! [B: $tType,A: $tType,C: $tType,D: $tType,R12: A > B > $o,R23: C > D > $o,A3: A,B2: C,C2: B,D3: D] :
( ( basic_rel_prod @ A @ B @ C @ D @ R12 @ R23 @ ( product_Pair @ A @ C @ A3 @ B2 ) @ ( product_Pair @ B @ D @ C2 @ D3 ) )
= ( ( R12 @ A3 @ C2 )
& ( R23 @ B2 @ D3 ) ) ) ).
% rel_prod_inject
thf(fact_5484_Heap_Opred__mono,axiom,
! [A: $tType,P: A > $o,Pa: A > $o] :
( ( ord_less_eq @ ( A > $o ) @ P @ Pa )
=> ( ord_less_eq @ ( ( heap_Time_Heap @ A ) > $o ) @ ( heap_Time_pred_Heap @ A @ P ) @ ( heap_Time_pred_Heap @ A @ Pa ) ) ) ).
% Heap.pred_mono
thf(fact_5485_rel__prod_Ointros,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,R12: A > B > $o,A3: A,B2: B,R23: C > D > $o,C2: C,D3: D] :
( ( R12 @ A3 @ B2 )
=> ( ( R23 @ C2 @ D3 )
=> ( basic_rel_prod @ A @ B @ C @ D @ R12 @ R23 @ ( product_Pair @ A @ C @ A3 @ C2 ) @ ( product_Pair @ B @ D @ B2 @ D3 ) ) ) ) ).
% rel_prod.intros
thf(fact_5486_rel__prod_Osimps,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType] :
( ( basic_rel_prod @ A @ B @ C @ D )
= ( ^ [R13: A > B > $o,R24: C > D > $o,A12: product_prod @ A @ C,A23: product_prod @ B @ D] :
? [A7: A,B5: B,C3: C,D5: D] :
( ( A12
= ( product_Pair @ A @ C @ A7 @ C3 ) )
& ( A23
= ( product_Pair @ B @ D @ B5 @ D5 ) )
& ( R13 @ A7 @ B5 )
& ( R24 @ C3 @ D5 ) ) ) ) ).
% rel_prod.simps
thf(fact_5487_rel__prod_Ocases,axiom,
! [B: $tType,A: $tType,C: $tType,D: $tType,R12: A > B > $o,R23: C > D > $o,A1: product_prod @ A @ C,A22: product_prod @ B @ D] :
( ( basic_rel_prod @ A @ B @ C @ D @ R12 @ R23 @ A1 @ A22 )
=> ~ ! [A5: A,B3: B,C4: C] :
( ( A1
= ( product_Pair @ A @ C @ A5 @ C4 ) )
=> ! [D2: D] :
( ( A22
= ( product_Pair @ B @ D @ B3 @ D2 ) )
=> ( ( R12 @ A5 @ B3 )
=> ~ ( R23 @ C4 @ D2 ) ) ) ) ) ).
% rel_prod.cases
thf(fact_5488_Heap_Opred__True,axiom,
! [A: $tType] :
( ( heap_Time_pred_Heap @ A
@ ^ [Uu: A] : $true )
= ( ^ [Uu: heap_Time_Heap @ A] : $true ) ) ).
% Heap.pred_True
thf(fact_5489_Heap_Opred__mono__strong,axiom,
! [A: $tType,P: A > $o,X: heap_Time_Heap @ A,Pa: A > $o] :
( ( heap_Time_pred_Heap @ A @ P @ X )
=> ( ! [Z3: A] :
( ( member @ A @ Z3 @ ( heap_Time_set_Heap @ A @ X ) )
=> ( ( P @ Z3 )
=> ( Pa @ Z3 ) ) )
=> ( heap_Time_pred_Heap @ A @ Pa @ X ) ) ) ).
% Heap.pred_mono_strong
thf(fact_5490_Heap_Opred__cong,axiom,
! [A: $tType,X: heap_Time_Heap @ A,Ya: heap_Time_Heap @ A,P: A > $o,Pa: A > $o] :
( ( X = Ya )
=> ( ! [Z3: A] :
( ( member @ A @ Z3 @ ( heap_Time_set_Heap @ A @ Ya ) )
=> ( ( P @ Z3 )
= ( Pa @ Z3 ) ) )
=> ( ( heap_Time_pred_Heap @ A @ P @ X )
= ( heap_Time_pred_Heap @ A @ Pa @ Ya ) ) ) ) ).
% Heap.pred_cong
thf(fact_5491_Heap_Octr__transfer,axiom,
! [A: $tType,B: $tType,R5: A > B > $o] :
( bNF_rel_fun @ ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ B )
@ ( bNF_rel_fun @ ( heap_ext @ product_unit ) @ ( heap_ext @ product_unit ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
@ ^ [Y5: heap_ext @ product_unit,Z4: heap_ext @ product_unit] : Y5 = Z4
@ ( rel_option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
@ ( basic_rel_prod @ A @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R5
@ ^ [Y5: product_prod @ ( heap_ext @ product_unit ) @ nat,Z4: product_prod @ ( heap_ext @ product_unit ) @ nat] : Y5 = Z4 ) ) )
@ ( heap_Time_rel_Heap @ A @ B @ R5 )
@ ( heap_Time_Heap2 @ A )
@ ( heap_Time_Heap2 @ B ) ) ).
% Heap.ctr_transfer
thf(fact_5492_Heap_Orel__intros,axiom,
! [A: $tType,B: $tType,R5: A > B > $o,X: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ),Y: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
( ( bNF_rel_fun @ ( heap_ext @ product_unit ) @ ( heap_ext @ product_unit ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
@ ^ [Y5: heap_ext @ product_unit,Z4: heap_ext @ product_unit] : Y5 = Z4
@ ( rel_option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
@ ( basic_rel_prod @ A @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R5
@ ^ [Y5: product_prod @ ( heap_ext @ product_unit ) @ nat,Z4: product_prod @ ( heap_ext @ product_unit ) @ nat] : Y5 = Z4 ) )
@ X
@ Y )
=> ( heap_Time_rel_Heap @ A @ B @ R5 @ ( heap_Time_Heap2 @ A @ X ) @ ( heap_Time_Heap2 @ B @ Y ) ) ) ).
% Heap.rel_intros
thf(fact_5493_Heap_Orel__inject,axiom,
! [A: $tType,B: $tType,R5: A > B > $o,X: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ),Y: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
( ( heap_Time_rel_Heap @ A @ B @ R5 @ ( heap_Time_Heap2 @ A @ X ) @ ( heap_Time_Heap2 @ B @ Y ) )
= ( bNF_rel_fun @ ( heap_ext @ product_unit ) @ ( heap_ext @ product_unit ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
@ ^ [Y5: heap_ext @ product_unit,Z4: heap_ext @ product_unit] : Y5 = Z4
@ ( rel_option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
@ ( basic_rel_prod @ A @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R5
@ ^ [Y5: product_prod @ ( heap_ext @ product_unit ) @ nat,Z4: product_prod @ ( heap_ext @ product_unit ) @ nat] : Y5 = Z4 ) )
@ X
@ Y ) ) ).
% Heap.rel_inject
thf(fact_5494_Heap_Orel__induct,axiom,
! [A: $tType,B: $tType,R5: A > B > $o,X: heap_Time_Heap @ A,Y: heap_Time_Heap @ B,Q: ( heap_Time_Heap @ A ) > ( heap_Time_Heap @ B ) > $o] :
( ( heap_Time_rel_Heap @ A @ B @ R5 @ X @ Y )
=> ( ! [A5: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ),B3: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
( ( bNF_rel_fun @ ( heap_ext @ product_unit ) @ ( heap_ext @ product_unit ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
@ ^ [Y5: heap_ext @ product_unit,Z4: heap_ext @ product_unit] : Y5 = Z4
@ ( rel_option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
@ ( basic_rel_prod @ A @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R5
@ ^ [Y5: product_prod @ ( heap_ext @ product_unit ) @ nat,Z4: product_prod @ ( heap_ext @ product_unit ) @ nat] : Y5 = Z4 ) )
@ A5
@ B3 )
=> ( Q @ ( heap_Time_Heap2 @ A @ A5 ) @ ( heap_Time_Heap2 @ B @ B3 ) ) )
=> ( Q @ X @ Y ) ) ) ).
% Heap.rel_induct
thf(fact_5495_Heap_Orel__cases,axiom,
! [A: $tType,B: $tType,R5: A > B > $o,A3: heap_Time_Heap @ A,B2: heap_Time_Heap @ B] :
( ( heap_Time_rel_Heap @ A @ B @ R5 @ A3 @ B2 )
=> ~ ! [Xa3: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
( ( A3
= ( heap_Time_Heap2 @ A @ Xa3 ) )
=> ! [Ya2: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
( ( B2
= ( heap_Time_Heap2 @ B @ Ya2 ) )
=> ~ ( bNF_rel_fun @ ( heap_ext @ product_unit ) @ ( heap_ext @ product_unit ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
@ ^ [Y5: heap_ext @ product_unit,Z4: heap_ext @ product_unit] : Y5 = Z4
@ ( rel_option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
@ ( basic_rel_prod @ A @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R5
@ ^ [Y5: product_prod @ ( heap_ext @ product_unit ) @ nat,Z4: product_prod @ ( heap_ext @ product_unit ) @ nat] : Y5 = Z4 ) )
@ Xa3
@ Ya2 ) ) ) ) ).
% Heap.rel_cases
thf(fact_5496_Heap_Omap__cong__pred,axiom,
! [B: $tType,A: $tType,X: heap_Time_Heap @ A,Ya: heap_Time_Heap @ A,F: A > B,G: A > B] :
( ( X = Ya )
=> ( ( heap_Time_pred_Heap @ A
@ ^ [Z5: A] :
( ( F @ Z5 )
= ( G @ Z5 ) )
@ Ya )
=> ( ( heap_Time_map_Heap @ A @ B @ F @ X )
= ( heap_Time_map_Heap @ A @ B @ G @ Ya ) ) ) ) ).
% Heap.map_cong_pred
thf(fact_5497_Heap_Orel__eq__onp,axiom,
! [A: $tType,P: A > $o] :
( ( heap_Time_rel_Heap @ A @ A @ ( bNF_eq_onp @ A @ P ) )
= ( bNF_eq_onp @ ( heap_Time_Heap @ A ) @ ( heap_Time_pred_Heap @ A @ P ) ) ) ).
% Heap.rel_eq_onp
thf(fact_5498_Heap_Opred__rel,axiom,
! [A: $tType] :
( ( heap_Time_pred_Heap @ A )
= ( ^ [P4: A > $o,X3: heap_Time_Heap @ A] : ( heap_Time_rel_Heap @ A @ A @ ( bNF_eq_onp @ A @ P4 ) @ X3 @ X3 ) ) ) ).
% Heap.pred_rel
thf(fact_5499_Heap_Opred__set,axiom,
! [A: $tType] :
( ( heap_Time_pred_Heap @ A )
= ( ^ [P4: A > $o,X3: heap_Time_Heap @ A] :
! [Y3: A] :
( ( member @ A @ Y3 @ ( heap_Time_set_Heap @ A @ X3 ) )
=> ( P4 @ Y3 ) ) ) ) ).
% Heap.pred_set
thf(fact_5500_Pair__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A4: A > B > $o,B6: C > D > $o] : ( bNF_rel_fun @ A @ B @ ( C > ( product_prod @ A @ C ) ) @ ( D > ( product_prod @ B @ D ) ) @ A4 @ ( bNF_rel_fun @ C @ D @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ B6 @ ( basic_rel_prod @ A @ B @ C @ D @ A4 @ B6 ) ) @ ( product_Pair @ A @ C ) @ ( product_Pair @ B @ D ) ) ).
% Pair_transfer
thf(fact_5501_Heap_Opred__map,axiom,
! [B: $tType,A: $tType,Q: B > $o,F: A > B,X: heap_Time_Heap @ A] :
( ( heap_Time_pred_Heap @ B @ Q @ ( heap_Time_map_Heap @ A @ B @ F @ X ) )
= ( heap_Time_pred_Heap @ A @ ( comp @ B @ $o @ A @ Q @ F ) @ X ) ) ).
% Heap.pred_map
thf(fact_5502_Heap_Ocase__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,R5: A > B > $o,S2: C > D > $o] :
( bNF_rel_fun @ ( ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > C ) @ ( ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > D ) @ ( ( heap_Time_Heap @ A ) > C ) @ ( ( heap_Time_Heap @ B ) > D )
@ ( bNF_rel_fun @ ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ C @ D
@ ( bNF_rel_fun @ ( heap_ext @ product_unit ) @ ( heap_ext @ product_unit ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
@ ^ [Y5: heap_ext @ product_unit,Z4: heap_ext @ product_unit] : Y5 = Z4
@ ( rel_option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
@ ( basic_rel_prod @ A @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R5
@ ^ [Y5: product_prod @ ( heap_ext @ product_unit ) @ nat,Z4: product_prod @ ( heap_ext @ product_unit ) @ nat] : Y5 = Z4 ) ) )
@ S2 )
@ ( bNF_rel_fun @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ B ) @ C @ D @ ( heap_Time_rel_Heap @ A @ B @ R5 ) @ S2 )
@ ( heap_Time_case_Heap @ A @ C )
@ ( heap_Time_case_Heap @ B @ D ) ) ).
% Heap.case_transfer
thf(fact_5503_sndOp__def,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( bNF_sndOp @ C @ A @ B )
= ( ^ [P4: C > A > $o,Q6: A > B > $o,Ac: product_prod @ C @ B] : ( product_Pair @ A @ B @ ( bNF_pick_middlep @ C @ A @ B @ P4 @ Q6 @ ( product_fst @ C @ B @ Ac ) @ ( product_snd @ C @ B @ Ac ) ) @ ( product_snd @ C @ B @ Ac ) ) ) ) ).
% sndOp_def
thf(fact_5504_fstOp__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( bNF_fstOp @ A @ B @ C )
= ( ^ [P4: A > B > $o,Q6: B > C > $o,Ac: product_prod @ A @ C] : ( product_Pair @ A @ B @ ( product_fst @ A @ C @ Ac ) @ ( bNF_pick_middlep @ A @ B @ C @ P4 @ Q6 @ ( product_fst @ A @ C @ Ac ) @ ( product_snd @ A @ C @ Ac ) ) ) ) ) ).
% fstOp_def
thf(fact_5505_Heap_Ocase__distrib,axiom,
! [C: $tType,B: $tType,A: $tType,H3: B > C,F: ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > B,Heap: heap_Time_Heap @ A] :
( ( H3 @ ( heap_Time_case_Heap @ A @ B @ F @ Heap ) )
= ( heap_Time_case_Heap @ A @ C
@ ^ [X3: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] : ( H3 @ ( F @ X3 ) )
@ Heap ) ) ).
% Heap.case_distrib
thf(fact_5506_Heap_Ocase,axiom,
! [B: $tType,A: $tType,F: ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > B,X: ( 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 ) )
= ( F @ X ) ) ).
% Heap.case
thf(fact_5507_trancl__def,axiom,
! [A: $tType] :
( ( transitive_trancl @ A )
= ( ^ [R3: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ( transitive_tranclp @ A
@ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 ) ) ) ) ) ) ).
% trancl_def
thf(fact_5508_Nitpick_Otranclp__unfold,axiom,
! [A: $tType] :
( ( transitive_tranclp @ A )
= ( ^ [R3: A > A > $o,A7: A,B5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B5 ) @ ( transitive_trancl @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R3 ) ) ) ) ) ) ).
% Nitpick.tranclp_unfold
thf(fact_5509_reflp__refl__eq,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( reflp @ A
@ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R ) )
= ( refl_on @ A @ ( top_top @ ( set @ A ) ) @ R ) ) ).
% reflp_refl_eq
thf(fact_5510_tranclp__induct2,axiom,
! [A: $tType,B: $tType,R: ( product_prod @ A @ B ) > ( product_prod @ A @ B ) > $o,Ax: A,Ay: B,Bx: A,By: B,P: A > B > $o] :
( ( transitive_tranclp @ ( product_prod @ A @ B ) @ R @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) )
=> ( ! [A5: A,B3: B] :
( ( R @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A5 @ B3 ) )
=> ( P @ A5 @ B3 ) )
=> ( ! [A5: A,B3: B,Aa3: A,Ba: B] :
( ( transitive_tranclp @ ( product_prod @ A @ B ) @ R @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A5 @ B3 ) )
=> ( ( R @ ( product_Pair @ A @ B @ A5 @ B3 ) @ ( product_Pair @ A @ B @ Aa3 @ Ba ) )
=> ( ( P @ A5 @ B3 )
=> ( P @ Aa3 @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% tranclp_induct2
thf(fact_5511_Heap_Orel__reflp,axiom,
! [A: $tType,R5: A > A > $o] :
( ( reflp @ A @ R5 )
=> ( reflp @ ( heap_Time_Heap @ A ) @ ( heap_Time_rel_Heap @ A @ A @ R5 ) ) ) ).
% Heap.rel_reflp
thf(fact_5512_option_Orel__reflp,axiom,
! [A: $tType,R5: A > A > $o] :
( ( reflp @ A @ R5 )
=> ( reflp @ ( option @ A ) @ ( rel_option @ A @ A @ R5 ) ) ) ).
% option.rel_reflp
thf(fact_5513_tranclp__trancl__eq,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( transitive_tranclp @ A
@ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R ) )
= ( ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ ( transitive_trancl @ A @ R ) ) ) ) ).
% tranclp_trancl_eq
thf(fact_5514_Lcm__fin__1__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: set @ A] :
( ( ( semiring_gcd_Lcm_fin @ A @ A4 )
= ( one_one @ A ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( dvd_dvd @ A @ X3 @ ( one_one @ A ) ) )
& ( finite_finite2 @ A @ A4 ) ) ) ) ).
% Lcm_fin_1_iff
thf(fact_5515_heap__interpretation,axiom,
! [A: $tType] : ( partia7178651479351089652itions @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap_ord @ A ) @ ( heap_Time_Heap_lub @ A ) ) ).
% heap_interpretation
thf(fact_5516_old_Orec__bool__def,axiom,
! [T: $tType] :
( ( product_rec_bool @ T )
= ( ^ [F12: T,F23: T,X3: $o] : ( the @ T @ ( product_rec_set_bool @ T @ F12 @ F23 @ X3 ) ) ) ) ).
% old.rec_bool_def
thf(fact_5517_old_Obool_Osimps_I6_J,axiom,
! [T: $tType,F1: T,F22: T] :
( ( product_rec_bool @ T @ F1 @ F22 @ $false )
= F22 ) ).
% old.bool.simps(6)
thf(fact_5518_old_Obool_Osimps_I5_J,axiom,
! [T: $tType,F1: T,F22: T] :
( ( product_rec_bool @ T @ F1 @ F22 @ $true )
= F1 ) ).
% old.bool.simps(5)
thf(fact_5519_Lcm__fin_Oempty,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( semiring_gcd_Lcm_fin @ A @ ( bot_bot @ ( set @ A ) ) )
= ( one_one @ A ) ) ) ).
% Lcm_fin.empty
thf(fact_5520_is__unit__Lcm__fin__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: set @ A] :
( ( dvd_dvd @ A @ ( semiring_gcd_Lcm_fin @ A @ A4 ) @ ( one_one @ A ) )
= ( ( semiring_gcd_Lcm_fin @ A @ A4 )
= ( one_one @ A ) ) ) ) ).
% is_unit_Lcm_fin_iff
thf(fact_5521_Lcm__fin_Oeq__fold,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( semiring_gcd_Lcm_fin @ A )
= ( ^ [A9: set @ A] : ( if @ A @ ( finite_finite2 @ A @ A9 ) @ ( finite_fold @ A @ A @ ( gcd_lcm @ A ) @ ( one_one @ A ) @ A9 ) @ ( zero_zero @ A ) ) ) ) ) ).
% Lcm_fin.eq_fold
thf(fact_5522_Lcm__fin__def,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( semiring_gcd_Lcm_fin @ A )
= ( bounde2362111253966948842tice_F @ A @ ( gcd_lcm @ A ) @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ) ).
% Lcm_fin_def
thf(fact_5523_fun_Opred__map,axiom,
! [B: $tType,A: $tType,D: $tType,Q: B > $o,F: A > B,X: D > A] :
( ( basic_pred_fun @ D @ B
@ ^ [Uu: D] : $true
@ Q
@ ( comp @ A @ B @ D @ F @ X ) )
= ( basic_pred_fun @ D @ A
@ ^ [Uu: D] : $true
@ ( comp @ B @ $o @ A @ Q @ F )
@ X ) ) ).
% fun.pred_map
thf(fact_5524_lcm__eq__1__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( ( gcd_lcm @ A @ A3 @ B2 )
= ( one_one @ A ) )
= ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
& ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) ) ) ) ).
% lcm_eq_1_iff
thf(fact_5525_unit__factor__lcm,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( ( ( A3
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_lcm @ A @ A3 @ B2 ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ( A3
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_lcm @ A @ A3 @ B2 ) )
= ( one_one @ A ) ) ) ) ) ).
% unit_factor_lcm
thf(fact_5526_fun_Omap__cong__pred,axiom,
! [B: $tType,A: $tType,D: $tType,X: D > A,Ya: D > A,F: A > B,G: A > B] :
( ( X = Ya )
=> ( ( basic_pred_fun @ D @ A
@ ^ [Uu: D] : $true
@ ^ [Z5: A] :
( ( F @ Z5 )
= ( G @ Z5 ) )
@ Ya )
=> ( ( comp @ A @ B @ D @ F @ X )
= ( comp @ A @ B @ D @ G @ Ya ) ) ) ) ).
% fun.map_cong_pred
thf(fact_5527_lcm__mult__unit2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( gcd_lcm @ A @ B2 @ ( times_times @ A @ C2 @ A3 ) )
= ( gcd_lcm @ A @ B2 @ C2 ) ) ) ) ).
% lcm_mult_unit2
thf(fact_5528_lcm__mult__unit1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( gcd_lcm @ A @ ( times_times @ A @ B2 @ A3 ) @ C2 )
= ( gcd_lcm @ A @ B2 @ C2 ) ) ) ) ).
% lcm_mult_unit1
thf(fact_5529_lcm__div__unit1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( gcd_lcm @ A @ ( divide_divide @ A @ B2 @ A3 ) @ C2 )
= ( gcd_lcm @ A @ B2 @ C2 ) ) ) ) ).
% lcm_div_unit1
thf(fact_5530_lcm__div__unit2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( gcd_lcm @ A @ B2 @ ( divide_divide @ A @ C2 @ A3 ) )
= ( gcd_lcm @ A @ B2 @ C2 ) ) ) ) ).
% lcm_div_unit2
thf(fact_5531_Lcm__nat__def,axiom,
( ( gcd_Lcm @ nat )
= ( ^ [M8: set @ nat] : ( if @ nat @ ( finite_finite2 @ nat @ M8 ) @ ( lattic5214292709420241887eutr_F @ nat @ ( gcd_lcm @ nat ) @ ( one_one @ nat ) @ M8 ) @ ( zero_zero @ nat ) ) ) ) ).
% Lcm_nat_def
thf(fact_5532_Heap_Opred__inject,axiom,
! [A: $tType,P: A > $o,A3: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
( ( heap_Time_pred_Heap @ A @ P @ ( heap_Time_Heap2 @ A @ A3 ) )
= ( basic_pred_fun @ ( heap_ext @ product_unit ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
@ ^ [Uu: heap_ext @ product_unit] : $true
@ ( pred_option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( basic_pred_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ P @ ( top_top @ ( ( product_prod @ ( heap_ext @ product_unit ) @ nat ) > $o ) ) ) )
@ A3 ) ) ).
% Heap.pred_inject
thf(fact_5533_Lcm__fin_Oset__eq__fold,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [Xs: list @ A] :
( ( semiring_gcd_Lcm_fin @ A @ ( set2 @ A @ Xs ) )
= ( fold @ A @ A @ ( gcd_lcm @ A ) @ Xs @ ( one_one @ A ) ) ) ) ).
% Lcm_fin.set_eq_fold
thf(fact_5534_pred__prod__inject,axiom,
! [A: $tType,B: $tType,P14: A > $o,P25: B > $o,A3: A,B2: B] :
( ( basic_pred_prod @ A @ B @ P14 @ P25 @ ( product_Pair @ A @ B @ A3 @ B2 ) )
= ( ( P14 @ A3 )
& ( P25 @ B2 ) ) ) ).
% pred_prod_inject
thf(fact_5535_fold__append,axiom,
! [A: $tType,B: $tType,F: B > A > A,Xs: list @ B,Ys2: list @ B] :
( ( fold @ B @ A @ F @ ( append @ B @ Xs @ Ys2 ) )
= ( comp @ A @ A @ A @ ( fold @ B @ A @ F @ Ys2 ) @ ( fold @ B @ A @ F @ Xs ) ) ) ).
% fold_append
thf(fact_5536_pred__prod__split,axiom,
! [B: $tType,A: $tType,P: $o > $o,Q: A > $o,R5: B > $o,Xy2: product_prod @ A @ B] :
( ( P @ ( basic_pred_prod @ A @ B @ Q @ R5 @ Xy2 ) )
= ( ! [X3: A,Y3: B] :
( ( Xy2
= ( product_Pair @ A @ B @ X3 @ Y3 ) )
=> ( P
@ ( ( Q @ X3 )
& ( R5 @ Y3 ) ) ) ) ) ) ).
% pred_prod_split
thf(fact_5537_pred__prod_Ocases,axiom,
! [A: $tType,B: $tType,P14: A > $o,P25: B > $o,A3: product_prod @ A @ B] :
( ( basic_pred_prod @ A @ B @ P14 @ P25 @ A3 )
=> ~ ! [A5: A,B3: B] :
( ( A3
= ( product_Pair @ A @ B @ A5 @ B3 ) )
=> ( ( P14 @ A5 )
=> ~ ( P25 @ B3 ) ) ) ) ).
% pred_prod.cases
thf(fact_5538_pred__prod_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( basic_pred_prod @ A @ B )
= ( ^ [P12: A > $o,P23: B > $o,A7: product_prod @ A @ B] :
? [B5: A,C3: B] :
( ( A7
= ( product_Pair @ A @ B @ B5 @ C3 ) )
& ( P12 @ B5 )
& ( P23 @ C3 ) ) ) ) ).
% pred_prod.simps
thf(fact_5539_pred__prod_Ointros,axiom,
! [A: $tType,B: $tType,P14: A > $o,A3: A,P25: B > $o,B2: B] :
( ( P14 @ A3 )
=> ( ( P25 @ B2 )
=> ( basic_pred_prod @ A @ B @ P14 @ P25 @ ( product_Pair @ A @ B @ A3 @ B2 ) ) ) ) ).
% pred_prod.intros
thf(fact_5540_fold__Cons,axiom,
! [B: $tType,A: $tType,F: A > B > B,X: A,Xs: list @ A] :
( ( fold @ A @ B @ F @ ( cons @ A @ X @ Xs ) )
= ( comp @ B @ B @ B @ ( fold @ A @ B @ F @ Xs ) @ ( F @ X ) ) ) ).
% fold_Cons
thf(fact_5541_fold__commute__apply,axiom,
! [A: $tType,C: $tType,B: $tType,Xs: list @ A,H3: B > C,G: A > B > B,F: A > C > C,S: B] :
( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( ( comp @ B @ C @ B @ H3 @ ( G @ X2 ) )
= ( comp @ C @ C @ B @ ( F @ X2 ) @ H3 ) ) )
=> ( ( H3 @ ( fold @ A @ B @ G @ Xs @ S ) )
= ( fold @ A @ C @ F @ Xs @ ( H3 @ S ) ) ) ) ).
% fold_commute_apply
thf(fact_5542_fold__commute,axiom,
! [A: $tType,C: $tType,B: $tType,Xs: list @ A,H3: B > C,G: A > B > B,F: A > C > C] :
( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( ( comp @ B @ C @ B @ H3 @ ( G @ X2 ) )
= ( comp @ C @ C @ B @ ( F @ X2 ) @ H3 ) ) )
=> ( ( comp @ B @ C @ B @ H3 @ ( fold @ A @ B @ G @ Xs ) )
= ( comp @ C @ C @ B @ ( fold @ A @ C @ F @ Xs ) @ H3 ) ) ) ).
% fold_commute
thf(fact_5543_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
thf(fact_5544_fold__multiset__equiv,axiom,
! [B: $tType,A: $tType,Xs: list @ A,F: A > B > B,Ys2: list @ A] :
( ! [X2: A,Y4: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( ( member @ A @ Y4 @ ( set2 @ A @ Xs ) )
=> ( ( comp @ B @ B @ B @ ( F @ X2 ) @ ( F @ Y4 ) )
= ( comp @ B @ B @ B @ ( F @ Y4 ) @ ( F @ X2 ) ) ) ) )
=> ( ( ( mset @ A @ Xs )
= ( mset @ A @ Ys2 ) )
=> ( ( fold @ A @ B @ F @ Xs )
= ( fold @ A @ B @ F @ Ys2 ) ) ) ) ).
% fold_multiset_equiv
thf(fact_5545_fold__rev,axiom,
! [B: $tType,A: $tType,Xs: list @ A,F: A > B > B] :
( ! [X2: A,Y4: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( ( member @ A @ Y4 @ ( set2 @ A @ Xs ) )
=> ( ( comp @ B @ B @ B @ ( F @ Y4 ) @ ( F @ X2 ) )
= ( comp @ B @ B @ B @ ( F @ X2 ) @ ( F @ Y4 ) ) ) ) )
=> ( ( fold @ A @ B @ F @ ( rev @ A @ Xs ) )
= ( fold @ A @ B @ F @ Xs ) ) ) ).
% fold_rev
thf(fact_5546_foldr__fold,axiom,
! [B: $tType,A: $tType,Xs: list @ A,F: A > B > B] :
( ! [X2: A,Y4: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( ( member @ A @ Y4 @ ( set2 @ A @ Xs ) )
=> ( ( comp @ B @ B @ B @ ( F @ Y4 ) @ ( F @ X2 ) )
= ( comp @ B @ B @ B @ ( F @ X2 ) @ ( F @ Y4 ) ) ) ) )
=> ( ( foldr @ A @ B @ F @ Xs )
= ( fold @ A @ B @ F @ Xs ) ) ) ).
% foldr_fold
thf(fact_5547_fold__plus__sum__list__rev,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [Xs: list @ A] :
( ( fold @ A @ A @ ( plus_plus @ A ) @ Xs )
= ( plus_plus @ A @ ( groups8242544230860333062m_list @ A @ ( rev @ A @ Xs ) ) ) ) ) ).
% fold_plus_sum_list_rev
thf(fact_5548_fold__remove1__split,axiom,
! [B: $tType,A: $tType,Xs: list @ A,F: A > B > B,X: A] :
( ! [X2: A,Y4: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( ( member @ A @ Y4 @ ( set2 @ A @ Xs ) )
=> ( ( comp @ B @ B @ B @ ( F @ X2 ) @ ( F @ Y4 ) )
= ( comp @ B @ B @ B @ ( F @ Y4 ) @ ( F @ X2 ) ) ) ) )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( fold @ A @ B @ F @ Xs )
= ( comp @ B @ B @ B @ ( fold @ A @ B @ F @ ( remove1 @ A @ X @ Xs ) ) @ ( F @ X ) ) ) ) ) ).
% fold_remove1_split
thf(fact_5549_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
thf(fact_5550_subset__mset_Omax__add__distrib__left,axiom,
! [A: $tType,X: multiset @ A,Y: multiset @ A,Z2: multiset @ A] :
( ( plus_plus @ ( multiset @ A ) @ ( max @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ X @ Y ) @ Z2 )
= ( max @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( plus_plus @ ( multiset @ A ) @ X @ Z2 ) @ ( plus_plus @ ( multiset @ A ) @ Y @ Z2 ) ) ) ).
% subset_mset.max_add_distrib_left
thf(fact_5551_subset__mset_Omax__add__distrib__right,axiom,
! [A: $tType,X: multiset @ A,Y: multiset @ A,Z2: multiset @ A] :
( ( plus_plus @ ( multiset @ A ) @ X @ ( max @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ Y @ Z2 ) )
= ( max @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( plus_plus @ ( multiset @ A ) @ X @ Y ) @ ( plus_plus @ ( multiset @ A ) @ X @ Z2 ) ) ) ).
% subset_mset.max_add_distrib_right
thf(fact_5552_prod__list_Ocomm__monoid__list__axioms,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ( groups1828464146339083142d_list @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ).
% prod_list.comm_monoid_list_axioms
thf(fact_5553_sum__list_Ocomm__monoid__list__axioms,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ( groups1828464146339083142d_list @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ).
% sum_list.comm_monoid_list_axioms
thf(fact_5554_lazI,axiom,
! [A: $tType,B: $tType,A3: list @ A,B2: list @ B,P: A > B > $o] :
( ( ( size_size @ ( list @ A ) @ A3 )
= ( size_size @ ( list @ B ) @ B2 ) )
=> ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ B ) @ B2 ) )
=> ( P @ ( nth @ A @ A3 @ I2 ) @ ( nth @ B @ B2 @ I2 ) ) )
=> ( list_all_zip @ A @ B @ P @ A3 @ B2 ) ) ) ).
% lazI
thf(fact_5555_list__all__zip__alt,axiom,
! [B: $tType,A: $tType] :
( ( list_all_zip @ A @ B )
= ( ^ [P4: A > B > $o,As5: list @ A,Bs3: list @ B] :
( ( ( size_size @ ( list @ A ) @ As5 )
= ( size_size @ ( list @ B ) @ Bs3 ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ As5 ) )
=> ( P4 @ ( nth @ A @ As5 @ I3 ) @ ( nth @ B @ Bs3 @ I3 ) ) ) ) ) ) ).
% list_all_zip_alt
thf(fact_5556_is__none__bind,axiom,
! [A: $tType,B: $tType,F: option @ B,G: B > ( option @ A )] :
( ( is_none @ A @ ( bind @ B @ A @ F @ G ) )
= ( ( is_none @ B @ F )
| ( is_none @ A @ ( G @ ( the2 @ B @ F ) ) ) ) ) ).
% is_none_bind
thf(fact_5557_laz__conj,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,A3: list @ A,B2: list @ B] :
( ( list_all_zip @ A @ B
@ ^ [X3: A,Y3: B] :
( ( P @ X3 @ Y3 )
& ( Q @ X3 @ Y3 ) )
@ A3
@ B2 )
= ( ( list_all_zip @ A @ B @ P @ A3 @ B2 )
& ( list_all_zip @ A @ B @ Q @ A3 @ B2 ) ) ) ).
% laz_conj
thf(fact_5558_is__none__code_I2_J,axiom,
! [B: $tType,X: B] :
~ ( is_none @ B @ ( some @ B @ X ) ) ).
% is_none_code(2)
thf(fact_5559_is__none__code_I1_J,axiom,
! [A: $tType] : ( is_none @ A @ ( none @ A ) ) ).
% is_none_code(1)
thf(fact_5560_is__none__map__option,axiom,
! [A: $tType,B: $tType,F: B > A,X: option @ B] :
( ( is_none @ A @ ( map_option @ B @ A @ F @ X ) )
= ( is_none @ B @ X ) ) ).
% is_none_map_option
thf(fact_5561_laz__weak__Pb,axiom,
! [A: $tType,B: $tType,P: B > $o,A4: list @ A,B6: list @ B] :
( ( list_all_zip @ A @ B
@ ^ [A7: A] : P
@ A4
@ B6 )
= ( ( ( size_size @ ( list @ A ) @ A4 )
= ( size_size @ ( list @ B ) @ B6 ) )
& ! [X3: B] :
( ( member @ B @ X3 @ ( set2 @ B @ B6 ) )
=> ( P @ X3 ) ) ) ) ).
% laz_weak_Pb
thf(fact_5562_laz__weak__Pa,axiom,
! [B: $tType,A: $tType,P: A > $o,A4: list @ A,B6: list @ B] :
( ( list_all_zip @ A @ B
@ ^ [A7: A,B5: B] : ( P @ A7 )
@ A4
@ B6 )
= ( ( ( size_size @ ( list @ A ) @ A4 )
= ( size_size @ ( list @ B ) @ B6 ) )
& ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ A4 ) )
=> ( P @ X3 ) ) ) ) ).
% laz_weak_Pa
thf(fact_5563_list__all__zip_Osimps_I1_J,axiom,
! [A: $tType,B: $tType,P: A > B > $o] : ( list_all_zip @ A @ B @ P @ ( nil @ A ) @ ( nil @ B ) ) ).
% list_all_zip.simps(1)
thf(fact_5564_laz__eq,axiom,
! [A: $tType,A3: list @ A,B2: list @ A] :
( ( list_all_zip @ A @ A
@ ^ [Y5: A,Z4: A] : Y5 = Z4
@ A3
@ B2 )
= ( A3 = B2 ) ) ).
% laz_eq
thf(fact_5565_laz__len,axiom,
! [A: $tType,B: $tType,P: A > B > $o,A3: list @ A,B2: list @ B] :
( ( list_all_zip @ A @ B @ P @ A3 @ B2 )
=> ( ( size_size @ ( list @ A ) @ A3 )
= ( size_size @ ( list @ B ) @ B2 ) ) ) ).
% laz_len
thf(fact_5566_laz__swap__ex,axiom,
! [B: $tType,A: $tType,C: $tType,P: A > B > C > $o,A4: list @ A,B6: list @ B] :
( ( list_all_zip @ A @ B
@ ^ [A7: A,B5: B] :
? [X8: C] : ( P @ A7 @ B5 @ X8 )
@ A4
@ B6 )
=> ~ ! [C8: list @ C] :
( ( list_all_zip @ A @ C
@ ^ [A7: A,C3: C] :
? [B5: B] : ( P @ A7 @ B5 @ C3 )
@ A4
@ C8 )
=> ~ ( list_all_zip @ B @ C
@ ^ [B5: B,C3: C] :
? [A7: A] : ( P @ A7 @ B5 @ C3 )
@ B6
@ C8 ) ) ) ).
% laz_swap_ex
thf(fact_5567_is__none__simps_I2_J,axiom,
! [B: $tType,X: B] :
~ ( is_none @ B @ ( some @ B @ X ) ) ).
% is_none_simps(2)
thf(fact_5568_list__all__zip__map2,axiom,
! [A: $tType,B: $tType,C: $tType,P: A > B > $o,As2: list @ A,F: C > B,Bs2: list @ C] :
( ( list_all_zip @ A @ B @ P @ As2 @ ( map @ C @ B @ F @ Bs2 ) )
= ( list_all_zip @ A @ C
@ ^ [A7: A,B5: C] : ( P @ A7 @ ( F @ B5 ) )
@ As2
@ Bs2 ) ) ).
% list_all_zip_map2
thf(fact_5569_list__all__zip__map1,axiom,
! [C: $tType,A: $tType,B: $tType,P: A > B > $o,F: C > A,As2: list @ C,Bs2: list @ B] :
( ( list_all_zip @ A @ B @ P @ ( map @ C @ A @ F @ As2 ) @ Bs2 )
= ( list_all_zip @ C @ B
@ ^ [A7: C] : ( P @ ( F @ A7 ) )
@ As2
@ Bs2 ) ) ).
% list_all_zip_map1
thf(fact_5570_is__none__simps_I1_J,axiom,
! [A: $tType] : ( is_none @ A @ ( none @ A ) ) ).
% is_none_simps(1)
thf(fact_5571_Option_Ois__none__def,axiom,
! [A: $tType] :
( ( is_none @ A )
= ( ^ [X3: option @ A] :
( X3
= ( none @ A ) ) ) ) ).
% Option.is_none_def
thf(fact_5572_list__all__zip_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,P: A > B > $o,A3: A,As2: list @ A,B2: B,Bs2: list @ B] :
( ( list_all_zip @ A @ B @ P @ ( cons @ A @ A3 @ As2 ) @ ( cons @ B @ B2 @ Bs2 ) )
= ( ( P @ A3 @ B2 )
& ( list_all_zip @ A @ B @ P @ As2 @ Bs2 ) ) ) ).
% list_all_zip.simps(2)
thf(fact_5573_list__all__zip_Oelims_I3_J,axiom,
! [A: $tType,B: $tType,X: A > B > $o,Xa: list @ A,Xb: list @ B] :
( ~ ( list_all_zip @ A @ B @ X @ Xa @ Xb )
=> ( ! [A5: A,As: list @ A] :
( ( Xa
= ( cons @ A @ A5 @ As ) )
=> ! [B3: B,Bs: list @ B] :
( ( Xb
= ( cons @ B @ B3 @ Bs ) )
=> ( ( X @ A5 @ B3 )
& ( list_all_zip @ A @ B @ X @ As @ Bs ) ) ) )
=> ( ( ? [V4: A,Va: list @ A] :
( Xa
= ( cons @ A @ V4 @ Va ) )
=> ( Xb
!= ( nil @ B ) ) )
=> ~ ( ( Xa
= ( nil @ A ) )
=> ! [V4: B,Va: list @ B] :
( Xb
!= ( cons @ B @ V4 @ Va ) ) ) ) ) ) ).
% list_all_zip.elims(3)
thf(fact_5574_list__all__zip_Oelims_I2_J,axiom,
! [A: $tType,B: $tType,X: A > B > $o,Xa: list @ A,Xb: list @ B] :
( ( list_all_zip @ A @ B @ X @ Xa @ Xb )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( Xb
!= ( nil @ B ) ) )
=> ~ ! [A5: A,As: list @ A] :
( ( Xa
= ( cons @ A @ A5 @ As ) )
=> ! [B3: B,Bs: list @ B] :
( ( Xb
= ( cons @ B @ B3 @ Bs ) )
=> ~ ( ( X @ A5 @ B3 )
& ( list_all_zip @ A @ B @ X @ As @ Bs ) ) ) ) ) ) ).
% list_all_zip.elims(2)
thf(fact_5575_list__all__zip_Oelims_I1_J,axiom,
! [B: $tType,A: $tType,X: A > B > $o,Xa: list @ A,Xb: list @ B,Y: $o] :
( ( ( list_all_zip @ A @ B @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( ( Xb
= ( nil @ B ) )
=> ~ Y ) )
=> ( ! [A5: A,As: list @ A] :
( ( Xa
= ( cons @ A @ A5 @ As ) )
=> ! [B3: B,Bs: list @ B] :
( ( Xb
= ( cons @ B @ B3 @ Bs ) )
=> ( Y
= ( ~ ( ( X @ A5 @ B3 )
& ( list_all_zip @ A @ B @ X @ As @ Bs ) ) ) ) ) )
=> ( ( ? [V4: A,Va: list @ A] :
( Xa
= ( cons @ A @ V4 @ Va ) )
=> ( ( Xb
= ( nil @ B ) )
=> Y ) )
=> ~ ( ( Xa
= ( nil @ A ) )
=> ( ? [V4: B,Va: list @ B] :
( Xb
= ( cons @ B @ V4 @ Va ) )
=> Y ) ) ) ) ) ) ).
% list_all_zip.elims(1)
thf(fact_5576_list__all__zip_Osimps_I4_J,axiom,
! [A: $tType,B: $tType,P: A > B > $o,V: B,Va2: list @ B] :
~ ( list_all_zip @ A @ B @ P @ ( nil @ A ) @ ( cons @ B @ V @ Va2 ) ) ).
% list_all_zip.simps(4)
thf(fact_5577_list__all__zip_Osimps_I3_J,axiom,
! [A: $tType,B: $tType,P: A > B > $o,V: A,Va2: list @ A] :
~ ( list_all_zip @ A @ B @ P @ ( cons @ A @ V @ Va2 ) @ ( nil @ B ) ) ).
% list_all_zip.simps(3)
thf(fact_5578_the__map__option,axiom,
! [B: $tType,A: $tType,X: option @ A,F: A > B] :
( ~ ( is_none @ A @ X )
=> ( ( the2 @ B @ ( map_option @ A @ B @ F @ X ) )
= ( F @ ( the2 @ A @ X ) ) ) ) ).
% the_map_option
thf(fact_5579_rel__optionI,axiom,
! [A: $tType,B: $tType,X: option @ A,Y: option @ B,P: A > B > $o] :
( ( ( is_none @ A @ X )
= ( is_none @ B @ Y ) )
=> ( ( ~ ( is_none @ A @ X )
=> ( ~ ( is_none @ B @ Y )
=> ( P @ ( the2 @ A @ X ) @ ( the2 @ B @ Y ) ) ) )
=> ( rel_option @ A @ B @ P @ X @ Y ) ) ) ).
% rel_optionI
thf(fact_5580_rel__option__unfold,axiom,
! [B: $tType,A: $tType] :
( ( rel_option @ A @ B )
= ( ^ [R6: A > B > $o,X3: option @ A,Y3: option @ B] :
( ( ( is_none @ A @ X3 )
= ( is_none @ B @ Y3 ) )
& ( ~ ( is_none @ A @ X3 )
=> ( ~ ( is_none @ B @ Y3 )
=> ( R6 @ ( the2 @ A @ X3 ) @ ( the2 @ B @ Y3 ) ) ) ) ) ) ) ).
% rel_option_unfold
thf(fact_5581_list__all__zip_Opelims_I1_J,axiom,
! [A: $tType,B: $tType,X: A > B > $o,Xa: list @ A,Xb: list @ B,Y: $o] :
( ( ( list_all_zip @ A @ B @ X @ Xa @ Xb )
= Y )
=> ( ( accp @ ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( list_all_zip_rel @ A @ B ) @ ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xa @ Xb ) ) )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( ( Xb
= ( nil @ B ) )
=> ( Y
=> ~ ( accp @ ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( list_all_zip_rel @ A @ B ) @ ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( nil @ B ) ) ) ) ) ) )
=> ( ! [A5: A,As: list @ A] :
( ( Xa
= ( cons @ A @ A5 @ As ) )
=> ! [B3: B,Bs: list @ B] :
( ( Xb
= ( cons @ B @ B3 @ Bs ) )
=> ( ( Y
= ( ( X @ A5 @ B3 )
& ( list_all_zip @ A @ B @ X @ As @ Bs ) ) )
=> ~ ( accp @ ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( list_all_zip_rel @ A @ B ) @ ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A5 @ As ) @ ( cons @ B @ B3 @ Bs ) ) ) ) ) ) )
=> ( ! [V4: A,Va: list @ A] :
( ( Xa
= ( cons @ A @ V4 @ Va ) )
=> ( ( Xb
= ( nil @ B ) )
=> ( ~ Y
=> ~ ( accp @ ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( list_all_zip_rel @ A @ B ) @ ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ V4 @ Va ) @ ( nil @ B ) ) ) ) ) ) )
=> ~ ( ( Xa
= ( nil @ A ) )
=> ! [V4: B,Va: list @ B] :
( ( Xb
= ( cons @ B @ V4 @ Va ) )
=> ( ~ Y
=> ~ ( accp @ ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( list_all_zip_rel @ A @ B ) @ ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( cons @ B @ V4 @ Va ) ) ) ) ) ) ) ) ) ) ) ) ).
% list_all_zip.pelims(1)
thf(fact_5582_list__all__zip_Opelims_I2_J,axiom,
! [A: $tType,B: $tType,X: A > B > $o,Xa: list @ A,Xb: list @ B] :
( ( list_all_zip @ A @ B @ X @ Xa @ Xb )
=> ( ( accp @ ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( list_all_zip_rel @ A @ B ) @ ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xa @ Xb ) ) )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( ( Xb
= ( nil @ B ) )
=> ~ ( accp @ ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( list_all_zip_rel @ A @ B ) @ ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( nil @ B ) ) ) ) ) )
=> ~ ! [A5: A,As: list @ A] :
( ( Xa
= ( cons @ A @ A5 @ As ) )
=> ! [B3: B,Bs: list @ B] :
( ( Xb
= ( cons @ B @ B3 @ Bs ) )
=> ( ( accp @ ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( list_all_zip_rel @ A @ B ) @ ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A5 @ As ) @ ( cons @ B @ B3 @ Bs ) ) ) )
=> ~ ( ( X @ A5 @ B3 )
& ( list_all_zip @ A @ B @ X @ As @ Bs ) ) ) ) ) ) ) ) ).
% list_all_zip.pelims(2)
thf(fact_5583_list__all__zip_Opelims_I3_J,axiom,
! [A: $tType,B: $tType,X: A > B > $o,Xa: list @ A,Xb: list @ B] :
( ~ ( list_all_zip @ A @ B @ X @ Xa @ Xb )
=> ( ( accp @ ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( list_all_zip_rel @ A @ B ) @ ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xa @ Xb ) ) )
=> ( ! [A5: A,As: list @ A] :
( ( Xa
= ( cons @ A @ A5 @ As ) )
=> ! [B3: B,Bs: list @ B] :
( ( Xb
= ( cons @ B @ B3 @ Bs ) )
=> ( ( accp @ ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( list_all_zip_rel @ A @ B ) @ ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A5 @ As ) @ ( cons @ B @ B3 @ Bs ) ) ) )
=> ( ( X @ A5 @ B3 )
& ( list_all_zip @ A @ B @ X @ As @ Bs ) ) ) ) )
=> ( ! [V4: A,Va: list @ A] :
( ( Xa
= ( cons @ A @ V4 @ Va ) )
=> ( ( Xb
= ( nil @ B ) )
=> ~ ( accp @ ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( list_all_zip_rel @ A @ B ) @ ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ V4 @ Va ) @ ( nil @ B ) ) ) ) ) )
=> ~ ( ( Xa
= ( nil @ A ) )
=> ! [V4: B,Va: list @ B] :
( ( Xb
= ( cons @ B @ V4 @ Va ) )
=> ~ ( accp @ ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( list_all_zip_rel @ A @ B ) @ ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( cons @ B @ V4 @ Va ) ) ) ) ) ) ) ) ) ) ).
% list_all_zip.pelims(3)
thf(fact_5584_plus__multiset_Otransfer,axiom,
! [A: $tType] :
( bNF_rel_fun @ ( A > nat ) @ ( multiset @ A ) @ ( ( A > nat ) > A > nat ) @ ( ( multiset @ A ) > ( multiset @ A ) )
@ ( pcr_multiset @ A @ A
@ ^ [Y5: A,Z4: A] : Y5 = Z4 )
@ ( bNF_rel_fun @ ( A > nat ) @ ( multiset @ A ) @ ( A > nat ) @ ( multiset @ A )
@ ( pcr_multiset @ A @ A
@ ^ [Y5: A,Z4: A] : Y5 = Z4 )
@ ( pcr_multiset @ A @ A
@ ^ [Y5: A,Z4: A] : Y5 = Z4 ) )
@ ^ [M8: A > nat,N9: A > nat,A7: A] : ( plus_plus @ nat @ ( M8 @ A7 ) @ ( N9 @ A7 ) )
@ ( plus_plus @ ( multiset @ A ) ) ) ).
% plus_multiset.transfer
thf(fact_5585_map__option__o__case__sum,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F: D > C,G: A > ( option @ D ),H3: B > ( option @ D )] :
( ( comp @ ( option @ D ) @ ( option @ C ) @ ( sum_sum @ A @ B ) @ ( map_option @ D @ C @ F ) @ ( sum_case_sum @ A @ ( option @ D ) @ B @ G @ H3 ) )
= ( sum_case_sum @ A @ ( option @ C ) @ B @ ( comp @ ( option @ D ) @ ( option @ C ) @ A @ ( map_option @ D @ C @ F ) @ G ) @ ( comp @ ( option @ D ) @ ( option @ C ) @ B @ ( map_option @ D @ C @ F ) @ H3 ) ) ) ).
% map_option_o_case_sum
thf(fact_5586_semilattice__set_Oeq__fold_H,axiom,
! [A: $tType,F: A > A > A,A4: set @ A] :
( ( lattic149705377957585745ce_set @ A @ F )
=> ( ( lattic1715443433743089157tice_F @ A @ F @ A4 )
= ( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X3: A,Y3: option @ A] : ( some @ A @ ( case_option @ A @ A @ X3 @ ( F @ X3 ) @ Y3 ) )
@ ( none @ A )
@ A4 ) ) ) ) ).
% semilattice_set.eq_fold'
thf(fact_5587_o__case__sum,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,H3: D > C,F: A > D,G: B > D] :
( ( comp @ D @ C @ ( sum_sum @ A @ B ) @ H3 @ ( sum_case_sum @ A @ D @ B @ F @ G ) )
= ( sum_case_sum @ A @ C @ B @ ( comp @ D @ C @ A @ H3 @ F ) @ ( comp @ D @ C @ B @ H3 @ G ) ) ) ).
% o_case_sum
thf(fact_5588_semilattice__set_Oinfinite,axiom,
! [A: $tType,F: A > A > A,A4: set @ A] :
( ( lattic149705377957585745ce_set @ A @ F )
=> ( ~ ( finite_finite2 @ A @ A4 )
=> ( ( lattic1715443433743089157tice_F @ A @ F @ A4 )
= ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% semilattice_set.infinite
thf(fact_5589_group_Oaxioms_I2_J,axiom,
! [A: $tType,F: A > A > A,Z2: A,Inverse: A > A] :
( ( group @ A @ F @ Z2 @ Inverse )
=> ( group_axioms @ A @ F @ Z2 @ Inverse ) ) ).
% group.axioms(2)
thf(fact_5590_normalize__div,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ A3 )
= ( divide_divide @ A @ ( one_one @ A ) @ ( unit_f5069060285200089521factor @ A @ A3 ) ) ) ) ).
% normalize_div
thf(fact_5591_strict__mono__inv,axiom,
! [A: $tType,B: $tType] :
( ( ( linorder @ B )
& ( linorder @ A ) )
=> ! [F: A > B,G: B > A] :
( ( order_strict_mono @ A @ B @ F )
=> ( ( ( image2 @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ B ) ) )
=> ( ! [X2: A] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( order_strict_mono @ B @ A @ G ) ) ) ) ) ).
% strict_mono_inv
thf(fact_5592_gcd_Onormalize__bottom,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( normal6383669964737779283malize @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% gcd.normalize_bottom
thf(fact_5593_normalize__1,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ( ( normal6383669964737779283malize @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% normalize_1
thf(fact_5594_lcm_Otop__left__normalize,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A] :
( ( gcd_lcm @ A @ ( one_one @ A ) @ A3 )
= ( normal6383669964737779283malize @ A @ A3 ) ) ) ).
% lcm.top_left_normalize
thf(fact_5595_lcm_Otop__right__normalize,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A] :
( ( gcd_lcm @ A @ A3 @ ( one_one @ A ) )
= ( normal6383669964737779283malize @ A @ A3 ) ) ) ).
% lcm.top_right_normalize
thf(fact_5596_normalize__mult__unit__right,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [B2: A,A3: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( normal6383669964737779283malize @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( normal6383669964737779283malize @ A @ A3 ) ) ) ) ).
% normalize_mult_unit_right
thf(fact_5597_normalize__mult__unit__left,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( normal6383669964737779283malize @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( normal6383669964737779283malize @ A @ B2 ) ) ) ) ).
% normalize_mult_unit_left
thf(fact_5598_unit__factor__normalize,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( normal6383669964737779283malize @ A @ A3 ) )
= ( one_one @ A ) ) ) ) ).
% unit_factor_normalize
thf(fact_5599_normalize__unit__factor,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( normal6383669964737779283malize @ A @ ( unit_f5069060285200089521factor @ A @ A3 ) )
= ( one_one @ A ) ) ) ) ).
% normalize_unit_factor
thf(fact_5600_associated__unit,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: A] :
( ( ( normal6383669964737779283malize @ A @ A3 )
= ( normal6383669964737779283malize @ A @ B2 ) )
=> ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) ) ) ) ).
% associated_unit
thf(fact_5601_normalize__1__iff,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( ( normal6383669964737779283malize @ A @ A3 )
= ( one_one @ A ) )
= ( dvd_dvd @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% normalize_1_iff
thf(fact_5602_is__unit__normalize,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( normal6383669964737779283malize @ A @ A3 )
= ( one_one @ A ) ) ) ) ).
% is_unit_normalize
thf(fact_5603_normalize__idem__imp__is__unit__iff,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( ( normal6383669964737779283malize @ A @ A3 )
= A3 )
=> ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
= ( A3
= ( one_one @ A ) ) ) ) ) ).
% normalize_idem_imp_is_unit_iff
thf(fact_5604_group__axioms__def,axiom,
! [A: $tType] :
( ( group_axioms @ A )
= ( ^ [F2: A > A > A,Z5: A,Inverse2: A > A] :
( ! [A7: A] :
( ( F2 @ Z5 @ A7 )
= A7 )
& ! [A7: A] :
( ( F2 @ ( Inverse2 @ A7 ) @ A7 )
= Z5 ) ) ) ) ).
% group_axioms_def
thf(fact_5605_group__axioms_Ointro,axiom,
! [A: $tType,F: A > A > A,Z2: A,Inverse: A > A] :
( ! [A5: A] :
( ( F @ Z2 @ A5 )
= A5 )
=> ( ! [A5: A] :
( ( F @ ( Inverse @ A5 ) @ A5 )
= Z2 )
=> ( group_axioms @ A @ F @ Z2 @ Inverse ) ) ) ).
% group_axioms.intro
thf(fact_5606_strict__mono__imp__inj__on,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F: A > B,A4: set @ A] :
( ( order_strict_mono @ A @ B @ F )
=> ( inj_on @ A @ B @ F @ A4 ) ) ) ).
% strict_mono_imp_inj_on
thf(fact_5607_Lcm__fin_Obounded__quasi__semilattice__set__axioms,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( bounde6485984586167503788ce_set @ A @ ( gcd_lcm @ A ) @ ( one_one @ A ) @ ( zero_zero @ A ) @ ( normal6383669964737779283malize @ A ) ) ) ).
% Lcm_fin.bounded_quasi_semilattice_set_axioms
thf(fact_5608_Gcd__fin_Obounded__quasi__semilattice__set__axioms,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( bounde6485984586167503788ce_set @ A @ ( gcd_gcd @ A ) @ ( zero_zero @ A ) @ ( one_one @ A ) @ ( normal6383669964737779283malize @ A ) ) ) ).
% Gcd_fin.bounded_quasi_semilattice_set_axioms
thf(fact_5609_lcm_Obounded__quasi__semilattice__axioms,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( bounde8507323023520639062attice @ A @ ( gcd_lcm @ A ) @ ( one_one @ A ) @ ( zero_zero @ A ) @ ( normal6383669964737779283malize @ A ) ) ) ).
% lcm.bounded_quasi_semilattice_axioms
thf(fact_5610_gcd_Obounded__quasi__semilattice__axioms,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( bounde8507323023520639062attice @ A @ ( gcd_gcd @ A ) @ ( zero_zero @ A ) @ ( one_one @ A ) @ ( normal6383669964737779283malize @ A ) ) ) ).
% gcd.bounded_quasi_semilattice_axioms
thf(fact_5611_group__def,axiom,
! [A: $tType] :
( ( group @ A )
= ( ^ [F2: A > A > A,Z5: A,Inverse2: A > A] :
( ( semigroup @ A @ F2 )
& ( group_axioms @ A @ F2 @ Z5 @ Inverse2 ) ) ) ) ).
% group_def
thf(fact_5612_group_Ointro,axiom,
! [A: $tType,F: A > A > A,Z2: A,Inverse: A > A] :
( ( semigroup @ A @ F )
=> ( ( group_axioms @ A @ F @ Z2 @ Inverse )
=> ( group @ A @ F @ Z2 @ Inverse ) ) ) ).
% group.intro
thf(fact_5613_Predicate_Oiterate__upto_Opsimps,axiom,
! [A: $tType,F: code_natural > A,N: code_natural,M2: code_natural] :
( ( accp @ ( product_prod @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) ) @ ( iterate_upto_rel @ A ) @ ( product_Pair @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) @ F @ ( product_Pair @ code_natural @ code_natural @ N @ M2 ) ) )
=> ( ( iterate_upto @ A @ F @ N @ M2 )
= ( seq2 @ A
@ ^ [U2: product_unit] : ( if @ ( seq @ A ) @ ( ord_less @ code_natural @ M2 @ N ) @ ( empty @ A ) @ ( insert @ A @ ( F @ N ) @ ( iterate_upto @ A @ F @ ( plus_plus @ code_natural @ N @ ( one_one @ code_natural ) ) @ M2 ) ) ) ) ) ) ).
% Predicate.iterate_upto.psimps
thf(fact_5614_group_Oaxioms_I1_J,axiom,
! [A: $tType,F: A > A > A,Z2: A,Inverse: A > A] :
( ( group @ A @ F @ Z2 @ Inverse )
=> ( semigroup @ A @ F ) ) ).
% group.axioms(1)
thf(fact_5615_monoid_Oaxioms_I1_J,axiom,
! [A: $tType,F: A > A > A,Z2: A] :
( ( monoid @ A @ F @ Z2 )
=> ( semigroup @ A @ F ) ) ).
% monoid.axioms(1)
thf(fact_5616_semigroup__def,axiom,
! [A: $tType] :
( ( semigroup @ A )
= ( ^ [F2: A > A > A] :
! [A7: A,B5: A,C3: A] :
( ( F2 @ ( F2 @ A7 @ B5 ) @ C3 )
= ( F2 @ A7 @ ( F2 @ B5 @ C3 ) ) ) ) ) ).
% semigroup_def
thf(fact_5617_semigroup_Ointro,axiom,
! [A: $tType,F: A > A > A] :
( ! [A5: A,B3: A,C4: A] :
( ( F @ ( F @ A5 @ B3 ) @ C4 )
= ( F @ A5 @ ( F @ B3 @ C4 ) ) )
=> ( semigroup @ A @ F ) ) ).
% semigroup.intro
thf(fact_5618_semigroup_Oassoc,axiom,
! [A: $tType,F: A > A > A,A3: A,B2: A,C2: A] :
( ( semigroup @ A @ F )
=> ( ( F @ ( F @ A3 @ B2 ) @ C2 )
= ( F @ A3 @ ( F @ B2 @ C2 ) ) ) ) ).
% semigroup.assoc
thf(fact_5619_mult_Osemigroup__axioms,axiom,
! [A: $tType] :
( ( semigroup_mult @ A )
=> ( semigroup @ A @ ( times_times @ A ) ) ) ).
% mult.semigroup_axioms
thf(fact_5620_Predicate_Oiterate__upto_Osimps,axiom,
! [A: $tType] :
( ( iterate_upto @ A )
= ( ^ [F2: code_natural > A,N5: code_natural,M: code_natural] :
( seq2 @ A
@ ^ [U2: product_unit] : ( if @ ( seq @ A ) @ ( ord_less @ code_natural @ M @ N5 ) @ ( empty @ A ) @ ( insert @ A @ ( F2 @ N5 ) @ ( iterate_upto @ A @ F2 @ ( plus_plus @ code_natural @ N5 @ ( one_one @ code_natural ) ) @ M ) ) ) ) ) ) ).
% Predicate.iterate_upto.simps
thf(fact_5621_Predicate_Oiterate__upto_Oelims,axiom,
! [A: $tType,X: code_natural > A,Xa: code_natural,Xb: code_natural,Y: pred @ A] :
( ( ( iterate_upto @ A @ X @ Xa @ Xb )
= Y )
=> ( Y
= ( seq2 @ A
@ ^ [U2: product_unit] : ( if @ ( seq @ A ) @ ( ord_less @ code_natural @ Xb @ Xa ) @ ( empty @ A ) @ ( insert @ A @ ( X @ Xa ) @ ( iterate_upto @ A @ X @ ( plus_plus @ code_natural @ Xa @ ( one_one @ code_natural ) ) @ Xb ) ) ) ) ) ) ).
% Predicate.iterate_upto.elims
thf(fact_5622_add_Osemigroup__axioms,axiom,
! [A: $tType] :
( ( semigroup_add @ A )
=> ( semigroup @ A @ ( plus_plus @ A ) ) ) ).
% add.semigroup_axioms
thf(fact_5623_Predicate_Oiterate__upto_Opelims,axiom,
! [A: $tType,X: code_natural > A,Xa: code_natural,Xb: code_natural,Y: pred @ A] :
( ( ( iterate_upto @ A @ X @ Xa @ Xb )
= Y )
=> ( ( accp @ ( product_prod @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) ) @ ( iterate_upto_rel @ A ) @ ( product_Pair @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) @ X @ ( product_Pair @ code_natural @ code_natural @ Xa @ Xb ) ) )
=> ~ ( ( Y
= ( seq2 @ A
@ ^ [U2: product_unit] : ( if @ ( seq @ A ) @ ( ord_less @ code_natural @ Xb @ Xa ) @ ( empty @ A ) @ ( insert @ A @ ( X @ Xa ) @ ( iterate_upto @ A @ X @ ( plus_plus @ code_natural @ Xa @ ( one_one @ code_natural ) ) @ Xb ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) ) @ ( iterate_upto_rel @ A ) @ ( product_Pair @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) @ X @ ( product_Pair @ code_natural @ code_natural @ Xa @ Xb ) ) ) ) ) ) ).
% Predicate.iterate_upto.pelims
thf(fact_5624_Random__Pred_Oiterate__upto__def,axiom,
! [A: $tType] :
( ( random_iterate_upto @ A )
= ( ^ [F2: code_natural > A,N5: code_natural,M: code_natural] : ( product_Pair @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) @ ( iterate_upto @ A @ F2 @ N5 @ M ) ) ) ) ).
% Random_Pred.iterate_upto_def
thf(fact_5625_monoid_Ointro,axiom,
! [A: $tType,F: A > A > A,Z2: A] :
( ( semigroup @ A @ F )
=> ( ( monoid_axioms @ A @ F @ Z2 )
=> ( monoid @ A @ F @ Z2 ) ) ) ).
% monoid.intro
thf(fact_5626_monoid__def,axiom,
! [A: $tType] :
( ( monoid @ A )
= ( ^ [F2: A > A > A,Z5: A] :
( ( semigroup @ A @ F2 )
& ( monoid_axioms @ A @ F2 @ Z5 ) ) ) ) ).
% monoid_def
thf(fact_5627_monoid__axioms_Ointro,axiom,
! [A: $tType,F: A > A > A,Z2: A] :
( ! [A5: A] :
( ( F @ Z2 @ A5 )
= A5 )
=> ( ! [A5: A] :
( ( F @ A5 @ Z2 )
= A5 )
=> ( monoid_axioms @ A @ F @ Z2 ) ) ) ).
% monoid_axioms.intro
thf(fact_5628_monoid__axioms__def,axiom,
! [A: $tType] :
( ( monoid_axioms @ A )
= ( ^ [F2: A > A > A,Z5: A] :
( ! [A7: A] :
( ( F2 @ Z5 @ A7 )
= A7 )
& ! [A7: A] :
( ( F2 @ A7 @ Z5 )
= A7 ) ) ) ) ).
% monoid_axioms_def
thf(fact_5629_monoid_Oaxioms_I2_J,axiom,
! [A: $tType,F: A > A > A,Z2: A] :
( ( monoid @ A @ F @ Z2 )
=> ( monoid_axioms @ A @ F @ Z2 ) ) ).
% monoid.axioms(2)
thf(fact_5630_mod__h__bot__normalize,axiom,
! [A: $tType,H3: heap_ext @ product_unit,P: assn] :
( ( syntax7388354845996824322omatch @ A @ ( heap_ext @ product_unit ) @ ( undefined @ A ) @ H3 )
=> ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) )
= ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ ( undefined @ ( heap_ext @ product_unit ) ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% mod_h_bot_normalize
thf(fact_5631_image__bind,axiom,
! [A: $tType,B: $tType,C: $tType,F: B > A,A4: set @ C,G: C > ( set @ B )] :
( ( image2 @ B @ A @ F @ ( bind3 @ C @ B @ A4 @ G ) )
= ( bind3 @ C @ A @ A4 @ ( comp @ ( set @ B ) @ ( set @ A ) @ C @ ( image2 @ B @ A @ F ) @ G ) ) ) ).
% image_bind
thf(fact_5632_pure__assn__raw_Ocases,axiom,
! [A: $tType,B: $tType,X: product_prod @ $o @ ( product_prod @ A @ ( set @ B ) )] :
~ ! [B3: $o,H2: A,As: set @ B] :
( X
!= ( product_Pair @ $o @ ( product_prod @ A @ ( set @ B ) ) @ B3 @ ( product_Pair @ A @ ( set @ B ) @ H2 @ As ) ) ) ).
% pure_assn_raw.cases
thf(fact_5633_mod__h__bot__iff_I7_J,axiom,
! [P: assn,Q: assn,H3: heap_ext @ product_unit] :
( ( rep_assn @ ( sup_sup @ assn @ P @ Q ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) )
= ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) )
| ( rep_assn @ Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% mod_h_bot_iff(7)
thf(fact_5634_mod__h__bot__iff_I5_J,axiom,
! [P: assn,Q: assn,H3: heap_ext @ product_unit] :
( ( rep_assn @ ( times_times @ assn @ P @ Q ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) )
= ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) )
& ( rep_assn @ Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% mod_h_bot_iff(5)
thf(fact_5635_mod__h__bot__iff_I6_J,axiom,
! [P: assn,Q: assn,H3: heap_ext @ product_unit] :
( ( rep_assn @ ( inf_inf @ assn @ P @ Q ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) )
= ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) )
& ( rep_assn @ Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% mod_h_bot_iff(6)
thf(fact_5636_mod__h__bot__iff_I2_J,axiom,
! [H3: heap_ext @ product_unit] : ( rep_assn @ ( top_top @ assn ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% mod_h_bot_iff(2)
thf(fact_5637_times__assn__raw_Ocases,axiom,
! [X: product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) )] :
~ ! [P7: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Q7: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,H2: heap_ext @ product_unit,As: set @ nat] :
( X
!= ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ P7 @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Q7 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) ) ) ).
% times_assn_raw.cases
thf(fact_5638_sngr__assn__raw_Ocases,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [X: product_prod @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) )] :
~ ! [R2: ref @ A,X2: A,H2: heap_ext @ product_unit,As: set @ nat] :
( X
!= ( product_Pair @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ R2 @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ X2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) ) ) ) ).
% sngr_assn_raw.cases
thf(fact_5639_snga__assn__raw_Ocases,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [X: product_prod @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) )] :
~ ! [R2: array @ A,X2: list @ A,H2: heap_ext @ product_unit,As: set @ nat] :
( X
!= ( product_Pair @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ R2 @ ( product_Pair @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ X2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) ) ) ) ).
% snga_assn_raw.cases
thf(fact_5640_one__assn__raw_Ocases,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
( X
!= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) ).
% one_assn_raw.cases
thf(fact_5641_mod__star__conv,axiom,
! [A4: assn,B6: assn,H3: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( rep_assn @ ( times_times @ assn @ A4 @ B6 ) @ H3 )
= ( ? [Hr: heap_ext @ product_unit,As1: set @ nat,As22: set @ nat] :
( ( H3
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ Hr @ ( sup_sup @ ( set @ nat ) @ As1 @ As22 ) ) )
& ( ( inf_inf @ ( set @ nat ) @ As1 @ As22 )
= ( bot_bot @ ( set @ nat ) ) )
& ( rep_assn @ A4 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ Hr @ As1 ) )
& ( rep_assn @ B6 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ Hr @ As22 ) ) ) ) ) ).
% mod_star_conv
thf(fact_5642_star__assnI,axiom,
! [P: assn,H3: heap_ext @ product_unit,As2: set @ nat,Q: assn,As6: set @ nat] :
( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) )
=> ( ( rep_assn @ Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As6 ) )
=> ( ( ( inf_inf @ ( set @ nat ) @ As2 @ As6 )
= ( bot_bot @ ( set @ nat ) ) )
=> ( rep_assn @ ( times_times @ assn @ P @ Q ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( sup_sup @ ( set @ nat ) @ As2 @ As6 ) ) ) ) ) ) ).
% star_assnI
thf(fact_5643_mod__emp__simp,axiom,
! [H3: heap_ext @ product_unit] : ( rep_assn @ ( one_one @ assn ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% mod_emp_simp
thf(fact_5644_mod__h__bot__indep,axiom,
! [P: assn,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit] :
( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) )
= ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% mod_h_bot_indep
thf(fact_5645_bind__image,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > B,A4: set @ C,G: B > ( set @ A )] :
( ( bind3 @ B @ A @ ( image2 @ C @ B @ F @ A4 ) @ G )
= ( bind3 @ C @ A @ A4 @ ( comp @ B @ ( set @ A ) @ C @ G @ F ) ) ) ).
% bind_image
thf(fact_5646_mod__h__bot__iff_I3_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [P5: ref @ A,X: A,H3: heap_ext @ product_unit] :
~ ( rep_assn @ ( sngr_assn @ A @ P5 @ X ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% mod_h_bot_iff(3)
thf(fact_5647_mod__h__bot__iff_I4_J,axiom,
! [B: $tType] :
( ( heap @ B )
=> ! [Q3: array @ B,Y: list @ B,H3: heap_ext @ product_unit] :
~ ( rep_assn @ ( snga_assn @ B @ Q3 @ Y ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% mod_h_bot_iff(4)
thf(fact_5648_times__assn__raw_Osimps,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Q: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,H3: heap_ext @ product_unit,As2: set @ nat] :
( ( times_assn_raw @ P @ Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) )
= ( ? [As1: set @ nat,As22: set @ nat] :
( ( As2
= ( sup_sup @ ( set @ nat ) @ As1 @ As22 ) )
& ( ( inf_inf @ ( set @ nat ) @ As1 @ As22 )
= ( bot_bot @ ( set @ nat ) ) )
& ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As1 ) )
& ( Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As22 ) ) ) ) ) ).
% times_assn_raw.simps
thf(fact_5649_times__assn__raw_Oelims_I3_J,axiom,
! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ~ ( times_assn_raw @ X @ Xa @ Xb )
=> ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
=> ? [As12: set @ nat,As23: set @ nat] :
( ( As
= ( sup_sup @ ( set @ nat ) @ As12 @ As23 ) )
& ( ( inf_inf @ ( set @ nat ) @ As12 @ As23 )
= ( bot_bot @ ( set @ nat ) ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As12 ) )
& ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As23 ) ) ) ) ) ).
% times_assn_raw.elims(3)
thf(fact_5650_times__assn__raw_Oelims_I2_J,axiom,
! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( times_assn_raw @ X @ Xa @ Xb )
=> ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
=> ~ ? [As13: set @ nat,As24: set @ nat] :
( ( As
= ( sup_sup @ ( set @ nat ) @ As13 @ As24 ) )
& ( ( inf_inf @ ( set @ nat ) @ As13 @ As24 )
= ( bot_bot @ ( set @ nat ) ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As13 ) )
& ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As24 ) ) ) ) ) ).
% times_assn_raw.elims(2)
thf(fact_5651_times__assn__raw_Oelims_I1_J,axiom,
! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
( ( ( times_assn_raw @ X @ Xa @ Xb )
= Y )
=> ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
=> ( Y
= ( ~ ? [As1: set @ nat,As22: set @ nat] :
( ( As
= ( sup_sup @ ( set @ nat ) @ As1 @ As22 ) )
& ( ( inf_inf @ ( set @ nat ) @ As1 @ As22 )
= ( bot_bot @ ( set @ nat ) ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As1 ) )
& ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As22 ) ) ) ) ) ) ) ).
% times_assn_raw.elims(1)
thf(fact_5652_times__assn__raw_Opelims_I1_J,axiom,
! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
( ( ( times_assn_raw @ X @ Xa @ Xb )
= Y )
=> ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ times_assn_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
=> ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
=> ( ( Y
= ( ? [As1: set @ nat,As22: set @ nat] :
( ( As
= ( sup_sup @ ( set @ nat ) @ As1 @ As22 ) )
& ( ( inf_inf @ ( set @ nat ) @ As1 @ As22 )
= ( bot_bot @ ( set @ nat ) ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As1 ) )
& ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As22 ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ times_assn_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) ) ) ) ) ) ) ).
% times_assn_raw.pelims(1)
thf(fact_5653_times__assn__raw_Opelims_I2_J,axiom,
! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( times_assn_raw @ X @ Xa @ Xb )
=> ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ times_assn_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
=> ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
=> ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ times_assn_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) ) )
=> ~ ? [As13: set @ nat,As24: set @ nat] :
( ( As
= ( sup_sup @ ( set @ nat ) @ As13 @ As24 ) )
& ( ( inf_inf @ ( set @ nat ) @ As13 @ As24 )
= ( bot_bot @ ( set @ nat ) ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As13 ) )
& ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As24 ) ) ) ) ) ) ) ).
% times_assn_raw.pelims(2)
thf(fact_5654_times__assn__raw_Opelims_I3_J,axiom,
! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ~ ( times_assn_raw @ X @ Xa @ Xb )
=> ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ times_assn_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
=> ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
=> ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ times_assn_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) ) )
=> ? [As12: set @ nat,As23: set @ nat] :
( ( As
= ( sup_sup @ ( set @ nat ) @ As12 @ As23 ) )
& ( ( inf_inf @ ( set @ nat ) @ As12 @ As23 )
= ( bot_bot @ ( set @ nat ) ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As12 ) )
& ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As23 ) ) ) ) ) ) ) ).
% times_assn_raw.pelims(3)
thf(fact_5655_mod__h__bot__iff_I8_J,axiom,
! [C: $tType,R5: C > assn,H3: heap_ext @ product_unit] :
( ( rep_assn @ ( ex_assn @ C @ R5 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) )
= ( ? [X3: C] : ( rep_assn @ ( R5 @ X3 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% mod_h_bot_iff(8)
thf(fact_5656_make__rule,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [N: nat,F: nat > A] :
( hoare_hoare_triple @ ( array @ A ) @ ( one_one @ assn ) @ ( array_make @ A @ N @ F )
@ ^ [R3: array @ A] : ( snga_assn @ A @ R3 @ ( map @ nat @ A @ F @ ( upt @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% make_rule
thf(fact_5657_mod__h__bot__iff_I1_J,axiom,
! [B2: $o,H3: heap_ext @ product_unit] :
( ( rep_assn @ ( pure_assn @ B2 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) )
= B2 ) ).
% mod_h_bot_iff(1)
thf(fact_5658_bind__rule,axiom,
! [A: $tType,B: $tType,P: assn,F: heap_Time_Heap @ A,R5: A > assn,G: A > ( heap_Time_Heap @ B ),Q: B > assn] :
( ( hoare_hoare_triple @ A @ P @ F @ R5 )
=> ( ! [X2: A] : ( hoare_hoare_triple @ B @ ( R5 @ X2 ) @ ( G @ X2 ) @ Q )
=> ( hoare_hoare_triple @ B @ P @ ( heap_Time_bind @ A @ B @ F @ G ) @ Q ) ) ) ).
% bind_rule
thf(fact_5659_raise__iff,axiom,
! [A: $tType,P: assn,S: list @ char,Q: A > assn] :
( ( hoare_hoare_triple @ A @ P @ ( heap_Time_raise @ A @ S ) @ Q )
= ( P
= ( bot_bot @ assn ) ) ) ).
% raise_iff
thf(fact_5660_raise__rule,axiom,
! [A: $tType,S: list @ char,Q: A > assn] : ( hoare_hoare_triple @ A @ ( bot_bot @ assn ) @ ( heap_Time_raise @ A @ S ) @ Q ) ).
% raise_rule
thf(fact_5661_bind__rule_H,axiom,
! [A: $tType,B: $tType,C: $tType,P: assn,F: heap_Time_Heap @ B,R5: B > assn,G: B > ( heap_Time_Heap @ C ),Q: C > assn] :
( ! [R2: A] :
( hoare_hoare_triple @ B @ P @ F
@ ^ [S4: B] : ( times_times @ assn @ P @ ( R5 @ S4 ) ) )
=> ( ! [X2: B] : ( hoare_hoare_triple @ C @ ( times_times @ assn @ ( R5 @ X2 ) @ P ) @ ( G @ X2 ) @ Q )
=> ( hoare_hoare_triple @ C @ P @ ( heap_Time_bind @ B @ C @ F @ G ) @ Q ) ) ) ).
% bind_rule'
thf(fact_5662_case__prod__rule,axiom,
! [A: $tType,B: $tType,C: $tType,X: product_prod @ A @ B,P: assn,F: A > B > ( heap_Time_Heap @ C ),Q: C > assn] :
( ! [A5: A,B3: B] :
( ( X
= ( product_Pair @ A @ B @ A5 @ B3 ) )
=> ( hoare_hoare_triple @ C @ P @ ( F @ A5 @ B3 ) @ Q ) )
=> ( hoare_hoare_triple @ C @ P @ ( product_case_prod @ A @ B @ ( heap_Time_Heap @ C ) @ F @ X ) @ Q ) ) ).
% case_prod_rule
thf(fact_5663_hoare__triple__success,axiom,
! [A: $tType,P: assn,C2: heap_Time_Heap @ A,Q: A > assn,H3: heap_ext @ product_unit,As2: set @ nat] :
( ( hoare_hoare_triple @ A @ P @ C2 @ Q )
=> ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) )
=> ( heap_Time_success @ A @ C2 @ H3 ) ) ) ).
% hoare_triple_success
thf(fact_5664_case__option__rule,axiom,
! [A: $tType,B: $tType,V: option @ A,P: assn,Fn: heap_Time_Heap @ B,Q: B > assn,Fs: A > ( heap_Time_Heap @ B )] :
( ( ( V
= ( none @ A ) )
=> ( hoare_hoare_triple @ B @ P @ Fn @ Q ) )
=> ( ! [X2: A] :
( ( V
= ( some @ A @ X2 ) )
=> ( hoare_hoare_triple @ B @ P @ ( Fs @ X2 ) @ Q ) )
=> ( hoare_hoare_triple @ B @ P @ ( case_option @ ( heap_Time_Heap @ B ) @ A @ Fn @ Fs @ V ) @ Q ) ) ) ).
% case_option_rule
thf(fact_5665_of__list__rule,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [Xs: list @ A] :
( hoare_hoare_triple @ ( array @ A ) @ ( one_one @ assn ) @ ( array_of_list @ A @ Xs )
@ ^ [R3: array @ A] : ( snga_assn @ A @ R3 @ Xs ) ) ) ).
% of_list_rule
thf(fact_5666_new__rule,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [N: nat,X: A] :
( hoare_hoare_triple @ ( array @ A ) @ ( one_one @ assn ) @ ( array_new @ A @ N @ X )
@ ^ [R3: array @ A] : ( snga_assn @ A @ R3 @ ( replicate @ A @ N @ X ) ) ) ) ).
% new_rule
thf(fact_5667_hoare__triple__effect,axiom,
! [A: $tType,P: assn,C2: heap_Time_Heap @ A,Q: A > assn,H3: heap_ext @ product_unit,As2: set @ nat] :
( ( hoare_hoare_triple @ A @ P @ C2 @ Q )
=> ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) )
=> ? [H6: heap_ext @ product_unit,R2: A,T7: nat] :
( ( heap_Time_effect @ A @ C2 @ H3 @ H6 @ R2 @ T7 )
& ( rep_assn @ ( Q @ R2 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ ( hoare_new_addrs @ H3 @ As2 @ H6 ) ) ) ) ) ) ).
% hoare_triple_effect
thf(fact_5668_hoare__triple__def,axiom,
! [A: $tType] :
( ( hoare_hoare_triple @ A )
= ( ^ [P4: assn,C3: heap_Time_Heap @ A,Q6: A > assn] :
! [H: heap_ext @ product_unit,As5: set @ nat] :
( ( rep_assn @ P4 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As5 ) )
=> ? [R3: A,H7: heap_ext @ product_unit] :
( ? [T3: nat] :
( ( heap_Time_execute @ A @ C3 @ H )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H7 @ T3 ) ) ) )
& ( rep_assn @ ( Q6 @ R3 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ ( hoare_new_addrs @ H @ As5 @ H7 ) ) )
& ( relH
@ ( collect @ nat
@ ^ [A7: nat] :
( ( ord_less @ nat @ A7 @ ( lim @ product_unit @ H ) )
& ~ ( member @ nat @ A7 @ As5 ) ) )
@ H
@ H7 )
& ( ord_less_eq @ nat @ ( lim @ product_unit @ H ) @ ( lim @ product_unit @ H7 ) ) ) ) ) ) ).
% hoare_triple_def
thf(fact_5669_hoare__tripleI,axiom,
! [A: $tType,P: assn,C2: heap_Time_Heap @ A,Q: A > assn] :
( ! [H2: heap_ext @ product_unit,As: set @ nat] :
( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
=> ? [R15: A,H9: heap_ext @ product_unit,T8: nat] :
( ( ( heap_Time_execute @ A @ C2 @ H2 )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R15 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H9 @ T8 ) ) ) )
& ( rep_assn @ ( Q @ R15 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H9 @ ( hoare_new_addrs @ H2 @ As @ H9 ) ) )
& ( relH
@ ( collect @ nat
@ ^ [A7: nat] :
( ( ord_less @ nat @ A7 @ ( lim @ product_unit @ H2 ) )
& ~ ( member @ nat @ A7 @ As ) ) )
@ H2
@ H9 )
& ( ord_less_eq @ nat @ ( lim @ product_unit @ H2 ) @ ( lim @ product_unit @ H9 ) ) ) )
=> ( hoare_hoare_triple @ A @ P @ C2 @ Q ) ) ).
% hoare_tripleI
thf(fact_5670_mod__relH,axiom,
! [As2: set @ nat,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,P: assn] :
( ( relH @ As2 @ H3 @ H4 )
=> ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) )
= ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As2 ) ) ) ) ).
% mod_relH
thf(fact_5671_hoare__tripleE,axiom,
! [A: $tType,P: assn,C2: heap_Time_Heap @ A,Q: A > assn,H3: heap_ext @ product_unit,As2: set @ nat] :
( ( hoare_hoare_triple @ A @ P @ C2 @ Q )
=> ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) )
=> ~ ! [R2: A,H6: heap_ext @ product_unit] :
( ? [T7: nat] :
( ( heap_Time_execute @ A @ C2 @ H3 )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H6 @ T7 ) ) ) )
=> ( ( rep_assn @ ( Q @ R2 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ ( hoare_new_addrs @ H3 @ As2 @ H6 ) ) )
=> ( ( relH
@ ( collect @ nat
@ ^ [A7: nat] :
( ( ord_less @ nat @ A7 @ ( lim @ product_unit @ H3 ) )
& ~ ( member @ nat @ A7 @ As2 ) ) )
@ H3
@ H6 )
=> ~ ( ord_less_eq @ nat @ ( lim @ product_unit @ H3 ) @ ( lim @ product_unit @ H6 ) ) ) ) ) ) ) ).
% hoare_tripleE
thf(fact_5672_Array__Time_Oalloc__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_alloc @ A )
= ( ^ [Xs3: list @ A,H: heap_ext @ product_unit] :
( product_Pair @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array2 @ A @ ( lim @ product_unit @ H ) )
@ ( array_set @ A @ ( array2 @ A @ ( lim @ product_unit @ H ) ) @ Xs3
@ ( lim_update @ product_unit
@ ^ [Uu: nat] : ( plus_plus @ nat @ ( lim @ product_unit @ H ) @ ( one_one @ nat ) )
@ H ) ) ) ) ) ) ).
% Array_Time.alloc_def
thf(fact_5673_snga__assn__raw_Opelims_I1_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [X: array @ A,Xa: list @ A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
( ( ( snga_assn_raw @ A @ X @ Xa @ Xb )
= Y )
=> ( ( accp @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ ( snga_assn_raw_rel @ A ) @ ( product_Pair @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
=> ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
=> ( ( Y
= ( ( ( array_get @ A @ H2 @ X )
= Xa )
& ( As
= ( insert2 @ nat @ ( addr_of_array @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_array @ A @ X ) @ ( lim @ product_unit @ H2 ) ) ) )
=> ~ ( accp @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ ( snga_assn_raw_rel @ A ) @ ( product_Pair @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) ) ) ) ) ) ) ) ).
% snga_assn_raw.pelims(1)
thf(fact_5674_snga__assn__raw_Opelims_I2_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [X: array @ A,Xa: list @ A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( snga_assn_raw @ A @ X @ Xa @ Xb )
=> ( ( accp @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ ( snga_assn_raw_rel @ A ) @ ( product_Pair @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
=> ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
=> ( ( accp @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ ( snga_assn_raw_rel @ A ) @ ( product_Pair @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) ) )
=> ~ ( ( ( array_get @ A @ H2 @ X )
= Xa )
& ( As
= ( insert2 @ nat @ ( addr_of_array @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_array @ A @ X ) @ ( lim @ product_unit @ H2 ) ) ) ) ) ) ) ) ).
% snga_assn_raw.pelims(2)
thf(fact_5675_Array__Time_Opresent__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_present @ A )
= ( ^ [H: heap_ext @ product_unit,A7: array @ A] : ( ord_less @ nat @ ( addr_of_array @ A @ A7 ) @ ( lim @ product_unit @ H ) ) ) ) ) ).
% Array_Time.present_def
thf(fact_5676_snga__assn__raw_Osimps,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R: array @ A,X: list @ A,H3: heap_ext @ product_unit,As2: set @ nat] :
( ( snga_assn_raw @ A @ R @ X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) )
= ( ( ( array_get @ A @ H3 @ R )
= X )
& ( As2
= ( insert2 @ nat @ ( addr_of_array @ A @ R ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_array @ A @ R ) @ ( lim @ product_unit @ H3 ) ) ) ) ) ).
% snga_assn_raw.simps
thf(fact_5677_snga__assn__raw_Oelims_I1_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [X: array @ A,Xa: list @ A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
( ( ( snga_assn_raw @ A @ X @ Xa @ Xb )
= Y )
=> ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
=> ( Y
= ( ~ ( ( ( array_get @ A @ H2 @ X )
= Xa )
& ( As
= ( insert2 @ nat @ ( addr_of_array @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_array @ A @ X ) @ ( lim @ product_unit @ H2 ) ) ) ) ) ) ) ) ).
% snga_assn_raw.elims(1)
thf(fact_5678_snga__assn__raw_Oelims_I2_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [X: array @ A,Xa: list @ A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( snga_assn_raw @ A @ X @ Xa @ Xb )
=> ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
=> ~ ( ( ( array_get @ A @ H2 @ X )
= Xa )
& ( As
= ( insert2 @ nat @ ( addr_of_array @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_array @ A @ X ) @ ( lim @ product_unit @ H2 ) ) ) ) ) ) ).
% snga_assn_raw.elims(2)
thf(fact_5679_snga__assn__raw_Oelims_I3_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [X: array @ A,Xa: list @ A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ~ ( snga_assn_raw @ A @ X @ Xa @ Xb )
=> ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
=> ( ( ( array_get @ A @ H2 @ X )
= Xa )
& ( As
= ( insert2 @ nat @ ( addr_of_array @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_array @ A @ X ) @ ( lim @ product_unit @ H2 ) ) ) ) ) ) ).
% snga_assn_raw.elims(3)
thf(fact_5680_snga__assn__raw_Opelims_I3_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [X: array @ A,Xa: list @ A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ~ ( snga_assn_raw @ A @ X @ Xa @ Xb )
=> ( ( accp @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ ( snga_assn_raw_rel @ A ) @ ( product_Pair @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
=> ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
=> ( ( accp @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ ( snga_assn_raw_rel @ A ) @ ( product_Pair @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) ) )
=> ( ( ( array_get @ A @ H2 @ X )
= Xa )
& ( As
= ( insert2 @ nat @ ( addr_of_array @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_array @ A @ X ) @ ( lim @ product_unit @ H2 ) ) ) ) ) ) ) ) ).
% snga_assn_raw.pelims(3)
thf(fact_5681_relH__set__array,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R: array @ A,As2: set @ nat,H3: heap_ext @ product_unit,X: list @ A] :
( ~ ( member @ nat @ ( addr_of_array @ A @ R ) @ As2 )
=> ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) )
=> ( relH @ As2 @ H3 @ ( array_set @ A @ R @ X @ H3 ) ) ) ) ) ).
% relH_set_array
thf(fact_5682_sngr__assn__raw_Opelims_I1_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [X: ref @ A,Xa: A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
( ( ( sngr_assn_raw @ A @ X @ Xa @ Xb )
= Y )
=> ( ( accp @ ( product_prod @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ ( sngr_assn_raw_rel @ A ) @ ( product_Pair @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
=> ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
=> ( ( Y
= ( ( ( ref_get @ A @ H2 @ X )
= Xa )
& ( As
= ( insert2 @ nat @ ( addr_of_ref @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_ref @ A @ X ) @ ( lim @ product_unit @ H2 ) ) ) )
=> ~ ( accp @ ( product_prod @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ ( sngr_assn_raw_rel @ A ) @ ( product_Pair @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) ) ) ) ) ) ) ) ).
% sngr_assn_raw.pelims(1)
thf(fact_5683_sngr__assn__raw_Opelims_I2_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [X: ref @ A,Xa: A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( sngr_assn_raw @ A @ X @ Xa @ Xb )
=> ( ( accp @ ( product_prod @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ ( sngr_assn_raw_rel @ A ) @ ( product_Pair @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
=> ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
=> ( ( accp @ ( product_prod @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ ( sngr_assn_raw_rel @ A ) @ ( product_Pair @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) ) )
=> ~ ( ( ( ref_get @ A @ H2 @ X )
= Xa )
& ( As
= ( insert2 @ nat @ ( addr_of_ref @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_ref @ A @ X ) @ ( lim @ product_unit @ H2 ) ) ) ) ) ) ) ) ).
% sngr_assn_raw.pelims(2)
thf(fact_5684_in__range__empty,axiom,
! [H3: heap_ext @ product_unit] : ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% in_range_empty
thf(fact_5685_in__range__dist__union,axiom,
! [H3: heap_ext @ product_unit,As2: set @ nat,As6: set @ nat] :
( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( sup_sup @ ( set @ nat ) @ As2 @ As6 ) ) )
= ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As6 ) ) ) ) ).
% in_range_dist_union
thf(fact_5686_relH__set__ref,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R: ref @ A,As2: set @ nat,H3: heap_ext @ product_unit,X: A] :
( ~ ( member @ nat @ ( addr_of_ref @ A @ R ) @ As2 )
=> ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) )
=> ( relH @ As2 @ H3 @ ( ref_set @ A @ R @ X @ H3 ) ) ) ) ) ).
% relH_set_ref
thf(fact_5687_relH__refl,axiom,
! [H3: heap_ext @ product_unit,As2: set @ nat] :
( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) )
=> ( relH @ As2 @ H3 @ H3 ) ) ).
% relH_refl
thf(fact_5688_relH__in__rangeI_I1_J,axiom,
! [As2: set @ nat,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit] :
( ( relH @ As2 @ H3 @ H4 )
=> ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) ) ) ).
% relH_in_rangeI(1)
thf(fact_5689_relH__in__rangeI_I2_J,axiom,
! [As2: set @ nat,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit] :
( ( relH @ As2 @ H3 @ H4 )
=> ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As2 ) ) ) ).
% relH_in_rangeI(2)
thf(fact_5690_in__range__subset,axiom,
! [As2: set @ nat,As6: set @ nat,H3: heap_ext @ product_unit] :
( ( ord_less_eq @ ( set @ nat ) @ As2 @ As6 )
=> ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As6 ) )
=> ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) ) ) ) ).
% in_range_subset
thf(fact_5691_in__range_Oelims_I3_J,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ~ ( in_range @ X )
=> ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
=> ! [X2: nat] :
( ( member @ nat @ X2 @ As )
=> ( ord_less @ nat @ X2 @ ( lim @ product_unit @ H2 ) ) ) ) ) ).
% in_range.elims(3)
thf(fact_5692_in__range_Oelims_I2_J,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( in_range @ X )
=> ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
=> ~ ! [X7: nat] :
( ( member @ nat @ X7 @ As )
=> ( ord_less @ nat @ X7 @ ( lim @ product_unit @ H2 ) ) ) ) ) ).
% in_range.elims(2)
thf(fact_5693_in__range_Oelims_I1_J,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
( ( ( in_range @ X )
= Y )
=> ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
=> ( Y
= ( ~ ! [X3: nat] :
( ( member @ nat @ X3 @ As )
=> ( ord_less @ nat @ X3 @ ( lim @ product_unit @ H2 ) ) ) ) ) ) ) ).
% in_range.elims(1)
thf(fact_5694_in__range_Osimps,axiom,
! [H3: heap_ext @ product_unit,As2: set @ nat] :
( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) )
= ( ! [X3: nat] :
( ( member @ nat @ X3 @ As2 )
=> ( ord_less @ nat @ X3 @ ( lim @ product_unit @ H3 ) ) ) ) ) ).
% in_range.simps
thf(fact_5695_sngr__assn__raw_Oelims_I3_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [X: ref @ A,Xa: A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ~ ( sngr_assn_raw @ A @ X @ Xa @ Xb )
=> ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
=> ( ( ( ref_get @ A @ H2 @ X )
= Xa )
& ( As
= ( insert2 @ nat @ ( addr_of_ref @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_ref @ A @ X ) @ ( lim @ product_unit @ H2 ) ) ) ) ) ) ).
% sngr_assn_raw.elims(3)
thf(fact_5696_sngr__assn__raw_Oelims_I2_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [X: ref @ A,Xa: A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( sngr_assn_raw @ A @ X @ Xa @ Xb )
=> ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
=> ~ ( ( ( ref_get @ A @ H2 @ X )
= Xa )
& ( As
= ( insert2 @ nat @ ( addr_of_ref @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_ref @ A @ X ) @ ( lim @ product_unit @ H2 ) ) ) ) ) ) ).
% sngr_assn_raw.elims(2)
thf(fact_5697_sngr__assn__raw_Oelims_I1_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [X: ref @ A,Xa: A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
( ( ( sngr_assn_raw @ A @ X @ Xa @ Xb )
= Y )
=> ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
=> ( Y
= ( ~ ( ( ( ref_get @ A @ H2 @ X )
= Xa )
& ( As
= ( insert2 @ nat @ ( addr_of_ref @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_ref @ A @ X ) @ ( lim @ product_unit @ H2 ) ) ) ) ) ) ) ) ).
% sngr_assn_raw.elims(1)
thf(fact_5698_sngr__assn__raw_Osimps,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R: ref @ A,X: A,H3: heap_ext @ product_unit,As2: set @ nat] :
( ( sngr_assn_raw @ A @ R @ X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) )
= ( ( ( ref_get @ A @ H3 @ R )
= X )
& ( As2
= ( insert2 @ nat @ ( addr_of_ref @ A @ R ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_ref @ A @ R ) @ ( lim @ product_unit @ H3 ) ) ) ) ) ).
% sngr_assn_raw.simps
thf(fact_5699_sngr__assn__raw_Opelims_I3_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [X: ref @ A,Xa: A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ~ ( sngr_assn_raw @ A @ X @ Xa @ Xb )
=> ( ( accp @ ( product_prod @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ ( sngr_assn_raw_rel @ A ) @ ( product_Pair @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
=> ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
=> ( ( accp @ ( product_prod @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ ( sngr_assn_raw_rel @ A ) @ ( product_Pair @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) ) )
=> ( ( ( ref_get @ A @ H2 @ X )
= Xa )
& ( As
= ( insert2 @ nat @ ( addr_of_ref @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
& ( ord_less @ nat @ ( addr_of_ref @ A @ X ) @ ( lim @ product_unit @ H2 ) ) ) ) ) ) ) ) ).
% sngr_assn_raw.pelims(3)
thf(fact_5700_wand__assnI,axiom,
! [H3: heap_ext @ product_unit,As2: set @ nat,Q: assn,R5: assn] :
( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) )
=> ( ! [H6: heap_ext @ product_unit,As7: set @ nat] :
( ( ( inf_inf @ ( set @ nat ) @ As2 @ As7 )
= ( bot_bot @ ( set @ nat ) ) )
=> ( ( relH @ As2 @ H3 @ H6 )
=> ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ As2 ) )
=> ( ( rep_assn @ Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ As7 ) )
=> ( rep_assn @ R5 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ ( sup_sup @ ( set @ nat ) @ As2 @ As7 ) ) ) ) ) ) )
=> ( rep_assn @ ( wand_assn @ Q @ R5 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) ) ) ) ).
% wand_assnI
thf(fact_5701_wand__raw_Osimps,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Q: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,H3: heap_ext @ product_unit,As2: set @ nat] :
( ( wand_raw @ P @ Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) )
= ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) )
& ! [H7: heap_ext @ product_unit,As8: set @ nat] :
( ( ( ( inf_inf @ ( set @ nat ) @ As2 @ As8 )
= ( bot_bot @ ( set @ nat ) ) )
& ( relH @ As2 @ H3 @ H7 )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ As2 ) )
& ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ As8 ) ) )
=> ( Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ ( sup_sup @ ( set @ nat ) @ As2 @ As8 ) ) ) ) ) ) ).
% wand_raw.simps
thf(fact_5702_wand__raw_Oelims_I1_J,axiom,
! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
( ( ( wand_raw @ X @ Xa @ Xb )
= Y )
=> ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
=> ( Y
= ( ~ ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
& ! [H7: heap_ext @ product_unit,As8: set @ nat] :
( ( ( ( inf_inf @ ( set @ nat ) @ As @ As8 )
= ( bot_bot @ ( set @ nat ) ) )
& ( relH @ As @ H2 @ H7 )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ As ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ As8 ) ) )
=> ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ ( sup_sup @ ( set @ nat ) @ As @ As8 ) ) ) ) ) ) ) ) ) ).
% wand_raw.elims(1)
thf(fact_5703_wand__raw_Oelims_I3_J,axiom,
! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ~ ( wand_raw @ X @ Xa @ Xb )
=> ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
=> ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
& ! [H6: heap_ext @ product_unit,As7: set @ nat] :
( ( ( ( inf_inf @ ( set @ nat ) @ As @ As7 )
= ( bot_bot @ ( set @ nat ) ) )
& ( relH @ As @ H2 @ H6 )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ As ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ As7 ) ) )
=> ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ ( sup_sup @ ( set @ nat ) @ As @ As7 ) ) ) ) ) ) ) ).
% wand_raw.elims(3)
thf(fact_5704_wand__raw_Oelims_I2_J,axiom,
! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( wand_raw @ X @ Xa @ Xb )
=> ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
=> ~ ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
& ! [H9: heap_ext @ product_unit,As9: set @ nat] :
( ( ( ( inf_inf @ ( set @ nat ) @ As @ As9 )
= ( bot_bot @ ( set @ nat ) ) )
& ( relH @ As @ H2 @ H9 )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H9 @ As ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H9 @ As9 ) ) )
=> ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H9 @ ( sup_sup @ ( set @ nat ) @ As @ As9 ) ) ) ) ) ) ) ).
% wand_raw.elims(2)
thf(fact_5705_wand__raw_Opelims_I3_J,axiom,
! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ~ ( wand_raw @ X @ Xa @ Xb )
=> ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ wand_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
=> ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
=> ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ wand_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) ) )
=> ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
& ! [H6: heap_ext @ product_unit,As7: set @ nat] :
( ( ( ( inf_inf @ ( set @ nat ) @ As @ As7 )
= ( bot_bot @ ( set @ nat ) ) )
& ( relH @ As @ H2 @ H6 )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ As ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ As7 ) ) )
=> ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ ( sup_sup @ ( set @ nat ) @ As @ As7 ) ) ) ) ) ) ) ) ) ).
% wand_raw.pelims(3)
thf(fact_5706_wand__raw_Opelims_I2_J,axiom,
! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( wand_raw @ X @ Xa @ Xb )
=> ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ wand_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
=> ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
=> ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ wand_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) ) )
=> ~ ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
& ! [H9: heap_ext @ product_unit,As9: set @ nat] :
( ( ( ( inf_inf @ ( set @ nat ) @ As @ As9 )
= ( bot_bot @ ( set @ nat ) ) )
& ( relH @ As @ H2 @ H9 )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H9 @ As ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H9 @ As9 ) ) )
=> ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H9 @ ( sup_sup @ ( set @ nat ) @ As @ As9 ) ) ) ) ) ) ) ) ) ).
% wand_raw.pelims(2)
thf(fact_5707_wand__raw_Opelims_I1_J,axiom,
! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
( ( ( wand_raw @ X @ Xa @ Xb )
= Y )
=> ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ wand_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
=> ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
( ( Xb
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
=> ( ( Y
= ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
& ! [H7: heap_ext @ product_unit,As8: set @ nat] :
( ( ( ( inf_inf @ ( set @ nat ) @ As @ As8 )
= ( bot_bot @ ( set @ nat ) ) )
& ( relH @ As @ H2 @ H7 )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ As ) )
& ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ As8 ) ) )
=> ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ ( sup_sup @ ( set @ nat ) @ As @ As8 ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ wand_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) ) ) ) ) ) ) ).
% wand_raw.pelims(1)
thf(fact_5708_in__range_Opelims_I3_J,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ~ ( in_range @ X )
=> ( ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ in_range_rel @ X )
=> ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
=> ( ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ in_range_rel @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
=> ! [X2: nat] :
( ( member @ nat @ X2 @ As )
=> ( ord_less @ nat @ X2 @ ( lim @ product_unit @ H2 ) ) ) ) ) ) ) ).
% in_range.pelims(3)
thf(fact_5709_in__range_Opelims_I2_J,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( in_range @ X )
=> ( ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ in_range_rel @ X )
=> ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
=> ( ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ in_range_rel @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
=> ~ ! [X7: nat] :
( ( member @ nat @ X7 @ As )
=> ( ord_less @ nat @ X7 @ ( lim @ product_unit @ H2 ) ) ) ) ) ) ) ).
% in_range.pelims(2)
thf(fact_5710_in__range_Opelims_I1_J,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
( ( ( in_range @ X )
= Y )
=> ( ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ in_range_rel @ X )
=> ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
=> ( ( Y
= ( ! [X3: nat] :
( ( member @ nat @ X3 @ As )
=> ( ord_less @ nat @ X3 @ ( lim @ product_unit @ H2 ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ in_range_rel @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) ) ) ) ) ).
% in_range.pelims(1)
thf(fact_5711_Ref__Time_Oalloc__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( ref_alloc @ A )
= ( ^ [X3: A,H: heap_ext @ product_unit] :
( product_Pair @ ( ref @ A ) @ ( heap_ext @ product_unit ) @ ( ref2 @ A @ ( lim @ product_unit @ H ) )
@ ( ref_set @ A @ ( ref2 @ A @ ( lim @ product_unit @ H ) ) @ X3
@ ( lim_update @ product_unit
@ ^ [Uu: nat] : ( plus_plus @ nat @ ( lim @ product_unit @ H ) @ ( one_one @ nat ) )
@ H ) ) ) ) ) ) ).
% Ref_Time.alloc_def
thf(fact_5712_pure__assn__raw_Osimps,axiom,
! [A: $tType,B: $tType,B2: $o,H3: A,As2: set @ B] :
( ( pure_assn_raw @ A @ B @ B2 @ ( product_Pair @ A @ ( set @ B ) @ H3 @ As2 ) )
= ( ( As2
= ( bot_bot @ ( set @ B ) ) )
& B2 ) ) ).
% pure_assn_raw.simps
thf(fact_5713_pure__assn__raw_Oelims_I1_J,axiom,
! [A: $tType,B: $tType,X: $o,Xa: product_prod @ A @ ( set @ B ),Y: $o] :
( ( ( pure_assn_raw @ A @ B @ X @ Xa )
= Y )
=> ~ ! [H2: A,As: set @ B] :
( ( Xa
= ( product_Pair @ A @ ( set @ B ) @ H2 @ As ) )
=> ( Y
= ( ~ ( ( As
= ( bot_bot @ ( set @ B ) ) )
& X ) ) ) ) ) ).
% pure_assn_raw.elims(1)
thf(fact_5714_pure__assn__raw_Oelims_I3_J,axiom,
! [B: $tType,A: $tType,X: $o,Xa: product_prod @ A @ ( set @ B )] :
( ~ ( pure_assn_raw @ A @ B @ X @ Xa )
=> ~ ! [H2: A,As: set @ B] :
( ( Xa
= ( product_Pair @ A @ ( set @ B ) @ H2 @ As ) )
=> ( ( As
= ( bot_bot @ ( set @ B ) ) )
& X ) ) ) ).
% pure_assn_raw.elims(3)
thf(fact_5715_pure__assn__raw_Oelims_I2_J,axiom,
! [B: $tType,A: $tType,X: $o,Xa: product_prod @ A @ ( set @ B )] :
( ( pure_assn_raw @ A @ B @ X @ Xa )
=> ~ ! [H2: A,As: set @ B] :
( ( Xa
= ( product_Pair @ A @ ( set @ B ) @ H2 @ As ) )
=> ~ ( ( As
= ( bot_bot @ ( set @ B ) ) )
& X ) ) ) ).
% pure_assn_raw.elims(2)
thf(fact_5716_pure__assn__raw_Opelims_I3_J,axiom,
! [B: $tType,A: $tType,X: $o,Xa: product_prod @ A @ ( set @ B )] :
( ~ ( pure_assn_raw @ A @ B @ X @ Xa )
=> ( ( accp @ ( product_prod @ $o @ ( product_prod @ A @ ( set @ B ) ) ) @ ( pure_assn_raw_rel @ A @ B ) @ ( product_Pair @ $o @ ( product_prod @ A @ ( set @ B ) ) @ X @ Xa ) )
=> ~ ! [H2: A,As: set @ B] :
( ( Xa
= ( product_Pair @ A @ ( set @ B ) @ H2 @ As ) )
=> ( ( accp @ ( product_prod @ $o @ ( product_prod @ A @ ( set @ B ) ) ) @ ( pure_assn_raw_rel @ A @ B ) @ ( product_Pair @ $o @ ( product_prod @ A @ ( set @ B ) ) @ X @ ( product_Pair @ A @ ( set @ B ) @ H2 @ As ) ) )
=> ( ( As
= ( bot_bot @ ( set @ B ) ) )
& X ) ) ) ) ) ).
% pure_assn_raw.pelims(3)
thf(fact_5717_pure__assn__raw_Opelims_I2_J,axiom,
! [B: $tType,A: $tType,X: $o,Xa: product_prod @ A @ ( set @ B )] :
( ( pure_assn_raw @ A @ B @ X @ Xa )
=> ( ( accp @ ( product_prod @ $o @ ( product_prod @ A @ ( set @ B ) ) ) @ ( pure_assn_raw_rel @ A @ B ) @ ( product_Pair @ $o @ ( product_prod @ A @ ( set @ B ) ) @ X @ Xa ) )
=> ~ ! [H2: A,As: set @ B] :
( ( Xa
= ( product_Pair @ A @ ( set @ B ) @ H2 @ As ) )
=> ( ( accp @ ( product_prod @ $o @ ( product_prod @ A @ ( set @ B ) ) ) @ ( pure_assn_raw_rel @ A @ B ) @ ( product_Pair @ $o @ ( product_prod @ A @ ( set @ B ) ) @ X @ ( product_Pair @ A @ ( set @ B ) @ H2 @ As ) ) )
=> ~ ( ( As
= ( bot_bot @ ( set @ B ) ) )
& X ) ) ) ) ) ).
% pure_assn_raw.pelims(2)
thf(fact_5718_pure__assn__raw_Opelims_I1_J,axiom,
! [A: $tType,B: $tType,X: $o,Xa: product_prod @ A @ ( set @ B ),Y: $o] :
( ( ( pure_assn_raw @ A @ B @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ $o @ ( product_prod @ A @ ( set @ B ) ) ) @ ( pure_assn_raw_rel @ A @ B ) @ ( product_Pair @ $o @ ( product_prod @ A @ ( set @ B ) ) @ X @ Xa ) )
=> ~ ! [H2: A,As: set @ B] :
( ( Xa
= ( product_Pair @ A @ ( set @ B ) @ H2 @ As ) )
=> ( ( Y
= ( ( As
= ( bot_bot @ ( set @ B ) ) )
& X ) )
=> ~ ( accp @ ( product_prod @ $o @ ( product_prod @ A @ ( set @ B ) ) ) @ ( pure_assn_raw_rel @ A @ B ) @ ( product_Pair @ $o @ ( product_prod @ A @ ( set @ B ) ) @ X @ ( product_Pair @ A @ ( set @ B ) @ H2 @ As ) ) ) ) ) ) ) ).
% pure_assn_raw.pelims(1)
thf(fact_5719_effect__refI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R: ref @ A,H4: heap_ext @ product_unit,V: A,H3: heap_ext @ product_unit,N: nat] :
( ( ( product_Pair @ ( ref @ A ) @ ( heap_ext @ product_unit ) @ R @ H4 )
= ( ref_alloc @ A @ V @ H3 ) )
=> ( ( N
= ( one_one @ nat ) )
=> ( heap_Time_effect @ ( ref @ A ) @ ( ref_ref @ A @ V ) @ H3 @ H4 @ R @ N ) ) ) ) ).
% effect_refI
thf(fact_5720_ref__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( ref_ref @ A )
= ( ^ [V2: A] :
( heap_Time_heap @ ( ref @ A )
@ ^ [H: heap_ext @ product_unit] :
( product_case_prod @ ( ref @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( ref @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
@ ^ [R3: ref @ A,H7: heap_ext @ product_unit] : ( product_Pair @ ( ref @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H7 @ ( one_one @ nat ) ) )
@ ( ref_alloc @ A @ V2 @ H ) ) ) ) ) ) ).
% ref_def
thf(fact_5721_execute__ref,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [V: A,H3: heap_ext @ product_unit] :
( ( heap_Time_execute @ ( ref @ A ) @ ( ref_ref @ A @ V ) @ H3 )
= ( some @ ( product_prod @ ( ref @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
@ ( product_case_prod @ ( ref @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( ref @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
@ ^ [R3: ref @ A,H7: heap_ext @ product_unit] : ( product_Pair @ ( ref @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H7 @ ( one_one @ nat ) ) )
@ ( ref_alloc @ A @ V @ H3 ) ) ) ) ) ).
% execute_ref
thf(fact_5722_effect__refE,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [V: A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: ref @ A,N: nat] :
( ( heap_Time_effect @ ( ref @ A ) @ ( ref_ref @ A @ V ) @ H3 @ H4 @ R @ N )
=> ~ ( ( ( ref_get @ A @ H4 @ R )
= V )
=> ( ( ref_present @ A @ H4 @ R )
=> ( ~ ( ref_present @ A @ H3 @ R )
=> ( N
!= ( one_one @ nat ) ) ) ) ) ) ) ).
% effect_refE
thf(fact_5723_next__fresh,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R: ref @ A,H4: heap_ext @ product_unit,X: A,H3: heap_ext @ product_unit] :
( ( ( product_Pair @ ( ref @ A ) @ ( heap_ext @ product_unit ) @ R @ H4 )
= ( ref_alloc @ A @ X @ H3 ) )
=> ~ ( ref_present @ A @ H3 @ R ) ) ) ).
% next_fresh
thf(fact_5724_next__present,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R: ref @ A,H4: heap_ext @ product_unit,X: A,H3: heap_ext @ product_unit] :
( ( ( product_Pair @ ( ref @ A ) @ ( heap_ext @ product_unit ) @ R @ H4 )
= ( ref_alloc @ A @ X @ H3 ) )
=> ( ref_present @ A @ H4 @ R ) ) ) ).
% next_present
thf(fact_5725_one__assn__raw_Oelims_I3_J,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ~ ( one_assn_raw @ X )
=> ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
=> ( As
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% one_assn_raw.elims(3)
thf(fact_5726_one__assn__raw_Oelims_I2_J,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( one_assn_raw @ X )
=> ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
=> ( As
!= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% one_assn_raw.elims(2)
thf(fact_5727_one__assn__raw_Oelims_I1_J,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
( ( ( one_assn_raw @ X )
= Y )
=> ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
=> ( Y
= ( As
!= ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% one_assn_raw.elims(1)
thf(fact_5728_one__assn__raw_Osimps,axiom,
! [H3: heap_ext @ product_unit,As2: set @ nat] :
( ( one_assn_raw @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) )
= ( As2
= ( bot_bot @ ( set @ nat ) ) ) ) ).
% one_assn_raw.simps
thf(fact_5729_one__assn__raw_Opelims_I3_J,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ~ ( one_assn_raw @ X )
=> ( ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ one_assn_raw_rel @ X )
=> ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
=> ( ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ one_assn_raw_rel @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
=> ( As
= ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% one_assn_raw.pelims(3)
thf(fact_5730_one__assn__raw_Opelims_I2_J,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
( ( one_assn_raw @ X )
=> ( ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ one_assn_raw_rel @ X )
=> ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
=> ( ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ one_assn_raw_rel @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
=> ( As
!= ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% one_assn_raw.pelims(2)
thf(fact_5731_one__assn__raw_Opelims_I1_J,axiom,
! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
( ( ( one_assn_raw @ X )
= Y )
=> ( ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ one_assn_raw_rel @ X )
=> ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
( ( X
= ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
=> ( ( Y
= ( As
= ( bot_bot @ ( set @ nat ) ) ) )
=> ~ ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ one_assn_raw_rel @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) ) ) ) ) ).
% one_assn_raw.pelims(1)
thf(fact_5732_proper__iff,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,As2: set @ nat,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit] :
( ( proper @ P )
=> ( ( relH @ As2 @ H3 @ H4 )
=> ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As2 ) )
=> ( ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) )
= ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As2 ) ) ) ) ) ) ).
% proper_iff
thf(fact_5733_proper__def,axiom,
( proper
= ( ^ [P4: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o] :
! [H: heap_ext @ product_unit,H7: heap_ext @ product_unit,As5: set @ nat] :
( ( ( P4 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As5 ) )
=> ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As5 ) ) )
& ( ( ( P4 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As5 ) )
& ( relH @ As5 @ H @ H7 )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ As5 ) ) )
=> ( P4 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ As5 ) ) ) ) ) ) ).
% proper_def
thf(fact_5734_properD2,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,H3: heap_ext @ product_unit,As2: set @ nat,H4: heap_ext @ product_unit] :
( ( proper @ P )
=> ( ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) )
=> ( ( relH @ As2 @ H3 @ H4 )
=> ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As2 ) )
=> ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As2 ) ) ) ) ) ) ).
% properD2
thf(fact_5735_properD1,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,H3: heap_ext @ product_unit,As2: set @ nat] :
( ( proper @ P )
=> ( ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) )
=> ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) ) ) ) ).
% properD1
thf(fact_5736_properI,axiom,
! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o] :
( ! [As: set @ nat,H2: heap_ext @ product_unit] :
( ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
=> ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) )
=> ( ! [As: set @ nat,H2: heap_ext @ product_unit,H6: heap_ext @ product_unit] :
( ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
=> ( ( relH @ As @ H2 @ H6 )
=> ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ As ) )
=> ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ As ) ) ) ) )
=> ( proper @ P ) ) ) ).
% properI
thf(fact_5737_Array__Time_Onoteq__def,axiom,
! [B: $tType,A: $tType] :
( ( ( heap @ A )
& ( heap @ B ) )
=> ( ( array_noteq @ A @ B )
= ( ^ [R3: array @ A,S4: array @ B] :
( ( ( typerep_typerep @ A @ ( type2 @ A ) )
!= ( typerep_typerep @ B @ ( type2 @ B ) ) )
| ( ( addr_of_array @ A @ R3 )
!= ( addr_of_array @ B @ S4 ) ) ) ) ) ) ).
% Array_Time.noteq_def
thf(fact_5738_comm__monoid__mset_Ounion,axiom,
! [A: $tType,F: A > A > A,Z2: A,M3: multiset @ A,N6: multiset @ A] :
( ( comm_monoid_mset @ A @ F @ Z2 )
=> ( ( comm_monoid_F @ A @ F @ Z2 @ ( plus_plus @ ( multiset @ A ) @ M3 @ N6 ) )
= ( F @ ( comm_monoid_F @ A @ F @ Z2 @ M3 ) @ ( comm_monoid_F @ A @ F @ Z2 @ N6 ) ) ) ) ).
% comm_monoid_mset.union
thf(fact_5739_prod__list_Omonoid__list__axioms,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ( groups_monoid_list @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ).
% prod_list.monoid_list_axioms
thf(fact_5740_sum__mset_Ocomm__monoid__mset__axioms,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ( comm_monoid_mset @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ).
% sum_mset.comm_monoid_mset_axioms
thf(fact_5741_prod__mset_Ocomm__monoid__mset__axioms,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ( comm_monoid_mset @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ).
% prod_mset.comm_monoid_mset_axioms
thf(fact_5742_sum__list_Omonoid__list__axioms,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ( groups_monoid_list @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ).
% sum_list.monoid_list_axioms
thf(fact_5743_Array__Time_Oset__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_set @ A )
= ( ^ [A7: array @ A,X3: list @ A] :
( arrays_update @ product_unit
@ ^ [H: typerep > nat > ( list @ nat )] : ( fun_upd @ typerep @ ( nat > ( list @ nat ) ) @ H @ ( typerep_typerep @ A @ ( type2 @ A ) ) @ ( fun_upd @ nat @ ( list @ nat ) @ ( H @ ( typerep_typerep @ A @ ( type2 @ A ) ) ) @ ( addr_of_array @ A @ A7 ) @ ( map @ A @ nat @ ( to_nat @ A ) @ X3 ) ) ) ) ) ) ) ).
% Array_Time.set_def
thf(fact_5744_Array__Time_Oget__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_get @ A )
= ( ^ [H: heap_ext @ product_unit,A7: array @ A] : ( map @ nat @ A @ ( from_nat @ A ) @ ( arrays @ product_unit @ H @ ( typerep_typerep @ A @ ( type2 @ A ) ) @ ( addr_of_array @ A @ A7 ) ) ) ) ) ) ).
% Array_Time.get_def
thf(fact_5745_Int_Osub__code_I8_J,axiom,
! [M2: num,N: num] :
( ( sub @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
= ( plus_plus @ int @ ( dup @ ( sub @ M2 @ N ) ) @ ( one_one @ int ) ) ) ).
% Int.sub_code(8)
thf(fact_5746_Int_Odup__def,axiom,
( dup
= ( ^ [K4: int] : ( plus_plus @ int @ K4 @ K4 ) ) ) ).
% Int.dup_def
thf(fact_5747_relH__def,axiom,
( relH
= ( ^ [As5: set @ nat,H: heap_ext @ product_unit,H7: heap_ext @ product_unit] :
( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As5 ) )
& ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ As5 ) )
& ! [T3: typerep,X3: nat] :
( ( member @ nat @ X3 @ As5 )
=> ( ( ( refs @ product_unit @ H @ T3 @ X3 )
= ( refs @ product_unit @ H7 @ T3 @ X3 ) )
& ( ( arrays @ product_unit @ H @ T3 @ X3 )
= ( arrays @ product_unit @ H7 @ T3 @ X3 ) ) ) ) ) ) ) ).
% relH_def
thf(fact_5748_case__sum__o__map__sum__id,axiom,
! [A: $tType,B: $tType,C: $tType,G: B > A,F: C > A,X: sum_sum @ C @ B] :
( ( comp @ ( sum_sum @ A @ B ) @ A @ ( sum_sum @ C @ B ) @ ( sum_case_sum @ A @ A @ B @ ( id @ A ) @ G ) @ ( sum_map_sum @ C @ A @ B @ B @ F @ ( id @ B ) ) @ X )
= ( sum_case_sum @ C @ A @ B @ ( comp @ C @ A @ C @ F @ ( id @ C ) ) @ G @ X ) ) ).
% case_sum_o_map_sum_id
thf(fact_5749_comm__monoid_Oaxioms_I2_J,axiom,
! [A: $tType,F: A > A > A,Z2: A] :
( ( comm_monoid @ A @ F @ Z2 )
=> ( comm_monoid_axioms @ A @ F @ Z2 ) ) ).
% comm_monoid.axioms(2)
thf(fact_5750_sum_Omap__comp,axiom,
! [D: $tType,F5: $tType,E: $tType,C: $tType,B: $tType,A: $tType,G1: C > E,G22: D > F5,F1: A > C,F22: B > D,V: sum_sum @ A @ B] :
( ( sum_map_sum @ C @ E @ D @ F5 @ G1 @ G22 @ ( sum_map_sum @ A @ C @ B @ D @ F1 @ F22 @ V ) )
= ( sum_map_sum @ A @ E @ B @ F5 @ ( comp @ C @ E @ A @ G1 @ F1 ) @ ( comp @ D @ F5 @ B @ G22 @ F22 ) @ V ) ) ).
% sum.map_comp
thf(fact_5751_comm__monoid__axioms_Ointro,axiom,
! [A: $tType,F: A > A > A,Z2: A] :
( ! [A5: A] :
( ( F @ A5 @ Z2 )
= A5 )
=> ( comm_monoid_axioms @ A @ F @ Z2 ) ) ).
% comm_monoid_axioms.intro
thf(fact_5752_comm__monoid__axioms__def,axiom,
! [A: $tType] :
( ( comm_monoid_axioms @ A )
= ( ^ [F2: A > A > A,Z5: A] :
! [A7: A] :
( ( F2 @ A7 @ Z5 )
= A7 ) ) ) ).
% comm_monoid_axioms_def
thf(fact_5753_case__sum__map__sum,axiom,
! [C: $tType,A: $tType,B: $tType,E: $tType,D: $tType,L: B > A,R: C > A,F: D > B,G: E > C,X: sum_sum @ D @ E] :
( ( sum_case_sum @ B @ A @ C @ L @ R @ ( sum_map_sum @ D @ B @ E @ C @ F @ G @ X ) )
= ( sum_case_sum @ D @ A @ E @ ( comp @ B @ A @ D @ L @ F ) @ ( comp @ C @ A @ E @ R @ G ) @ X ) ) ).
% case_sum_map_sum
thf(fact_5754_case__sum__o__map__sum,axiom,
! [A: $tType,D: $tType,C: $tType,E: $tType,B: $tType,F: D > C,G: E > C,H1: A > D,H22: B > E] :
( ( comp @ ( sum_sum @ D @ E ) @ C @ ( sum_sum @ A @ B ) @ ( sum_case_sum @ D @ C @ E @ F @ G ) @ ( sum_map_sum @ A @ D @ B @ E @ H1 @ H22 ) )
= ( sum_case_sum @ A @ C @ B @ ( comp @ D @ C @ A @ F @ H1 ) @ ( comp @ E @ C @ B @ G @ H22 ) ) ) ).
% case_sum_o_map_sum
thf(fact_5755_map__sum_Ocomp,axiom,
! [A: $tType,C: $tType,E: $tType,F5: $tType,D: $tType,B: $tType,F: C > E,G: D > F5,H3: A > C,I: B > D] :
( ( comp @ ( sum_sum @ C @ D ) @ ( sum_sum @ E @ F5 ) @ ( sum_sum @ A @ B ) @ ( sum_map_sum @ C @ E @ D @ F5 @ F @ G ) @ ( sum_map_sum @ A @ C @ B @ D @ H3 @ I ) )
= ( sum_map_sum @ A @ E @ B @ F5 @ ( comp @ C @ E @ A @ F @ H3 ) @ ( comp @ D @ F5 @ B @ G @ I ) ) ) ).
% map_sum.comp
thf(fact_5756_map__sum_Ocompositionality,axiom,
! [D: $tType,F5: $tType,E: $tType,C: $tType,B: $tType,A: $tType,F: C > E,G: D > F5,H3: A > C,I: B > D,Sum: sum_sum @ A @ B] :
( ( sum_map_sum @ C @ E @ D @ F5 @ F @ G @ ( sum_map_sum @ A @ C @ B @ D @ H3 @ I @ Sum ) )
= ( sum_map_sum @ A @ E @ B @ F5 @ ( comp @ C @ E @ A @ F @ H3 ) @ ( comp @ D @ F5 @ B @ G @ I ) @ Sum ) ) ).
% map_sum.compositionality
thf(fact_5757_comm__monoid__def,axiom,
! [A: $tType] :
( ( comm_monoid @ A )
= ( ^ [F2: A > A > A,Z5: A] :
( ( abel_semigroup @ A @ F2 )
& ( comm_monoid_axioms @ A @ F2 @ Z5 ) ) ) ) ).
% comm_monoid_def
thf(fact_5758_mult_Oabel__semigroup__axioms,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ( abel_semigroup @ A @ ( times_times @ A ) ) ) ).
% mult.abel_semigroup_axioms
thf(fact_5759_comm__monoid_Oaxioms_I1_J,axiom,
! [A: $tType,F: A > A > A,Z2: A] :
( ( comm_monoid @ A @ F @ Z2 )
=> ( abel_semigroup @ A @ F ) ) ).
% comm_monoid.axioms(1)
thf(fact_5760_abel__semigroup_Oleft__commute,axiom,
! [A: $tType,F: A > A > A,B2: A,A3: A,C2: A] :
( ( abel_semigroup @ A @ F )
=> ( ( F @ B2 @ ( F @ A3 @ C2 ) )
= ( F @ A3 @ ( F @ B2 @ C2 ) ) ) ) ).
% abel_semigroup.left_commute
thf(fact_5761_abel__semigroup_Ocommute,axiom,
! [A: $tType,F: A > A > A,A3: A,B2: A] :
( ( abel_semigroup @ A @ F )
=> ( ( F @ A3 @ B2 )
= ( F @ B2 @ A3 ) ) ) ).
% abel_semigroup.commute
thf(fact_5762_add_Oabel__semigroup__axioms,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ( abel_semigroup @ A @ ( plus_plus @ A ) ) ) ).
% add.abel_semigroup_axioms
thf(fact_5763_abel__semigroup_Oaxioms_I1_J,axiom,
! [A: $tType,F: A > A > A] :
( ( abel_semigroup @ A @ F )
=> ( semigroup @ A @ F ) ) ).
% abel_semigroup.axioms(1)
thf(fact_5764_comm__monoid_Ointro,axiom,
! [A: $tType,F: A > A > A,Z2: A] :
( ( abel_semigroup @ A @ F )
=> ( ( comm_monoid_axioms @ A @ F @ Z2 )
=> ( comm_monoid @ A @ F @ Z2 ) ) ) ).
% comm_monoid.intro
thf(fact_5765_abel__semigroup_Ointro,axiom,
! [A: $tType,F: A > A > A] :
( ( semigroup @ A @ F )
=> ( ( abel_s757365448890700780axioms @ A @ F )
=> ( abel_semigroup @ A @ F ) ) ) ).
% abel_semigroup.intro
thf(fact_5766_abel__semigroup__def,axiom,
! [A: $tType] :
( ( abel_semigroup @ A )
= ( ^ [F2: A > A > A] :
( ( semigroup @ A @ F2 )
& ( abel_s757365448890700780axioms @ A @ F2 ) ) ) ) ).
% abel_semigroup_def
thf(fact_5767_curr__def,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( bNF_Wellorder_curr @ A @ B @ C )
= ( ^ [A9: set @ A,F2: ( product_prod @ A @ B ) > C,A7: A] :
( if @ ( B > C ) @ ( member @ A @ A7 @ A9 )
@ ^ [B5: B] : ( F2 @ ( product_Pair @ A @ B @ A7 @ B5 ) )
@ ( undefined @ ( B > C ) ) ) ) ) ).
% curr_def
thf(fact_5768_abel__semigroup__axioms_Ointro,axiom,
! [A: $tType,F: A > A > A] :
( ! [A5: A,B3: A] :
( ( F @ A5 @ B3 )
= ( F @ B3 @ A5 ) )
=> ( abel_s757365448890700780axioms @ A @ F ) ) ).
% abel_semigroup_axioms.intro
thf(fact_5769_abel__semigroup__axioms__def,axiom,
! [A: $tType] :
( ( abel_s757365448890700780axioms @ A )
= ( ^ [F2: A > A > A] :
! [A7: A,B5: A] :
( ( F2 @ A7 @ B5 )
= ( F2 @ B5 @ A7 ) ) ) ) ).
% abel_semigroup_axioms_def
thf(fact_5770_abel__semigroup_Oaxioms_I2_J,axiom,
! [A: $tType,F: A > A > A] :
( ( abel_semigroup @ A @ F )
=> ( abel_s757365448890700780axioms @ A @ F ) ) ).
% abel_semigroup.axioms(2)
thf(fact_5771_comm__monoid__set_Ozero__middle,axiom,
! [A: $tType,F: A > A > A,Z2: A,P5: nat,K: nat,G: nat > A,H3: nat > A] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P5 )
=> ( ( ord_less_eq @ nat @ K @ P5 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2
@ ^ [J2: nat] : ( if @ A @ ( ord_less @ nat @ J2 @ K ) @ ( G @ J2 ) @ ( if @ A @ ( J2 = K ) @ Z2 @ ( H3 @ ( minus_minus @ nat @ J2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( set_ord_atMost @ nat @ P5 ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z2
@ ^ [J2: nat] : ( if @ A @ ( ord_less @ nat @ J2 @ K ) @ ( G @ J2 ) @ ( H3 @ J2 ) )
@ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P5 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% comm_monoid_set.zero_middle
thf(fact_5772_iterate__add__diff__add__eq2,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [I: nat,J: nat,U: A,M2: A,N: A] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( minus_minus @ A @ ( plus_plus @ A @ ( iterate_add @ A @ I @ U ) @ M2 ) @ ( plus_plus @ A @ ( iterate_add @ A @ J @ U ) @ N ) )
= ( minus_minus @ A @ M2 @ ( plus_plus @ A @ ( iterate_add @ A @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ) ).
% iterate_add_diff_add_eq2
thf(fact_5773_iterate__add__add__eq1,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [J: nat,I: nat,U: A,M2: A,N: A] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( minus_minus @ A @ ( plus_plus @ A @ ( iterate_add @ A @ I @ U ) @ M2 ) @ ( plus_plus @ A @ ( iterate_add @ A @ J @ U ) @ N ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( iterate_add @ A @ ( minus_minus @ nat @ I @ J ) @ U ) @ M2 ) @ N ) ) ) ) ).
% iterate_add_add_eq1
thf(fact_5774_iterate__add__simps_I2_J,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [N: nat,A3: A] :
( ( iterate_add @ A @ ( suc @ N ) @ A3 )
= ( plus_plus @ A @ A3 @ ( iterate_add @ A @ N @ A3 ) ) ) ) ).
% iterate_add_simps(2)
thf(fact_5775_iterate__add__distrib,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [M2: nat,N: nat,A3: A] :
( ( iterate_add @ A @ ( plus_plus @ nat @ M2 @ N ) @ A3 )
= ( plus_plus @ A @ ( iterate_add @ A @ M2 @ A3 ) @ ( iterate_add @ A @ N @ A3 ) ) ) ) ).
% iterate_add_distrib
thf(fact_5776_iterate__add__1,axiom,
! [A: $tType] :
( ( ( cancel1802427076303600483id_add @ A )
& ( semiring_1 @ A ) )
=> ! [N: nat] :
( ( iterate_add @ A @ N @ ( one_one @ A ) )
= ( semiring_1_of_nat @ A @ N ) ) ) ).
% iterate_add_1
thf(fact_5777_prod_Ocomm__monoid__set__axioms,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ( groups778175481326437816id_set @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ).
% prod.comm_monoid_set_axioms
thf(fact_5778_comm__monoid__set_Onat__group,axiom,
! [A: $tType,F: A > A > A,Z2: A,G: nat > A,K: nat,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2
@ ^ [M: nat] : ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ M @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ M @ K ) @ K ) ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ N @ K ) ) ) ) ) ).
% comm_monoid_set.nat_group
thf(fact_5779_comm__monoid__set_Oub__add__nat,axiom,
! [A: $tType,F: A > A > A,Z2: A,M2: nat,N: nat,G: nat > A,P5: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( ord_less_eq @ nat @ M2 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( plus_plus @ nat @ N @ P5 ) ) )
= ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N @ P5 ) ) ) ) ) ) ) ).
% comm_monoid_set.ub_add_nat
thf(fact_5780_comm__monoid__set_Oivl__cong,axiom,
! [A: $tType,B: $tType] :
( ( ord @ B )
=> ! [F: A > A > A,Z2: A,A3: B,C2: B,B2: B,D3: B,G: B > A,H3: B > A] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( A3 = C2 )
=> ( ( B2 = D3 )
=> ( ! [X2: B] :
( ( ord_less_eq @ B @ C2 @ X2 )
=> ( ( ord_less @ B @ X2 @ D3 )
=> ( ( G @ X2 )
= ( H3 @ X2 ) ) ) )
=> ( ( groups_comm_monoid_F @ A @ B @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ B @ A3 @ B2 ) )
= ( groups_comm_monoid_F @ A @ B @ F @ Z2 @ H3 @ ( set_or7035219750837199246ssThan @ B @ C2 @ D3 ) ) ) ) ) ) ) ) ).
% comm_monoid_set.ivl_cong
thf(fact_5781_comm__monoid__set_Ohead,axiom,
! [A: $tType,F: A > A > A,Z2: A,M2: nat,N: nat,G: nat > A] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( F @ ( G @ M2 ) @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or3652927894154168847AtMost @ nat @ M2 @ N ) ) ) ) ) ) ).
% comm_monoid_set.head
thf(fact_5782_comm__monoid__set_Olast__plus,axiom,
! [A: $tType,F: A > A > A,Z2: A,M2: nat,N: nat,G: nat > A] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( F @ ( G @ N ) @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ) ) ).
% comm_monoid_set.last_plus
thf(fact_5783_comm__monoid__set_OatLeast__Suc__atMost,axiom,
! [A: $tType,F: A > A > A,Z2: A,M2: nat,N: nat,G: nat > A] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( F @ ( G @ M2 ) @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ N ) ) ) ) ) ) ).
% comm_monoid_set.atLeast_Suc_atMost
thf(fact_5784_comm__monoid__set_Onat__ivl__Suc_H,axiom,
! [A: $tType,F: A > A > A,Z2: A,M2: nat,N: nat,G: nat > A] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( ord_less_eq @ nat @ M2 @ ( suc @ N ) )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) ) )
= ( F @ ( G @ ( suc @ N ) ) @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ) ) ).
% comm_monoid_set.nat_ivl_Suc'
thf(fact_5785_comm__monoid__set_OatLeastLessThan__Suc,axiom,
! [A: $tType,F: A > A > A,Z2: A,A3: nat,B2: nat,G: nat > A] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( ord_less_eq @ nat @ A3 @ B2 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ A3 @ ( suc @ B2 ) ) )
= ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ A3 @ B2 ) ) @ ( G @ B2 ) ) ) ) ) ).
% comm_monoid_set.atLeastLessThan_Suc
thf(fact_5786_comm__monoid__set_OatLeastLessThan__concat,axiom,
! [A: $tType,F: A > A > A,Z2: A,M2: nat,N: nat,P5: nat,G: nat > A] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( ord_less_eq @ nat @ N @ P5 )
=> ( ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ P5 ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ P5 ) ) ) ) ) ) ).
% comm_monoid_set.atLeastLessThan_concat
thf(fact_5787_comm__monoid__set_OSuc__reindex__ivl,axiom,
! [A: $tType,F: A > A > A,Z2: A,M2: nat,N: nat,G: nat > A] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) )
= ( F @ ( G @ M2 )
@ ( groups_comm_monoid_F @ A @ nat @ F @ Z2
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ) ) ).
% comm_monoid_set.Suc_reindex_ivl
thf(fact_5788_comm__monoid__set_OatLeast1__atMost__eq,axiom,
! [A: $tType,F: A > A > A,Z2: A,G: nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z2
@ ^ [K4: nat] : ( G @ ( suc @ K4 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% comm_monoid_set.atLeast1_atMost_eq
thf(fact_5789_comm__monoid__set_OatLeastLessThan__rev,axiom,
! [A: $tType,F: A > A > A,Z2: A,G: nat > A,N: nat,M2: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ M2 ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z2
@ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ ( suc @ I3 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ N @ M2 ) ) ) ) ).
% comm_monoid_set.atLeastLessThan_rev
thf(fact_5790_comm__monoid__set_Onested__swap,axiom,
! [A: $tType,F: A > A > A,Z2: A,A3: nat > nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2
@ ^ [I3: nat] : ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ ( A3 @ I3 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z2
@ ^ [J2: nat] :
( groups_comm_monoid_F @ A @ nat @ F @ Z2
@ ^ [I3: nat] : ( A3 @ I3 @ J2 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J2 ) @ N ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% comm_monoid_set.nested_swap
thf(fact_5791_comm__monoid__set_Onat__diff__reindex,axiom,
! [A: $tType,F: A > A > A,Z2: A,G: nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2
@ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ N @ ( suc @ I3 ) ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% comm_monoid_set.nat_diff_reindex
thf(fact_5792_comm__monoid__set_Oshift__bounds__cl__Suc__ivl,axiom,
! [A: $tType,F: A > A > A,Z2: A,G: nat > A,M2: nat,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z2
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% comm_monoid_set.shift_bounds_cl_Suc_ivl
thf(fact_5793_comm__monoid__set_Oshift__bounds__Suc__ivl,axiom,
! [A: $tType,F: A > A > A,Z2: A,G: nat > A,M2: nat,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z2
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% comm_monoid_set.shift_bounds_Suc_ivl
thf(fact_5794_comm__monoid__set_OlessThan__Suc,axiom,
! [A: $tType,F: A > A > A,Z2: A,G: nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
= ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_ord_lessThan @ nat @ N ) ) @ ( G @ N ) ) ) ) ).
% comm_monoid_set.lessThan_Suc
thf(fact_5795_comm__monoid__set_OatMost__Suc,axiom,
! [A: $tType,F: A > A > A,Z2: A,G: nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
= ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_ord_atMost @ nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ).
% comm_monoid_set.atMost_Suc
thf(fact_5796_comm__monoid__set_Ohead__if,axiom,
! [A: $tType,F: A > A > A,Z2: A,N: nat,M2: nat,G: nat > A] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( ( ord_less @ nat @ N @ M2 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= Z2 ) )
& ( ~ ( ord_less @ nat @ N @ M2 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) @ ( G @ N ) ) ) ) ) ) ).
% comm_monoid_set.head_if
thf(fact_5797_comm__monoid__set_Ocl__ivl__Suc,axiom,
! [A: $tType,F: A > A > A,Z2: A,N: nat,M2: nat,G: nat > A] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( ( ord_less @ nat @ ( suc @ N ) @ M2 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) ) )
= Z2 ) )
& ( ~ ( ord_less @ nat @ ( suc @ N ) @ M2 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) ) )
= ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ) ).
% comm_monoid_set.cl_ivl_Suc
thf(fact_5798_comm__monoid__set_Oop__ivl__Suc,axiom,
! [A: $tType,F: A > A > A,Z2: A,N: nat,M2: nat,G: nat > A] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( ( ord_less @ nat @ N @ M2 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ N ) ) )
= Z2 ) )
& ( ~ ( ord_less @ nat @ N @ M2 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ N ) ) )
= ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) @ ( G @ N ) ) ) ) ) ) ).
% comm_monoid_set.op_ivl_Suc
thf(fact_5799_comm__monoid__set_OatLeast__Suc__lessThan,axiom,
! [A: $tType,F: A > A > A,Z2: A,M2: nat,N: nat,G: nat > A] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( ord_less @ nat @ M2 @ N )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) )
= ( F @ ( G @ M2 ) @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ N ) ) ) ) ) ) ).
% comm_monoid_set.atLeast_Suc_lessThan
thf(fact_5800_comm__monoid__set_OatLeast0__lessThan__Suc,axiom,
! [A: $tType,F: A > A > A,Z2: A,G: nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G @ N ) ) ) ) ).
% comm_monoid_set.atLeast0_lessThan_Suc
thf(fact_5801_comm__monoid__set_OatLeast0__atMost__Suc,axiom,
! [A: $tType,F: A > A > A,Z2: A,G: nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ).
% comm_monoid_set.atLeast0_atMost_Suc
thf(fact_5802_comm__monoid__set_OatMost__Suc__shift,axiom,
! [A: $tType,F: A > A > A,Z2: A,G: nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
= ( F @ ( G @ ( zero_zero @ nat ) )
@ ( groups_comm_monoid_F @ A @ nat @ F @ Z2
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% comm_monoid_set.atMost_Suc_shift
thf(fact_5803_comm__monoid__set_OatLeastAtMost__rev,axiom,
! [A: $tType,F: A > A > A,Z2: A,G: nat > A,N: nat,M2: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ N @ M2 ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z2
@ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ N @ M2 ) ) ) ) ).
% comm_monoid_set.atLeastAtMost_rev
thf(fact_5804_comm__monoid__set_OlessThan__Suc__shift,axiom,
! [A: $tType,F: A > A > A,Z2: A,G: nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
= ( F @ ( G @ ( zero_zero @ nat ) )
@ ( groups_comm_monoid_F @ A @ nat @ F @ Z2
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% comm_monoid_set.lessThan_Suc_shift
thf(fact_5805_comm__monoid__set_Oshift__bounds__cl__nat__ivl,axiom,
! [A: $tType,F: A > A > A,Z2: A,G: nat > A,M2: nat,K: nat,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M2 @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z2
@ ^ [I3: nat] : ( G @ ( plus_plus @ nat @ I3 @ K ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% comm_monoid_set.shift_bounds_cl_nat_ivl
thf(fact_5806_comm__monoid__set_Oshift__bounds__nat__ivl,axiom,
! [A: $tType,F: A > A > A,Z2: A,G: nat > A,M2: nat,K: nat,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M2 @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z2
@ ^ [I3: nat] : ( G @ ( plus_plus @ nat @ I3 @ K ) )
@ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% comm_monoid_set.shift_bounds_nat_ivl
thf(fact_5807_sum_Ocomm__monoid__set__axioms,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ( groups778175481326437816id_set @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ).
% sum.comm_monoid_set_axioms
thf(fact_5808_comm__monoid__set_OatMost__shift,axiom,
! [A: $tType,F: A > A > A,Z2: A,G: nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_ord_atMost @ nat @ N ) )
= ( F @ ( G @ ( zero_zero @ nat ) )
@ ( groups_comm_monoid_F @ A @ nat @ F @ Z2
@ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% comm_monoid_set.atMost_shift
thf(fact_5809_comm__monoid__set_Onested__swap_H,axiom,
! [A: $tType,F: A > A > A,Z2: A,A3: nat > nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2
@ ^ [I3: nat] : ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ ( A3 @ I3 ) @ ( set_ord_lessThan @ nat @ I3 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z2
@ ^ [J2: nat] :
( groups_comm_monoid_F @ A @ nat @ F @ Z2
@ ^ [I3: nat] : ( A3 @ I3 @ J2 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J2 ) @ N ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% comm_monoid_set.nested_swap'
thf(fact_5810_comm__monoid__set_Oreindex__nontrivial,axiom,
! [C: $tType,A: $tType,B: $tType,F: A > A > A,Z2: A,A4: set @ B,H3: B > C,G: C > A] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( finite_finite2 @ B @ A4 )
=> ( ! [X2: B,Y4: B] :
( ( member @ B @ X2 @ A4 )
=> ( ( member @ B @ Y4 @ A4 )
=> ( ( X2 != Y4 )
=> ( ( ( H3 @ X2 )
= ( H3 @ Y4 ) )
=> ( ( G @ ( H3 @ X2 ) )
= Z2 ) ) ) ) )
=> ( ( groups_comm_monoid_F @ A @ C @ F @ Z2 @ G @ ( image2 @ B @ C @ H3 @ A4 ) )
= ( groups_comm_monoid_F @ A @ B @ F @ Z2 @ ( comp @ C @ A @ B @ G @ H3 ) @ A4 ) ) ) ) ) ).
% comm_monoid_set.reindex_nontrivial
thf(fact_5811_comm__monoid__set_Oreindex,axiom,
! [C: $tType,A: $tType,B: $tType,F: A > A > A,Z2: A,H3: B > C,A4: set @ B,G: C > A] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( inj_on @ B @ C @ H3 @ A4 )
=> ( ( groups_comm_monoid_F @ A @ C @ F @ Z2 @ G @ ( image2 @ B @ C @ H3 @ A4 ) )
= ( groups_comm_monoid_F @ A @ B @ F @ Z2 @ ( comp @ C @ A @ B @ G @ H3 ) @ A4 ) ) ) ) ).
% comm_monoid_set.reindex
thf(fact_5812_comm__monoid__set_OatLeast__Suc__lessThan__Suc__shift,axiom,
! [A: $tType,F: A > A > A,Z2: A,G: nat > A,M2: nat,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% comm_monoid_set.atLeast_Suc_lessThan_Suc_shift
thf(fact_5813_comm__monoid__set_OatLeast__Suc__atMost__Suc__shift,axiom,
! [A: $tType,F: A > A > A,Z2: A,G: nat > A,M2: nat,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% comm_monoid_set.atLeast_Suc_atMost_Suc_shift
thf(fact_5814_comm__monoid__set_OatLeastAtMost__shift__bounds,axiom,
! [A: $tType,F: A > A > A,Z2: A,G: nat > A,M2: nat,K: nat,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M2 @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ K ) ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% comm_monoid_set.atLeastAtMost_shift_bounds
thf(fact_5815_comm__monoid__set_OatLeastLessThan__shift__bounds,axiom,
! [A: $tType,F: A > A > A,Z2: A,G: nat > A,M2: nat,K: nat,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M2 @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ K ) ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% comm_monoid_set.atLeastLessThan_shift_bounds
thf(fact_5816_comm__monoid__set_OatLeastLessThan__reindex,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > A > A,Z2: A,H3: nat > B,M2: nat,N: nat,G: B > A] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( bij_betw @ nat @ B @ H3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) @ ( set_or7035219750837199246ssThan @ B @ ( H3 @ M2 ) @ ( H3 @ N ) ) )
=> ( ( groups_comm_monoid_F @ A @ B @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ B @ ( H3 @ M2 ) @ ( H3 @ N ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ ( comp @ B @ A @ nat @ G @ H3 ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ) ) ).
% comm_monoid_set.atLeastLessThan_reindex
thf(fact_5817_comm__monoid__set_OatLeastAtMost__reindex,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > A > A,Z2: A,H3: nat > B,M2: nat,N: nat,G: B > A] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( bij_betw @ nat @ B @ H3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) @ ( set_or1337092689740270186AtMost @ B @ ( H3 @ M2 ) @ ( H3 @ N ) ) )
=> ( ( groups_comm_monoid_F @ A @ B @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ B @ ( H3 @ M2 ) @ ( H3 @ N ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ ( comp @ B @ A @ nat @ G @ H3 ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ) ) ).
% comm_monoid_set.atLeastAtMost_reindex
thf(fact_5818_comm__monoid__set_OatLeast0__lessThan__Suc__shift,axiom,
! [A: $tType,F: A > A > A,Z2: A,G: nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( F @ ( G @ ( zero_zero @ nat ) ) @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% comm_monoid_set.atLeast0_lessThan_Suc_shift
thf(fact_5819_comm__monoid__set_OatLeast0__atMost__Suc__shift,axiom,
! [A: $tType,F: A > A > A,Z2: A,G: nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( F @ ( G @ ( zero_zero @ nat ) ) @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% comm_monoid_set.atLeast0_atMost_Suc_shift
thf(fact_5820_comm__monoid__set_OatLeastLessThan__shift__0,axiom,
! [A: $tType,F: A > A > A,Z2: A,G: nat > A,M2: nat,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M2 ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ).
% comm_monoid_set.atLeastLessThan_shift_0
thf(fact_5821_comm__monoid__set_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
! [A: $tType,F: A > A > A,Z2: A,G: nat > A,N: nat,M2: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ M2 ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z2
@ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ I3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M2 ) ) ) ) ).
% comm_monoid_set.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_5822_iterate__add__eq__add__iff1,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [I: nat,J: nat,U: A,M2: A,N: A] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ( plus_plus @ A @ ( iterate_add @ A @ J @ U ) @ M2 )
= ( plus_plus @ A @ ( iterate_add @ A @ I @ U ) @ N ) )
= ( ( plus_plus @ A @ ( iterate_add @ A @ ( minus_minus @ nat @ J @ I ) @ U ) @ M2 )
= N ) ) ) ) ).
% iterate_add_eq_add_iff1
thf(fact_5823_iterate__add__eq__add__iff2,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [I: nat,J: nat,U: A,M2: A,N: A] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ( plus_plus @ A @ ( iterate_add @ A @ I @ U ) @ M2 )
= ( plus_plus @ A @ ( iterate_add @ A @ J @ U ) @ N ) )
= ( M2
= ( plus_plus @ A @ ( iterate_add @ A @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ) ).
% iterate_add_eq_add_iff2
thf(fact_5824_iterate__add__def,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ( ( iterate_add @ A )
= ( ^ [N5: nat,A7: A] : ( compow @ ( A > A ) @ N5 @ ( plus_plus @ A @ A7 ) @ ( zero_zero @ A ) ) ) ) ) ).
% iterate_add_def
thf(fact_5825_comm__monoid__set_OatLeast__atMost__pred__shift,axiom,
! [A: $tType,F: A > A > A,Z2: A,G: nat > A,M2: nat,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2
@ ( comp @ nat @ A @ nat @ G
@ ^ [N5: nat] : ( minus_minus @ nat @ N5 @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% comm_monoid_set.atLeast_atMost_pred_shift
thf(fact_5826_comm__monoid__set_OatLeast__lessThan__pred__shift,axiom,
! [A: $tType,F: A > A > A,Z2: A,G: nat > A,M2: nat,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2
@ ( comp @ nat @ A @ nat @ G
@ ^ [N5: nat] : ( minus_minus @ nat @ N5 @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% comm_monoid_set.atLeast_lessThan_pred_shift
thf(fact_5827_comm__monoid__add__class_Osum__def,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ( ( groups7311177749621191930dd_sum @ B @ A )
= ( groups_comm_monoid_F @ A @ B @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ) ).
% comm_monoid_add_class.sum_def
thf(fact_5828_comm__monoid__mult__class_Oprod__def,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ( ( groups7121269368397514597t_prod @ B @ A )
= ( groups_comm_monoid_F @ A @ B @ ( times_times @ A ) @ ( one_one @ A ) ) ) ) ).
% comm_monoid_mult_class.prod_def
thf(fact_5829_comm__monoid__set_OatLeastAtMost__shift__0,axiom,
! [A: $tType,F: A > A > A,Z2: A,M2: nat,N: nat,G: nat > A] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M2 ) ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ) ).
% comm_monoid_set.atLeastAtMost_shift_0
thf(fact_5830_comm__monoid__set_Oin__pairs,axiom,
! [A: $tType,F: A > A > A,Z2: A,G: nat > A,M2: nat,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z2
@ ^ [I3: nat] : ( F @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% comm_monoid_set.in_pairs
thf(fact_5831_comm__monoid__set_Oin__pairs__0,axiom,
! [A: $tType,F: A > A > A,Z2: A,G: nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_ord_atMost @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z2
@ ^ [I3: nat] : ( F @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% comm_monoid_set.in_pairs_0
thf(fact_5832_iterate__add__less__eq__iff2,axiom,
! [A: $tType] :
( ( ( cancel1802427076303600483id_add @ A )
& ( ordere2412721322843649153imp_le @ A ) )
=> ! [I: nat,J: nat,U: A,M2: A,N: A] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ A @ ( plus_plus @ A @ ( iterate_add @ A @ I @ U ) @ M2 ) @ ( plus_plus @ A @ ( iterate_add @ A @ J @ U ) @ N ) )
= ( ord_less_eq @ A @ M2 @ ( plus_plus @ A @ ( iterate_add @ A @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ) ).
% iterate_add_less_eq_iff2
thf(fact_5833_iterate__add__less__eq__iff1,axiom,
! [A: $tType] :
( ( ( cancel1802427076303600483id_add @ A )
& ( ordere2412721322843649153imp_le @ A ) )
=> ! [J: nat,I: nat,U: A,M2: A,N: A] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( ord_less_eq @ A @ ( plus_plus @ A @ ( iterate_add @ A @ I @ U ) @ M2 ) @ ( plus_plus @ A @ ( iterate_add @ A @ J @ U ) @ N ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( iterate_add @ A @ ( minus_minus @ nat @ I @ J ) @ U ) @ M2 ) @ N ) ) ) ) ).
% iterate_add_less_eq_iff1
thf(fact_5834_iterate__add__less__iff1,axiom,
! [A: $tType] :
( ( ( cancel1802427076303600483id_add @ A )
& ( ordere2412721322843649153imp_le @ A ) )
=> ! [J: nat,I: nat,U: A,M2: A,N: A] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( ord_less @ A @ ( plus_plus @ A @ ( iterate_add @ A @ I @ U ) @ M2 ) @ ( plus_plus @ A @ ( iterate_add @ A @ J @ U ) @ N ) )
= ( ord_less @ A @ ( plus_plus @ A @ ( iterate_add @ A @ ( minus_minus @ nat @ I @ J ) @ U ) @ M2 ) @ N ) ) ) ) ).
% iterate_add_less_iff1
thf(fact_5835_iterate__add__less__iff2,axiom,
! [A: $tType] :
( ( ( cancel1802427076303600483id_add @ A )
& ( ordere2412721322843649153imp_le @ A ) )
=> ! [I: nat,J: nat,U: A,M2: A,N: A] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less @ A @ ( plus_plus @ A @ ( iterate_add @ A @ I @ U ) @ M2 ) @ ( plus_plus @ A @ ( iterate_add @ A @ J @ U ) @ N ) )
= ( ord_less @ A @ M2 @ ( plus_plus @ A @ ( iterate_add @ A @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ) ).
% iterate_add_less_iff2
thf(fact_5836_comm__monoid__set_Otriangle__reindex,axiom,
! [A: $tType,F: A > A > A,Z2: A,G: nat > nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( 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 @ $o
@ ^ [I3: nat,J2: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I3 @ J2 ) @ N ) ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z2
@ ^ [K4: nat] :
( groups_comm_monoid_F @ A @ nat @ F @ Z2
@ ^ [I3: nat] : ( G @ I3 @ ( minus_minus @ nat @ K4 @ I3 ) )
@ ( set_ord_atMost @ nat @ K4 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% comm_monoid_set.triangle_reindex
thf(fact_5837_comm__monoid__set_Otriangle__reindex__eq,axiom,
! [A: $tType,F: A > A > A,Z2: A,G: nat > nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( 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 @ $o
@ ^ [I3: nat,J2: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I3 @ J2 ) @ N ) ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z2
@ ^ [K4: nat] :
( groups_comm_monoid_F @ A @ nat @ F @ Z2
@ ^ [I3: nat] : ( G @ I3 @ ( minus_minus @ nat @ K4 @ I3 ) )
@ ( set_ord_atMost @ nat @ K4 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% comm_monoid_set.triangle_reindex_eq
thf(fact_5838_comm__monoid__set_OatLeast__int__lessThan__int__shift,axiom,
! [A: $tType,F: A > A > A,Z2: A,G: int > A,M2: nat,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( groups_comm_monoid_F @ A @ int @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ ( comp @ int @ A @ nat @ G @ ( semiring_1_of_nat @ int ) ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% comm_monoid_set.atLeast_int_lessThan_int_shift
thf(fact_5839_comm__monoid__set_OatLeast__int__atMost__int__shift,axiom,
! [A: $tType,F: A > A > A,Z2: A,G: int > A,M2: nat,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z2 )
=> ( ( groups_comm_monoid_F @ A @ int @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ ( comp @ int @ A @ nat @ G @ ( semiring_1_of_nat @ int ) ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% comm_monoid_set.atLeast_int_atMost_int_shift
thf(fact_5840_old_Ounit_Ocase,axiom,
! [A: $tType,F: A] :
( ( product_case_unit @ A @ F @ product_Unity )
= F ) ).
% old.unit.case
thf(fact_5841_wfP__wf__eq,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( wfP @ A
@ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R ) )
= ( wf @ A @ R ) ) ).
% wfP_wf_eq
thf(fact_5842_subset__mset_Omin__add__distrib__left,axiom,
! [A: $tType,X: multiset @ A,Y: multiset @ A,Z2: multiset @ A] :
( ( plus_plus @ ( multiset @ A ) @ ( min @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ X @ Y ) @ Z2 )
= ( min @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( plus_plus @ ( multiset @ A ) @ X @ Z2 ) @ ( plus_plus @ ( multiset @ A ) @ Y @ Z2 ) ) ) ).
% subset_mset.min_add_distrib_left
thf(fact_5843_subset__mset_Omin__arg__le_I1_J,axiom,
! [A: $tType,N: multiset @ A,M2: multiset @ A] :
( ( subseteq_mset @ A @ N @ ( min @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ M2 @ N ) )
= ( ( min @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ M2 @ N )
= N ) ) ).
% subset_mset.min_arg_le(1)
thf(fact_5844_subset__mset_Omin__arg__le_I2_J,axiom,
! [A: $tType,M2: multiset @ A,N: multiset @ A] :
( ( subseteq_mset @ A @ M2 @ ( min @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ M2 @ N ) )
= ( ( min @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ M2 @ N )
= M2 ) ) ).
% subset_mset.min_arg_le(2)
thf(fact_5845_wfP__accp__iff,axiom,
! [A: $tType] :
( ( wfP @ A )
= ( ^ [R3: A > A > $o] :
! [X8: A] : ( accp @ A @ R3 @ X8 ) ) ) ).
% wfP_accp_iff
thf(fact_5846_accp__wfPI,axiom,
! [A: $tType,R: A > A > $o] :
( ! [X_12: A] : ( accp @ A @ R @ X_12 )
=> ( wfP @ A @ R ) ) ).
% accp_wfPI
thf(fact_5847_accp__wfPD,axiom,
! [A: $tType,R: A > A > $o,X: A] :
( ( wfP @ A @ R )
=> ( accp @ A @ R @ X ) ) ).
% accp_wfPD
thf(fact_5848_unit_Ocase__distrib,axiom,
! [A: $tType,B: $tType,H3: A > B,F: A,Unit: product_unit] :
( ( H3 @ ( product_case_unit @ A @ F @ Unit ) )
= ( product_case_unit @ B @ ( H3 @ F ) @ Unit ) ) ).
% unit.case_distrib
thf(fact_5849_subset__mset_Omin__add__distrib__right,axiom,
! [A: $tType,X: multiset @ A,Y: multiset @ A,Z2: multiset @ A] :
( ( plus_plus @ ( multiset @ A ) @ X @ ( min @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ Y @ Z2 ) )
= ( min @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( plus_plus @ ( multiset @ A ) @ X @ Y ) @ ( plus_plus @ ( multiset @ A ) @ X @ Z2 ) ) ) ).
% subset_mset.min_add_distrib_right
thf(fact_5850_accp__eq__acc,axiom,
! [A: $tType] :
( ( accp @ A )
= ( ^ [R3: A > A > $o,X3: A] : ( member @ A @ X3 @ ( acc @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R3 ) ) ) ) ) ) ).
% accp_eq_acc
thf(fact_5851_rtrancl__def,axiom,
! [A: $tType] :
( ( transitive_rtrancl @ A )
= ( ^ [R3: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ( transitive_rtranclp @ A
@ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 ) ) ) ) ) ) ).
% rtrancl_def
thf(fact_5852_Enum_Ortranclp__rtrancl__eq,axiom,
! [A: $tType] :
( ( transitive_rtranclp @ A )
= ( ^ [R3: A > A > $o,X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ ( transitive_rtrancl @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R3 ) ) ) ) ) ) ).
% Enum.rtranclp_rtrancl_eq
thf(fact_5853_acc_Ocases,axiom,
! [A: $tType,A3: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ A @ A3 @ ( acc @ A @ R ) )
=> ! [Y6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ A3 ) @ R )
=> ( member @ A @ Y6 @ ( acc @ A @ R ) ) ) ) ).
% acc.cases
thf(fact_5854_acc_Osimps,axiom,
! [A: $tType,A3: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ A @ A3 @ ( acc @ A @ R ) )
= ( ? [X3: A] :
( ( A3 = X3 )
& ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R )
=> ( member @ A @ Y3 @ ( acc @ A @ R ) ) ) ) ) ) ).
% acc.simps
thf(fact_5855_acc_Ointros,axiom,
! [A: $tType,X: A,R: set @ ( product_prod @ A @ A )] :
( ! [Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X ) @ R )
=> ( member @ A @ Y4 @ ( acc @ A @ R ) ) )
=> ( member @ A @ X @ ( acc @ A @ R ) ) ) ).
% acc.intros
thf(fact_5856_acc__induct,axiom,
! [A: $tType,A3: A,R: set @ ( product_prod @ A @ A ),P: A > $o] :
( ( member @ A @ A3 @ ( acc @ A @ R ) )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ ( acc @ A @ R ) )
=> ( ! [Y6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X2 ) @ R )
=> ( P @ Y6 ) )
=> ( P @ X2 ) ) )
=> ( P @ A3 ) ) ) ).
% acc_induct
thf(fact_5857_acc__downward,axiom,
! [A: $tType,B2: A,R: set @ ( product_prod @ A @ A ),A3: A] :
( ( member @ A @ B2 @ ( acc @ A @ R ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
=> ( member @ A @ A3 @ ( acc @ A @ R ) ) ) ) ).
% acc_downward
thf(fact_5858_not__acc__down,axiom,
! [A: $tType,X: A,R5: set @ ( product_prod @ A @ A )] :
( ~ ( member @ A @ X @ ( acc @ A @ R5 ) )
=> ~ ! [Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z3 @ X ) @ R5 )
=> ( member @ A @ Z3 @ ( acc @ A @ R5 ) ) ) ) ).
% not_acc_down
thf(fact_5859_acc__induct__rule,axiom,
! [A: $tType,A3: A,R: set @ ( product_prod @ A @ A ),P: A > $o] :
( ( member @ A @ A3 @ ( acc @ A @ R ) )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ ( acc @ A @ R ) )
=> ( ! [Y6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X2 ) @ R )
=> ( P @ Y6 ) )
=> ( P @ X2 ) ) )
=> ( P @ A3 ) ) ) ).
% acc_induct_rule
thf(fact_5860_rtranclp__induct2,axiom,
! [A: $tType,B: $tType,R: ( product_prod @ A @ B ) > ( product_prod @ A @ B ) > $o,Ax: A,Ay: B,Bx: A,By: B,P: A > B > $o] :
( ( transitive_rtranclp @ ( product_prod @ A @ B ) @ R @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) )
=> ( ( P @ Ax @ Ay )
=> ( ! [A5: A,B3: B,Aa3: A,Ba: B] :
( ( transitive_rtranclp @ ( product_prod @ A @ B ) @ R @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A5 @ B3 ) )
=> ( ( R @ ( product_Pair @ A @ B @ A5 @ B3 ) @ ( product_Pair @ A @ B @ Aa3 @ Ba ) )
=> ( ( P @ A5 @ B3 )
=> ( P @ Aa3 @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% rtranclp_induct2
thf(fact_5861_converse__rtranclpE2,axiom,
! [A: $tType,B: $tType,R: ( product_prod @ A @ B ) > ( product_prod @ A @ B ) > $o,Xa: A,Xb: B,Za2: A,Zb: B] :
( ( transitive_rtranclp @ ( product_prod @ A @ B ) @ R @ ( product_Pair @ A @ B @ Xa @ Xb ) @ ( product_Pair @ A @ B @ Za2 @ Zb ) )
=> ( ( ( product_Pair @ A @ B @ Xa @ Xb )
!= ( product_Pair @ A @ B @ Za2 @ Zb ) )
=> ~ ! [A5: A,B3: B] :
( ( R @ ( product_Pair @ A @ B @ Xa @ Xb ) @ ( product_Pair @ A @ B @ A5 @ B3 ) )
=> ~ ( transitive_rtranclp @ ( product_prod @ A @ B ) @ R @ ( product_Pair @ A @ B @ A5 @ B3 ) @ ( product_Pair @ A @ B @ Za2 @ Zb ) ) ) ) ) ).
% converse_rtranclpE2
thf(fact_5862_converse__rtranclp__induct2,axiom,
! [A: $tType,B: $tType,R: ( product_prod @ A @ B ) > ( product_prod @ A @ B ) > $o,Ax: A,Ay: B,Bx: A,By: B,P: A > B > $o] :
( ( transitive_rtranclp @ ( product_prod @ A @ B ) @ R @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) )
=> ( ( P @ Bx @ By )
=> ( ! [A5: A,B3: B,Aa3: A,Ba: B] :
( ( R @ ( product_Pair @ A @ B @ A5 @ B3 ) @ ( product_Pair @ A @ B @ Aa3 @ Ba ) )
=> ( ( transitive_rtranclp @ ( product_prod @ A @ B ) @ R @ ( product_Pair @ A @ B @ Aa3 @ Ba ) @ ( product_Pair @ A @ B @ Bx @ By ) )
=> ( ( P @ Aa3 @ Ba )
=> ( P @ A5 @ B3 ) ) ) )
=> ( P @ Ax @ Ay ) ) ) ) ).
% converse_rtranclp_induct2
thf(fact_5863_accp__downwards__aux,axiom,
! [A: $tType,R: A > A > $o,B2: A,A3: A] :
( ( transitive_rtranclp @ A @ R @ B2 @ A3 )
=> ( ( accp @ A @ R @ A3 )
=> ( accp @ A @ R @ B2 ) ) ) ).
% accp_downwards_aux
thf(fact_5864_accp__downwards,axiom,
! [A: $tType,R: A > A > $o,A3: A,B2: A] :
( ( accp @ A @ R @ A3 )
=> ( ( transitive_rtranclp @ A @ R @ B2 @ A3 )
=> ( accp @ A @ R @ B2 ) ) ) ).
% accp_downwards
thf(fact_5865_Transitive__Closure_Ortranclp__rtrancl__eq,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( transitive_rtranclp @ A
@ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R ) )
= ( ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ).
% Transitive_Closure.rtranclp_rtrancl_eq
thf(fact_5866_acc__subset__induct,axiom,
! [A: $tType,D4: set @ A,R5: set @ ( product_prod @ A @ A ),X: A,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ D4 @ ( acc @ A @ R5 ) )
=> ( ! [X2: A,Z3: A] :
( ( member @ A @ X2 @ D4 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z3 @ X2 ) @ R5 )
=> ( member @ A @ Z3 @ D4 ) ) )
=> ( ( member @ A @ X @ D4 )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ D4 )
=> ( ! [Z6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z6 @ X2 ) @ R5 )
=> ( P @ Z6 ) )
=> ( P @ X2 ) ) )
=> ( P @ X ) ) ) ) ) ).
% acc_subset_induct
thf(fact_5867_acc__downwards,axiom,
! [A: $tType,A3: A,R: set @ ( product_prod @ A @ A ),B2: A] :
( ( member @ A @ A3 @ ( acc @ A @ R ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( member @ A @ B2 @ ( acc @ A @ R ) ) ) ) ).
% acc_downwards
thf(fact_5868_acc__downwards__aux,axiom,
! [A: $tType,B2: A,A3: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( member @ A @ A3 @ ( acc @ A @ R ) )
=> ( member @ A @ B2 @ ( acc @ A @ R ) ) ) ) ).
% acc_downwards_aux
thf(fact_5869_accp__acc__eq,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( accp @ A
@ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R ) )
= ( ^ [X3: A] : ( member @ A @ X3 @ ( acc @ A @ R ) ) ) ) ).
% accp_acc_eq
thf(fact_5870_acc__def,axiom,
! [A: $tType] :
( ( acc @ A )
= ( ^ [R3: set @ ( product_prod @ A @ A )] :
( collect @ A
@ ( accp @ A
@ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 ) ) ) ) ) ).
% acc_def
thf(fact_5871_scomp__fcomp,axiom,
! [A: $tType,C: $tType,B: $tType,E: $tType,D: $tType,F: A > ( product_prod @ D @ E ),G: D > E > C,H3: C > B] :
( ( fcomp @ A @ C @ B @ ( product_scomp @ A @ D @ E @ C @ F @ G ) @ H3 )
= ( product_scomp @ A @ D @ E @ B @ F
@ ^ [X3: D] : ( fcomp @ E @ C @ B @ ( G @ X3 ) @ H3 ) ) ) ).
% scomp_fcomp
thf(fact_5872_successively_Opelims_I2_J,axiom,
! [A: $tType,X: A > A > $o,Xa: list @ A] :
( ( successively @ A @ X @ Xa )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ Xa ) )
=> ( ( ( Xa
= ( nil @ A ) )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( nil @ A ) ) ) )
=> ( ! [X2: A] :
( ( Xa
= ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) )
=> ~ ! [X2: A,Y4: A,Xs2: list @ A] :
( ( Xa
= ( cons @ A @ X2 @ ( cons @ A @ Y4 @ Xs2 ) ) )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( cons @ A @ X2 @ ( cons @ A @ Y4 @ Xs2 ) ) ) )
=> ~ ( ( X @ X2 @ Y4 )
& ( successively @ A @ X @ ( cons @ A @ Y4 @ Xs2 ) ) ) ) ) ) ) ) ) ).
% successively.pelims(2)
thf(fact_5873_successively_Opelims_I1_J,axiom,
! [A: $tType,X: A > A > $o,Xa: list @ A,Y: $o] :
( ( ( successively @ A @ X @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ Xa ) )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( Y
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( nil @ A ) ) ) ) )
=> ( ! [X2: A] :
( ( Xa
= ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ( Y
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) ) )
=> ~ ! [X2: A,Y4: A,Xs2: list @ A] :
( ( Xa
= ( cons @ A @ X2 @ ( cons @ A @ Y4 @ Xs2 ) ) )
=> ( ( Y
= ( ( X @ X2 @ Y4 )
& ( successively @ A @ X @ ( cons @ A @ Y4 @ Xs2 ) ) ) )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( cons @ A @ X2 @ ( cons @ A @ Y4 @ Xs2 ) ) ) ) ) ) ) ) ) ) ).
% successively.pelims(1)
thf(fact_5874_fcomp__apply,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( fcomp @ B @ C @ A )
= ( ^ [F2: B > C,G2: C > A,X3: B] : ( G2 @ ( F2 @ X3 ) ) ) ) ).
% fcomp_apply
thf(fact_5875_id__fcomp,axiom,
! [B: $tType,A: $tType,G: A > B] :
( ( fcomp @ A @ A @ B @ ( id @ A ) @ G )
= G ) ).
% id_fcomp
thf(fact_5876_fcomp__id,axiom,
! [B: $tType,A: $tType,F: A > B] :
( ( fcomp @ A @ B @ B @ F @ ( id @ B ) )
= F ) ).
% fcomp_id
thf(fact_5877_fcomp__scomp,axiom,
! [A: $tType,E: $tType,B: $tType,D: $tType,C: $tType,F: A > E,G: E > ( product_prod @ C @ D ),H3: C > D > B] :
( ( product_scomp @ A @ C @ D @ B @ ( fcomp @ A @ E @ ( product_prod @ C @ D ) @ F @ G ) @ H3 )
= ( fcomp @ A @ E @ B @ F @ ( product_scomp @ E @ C @ D @ B @ G @ H3 ) ) ) ).
% fcomp_scomp
thf(fact_5878_fcomp__assoc,axiom,
! [A: $tType,D: $tType,B: $tType,C: $tType,F: A > D,G: D > C,H3: C > B] :
( ( fcomp @ A @ C @ B @ ( fcomp @ A @ D @ C @ F @ G ) @ H3 )
= ( fcomp @ A @ D @ B @ F @ ( fcomp @ D @ C @ B @ G @ H3 ) ) ) ).
% fcomp_assoc
thf(fact_5879_fcomp__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( fcomp @ A @ B @ C )
= ( ^ [F2: A > B,G2: B > C,X3: A] : ( G2 @ ( F2 @ X3 ) ) ) ) ).
% fcomp_def
thf(fact_5880_fcomp__comp,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( fcomp @ A @ C @ B )
= ( ^ [F2: A > C,G2: C > B] : ( comp @ C @ B @ A @ G2 @ F2 ) ) ) ).
% fcomp_comp
thf(fact_5881_successively_Opelims_I3_J,axiom,
! [A: $tType,X: A > A > $o,Xa: list @ A] :
( ~ ( successively @ A @ X @ Xa )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ Xa ) )
=> ~ ! [X2: A,Y4: A,Xs2: list @ A] :
( ( Xa
= ( cons @ A @ X2 @ ( cons @ A @ Y4 @ Xs2 ) ) )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( cons @ A @ X2 @ ( cons @ A @ Y4 @ Xs2 ) ) ) )
=> ( ( X @ X2 @ Y4 )
& ( successively @ A @ X @ ( cons @ A @ Y4 @ Xs2 ) ) ) ) ) ) ) ).
% successively.pelims(3)
thf(fact_5882_folding__idem__axioms_Ointro,axiom,
! [B: $tType,A: $tType,F: A > B > B] :
( ! [X2: A] :
( ( comp @ B @ B @ B @ ( F @ X2 ) @ ( F @ X2 ) )
= ( F @ X2 ) )
=> ( finite7837460588564673216axioms @ A @ B @ F ) ) ).
% folding_idem_axioms.intro
thf(fact_5883_comp__fun__idem__axioms_Ointro,axiom,
! [B: $tType,A: $tType,F: A > B > B] :
( ! [X2: A] :
( ( comp @ B @ B @ B @ ( F @ X2 ) @ ( F @ X2 ) )
= ( F @ X2 ) )
=> ( finite1981172056906289455axioms @ A @ B @ F ) ) ).
% comp_fun_idem_axioms.intro
thf(fact_5884_folding__idem__on__axioms_Ointro,axiom,
! [B: $tType,A: $tType,S2: set @ A,F: A > B > B] :
( ! [X2: A,Y4: A] :
( ( member @ A @ X2 @ S2 )
=> ( ( member @ A @ Y4 @ S2 )
=> ( ( comp @ B @ B @ B @ ( F @ X2 ) @ ( F @ X2 ) )
= ( F @ X2 ) ) ) )
=> ( finite6916993218817215295axioms @ A @ B @ S2 @ F ) ) ).
% folding_idem_on_axioms.intro
thf(fact_5885_folding__idem__on__axioms__def,axiom,
! [B: $tType,A: $tType] :
( ( finite6916993218817215295axioms @ A @ B )
= ( ^ [S5: set @ A,F2: A > B > B] :
! [X3: A,Y3: A] :
( ( member @ A @ X3 @ S5 )
=> ( ( member @ A @ Y3 @ S5 )
=> ( ( comp @ B @ B @ B @ ( F2 @ X3 ) @ ( F2 @ X3 ) )
= ( F2 @ X3 ) ) ) ) ) ) ).
% folding_idem_on_axioms_def
thf(fact_5886_comp__fun__idem__axioms__def,axiom,
! [B: $tType,A: $tType] :
( ( finite1981172056906289455axioms @ A @ B )
= ( ^ [F2: A > B > B] :
! [X3: A] :
( ( comp @ B @ B @ B @ ( F2 @ X3 ) @ ( F2 @ X3 ) )
= ( F2 @ X3 ) ) ) ) ).
% comp_fun_idem_axioms_def
thf(fact_5887_folding__idem__axioms__def,axiom,
! [B: $tType,A: $tType] :
( ( finite7837460588564673216axioms @ A @ B )
= ( ^ [F2: A > B > B] :
! [X3: A] :
( ( comp @ B @ B @ B @ ( F2 @ X3 ) @ ( F2 @ X3 ) )
= ( F2 @ X3 ) ) ) ) ).
% folding_idem_axioms_def
thf(fact_5888_comp__fun__idem__on__axioms__def,axiom,
! [B: $tType,A: $tType] :
( ( finite4980608107308702382axioms @ A @ B )
= ( ^ [S5: set @ A,F2: A > B > B] :
! [X3: A] :
( ( member @ A @ X3 @ S5 )
=> ( ( comp @ B @ B @ B @ ( F2 @ X3 ) @ ( F2 @ X3 ) )
= ( F2 @ X3 ) ) ) ) ) ).
% comp_fun_idem_on_axioms_def
thf(fact_5889_comp__fun__idem__on__axioms_Ointro,axiom,
! [B: $tType,A: $tType,S2: set @ A,F: A > B > B] :
( ! [X2: A] :
( ( member @ A @ X2 @ S2 )
=> ( ( comp @ B @ B @ B @ ( F @ X2 ) @ ( F @ X2 ) )
= ( F @ X2 ) ) )
=> ( finite4980608107308702382axioms @ A @ B @ S2 @ F ) ) ).
% comp_fun_idem_on_axioms.intro
thf(fact_5890_image__mset__If,axiom,
! [A: $tType,B: $tType,P: B > $o,F: B > A,G: B > A,A4: multiset @ B] :
( ( image_mset @ B @ A
@ ^ [X3: B] : ( if @ A @ ( P @ X3 ) @ ( F @ X3 ) @ ( G @ X3 ) )
@ A4 )
= ( plus_plus @ ( multiset @ A ) @ ( image_mset @ B @ A @ F @ ( filter_mset @ B @ P @ A4 ) )
@ ( image_mset @ B @ A @ G
@ ( filter_mset @ B
@ ^ [X3: B] :
~ ( P @ X3 )
@ A4 ) ) ) ) ).
% image_mset_If
thf(fact_5891_union__filter__mset__complement,axiom,
! [A: $tType,P: A > $o,Q: A > $o,M3: multiset @ A] :
( ! [X2: A] :
( ( P @ X2 )
= ( ~ ( Q @ X2 ) ) )
=> ( ( plus_plus @ ( multiset @ A ) @ ( filter_mset @ A @ P @ M3 ) @ ( filter_mset @ A @ Q @ M3 ) )
= M3 ) ) ).
% union_filter_mset_complement
thf(fact_5892_filter__union__mset,axiom,
! [A: $tType,P: A > $o,M3: multiset @ A,N6: multiset @ A] :
( ( filter_mset @ A @ P @ ( plus_plus @ ( multiset @ A ) @ M3 @ N6 ) )
= ( plus_plus @ ( multiset @ A ) @ ( filter_mset @ A @ P @ M3 ) @ ( filter_mset @ A @ P @ N6 ) ) ) ).
% filter_union_mset
thf(fact_5893_multiset__partition,axiom,
! [A: $tType,M3: multiset @ A,P: A > $o] :
( M3
= ( plus_plus @ ( multiset @ A ) @ ( filter_mset @ A @ P @ M3 )
@ ( filter_mset @ A
@ ^ [X3: A] :
~ ( P @ X3 )
@ M3 ) ) ) ).
% multiset_partition
thf(fact_5894_heap_Olub__upper,axiom,
! [A: $tType,A4: set @ ( heap_Time_Heap @ A ),X: heap_Time_Heap @ A] :
( ( comple1602240252501008431_chain @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap_ord @ A ) @ A4 )
=> ( ( member @ ( heap_Time_Heap @ A ) @ X @ A4 )
=> ( heap_Time_Heap_ord @ A @ X @ ( heap_Time_Heap_lub @ A @ A4 ) ) ) ) ).
% heap.lub_upper
thf(fact_5895_heap_Olub__least,axiom,
! [A: $tType,A4: set @ ( heap_Time_Heap @ A ),Z2: heap_Time_Heap @ A] :
( ( comple1602240252501008431_chain @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap_ord @ A ) @ A4 )
=> ( ! [X2: heap_Time_Heap @ A] :
( ( member @ ( heap_Time_Heap @ A ) @ X2 @ A4 )
=> ( heap_Time_Heap_ord @ A @ X2 @ Z2 ) )
=> ( heap_Time_Heap_ord @ A @ ( heap_Time_Heap_lub @ A @ A4 ) @ Z2 ) ) ) ).
% heap.lub_least
thf(fact_5896_plus__int__code_I6_J,axiom,
! [M2: num,N: num] :
( ( plus_plus @ int @ ( neg @ M2 ) @ ( neg @ N ) )
= ( neg @ ( plus_plus @ num @ M2 @ N ) ) ) ).
% plus_int_code(6)
thf(fact_5897_ccpo__Sup__mono,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [A4: set @ A,B6: set @ A] :
( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ A4 )
=> ( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ B6 )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ B6 )
& ( ord_less_eq @ A @ X2 @ Xa2 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A4 ) @ ( complete_Sup_Sup @ A @ B6 ) ) ) ) ) ) ).
% ccpo_Sup_mono
thf(fact_5898_plus__int__code_I5_J,axiom,
! [M2: num,N: num] :
( ( plus_plus @ int @ ( neg @ M2 ) @ ( pos @ N ) )
= ( sub @ N @ M2 ) ) ).
% plus_int_code(5)
thf(fact_5899_plus__int__code_I4_J,axiom,
! [M2: num,N: num] :
( ( plus_plus @ int @ ( pos @ M2 ) @ ( neg @ N ) )
= ( sub @ M2 @ N ) ) ).
% plus_int_code(4)
thf(fact_5900_minus__int__code_I4_J,axiom,
! [M2: num,N: num] :
( ( minus_minus @ int @ ( pos @ M2 ) @ ( neg @ N ) )
= ( pos @ ( plus_plus @ num @ M2 @ N ) ) ) ).
% minus_int_code(4)
thf(fact_5901_plus__int__code_I3_J,axiom,
! [M2: num,N: num] :
( ( plus_plus @ int @ ( pos @ M2 ) @ ( pos @ N ) )
= ( pos @ ( plus_plus @ num @ M2 @ N ) ) ) ).
% plus_int_code(3)
thf(fact_5902_minus__int__code_I5_J,axiom,
! [M2: num,N: num] :
( ( minus_minus @ int @ ( neg @ M2 ) @ ( pos @ N ) )
= ( neg @ ( plus_plus @ num @ M2 @ N ) ) ) ).
% minus_int_code(5)
thf(fact_5903_Unity__def,axiom,
( product_Unity
= ( product_Abs_unit @ $true ) ) ).
% Unity_def
thf(fact_5904_Domainp__Domain__eq,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B )] :
( ( domainp @ A @ B
@ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R ) )
= ( ^ [X3: A] : ( member @ A @ X3 @ ( domain @ A @ B @ R ) ) ) ) ).
% Domainp_Domain_eq
thf(fact_5905_Domain__def,axiom,
! [B: $tType,A: $tType] :
( ( domain @ A @ B )
= ( ^ [R3: set @ ( product_prod @ A @ B )] :
( collect @ A
@ ( domainp @ A @ B
@ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R3 ) ) ) ) ) ).
% Domain_def
thf(fact_5906_Abs__unit__inject,axiom,
! [X: $o,Y: $o] :
( ( member @ $o @ X @ ( insert2 @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) )
=> ( ( member @ $o @ Y @ ( insert2 @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) )
=> ( ( ( product_Abs_unit @ X )
= ( product_Abs_unit @ Y ) )
= ( X = Y ) ) ) ) ).
% Abs_unit_inject
thf(fact_5907_Abs__unit__induct,axiom,
! [P: product_unit > $o,X: product_unit] :
( ! [Y4: $o] :
( ( member @ $o @ Y4 @ ( insert2 @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) )
=> ( P @ ( product_Abs_unit @ Y4 ) ) )
=> ( P @ X ) ) ).
% Abs_unit_induct
thf(fact_5908_Abs__unit__cases,axiom,
! [X: product_unit] :
~ ! [Y4: $o] :
( ( X
= ( product_Abs_unit @ Y4 ) )
=> ~ ( member @ $o @ Y4 @ ( insert2 @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) ) ) ).
% Abs_unit_cases
thf(fact_5909_Heap_ODomainp__rel,axiom,
! [B: $tType,A: $tType,R5: A > B > $o] :
( ( domainp @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ B ) @ ( heap_Time_rel_Heap @ A @ B @ R5 ) )
= ( heap_Time_pred_Heap @ A @ ( domainp @ A @ B @ R5 ) ) ) ).
% Heap.Domainp_rel
thf(fact_5910_integer_Odomain,axiom,
! [P: int > $o] :
( ( ( domainp @ int @ int
@ ^ [Y5: int,Z4: int] : Y5 = Z4 )
= P )
=> ( ( domainp @ int @ code_integer @ code_pcr_integer )
= P ) ) ).
% integer.domain
thf(fact_5911_natural_Odomain,axiom,
! [P: nat > $o] :
( ( ( domainp @ nat @ nat
@ ^ [Y5: nat,Z4: nat] : Y5 = Z4 )
= P )
=> ( ( domainp @ nat @ code_natural @ code_pcr_natural )
= P ) ) ).
% natural.domain
thf(fact_5912_option_ODomainp__rel,axiom,
! [B: $tType,A: $tType,R5: A > B > $o] :
( ( domainp @ ( option @ A ) @ ( option @ B ) @ ( rel_option @ A @ B @ R5 ) )
= ( pred_option @ A @ ( domainp @ A @ B @ R5 ) ) ) ).
% option.Domainp_rel
thf(fact_5913_Abs__unit__inverse,axiom,
! [Y: $o] :
( ( member @ $o @ Y @ ( insert2 @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) )
=> ( ( product_Rep_unit @ ( product_Abs_unit @ Y ) )
= Y ) ) ).
% Abs_unit_inverse
thf(fact_5914_type__definition__unit,axiom,
type_definition @ product_unit @ $o @ product_Rep_unit @ product_Abs_unit @ ( insert2 @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) ).
% type_definition_unit
thf(fact_5915_Rep__unit__inverse,axiom,
! [X: product_unit] :
( ( product_Abs_unit @ ( product_Rep_unit @ X ) )
= X ) ).
% Rep_unit_inverse
thf(fact_5916_Rep__unit__inject,axiom,
! [X: product_unit,Y: product_unit] :
( ( ( product_Rep_unit @ X )
= ( product_Rep_unit @ Y ) )
= ( X = Y ) ) ).
% Rep_unit_inject
thf(fact_5917_Rep__unit,axiom,
! [X: product_unit] : ( member @ $o @ ( product_Rep_unit @ X ) @ ( insert2 @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) ) ).
% Rep_unit
thf(fact_5918_Rep__unit__cases,axiom,
! [Y: $o] :
( ( member @ $o @ Y @ ( insert2 @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) )
=> ~ ! [X2: product_unit] :
( Y
= ( ~ ( product_Rep_unit @ X2 ) ) ) ) ).
% Rep_unit_cases
thf(fact_5919_Rep__unit__induct,axiom,
! [Y: $o,P: $o > $o] :
( ( member @ $o @ Y @ ( insert2 @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) )
=> ( ! [X2: product_unit] : ( P @ ( product_Rep_unit @ X2 ) )
=> ( P @ Y ) ) ) ).
% Rep_unit_induct
thf(fact_5920_adm__wf__def,axiom,
! [B: $tType,A: $tType] :
( ( adm_wf @ A @ B )
= ( ^ [R6: set @ ( product_prod @ A @ A ),F7: ( A > B ) > A > B] :
! [F2: A > B,G2: A > B,X3: A] :
( ! [Z5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z5 @ X3 ) @ R6 )
=> ( ( F2 @ Z5 )
= ( G2 @ Z5 ) ) )
=> ( ( F7 @ F2 @ X3 )
= ( F7 @ G2 @ X3 ) ) ) ) ) ).
% adm_wf_def
thf(fact_5921_folding__idem_Ocomp__fun__idem,axiom,
! [B: $tType,A: $tType,F: A > B > B,X: A] :
( ( finite_folding_idem @ A @ B @ F )
=> ( ( comp @ B @ B @ B @ ( F @ X ) @ ( F @ X ) )
= ( F @ X ) ) ) ).
% folding_idem.comp_fun_idem
thf(fact_5922_compat__def,axiom,
! [A2: $tType,A: $tType] :
( ( bNF_Wellorder_compat @ A @ A2 )
= ( ^ [R3: set @ ( product_prod @ A @ A ),R11: set @ ( product_prod @ A2 @ A2 ),F2: A > A2] :
! [A7: A,B5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B5 ) @ R3 )
=> ( member @ ( product_prod @ A2 @ A2 ) @ ( product_Pair @ A2 @ A2 @ ( F2 @ A7 ) @ ( F2 @ B5 ) ) @ R11 ) ) ) ) ).
% compat_def
thf(fact_5923_positive__add,axiom,
! [X: rat,Y: rat] :
( ( positive @ X )
=> ( ( positive @ Y )
=> ( positive @ ( plus_plus @ rat @ X @ Y ) ) ) ) ).
% positive_add
thf(fact_5924_inj__on__iff__Uniq,axiom,
! [B: $tType,A: $tType] :
( ( inj_on @ A @ B )
= ( ^ [F2: A > B,A9: set @ A] :
! [X3: A] :
( ( member @ A @ X3 @ A9 )
=> ( uniq @ A
@ ^ [Y3: A] :
( ( member @ A @ Y3 @ A9 )
& ( ( F2 @ X3 )
= ( F2 @ Y3 ) ) ) ) ) ) ) ).
% inj_on_iff_Uniq
thf(fact_5925_mono__imp__mono__on,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F: A > B,A4: set @ A] :
( ( order_mono @ A @ B @ F )
=> ( mono_on @ A @ B @ F @ A4 ) ) ) ).
% mono_imp_mono_on
thf(fact_5926_mono__on__subset,axiom,
! [B: $tType,A: $tType] :
( ( ( ord @ A )
& ( ord @ B ) )
=> ! [F: A > B,A4: set @ A,B6: set @ A] :
( ( mono_on @ A @ B @ F @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ A4 )
=> ( mono_on @ A @ B @ F @ B6 ) ) ) ) ).
% mono_on_subset
thf(fact_5927_mono__on__greaterD,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [G: A > B,A4: set @ A,X: A,Y: A] :
( ( mono_on @ A @ B @ G @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( member @ A @ Y @ A4 )
=> ( ( ord_less @ B @ ( G @ Y ) @ ( G @ X ) )
=> ( ord_less @ A @ Y @ X ) ) ) ) ) ) ).
% mono_on_greaterD
thf(fact_5928_mono__onD,axiom,
! [B: $tType,A: $tType] :
( ( ( ord @ A )
& ( ord @ B ) )
=> ! [F: A > B,A4: set @ A,R: A,S: A] :
( ( mono_on @ A @ B @ F @ A4 )
=> ( ( member @ A @ R @ A4 )
=> ( ( member @ A @ S @ A4 )
=> ( ( ord_less_eq @ A @ R @ S )
=> ( ord_less_eq @ B @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ) ).
% mono_onD
thf(fact_5929_mono__onI,axiom,
! [B: $tType,A: $tType] :
( ( ( ord @ A )
& ( ord @ B ) )
=> ! [A4: set @ A,F: A > B] :
( ! [R2: A,S6: A] :
( ( member @ A @ R2 @ A4 )
=> ( ( member @ A @ S6 @ A4 )
=> ( ( ord_less_eq @ A @ R2 @ S6 )
=> ( ord_less_eq @ B @ ( F @ R2 ) @ ( F @ S6 ) ) ) ) )
=> ( mono_on @ A @ B @ F @ A4 ) ) ) ).
% mono_onI
thf(fact_5930_mono__on__def,axiom,
! [B: $tType,A: $tType] :
( ( ( ord @ A )
& ( ord @ B ) )
=> ( ( mono_on @ A @ B )
= ( ^ [F2: A > B,A9: set @ A] :
! [R3: A,S4: A] :
( ( ( member @ A @ R3 @ A9 )
& ( member @ A @ S4 @ A9 )
& ( ord_less_eq @ A @ R3 @ S4 ) )
=> ( ord_less_eq @ B @ ( F2 @ R3 ) @ ( F2 @ S4 ) ) ) ) ) ) ).
% mono_on_def
thf(fact_5931_strict__mono__on__imp__mono__on,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( preorder @ B ) )
=> ! [F: A > B,A4: set @ A] :
( ( strict_mono_on @ A @ B @ F @ A4 )
=> ( mono_on @ A @ B @ F @ A4 ) ) ) ).
% strict_mono_on_imp_mono_on
thf(fact_5932_folding_Ointro,axiom,
! [B: $tType,A: $tType,F: A > B > B] :
( ! [Y4: A,X2: A] :
( ( comp @ B @ B @ B @ ( F @ Y4 ) @ ( F @ X2 ) )
= ( comp @ B @ B @ B @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( finite_folding @ A @ B @ F ) ) ).
% folding.intro
thf(fact_5933_folding_Ocomp__fun__commute,axiom,
! [B: $tType,A: $tType,F: A > B > B,Y: A,X: A] :
( ( finite_folding @ A @ B @ F )
=> ( ( comp @ B @ B @ B @ ( F @ Y ) @ ( F @ X ) )
= ( comp @ B @ B @ B @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% folding.comp_fun_commute
thf(fact_5934_strict__mono__on__eqD,axiom,
! [D: $tType,C: $tType] :
( ( ( linorder @ C )
& ( preorder @ D ) )
=> ! [F: C > D,A4: set @ C,X: C,Y: C] :
( ( strict_mono_on @ C @ D @ F @ A4 )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( ( member @ C @ X @ A4 )
=> ( ( member @ C @ Y @ A4 )
=> ( Y = X ) ) ) ) ) ) ).
% strict_mono_on_eqD
thf(fact_5935_strict__mono__on__leD,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( preorder @ B ) )
=> ! [F: A > B,A4: set @ A,X: A,Y: A] :
( ( strict_mono_on @ A @ B @ F @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( member @ A @ Y @ A4 )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_5936_strict__mono__on__def,axiom,
! [B: $tType,A: $tType] :
( ( ( ord @ A )
& ( ord @ B ) )
=> ( ( strict_mono_on @ A @ B )
= ( ^ [F2: A > B,A9: set @ A] :
! [R3: A,S4: A] :
( ( ( member @ A @ R3 @ A9 )
& ( member @ A @ S4 @ A9 )
& ( ord_less @ A @ R3 @ S4 ) )
=> ( ord_less @ B @ ( F2 @ R3 ) @ ( F2 @ S4 ) ) ) ) ) ) ).
% strict_mono_on_def
thf(fact_5937_strict__mono__onI,axiom,
! [B: $tType,A: $tType] :
( ( ( ord @ A )
& ( ord @ B ) )
=> ! [A4: set @ A,F: A > B] :
( ! [R2: A,S6: A] :
( ( member @ A @ R2 @ A4 )
=> ( ( member @ A @ S6 @ A4 )
=> ( ( ord_less @ A @ R2 @ S6 )
=> ( ord_less @ B @ ( F @ R2 ) @ ( F @ S6 ) ) ) ) )
=> ( strict_mono_on @ A @ B @ F @ A4 ) ) ) ).
% strict_mono_onI
thf(fact_5938_strict__mono__onD,axiom,
! [B: $tType,A: $tType] :
( ( ( ord @ A )
& ( ord @ B ) )
=> ! [F: A > B,A4: set @ A,R: A,S: A] :
( ( strict_mono_on @ A @ B @ F @ A4 )
=> ( ( member @ A @ R @ A4 )
=> ( ( member @ A @ S @ A4 )
=> ( ( ord_less @ A @ R @ S )
=> ( ord_less @ B @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_5939_strict__mono__on__imp__inj__on,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( preorder @ B ) )
=> ! [F: A > B,A4: set @ A] :
( ( strict_mono_on @ A @ B @ F @ A4 )
=> ( inj_on @ A @ B @ F @ A4 ) ) ) ).
% strict_mono_on_imp_inj_on
thf(fact_5940_folding__def,axiom,
! [B: $tType,A: $tType] :
( ( finite_folding @ A @ B )
= ( ^ [F2: A > B > B] :
! [Y3: A,X3: A] :
( ( comp @ B @ B @ B @ ( F2 @ Y3 ) @ ( F2 @ X3 ) )
= ( comp @ B @ B @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) ) ) ) ).
% folding_def
thf(fact_5941_folding__idem__on_Ocomp__fun__idem__on,axiom,
! [B: $tType,A: $tType,S2: set @ A,F: A > B > B,X: A,Y: A] :
( ( finite1890593828518410140dem_on @ A @ B @ S2 @ F )
=> ( ( member @ A @ X @ S2 )
=> ( ( member @ A @ Y @ S2 )
=> ( ( comp @ B @ B @ B @ ( F @ X ) @ ( F @ X ) )
= ( F @ X ) ) ) ) ) ).
% folding_idem_on.comp_fun_idem_on
thf(fact_5942_sum_OPlus,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,B6: set @ C,G: ( sum_sum @ B @ C ) > A] :
( ( finite_finite2 @ B @ A4 )
=> ( ( finite_finite2 @ C @ B6 )
=> ( ( groups7311177749621191930dd_sum @ ( sum_sum @ B @ C ) @ A @ G @ ( sum_Plus @ B @ C @ A4 @ B6 ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ ( comp @ ( sum_sum @ B @ C ) @ A @ B @ G @ ( sum_Inl @ B @ C ) ) @ A4 ) @ ( groups7311177749621191930dd_sum @ C @ A @ ( comp @ ( sum_sum @ B @ C ) @ A @ C @ G @ ( sum_Inr @ C @ B ) ) @ B6 ) ) ) ) ) ) ).
% sum.Plus
thf(fact_5943_map__sum__Inl__conv,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,Fl: C > A,Fr: D > B,S: sum_sum @ C @ D,Y: A] :
( ( ( sum_map_sum @ C @ A @ D @ B @ Fl @ Fr @ S )
= ( sum_Inl @ A @ B @ Y ) )
= ( ? [X3: C] :
( ( S
= ( sum_Inl @ C @ D @ X3 ) )
& ( Y
= ( Fl @ X3 ) ) ) ) ) ).
% map_sum_Inl_conv
thf(fact_5944_map__sum__Inr__conv,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Fl: C > A,Fr: D > B,S: sum_sum @ C @ D,Y: B] :
( ( ( sum_map_sum @ C @ A @ D @ B @ Fl @ Fr @ S )
= ( sum_Inr @ B @ A @ Y ) )
= ( ? [X3: D] :
( ( S
= ( sum_Inr @ D @ C @ X3 ) )
& ( Y
= ( Fr @ X3 ) ) ) ) ) ).
% map_sum_Inr_conv
thf(fact_5945_Union__plus,axiom,
! [A: $tType,B: $tType,C: $tType,F: ( sum_sum @ B @ C ) > ( set @ A ),A4: set @ B,B6: set @ C] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( sum_sum @ B @ C ) @ ( set @ A ) @ F @ ( sum_Plus @ B @ C @ A4 @ B6 ) ) )
= ( sup_sup @ ( set @ A )
@ ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [A7: B] : ( F @ ( sum_Inl @ B @ C @ A7 ) )
@ A4 ) )
@ ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [B5: C] : ( F @ ( sum_Inr @ C @ B @ B5 ) )
@ B6 ) ) ) ) ).
% Union_plus
thf(fact_5946_Union__sum,axiom,
! [C: $tType,A: $tType,B: $tType,F: ( sum_sum @ A @ B ) > ( set @ C )] :
( ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ ( sum_sum @ A @ B ) @ ( set @ C ) @ F @ ( top_top @ ( set @ ( sum_sum @ A @ B ) ) ) ) )
= ( sup_sup @ ( set @ C )
@ ( complete_Sup_Sup @ ( set @ C )
@ ( image2 @ A @ ( set @ C )
@ ^ [L3: A] : ( F @ ( sum_Inl @ A @ B @ L3 ) )
@ ( top_top @ ( set @ A ) ) ) )
@ ( complete_Sup_Sup @ ( set @ C )
@ ( image2 @ B @ ( set @ C )
@ ^ [R3: B] : ( F @ ( sum_Inr @ B @ A @ R3 ) )
@ ( top_top @ ( set @ B ) ) ) ) ) ) ).
% Union_sum
thf(fact_5947_sum_Osize__gen_I1_J,axiom,
! [B: $tType,A: $tType,Xa: A > nat,X: B > nat,X1: A] :
( ( basic_BNF_size_sum @ A @ B @ Xa @ X @ ( sum_Inl @ A @ B @ X1 ) )
= ( plus_plus @ nat @ ( Xa @ X1 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% sum.size_gen(1)
thf(fact_5948_sum_Osize__gen_I2_J,axiom,
! [A: $tType,B: $tType,Xa: A > nat,X: B > nat,X22: B] :
( ( basic_BNF_size_sum @ A @ B @ Xa @ X @ ( sum_Inr @ B @ A @ X22 ) )
= ( plus_plus @ nat @ ( X @ X22 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% sum.size_gen(2)
thf(fact_5949_Node__def,axiom,
! [A: $tType,B: $tType] :
( ( old_Node @ B @ A )
= ( collect @ ( product_prod @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat ) )
@ ^ [P6: product_prod @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat )] :
? [F2: nat > ( sum_sum @ B @ nat ),X3: sum_sum @ A @ nat,K4: nat] :
( ( P6
= ( product_Pair @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat ) @ F2 @ X3 ) )
& ( ( F2 @ K4 )
= ( sum_Inr @ nat @ B @ ( zero_zero @ nat ) ) ) ) ) ) ).
% Node_def
thf(fact_5950_Node__K0__I,axiom,
! [B: $tType,A: $tType,A3: sum_sum @ B @ nat] :
( member @ ( product_prod @ ( nat > ( sum_sum @ A @ nat ) ) @ ( sum_sum @ B @ nat ) )
@ ( product_Pair @ ( nat > ( sum_sum @ A @ nat ) ) @ ( sum_sum @ B @ nat )
@ ^ [K4: nat] : ( sum_Inr @ nat @ A @ ( zero_zero @ nat ) )
@ A3 )
@ ( old_Node @ A @ B ) ) ).
% Node_K0_I
thf(fact_5951_nth__item_Opinduct,axiom,
! [A0: nat,P: nat > $o] :
( ( accp @ nat @ nth_item_rel @ A0 )
=> ( ( ( accp @ nat @ nth_item_rel @ ( zero_zero @ nat ) )
=> ( P @ ( zero_zero @ nat ) ) )
=> ( ! [N3: nat] :
( ( accp @ nat @ nth_item_rel @ ( suc @ N3 ) )
=> ( ! [A17: nat,Aa4: nat] :
( ( ( nat_sum_decode @ N3 )
= ( sum_Inl @ nat @ nat @ A17 ) )
=> ( ( ( nat_sum_decode @ A17 )
= ( sum_Inl @ nat @ nat @ Aa4 ) )
=> ( P @ Aa4 ) ) )
=> ( ! [A17: nat,B8: nat] :
( ( ( nat_sum_decode @ N3 )
= ( sum_Inl @ nat @ nat @ A17 ) )
=> ( ( ( nat_sum_decode @ A17 )
= ( sum_Inr @ nat @ nat @ B8 ) )
=> ( P @ B8 ) ) )
=> ( ! [B8: nat,Ba2: nat,X7: nat,Y6: nat] :
( ( ( nat_sum_decode @ N3 )
= ( sum_Inr @ nat @ nat @ B8 ) )
=> ( ( ( nat_sum_decode @ B8 )
= ( sum_Inr @ nat @ nat @ Ba2 ) )
=> ( ( ( product_Pair @ nat @ nat @ X7 @ Y6 )
= ( nat_prod_decode @ Ba2 ) )
=> ( P @ X7 ) ) ) )
=> ( ! [B8: nat,Ba2: nat,X7: nat,Y6: nat] :
( ( ( nat_sum_decode @ N3 )
= ( sum_Inr @ nat @ nat @ B8 ) )
=> ( ( ( nat_sum_decode @ B8 )
= ( sum_Inr @ nat @ nat @ Ba2 ) )
=> ( ( ( product_Pair @ nat @ nat @ X7 @ Y6 )
= ( nat_prod_decode @ Ba2 ) )
=> ( P @ Y6 ) ) ) )
=> ( P @ ( suc @ N3 ) ) ) ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% nth_item.pinduct
thf(fact_5952_ndepth__K0,axiom,
! [A: $tType,B: $tType,X: sum_sum @ A @ nat] :
( ( old_ndepth @ A @ B
@ ( old_Abs_Node @ B @ A
@ ( product_Pair @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat )
@ ^ [K4: nat] : ( sum_Inr @ nat @ B @ ( zero_zero @ nat ) )
@ X ) ) )
= ( zero_zero @ nat ) ) ).
% ndepth_K0
thf(fact_5953_Atom__def,axiom,
! [B: $tType,A: $tType] :
( ( old_Atom @ A @ B )
= ( ^ [X3: sum_sum @ A @ nat] :
( insert2 @ ( old_node @ A @ B )
@ ( old_Abs_Node @ B @ A
@ ( product_Pair @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat )
@ ^ [K4: nat] : ( sum_Inr @ nat @ B @ ( zero_zero @ nat ) )
@ X3 ) )
@ ( bot_bot @ ( set @ ( old_node @ A @ B ) ) ) ) ) ) ).
% Atom_def
thf(fact_5954_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)
thf(fact_5955_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
= ( suc @ N3 ) )
=> ( ( Y
= ( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
@ ^ [I3: nat] :
( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
@ ^ [J2: nat] : ( old_In0 @ A @ product_unit @ ( nth_item @ A @ J2 ) )
@ ^ [J2: nat] : ( old_In1 @ A @ product_unit @ ( nth_item @ A @ J2 ) )
@ ( nat_sum_decode @ I3 ) )
@ ^ [I3: nat] :
( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
@ ^ [J2: nat] : ( old_Leaf @ A @ product_unit @ ( from_nat @ A @ J2 ) )
@ ^ [J2: nat] :
( product_case_prod @ nat @ nat @ ( set @ ( old_node @ A @ product_unit ) )
@ ^ [A7: nat,B5: nat] : ( old_Scons @ A @ product_unit @ ( nth_item @ A @ A7 ) @ ( nth_item @ A @ B5 ) )
@ ( nat_prod_decode @ J2 ) )
@ ( nat_sum_decode @ I3 ) )
@ ( nat_sum_decode @ N3 ) ) )
=> ~ ( accp @ nat @ nth_item_rel @ ( suc @ N3 ) ) ) ) ) ) ) ) ).
% nth_item.pelims
thf(fact_5956_nth__item_Opsimps_I2_J,axiom,
! [A: $tType] :
( ( countable @ A )
=> ! [N: nat] :
( ( accp @ nat @ nth_item_rel @ ( suc @ N ) )
=> ( ( nth_item @ A @ ( suc @ N ) )
= ( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
@ ^ [I3: nat] :
( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
@ ^ [J2: nat] : ( old_In0 @ A @ product_unit @ ( nth_item @ A @ J2 ) )
@ ^ [J2: nat] : ( old_In1 @ A @ product_unit @ ( nth_item @ A @ J2 ) )
@ ( nat_sum_decode @ I3 ) )
@ ^ [I3: nat] :
( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
@ ^ [J2: nat] : ( old_Leaf @ A @ product_unit @ ( from_nat @ A @ J2 ) )
@ ^ [J2: nat] :
( product_case_prod @ nat @ nat @ ( set @ ( old_node @ A @ product_unit ) )
@ ^ [A7: nat,B5: nat] : ( old_Scons @ A @ product_unit @ ( nth_item @ A @ A7 ) @ ( nth_item @ A @ B5 ) )
@ ( nat_prod_decode @ J2 ) )
@ ( nat_sum_decode @ I3 ) )
@ ( nat_sum_decode @ N ) ) ) ) ) ).
% nth_item.psimps(2)
thf(fact_5957_Scons__def,axiom,
! [B: $tType,A: $tType] :
( ( old_Scons @ A @ B )
= ( ^ [M8: set @ ( old_node @ A @ B ),N9: set @ ( old_node @ A @ B )] : ( sup_sup @ ( set @ ( old_node @ A @ B ) ) @ ( image2 @ ( old_node @ A @ B ) @ ( old_node @ A @ B ) @ ( old_Push_Node @ B @ A @ ( sum_Inr @ nat @ B @ ( one_one @ nat ) ) ) @ M8 ) @ ( image2 @ ( old_node @ A @ B ) @ ( old_node @ A @ B ) @ ( old_Push_Node @ B @ A @ ( sum_Inr @ nat @ B @ ( suc @ ( one_one @ nat ) ) ) ) @ N9 ) ) ) ) ).
% Scons_def
thf(fact_5958_In1__def,axiom,
! [A: $tType,B: $tType] :
( ( old_In1 @ A @ B )
= ( old_Scons @ A @ B @ ( old_Numb @ A @ B @ ( one_one @ nat ) ) ) ) ).
% In1_def
thf(fact_5959_dsum__def,axiom,
! [B: $tType,A: $tType] :
( ( old_dsum @ A @ B )
= ( ^ [R3: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ),S4: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) )] :
( sup_sup @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) )
@ ( complete_Sup_Sup @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) )
@ ( image2 @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) )
@ ( product_case_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) )
@ ^ [X3: set @ ( old_node @ A @ B ),X9: set @ ( old_node @ A @ B )] : ( insert2 @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( old_In0 @ A @ B @ X3 ) @ ( old_In0 @ A @ B @ X9 ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) ) ) ) )
@ R3 ) )
@ ( complete_Sup_Sup @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) )
@ ( image2 @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) )
@ ( product_case_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) )
@ ^ [Y3: set @ ( old_node @ A @ B ),Y8: set @ ( old_node @ A @ B )] : ( insert2 @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( old_In1 @ A @ B @ Y3 ) @ ( old_In1 @ A @ B @ Y8 ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) ) ) ) )
@ S4 ) ) ) ) ) ).
% dsum_def
thf(fact_5960_dsumE,axiom,
! [B: $tType,A: $tType,W: product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ),R: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ),S: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) )] :
( ( member @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ W @ ( old_dsum @ A @ B @ R @ S ) )
=> ( ! [X2: set @ ( old_node @ A @ B ),X5: set @ ( old_node @ A @ B )] :
( ( member @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ X2 @ X5 ) @ R )
=> ( W
!= ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( old_In0 @ A @ B @ X2 ) @ ( old_In0 @ A @ B @ X5 ) ) ) )
=> ~ ! [Y4: set @ ( old_node @ A @ B ),Y14: set @ ( old_node @ A @ B )] :
( ( member @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ Y4 @ Y14 ) @ S )
=> ( W
!= ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( old_In1 @ A @ B @ Y4 ) @ ( old_In1 @ A @ B @ Y14 ) ) ) ) ) ) ).
% dsumE
thf(fact_5961_dsum__In0I,axiom,
! [B: $tType,A: $tType,M3: set @ ( old_node @ A @ B ),M10: set @ ( old_node @ A @ B ),R: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ),S: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) )] :
( ( member @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ M3 @ M10 ) @ R )
=> ( member @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( old_In0 @ A @ B @ M3 ) @ ( old_In0 @ A @ B @ M10 ) ) @ ( old_dsum @ A @ B @ R @ S ) ) ) ).
% dsum_In0I
thf(fact_5962_dsum__In1I,axiom,
! [B: $tType,A: $tType,N6: set @ ( old_node @ A @ B ),N11: set @ ( old_node @ A @ B ),S: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ),R: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) )] :
( ( member @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ N6 @ N11 ) @ S )
=> ( member @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( old_In1 @ A @ B @ N6 ) @ ( old_In1 @ A @ B @ N11 ) ) @ ( old_dsum @ A @ B @ R @ S ) ) ) ).
% dsum_In1I
thf(fact_5963_dprod__def,axiom,
! [B: $tType,A: $tType] :
( ( old_dprod @ A @ B )
= ( ^ [R3: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ),S4: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) )] :
( complete_Sup_Sup @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) )
@ ( image2 @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) )
@ ( product_case_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) )
@ ^ [X3: set @ ( old_node @ A @ B ),X9: set @ ( old_node @ A @ B )] :
( complete_Sup_Sup @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) )
@ ( image2 @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) )
@ ( product_case_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) )
@ ^ [Y3: set @ ( old_node @ A @ B ),Y8: set @ ( old_node @ A @ B )] : ( insert2 @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( old_Scons @ A @ B @ X3 @ Y3 ) @ ( old_Scons @ A @ B @ X9 @ Y8 ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) ) ) ) )
@ S4 ) ) )
@ R3 ) ) ) ) ).
% dprod_def
thf(fact_5964_dprodI,axiom,
! [B: $tType,A: $tType,M3: set @ ( old_node @ A @ B ),M10: set @ ( old_node @ A @ B ),R: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ),N6: set @ ( old_node @ A @ B ),N11: set @ ( old_node @ A @ B ),S: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) )] :
( ( member @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ M3 @ M10 ) @ R )
=> ( ( member @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ N6 @ N11 ) @ S )
=> ( member @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( old_Scons @ A @ B @ M3 @ N6 ) @ ( old_Scons @ A @ B @ M10 @ N11 ) ) @ ( old_dprod @ A @ B @ R @ S ) ) ) ) ).
% dprodI
thf(fact_5965_dprodE,axiom,
! [B: $tType,A: $tType,C2: product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ),R: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ),S: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) )] :
( ( member @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ C2 @ ( old_dprod @ A @ B @ R @ S ) )
=> ~ ! [X2: set @ ( old_node @ A @ B ),Y4: set @ ( old_node @ A @ B ),X5: set @ ( old_node @ A @ B )] :
( ( member @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ X2 @ X5 ) @ R )
=> ! [Y14: set @ ( old_node @ A @ B )] :
( ( member @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ Y4 @ Y14 ) @ S )
=> ( C2
!= ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( old_Scons @ A @ B @ X2 @ Y4 ) @ ( old_Scons @ A @ B @ X5 @ Y14 ) ) ) ) ) ) ).
% dprodE
thf(fact_5966_prop__matchD,axiom,
! [A: $tType,Al: list @ A,E2: A,Bl: list @ A,Al2: list @ A,E6: A,Bl2: list @ A,P: A > $o] :
( ( ( append @ A @ Al @ ( cons @ A @ E2 @ Bl ) )
= ( append @ A @ Al2 @ ( cons @ A @ E6 @ Bl2 ) ) )
=> ( ( list_all @ A @ P @ Al )
=> ( ~ ( P @ E2 )
=> ( ~ ( P @ E6 )
=> ( ( list_all @ A @ P @ Bl )
=> ( ( Al = Al2 )
& ( E2 = E6 )
& ( Bl = Bl2 ) ) ) ) ) ) ) ).
% prop_matchD
thf(fact_5967_prop__match,axiom,
! [A: $tType,P: A > $o,Al: list @ A,E2: A,E6: A,Bl: list @ A,Al2: list @ A,Bl2: list @ A] :
( ( list_all @ A @ P @ Al )
=> ( ~ ( P @ E2 )
=> ( ~ ( P @ E6 )
=> ( ( list_all @ A @ P @ Bl )
=> ( ( ( append @ A @ Al @ ( cons @ A @ E2 @ Bl ) )
= ( append @ A @ Al2 @ ( cons @ A @ E6 @ Bl2 ) ) )
= ( ( Al = Al2 )
& ( E2 = E6 )
& ( Bl = Bl2 ) ) ) ) ) ) ) ).
% prop_match
thf(fact_5968_Heap__lub__def,axiom,
! [A: $tType] :
( ( heap_Time_Heap_lub @ A )
= ( partial_img_lub @ ( heap_Time_Heap @ A ) @ ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ ( heap_Time_Heap @ A ) @ ( heap_Time_execute @ A ) @ ( heap_Time_Heap2 @ A ) @ ( partial_fun_lub @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( heap_ext @ product_unit ) @ ( partial_flat_lub @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ) ).
% Heap_lub_def
thf(fact_5969_heap__step__admissible,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: E > A > C > B > D > $o,X: E] :
( comple1908693960933563346ssible @ ( A > ( option @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) ) @ ( partial_fun_lub @ ( option @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) @ ( option @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) @ A @ ( partial_flat_lub @ ( option @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) @ ( none @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) ) ) @ ( partial_fun_ord @ ( option @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) @ ( option @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) @ A @ ( partial_flat_ord @ ( option @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) @ ( none @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) ) )
@ ^ [F2: A > ( option @ ( product_prod @ B @ ( product_prod @ C @ D ) ) )] :
! [H: A,H7: C,R3: B,N5: D] :
( ( ( F2 @ H )
= ( some @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ R3 @ ( product_Pair @ C @ D @ H7 @ N5 ) ) ) )
=> ( P @ X @ H @ H7 @ R3 @ N5 ) ) ) ).
% heap_step_admissible
thf(fact_5970_option__admissible,axiom,
! [B: $tType,A: $tType,P: A > B > $o] :
( comple1908693960933563346ssible @ ( A > ( option @ B ) ) @ ( partial_fun_lub @ ( option @ B ) @ ( option @ B ) @ A @ ( partial_flat_lub @ ( option @ B ) @ ( none @ B ) ) ) @ ( partial_fun_ord @ ( option @ B ) @ ( option @ B ) @ A @ ( partial_flat_ord @ ( option @ B ) @ ( none @ B ) ) )
@ ^ [F2: A > ( option @ B )] :
! [X3: A,Y3: B] :
( ( ( F2 @ X3 )
= ( some @ B @ Y3 ) )
=> ( P @ X3 @ Y3 ) ) ) ).
% option_admissible
thf(fact_5971_option_Olub__upper,axiom,
! [A: $tType,A4: set @ ( option @ A ),X: option @ A] :
( ( comple1602240252501008431_chain @ ( option @ A ) @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) ) @ A4 )
=> ( ( member @ ( option @ A ) @ X @ A4 )
=> ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) @ X @ ( partial_flat_lub @ ( option @ A ) @ ( none @ A ) @ A4 ) ) ) ) ).
% option.lub_upper
thf(fact_5972_option_Oleq__antisym,axiom,
! [A: $tType,X: option @ A,Y: option @ A] :
( ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) @ X @ Y )
=> ( ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) @ Y @ X )
=> ( X = Y ) ) ) ).
% option.leq_antisym
thf(fact_5973_option_Oleq__trans,axiom,
! [A: $tType,X: option @ A,Y: option @ A,Z2: option @ A] :
( ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) @ X @ Y )
=> ( ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) @ Y @ Z2 )
=> ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) @ X @ Z2 ) ) ) ).
% option.leq_trans
thf(fact_5974_option_Oleq__refl,axiom,
! [A: $tType,X: option @ A] : ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) @ X @ X ) ).
% option.leq_refl
thf(fact_5975_option_Olub__least,axiom,
! [A: $tType,A4: set @ ( option @ A ),Z2: option @ A] :
( ( comple1602240252501008431_chain @ ( option @ A ) @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) ) @ A4 )
=> ( ! [X2: option @ A] :
( ( member @ ( option @ A ) @ X2 @ A4 )
=> ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) @ X2 @ Z2 ) )
=> ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) @ ( partial_flat_lub @ ( option @ A ) @ ( none @ A ) @ A4 ) @ Z2 ) ) ) ).
% option.lub_least
thf(fact_5976_Heap__ord__def,axiom,
! [A: $tType] :
( ( heap_Time_Heap_ord @ A )
= ( partial_img_ord @ ( heap_Time_Heap @ A ) @ ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ $o @ ( heap_Time_execute @ A ) @ ( partial_fun_ord @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( heap_ext @ product_unit ) @ ( partial_flat_ord @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ) ).
% Heap_ord_def
thf(fact_5977_Partial__Function_Obind__mono,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,B6: ( A > ( option @ B ) ) > ( option @ C ),C5: C > ( A > ( option @ B ) ) > ( option @ D )] :
( ( comple7038119648293358887notone @ ( A > ( option @ B ) ) @ ( option @ C ) @ ( partial_fun_ord @ ( option @ B ) @ ( option @ B ) @ A @ ( partial_flat_ord @ ( option @ B ) @ ( none @ B ) ) ) @ ( partial_flat_ord @ ( option @ C ) @ ( none @ C ) ) @ B6 )
=> ( ! [Y4: C] : ( comple7038119648293358887notone @ ( A > ( option @ B ) ) @ ( option @ D ) @ ( partial_fun_ord @ ( option @ B ) @ ( option @ B ) @ A @ ( partial_flat_ord @ ( option @ B ) @ ( none @ B ) ) ) @ ( partial_flat_ord @ ( option @ D ) @ ( none @ D ) ) @ ( C5 @ Y4 ) )
=> ( comple7038119648293358887notone @ ( A > ( option @ B ) ) @ ( option @ D ) @ ( partial_fun_ord @ ( option @ B ) @ ( option @ B ) @ A @ ( partial_flat_ord @ ( option @ B ) @ ( none @ B ) ) ) @ ( partial_flat_ord @ ( option @ D ) @ ( none @ D ) )
@ ^ [F2: A > ( option @ B )] :
( bind @ C @ D @ ( B6 @ F2 )
@ ^ [Y3: C] : ( C5 @ Y3 @ F2 ) ) ) ) ) ).
% Partial_Function.bind_mono
thf(fact_5978_option_Oconst__mono,axiom,
! [A: $tType,B: $tType,Ord: B > B > $o,C2: option @ A] :
( comple7038119648293358887notone @ B @ ( option @ A ) @ Ord @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) )
@ ^ [F2: B] : C2 ) ).
% option.const_mono
thf(fact_5979_option_Ofixp__induct__uc,axiom,
! [B: $tType,A: $tType,C: $tType,U4: C > B > ( option @ A ),F6: C > C,C5: ( B > ( option @ A ) ) > C,F: C,P: ( B > ( option @ A ) ) > $o] :
( ! [X2: B] :
( comple7038119648293358887notone @ ( B > ( option @ A ) ) @ ( option @ A ) @ ( partial_fun_ord @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) ) ) @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) )
@ ^ [F2: B > ( option @ A )] : ( U4 @ ( F6 @ ( C5 @ F2 ) ) @ X2 ) )
=> ( ( F
= ( C5
@ ( comple187402453842119260l_fixp @ ( B > ( option @ A ) ) @ ( partial_fun_lub @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_lub @ ( option @ A ) @ ( none @ A ) ) ) @ ( partial_fun_ord @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) ) )
@ ^ [F2: B > ( option @ A )] : ( U4 @ ( F6 @ ( C5 @ F2 ) ) ) ) ) )
=> ( ! [F4: B > ( option @ A )] :
( ( U4 @ ( C5 @ F4 ) )
= F4 )
=> ( ( comple1908693960933563346ssible @ ( B > ( option @ A ) ) @ ( partial_fun_lub @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_lub @ ( option @ A ) @ ( none @ A ) ) ) @ ( partial_fun_ord @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) ) ) @ P )
=> ( ( P
@ ^ [Uu: B] : ( none @ A ) )
=> ( ! [F4: C] :
( ( P @ ( U4 @ F4 ) )
=> ( P @ ( U4 @ ( F6 @ F4 ) ) ) )
=> ( P @ ( U4 @ F ) ) ) ) ) ) ) ) ).
% option.fixp_induct_uc
thf(fact_5980_fixp__induct__option,axiom,
! [C: $tType,B: $tType,A: $tType,U4: C > B > ( option @ A ),F6: C > C,C5: ( B > ( option @ A ) ) > C,F: C,P: B > A > $o,X: B,Y: A] :
( ! [X2: B] :
( comple7038119648293358887notone @ ( B > ( option @ A ) ) @ ( option @ A ) @ ( partial_fun_ord @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) ) ) @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) )
@ ^ [F2: B > ( option @ A )] : ( U4 @ ( F6 @ ( C5 @ F2 ) ) @ X2 ) )
=> ( ( F
= ( C5
@ ( comple187402453842119260l_fixp @ ( B > ( option @ A ) ) @ ( partial_fun_lub @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_lub @ ( option @ A ) @ ( none @ A ) ) ) @ ( partial_fun_ord @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) ) )
@ ^ [F2: B > ( option @ A )] : ( U4 @ ( F6 @ ( C5 @ F2 ) ) ) ) ) )
=> ( ! [F4: B > ( option @ A )] :
( ( U4 @ ( C5 @ F4 ) )
= F4 )
=> ( ! [F4: C,X2: B,Y4: A] :
( ! [Xa2: B,Ya3: A] :
( ( ( U4 @ F4 @ Xa2 )
= ( some @ A @ Ya3 ) )
=> ( P @ Xa2 @ Ya3 ) )
=> ( ( ( U4 @ ( F6 @ F4 ) @ X2 )
= ( some @ A @ Y4 ) )
=> ( P @ X2 @ Y4 ) ) )
=> ( ( ( U4 @ F @ X )
= ( some @ A @ Y ) )
=> ( P @ X @ Y ) ) ) ) ) ) ).
% fixp_induct_option
thf(fact_5981_option_Ofixp__rule__uc,axiom,
! [B: $tType,A: $tType,C: $tType,U4: C > B > ( option @ A ),F6: C > C,C5: ( B > ( option @ A ) ) > C,F: C] :
( ! [X2: B] :
( comple7038119648293358887notone @ ( B > ( option @ A ) ) @ ( option @ A ) @ ( partial_fun_ord @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) ) ) @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) )
@ ^ [F2: B > ( option @ A )] : ( U4 @ ( F6 @ ( C5 @ F2 ) ) @ X2 ) )
=> ( ( F
= ( C5
@ ( comple187402453842119260l_fixp @ ( B > ( option @ A ) ) @ ( partial_fun_lub @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_lub @ ( option @ A ) @ ( none @ A ) ) ) @ ( partial_fun_ord @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) ) )
@ ^ [F2: B > ( option @ A )] : ( U4 @ ( F6 @ ( C5 @ F2 ) ) ) ) ) )
=> ( ! [F4: C] :
( ( C5 @ ( U4 @ F4 ) )
= F4 )
=> ( F
= ( F6 @ F ) ) ) ) ) ).
% option.fixp_rule_uc
thf(fact_5982_option_Omono__body__fixp,axiom,
! [A: $tType,B: $tType,F6: ( B > ( option @ A ) ) > B > ( option @ A )] :
( ! [X2: B] :
( comple7038119648293358887notone @ ( B > ( option @ A ) ) @ ( option @ A ) @ ( partial_fun_ord @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) ) ) @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) )
@ ^ [F2: B > ( option @ A )] : ( F6 @ F2 @ X2 ) )
=> ( ( comple187402453842119260l_fixp @ ( B > ( option @ A ) ) @ ( partial_fun_lub @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_lub @ ( option @ A ) @ ( none @ A ) ) ) @ ( partial_fun_ord @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) ) ) @ F6 )
= ( F6 @ ( comple187402453842119260l_fixp @ ( B > ( option @ A ) ) @ ( partial_fun_lub @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_lub @ ( option @ A ) @ ( none @ A ) ) ) @ ( partial_fun_ord @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) ) ) @ F6 ) ) ) ) ).
% option.mono_body_fixp
thf(fact_5983_option_Opartial__function__definitions__axioms,axiom,
! [A: $tType] : ( partia7178651479351089652itions @ ( option @ A ) @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) ) @ ( partial_flat_lub @ ( option @ A ) @ ( none @ A ) ) ) ).
% option.partial_function_definitions_axioms
thf(fact_5984_fixp__mono,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [F: A > A,G: A > A] :
( ( comple7038119648293358887notone @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less_eq @ A ) @ F )
=> ( ( comple7038119648293358887notone @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less_eq @ A ) @ G )
=> ( ! [Z11: A] : ( ord_less_eq @ A @ ( F @ Z11 ) @ ( G @ Z11 ) )
=> ( ord_less_eq @ A @ ( comple115746919287870866o_fixp @ A @ F ) @ ( comple115746919287870866o_fixp @ A @ G ) ) ) ) ) ) ).
% fixp_mono
thf(fact_5985_heap_Oconst__mono,axiom,
! [A: $tType,B: $tType,Ord: B > B > $o,C2: heap_Time_Heap @ A] :
( comple7038119648293358887notone @ B @ ( heap_Time_Heap @ A ) @ Ord @ ( heap_Time_Heap_ord @ A )
@ ^ [F2: B] : C2 ) ).
% heap.const_mono
thf(fact_5986_Heap__Time__Monad_Obind__mono,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,B6: ( A > ( heap_Time_Heap @ B ) ) > ( heap_Time_Heap @ C ),C5: C > ( A > ( heap_Time_Heap @ B ) ) > ( heap_Time_Heap @ D )] :
( ( comple7038119648293358887notone @ ( 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 ) @ B6 )
=> ( ! [Y4: C] : ( comple7038119648293358887notone @ ( 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 ) @ ( C5 @ Y4 ) )
=> ( comple7038119648293358887notone @ ( 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 )
@ ^ [F2: A > ( heap_Time_Heap @ B )] :
( heap_Time_bind @ C @ D @ ( B6 @ F2 )
@ ^ [Y3: C] : ( C5 @ Y3 @ F2 ) ) ) ) ) ).
% Heap_Time_Monad.bind_mono
thf(fact_5987_fixp__induct__heap,axiom,
! [C: $tType,A: $tType,B: $tType,U4: C > B > ( heap_Time_Heap @ A ),F6: C > C,C5: ( B > ( heap_Time_Heap @ A ) ) > C,F: C,P: B > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) > A > nat > $o,X: B,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A,N: nat] :
( ! [X2: B] :
( comple7038119648293358887notone @ ( 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 )
@ ^ [F2: B > ( heap_Time_Heap @ A )] : ( U4 @ ( F6 @ ( C5 @ F2 ) ) @ X2 ) )
=> ( ( F
= ( C5
@ ( comple187402453842119260l_fixp @ ( 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 ) )
@ ^ [F2: B > ( heap_Time_Heap @ A )] : ( U4 @ ( F6 @ ( C5 @ F2 ) ) ) ) ) )
=> ( ! [F4: B > ( heap_Time_Heap @ A )] :
( ( U4 @ ( C5 @ F4 ) )
= F4 )
=> ( ! [F4: C,X2: B,H2: heap_ext @ product_unit,H6: heap_ext @ product_unit,R2: A,N3: nat] :
( ! [Xa2: B,Ha: heap_ext @ product_unit,H_a: heap_ext @ product_unit,Ra3: A,Na: nat] :
( ( heap_Time_effect @ A @ ( U4 @ F4 @ Xa2 ) @ Ha @ H_a @ Ra3 @ Na )
=> ( P @ Xa2 @ Ha @ H_a @ Ra3 @ Na ) )
=> ( ( heap_Time_effect @ A @ ( U4 @ ( F6 @ F4 ) @ X2 ) @ H2 @ H6 @ R2 @ N3 )
=> ( P @ X2 @ H2 @ H6 @ R2 @ N3 ) ) )
=> ( ( heap_Time_effect @ A @ ( U4 @ F @ X ) @ H3 @ H4 @ R @ N )
=> ( P @ X @ H3 @ H4 @ R @ N ) ) ) ) ) ) ).
% fixp_induct_heap
thf(fact_5988_refines__adm,axiom,
! [B: $tType,A: $tType,T2: A > ( heap_Time_Heap @ B )] :
( comple1908693960933563346ssible @ ( 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 ) )
@ ^ [F2: A > ( heap_Time_Heap @ B )] :
! [X3: A] : ( refine_Imp_refines @ B @ ( T2 @ X3 ) @ ( F2 @ X3 ) ) ) ).
% refines_adm
thf(fact_5989_heap_Ofixp__induct__uc,axiom,
! [B: $tType,A: $tType,C: $tType,U4: C > B > ( heap_Time_Heap @ A ),F6: C > C,C5: ( B > ( heap_Time_Heap @ A ) ) > C,F: C,P: ( B > ( heap_Time_Heap @ A ) ) > $o] :
( ! [X2: B] :
( comple7038119648293358887notone @ ( 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 )
@ ^ [F2: B > ( heap_Time_Heap @ A )] : ( U4 @ ( F6 @ ( C5 @ F2 ) ) @ X2 ) )
=> ( ( F
= ( C5
@ ( comple187402453842119260l_fixp @ ( 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 ) )
@ ^ [F2: B > ( heap_Time_Heap @ A )] : ( U4 @ ( F6 @ ( C5 @ F2 ) ) ) ) ) )
=> ( ! [F4: B > ( heap_Time_Heap @ A )] :
( ( U4 @ ( C5 @ F4 ) )
= F4 )
=> ( ( comple1908693960933563346ssible @ ( 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 )
=> ( ( P
@ ^ [Uu: B] :
( heap_Time_Heap2 @ A
@ ^ [X3: heap_ext @ product_unit] : ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) )
=> ( ! [F4: C] :
( ( P @ ( U4 @ F4 ) )
=> ( P @ ( U4 @ ( F6 @ F4 ) ) ) )
=> ( P @ ( U4 @ F ) ) ) ) ) ) ) ) ).
% heap.fixp_induct_uc
thf(fact_5990_heap_Omono__body__fixp,axiom,
! [A: $tType,B: $tType,F6: ( B > ( heap_Time_Heap @ A ) ) > B > ( heap_Time_Heap @ A )] :
( ! [X2: B] :
( comple7038119648293358887notone @ ( 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 )
@ ^ [F2: B > ( heap_Time_Heap @ A )] : ( F6 @ F2 @ X2 ) )
=> ( ( comple187402453842119260l_fixp @ ( 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 ) ) @ F6 )
= ( F6 @ ( comple187402453842119260l_fixp @ ( 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 ) ) @ F6 ) ) ) ) ).
% heap.mono_body_fixp
thf(fact_5991_heap_Ofixp__rule__uc,axiom,
! [B: $tType,A: $tType,C: $tType,U4: C > B > ( heap_Time_Heap @ A ),F6: C > C,C5: ( B > ( heap_Time_Heap @ A ) ) > C,F: C] :
( ! [X2: B] :
( comple7038119648293358887notone @ ( 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 )
@ ^ [F2: B > ( heap_Time_Heap @ A )] : ( U4 @ ( F6 @ ( C5 @ F2 ) ) @ X2 ) )
=> ( ( F
= ( C5
@ ( comple187402453842119260l_fixp @ ( 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 ) )
@ ^ [F2: B > ( heap_Time_Heap @ A )] : ( U4 @ ( F6 @ ( C5 @ F2 ) ) ) ) ) )
=> ( ! [F4: C] :
( ( C5 @ ( U4 @ F4 ) )
= F4 )
=> ( F
= ( F6 @ F ) ) ) ) ) ).
% heap.fixp_rule_uc
thf(fact_5992_admissible__heap,axiom,
! [B: $tType,A: $tType,P: A > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) > B > nat > $o] :
( comple1908693960933563346ssible @ ( 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 ) )
@ ^ [F2: A > ( heap_Time_Heap @ B )] :
! [X3: A,H: heap_ext @ product_unit,H7: heap_ext @ product_unit,R3: B,N5: nat] :
( ( heap_Time_effect @ B @ ( F2 @ X3 ) @ H @ H7 @ R3 @ N5 )
=> ( P @ X3 @ H @ H7 @ R3 @ N5 ) ) ) ).
% admissible_heap
% Subclasses (3)
thf(subcl_Heap_Oheap___Typerep_Otyperep,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( typerep2 @ A ) ) ).
thf(subcl_Heap_Oheap___HOL_Otype,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( type @ A ) ) ).
thf(subcl_Heap_Oheap___Countable_Ocountable,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( countable @ A ) ) ).
% Type constructors (697)
thf(tcon_Heap__Time__Monad_OHeap___Code__Evaluation_Oterm__of,axiom,
! [A20: $tType] :
( ( typerep2 @ A20 )
=> ( code_term_of @ ( heap_Time_Heap @ A20 ) ) ) ).
thf(tcon_Heap__Time__Monad_OHeap___Typerep_Otyperep,axiom,
! [A20: $tType] :
( ( typerep2 @ A20 )
=> ( typerep2 @ ( heap_Time_Heap @ A20 ) ) ) ).
thf(tcon_Code__Numeral_Onatural___Code__Evaluation_Oterm__of_1,axiom,
code_term_of @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Typerep_Otyperep_2,axiom,
typerep2 @ code_natural ).
thf(tcon_Code__Numeral_Ointeger___Code__Evaluation_Oterm__of_3,axiom,
code_term_of @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Typerep_Otyperep_4,axiom,
typerep2 @ code_integer ).
thf(tcon_Code__Evaluation_Oterm___Code__Evaluation_Oterm__of_5,axiom,
code_term_of @ code_term ).
thf(tcon_Code__Evaluation_Oterm___Typerep_Otyperep_6,axiom,
typerep2 @ code_term ).
thf(tcon_Heap_Oheap_Oheap__ext___Code__Evaluation_Oterm__of_7,axiom,
! [A20: $tType] :
( ( typerep2 @ A20 )
=> ( code_term_of @ ( heap_ext @ A20 ) ) ) ).
thf(tcon_Heap_Oheap_Oheap__ext___Typerep_Otyperep_8,axiom,
! [A20: $tType] :
( ( typerep2 @ A20 )
=> ( typerep2 @ ( heap_ext @ A20 ) ) ) ).
thf(tcon_Product__Type_Ounit___Code__Evaluation_Oterm__of_9,axiom,
code_term_of @ product_unit ).
thf(tcon_Product__Type_Ounit___Enum_Oenum,axiom,
enum @ product_unit ).
thf(tcon_Product__Type_Ounit___Typerep_Otyperep_10,axiom,
typerep2 @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__top,axiom,
bounded_lattice_top @ product_unit ).
thf(tcon_Product__Type_Oprod___Code__Evaluation_Oterm__of_11,axiom,
! [A20: $tType,A21: $tType] :
( ( ( typerep2 @ A20 )
& ( typerep2 @ A21 ) )
=> ( code_term_of @ ( product_prod @ A20 @ A21 ) ) ) ).
thf(tcon_Product__Type_Oprod___Enum_Oenum_12,axiom,
! [A20: $tType,A21: $tType] :
( ( ( enum @ A20 )
& ( enum @ A21 ) )
=> ( enum @ ( product_prod @ A20 @ A21 ) ) ) ).
thf(tcon_Product__Type_Oprod___Typerep_Otyperep_13,axiom,
! [A20: $tType,A21: $tType] :
( ( ( typerep2 @ A20 )
& ( typerep2 @ A21 ) )
=> ( typerep2 @ ( product_prod @ A20 @ A21 ) ) ) ).
thf(tcon_Old__Datatype_Onode___Typerep_Otyperep_14,axiom,
! [A20: $tType,A21: $tType] :
( ( ( typerep2 @ A20 )
& ( typerep2 @ A21 ) )
=> ( typerep2 @ ( old_node @ A20 @ A21 ) ) ) ).
thf(tcon_Multiset_Omultiset___Code__Evaluation_Oterm__of_15,axiom,
! [A20: $tType] :
( ( typerep2 @ A20 )
=> ( code_term_of @ ( multiset @ A20 ) ) ) ).
thf(tcon_Multiset_Omultiset___Typerep_Otyperep_16,axiom,
! [A20: $tType] :
( ( typerep2 @ A20 )
=> ( typerep2 @ ( multiset @ A20 ) ) ) ).
thf(tcon_Typerep_Otyperep___Code__Evaluation_Oterm__of_17,axiom,
code_term_of @ typerep ).
thf(tcon_Typerep_Otyperep___Typerep_Otyperep_18,axiom,
typerep2 @ typerep ).
thf(tcon_Assertions_Oassn___Typerep_Otyperep_19,axiom,
typerep2 @ assn ).
thf(tcon_Assertions_Oassn___Lattices_Obounded__lattice__top_20,axiom,
bounded_lattice_top @ assn ).
thf(tcon_Predicate_Opred___Code__Evaluation_Oterm__of_21,axiom,
! [A20: $tType] :
( ( typerep2 @ A20 )
=> ( code_term_of @ ( pred @ A20 ) ) ) ).
thf(tcon_Predicate_Opred___Typerep_Otyperep_22,axiom,
! [A20: $tType] :
( ( typerep2 @ A20 )
=> ( typerep2 @ ( pred @ A20 ) ) ) ).
thf(tcon_Predicate_Opred___Lattices_Obounded__lattice__top_23,axiom,
! [A20: $tType] : ( bounded_lattice_top @ ( pred @ A20 ) ) ).
thf(tcon_Predicate_Oseq___Code__Evaluation_Oterm__of_24,axiom,
! [A20: $tType] :
( ( typerep2 @ A20 )
=> ( code_term_of @ ( seq @ A20 ) ) ) ).
thf(tcon_Predicate_Oseq___Typerep_Otyperep_25,axiom,
! [A20: $tType] :
( ( typerep2 @ A20 )
=> ( typerep2 @ ( seq @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Code__Evaluation_Oterm__of_26,axiom,
! [A20: $tType] :
( ( typerep2 @ A20 )
=> ( code_term_of @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Enum_Oenum_27,axiom,
! [A20: $tType] :
( ( enum @ A20 )
=> ( enum @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Typerep_Otyperep_28,axiom,
! [A20: $tType] :
( ( typerep2 @ A20 )
=> ( typerep2 @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Obounded__lattice__top_29,axiom,
! [A20: $tType] :
( ( bounded_lattice_top @ A20 )
=> ( bounded_lattice_top @ ( option @ A20 ) ) ) ).
thf(tcon_Sum__Type_Osum___Code__Evaluation_Oterm__of_30,axiom,
! [A20: $tType,A21: $tType] :
( ( ( typerep2 @ A20 )
& ( typerep2 @ A21 ) )
=> ( code_term_of @ ( sum_sum @ A20 @ A21 ) ) ) ).
thf(tcon_Sum__Type_Osum___Enum_Oenum_31,axiom,
! [A20: $tType,A21: $tType] :
( ( ( enum @ A20 )
& ( enum @ A21 ) )
=> ( enum @ ( sum_sum @ A20 @ A21 ) ) ) ).
thf(tcon_Sum__Type_Osum___Typerep_Otyperep_32,axiom,
! [A20: $tType,A21: $tType] :
( ( ( typerep2 @ A20 )
& ( typerep2 @ A21 ) )
=> ( typerep2 @ ( sum_sum @ A20 @ A21 ) ) ) ).
thf(tcon_String_Ochar___Code__Evaluation_Oterm__of_33,axiom,
code_term_of @ char ).
thf(tcon_String_Ochar___Enum_Oenum_34,axiom,
enum @ char ).
thf(tcon_String_Ochar___Typerep_Otyperep_35,axiom,
typerep2 @ char ).
thf(tcon_Heap_Oarray___Code__Evaluation_Oterm__of_36,axiom,
! [A20: $tType] :
( ( typerep2 @ A20 )
=> ( code_term_of @ ( array @ A20 ) ) ) ).
thf(tcon_Heap_Oarray___Typerep_Otyperep_37,axiom,
! [A20: $tType] : ( typerep2 @ ( array @ A20 ) ) ).
thf(tcon_List_Olist___Code__Evaluation_Oterm__of_38,axiom,
! [A20: $tType] :
( ( typerep2 @ A20 )
=> ( code_term_of @ ( list @ A20 ) ) ) ).
thf(tcon_List_Olist___Typerep_Otyperep_39,axiom,
! [A20: $tType] :
( ( typerep2 @ A20 )
=> ( typerep2 @ ( list @ A20 ) ) ) ).
thf(tcon_Heap_Oref___Code__Evaluation_Oterm__of_40,axiom,
! [A20: $tType] :
( ( typerep2 @ A20 )
=> ( code_term_of @ ( ref @ A20 ) ) ) ).
thf(tcon_Heap_Oref___Typerep_Otyperep_41,axiom,
! [A20: $tType] : ( typerep2 @ ( ref @ A20 ) ) ).
thf(tcon_HOL_Obool___Code__Evaluation_Oterm__of_42,axiom,
code_term_of @ $o ).
thf(tcon_HOL_Obool___Enum_Oenum_43,axiom,
enum @ $o ).
thf(tcon_HOL_Obool___Typerep_Otyperep_44,axiom,
typerep2 @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice__top_45,axiom,
bounded_lattice_top @ $o ).
thf(tcon_Set_Oset___Code__Evaluation_Oterm__of_46,axiom,
! [A20: $tType] :
( ( typerep2 @ A20 )
=> ( code_term_of @ ( set @ A20 ) ) ) ).
thf(tcon_Set_Oset___Enum_Oenum_47,axiom,
! [A20: $tType] :
( ( enum @ A20 )
=> ( enum @ ( set @ A20 ) ) ) ).
thf(tcon_Set_Oset___Typerep_Otyperep_48,axiom,
! [A20: $tType] :
( ( typerep2 @ A20 )
=> ( typerep2 @ ( set @ A20 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice__top_49,axiom,
! [A20: $tType] : ( bounded_lattice_top @ ( set @ A20 ) ) ).
thf(tcon_Rat_Orat___Code__Evaluation_Oterm__of_50,axiom,
code_term_of @ rat ).
thf(tcon_Rat_Orat___Typerep_Otyperep_51,axiom,
typerep2 @ rat ).
thf(tcon_Num_Onum___Code__Evaluation_Oterm__of_52,axiom,
code_term_of @ num ).
thf(tcon_Num_Onum___Typerep_Otyperep_53,axiom,
typerep2 @ num ).
thf(tcon_Nat_Onat___Code__Evaluation_Oterm__of_54,axiom,
code_term_of @ nat ).
thf(tcon_Nat_Onat___Typerep_Otyperep_55,axiom,
typerep2 @ nat ).
thf(tcon_Int_Oint___Code__Evaluation_Oterm__of_56,axiom,
code_term_of @ int ).
thf(tcon_Int_Oint___Typerep_Otyperep_57,axiom,
typerep2 @ int ).
thf(tcon_itself___Code__Evaluation_Oterm__of_58,axiom,
! [A20: $tType] :
( ( typerep2 @ A20 )
=> ( code_term_of @ ( itself @ A20 ) ) ) ).
thf(tcon_itself___Typerep_Otyperep_59,axiom,
! [A20: $tType] :
( ( typerep2 @ A20 )
=> ( typerep2 @ ( itself @ A20 ) ) ) ).
thf(tcon_fun___Code__Evaluation_Oterm__of_60,axiom,
! [A20: $tType,A21: $tType] :
( ( ( typerep2 @ A20 )
& ( typerep2 @ A21 ) )
=> ( code_term_of @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Enum_Oenum_61,axiom,
! [A20: $tType,A21: $tType] :
( ( ( enum @ A20 )
& ( enum @ A21 ) )
=> ( enum @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Typerep_Otyperep_62,axiom,
! [A20: $tType,A21: $tType] :
( ( ( typerep2 @ A20 )
& ( typerep2 @ A21 ) )
=> ( typerep2 @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice__top_63,axiom,
! [A20: $tType,A21: $tType] :
( ( bounded_lattice @ A21 )
=> ( bounded_lattice_top @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Quickcheck__Exhaustive_Ofull__exhaustive,axiom,
! [A20: $tType,A21: $tType] :
( ( ( cl_HOL_Oequal @ A20 )
& ( quickc3360725361186068524ustive @ A20 )
& ( quickc3360725361186068524ustive @ A21 ) )
=> ( quickc3360725361186068524ustive @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
! [A20: $tType,A21: $tType] :
( ( comple6319245703460814977attice @ A21 )
=> ( comple6319245703460814977attice @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Quickcheck__Exhaustive_Oexhaustive,axiom,
! [A20: $tType,A21: $tType] :
( ( ( cl_HOL_Oequal @ A20 )
& ( quickc658316121487927005ustive @ A20 )
& ( quickc658316121487927005ustive @ A21 ) )
=> ( quickc658316121487927005ustive @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
! [A20: $tType,A21: $tType] :
( ( boolea8198339166811842893lgebra @ A21 )
=> ( boolea8198339166811842893lgebra @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Complete__Partial__Order_Occpo,axiom,
! [A20: $tType,A21: $tType] :
( ( comple6319245703460814977attice @ A21 )
=> ( comple9053668089753744459l_ccpo @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Quickcheck__Random_Orandom,axiom,
! [A20: $tType,A21: $tType] :
( ( ( code_term_of @ A20 )
& ( cl_HOL_Oequal @ A20 )
& ( quickcheck_random @ A21 ) )
=> ( quickcheck_random @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A20: $tType,A21: $tType] :
( ( semilattice_sup @ A21 )
=> ( semilattice_sup @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__inf,axiom,
! [A20: $tType,A21: $tType] :
( ( semilattice_inf @ A21 )
=> ( semilattice_inf @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice,axiom,
! [A20: $tType,A21: $tType] :
( ( bounded_lattice @ A21 )
=> ( bounded_lattice @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Complete__Lattices_OSup,axiom,
! [A20: $tType,A21: $tType] :
( ( complete_Sup @ A21 )
=> ( complete_Sup @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Complete__Lattices_OInf,axiom,
! [A20: $tType,A21: $tType] :
( ( complete_Inf @ A21 )
=> ( complete_Inf @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__top,axiom,
! [A20: $tType,A21: $tType] :
( ( order_top @ A21 )
=> ( order_top @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__bot,axiom,
! [A20: $tType,A21: $tType] :
( ( order_bot @ A21 )
=> ( order_bot @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Countable_Ocountable,axiom,
! [A20: $tType,A21: $tType] :
( ( ( finite_finite @ A20 )
& ( countable @ A21 ) )
=> ( countable @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A20: $tType,A21: $tType] :
( ( preorder @ A21 )
=> ( preorder @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Finite__Set_Ofinite,axiom,
! [A20: $tType,A21: $tType] :
( ( ( finite_finite @ A20 )
& ( finite_finite @ A21 ) )
=> ( finite_finite @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A20: $tType,A21: $tType] :
( ( order @ A21 )
=> ( order @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A20: $tType,A21: $tType] :
( ( ord @ A21 )
=> ( ord @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Groups_Ouminus,axiom,
! [A20: $tType,A21: $tType] :
( ( uminus @ A21 )
=> ( uminus @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Lattices_Osup,axiom,
! [A20: $tType,A21: $tType] :
( ( semilattice_sup @ A21 )
=> ( sup @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___Lattices_Oinf,axiom,
! [A20: $tType,A21: $tType] :
( ( semilattice_inf @ A21 )
=> ( inf @ ( A20 > A21 ) ) ) ).
thf(tcon_fun___HOL_Oequal,axiom,
! [A20: $tType,A21: $tType] :
( ( ( enum @ A20 )
& ( cl_HOL_Oequal @ A21 ) )
=> ( cl_HOL_Oequal @ ( A20 > A21 ) ) ) ).
thf(tcon_itself___Quickcheck__Random_Orandom_64,axiom,
! [A20: $tType] :
( ( typerep2 @ A20 )
=> ( quickcheck_random @ ( itself @ A20 ) ) ) ).
thf(tcon_itself___HOL_Oequal_65,axiom,
! [A20: $tType] : ( cl_HOL_Oequal @ ( itself @ A20 ) ) ).
thf(tcon_Int_Oint___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations,axiom,
bit_un5681908812861735899ations @ int ).
thf(tcon_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
semiri1453513574482234551roduct @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring__with__nat,axiom,
euclid5411537665997757685th_nat @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__ring__with__nat,axiom,
euclid8789492081693882211th_nat @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
ordere1937475149494474687imp_le @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring,axiom,
euclid3128863361964157862miring @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring__cancel,axiom,
euclid4440199948858584721cancel @ int ).
thf(tcon_Int_Oint___Divides_Ounique__euclidean__semiring__numeral,axiom,
unique1627219031080169319umeral @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__ring__cancel,axiom,
euclid8851590272496341667cancel @ int ).
thf(tcon_Int_Oint___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
strict9044650504122735259up_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add,axiom,
ordere580206878836729694up_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere2412721322843649153imp_le @ int ).
thf(tcon_Int_Oint___Bit__Operations_Osemiring__bit__operations,axiom,
bit_se359711467146920520ations @ int ).
thf(tcon_Int_Oint___Quickcheck__Exhaustive_Ofull__exhaustive_66,axiom,
quickc3360725361186068524ustive @ int ).
thf(tcon_Int_Oint___Groups_Ostrict__ordered__comm__monoid__add,axiom,
strict7427464778891057005id_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__cancel__comm__monoid__add,axiom,
ordere8940638589300402666id_add @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring,axiom,
euclid3725896446679973847miring @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semiring__1__strict,axiom,
linord715952674999750819strict @ int ).
thf(tcon_Int_Oint___Groups_Olinordered__ab__semigroup__add,axiom,
linord4140545234300271783up_add @ int ).
thf(tcon_Int_Oint___Bit__Operations_Oring__bit__operations,axiom,
bit_ri3973907225187159222ations @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring,axiom,
linord181362715937106298miring @ int ).
thf(tcon_Int_Oint___Rings_Osemidom__divide__unit__factor,axiom,
semido2269285787275462019factor @ int ).
thf(tcon_Int_Oint___Quickcheck__Exhaustive_Oexhaustive_67,axiom,
quickc658316121487927005ustive @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors,axiom,
semiri3467727345109120633visors @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add,axiom,
ordere6658533253407199908up_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__group__add__abs,axiom,
ordere166539214618696060dd_abs @ int ).
thf(tcon_Int_Oint___Groups_Oordered__comm__monoid__add,axiom,
ordere6911136660526730532id_add @ int ).
thf(tcon_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
linord5086331880401160121up_add @ int ).
thf(tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
cancel2418104881723323429up_add @ int ).
thf(tcon_Int_Oint___Rings_Oring__1__no__zero__divisors,axiom,
ring_15535105094025558882visors @ int ).
thf(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add,axiom,
cancel1802427076303600483id_add @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__ring__strict,axiom,
linord4710134922213307826strict @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__semiring__1__cancel,axiom,
comm_s4317794764714335236cancel @ int ).
thf(tcon_Int_Oint___Bit__Operations_Osemiring__bits,axiom,
bit_semiring_bits @ int ).
thf(tcon_Int_Oint___Rings_Onormalization__semidom,axiom,
normal8620421768224518004emidom @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semiring__1,axiom,
linord6961819062388156250ring_1 @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__group__add,axiom,
ordered_ab_group_add @ int ).
thf(tcon_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
linordered_semidom @ int ).
thf(tcon_Int_Oint___Quickcheck__Random_Orandom_68,axiom,
quickcheck_random @ int ).
thf(tcon_Int_Oint___Lattices_Osemilattice__sup_69,axiom,
semilattice_sup @ int ).
thf(tcon_Int_Oint___Lattices_Osemilattice__inf_70,axiom,
semilattice_inf @ int ).
thf(tcon_Int_Oint___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult @ int ).
thf(tcon_Int_Oint___Rings_Oalgebraic__semidom,axiom,
algebraic_semidom @ int ).
thf(tcon_Int_Oint___Groups_Ocomm__monoid__mult,axiom,
comm_monoid_mult @ int ).
thf(tcon_Int_Oint___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add @ int ).
thf(tcon_Int_Oint___Parity_Osemiring__parity,axiom,
semiring_parity @ int ).
thf(tcon_Int_Oint___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__modulo,axiom,
semiring_modulo @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__ring,axiom,
linordered_ring @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
linordered_idom @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1 @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__semiring__0,axiom,
comm_semiring_0 @ int ).
thf(tcon_Int_Oint___Groups_Osemigroup__mult,axiom,
semigroup_mult @ int ).
thf(tcon_Int_Oint___Complete__Lattices_OSup_71,axiom,
complete_Sup @ int ).
thf(tcon_Int_Oint___Complete__Lattices_OInf_72,axiom,
complete_Inf @ int ).
thf(tcon_Int_Oint___Rings_Osemidom__modulo,axiom,
semidom_modulo @ int ).
thf(tcon_Int_Oint___Rings_Osemidom__divide,axiom,
semidom_divide @ int ).
thf(tcon_Int_Oint___Num_Osemiring__numeral,axiom,
semiring_numeral @ int ).
thf(tcon_Int_Oint___Groups_Osemigroup__add,axiom,
semigroup_add @ int ).
thf(tcon_Int_Oint___Rings_Ozero__less__one,axiom,
zero_less_one @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__semiring,axiom,
comm_semiring @ int ).
thf(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
semiring_char_0 @ int ).
thf(tcon_Int_Oint___Groups_Oab__group__add,axiom,
ab_group_add @ int ).
thf(tcon_Int_Oint___Countable_Ocountable_73,axiom,
countable @ int ).
thf(tcon_Int_Oint___Rings_Ozero__neq__one,axiom,
zero_neq_one @ int ).
thf(tcon_Int_Oint___Rings_Oordered__ring,axiom,
ordered_ring @ int ).
thf(tcon_Int_Oint___Rings_Oidom__abs__sgn,axiom,
idom_abs_sgn @ int ).
thf(tcon_Int_Oint___Parity_Oring__parity,axiom,
ring_parity @ int ).
thf(tcon_Int_Oint___Orderings_Opreorder_74,axiom,
preorder @ int ).
thf(tcon_Int_Oint___Orderings_Olinorder,axiom,
linorder @ int ).
thf(tcon_Int_Oint___Groups_Omonoid__mult,axiom,
monoid_mult @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__ring__1,axiom,
comm_ring_1 @ int ).
thf(tcon_Int_Oint___Groups_Omonoid__add,axiom,
monoid_add @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__1,axiom,
semiring_1 @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__0,axiom,
semiring_0 @ int ).
thf(tcon_Int_Oint___Orderings_Ono__top,axiom,
no_top @ int ).
thf(tcon_Int_Oint___Orderings_Ono__bot,axiom,
no_bot @ int ).
thf(tcon_Int_Oint___Groups_Ogroup__add,axiom,
group_add @ int ).
thf(tcon_Int_Oint___GCD_Osemiring__gcd,axiom,
semiring_gcd @ int ).
thf(tcon_Int_Oint___GCD_Osemiring__Gcd,axiom,
semiring_Gcd @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__ring,axiom,
comm_ring @ int ).
thf(tcon_Int_Oint___Orderings_Oorder_75,axiom,
order @ int ).
thf(tcon_Int_Oint___Num_Oneg__numeral,axiom,
neg_numeral @ int ).
thf(tcon_Int_Oint___Nat_Oring__char__0,axiom,
ring_char_0 @ int ).
thf(tcon_Int_Oint___Rings_Osemiring,axiom,
semiring @ int ).
thf(tcon_Int_Oint___Orderings_Oord_76,axiom,
ord @ int ).
thf(tcon_Int_Oint___Groups_Ouminus_77,axiom,
uminus @ int ).
thf(tcon_Int_Oint___Rings_Oring__1,axiom,
ring_1 @ int ).
thf(tcon_Int_Oint___Lattices_Osup_78,axiom,
sup @ int ).
thf(tcon_Int_Oint___Lattices_Oinf_79,axiom,
inf @ int ).
thf(tcon_Int_Oint___GCD_Oring__gcd,axiom,
ring_gcd @ int ).
thf(tcon_Int_Oint___Power_Opower,axiom,
power @ int ).
thf(tcon_Int_Oint___Num_Onumeral,axiom,
numeral @ int ).
thf(tcon_Int_Oint___Groups_Ozero,axiom,
zero @ int ).
thf(tcon_Int_Oint___Groups_Oplus,axiom,
plus @ int ).
thf(tcon_Int_Oint___Rings_Oring,axiom,
ring @ int ).
thf(tcon_Int_Oint___Groups_Oone,axiom,
one @ int ).
thf(tcon_Int_Oint___Rings_Odvd,axiom,
dvd @ int ).
thf(tcon_Int_Oint___Heap_Oheap,axiom,
heap @ int ).
thf(tcon_Int_Oint___HOL_Oequal_80,axiom,
cl_HOL_Oequal @ int ).
thf(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_81,axiom,
bit_un5681908812861735899ations @ nat ).
thf(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_82,axiom,
semiri1453513574482234551roduct @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_83,axiom,
euclid5411537665997757685th_nat @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_84,axiom,
ordere1937475149494474687imp_le @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring_85,axiom,
euclid3128863361964157862miring @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_86,axiom,
euclid4440199948858584721cancel @ nat ).
thf(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_87,axiom,
unique1627219031080169319umeral @ nat ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_88,axiom,
strict9044650504122735259up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__diff,axiom,
ordere1170586879665033532d_diff @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add_89,axiom,
ordere580206878836729694up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_90,axiom,
ordere2412721322843649153imp_le @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_91,axiom,
bit_se359711467146920520ations @ nat ).
thf(tcon_Nat_Onat___Quickcheck__Exhaustive_Ofull__exhaustive_92,axiom,
quickc3360725361186068524ustive @ nat ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_93,axiom,
strict7427464778891057005id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_94,axiom,
ordere8940638589300402666id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
canoni5634975068530333245id_add @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring_95,axiom,
euclid3725896446679973847miring @ nat ).
thf(tcon_Nat_Onat___Groups_Olinordered__ab__semigroup__add_96,axiom,
linord4140545234300271783up_add @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_97,axiom,
linord181362715937106298miring @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom__divide__unit__factor_98,axiom,
semido2269285787275462019factor @ nat ).
thf(tcon_Nat_Onat___Quickcheck__Exhaustive_Oexhaustive_99,axiom,
quickc658316121487927005ustive @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_100,axiom,
semiri3467727345109120633visors @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_101,axiom,
ordere6658533253407199908up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_102,axiom,
ordere6911136660526730532id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_103,axiom,
cancel2418104881723323429up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_104,axiom,
cancel1802427076303600483id_add @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel_105,axiom,
comm_s4317794764714335236cancel @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_106,axiom,
bit_semiring_bits @ nat ).
thf(tcon_Nat_Onat___Rings_Onormalization__semidom_107,axiom,
normal8620421768224518004emidom @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_108,axiom,
cancel_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semidom_109,axiom,
linordered_semidom @ nat ).
thf(tcon_Nat_Onat___Quickcheck__Random_Orandom_110,axiom,
quickcheck_random @ nat ).
thf(tcon_Nat_Onat___Lattices_Osemilattice__sup_111,axiom,
semilattice_sup @ nat ).
thf(tcon_Nat_Onat___Lattices_Osemilattice__inf_112,axiom,
semilattice_inf @ nat ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__mult_113,axiom,
ab_semigroup_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Oalgebraic__semidom_114,axiom,
algebraic_semidom @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_115,axiom,
comm_monoid_mult @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__diff,axiom,
comm_monoid_diff @ nat ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__add_116,axiom,
ab_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Parity_Osemiring__parity_117,axiom,
semiring_parity @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add_118,axiom,
comm_monoid_add @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__modulo_119,axiom,
semiring_modulo @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1_120,axiom,
comm_semiring_1 @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__0_121,axiom,
comm_semiring_0 @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__mult_122,axiom,
semigroup_mult @ nat ).
thf(tcon_Nat_Onat___Complete__Lattices_OSup_123,axiom,
complete_Sup @ nat ).
thf(tcon_Nat_Onat___Complete__Lattices_OInf_124,axiom,
complete_Inf @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom__modulo_125,axiom,
semidom_modulo @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom__divide_126,axiom,
semidom_divide @ nat ).
thf(tcon_Nat_Onat___Num_Osemiring__numeral_127,axiom,
semiring_numeral @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__add_128,axiom,
semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Ozero__less__one_129,axiom,
zero_less_one @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring_130,axiom,
comm_semiring @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder__bot_131,axiom,
order_bot @ nat ).
thf(tcon_Nat_Onat___Nat_Osemiring__char__0_132,axiom,
semiring_char_0 @ nat ).
thf(tcon_Nat_Onat___Countable_Ocountable_133,axiom,
countable @ nat ).
thf(tcon_Nat_Onat___Rings_Ozero__neq__one_134,axiom,
zero_neq_one @ nat ).
thf(tcon_Nat_Onat___Orderings_Opreorder_135,axiom,
preorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Olinorder_136,axiom,
linorder @ nat ).
thf(tcon_Nat_Onat___Groups_Omonoid__mult_137,axiom,
monoid_mult @ nat ).
thf(tcon_Nat_Onat___Groups_Omonoid__add_138,axiom,
monoid_add @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1_139,axiom,
semiring_1 @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__0_140,axiom,
semiring_0 @ nat ).
thf(tcon_Nat_Onat___Orderings_Ono__top_141,axiom,
no_top @ nat ).
thf(tcon_Nat_Onat___GCD_Osemiring__gcd_142,axiom,
semiring_gcd @ nat ).
thf(tcon_Nat_Onat___GCD_Osemiring__Gcd_143,axiom,
semiring_Gcd @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder_144,axiom,
order @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring_145,axiom,
semiring @ nat ).
thf(tcon_Nat_Onat___Orderings_Oord_146,axiom,
ord @ nat ).
thf(tcon_Nat_Onat___Lattices_Osup_147,axiom,
sup @ nat ).
thf(tcon_Nat_Onat___Lattices_Oinf_148,axiom,
inf @ nat ).
thf(tcon_Nat_Onat___Power_Opower_149,axiom,
power @ nat ).
thf(tcon_Nat_Onat___Num_Onumeral_150,axiom,
numeral @ nat ).
thf(tcon_Nat_Onat___Groups_Ozero_151,axiom,
zero @ nat ).
thf(tcon_Nat_Onat___Groups_Oplus_152,axiom,
plus @ nat ).
thf(tcon_Nat_Onat___Groups_Oone_153,axiom,
one @ nat ).
thf(tcon_Nat_Onat___Rings_Odvd_154,axiom,
dvd @ nat ).
thf(tcon_Nat_Onat___Heap_Oheap_155,axiom,
heap @ nat ).
thf(tcon_Nat_Onat___HOL_Oequal_156,axiom,
cl_HOL_Oequal @ nat ).
thf(tcon_Num_Onum___Quickcheck__Exhaustive_Ofull__exhaustive_157,axiom,
quickc3360725361186068524ustive @ num ).
thf(tcon_Num_Onum___Quickcheck__Random_Orandom_158,axiom,
quickcheck_random @ num ).
thf(tcon_Num_Onum___Orderings_Opreorder_159,axiom,
preorder @ num ).
thf(tcon_Num_Onum___Orderings_Olinorder_160,axiom,
linorder @ num ).
thf(tcon_Num_Onum___Orderings_Oorder_161,axiom,
order @ num ).
thf(tcon_Num_Onum___Orderings_Oord_162,axiom,
ord @ num ).
thf(tcon_Num_Onum___Groups_Oplus_163,axiom,
plus @ num ).
thf(tcon_Num_Onum___HOL_Oequal_164,axiom,
cl_HOL_Oequal @ num ).
thf(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_165,axiom,
semiri1453513574482234551roduct @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_166,axiom,
ordere1937475149494474687imp_le @ rat ).
thf(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_167,axiom,
strict9044650504122735259up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_168,axiom,
ordere580206878836729694up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_169,axiom,
ordere2412721322843649153imp_le @ rat ).
thf(tcon_Rat_Orat___Quickcheck__Exhaustive_Ofull__exhaustive_170,axiom,
quickc3360725361186068524ustive @ rat ).
thf(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_171,axiom,
strict7427464778891057005id_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_172,axiom,
ordere8940638589300402666id_add @ rat ).
thf(tcon_Rat_Orat___Archimedean__Field_Oarchimedean__field,axiom,
archim462609752435547400_field @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring__1__strict_173,axiom,
linord715952674999750819strict @ rat ).
thf(tcon_Rat_Orat___Groups_Olinordered__ab__semigroup__add_174,axiom,
linord4140545234300271783up_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_175,axiom,
linord181362715937106298miring @ rat ).
thf(tcon_Rat_Orat___Quickcheck__Exhaustive_Oexhaustive_176,axiom,
quickc658316121487927005ustive @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_177,axiom,
semiri3467727345109120633visors @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_178,axiom,
ordere6658533253407199908up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_179,axiom,
ordere166539214618696060dd_abs @ rat ).
thf(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
archim2362893244070406136eiling @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_180,axiom,
ordere6911136660526730532id_add @ rat ).
thf(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_181,axiom,
linord5086331880401160121up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_182,axiom,
cancel2418104881723323429up_add @ rat ).
thf(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_183,axiom,
ring_15535105094025558882visors @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_184,axiom,
cancel1802427076303600483id_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__ring__strict_185,axiom,
linord4710134922213307826strict @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1__cancel_186,axiom,
comm_s4317794764714335236cancel @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring__1_187,axiom,
linord6961819062388156250ring_1 @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add_188,axiom,
ordered_ab_group_add @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_189,axiom,
cancel_semigroup_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semidom_190,axiom,
linordered_semidom @ rat ).
thf(tcon_Rat_Orat___Quickcheck__Random_Orandom_191,axiom,
quickcheck_random @ rat ).
thf(tcon_Rat_Orat___Orderings_Odense__linorder,axiom,
dense_linorder @ rat ).
thf(tcon_Rat_Orat___Lattices_Osemilattice__sup_192,axiom,
semilattice_sup @ rat ).
thf(tcon_Rat_Orat___Lattices_Osemilattice__inf_193,axiom,
semilattice_inf @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__semigroup__mult_194,axiom,
ab_semigroup_mult @ rat ).
thf(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_195,axiom,
comm_monoid_mult @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__semigroup__add_196,axiom,
ab_semigroup_add @ rat ).
thf(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
linordered_field @ rat ).
thf(tcon_Rat_Orat___Groups_Ocomm__monoid__add_197,axiom,
comm_monoid_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__ring_198,axiom,
linordered_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__idom_199,axiom,
linordered_idom @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1_200,axiom,
comm_semiring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__0_201,axiom,
comm_semiring_0 @ rat ).
thf(tcon_Rat_Orat___Groups_Osemigroup__mult_202,axiom,
semigroup_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Osemidom__divide_203,axiom,
semidom_divide @ rat ).
thf(tcon_Rat_Orat___Num_Osemiring__numeral_204,axiom,
semiring_numeral @ rat ).
thf(tcon_Rat_Orat___Groups_Osemigroup__add_205,axiom,
semigroup_add @ rat ).
thf(tcon_Rat_Orat___Fields_Odivision__ring,axiom,
division_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Ozero__less__one_206,axiom,
zero_less_one @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring_207,axiom,
comm_semiring @ rat ).
thf(tcon_Rat_Orat___Nat_Osemiring__char__0_208,axiom,
semiring_char_0 @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__group__add_209,axiom,
ab_group_add @ rat ).
thf(tcon_Rat_Orat___Fields_Ofield__char__0,axiom,
field_char_0 @ rat ).
thf(tcon_Rat_Orat___Countable_Ocountable_210,axiom,
countable @ rat ).
thf(tcon_Rat_Orat___Rings_Ozero__neq__one_211,axiom,
zero_neq_one @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__ring_212,axiom,
ordered_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Oidom__abs__sgn_213,axiom,
idom_abs_sgn @ rat ).
thf(tcon_Rat_Orat___Orderings_Opreorder_214,axiom,
preorder @ rat ).
thf(tcon_Rat_Orat___Orderings_Olinorder_215,axiom,
linorder @ rat ).
thf(tcon_Rat_Orat___Groups_Omonoid__mult_216,axiom,
monoid_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__ring__1_217,axiom,
comm_ring_1 @ rat ).
thf(tcon_Rat_Orat___Groups_Omonoid__add_218,axiom,
monoid_add @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1_219,axiom,
semiring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__0_220,axiom,
semiring_0 @ rat ).
thf(tcon_Rat_Orat___Orderings_Ono__top_221,axiom,
no_top @ rat ).
thf(tcon_Rat_Orat___Orderings_Ono__bot_222,axiom,
no_bot @ rat ).
thf(tcon_Rat_Orat___Groups_Ogroup__add_223,axiom,
group_add @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__ring_224,axiom,
comm_ring @ rat ).
thf(tcon_Rat_Orat___Orderings_Oorder_225,axiom,
order @ rat ).
thf(tcon_Rat_Orat___Num_Oneg__numeral_226,axiom,
neg_numeral @ rat ).
thf(tcon_Rat_Orat___Nat_Oring__char__0_227,axiom,
ring_char_0 @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring_228,axiom,
semiring @ rat ).
thf(tcon_Rat_Orat___Fields_Oinverse,axiom,
inverse @ rat ).
thf(tcon_Rat_Orat___Orderings_Oord_229,axiom,
ord @ rat ).
thf(tcon_Rat_Orat___Groups_Ouminus_230,axiom,
uminus @ rat ).
thf(tcon_Rat_Orat___Rings_Oring__1_231,axiom,
ring_1 @ rat ).
thf(tcon_Rat_Orat___Lattices_Osup_232,axiom,
sup @ rat ).
thf(tcon_Rat_Orat___Lattices_Oinf_233,axiom,
inf @ rat ).
thf(tcon_Rat_Orat___Fields_Ofield,axiom,
field @ rat ).
thf(tcon_Rat_Orat___Power_Opower_234,axiom,
power @ rat ).
thf(tcon_Rat_Orat___Num_Onumeral_235,axiom,
numeral @ rat ).
thf(tcon_Rat_Orat___Groups_Ozero_236,axiom,
zero @ rat ).
thf(tcon_Rat_Orat___Groups_Oplus_237,axiom,
plus @ rat ).
thf(tcon_Rat_Orat___Rings_Oring_238,axiom,
ring @ rat ).
thf(tcon_Rat_Orat___Groups_Oone_239,axiom,
one @ rat ).
thf(tcon_Rat_Orat___Rings_Odvd_240,axiom,
dvd @ rat ).
thf(tcon_Rat_Orat___HOL_Oequal_241,axiom,
cl_HOL_Oequal @ rat ).
thf(tcon_Set_Oset___Quickcheck__Exhaustive_Ofull__exhaustive_242,axiom,
! [A20: $tType] :
( ( quickc3360725361186068524ustive @ A20 )
=> ( quickc3360725361186068524ustive @ ( set @ A20 ) ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_243,axiom,
! [A20: $tType] : ( comple6319245703460814977attice @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Quickcheck__Exhaustive_Oexhaustive_244,axiom,
! [A20: $tType] :
( ( quickc658316121487927005ustive @ A20 )
=> ( quickc658316121487927005ustive @ ( set @ A20 ) ) ) ).
thf(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_245,axiom,
! [A20: $tType] : ( boolea8198339166811842893lgebra @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Complete__Partial__Order_Occpo_246,axiom,
! [A20: $tType] : ( comple9053668089753744459l_ccpo @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Quickcheck__Random_Orandom_247,axiom,
! [A20: $tType] :
( ( quickcheck_random @ A20 )
=> ( quickcheck_random @ ( set @ A20 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__sup_248,axiom,
! [A20: $tType] : ( semilattice_sup @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__inf_249,axiom,
! [A20: $tType] : ( semilattice_inf @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice_250,axiom,
! [A20: $tType] : ( bounded_lattice @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_OSup_251,axiom,
! [A20: $tType] : ( complete_Sup @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_OInf_252,axiom,
! [A20: $tType] : ( complete_Inf @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__top_253,axiom,
! [A20: $tType] : ( order_top @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_254,axiom,
! [A20: $tType] : ( order_bot @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Countable_Ocountable_255,axiom,
! [A20: $tType] :
( ( finite_finite @ A20 )
=> ( countable @ ( set @ A20 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_256,axiom,
! [A20: $tType] : ( preorder @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_257,axiom,
! [A20: $tType] :
( ( finite_finite @ A20 )
=> ( finite_finite @ ( set @ A20 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_258,axiom,
! [A20: $tType] : ( order @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_259,axiom,
! [A20: $tType] : ( ord @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Groups_Ouminus_260,axiom,
! [A20: $tType] : ( uminus @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Lattices_Osup_261,axiom,
! [A20: $tType] : ( sup @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___Lattices_Oinf_262,axiom,
! [A20: $tType] : ( inf @ ( set @ A20 ) ) ).
thf(tcon_Set_Oset___HOL_Oequal_263,axiom,
! [A20: $tType] :
( ( cl_HOL_Oequal @ A20 )
=> ( cl_HOL_Oequal @ ( set @ A20 ) ) ) ).
thf(tcon_HOL_Obool___Quickcheck__Exhaustive_Ofull__exhaustive_264,axiom,
quickc3360725361186068524ustive @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_265,axiom,
comple6319245703460814977attice @ $o ).
thf(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_266,axiom,
boolea8198339166811842893lgebra @ $o ).
thf(tcon_HOL_Obool___Complete__Partial__Order_Occpo_267,axiom,
comple9053668089753744459l_ccpo @ $o ).
thf(tcon_HOL_Obool___Quickcheck__Random_Orandom_268,axiom,
quickcheck_random @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_269,axiom,
semilattice_sup @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_270,axiom,
semilattice_inf @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice_271,axiom,
bounded_lattice @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_OSup_272,axiom,
complete_Sup @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_OInf_273,axiom,
complete_Inf @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__top_274,axiom,
order_top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_275,axiom,
order_bot @ $o ).
thf(tcon_HOL_Obool___Countable_Ocountable_276,axiom,
countable @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_277,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_278,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_279,axiom,
finite_finite @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_280,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_281,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Groups_Ouminus_282,axiom,
uminus @ $o ).
thf(tcon_HOL_Obool___Lattices_Osup_283,axiom,
sup @ $o ).
thf(tcon_HOL_Obool___Lattices_Oinf_284,axiom,
inf @ $o ).
thf(tcon_HOL_Obool___Heap_Oheap_285,axiom,
heap @ $o ).
thf(tcon_HOL_Obool___HOL_Oequal_286,axiom,
cl_HOL_Oequal @ $o ).
thf(tcon_Heap_Oref___Quickcheck__Exhaustive_Ofull__exhaustive_287,axiom,
! [A20: $tType] :
( ( typerep2 @ A20 )
=> ( quickc3360725361186068524ustive @ ( ref @ A20 ) ) ) ).
thf(tcon_Heap_Oref___Quickcheck__Random_Orandom_288,axiom,
! [A20: $tType] :
( ( typerep2 @ A20 )
=> ( quickcheck_random @ ( ref @ A20 ) ) ) ).
thf(tcon_Heap_Oref___Countable_Ocountable_289,axiom,
! [A20: $tType] : ( countable @ ( ref @ A20 ) ) ).
thf(tcon_Heap_Oref___Heap_Oheap_290,axiom,
! [A20: $tType] : ( heap @ ( ref @ A20 ) ) ).
thf(tcon_Heap_Oref___HOL_Oequal_291,axiom,
! [A20: $tType] : ( cl_HOL_Oequal @ ( ref @ A20 ) ) ).
thf(tcon_List_Olist___Quickcheck__Exhaustive_Ofull__exhaustive_292,axiom,
! [A20: $tType] :
( ( quickc3360725361186068524ustive @ A20 )
=> ( quickc3360725361186068524ustive @ ( list @ A20 ) ) ) ).
thf(tcon_List_Olist___Quickcheck__Random_Orandom_293,axiom,
! [A20: $tType] :
( ( quickcheck_random @ A20 )
=> ( quickcheck_random @ ( list @ A20 ) ) ) ).
thf(tcon_List_Olist___Countable_Ocountable_294,axiom,
! [A20: $tType] :
( ( countable @ A20 )
=> ( countable @ ( list @ A20 ) ) ) ).
thf(tcon_List_Olist___Heap_Oheap_295,axiom,
! [A20: $tType] :
( ( heap @ A20 )
=> ( heap @ ( list @ A20 ) ) ) ).
thf(tcon_List_Olist___HOL_Oequal_296,axiom,
! [A20: $tType] : ( cl_HOL_Oequal @ ( list @ A20 ) ) ).
thf(tcon_Heap_Oarray___Quickcheck__Exhaustive_Ofull__exhaustive_297,axiom,
! [A20: $tType] :
( ( typerep2 @ A20 )
=> ( quickc3360725361186068524ustive @ ( array @ A20 ) ) ) ).
thf(tcon_Heap_Oarray___Quickcheck__Random_Orandom_298,axiom,
! [A20: $tType] :
( ( typerep2 @ A20 )
=> ( quickcheck_random @ ( array @ A20 ) ) ) ).
thf(tcon_Heap_Oarray___Countable_Ocountable_299,axiom,
! [A20: $tType] : ( countable @ ( array @ A20 ) ) ).
thf(tcon_Heap_Oarray___Heap_Oheap_300,axiom,
! [A20: $tType] : ( heap @ ( array @ A20 ) ) ).
thf(tcon_Heap_Oarray___HOL_Oequal_301,axiom,
! [A20: $tType] : ( cl_HOL_Oequal @ ( array @ A20 ) ) ).
thf(tcon_String_Ochar___Quickcheck__Exhaustive_Ofull__exhaustive_302,axiom,
quickc3360725361186068524ustive @ char ).
thf(tcon_String_Ochar___Quickcheck__Random_Orandom_303,axiom,
quickcheck_random @ char ).
thf(tcon_String_Ochar___Countable_Ocountable_304,axiom,
countable @ char ).
thf(tcon_String_Ochar___Finite__Set_Ofinite_305,axiom,
finite_finite @ char ).
thf(tcon_String_Ochar___Heap_Oheap_306,axiom,
heap @ char ).
thf(tcon_String_Ochar___HOL_Oequal_307,axiom,
cl_HOL_Oequal @ char ).
thf(tcon_Sum__Type_Osum___Quickcheck__Exhaustive_Ofull__exhaustive_308,axiom,
! [A20: $tType,A21: $tType] :
( ( ( quickc3360725361186068524ustive @ A20 )
& ( quickc3360725361186068524ustive @ A21 ) )
=> ( quickc3360725361186068524ustive @ ( sum_sum @ A20 @ A21 ) ) ) ).
thf(tcon_Sum__Type_Osum___Quickcheck__Random_Orandom_309,axiom,
! [A20: $tType,A21: $tType] :
( ( ( quickcheck_random @ A20 )
& ( quickcheck_random @ A21 ) )
=> ( quickcheck_random @ ( sum_sum @ A20 @ A21 ) ) ) ).
thf(tcon_Sum__Type_Osum___Countable_Ocountable_310,axiom,
! [A20: $tType,A21: $tType] :
( ( ( countable @ A20 )
& ( countable @ A21 ) )
=> ( countable @ ( sum_sum @ A20 @ A21 ) ) ) ).
thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_311,axiom,
! [A20: $tType,A21: $tType] :
( ( ( finite_finite @ A20 )
& ( finite_finite @ A21 ) )
=> ( finite_finite @ ( sum_sum @ A20 @ A21 ) ) ) ).
thf(tcon_Sum__Type_Osum___Heap_Oheap_312,axiom,
! [A20: $tType,A21: $tType] :
( ( ( heap @ A20 )
& ( heap @ A21 ) )
=> ( heap @ ( sum_sum @ A20 @ A21 ) ) ) ).
thf(tcon_Sum__Type_Osum___HOL_Oequal_313,axiom,
! [A20: $tType,A21: $tType] : ( cl_HOL_Oequal @ ( sum_sum @ A20 @ A21 ) ) ).
thf(tcon_Option_Ooption___Quickcheck__Exhaustive_Ofull__exhaustive_314,axiom,
! [A20: $tType] :
( ( quickc3360725361186068524ustive @ A20 )
=> ( quickc3360725361186068524ustive @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Complete__Lattices_Ocomplete__lattice_315,axiom,
! [A20: $tType] :
( ( comple6319245703460814977attice @ A20 )
=> ( comple6319245703460814977attice @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Complete__Partial__Order_Occpo_316,axiom,
! [A20: $tType] :
( ( comple6319245703460814977attice @ A20 )
=> ( comple9053668089753744459l_ccpo @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Quickcheck__Random_Orandom_317,axiom,
! [A20: $tType] :
( ( quickcheck_random @ A20 )
=> ( quickcheck_random @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Osemilattice__sup_318,axiom,
! [A20: $tType] :
( ( semilattice_sup @ A20 )
=> ( semilattice_sup @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Osemilattice__inf_319,axiom,
! [A20: $tType] :
( ( semilattice_inf @ A20 )
=> ( semilattice_inf @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Obounded__lattice_320,axiom,
! [A20: $tType] :
( ( bounded_lattice_top @ A20 )
=> ( bounded_lattice @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Complete__Lattices_OSup_321,axiom,
! [A20: $tType] :
( ( comple6319245703460814977attice @ A20 )
=> ( complete_Sup @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Complete__Lattices_OInf_322,axiom,
! [A20: $tType] :
( ( comple6319245703460814977attice @ A20 )
=> ( complete_Inf @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Oorder__top_323,axiom,
! [A20: $tType] :
( ( order_top @ A20 )
=> ( order_top @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Oorder__bot_324,axiom,
! [A20: $tType] :
( ( order @ A20 )
=> ( order_bot @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Countable_Ocountable_325,axiom,
! [A20: $tType] :
( ( countable @ A20 )
=> ( countable @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Opreorder_326,axiom,
! [A20: $tType] :
( ( preorder @ A20 )
=> ( preorder @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Olinorder_327,axiom,
! [A20: $tType] :
( ( linorder @ A20 )
=> ( linorder @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Finite__Set_Ofinite_328,axiom,
! [A20: $tType] :
( ( finite_finite @ A20 )
=> ( finite_finite @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Oorder_329,axiom,
! [A20: $tType] :
( ( order @ A20 )
=> ( order @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Oord_330,axiom,
! [A20: $tType] :
( ( preorder @ A20 )
=> ( ord @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Osup_331,axiom,
! [A20: $tType] :
( ( sup @ A20 )
=> ( sup @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Oinf_332,axiom,
! [A20: $tType] :
( ( inf @ A20 )
=> ( inf @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___Heap_Oheap_333,axiom,
! [A20: $tType] :
( ( heap @ A20 )
=> ( heap @ ( option @ A20 ) ) ) ).
thf(tcon_Option_Ooption___HOL_Oequal_334,axiom,
! [A20: $tType] : ( cl_HOL_Oequal @ ( option @ A20 ) ) ).
thf(tcon_Predicate_Oseq___HOL_Oequal_335,axiom,
! [A20: $tType] : ( cl_HOL_Oequal @ ( seq @ A20 ) ) ).
thf(tcon_Predicate_Opred___Complete__Lattices_Ocomplete__lattice_336,axiom,
! [A20: $tType] : ( comple6319245703460814977attice @ ( pred @ A20 ) ) ).
thf(tcon_Predicate_Opred___Boolean__Algebras_Oboolean__algebra_337,axiom,
! [A20: $tType] : ( boolea8198339166811842893lgebra @ ( pred @ A20 ) ) ).
thf(tcon_Predicate_Opred___Complete__Partial__Order_Occpo_338,axiom,
! [A20: $tType] : ( comple9053668089753744459l_ccpo @ ( pred @ A20 ) ) ).
thf(tcon_Predicate_Opred___Lattices_Osemilattice__sup_339,axiom,
! [A20: $tType] : ( semilattice_sup @ ( pred @ A20 ) ) ).
thf(tcon_Predicate_Opred___Lattices_Osemilattice__inf_340,axiom,
! [A20: $tType] : ( semilattice_inf @ ( pred @ A20 ) ) ).
thf(tcon_Predicate_Opred___Lattices_Obounded__lattice_341,axiom,
! [A20: $tType] : ( bounded_lattice @ ( pred @ A20 ) ) ).
thf(tcon_Predicate_Opred___Complete__Lattices_OSup_342,axiom,
! [A20: $tType] : ( complete_Sup @ ( pred @ A20 ) ) ).
thf(tcon_Predicate_Opred___Complete__Lattices_OInf_343,axiom,
! [A20: $tType] : ( complete_Inf @ ( pred @ A20 ) ) ).
thf(tcon_Predicate_Opred___Orderings_Oorder__top_344,axiom,
! [A20: $tType] : ( order_top @ ( pred @ A20 ) ) ).
thf(tcon_Predicate_Opred___Orderings_Oorder__bot_345,axiom,
! [A20: $tType] : ( order_bot @ ( pred @ A20 ) ) ).
thf(tcon_Predicate_Opred___Orderings_Opreorder_346,axiom,
! [A20: $tType] : ( preorder @ ( pred @ A20 ) ) ).
thf(tcon_Predicate_Opred___Orderings_Oorder_347,axiom,
! [A20: $tType] : ( order @ ( pred @ A20 ) ) ).
thf(tcon_Predicate_Opred___Orderings_Oord_348,axiom,
! [A20: $tType] : ( ord @ ( pred @ A20 ) ) ).
thf(tcon_Predicate_Opred___Groups_Ouminus_349,axiom,
! [A20: $tType] : ( uminus @ ( pred @ A20 ) ) ).
thf(tcon_Predicate_Opred___Lattices_Osup_350,axiom,
! [A20: $tType] : ( sup @ ( pred @ A20 ) ) ).
thf(tcon_Predicate_Opred___Lattices_Oinf_351,axiom,
! [A20: $tType] : ( inf @ ( pred @ A20 ) ) ).
thf(tcon_Predicate_Opred___HOL_Oequal_352,axiom,
! [A20: $tType] : ( cl_HOL_Oequal @ ( pred @ A20 ) ) ).
thf(tcon_Assertions_Oassn___Boolean__Algebras_Oboolean__algebra_353,axiom,
boolea8198339166811842893lgebra @ assn ).
thf(tcon_Assertions_Oassn___Lattices_Osemilattice__sup_354,axiom,
semilattice_sup @ assn ).
thf(tcon_Assertions_Oassn___Lattices_Osemilattice__inf_355,axiom,
semilattice_inf @ assn ).
thf(tcon_Assertions_Oassn___Lattices_Obounded__lattice_356,axiom,
bounded_lattice @ assn ).
thf(tcon_Assertions_Oassn___Groups_Oab__semigroup__mult_357,axiom,
ab_semigroup_mult @ assn ).
thf(tcon_Assertions_Oassn___Groups_Ocomm__monoid__mult_358,axiom,
comm_monoid_mult @ assn ).
thf(tcon_Assertions_Oassn___Groups_Osemigroup__mult_359,axiom,
semigroup_mult @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Oorder__top_360,axiom,
order_top @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Oorder__bot_361,axiom,
order_bot @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Opreorder_362,axiom,
preorder @ assn ).
thf(tcon_Assertions_Oassn___Groups_Omonoid__mult_363,axiom,
monoid_mult @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Oorder_364,axiom,
order @ assn ).
thf(tcon_Assertions_Oassn___Orderings_Oord_365,axiom,
ord @ assn ).
thf(tcon_Assertions_Oassn___Groups_Ouminus_366,axiom,
uminus @ assn ).
thf(tcon_Assertions_Oassn___Lattices_Osup_367,axiom,
sup @ assn ).
thf(tcon_Assertions_Oassn___Lattices_Oinf_368,axiom,
inf @ assn ).
thf(tcon_Assertions_Oassn___Power_Opower_369,axiom,
power @ assn ).
thf(tcon_Assertions_Oassn___Groups_Oone_370,axiom,
one @ assn ).
thf(tcon_Assertions_Oassn___Rings_Odvd_371,axiom,
dvd @ assn ).
thf(tcon_Typerep_Otyperep___Countable_Ocountable_372,axiom,
countable @ typerep ).
thf(tcon_Typerep_Otyperep___Heap_Oheap_373,axiom,
heap @ typerep ).
thf(tcon_Typerep_Otyperep___HOL_Oequal_374,axiom,
cl_HOL_Oequal @ typerep ).
thf(tcon_Multiset_Omultiset___Quickcheck__Exhaustive_Ofull__exhaustive_375,axiom,
! [A20: $tType] :
( ( quickc3360725361186068524ustive @ A20 )
=> ( quickc3360725361186068524ustive @ ( multiset @ A20 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Oordered__ab__semigroup__add_376,axiom,
! [A20: $tType] :
( ( preorder @ A20 )
=> ( ordere6658533253407199908up_add @ ( multiset @ A20 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocancel__ab__semigroup__add_377,axiom,
! [A20: $tType] : ( cancel2418104881723323429up_add @ ( multiset @ A20 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocancel__comm__monoid__add_378,axiom,
! [A20: $tType] : ( cancel1802427076303600483id_add @ ( multiset @ A20 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocancel__semigroup__add_379,axiom,
! [A20: $tType] : ( cancel_semigroup_add @ ( multiset @ A20 ) ) ).
thf(tcon_Multiset_Omultiset___Quickcheck__Random_Orandom_380,axiom,
! [A20: $tType] :
( ( quickcheck_random @ A20 )
=> ( quickcheck_random @ ( multiset @ A20 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__diff_381,axiom,
! [A20: $tType] : ( comm_monoid_diff @ ( multiset @ A20 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Oab__semigroup__add_382,axiom,
! [A20: $tType] : ( ab_semigroup_add @ ( multiset @ A20 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__add_383,axiom,
! [A20: $tType] : ( comm_monoid_add @ ( multiset @ A20 ) ) ).
thf(tcon_Multiset_Omultiset___Complete__Lattices_OSup_384,axiom,
! [A20: $tType] : ( complete_Sup @ ( multiset @ A20 ) ) ).
thf(tcon_Multiset_Omultiset___Complete__Lattices_OInf_385,axiom,
! [A20: $tType] : ( complete_Inf @ ( multiset @ A20 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Osemigroup__add_386,axiom,
! [A20: $tType] : ( semigroup_add @ ( multiset @ A20 ) ) ).
thf(tcon_Multiset_Omultiset___Orderings_Opreorder_387,axiom,
! [A20: $tType] :
( ( preorder @ A20 )
=> ( preorder @ ( multiset @ A20 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Omonoid__add_388,axiom,
! [A20: $tType] : ( monoid_add @ ( multiset @ A20 ) ) ).
thf(tcon_Multiset_Omultiset___Orderings_Oorder_389,axiom,
! [A20: $tType] :
( ( preorder @ A20 )
=> ( order @ ( multiset @ A20 ) ) ) ).
thf(tcon_Multiset_Omultiset___Orderings_Oord_390,axiom,
! [A20: $tType] :
( ( preorder @ A20 )
=> ( ord @ ( multiset @ A20 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ozero_391,axiom,
! [A20: $tType] : ( zero @ ( multiset @ A20 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Oplus_392,axiom,
! [A20: $tType] : ( plus @ ( multiset @ A20 ) ) ).
thf(tcon_Multiset_Omultiset___HOL_Oequal_393,axiom,
! [A20: $tType] :
( ( cl_HOL_Oequal @ A20 )
=> ( cl_HOL_Oequal @ ( multiset @ A20 ) ) ) ).
thf(tcon_Product__Type_Oprod___Quickcheck__Exhaustive_Ofull__exhaustive_394,axiom,
! [A20: $tType,A21: $tType] :
( ( ( quickc3360725361186068524ustive @ A20 )
& ( quickc3360725361186068524ustive @ A21 ) )
=> ( quickc3360725361186068524ustive @ ( product_prod @ A20 @ A21 ) ) ) ).
thf(tcon_Product__Type_Oprod___Quickcheck__Exhaustive_Oexhaustive_395,axiom,
! [A20: $tType,A21: $tType] :
( ( ( quickc658316121487927005ustive @ A20 )
& ( quickc658316121487927005ustive @ A21 ) )
=> ( quickc658316121487927005ustive @ ( product_prod @ A20 @ A21 ) ) ) ).
thf(tcon_Product__Type_Oprod___Quickcheck__Random_Orandom_396,axiom,
! [A20: $tType,A21: $tType] :
( ( ( quickcheck_random @ A20 )
& ( quickcheck_random @ A21 ) )
=> ( quickcheck_random @ ( product_prod @ A20 @ A21 ) ) ) ).
thf(tcon_Product__Type_Oprod___Countable_Ocountable_397,axiom,
! [A20: $tType,A21: $tType] :
( ( ( countable @ A20 )
& ( countable @ A21 ) )
=> ( countable @ ( product_prod @ A20 @ A21 ) ) ) ).
thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_398,axiom,
! [A20: $tType,A21: $tType] :
( ( ( finite_finite @ A20 )
& ( finite_finite @ A21 ) )
=> ( finite_finite @ ( product_prod @ A20 @ A21 ) ) ) ).
thf(tcon_Product__Type_Oprod___Heap_Oheap_399,axiom,
! [A20: $tType,A21: $tType] :
( ( ( heap @ A20 )
& ( heap @ A21 ) )
=> ( heap @ ( product_prod @ A20 @ A21 ) ) ) ).
thf(tcon_Product__Type_Oprod___HOL_Oequal_400,axiom,
! [A20: $tType,A21: $tType] : ( cl_HOL_Oequal @ ( product_prod @ A20 @ A21 ) ) ).
thf(tcon_Product__Type_Ounit___Quickcheck__Exhaustive_Ofull__exhaustive_401,axiom,
quickc3360725361186068524ustive @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_402,axiom,
comple6319245703460814977attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_403,axiom,
boolea8198339166811842893lgebra @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Partial__Order_Occpo_404,axiom,
comple9053668089753744459l_ccpo @ product_unit ).
thf(tcon_Product__Type_Ounit___Quickcheck__Random_Orandom_405,axiom,
quickcheck_random @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_406,axiom,
semilattice_sup @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__inf_407,axiom,
semilattice_inf @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice_408,axiom,
bounded_lattice @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_OSup_409,axiom,
complete_Sup @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_OInf_410,axiom,
complete_Inf @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder__top_411,axiom,
order_top @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder__bot_412,axiom,
order_bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Countable_Ocountable_413,axiom,
countable @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Opreorder_414,axiom,
preorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Olinorder_415,axiom,
linorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Finite__Set_Ofinite_416,axiom,
finite_finite @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder_417,axiom,
order @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oord_418,axiom,
ord @ product_unit ).
thf(tcon_Product__Type_Ounit___Groups_Ouminus_419,axiom,
uminus @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Osup_420,axiom,
sup @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Oinf_421,axiom,
inf @ product_unit ).
thf(tcon_Product__Type_Ounit___Heap_Oheap_422,axiom,
heap @ product_unit ).
thf(tcon_Product__Type_Ounit___HOL_Oequal_423,axiom,
cl_HOL_Oequal @ product_unit ).
thf(tcon_Heap_Oheap_Oheap__ext___Quickcheck__Random_Orandom_424,axiom,
! [A20: $tType] :
( ( quickcheck_random @ A20 )
=> ( quickcheck_random @ ( heap_ext @ A20 ) ) ) ).
thf(tcon_Heap_Oheap_Oheap__ext___HOL_Oequal_425,axiom,
! [A20: $tType] : ( cl_HOL_Oequal @ ( heap_ext @ A20 ) ) ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_426,axiom,
bit_un5681908812861735899ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_427,axiom,
semiri1453513574482234551roduct @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_428,axiom,
euclid5411537665997757685th_nat @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__ring__with__nat_429,axiom,
euclid8789492081693882211th_nat @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_430,axiom,
ordere1937475149494474687imp_le @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring_431,axiom,
euclid3128863361964157862miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_432,axiom,
euclid4440199948858584721cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_433,axiom,
unique1627219031080169319umeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_434,axiom,
euclid8851590272496341667cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_435,axiom,
strict9044650504122735259up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_436,axiom,
ordere580206878836729694up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_437,axiom,
ordere2412721322843649153imp_le @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_438,axiom,
bit_se359711467146920520ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Quickcheck__Exhaustive_Ofull__exhaustive_439,axiom,
quickc3360725361186068524ustive @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_440,axiom,
strict7427464778891057005id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_441,axiom,
ordere8940638589300402666id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_442,axiom,
euclid3725896446679973847miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_443,axiom,
linord715952674999750819strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__semigroup__add_444,axiom,
linord4140545234300271783up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_445,axiom,
bit_ri3973907225187159222ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_446,axiom,
linord181362715937106298miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Quickcheck__Exhaustive_Oexhaustive_447,axiom,
quickc658316121487927005ustive @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_448,axiom,
semiri3467727345109120633visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_449,axiom,
ordere6658533253407199908up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_450,axiom,
ordere166539214618696060dd_abs @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_451,axiom,
ordere6911136660526730532id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_452,axiom,
linord5086331880401160121up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__ab__semigroup__add_453,axiom,
cancel2418104881723323429up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_454,axiom,
ring_15535105094025558882visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__comm__monoid__add_455,axiom,
cancel1802427076303600483id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_456,axiom,
linord4710134922213307826strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1__cancel_457,axiom,
comm_s4317794764714335236cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_458,axiom,
bit_semiring_bits @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_459,axiom,
linord6961819062388156250ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_460,axiom,
ordered_ab_group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_461,axiom,
cancel_semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_462,axiom,
linordered_semidom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Quickcheck__Random_Orandom_463,axiom,
quickcheck_random @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_464,axiom,
ab_semigroup_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_465,axiom,
algebraic_semidom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_466,axiom,
comm_monoid_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__add_467,axiom,
ab_semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_468,axiom,
semiring_parity @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_469,axiom,
comm_monoid_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_470,axiom,
semiring_modulo @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_471,axiom,
linordered_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_472,axiom,
linordered_idom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_473,axiom,
comm_semiring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__0_474,axiom,
comm_semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__mult_475,axiom,
semigroup_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_476,axiom,
semidom_modulo @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_477,axiom,
semidom_divide @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_478,axiom,
semiring_numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__add_479,axiom,
semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_480,axiom,
zero_less_one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring_481,axiom,
comm_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_482,axiom,
semiring_char_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_483,axiom,
ab_group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_484,axiom,
zero_neq_one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_485,axiom,
ordered_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__abs__sgn_486,axiom,
idom_abs_sgn @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Parity_Oring__parity_487,axiom,
ring_parity @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_488,axiom,
preorder @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_489,axiom,
linorder @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_490,axiom,
monoid_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_491,axiom,
comm_ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_492,axiom,
monoid_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_493,axiom,
semiring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_494,axiom,
semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_495,axiom,
group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_496,axiom,
comm_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Oorder_497,axiom,
order @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Oneg__numeral_498,axiom,
neg_numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_499,axiom,
ring_char_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring_500,axiom,
semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Oord_501,axiom,
ord @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ouminus_502,axiom,
uminus @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1_503,axiom,
ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Power_Opower_504,axiom,
power @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Onumeral_505,axiom,
numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ozero_506,axiom,
zero @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oplus_507,axiom,
plus @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring_508,axiom,
ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oone_509,axiom,
one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Odvd_510,axiom,
dvd @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___HOL_Oequal_511,axiom,
cl_HOL_Oequal @ code_integer ).
thf(tcon_Code__Numeral_Onatural___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_512,axiom,
bit_un5681908812861735899ations @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring__with__nat_513,axiom,
euclid5411537665997757685th_nat @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__monoid__add__imp__le_514,axiom,
ordere1937475149494474687imp_le @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring_515,axiom,
euclid3128863361964157862miring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring__cancel_516,axiom,
euclid4440199948858584721cancel @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__ab__semigroup__add_517,axiom,
strict9044650504122735259up_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__ab__semigroup__add_518,axiom,
ordere580206878836729694up_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add__imp__le_519,axiom,
ordere2412721322843649153imp_le @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bit__operations_520,axiom,
bit_se359711467146920520ations @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Quickcheck__Exhaustive_Ofull__exhaustive_521,axiom,
quickc3360725361186068524ustive @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__comm__monoid__add_522,axiom,
strict7427464778891057005id_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__comm__monoid__add_523,axiom,
ordere8940638589300402666id_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring_524,axiom,
euclid3725896446679973847miring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Olinordered__ab__semigroup__add_525,axiom,
linord4140545234300271783up_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__nonzero__semiring_526,axiom,
linord181362715937106298miring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Quickcheck__Exhaustive_Oexhaustive_527,axiom,
quickc658316121487927005ustive @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__no__zero__divisors_528,axiom,
semiri3467727345109120633visors @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add_529,axiom,
ordere6658533253407199908up_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__comm__monoid__add_530,axiom,
ordere6911136660526730532id_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__ab__semigroup__add_531,axiom,
cancel2418104881723323429up_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__comm__monoid__add_532,axiom,
cancel1802427076303600483id_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1__cancel_533,axiom,
comm_s4317794764714335236cancel @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bits_534,axiom,
bit_semiring_bits @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__semigroup__add_535,axiom,
cancel_semigroup_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__semidom_536,axiom,
linordered_semidom @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Quickcheck__Random_Orandom_537,axiom,
quickcheck_random @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__mult_538,axiom,
ab_semigroup_mult @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Oalgebraic__semidom_539,axiom,
algebraic_semidom @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__mult_540,axiom,
comm_monoid_mult @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__diff_541,axiom,
comm_monoid_diff @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__add_542,axiom,
ab_semigroup_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Parity_Osemiring__parity_543,axiom,
semiring_parity @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__add_544,axiom,
comm_monoid_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__modulo_545,axiom,
semiring_modulo @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1_546,axiom,
comm_semiring_1 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__0_547,axiom,
comm_semiring_0 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Osemigroup__mult_548,axiom,
semigroup_mult @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemidom__modulo_549,axiom,
semidom_modulo @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemidom__divide_550,axiom,
semidom_divide @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Num_Osemiring__numeral_551,axiom,
semiring_numeral @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Osemigroup__add_552,axiom,
semigroup_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ozero__less__one_553,axiom,
zero_less_one @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring_554,axiom,
comm_semiring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Nat_Osemiring__char__0_555,axiom,
semiring_char_0 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ozero__neq__one_556,axiom,
zero_neq_one @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Orderings_Opreorder_557,axiom,
preorder @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Orderings_Olinorder_558,axiom,
linorder @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Omonoid__mult_559,axiom,
monoid_mult @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Omonoid__add_560,axiom,
monoid_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__1_561,axiom,
semiring_1 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__0_562,axiom,
semiring_0 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Orderings_Oorder_563,axiom,
order @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring_564,axiom,
semiring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Orderings_Oord_565,axiom,
ord @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Power_Opower_566,axiom,
power @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Num_Onumeral_567,axiom,
numeral @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ozero_568,axiom,
zero @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oplus_569,axiom,
plus @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oone_570,axiom,
one @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Odvd_571,axiom,
dvd @ code_natural ).
thf(tcon_Code__Numeral_Onatural___HOL_Oequal_572,axiom,
cl_HOL_Oequal @ code_natural ).
thf(tcon_Heap__Time__Monad_OHeap___Quickcheck__Random_Orandom_573,axiom,
! [A20: $tType] :
( ( quickcheck_random @ A20 )
=> ( quickcheck_random @ ( heap_Time_Heap @ A20 ) ) ) ).
thf(tcon_Heap__Time__Monad_OHeap___HOL_Oequal_574,axiom,
! [A20: $tType] : ( cl_HOL_Oequal @ ( heap_Time_Heap @ A20 ) ) ).
% Helper facts (4)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $true @ X @ Y )
= X ) ).
thf(help_fChoice_1_1_T,axiom,
! [A: $tType,P: A > $o] :
( ( P @ ( fChoice @ A @ P ) )
= ( ? [X8: A] : ( P @ X8 ) ) ) ).
% Free types (1)
thf(tfree_0,hypothesis,
heap @ a ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( heap_Time_execute @ ( array @ a ) @ ( array_of_list @ a @ xs ) @ h )
!= ( none @ ( product_prod @ ( array @ a ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ).
%------------------------------------------------------------------------------