TPTP Problem File: ITP206^4.p
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%------------------------------------------------------------------------------
% File : ITP206^4 : TPTP v8.2.0. Released v8.0.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer problem Imperative_HOL_Add 00014_000455
% 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 : 0017_Imperative_HOL_Add_00014_000455 [Des22]
% Status : Theorem
% Rating : 1.00 v8.1.0
% Syntax : Number of formulae : 3694 (1422 unt; 687 typ; 0 def)
% Number of atoms : 8673 (3113 equ; 6 cnn)
% Maximal formula atoms : 17 ( 2 avg)
% Number of connectives : 77021 ( 559 ~; 55 |; 559 &;72384 @)
% ( 0 <=>;3464 =>; 0 <=; 0 <~>)
% Maximal formula depth : 34 ( 9 avg)
% Number of types : 13 ( 12 usr)
% Number of type conns : 5879 (5879 >; 0 *; 0 +; 0 <<)
% Number of symbols : 679 ( 675 usr; 21 con; 0-10 aty)
% Number of variables : 15727 (2254 ^;12328 !; 262 ?;15727 :)
% ( 883 !>; 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 13:38:33.285
%------------------------------------------------------------------------------
% Could-be-implicit typings (27)
thf(ty_t_Record_Otuple__isomorphism,type,
tuple_isomorphism: $tType > $tType > $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_Heap__Monad_OHeap,type,
heap_Heap: $tType > $tType ).
thf(ty_t_Typerep_Otyperep,type,
typerep: $tType ).
thf(ty_t_String_Oliteral,type,
literal: $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_Filter_Ofilter,type,
filter: $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 (660)
thf(sy_cl_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : $o ).
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Nat_Osize,type,
size:
!>[A: $tType] : $o ).
thf(sy_cl_Heap_Oheap,type,
heap:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Ofield,type,
field:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oring__1,type,
ring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Typerep_Otyperep,type,
typerep2:
!>[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_Lattices_Olattice,type,
lattice:
!>[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_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[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_Owellorder,type,
wellorder:
!>[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_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__ring,type,
linordered_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__add,type,
comm_monoid_add:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Olinordered__field,type,
linordered_field:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__mult,type,
comm_monoid_mult:
!>[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_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[A: $tType] : $o ).
thf(sy_cl_Enum_Ofinite__distrib__lattice,type,
finite8700451911770168679attice:
!>[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_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_Boolean__Algebras_Oboolean__algebra,type,
boolea8198339166811842893lgebra:
!>[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_Complete__Lattices_Ocomplete__linorder,type,
comple5582772986160207858norder:
!>[A: $tType] : $o ).
thf(sy_cl_Divides_Ounique__euclidean__semiring__numeral,type,
unique1627219031080169319umeral:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_Ocomplete__boolean__algebra,type,
comple489889107523837845lgebra:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_Ocomplete__distrib__lattice,type,
comple592849572758109894attice:
!>[A: $tType] : $o ).
thf(sy_cl_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations,type,
bit_un5681908812861735899ations:
!>[A: $tType] : $o ).
thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__lattice,type,
condit1219197933456340205attice:
!>[A: $tType] : $o ).
thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
condit6923001295902523014norder:
!>[A: $tType] : $o ).
thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor,type,
archim6421214686448440834_floor:
!>[A: $tType] : ( A > int ) ).
thf(sy_c_Array_Oalloc,type,
alloc:
!>[A: $tType] : ( ( list @ A ) > ( heap_ext @ product_unit ) > ( product_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) ) ) ).
thf(sy_c_Array_Ofreeze,type,
freeze:
!>[A: $tType] : ( ( array @ A ) > ( heap_Heap @ ( list @ A ) ) ) ).
thf(sy_c_Array_Oget,type,
get:
!>[A: $tType] : ( ( heap_ext @ product_unit ) > ( array @ A ) > ( list @ A ) ) ).
thf(sy_c_Array_Olen,type,
len:
!>[A: $tType] : ( ( array @ A ) > ( heap_Heap @ nat ) ) ).
thf(sy_c_Array_Olen_H,type,
len2:
!>[B: $tType,A: $tType] : ( ( array @ B ) > ( heap_Heap @ A ) ) ).
thf(sy_c_Array_Olength,type,
length:
!>[A: $tType] : ( ( heap_ext @ product_unit ) > ( array @ A ) > nat ) ).
thf(sy_c_Array_Omake,type,
make:
!>[A: $tType] : ( nat > ( nat > A ) > ( heap_Heap @ ( array @ A ) ) ) ).
thf(sy_c_Array_Omake_H,type,
make2:
!>[B: $tType,A: $tType] : ( code_integer > ( B > A ) > ( heap_Heap @ ( array @ A ) ) ) ).
thf(sy_c_Array_Omap__entry,type,
map_entry:
!>[A: $tType] : ( nat > ( A > A ) > ( array @ A ) > ( heap_Heap @ ( array @ A ) ) ) ).
thf(sy_c_Array_Onew,type,
new:
!>[A: $tType] : ( nat > A > ( heap_Heap @ ( array @ A ) ) ) ).
thf(sy_c_Array_Onew_H,type,
new2:
!>[A: $tType] : ( code_integer > A > ( heap_Heap @ ( array @ A ) ) ) ).
thf(sy_c_Array_Onoteq,type,
noteq:
!>[A: $tType,B: $tType] : ( ( array @ A ) > ( array @ B ) > $o ) ).
thf(sy_c_Array_Onth,type,
nth:
!>[A: $tType] : ( ( array @ A ) > nat > ( heap_Heap @ A ) ) ).
thf(sy_c_Array_Onth_H,type,
nth2:
!>[A: $tType] : ( ( array @ A ) > code_integer > ( heap_Heap @ A ) ) ).
thf(sy_c_Array_Oof__list,type,
of_list:
!>[A: $tType] : ( ( list @ A ) > ( heap_Heap @ ( array @ A ) ) ) ).
thf(sy_c_Array_Opresent,type,
present:
!>[A: $tType] : ( ( heap_ext @ product_unit ) > ( array @ A ) > $o ) ).
thf(sy_c_Array_Oset,type,
set2:
!>[A: $tType] : ( ( array @ A ) > ( list @ A ) > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) ) ).
thf(sy_c_Array_Oswap,type,
swap:
!>[A: $tType] : ( nat > A > ( array @ A ) > ( heap_Heap @ A ) ) ).
thf(sy_c_Array_Oupd,type,
upd:
!>[A: $tType] : ( nat > A > ( array @ A ) > ( heap_Heap @ ( array @ A ) ) ) ).
thf(sy_c_Array_Oupd_H,type,
upd2:
!>[A: $tType] : ( ( array @ A ) > code_integer > A > ( heap_Heap @ product_unit ) ) ).
thf(sy_c_Array_Oupdate,type,
update:
!>[A: $tType] : ( ( array @ A ) > nat > A > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) ) ).
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_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__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_OnatLeq,type,
bNF_Ca8665028551170535155natLeq: set @ ( product_prod @ nat @ nat ) ).
thf(sy_c_BNF__Cardinal__Order__Relation_OnatLess,type,
bNF_Ca8459412986667044542atLess: 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__Composition_Oid__bnf,type,
bNF_id_bnf:
!>[A: $tType] : ( A > A ) ).
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_Ocsquare,type,
bNF_csquare:
!>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( set @ A ) > ( B > C ) > ( D > C ) > ( A > B ) > ( A > D ) > $o ) ).
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_Orel__set,type,
bNF_rel_set:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( set @ A ) > ( set @ B ) > $o ) ).
thf(sy_c_BNF__Def_Orel__sum,type,
bNF_rel_sum:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C > $o ) > ( B > D > $o ) > ( sum_sum @ A @ B ) > ( sum_sum @ 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_OShift,type,
bNF_Greatest_Shift:
!>[A: $tType] : ( ( set @ ( list @ A ) ) > A > ( set @ ( list @ A ) ) ) ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc,type,
bNF_Greatest_Succ:
!>[A: $tType] : ( ( set @ ( list @ A ) ) > ( list @ A ) > ( set @ A ) ) ).
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_Oimage2p,type,
bNF_Greatest_image2p:
!>[C: $tType,A: $tType,D: $tType,B: $tType] : ( ( C > A ) > ( D > B ) > ( C > D > $o ) > A > B > $o ) ).
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_Oshift,type,
bNF_Greatest_shift:
!>[A: $tType,B: $tType] : ( ( ( list @ A ) > B ) > A > ( list @ A ) > B ) ).
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__Greatest__Fixpoint_Ouniv,type,
bNF_Greatest_univ:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > A ) ).
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_OFunc__map,type,
bNF_We4925052301507509544nc_map:
!>[B: $tType,C: $tType,A: $tType,D: $tType] : ( ( set @ B ) > ( C > A ) > ( B > D ) > ( D > C ) > B > A ) ).
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__Constructions_Oord__to__filter,type,
bNF_We8469521843155493636filter:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) ) ).
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_Octor__rec,type,
basic_BNF_ctor_rec:
!>[A: $tType] : ( 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__BNF__LFPs_Oxtor,type,
basic_BNF_xtor:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Basic__BNFs_Ofsts,type,
basic_fsts:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( set @ A ) ) ).
thf(sy_c_Basic__BNFs_Ofstsp,type,
basic_fstsp:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A > $o ) ).
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_Opred__sum,type,
basic_pred_sum:
!>[A: $tType,B: $tType] : ( ( A > $o ) > ( B > $o ) > ( sum_sum @ 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_Osetl,type,
basic_setl:
!>[A: $tType,B: $tType] : ( ( sum_sum @ A @ B ) > ( set @ A ) ) ).
thf(sy_c_Basic__BNFs_Osetr,type,
basic_setr:
!>[A: $tType,B: $tType] : ( ( sum_sum @ A @ B ) > ( set @ B ) ) ).
thf(sy_c_Basic__BNFs_Osnds,type,
basic_snds:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( set @ B ) ) ).
thf(sy_c_Basic__BNFs_Osndsp,type,
basic_sndsp:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B > $o ) ).
thf(sy_c_Bit__Operations_Otake__bit__num,type,
bit_take_bit_num: nat > num > ( option @ num ) ).
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_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_Ointeger_Oint__of__integer,type,
code_int_of_integer: code_integer > int ).
thf(sy_c_Code__Numeral_Onat__of__integer,type,
code_nat_of_integer: code_integer > nat ).
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_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_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_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_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above,type,
condit941137186595557371_above:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__below,type,
condit1013018076250108175_below:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Countable_Ofrom__nat,type,
from_nat:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Countable_Onat__to__rat__surj,type,
nat_to_rat_surj: nat > rat ).
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_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_Oequivclp,type,
equiv_equivclp:
!>[A: $tType] : ( ( A > A > $o ) > 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_Fields_Oinverse__class_Oinverse,type,
inverse_inverse:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Filter_Oat__bot,type,
at_bot:
!>[A: $tType] : ( filter @ A ) ).
thf(sy_c_Filter_Oat__top,type,
at_top:
!>[A: $tType] : ( filter @ A ) ).
thf(sy_c_Filter_Oeventually,type,
eventually:
!>[A: $tType] : ( ( A > $o ) > ( filter @ A ) > $o ) ).
thf(sy_c_Filter_Ofiltercomap,type,
filtercomap:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ B ) > ( filter @ A ) ) ).
thf(sy_c_Filter_Ofilterlim,type,
filterlim:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ B ) > ( filter @ A ) > $o ) ).
thf(sy_c_Filter_Ofiltermap,type,
filtermap:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ A ) > ( filter @ B ) ) ).
thf(sy_c_Filter_Ofinite__subsets__at__top,type,
finite5375528669736107172at_top:
!>[A: $tType] : ( ( set @ A ) > ( filter @ ( set @ A ) ) ) ).
thf(sy_c_Filter_Omap__filter__on,type,
map_filter_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > ( filter @ A ) > ( filter @ B ) ) ).
thf(sy_c_Filter_Oprincipal,type,
principal:
!>[A: $tType] : ( ( set @ A ) > ( filter @ A ) ) ).
thf(sy_c_Filter_Oprod__filter,type,
prod_filter:
!>[A: $tType,B: $tType] : ( ( filter @ A ) > ( filter @ B ) > ( filter @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Filter_Orel__filter,type,
rel_filter:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( filter @ A ) > ( filter @ B ) > $o ) ).
thf(sy_c_Finite__Set_OFpow,type,
finite_Fpow:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ 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__commute__on,type,
finite4664212375090638736ute_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ocomp__fun__idem,type,
finite_comp_fun_idem:
!>[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,type,
finite673082921795544331dem_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( 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_Ofold__graph,type,
finite_fold_graph:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( set @ A ) > B > $o ) ).
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_Finite__Set_Ofolding__on,type,
finite_folding_on:
!>[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_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_OTHE__default,type,
fun_THE_default:
!>[A: $tType] : ( A > ( A > $o ) > A ) ).
thf(sy_c_Fun__Def_Oin__rel,type,
fun_in_rel:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > A > B > $o ) ).
thf(sy_c_Fun__Def_Ois__measure,type,
fun_is_measure:
!>[A: $tType] : ( ( A > nat ) > $o ) ).
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_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_Oabs__class_Oabs,type,
abs_abs:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( A > A > A ) ).
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_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__class_Osum,type,
groups7311177749621191930dd_sum:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > 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__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__List_Omonoid__add__class_Osum__list,type,
groups8242544230860333062m_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_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_Oarray_Omap__array,type,
map_array:
!>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( array @ A ) > ( array @ Aa ) ) ).
thf(sy_c_Heap_Oarray_Opred__array,type,
pred_array:
!>[A: $tType] : ( ( A > $o ) > ( array @ A ) > $o ) ).
thf(sy_c_Heap_Oarray_Orec__array,type,
rec_array:
!>[C: $tType,A: $tType] : ( ( nat > C ) > ( array @ A ) > C ) ).
thf(sy_c_Heap_Oarray_Orel__array,type,
rel_array:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( array @ A ) > ( array @ B ) > $o ) ).
thf(sy_c_Heap_Oarray_Oset__array,type,
set_array:
!>[A: $tType] : ( ( array @ A ) > ( set @ A ) ) ).
thf(sy_c_Heap_Oarray_Osize__array,type,
size_array:
!>[A: $tType] : ( ( A > nat ) > ( array @ A ) > nat ) ).
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_Oheap__ext,type,
heap_ext2:
!>[Z: $tType] : ( ( typerep > nat > ( list @ nat ) ) > ( typerep > nat > nat ) > nat > Z > ( heap_ext @ Z ) ) ).
thf(sy_c_Heap_Oheap_Oheap__ext_Oheap__ext__Tuple__Iso,type,
heap_h6279449762549114721le_Iso:
!>[Z: $tType] : ( tuple_isomorphism @ ( heap_ext @ Z ) @ ( product_prod @ ( typerep > nat > ( list @ nat ) ) @ ( product_prod @ ( typerep > nat > nat ) @ nat ) ) @ 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_Omore,type,
more:
!>[Z: $tType] : ( ( heap_ext @ Z ) > Z ) ).
thf(sy_c_Heap_Oheap_Orefs,type,
refs:
!>[Z: $tType] : ( ( heap_ext @ Z ) > typerep > nat > nat ) ).
thf(sy_c_Heap_Oheap_Orefs__update,type,
refs_update:
!>[Z: $tType] : ( ( ( typerep > nat > nat ) > typerep > nat > nat ) > ( heap_ext @ Z ) > ( heap_ext @ Z ) ) ).
thf(sy_c_Heap_Oref_ORef,type,
ref2:
!>[A: $tType] : ( nat > ( ref @ A ) ) ).
thf(sy_c_Heap_Oref_Omap__ref,type,
map_ref:
!>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( ref @ A ) > ( ref @ Aa ) ) ).
thf(sy_c_Heap_Oref_Opred__ref,type,
pred_ref:
!>[A: $tType] : ( ( A > $o ) > ( ref @ A ) > $o ) ).
thf(sy_c_Heap_Oref_Orec__ref,type,
rec_ref:
!>[C: $tType,A: $tType] : ( ( nat > C ) > ( ref @ A ) > C ) ).
thf(sy_c_Heap_Oref_Orel__ref,type,
rel_ref:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( ref @ A ) > ( ref @ B ) > $o ) ).
thf(sy_c_Heap_Oref_Oset__ref,type,
set_ref:
!>[A: $tType] : ( ( ref @ A ) > ( set @ A ) ) ).
thf(sy_c_Heap_Oref_Osize__ref,type,
size_ref:
!>[A: $tType] : ( ( A > nat ) > ( ref @ A ) > nat ) ).
thf(sy_c_Heap__Monad_OHeap_OHeap,type,
heap_Heap2:
!>[A: $tType] : ( ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) ) > ( heap_Heap @ A ) ) ).
thf(sy_c_Heap__Monad_OHeap_Ocase__Heap,type,
heap_case_Heap:
!>[A: $tType,B: $tType] : ( ( ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) ) > B ) > ( heap_Heap @ A ) > B ) ).
thf(sy_c_Heap__Monad_OHeap_Omap__Heap,type,
heap_map_Heap:
!>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( heap_Heap @ A ) > ( heap_Heap @ Aa ) ) ).
thf(sy_c_Heap__Monad_OHeap_Opred__Heap,type,
heap_pred_Heap:
!>[A: $tType] : ( ( A > $o ) > ( heap_Heap @ A ) > $o ) ).
thf(sy_c_Heap__Monad_OHeap_Orec__Heap,type,
heap_rec_Heap:
!>[A: $tType,C: $tType] : ( ( ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) ) > C ) > ( heap_Heap @ A ) > C ) ).
thf(sy_c_Heap__Monad_OHeap_Orel__Heap,type,
heap_rel_Heap:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( heap_Heap @ A ) > ( heap_Heap @ B ) > $o ) ).
thf(sy_c_Heap__Monad_OHeap_Oset__Heap,type,
heap_set_Heap:
!>[A: $tType] : ( ( heap_Heap @ A ) > ( set @ A ) ) ).
thf(sy_c_Heap__Monad_OHeap_Osize__Heap,type,
heap_size_Heap:
!>[A: $tType] : ( ( A > nat ) > ( heap_Heap @ A ) > nat ) ).
thf(sy_c_Heap__Monad_OHeap__lub,type,
heap_Heap_lub:
!>[A: $tType] : ( ( set @ ( heap_Heap @ A ) ) > ( heap_Heap @ A ) ) ).
thf(sy_c_Heap__Monad_OHeap__ord,type,
heap_Heap_ord:
!>[A: $tType] : ( ( heap_Heap @ A ) > ( heap_Heap @ A ) > $o ) ).
thf(sy_c_Heap__Monad_Oassert,type,
heap_assert:
!>[A: $tType] : ( ( A > $o ) > A > ( heap_Heap @ A ) ) ).
thf(sy_c_Heap__Monad_Obind,type,
heap_bind:
!>[A: $tType,B: $tType] : ( ( heap_Heap @ A ) > ( A > ( heap_Heap @ B ) ) > ( heap_Heap @ B ) ) ).
thf(sy_c_Heap__Monad_Oeffect,type,
heap_effect:
!>[A: $tType] : ( ( heap_Heap @ A ) > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) > A > $o ) ).
thf(sy_c_Heap__Monad_Oexecute,type,
heap_execute:
!>[A: $tType] : ( ( heap_Heap @ A ) > ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) ) ).
thf(sy_c_Heap__Monad_Ofold__map,type,
heap_fold_map:
!>[A: $tType,B: $tType] : ( ( A > ( heap_Heap @ B ) ) > ( list @ A ) > ( heap_Heap @ ( list @ B ) ) ) ).
thf(sy_c_Heap__Monad_Oguard,type,
heap_guard:
!>[A: $tType] : ( ( ( heap_ext @ product_unit ) > $o ) > ( ( heap_ext @ product_unit ) > ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) > ( heap_Heap @ A ) ) ).
thf(sy_c_Heap__Monad_Oheap,type,
heap_heap:
!>[A: $tType] : ( ( ( heap_ext @ product_unit ) > ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) > ( heap_Heap @ A ) ) ).
thf(sy_c_Heap__Monad_Olift,type,
heap_lift:
!>[A: $tType,B: $tType] : ( ( A > B ) > A > ( heap_Heap @ B ) ) ).
thf(sy_c_Heap__Monad_Oraise,type,
heap_raise:
!>[A: $tType] : ( literal > ( heap_Heap @ A ) ) ).
thf(sy_c_Heap__Monad_Oreturn,type,
heap_return:
!>[A: $tType] : ( A > ( heap_Heap @ A ) ) ).
thf(sy_c_Heap__Monad_Osuccess,type,
heap_success:
!>[A: $tType] : ( ( heap_Heap @ A ) > ( heap_ext @ product_unit ) > $o ) ).
thf(sy_c_Heap__Monad_Otap,type,
heap_tap:
!>[A: $tType] : ( ( ( heap_ext @ product_unit ) > A ) > ( heap_Heap @ A ) ) ).
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_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Inductive_Ocomplete__lattice__class_Ogfp,type,
complete_lattice_gfp:
!>[A: $tType] : ( ( 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_ORep__Integ,type,
rep_Integ: int > ( product_prod @ nat @ nat ) ).
thf(sy_c_Int_Ocr__int,type,
cr_int: ( product_prod @ nat @ nat ) > int > $o ).
thf(sy_c_Int_Oint__ge__less__than,type,
int_ge_less_than: int > ( set @ ( product_prod @ int @ int ) ) ).
thf(sy_c_Int_Oint__ge__less__than2,type,
int_ge_less_than2: int > ( set @ ( product_prod @ 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_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_Lattices_Oinf__class_Oinf,type,
inf_inf:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Lattices_Osemilattice,type,
semilattice:
!>[A: $tType] : ( ( 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_Oord__class_Oarg__min__on,type,
lattic7623131987881927897min_on:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > B ) ).
thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin,type,
lattic7752659483105999362nf_fin:
!>[A: $tType] : ( ( 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_ONEG,type,
neg:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( A > B > $o ) > $o ) ).
thf(sy_c_Lifting_OPOS,type,
pos:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( A > B > $o ) > $o ) ).
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_Oall__interval__nat,type,
all_interval_nat: ( nat > $o ) > nat > nat > $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_Obind,type,
bind:
!>[A: $tType,B: $tType] : ( ( list @ A ) > ( A > ( list @ B ) ) > ( list @ B ) ) ).
thf(sy_c_List_Oconcat,type,
concat:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ A ) ) ).
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_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,
filter2:
!>[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_Ofolding__insort__key,type,
folding_insort_key:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > ( set @ B ) > ( B > A ) > $o ) ).
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_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_Oinsort__key,type,
insort_key:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > A ) > B > ( list @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Olinorder__class_Oinsort__insert__key,type,
linord329482645794927042rt_key:
!>[B: $tType,A: $tType] : ( ( B > A ) > B > ( list @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Olinorder__class_Oinsort__key,type,
linorder_insort_key:
!>[B: $tType,A: $tType] : ( ( B > A ) > B > ( list @ B ) > ( list @ B ) ) ).
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_Orec__list,type,
rec_list:
!>[C: $tType,A: $tType] : ( C > ( A > ( list @ A ) > C > C ) > ( list @ A ) > C ) ).
thf(sy_c_List_Olist_Oset,type,
set3:
!>[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__ex,type,
list_ex:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
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_Omeasures,type,
measures:
!>[A: $tType] : ( ( list @ ( A > nat ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_List_Onth,type,
nth3:
!>[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_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_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_Oset__Cons,type,
set_Cons:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( list @ A ) ) > ( set @ ( 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_Osubseqs,type,
subseqs:
!>[A: $tType] : ( ( list @ A ) > ( list @ ( list @ A ) ) ) ).
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_Oupt,type,
upt: nat > nat > ( list @ nat ) ).
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__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_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_ONats,type,
semiring_1_Nats:
!>[A: $tType] : ( set @ A ) ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
semiring_1_of_nat:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : ( A > nat ) ).
thf(sy_c_Nat__Bijection_Oint__decode,type,
nat_int_decode: nat > int ).
thf(sy_c_Nat__Bijection_Oint__encode,type,
nat_int_encode: int > 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_Oprod__decode,type,
nat_prod_decode: nat > ( product_prod @ nat @ nat ) ).
thf(sy_c_Nat__Bijection_Oprod__encode,type,
nat_prod_encode: ( product_prod @ nat @ nat ) > nat ).
thf(sy_c_Nat__Bijection_Osum__decode,type,
nat_sum_decode: nat > ( sum_sum @ nat @ nat ) ).
thf(sy_c_Nat__Bijection_Osum__encode,type,
nat_sum_encode: ( sum_sum @ nat @ nat ) > nat ).
thf(sy_c_Nat__Bijection_Otriangle,type,
nat_triangle: nat > nat ).
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,
one: 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__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_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_OLim,type,
old_Lim:
!>[B: $tType,A: $tType] : ( ( B > ( set @ ( old_node @ A @ B ) ) ) > ( set @ ( old_node @ A @ B ) ) ) ).
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_ORep__Node,type,
old_Rep_Node:
!>[A: $tType,B: $tType] : ( ( old_node @ A @ B ) > ( product_prod @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat ) ) ) ).
thf(sy_c_Old__Datatype_Ontrunc,type,
old_ntrunc:
!>[A: $tType,B: $tType] : ( nat > ( set @ ( old_node @ A @ B ) ) > ( set @ ( old_node @ A @ B ) ) ) ).
thf(sy_c_Old__Datatype_Ouprod,type,
old_uprod:
!>[A: $tType,B: $tType] : ( ( set @ ( set @ ( old_node @ A @ B ) ) ) > ( set @ ( set @ ( old_node @ A @ B ) ) ) > ( set @ ( set @ ( old_node @ A @ B ) ) ) ) ).
thf(sy_c_Old__Datatype_Ousum,type,
old_usum:
!>[A: $tType,B: $tType] : ( ( set @ ( set @ ( old_node @ A @ B ) ) ) > ( set @ ( set @ ( old_node @ A @ B ) ) ) > ( set @ ( set @ ( old_node @ A @ B ) ) ) ) ).
thf(sy_c_Option_Obind,type,
bind2:
!>[A: $tType,B: $tType] : ( ( option @ A ) > ( A > ( option @ B ) ) > ( option @ B ) ) ).
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__class_OLeast,type,
ord_Least:
!>[A: $tType] : ( ( A > $o ) > 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_Oantimono,type,
order_antimono:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
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_Predicate_Obind,type,
bind3:
!>[A: $tType,B: $tType] : ( ( pred @ A ) > ( A > ( pred @ B ) ) > ( pred @ B ) ) ).
thf(sy_c_Predicate_Omap,type,
map2:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( pred @ A ) > ( pred @ B ) ) ).
thf(sy_c_Predicate_Opred_OPred,type,
pred2:
!>[A: $tType] : ( ( A > $o ) > ( pred @ A ) ) ).
thf(sy_c_Predicate_Opred_Oeval,type,
eval:
!>[A: $tType] : ( ( pred @ A ) > A > $o ) ).
thf(sy_c_Predicate_Opred__of__seq,type,
pred_of_seq:
!>[A: $tType] : ( ( seq @ A ) > ( pred @ A ) ) ).
thf(sy_c_Predicate_Opred__of__set,type,
pred_of_set:
!>[A: $tType] : ( ( set @ A ) > ( pred @ A ) ) ).
thf(sy_c_Predicate_Oset__of__pred,type,
set_of_pred:
!>[A: $tType] : ( ( pred @ A ) > ( set @ A ) ) ).
thf(sy_c_Predicate_Oset__of__seq,type,
set_of_seq:
!>[A: $tType] : ( ( seq @ A ) > ( set @ A ) ) ).
thf(sy_c_Predicate_Osingle,type,
single:
!>[A: $tType] : ( A > ( pred @ 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_OPair__Rep,type,
product_Pair_Rep:
!>[A: $tType,B: $tType] : ( A > B > A > B > $o ) ).
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,type,
product_prod2:
!>[A: $tType,B: $tType] : ( set @ ( A > B > $o ) ) ).
thf(sy_c_Product__Type_Oprod_OAbs__prod,type,
product_Abs_prod:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( product_prod @ A @ B ) ) ).
thf(sy_c_Product__Type_Oprod_ORep__prod,type,
product_Rep_prod:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A > B > $o ) ).
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_Quotient_OQuotient3,type,
quotient3:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > ( B > A ) > $o ) ).
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_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_ORandom,type,
random_Random:
!>[A: $tType] : ( ( ( 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 @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( 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_Omap,type,
random_map:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( ( 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 @ B ) @ ( product_prod @ code_natural @ code_natural ) ) ) ).
thf(sy_c_Random__Pred_Onot__randompred,type,
random6974930770145893639ompred: ( ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ product_unit ) @ ( product_prod @ code_natural @ code_natural ) ) ) > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ product_unit ) @ ( product_prod @ code_natural @ code_natural ) ) ).
thf(sy_c_Random__Pred_Osingle,type,
random_single:
!>[A: $tType] : ( A > ( 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_OFract,type,
fract: int > int > rat ).
thf(sy_c_Rat_Ofield__char__0__class_ORats,type,
field_char_0_Rats:
!>[A: $tType] : ( set @ A ) ).
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_Oquotient__of,type,
quotient_of: rat > ( product_prod @ int @ int ) ).
thf(sy_c_Record_Oabst,type,
abst:
!>[A: $tType,B: $tType,C: $tType] : ( ( tuple_isomorphism @ A @ B @ C ) > ( product_prod @ B @ C ) > A ) ).
thf(sy_c_Record_Oiso__tuple__cons,type,
iso_tuple_cons:
!>[A: $tType,B: $tType,C: $tType] : ( ( tuple_isomorphism @ A @ B @ C ) > B > C > A ) ).
thf(sy_c_Record_Oiso__tuple__fst,type,
iso_tuple_fst:
!>[A: $tType,B: $tType,C: $tType] : ( ( tuple_isomorphism @ A @ B @ C ) > A > B ) ).
thf(sy_c_Record_Oiso__tuple__fst__update,type,
iso_tuple_fst_update:
!>[A: $tType,B: $tType,C: $tType] : ( ( tuple_isomorphism @ A @ B @ C ) > ( B > B ) > A > A ) ).
thf(sy_c_Record_Oiso__tuple__snd,type,
iso_tuple_snd:
!>[A: $tType,B: $tType,C: $tType] : ( ( tuple_isomorphism @ A @ B @ C ) > A > C ) ).
thf(sy_c_Record_Oiso__tuple__snd__update,type,
iso_tuple_snd_update:
!>[A: $tType,B: $tType,C: $tType] : ( ( tuple_isomorphism @ A @ B @ C ) > ( C > C ) > A > A ) ).
thf(sy_c_Record_Oiso__tuple__surjective__proof__assist,type,
iso_tu562120763810524177assist:
!>[A: $tType,B: $tType] : ( A > B > ( A > B ) > $o ) ).
thf(sy_c_Record_Oiso__tuple__update__accessor__cong__assist,type,
iso_tu6617205062836021118assist:
!>[B: $tType,A: $tType] : ( ( ( B > B ) > A > A ) > ( A > B ) > $o ) ).
thf(sy_c_Record_Oiso__tuple__update__accessor__eq__assist,type,
iso_tu8710522896132340869assist:
!>[B: $tType,A: $tType] : ( ( ( B > B ) > A > A ) > ( A > B ) > A > ( B > B ) > A > B > $o ) ).
thf(sy_c_Record_Oisomorphic__tuple,type,
isomorphic_tuple:
!>[A: $tType,B: $tType,C: $tType] : ( ( tuple_isomorphism @ A @ B @ C ) > $o ) ).
thf(sy_c_Record_Orepr,type,
repr:
!>[A: $tType,B: $tType,C: $tType] : ( ( tuple_isomorphism @ A @ B @ C ) > A > ( product_prod @ B @ C ) ) ).
thf(sy_c_Record_Otuple__iso__tuple,type,
tuple_iso_tuple:
!>[A: $tType,B: $tType] : ( tuple_isomorphism @ ( product_prod @ A @ B ) @ A @ B ) ).
thf(sy_c_Ref_Oalloc,type,
alloc2:
!>[A: $tType] : ( A > ( heap_ext @ product_unit ) > ( product_prod @ ( ref @ A ) @ ( heap_ext @ product_unit ) ) ) ).
thf(sy_c_Ref_Ochange,type,
change:
!>[A: $tType] : ( ( A > A ) > ( ref @ A ) > ( heap_Heap @ A ) ) ).
thf(sy_c_Ref_Oget,type,
get2:
!>[A: $tType] : ( ( heap_ext @ product_unit ) > ( ref @ A ) > A ) ).
thf(sy_c_Ref_Olookup,type,
lookup:
!>[A: $tType] : ( ( ref @ A ) > ( heap_Heap @ A ) ) ).
thf(sy_c_Ref_Onoteq,type,
noteq2:
!>[A: $tType,B: $tType] : ( ( ref @ A ) > ( ref @ B ) > $o ) ).
thf(sy_c_Ref_Opresent,type,
present2:
!>[A: $tType] : ( ( heap_ext @ product_unit ) > ( ref @ A ) > $o ) ).
thf(sy_c_Ref_Oref,type,
ref3:
!>[A: $tType] : ( A > ( heap_Heap @ ( ref @ A ) ) ) ).
thf(sy_c_Ref_Oref_H,type,
ref4:
!>[A: $tType] : ( A > ( heap_Heap @ ( ref @ A ) ) ) ).
thf(sy_c_Ref_Oset,type,
set4:
!>[A: $tType] : ( ( ref @ A ) > A > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) ) ).
thf(sy_c_Ref_Oupdate,type,
update2:
!>[A: $tType] : ( ( ref @ A ) > A > ( heap_Heap @ product_unit ) ) ).
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_Osym,type,
sym:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Relation_Osymp,type,
symp:
!>[A: $tType] : ( ( A > A > $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_Ozero__neq__one__class_Oof__bool,type,
zero_neq_one_of_bool:
!>[A: $tType] : ( $o > A ) ).
thf(sy_c_Set_OBall,type,
ball:
!>[A: $tType] : ( ( set @ A ) > ( A > $o ) > $o ) ).
thf(sy_c_Set_OBex,type,
bex:
!>[A: $tType] : ( ( set @ A ) > ( A > $o ) > $o ) ).
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,
bind4:
!>[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,
insert:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Set_Opairwise,type,
pairwise:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Set_Othe__elem,type,
the_elem:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Set_Ovimage,type,
vimage:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ B ) > ( set @ A ) ) ).
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_OLiteral,type,
literal2: $o > $o > $o > $o > $o > $o > $o > literal > literal ).
thf(sy_c_String_Ocomm__semiring__1__class_Oof__char,type,
comm_s6883823935334413003f_char:
!>[A: $tType] : ( char > A ) ).
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_OInl__Rep,type,
sum_Inl_Rep:
!>[A: $tType,B: $tType] : ( A > A > B > $o > $o ) ).
thf(sy_c_Sum__Type_OInr,type,
sum_Inr:
!>[B: $tType,A: $tType] : ( B > ( sum_sum @ A @ B ) ) ).
thf(sy_c_Sum__Type_OInr__Rep,type,
sum_Inr_Rep:
!>[B: $tType,A: $tType] : ( B > A > B > $o > $o ) ).
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_OAbs__sum,type,
sum_Abs_sum:
!>[A: $tType,B: $tType] : ( ( A > B > $o > $o ) > ( sum_sum @ A @ B ) ) ).
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_Sum__Type_Osum_Oisl,type,
sum_isl:
!>[A: $tType,B: $tType] : ( ( sum_sum @ A @ B ) > $o ) ).
thf(sy_c_Sum__Type_Osum_Oprojl,type,
sum_projl:
!>[A: $tType,B: $tType] : ( ( sum_sum @ A @ B ) > A ) ).
thf(sy_c_Sum__Type_Osum_Oprojr,type,
sum_projr:
!>[A: $tType,B: $tType] : ( ( sum_sum @ A @ B ) > B ) ).
thf(sy_c_Transfer_Obi__total,type,
bi_total:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
thf(sy_c_Transfer_Obi__unique,type,
bi_unique:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
thf(sy_c_Transfer_Oleft__total,type,
left_total:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
thf(sy_c_Transfer_Oleft__unique,type,
left_unique:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
thf(sy_c_Transfer_Oright__total,type,
right_total:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
thf(sy_c_Transfer_Oright__unique,type,
right_unique:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
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_OTyperep,type,
typerep3: literal > ( list @ typerep ) > typerep ).
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_Wfrec_Owfrec__rel,type,
wfrec_rel:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( ( A > B ) > A > B ) > A > B > $o ) ).
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_h_H,type,
h2: heap_ext @ product_unit ).
thf(sy_v_r,type,
r: ref @ a ).
thf(sy_v_thesis,type,
thesis: $o ).
thf(sy_v_x,type,
x: a ).
% Relevant facts (2715)
thf(fact_0_that,axiom,
( ( r
= ( product_fst @ ( ref @ a ) @ ( heap_ext @ product_unit ) @ ( alloc2 @ a @ x @ h ) ) )
=> ( ( h2
= ( product_snd @ ( ref @ a ) @ ( heap_ext @ product_unit ) @ ( alloc2 @ a @ x @ h ) ) )
=> thesis ) ) ).
% that
thf(fact_1_assms,axiom,
heap_effect @ ( ref @ a ) @ ( ref3 @ a @ x ) @ h @ h2 @ r ).
% assms
thf(fact_2_prod_Oexpand,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B,Prod2: product_prod @ A @ B] :
( ( ( ( product_fst @ A @ B @ Prod )
= ( product_fst @ A @ B @ Prod2 ) )
& ( ( product_snd @ A @ B @ Prod )
= ( product_snd @ A @ B @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_3_prod__eqI,axiom,
! [B: $tType,A: $tType,P: product_prod @ A @ B,Q: product_prod @ A @ B] :
( ( ( product_fst @ A @ B @ P )
= ( product_fst @ A @ B @ Q ) )
=> ( ( ( product_snd @ A @ B @ P )
= ( product_snd @ A @ B @ Q ) )
=> ( P = Q ) ) ) ).
% prod_eqI
thf(fact_4_exE__realizer_H,axiom,
! [A: $tType,B: $tType,P2: A > B > $o,P: product_prod @ B @ A] :
( ( P2 @ ( product_snd @ B @ A @ P ) @ ( product_fst @ B @ A @ P ) )
=> ~ ! [X: B,Y: A] :
~ ( P2 @ Y @ X ) ) ).
% exE_realizer'
thf(fact_5_prod__eq__iff,axiom,
! [B: $tType,A: $tType] :
( ( ^ [Y2: product_prod @ A @ B,Z2: product_prod @ A @ B] : Y2 = Z2 )
= ( ^ [S: product_prod @ A @ B,T2: product_prod @ A @ B] :
( ( ( product_fst @ A @ B @ S )
= ( product_fst @ A @ B @ T2 ) )
& ( ( product_snd @ A @ B @ S )
= ( product_snd @ A @ B @ T2 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_6_Ref_Oget__alloc,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [X2: A,H: heap_ext @ product_unit,X3: A] :
( ( get2 @ A @ ( product_snd @ ( ref @ A ) @ ( heap_ext @ product_unit ) @ ( alloc2 @ A @ X2 @ H ) ) @ ( product_fst @ ( ref @ A ) @ ( heap_ext @ product_unit ) @ ( alloc2 @ A @ X3 @ H ) ) )
= X2 ) ) ).
% Ref.get_alloc
thf(fact_7_set__alloc,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [V: A,H: heap_ext @ product_unit,V2: A] :
( ( set4 @ A @ ( product_fst @ ( ref @ A ) @ ( heap_ext @ product_unit ) @ ( alloc2 @ A @ V @ H ) ) @ V2 @ ( product_snd @ ( ref @ A ) @ ( heap_ext @ product_unit ) @ ( alloc2 @ A @ V @ H ) ) )
= ( product_snd @ ( ref @ A ) @ ( heap_ext @ product_unit ) @ ( alloc2 @ A @ V2 @ H ) ) ) ) ).
% set_alloc
thf(fact_8_Ref_Opresent__alloc,axiom,
! [B: $tType,A: $tType] :
( ( ( heap @ A )
& ( heap @ B ) )
=> ! [H: heap_ext @ product_unit,R: ref @ A,V: B] :
( ( present2 @ A @ H @ R )
=> ( present2 @ A @ ( product_snd @ ( ref @ B ) @ ( heap_ext @ product_unit ) @ ( alloc2 @ B @ V @ H ) ) @ R ) ) ) ).
% Ref.present_alloc
thf(fact_9_Ref_Onot__present__alloc,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [H: heap_ext @ product_unit,V: A] :
~ ( present2 @ A @ H @ ( product_fst @ ( ref @ A ) @ ( heap_ext @ product_unit ) @ ( alloc2 @ A @ V @ H ) ) ) ) ).
% Ref.not_present_alloc
thf(fact_10_array__present__alloc,axiom,
! [B: $tType,A: $tType] :
( ( ( heap @ A )
& ( heap @ B ) )
=> ! [H: heap_ext @ product_unit,A3: array @ A,V: B] :
( ( present @ A @ H @ A3 )
=> ( present @ A @ ( product_snd @ ( ref @ B ) @ ( heap_ext @ product_unit ) @ ( alloc2 @ B @ V @ H ) ) @ A3 ) ) ) ).
% array_present_alloc
thf(fact_11_Ref_Olength__alloc,axiom,
! [B: $tType,A: $tType] :
( ( ( heap @ A )
& ( heap @ B ) )
=> ! [V: B,H: heap_ext @ product_unit,A3: array @ A] :
( ( length @ A @ ( product_snd @ ( ref @ B ) @ ( heap_ext @ product_unit ) @ ( alloc2 @ B @ V @ H ) ) @ A3 )
= ( length @ A @ H @ A3 ) ) ) ).
% Ref.length_alloc
thf(fact_12_array__get__alloc,axiom,
! [B: $tType,A: $tType] :
( ( ( heap @ A )
& ( heap @ B ) )
=> ! [V: B,H: heap_ext @ product_unit] :
( ( get @ A @ ( product_snd @ ( ref @ B ) @ ( heap_ext @ product_unit ) @ ( alloc2 @ B @ V @ H ) ) )
= ( get @ A @ H ) ) ) ).
% array_get_alloc
thf(fact_13_Ref_Oset__same,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R: ref @ A,X2: A,Y3: A,H: heap_ext @ product_unit] :
( ( set4 @ A @ R @ X2 @ ( set4 @ A @ R @ Y3 @ H ) )
= ( set4 @ A @ R @ X2 @ H ) ) ) ).
% Ref.set_same
thf(fact_14_array__get__set,axiom,
! [B: $tType,A: $tType] :
( ( ( heap @ A )
& ( heap @ B ) )
=> ! [R: ref @ B,V: B,H: heap_ext @ product_unit] :
( ( get @ A @ ( set4 @ B @ R @ V @ H ) )
= ( get @ A @ H ) ) ) ).
% array_get_set
thf(fact_15_Ref_Oget__set__eq,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R: ref @ A,X2: A,H: heap_ext @ product_unit] :
( ( get2 @ A @ ( set4 @ A @ R @ X2 @ H ) @ R )
= X2 ) ) ).
% Ref.get_set_eq
thf(fact_16_array__present__set,axiom,
! [B: $tType,A: $tType] :
( ( ( heap @ A )
& ( heap @ B ) )
=> ! [R: ref @ B,V: B,H: heap_ext @ product_unit] :
( ( present @ A @ ( set4 @ B @ R @ V @ H ) )
= ( present @ A @ H ) ) ) ).
% array_present_set
thf(fact_17_present__set,axiom,
! [B: $tType,A: $tType] :
( ( ( heap @ A )
& ( heap @ B ) )
=> ! [R: ref @ B,V: B,H: heap_ext @ product_unit] :
( ( present2 @ A @ ( set4 @ B @ R @ V @ H ) )
= ( present2 @ A @ H ) ) ) ).
% present_set
thf(fact_18_effect__refE,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [V: A,H: heap_ext @ product_unit,H2: heap_ext @ product_unit,R: ref @ A] :
( ( heap_effect @ ( ref @ A ) @ ( ref3 @ A @ V ) @ H @ H2 @ R )
=> ~ ( ( ( get2 @ A @ H2 @ R )
= V )
=> ( ( present2 @ A @ H2 @ R )
=> ( present2 @ A @ H @ R ) ) ) ) ) ).
% effect_refE
thf(fact_19_alloc__set,axiom,
! [B: $tType,A: $tType] :
( ( ( heap @ A )
& ( heap @ B ) )
=> ! [X2: A,R: ref @ B,X3: B,H: heap_ext @ product_unit] :
( ( product_fst @ ( ref @ A ) @ ( heap_ext @ product_unit ) @ ( alloc2 @ A @ X2 @ ( set4 @ B @ R @ X3 @ H ) ) )
= ( product_fst @ ( ref @ A ) @ ( heap_ext @ product_unit ) @ ( alloc2 @ A @ X2 @ H ) ) ) ) ).
% alloc_set
thf(fact_20_ref_H__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( ref4 @ A )
= ( ref3 @ A ) ) ) ).
% ref'_def
thf(fact_21_get__alloc__neq,axiom,
! [B: $tType,A: $tType] :
( ( ( heap @ A )
& ( heap @ B ) )
=> ! [R: ref @ A,V: B,H: heap_ext @ product_unit] :
( ( noteq2 @ A @ B @ R @ ( product_fst @ ( ref @ B ) @ ( heap_ext @ product_unit ) @ ( alloc2 @ B @ V @ H ) ) )
=> ( ( get2 @ A @ ( product_snd @ ( ref @ B ) @ ( heap_ext @ product_unit ) @ ( alloc2 @ B @ V @ H ) ) @ R )
= ( get2 @ A @ H @ R ) ) ) ) ).
% get_alloc_neq
thf(fact_22_effect__changeE,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [F: A > A,R2: ref @ A,H: heap_ext @ product_unit,H2: heap_ext @ product_unit,R: A] :
( ( heap_effect @ A @ ( change @ A @ F @ R2 ) @ H @ H2 @ R )
=> ~ ( ( H2
= ( set4 @ A @ R2 @ ( F @ ( get2 @ A @ H @ R2 ) ) @ H ) )
=> ( R
!= ( F @ ( get2 @ A @ H @ R2 ) ) ) ) ) ) ).
% effect_changeE
thf(fact_23_effect__changeI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [H2: heap_ext @ product_unit,R: ref @ A,F: A > A,H: heap_ext @ product_unit,X2: A] :
( ( H2
= ( set4 @ A @ R @ ( F @ ( get2 @ A @ H @ R ) ) @ H ) )
=> ( ( X2
= ( F @ ( get2 @ A @ H @ R ) ) )
=> ( heap_effect @ A @ ( change @ A @ F @ R ) @ H @ H2 @ X2 ) ) ) ) ).
% effect_changeI
thf(fact_24_present__alloc__neq,axiom,
! [A: $tType,B: $tType] :
( ( ( heap @ B )
& ( heap @ A ) )
=> ! [H: heap_ext @ product_unit,R: ref @ A,V: B] :
( ( present2 @ A @ H @ R )
=> ( noteq2 @ A @ B @ R @ ( product_fst @ ( ref @ B ) @ ( heap_ext @ product_unit ) @ ( alloc2 @ B @ V @ H ) ) ) ) ) ).
% present_alloc_neq
thf(fact_25_Ref_Oget__set__neq,axiom,
! [B: $tType,A: $tType] :
( ( ( heap @ A )
& ( heap @ B ) )
=> ! [R: ref @ A,S2: ref @ B,X2: B,H: heap_ext @ product_unit] :
( ( noteq2 @ A @ B @ R @ S2 )
=> ( ( get2 @ A @ ( set4 @ B @ S2 @ X2 @ H ) @ R )
= ( get2 @ A @ H @ R ) ) ) ) ).
% Ref.get_set_neq
thf(fact_26_ref__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( ref3 @ A )
= ( ^ [V3: A] : ( heap_heap @ ( ref @ A ) @ ( alloc2 @ A @ V3 ) ) ) ) ) ).
% ref_def
thf(fact_27_effect__lookupE,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R: ref @ A,H: heap_ext @ product_unit,H2: heap_ext @ product_unit,X2: A] :
( ( heap_effect @ A @ ( lookup @ A @ R ) @ H @ H2 @ X2 )
=> ~ ( ( H2 = H )
=> ( X2
!= ( get2 @ A @ H @ R ) ) ) ) ) ).
% effect_lookupE
thf(fact_28_effect__lookupI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [H2: heap_ext @ product_unit,H: heap_ext @ product_unit,X2: A,R: ref @ A] :
( ( H2 = H )
=> ( ( X2
= ( get2 @ A @ H @ R ) )
=> ( heap_effect @ A @ ( lookup @ A @ R ) @ H @ H2 @ X2 ) ) ) ) ).
% effect_lookupI
thf(fact_29_effect__refI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R: ref @ A,H2: heap_ext @ product_unit,V: A,H: heap_ext @ product_unit] :
( ( ( product_Pair @ ( ref @ A ) @ ( heap_ext @ product_unit ) @ R @ H2 )
= ( alloc2 @ A @ V @ H ) )
=> ( heap_effect @ ( ref @ A ) @ ( ref3 @ A @ V ) @ H @ H2 @ R ) ) ) ).
% effect_refI
thf(fact_30_snd__swap,axiom,
! [B: $tType,A: $tType,X2: product_prod @ A @ B] :
( ( product_snd @ B @ A @ ( product_swap @ A @ B @ X2 ) )
= ( product_fst @ A @ B @ X2 ) ) ).
% snd_swap
thf(fact_31_fst__swap,axiom,
! [A: $tType,B: $tType,X2: product_prod @ B @ A] :
( ( product_fst @ A @ B @ ( product_swap @ B @ A @ X2 ) )
= ( product_snd @ B @ A @ X2 ) ) ).
% fst_swap
thf(fact_32_prod_Oinject,axiom,
! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y22: B] :
( ( ( product_Pair @ A @ B @ X1 @ X22 )
= ( product_Pair @ A @ B @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_33_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,A4: A,B3: B] :
( ( ( product_Pair @ A @ B @ A3 @ B2 )
= ( product_Pair @ A @ B @ A4 @ B3 ) )
= ( ( A3 = A4 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_34_swap__swap,axiom,
! [B: $tType,A: $tType,P: product_prod @ A @ B] :
( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P ) )
= P ) ).
% swap_swap
thf(fact_35_Ref_Ounequal,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R: ref @ A,R2: ref @ A] :
( ( R != R2 )
= ( noteq2 @ A @ A @ R @ R2 ) ) ) ).
% Ref.unequal
thf(fact_36_swap__simp,axiom,
! [A: $tType,B: $tType,X2: B,Y3: A] :
( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X2 @ Y3 ) )
= ( product_Pair @ A @ B @ Y3 @ X2 ) ) ).
% swap_simp
thf(fact_37_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_38_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y3: product_prod @ A @ B] :
~ ! [A5: A,B4: B] :
( Y3
!= ( product_Pair @ A @ B @ A5 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_39_surj__pair,axiom,
! [A: $tType,B: $tType,P: product_prod @ A @ B] :
? [X: A,Y: B] :
( P
= ( product_Pair @ A @ B @ X @ Y ) ) ).
% surj_pair
thf(fact_40_prod__cases,axiom,
! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,P: product_prod @ A @ B] :
( ! [A5: A,B4: B] : ( P2 @ ( product_Pair @ A @ B @ A5 @ B4 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_41_Pair__inject,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,A4: A,B3: B] :
( ( ( product_Pair @ A @ B @ A3 @ B2 )
= ( product_Pair @ A @ B @ A4 @ B3 ) )
=> ~ ( ( A3 = A4 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_42_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y3: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A5: A,B4: B,C2: C] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) ) ).
% prod_cases3
thf(fact_43_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
~ ! [A5: A,B4: B,C2: C,D2: D] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) ) ).
% prod_cases4
thf(fact_44_mem__Collect__eq,axiom,
! [A: $tType,A3: A,P2: A > $o] :
( ( member @ A @ A3 @ ( collect @ A @ P2 ) )
= ( P2 @ A3 ) ) ).
% mem_Collect_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A: $tType,A6: set @ A] :
( ( collect @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A6 ) )
= A6 ) ).
% Collect_mem_eq
thf(fact_46_Collect__cong,axiom,
! [A: $tType,P2: A > $o,Q2: A > $o] :
( ! [X: A] :
( ( P2 @ X )
= ( Q2 @ X ) )
=> ( ( collect @ A @ P2 )
= ( collect @ A @ Q2 ) ) ) ).
% Collect_cong
thf(fact_47_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X: A] :
( ( F @ X )
= ( G @ X ) )
=> ( F = G ) ) ).
% ext
thf(fact_48_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
~ ! [A5: A,B4: B,C2: C,D2: D,E2: E] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_49_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F2: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) )] :
~ ! [A5: A,B4: B,C2: C,D2: D,E2: E,F3: F2] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F2 ) @ D2 @ ( product_Pair @ E @ F2 @ E2 @ F3 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_50_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F2: $tType,G2: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) )] :
~ ! [A5: A,B4: B,C2: C,D2: D,E2: E,F3: F2,G3: G2] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F2 @ G2 ) @ E2 @ ( product_Pair @ F2 @ G2 @ F3 @ G3 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_51_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A5: A,B4: B,C2: C] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) )
=> ( P2 @ X2 ) ) ).
% prod_induct3
thf(fact_52_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
( ! [A5: A,B4: B,C2: C,D2: D] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) )
=> ( P2 @ X2 ) ) ).
% prod_induct4
thf(fact_53_prod__induct5,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
( ! [A5: A,B4: B,C2: C,D2: D,E2: E] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
=> ( P2 @ X2 ) ) ).
% prod_induct5
thf(fact_54_prod__induct6,axiom,
! [F2: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) )] :
( ! [A5: A,B4: B,C2: C,D2: D,E2: E,F3: F2] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F2 ) @ D2 @ ( product_Pair @ E @ F2 @ E2 @ F3 ) ) ) ) ) )
=> ( P2 @ X2 ) ) ).
% prod_induct6
thf(fact_55_prod__induct7,axiom,
! [G2: $tType,F2: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) )] :
( ! [A5: A,B4: B,C2: C,D2: D,E2: E,F3: F2,G3: G2] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F2 @ G2 ) @ E2 @ ( product_Pair @ F2 @ G2 @ F3 @ G3 ) ) ) ) ) ) )
=> ( P2 @ X2 ) ) ).
% prod_induct7
thf(fact_56_Ref_Onoteq__irrefl,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R: ref @ A] :
~ ( noteq2 @ A @ A @ R @ R ) ) ).
% Ref.noteq_irrefl
thf(fact_57_Ref_Onoteq__sym,axiom,
! [B: $tType,A: $tType] :
( ( ( heap @ A )
& ( heap @ B ) )
=> ! [R: ref @ A,S2: ref @ B] :
( ( noteq2 @ A @ B @ R @ S2 )
=> ( noteq2 @ B @ A @ S2 @ R ) ) ) ).
% Ref.noteq_sym
thf(fact_58_fst__eqD,axiom,
! [B: $tType,A: $tType,X2: A,Y3: B,A3: A] :
( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X2 @ Y3 ) )
= A3 )
=> ( X2 = A3 ) ) ).
% fst_eqD
thf(fact_59_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_60_snd__eqD,axiom,
! [B: $tType,A: $tType,X2: B,Y3: A,A3: A] :
( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X2 @ Y3 ) )
= A3 )
=> ( Y3 = A3 ) ) ).
% snd_eqD
thf(fact_61_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_62_prod_Oswap__def,axiom,
! [B: $tType,A: $tType] :
( ( product_swap @ A @ B )
= ( ^ [P3: product_prod @ A @ B] : ( product_Pair @ B @ A @ ( product_snd @ A @ B @ P3 ) @ ( product_fst @ A @ B @ P3 ) ) ) ) ).
% prod.swap_def
thf(fact_63_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_64_exI__realizer,axiom,
! [B: $tType,A: $tType,P2: A > B > $o,Y3: A,X2: B] :
( ( P2 @ Y3 @ X2 )
=> ( P2 @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X2 @ Y3 ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X2 @ Y3 ) ) ) ) ).
% exI_realizer
thf(fact_65_conjI__realizer,axiom,
! [A: $tType,B: $tType,P2: A > $o,P: A,Q2: B > $o,Q: B] :
( ( P2 @ P )
=> ( ( Q2 @ Q )
=> ( ( P2 @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P @ Q ) ) )
& ( Q2 @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P @ Q ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_66_surjective__pairing,axiom,
! [B: $tType,A: $tType,T3: product_prod @ A @ B] :
( T3
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T3 ) @ ( product_snd @ A @ B @ T3 ) ) ) ).
% surjective_pairing
thf(fact_67_Ref_Oset__set__swap,axiom,
! [B: $tType,A: $tType] :
( ( ( heap @ A )
& ( heap @ B ) )
=> ! [R: ref @ A,R2: ref @ B,X2: A,X3: B,H: heap_ext @ product_unit] :
( ( noteq2 @ A @ B @ R @ R2 )
=> ( ( set4 @ A @ R @ X2 @ ( set4 @ B @ R2 @ X3 @ H ) )
= ( set4 @ B @ R2 @ X3 @ ( set4 @ A @ R @ X2 @ H ) ) ) ) ) ).
% Ref.set_set_swap
thf(fact_68_next__fresh,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R: ref @ A,H2: heap_ext @ product_unit,X2: A,H: heap_ext @ product_unit] :
( ( ( product_Pair @ ( ref @ A ) @ ( heap_ext @ product_unit ) @ R @ H2 )
= ( alloc2 @ A @ X2 @ H ) )
=> ~ ( present2 @ A @ H @ R ) ) ) ).
% next_fresh
thf(fact_69_next__present,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R: ref @ A,H2: heap_ext @ product_unit,X2: A,H: heap_ext @ product_unit] :
( ( ( product_Pair @ ( ref @ A ) @ ( heap_ext @ product_unit ) @ R @ H2 )
= ( alloc2 @ A @ X2 @ H ) )
=> ( present2 @ A @ H2 @ R ) ) ) ).
% next_present
thf(fact_70_noteq__I,axiom,
! [A: $tType,B: $tType] :
( ( ( heap @ B )
& ( heap @ A ) )
=> ! [H: heap_ext @ product_unit,R: ref @ A,R2: ref @ B] :
( ( present2 @ A @ H @ R )
=> ( ~ ( present2 @ B @ H @ R2 )
=> ( noteq2 @ A @ B @ R @ R2 ) ) ) ) ).
% noteq_I
thf(fact_71_effect__heapE,axiom,
! [A: $tType,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( heap_ext @ product_unit ) ),H: heap_ext @ product_unit,H2: heap_ext @ product_unit,R: A] :
( ( heap_effect @ A @ ( heap_heap @ A @ F ) @ H @ H2 @ R )
=> ~ ( ( H2
= ( product_snd @ A @ ( heap_ext @ product_unit ) @ ( F @ H ) ) )
=> ( R
!= ( product_fst @ A @ ( heap_ext @ product_unit ) @ ( F @ H ) ) ) ) ) ).
% effect_heapE
thf(fact_72_effect__heapI,axiom,
! [A: $tType,H2: heap_ext @ product_unit,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( heap_ext @ product_unit ) ),H: heap_ext @ product_unit,R: A] :
( ( H2
= ( product_snd @ A @ ( heap_ext @ product_unit ) @ ( F @ H ) ) )
=> ( ( R
= ( product_fst @ A @ ( heap_ext @ product_unit ) @ ( F @ H ) ) )
=> ( heap_effect @ A @ ( heap_heap @ A @ F ) @ H @ H2 @ R ) ) ) ).
% effect_heapI
thf(fact_73_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_74_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [B: $tType,A: $tType,P2: A > B > $o,X2: A,Y3: B,A3: product_prod @ A @ B] :
( ( P2 @ X2 @ Y3 )
=> ( ( A3
= ( product_Pair @ A @ B @ X2 @ Y3 ) )
=> ( P2 @ ( product_fst @ A @ B @ A3 ) @ ( product_snd @ A @ B @ A3 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_75_sndI,axiom,
! [A: $tType,B: $tType,X2: product_prod @ A @ B,Y3: A,Z3: B] :
( ( X2
= ( product_Pair @ A @ B @ Y3 @ Z3 ) )
=> ( ( product_snd @ A @ B @ X2 )
= Z3 ) ) ).
% sndI
thf(fact_76_eq__snd__iff,axiom,
! [A: $tType,B: $tType,B2: A,P: product_prod @ B @ A] :
( ( B2
= ( product_snd @ B @ A @ P ) )
= ( ? [A7: B] :
( P
= ( product_Pair @ B @ A @ A7 @ B2 ) ) ) ) ).
% eq_snd_iff
thf(fact_77_fstI,axiom,
! [B: $tType,A: $tType,X2: product_prod @ A @ B,Y3: A,Z3: B] :
( ( X2
= ( product_Pair @ A @ B @ Y3 @ Z3 ) )
=> ( ( product_fst @ A @ B @ X2 )
= Y3 ) ) ).
% fstI
thf(fact_78_eq__fst__iff,axiom,
! [A: $tType,B: $tType,A3: A,P: product_prod @ A @ B] :
( ( A3
= ( product_fst @ A @ B @ P ) )
= ( ? [B5: B] :
( P
= ( product_Pair @ A @ B @ A3 @ B5 ) ) ) ) ).
% eq_fst_iff
thf(fact_79_internal__case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,C3: B > C > A,A3: B,B2: C] :
( ( produc5280177257484947105e_prod @ B @ C @ A @ C3 @ ( product_Pair @ B @ C @ A3 @ B2 ) )
= ( C3 @ A3 @ B2 ) ) ).
% internal_case_prod_conv
thf(fact_80_effect__guardE,axiom,
! [A: $tType,P2: ( heap_ext @ product_unit ) > $o,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( heap_ext @ product_unit ) ),H: heap_ext @ product_unit,H2: heap_ext @ product_unit,R: A] :
( ( heap_effect @ A @ ( heap_guard @ A @ P2 @ F ) @ H @ H2 @ R )
=> ~ ( ( H2
= ( product_snd @ A @ ( heap_ext @ product_unit ) @ ( F @ H ) ) )
=> ( ( R
= ( product_fst @ A @ ( heap_ext @ product_unit ) @ ( F @ H ) ) )
=> ~ ( P2 @ H ) ) ) ) ).
% effect_guardE
thf(fact_81_effect__guardI,axiom,
! [A: $tType,P2: ( heap_ext @ product_unit ) > $o,H: heap_ext @ product_unit,H2: heap_ext @ product_unit,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( heap_ext @ product_unit ) ),R: A] :
( ( P2 @ H )
=> ( ( H2
= ( product_snd @ A @ ( heap_ext @ product_unit ) @ ( F @ H ) ) )
=> ( ( R
= ( product_fst @ A @ ( heap_ext @ product_unit ) @ ( F @ H ) ) )
=> ( heap_effect @ A @ ( heap_guard @ A @ P2 @ F ) @ H @ H2 @ R ) ) ) ) ).
% effect_guardI
thf(fact_82_effect__ifI,axiom,
! [A: $tType,C3: $o,T3: heap_Heap @ A,H: heap_ext @ product_unit,H2: heap_ext @ product_unit,R: A,E3: heap_Heap @ A] :
( ( C3
=> ( heap_effect @ A @ T3 @ H @ H2 @ R ) )
=> ( ( ~ C3
=> ( heap_effect @ A @ E3 @ H @ H2 @ R ) )
=> ( heap_effect @ A @ ( if @ ( heap_Heap @ A ) @ C3 @ T3 @ E3 ) @ H @ H2 @ R ) ) ) ).
% effect_ifI
thf(fact_83_effect__deterministic_I2_J,axiom,
! [A: $tType,F: heap_Heap @ A,H: heap_ext @ product_unit,H2: heap_ext @ product_unit,A3: A,H3: heap_ext @ product_unit,B2: A] :
( ( heap_effect @ A @ F @ H @ H2 @ A3 )
=> ( ( heap_effect @ A @ F @ H @ H3 @ B2 )
=> ( H2 = H3 ) ) ) ).
% effect_deterministic(2)
thf(fact_84_effect__deterministic_I1_J,axiom,
! [A: $tType,F: heap_Heap @ A,H: heap_ext @ product_unit,H2: heap_ext @ product_unit,A3: A,H3: heap_ext @ product_unit,B2: A] :
( ( heap_effect @ A @ F @ H @ H2 @ A3 )
=> ( ( heap_effect @ A @ F @ H @ H3 @ B2 )
=> ( A3 = B2 ) ) ) ).
% effect_deterministic(1)
thf(fact_85_effect__ifE,axiom,
! [A: $tType,C3: $o,T3: heap_Heap @ A,E3: heap_Heap @ A,H: heap_ext @ product_unit,H2: heap_ext @ product_unit,R: A] :
( ( heap_effect @ A @ ( if @ ( heap_Heap @ A ) @ C3 @ T3 @ E3 ) @ H @ H2 @ R )
=> ( ( C3
=> ~ ( heap_effect @ A @ T3 @ H @ H2 @ R ) )
=> ~ ( ~ C3
=> ~ ( heap_effect @ A @ E3 @ H @ H2 @ R ) ) ) ) ).
% effect_ifE
thf(fact_86_return__def,axiom,
! [A: $tType] :
( ( heap_return @ A )
= ( ^ [X4: A] : ( heap_heap @ A @ ( product_Pair @ A @ ( heap_ext @ product_unit ) @ X4 ) ) ) ) ).
% return_def
thf(fact_87_sndOp__def,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( bNF_sndOp @ C @ A @ B )
= ( ^ [P4: C > A > $o,Q3: A > B > $o,Ac: product_prod @ C @ B] : ( product_Pair @ A @ B @ ( bNF_pick_middlep @ C @ A @ B @ P4 @ Q3 @ ( product_fst @ C @ B @ Ac ) @ ( product_snd @ C @ B @ Ac ) ) @ ( product_snd @ C @ B @ Ac ) ) ) ) ).
% sndOp_def
thf(fact_88_fstOp__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( bNF_fstOp @ A @ B @ C )
= ( ^ [P4: A > B > $o,Q3: B > C > $o,Ac: product_prod @ A @ C] : ( product_Pair @ A @ B @ ( product_fst @ A @ C @ Ac ) @ ( bNF_pick_middlep @ A @ B @ C @ P4 @ Q3 @ ( product_fst @ A @ C @ Ac ) @ ( product_snd @ A @ C @ Ac ) ) ) ) ) ).
% fstOp_def
thf(fact_89_image2__eqI,axiom,
! [A: $tType,C: $tType,B: $tType,B2: A,F: B > A,X2: B,C3: C,G: B > C,A6: set @ B] :
( ( B2
= ( F @ X2 ) )
=> ( ( C3
= ( G @ X2 ) )
=> ( ( member @ B @ X2 @ A6 )
=> ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ B2 @ C3 ) @ ( bNF_Greatest_image2 @ B @ A @ C @ A6 @ F @ G ) ) ) ) ) ).
% image2_eqI
thf(fact_90_alloc__update,axiom,
! [B: $tType,A: $tType] :
( ( ( heap @ A )
& ( heap @ B ) )
=> ! [V: A,A3: array @ B,I: nat,V2: B,H: heap_ext @ product_unit] :
( ( product_fst @ ( ref @ A ) @ ( heap_ext @ product_unit ) @ ( alloc2 @ A @ V @ ( update @ B @ A3 @ I @ V2 @ H ) ) )
= ( product_fst @ ( ref @ A ) @ ( heap_ext @ product_unit ) @ ( alloc2 @ A @ V @ H ) ) ) ) ).
% alloc_update
thf(fact_91_lim__set,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R: ref @ A,V: A,H: heap_ext @ product_unit] :
( ( lim @ product_unit @ ( set4 @ A @ R @ V @ H ) )
= ( lim @ product_unit @ H ) ) ) ).
% lim_set
thf(fact_92_prod_Osplit__sel__asm,axiom,
! [C: $tType,B: $tType,A: $tType,P2: C > $o,F: A > B > C,Prod: product_prod @ A @ B] :
( ( P2 @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
= ( ~ ( ( Prod
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
& ~ ( P2 @ ( F @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ) ).
% prod.split_sel_asm
thf(fact_93_prod_Osplit__sel,axiom,
! [C: $tType,B: $tType,A: $tType,P2: C > $o,F: A > B > C,Prod: product_prod @ A @ B] :
( ( P2 @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
= ( ( Prod
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
=> ( P2 @ ( F @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ).
% prod.split_sel
thf(fact_94_ssubst__Pair__rhs,axiom,
! [B: $tType,A: $tType,R: A,S2: B,R3: set @ ( product_prod @ A @ B ),S3: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S2 ) @ R3 )
=> ( ( S3 = S2 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S3 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_95_curry__conv,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( product_curry @ B @ C @ A )
= ( ^ [F4: ( product_prod @ B @ C ) > A,A7: B,B5: C] : ( F4 @ ( product_Pair @ B @ C @ A7 @ B5 ) ) ) ) ).
% curry_conv
thf(fact_96_effect__raiseE,axiom,
! [A: $tType,X2: literal,H: heap_ext @ product_unit,H2: heap_ext @ product_unit,R: A] :
~ ( heap_effect @ A @ ( heap_raise @ A @ X2 ) @ H @ H2 @ R ) ).
% effect_raiseE
thf(fact_97_effect__updateE,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R2: ref @ A,V: A,H: heap_ext @ product_unit,H2: heap_ext @ product_unit,R: product_unit] :
( ( heap_effect @ product_unit @ ( update2 @ A @ R2 @ V ) @ H @ H2 @ R )
=> ( H2
= ( set4 @ A @ R2 @ V @ H ) ) ) ) ).
% effect_updateE
thf(fact_98_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_99_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_100_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_101_get__update,axiom,
! [B: $tType,A: $tType] :
( ( ( heap @ A )
& ( heap @ B ) )
=> ! [A3: array @ B,I: nat,V: B,H: heap_ext @ product_unit,R: ref @ A] :
( ( get2 @ A @ ( update @ B @ A3 @ I @ V @ H ) @ R )
= ( get2 @ A @ H @ R ) ) ) ).
% get_update
thf(fact_102_Ref_Opresent__update,axiom,
! [B: $tType,A: $tType] :
( ( ( heap @ A )
& ( heap @ B ) )
=> ! [A3: array @ B,I: nat,V: B,H: heap_ext @ product_unit] :
( ( present2 @ A @ ( update @ B @ A3 @ I @ V @ H ) )
= ( present2 @ A @ H ) ) ) ).
% Ref.present_update
thf(fact_103_mem__case__prodE,axiom,
! [B: $tType,A: $tType,C: $tType,Z3: A,C3: B > C > ( set @ A ),P: product_prod @ B @ C] :
( ( member @ A @ Z3 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C3 @ P ) )
=> ~ ! [X: B,Y: C] :
( ( P
= ( product_Pair @ B @ C @ X @ Y ) )
=> ~ ( member @ A @ Z3 @ ( C3 @ X @ Y ) ) ) ) ).
% mem_case_prodE
thf(fact_104_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_105_update__set__swap,axiom,
! [B: $tType,A: $tType] :
( ( ( heap @ A )
& ( heap @ B ) )
=> ! [A3: array @ A,I: nat,V: A,R: ref @ B,V2: B,H: heap_ext @ product_unit] :
( ( update @ A @ A3 @ I @ V @ ( set4 @ B @ R @ V2 @ H ) )
= ( set4 @ B @ R @ V2 @ ( update @ A @ A3 @ I @ V @ H ) ) ) ) ).
% update_set_swap
thf(fact_106_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_107_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_108_Product__Type_OCollect__case__prodD,axiom,
! [B: $tType,A: $tType,X2: product_prod @ A @ B,A6: A > B > $o] :
( ( member @ ( product_prod @ A @ B ) @ X2 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A6 ) ) )
=> ( A6 @ ( product_fst @ A @ B @ X2 ) @ ( product_snd @ A @ B @ X2 ) ) ) ).
% Product_Type.Collect_case_prodD
thf(fact_109_effect__returnI,axiom,
! [A: $tType,H: heap_ext @ product_unit,H2: heap_ext @ product_unit,X2: A] :
( ( H = H2 )
=> ( heap_effect @ A @ ( heap_return @ A @ X2 ) @ H @ H2 @ X2 ) ) ).
% effect_returnI
thf(fact_110_effect__returnE,axiom,
! [A: $tType,X2: A,H: heap_ext @ product_unit,H2: heap_ext @ product_unit,R: A] :
( ( heap_effect @ A @ ( heap_return @ A @ X2 ) @ H @ H2 @ R )
=> ~ ( ( R = X2 )
=> ( H2 != H ) ) ) ).
% effect_returnE
thf(fact_111_split__beta,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ C )
= ( ^ [F4: A > B > C,Prod3: product_prod @ A @ B] : ( F4 @ ( product_fst @ A @ B @ Prod3 ) @ ( product_snd @ A @ B @ Prod3 ) ) ) ) ).
% split_beta
thf(fact_112_case__prod__beta,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( product_case_prod @ B @ C @ A )
= ( ^ [F4: B > C > A,P3: product_prod @ B @ C] : ( F4 @ ( product_fst @ B @ C @ P3 ) @ ( product_snd @ B @ C @ P3 ) ) ) ) ).
% case_prod_beta
thf(fact_113_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_114_effect__updateI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [H2: heap_ext @ product_unit,R: ref @ A,V: A,H: heap_ext @ product_unit,X2: product_unit] :
( ( H2
= ( set4 @ A @ R @ V @ H ) )
=> ( heap_effect @ product_unit @ ( update2 @ A @ R @ V ) @ H @ H2 @ X2 ) ) ) ).
% effect_updateI
thf(fact_115_Array_Opresent__update,axiom,
! [B: $tType,A: $tType] :
( ( ( heap @ A )
& ( heap @ B ) )
=> ! [B2: array @ B,I: nat,V: B,H: heap_ext @ product_unit] :
( ( present @ A @ ( update @ B @ B2 @ I @ V @ H ) )
= ( present @ A @ H ) ) ) ).
% Array.present_update
thf(fact_116_length__update,axiom,
! [B: $tType,A: $tType] :
( ( ( heap @ A )
& ( heap @ B ) )
=> ! [B2: array @ B,I: nat,V: B,H: heap_ext @ product_unit] :
( ( length @ A @ ( update @ B @ B2 @ I @ V @ H ) )
= ( length @ A @ H ) ) ) ).
% length_update
thf(fact_117_lift__collapse,axiom,
! [A: $tType,B: $tType] :
( ( heap_lift @ B @ A )
= ( ^ [F4: B > A,X4: B] : ( heap_return @ A @ ( F4 @ X4 ) ) ) ) ).
% lift_collapse
thf(fact_118_split__cong,axiom,
! [C: $tType,B: $tType,A: $tType,Q: product_prod @ A @ B,F: A > B > C,G: A > B > C,P: product_prod @ A @ B] :
( ! [X: A,Y: B] :
( ( ( product_Pair @ A @ B @ X @ Y )
= Q )
=> ( ( F @ X @ Y )
= ( G @ X @ Y ) ) )
=> ( ( P = Q )
=> ( ( product_case_prod @ A @ B @ C @ F @ P )
= ( product_case_prod @ A @ B @ C @ G @ Q ) ) ) ) ).
% split_cong
thf(fact_119_update__swap__neqIndex,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,I2: nat,A3: array @ A,V: A,V2: A,H: heap_ext @ product_unit] :
( ( I != I2 )
=> ( ( update @ A @ A3 @ I @ V @ ( update @ A @ A3 @ I2 @ V2 @ H ) )
= ( update @ A @ A3 @ I2 @ V2 @ ( update @ A @ A3 @ I @ V @ H ) ) ) ) ) ).
% update_swap_neqIndex
thf(fact_120_csquare__fstOp__sndOp,axiom,
! [A: $tType,B: $tType,C: $tType,F: ( A > B > $o ) > ( product_prod @ A @ B ) > $o,P2: A > C > $o,Q2: C > B > $o] : ( bNF_csquare @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) @ C @ ( product_prod @ C @ B ) @ ( collect @ ( product_prod @ A @ B ) @ ( F @ ( relcompp @ A @ C @ B @ P2 @ Q2 ) ) ) @ ( product_snd @ A @ C ) @ ( product_fst @ C @ B ) @ ( bNF_fstOp @ A @ C @ B @ P2 @ Q2 ) @ ( bNF_sndOp @ A @ C @ B @ P2 @ Q2 ) ) ).
% csquare_fstOp_sndOp
thf(fact_121_effect__tapI,axiom,
! [A: $tType,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,R: A,F: ( heap_ext @ product_unit ) > A] :
( ( H2 = H )
=> ( ( R
= ( F @ H ) )
=> ( heap_effect @ A @ ( heap_tap @ A @ F ) @ H @ H2 @ R ) ) ) ).
% effect_tapI
thf(fact_122_effect__tapE,axiom,
! [A: $tType,F: ( heap_ext @ product_unit ) > A,H: heap_ext @ product_unit,H2: heap_ext @ product_unit,R: A] :
( ( heap_effect @ A @ ( heap_tap @ A @ F ) @ H @ H2 @ R )
=> ~ ( ( H2 = H )
=> ( R
!= ( F @ H ) ) ) ) ).
% effect_tapE
thf(fact_123_scomp__apply,axiom,
! [A: $tType,D: $tType,C: $tType,B: $tType] :
( ( product_scomp @ B @ C @ D @ A )
= ( ^ [F4: B > ( product_prod @ C @ D ),G4: C > D > A,X4: B] : ( product_case_prod @ C @ D @ A @ G4 @ ( F4 @ X4 ) ) ) ) ).
% scomp_apply
thf(fact_124_pair__in__swap__image,axiom,
! [A: $tType,B: $tType,Y3: A,X2: B,A6: set @ ( product_prod @ B @ A )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y3 @ X2 ) @ ( image2 @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ A6 ) )
= ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X2 @ Y3 ) @ A6 ) ) ).
% pair_in_swap_image
thf(fact_125_raise__bind,axiom,
! [B: $tType,A: $tType,E3: literal,F: B > ( heap_Heap @ A )] :
( ( heap_bind @ B @ A @ ( heap_raise @ B @ E3 ) @ F )
= ( heap_raise @ A @ E3 ) ) ).
% raise_bind
thf(fact_126_snd__sndOp,axiom,
! [B: $tType,A: $tType,C: $tType,P2: B > C > $o,Q2: C > A > $o] :
( ( product_snd @ B @ A )
= ( comp @ ( product_prod @ C @ A ) @ A @ ( product_prod @ B @ A ) @ ( product_snd @ C @ A ) @ ( bNF_sndOp @ B @ C @ A @ P2 @ Q2 ) ) ) ).
% snd_sndOp
thf(fact_127_return__bind,axiom,
! [A: $tType,B: $tType,X2: B,F: B > ( heap_Heap @ A )] :
( ( heap_bind @ B @ A @ ( heap_return @ B @ X2 ) @ F )
= ( F @ X2 ) ) ).
% return_bind
thf(fact_128_bind__return,axiom,
! [A: $tType,F: heap_Heap @ A] :
( ( heap_bind @ A @ A @ F @ ( heap_return @ A ) )
= F ) ).
% bind_return
thf(fact_129_nchotomy__relcomppE,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType,F: B > A,R: C > A > $o,S2: A > D > $o,A3: C,C3: D] :
( ! [Y: A] :
? [X5: B] :
( Y
= ( F @ X5 ) )
=> ( ( relcompp @ C @ A @ D @ R @ S2 @ A3 @ C3 )
=> ~ ! [B4: B] :
( ( R @ A3 @ ( F @ B4 ) )
=> ~ ( S2 @ ( F @ B4 ) @ C3 ) ) ) ) ).
% nchotomy_relcomppE
thf(fact_130_comp__cong,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,E: $tType,F: B > A,G: C > B,X2: C,F5: D > A,G5: E > D,X3: E] :
( ( ( F @ ( G @ X2 ) )
= ( F5 @ ( G5 @ X3 ) ) )
=> ( ( comp @ B @ A @ C @ F @ G @ X2 )
= ( comp @ D @ A @ E @ F5 @ G5 @ X3 ) ) ) ).
% comp_cong
thf(fact_131_comp__apply__eq,axiom,
! [B: $tType,D: $tType,A: $tType,C: $tType,F: B > A,G: C > B,X2: C,H: D > A,K: C > D] :
( ( ( F @ ( G @ X2 ) )
= ( H @ ( K @ X2 ) ) )
=> ( ( comp @ B @ A @ C @ F @ G @ X2 )
= ( comp @ D @ A @ C @ H @ K @ X2 ) ) ) ).
% comp_apply_eq
thf(fact_132_pick__middlep,axiom,
! [B: $tType,A: $tType,C: $tType,P2: A > B > $o,Q2: B > C > $o,A3: A,C3: C] :
( ( relcompp @ A @ B @ C @ P2 @ Q2 @ A3 @ C3 )
=> ( ( P2 @ A3 @ ( bNF_pick_middlep @ A @ B @ C @ P2 @ Q2 @ A3 @ C3 ) )
& ( Q2 @ ( bNF_pick_middlep @ A @ B @ C @ P2 @ Q2 @ A3 @ C3 ) @ C3 ) ) ) ).
% pick_middlep
thf(fact_133_sndOp__in,axiom,
! [A: $tType,B: $tType,C: $tType,Ac2: product_prod @ A @ B,P2: A > C > $o,Q2: C > B > $o] :
( ( member @ ( product_prod @ A @ B ) @ Ac2 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( relcompp @ A @ C @ B @ P2 @ Q2 ) ) ) )
=> ( member @ ( product_prod @ C @ B ) @ ( bNF_sndOp @ A @ C @ B @ P2 @ Q2 @ Ac2 ) @ ( collect @ ( product_prod @ C @ B ) @ ( product_case_prod @ C @ B @ $o @ Q2 ) ) ) ) ).
% sndOp_in
thf(fact_134_fstOp__in,axiom,
! [B: $tType,C: $tType,A: $tType,Ac2: product_prod @ A @ B,P2: A > C > $o,Q2: C > B > $o] :
( ( member @ ( product_prod @ A @ B ) @ Ac2 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( relcompp @ A @ C @ B @ P2 @ Q2 ) ) ) )
=> ( member @ ( product_prod @ A @ C ) @ ( bNF_fstOp @ A @ C @ B @ P2 @ Q2 @ Ac2 ) @ ( collect @ ( product_prod @ A @ C ) @ ( product_case_prod @ A @ C @ $o @ P2 ) ) ) ) ).
% fstOp_in
thf(fact_135_effect__bindE,axiom,
! [B: $tType,A: $tType,F: heap_Heap @ B,G: B > ( heap_Heap @ A ),H: heap_ext @ product_unit,H3: heap_ext @ product_unit,R2: A] :
( ( heap_effect @ A @ ( heap_bind @ B @ A @ F @ G ) @ H @ H3 @ R2 )
=> ~ ! [H4: heap_ext @ product_unit,R4: B] :
( ( heap_effect @ B @ F @ H @ H4 @ R4 )
=> ~ ( heap_effect @ A @ ( G @ R4 ) @ H4 @ H3 @ R2 ) ) ) ).
% effect_bindE
thf(fact_136_effect__bindI,axiom,
! [A: $tType,B: $tType,F: heap_Heap @ A,H: heap_ext @ product_unit,H2: heap_ext @ product_unit,R: A,G: A > ( heap_Heap @ B ),H3: heap_ext @ product_unit,R2: B] :
( ( heap_effect @ A @ F @ H @ H2 @ R )
=> ( ( heap_effect @ B @ ( G @ R ) @ H2 @ H3 @ R2 )
=> ( heap_effect @ B @ ( heap_bind @ A @ B @ F @ G ) @ H @ H3 @ R2 ) ) ) ).
% effect_bindI
thf(fact_137_scomp__Pair,axiom,
! [C: $tType,B: $tType,A: $tType,X2: A > ( product_prod @ B @ C )] :
( ( product_scomp @ A @ B @ C @ ( product_prod @ B @ C ) @ X2 @ ( product_Pair @ B @ C ) )
= X2 ) ).
% scomp_Pair
thf(fact_138_Pair__scomp,axiom,
! [A: $tType,B: $tType,C: $tType,X2: C,F: C > A > B] :
( ( product_scomp @ A @ C @ A @ B @ ( product_Pair @ C @ A @ X2 ) @ F )
= ( F @ X2 ) ) ).
% Pair_scomp
thf(fact_139_csquare__def,axiom,
! [B: $tType,C: $tType,D: $tType,A: $tType] :
( ( bNF_csquare @ A @ B @ C @ D )
= ( ^ [A8: set @ A,F12: B > C,F22: D > C,P1: A > B,P22: A > D] :
! [X4: A] :
( ( member @ A @ X4 @ A8 )
=> ( ( F12 @ ( P1 @ X4 ) )
= ( F22 @ ( P22 @ X4 ) ) ) ) ) ) ).
% csquare_def
thf(fact_140_scomp__def,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType] :
( ( product_scomp @ A @ B @ C @ D )
= ( ^ [F4: A > ( product_prod @ B @ C ),G4: B > C > D,X4: A] : ( product_case_prod @ B @ C @ D @ G4 @ ( F4 @ X4 ) ) ) ) ).
% scomp_def
thf(fact_141_fst__fstOp,axiom,
! [A: $tType,B: $tType,C: $tType,P2: A > C > $o,Q2: C > B > $o] :
( ( product_fst @ A @ B )
= ( comp @ ( product_prod @ A @ C ) @ A @ ( product_prod @ A @ B ) @ ( product_fst @ A @ C ) @ ( bNF_fstOp @ A @ C @ B @ P2 @ Q2 ) ) ) ).
% fst_fstOp
thf(fact_142_comp__apply,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comp @ B @ A @ C )
= ( ^ [F4: B > A,G4: C > B,X4: C] : ( F4 @ ( G4 @ X4 ) ) ) ) ).
% comp_apply
thf(fact_143_image__eqI,axiom,
! [A: $tType,B: $tType,B2: A,F: B > A,X2: B,A6: set @ B] :
( ( B2
= ( F @ X2 ) )
=> ( ( member @ B @ X2 @ A6 )
=> ( member @ A @ B2 @ ( image2 @ B @ A @ F @ A6 ) ) ) ) ).
% image_eqI
thf(fact_144_Inf_OINF__image,axiom,
! [B: $tType,A: $tType,C: $tType,Inf: ( set @ A ) > A,G: B > A,F: C > B,A6: set @ C] :
( ( Inf @ ( image2 @ B @ A @ G @ ( image2 @ C @ B @ F @ A6 ) ) )
= ( Inf @ ( image2 @ C @ A @ ( comp @ B @ A @ C @ G @ F ) @ A6 ) ) ) ).
% Inf.INF_image
thf(fact_145_Sup_OSUP__image,axiom,
! [B: $tType,A: $tType,C: $tType,Sup: ( set @ A ) > A,G: B > A,F: C > B,A6: set @ C] :
( ( Sup @ ( image2 @ B @ A @ G @ ( image2 @ C @ B @ F @ A6 ) ) )
= ( Sup @ ( image2 @ C @ A @ ( comp @ B @ A @ C @ G @ F ) @ A6 ) ) ) ).
% Sup.SUP_image
thf(fact_146_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_147_image__eq__imp__comp,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F: B > A,A6: set @ B,G: C > A,B6: set @ C,H: A > D] :
( ( ( image2 @ B @ A @ F @ A6 )
= ( image2 @ C @ A @ G @ B6 ) )
=> ( ( image2 @ B @ D @ ( comp @ A @ D @ B @ H @ F ) @ A6 )
= ( image2 @ C @ D @ ( comp @ A @ D @ C @ H @ G ) @ B6 ) ) ) ).
% image_eq_imp_comp
thf(fact_148_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_149_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_150_Collect__case__prod__Grp__eqD,axiom,
! [B: $tType,A: $tType,Z3: product_prod @ A @ B,A6: set @ A,F: A > B] :
( ( member @ ( product_prod @ A @ B ) @ Z3 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( bNF_Grp @ A @ B @ A6 @ F ) ) ) )
=> ( ( comp @ A @ B @ ( product_prod @ A @ B ) @ F @ ( product_fst @ A @ B ) @ Z3 )
= ( product_snd @ A @ B @ Z3 ) ) ) ).
% Collect_case_prod_Grp_eqD
thf(fact_151_Collect__split__mono__strong,axiom,
! [B: $tType,A: $tType,X6: set @ A,A6: set @ ( product_prod @ A @ B ),Y4: set @ B,P2: A > B > $o,Q2: A > B > $o] :
( ( X6
= ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A6 ) )
=> ( ( Y4
= ( image2 @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A6 ) )
=> ( ! [X: A] :
( ( member @ A @ X @ X6 )
=> ! [Xa: B] :
( ( member @ B @ Xa @ Y4 )
=> ( ( P2 @ X @ Xa )
=> ( Q2 @ X @ Xa ) ) ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A6 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P2 ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A6 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ Q2 ) ) ) ) ) ) ) ).
% Collect_split_mono_strong
thf(fact_152_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_153_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_154_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_155_fst__map__prod,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F: C > A,G: D > B,X2: product_prod @ C @ D] :
( ( product_fst @ A @ B @ ( product_map_prod @ C @ A @ D @ B @ F @ G @ X2 ) )
= ( F @ ( product_fst @ C @ D @ X2 ) ) ) ).
% fst_map_prod
thf(fact_156_snd__map__prod,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F: C > B,G: D > A,X2: product_prod @ C @ D] :
( ( product_snd @ B @ A @ ( product_map_prod @ C @ B @ D @ A @ F @ G @ X2 ) )
= ( G @ ( product_snd @ C @ D @ X2 ) ) ) ).
% snd_map_prod
thf(fact_157_apfst__conv,axiom,
! [C: $tType,A: $tType,B: $tType,F: C > A,X2: C,Y3: B] :
( ( product_apfst @ C @ A @ B @ F @ ( product_Pair @ C @ B @ X2 @ Y3 ) )
= ( product_Pair @ A @ B @ ( F @ X2 ) @ Y3 ) ) ).
% apfst_conv
thf(fact_158_apsnd__conv,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > B,X2: A,Y3: C] :
( ( product_apsnd @ C @ B @ A @ F @ ( product_Pair @ A @ C @ X2 @ Y3 ) )
= ( product_Pair @ A @ B @ X2 @ ( F @ Y3 ) ) ) ).
% apsnd_conv
thf(fact_159_fst__apfst,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > A,X2: product_prod @ C @ B] :
( ( product_fst @ A @ B @ ( product_apfst @ C @ A @ B @ F @ X2 ) )
= ( F @ ( product_fst @ C @ B @ X2 ) ) ) ).
% fst_apfst
thf(fact_160_apfst__eq__conv,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > A,X2: product_prod @ C @ B,G: C > A] :
( ( ( product_apfst @ C @ A @ B @ F @ X2 )
= ( product_apfst @ C @ A @ B @ G @ X2 ) )
= ( ( F @ ( product_fst @ C @ B @ X2 ) )
= ( G @ ( product_fst @ C @ B @ X2 ) ) ) ) ).
% apfst_eq_conv
thf(fact_161_snd__apfst,axiom,
! [B: $tType,A: $tType,C: $tType,F: C > B,X2: product_prod @ C @ A] :
( ( product_snd @ B @ A @ ( product_apfst @ C @ B @ A @ F @ X2 ) )
= ( product_snd @ C @ A @ X2 ) ) ).
% snd_apfst
thf(fact_162_fst__apsnd,axiom,
! [B: $tType,C: $tType,A: $tType,F: C > B,X2: product_prod @ A @ C] :
( ( product_fst @ A @ B @ ( product_apsnd @ C @ B @ A @ F @ X2 ) )
= ( product_fst @ A @ C @ X2 ) ) ).
% fst_apsnd
thf(fact_163_snd__apsnd,axiom,
! [A: $tType,C: $tType,B: $tType,F: C > A,X2: product_prod @ B @ C] :
( ( product_snd @ B @ A @ ( product_apsnd @ C @ A @ B @ F @ X2 ) )
= ( F @ ( product_snd @ B @ C @ X2 ) ) ) ).
% snd_apsnd
thf(fact_164_apsnd__eq__conv,axiom,
! [B: $tType,C: $tType,A: $tType,F: C > B,X2: product_prod @ A @ C,G: C > B] :
( ( ( product_apsnd @ C @ B @ A @ F @ X2 )
= ( product_apsnd @ C @ B @ A @ G @ X2 ) )
= ( ( F @ ( product_snd @ A @ C @ X2 ) )
= ( G @ ( product_snd @ A @ C @ X2 ) ) ) ) ).
% apsnd_eq_conv
thf(fact_165_map__prod__imageI,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,A3: A,B2: B,R3: set @ ( product_prod @ A @ B ),F: A > C,G: B > D] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R3 )
=> ( 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 ) @ R3 ) ) ) ).
% map_prod_imageI
thf(fact_166_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_167_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_168_apfst__apsnd,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,F: C > A,G: D > B,X2: product_prod @ C @ D] :
( ( product_apfst @ C @ A @ B @ F @ ( product_apsnd @ D @ B @ C @ G @ X2 ) )
= ( product_Pair @ A @ B @ ( F @ ( product_fst @ C @ D @ X2 ) ) @ ( G @ ( product_snd @ C @ D @ X2 ) ) ) ) ).
% apfst_apsnd
thf(fact_169_apsnd__apfst,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,F: C > B,G: D > A,X2: product_prod @ D @ C] :
( ( product_apsnd @ C @ B @ A @ F @ ( product_apfst @ D @ A @ C @ G @ X2 ) )
= ( product_Pair @ A @ B @ ( G @ ( product_fst @ D @ C @ X2 ) ) @ ( F @ ( product_snd @ D @ C @ X2 ) ) ) ) ).
% apsnd_apfst
thf(fact_170_GrpE,axiom,
! [B: $tType,A: $tType,A6: set @ A,F: A > B,X2: A,Y3: B] :
( ( bNF_Grp @ A @ B @ A6 @ F @ X2 @ Y3 )
=> ~ ( ( ( F @ X2 )
= Y3 )
=> ~ ( member @ A @ X2 @ A6 ) ) ) ).
% GrpE
thf(fact_171_GrpI,axiom,
! [B: $tType,A: $tType,F: B > A,X2: B,Y3: A,A6: set @ B] :
( ( ( F @ X2 )
= Y3 )
=> ( ( member @ B @ X2 @ A6 )
=> ( bNF_Grp @ B @ A @ A6 @ F @ X2 @ Y3 ) ) ) ).
% GrpI
thf(fact_172_apsnd__apfst__commute,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,F: C > B,G: D > A,P: product_prod @ D @ C] :
( ( product_apsnd @ C @ B @ A @ F @ ( product_apfst @ D @ A @ C @ G @ P ) )
= ( product_apfst @ D @ A @ B @ G @ ( product_apsnd @ C @ B @ D @ F @ P ) ) ) ).
% apsnd_apfst_commute
thf(fact_173_subset__image__iff,axiom,
! [A: $tType,B: $tType,B6: set @ A,F: B > A,A6: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B6 @ ( image2 @ B @ A @ F @ A6 ) )
= ( ? [AA: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ AA @ A6 )
& ( B6
= ( image2 @ B @ A @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_174_image__subset__iff,axiom,
! [A: $tType,B: $tType,F: B > A,A6: set @ B,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F @ A6 ) @ B6 )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A6 )
=> ( member @ A @ ( F @ X4 ) @ B6 ) ) ) ) ).
% image_subset_iff
thf(fact_175_subset__imageE,axiom,
! [A: $tType,B: $tType,B6: set @ A,F: B > A,A6: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B6 @ ( image2 @ B @ A @ F @ A6 ) )
=> ~ ! [C4: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ C4 @ A6 )
=> ( B6
!= ( image2 @ B @ A @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_176_image__subsetI,axiom,
! [A: $tType,B: $tType,A6: set @ A,F: A > B,B6: set @ B] :
( ! [X: A] :
( ( member @ A @ X @ A6 )
=> ( member @ B @ ( F @ X ) @ B6 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F @ A6 ) @ B6 ) ) ).
% image_subsetI
thf(fact_177_image__mono,axiom,
! [B: $tType,A: $tType,A6: set @ A,B6: set @ A,F: A > B] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F @ A6 ) @ ( image2 @ A @ B @ F @ B6 ) ) ) ).
% image_mono
thf(fact_178_map__prod_Ocomp,axiom,
! [A: $tType,C: $tType,E: $tType,F2: $tType,D: $tType,B: $tType,F: C > E,G: D > F2,H: A > C,I: B > D] :
( ( comp @ ( product_prod @ C @ D ) @ ( product_prod @ E @ F2 ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ E @ D @ F2 @ F @ G ) @ ( product_map_prod @ A @ C @ B @ D @ H @ I ) )
= ( product_map_prod @ A @ E @ B @ F2 @ ( comp @ C @ E @ A @ F @ H ) @ ( comp @ D @ F2 @ B @ G @ I ) ) ) ).
% map_prod.comp
thf(fact_179_map__prod_Ocompositionality,axiom,
! [D: $tType,F2: $tType,E: $tType,C: $tType,B: $tType,A: $tType,F: C > E,G: D > F2,H: A > C,I: B > D,Prod: product_prod @ A @ B] :
( ( product_map_prod @ C @ E @ D @ F2 @ F @ G @ ( product_map_prod @ A @ C @ B @ D @ H @ I @ Prod ) )
= ( product_map_prod @ A @ E @ B @ F2 @ ( comp @ C @ E @ A @ F @ H ) @ ( comp @ D @ F2 @ B @ G @ I ) @ Prod ) ) ).
% map_prod.compositionality
thf(fact_180_map__prod__compose,axiom,
! [D: $tType,C: $tType,A: $tType,E: $tType,F2: $tType,B: $tType,F1: E > C,F23: A > E,G1: F2 > D,G22: B > F2] :
( ( product_map_prod @ A @ C @ B @ D @ ( comp @ E @ C @ A @ F1 @ F23 ) @ ( comp @ F2 @ D @ B @ G1 @ G22 ) )
= ( comp @ ( product_prod @ E @ F2 ) @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ E @ C @ F2 @ D @ F1 @ G1 ) @ ( product_map_prod @ A @ E @ B @ F2 @ F23 @ G22 ) ) ) ).
% map_prod_compose
thf(fact_181_apfst__compose,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F: C > A,G: D > C,X2: product_prod @ D @ B] :
( ( product_apfst @ C @ A @ B @ F @ ( product_apfst @ D @ C @ B @ G @ X2 ) )
= ( product_apfst @ D @ A @ B @ ( comp @ C @ A @ D @ F @ G ) @ X2 ) ) ).
% apfst_compose
thf(fact_182_apsnd__compose,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,F: C > B,G: D > C,X2: product_prod @ A @ D] :
( ( product_apsnd @ C @ B @ A @ F @ ( product_apsnd @ D @ C @ A @ G @ X2 ) )
= ( product_apsnd @ D @ B @ A @ ( comp @ C @ B @ D @ F @ G ) @ X2 ) ) ).
% apsnd_compose
thf(fact_183_Collect__case__prod__Grp__in,axiom,
! [B: $tType,A: $tType,Z3: product_prod @ A @ B,A6: set @ A,F: A > B] :
( ( member @ ( product_prod @ A @ B ) @ Z3 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( bNF_Grp @ A @ B @ A6 @ F ) ) ) )
=> ( member @ A @ ( product_fst @ A @ B @ Z3 ) @ A6 ) ) ).
% Collect_case_prod_Grp_in
thf(fact_184_prod__fun__imageE,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,C3: product_prod @ A @ B,F: C > A,G: D > B,R3: set @ ( product_prod @ C @ D )] :
( ( member @ ( product_prod @ A @ B ) @ C3 @ ( image2 @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ A @ D @ B @ F @ G ) @ R3 ) )
=> ~ ! [X: C,Y: D] :
( ( C3
= ( product_Pair @ A @ B @ ( F @ X ) @ ( G @ Y ) ) )
=> ~ ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ X @ Y ) @ R3 ) ) ) ).
% prod_fun_imageE
thf(fact_185_lift__def,axiom,
! [A: $tType,B: $tType] :
( ( heap_lift @ A @ B )
= ( comp @ B @ ( heap_Heap @ B ) @ A @ ( heap_return @ B ) ) ) ).
% lift_def
thf(fact_186_rev__image__eqI,axiom,
! [B: $tType,A: $tType,X2: A,A6: set @ A,B2: B,F: A > B] :
( ( member @ A @ X2 @ A6 )
=> ( ( B2
= ( F @ X2 ) )
=> ( member @ B @ B2 @ ( image2 @ A @ B @ F @ A6 ) ) ) ) ).
% rev_image_eqI
thf(fact_187_ball__imageD,axiom,
! [A: $tType,B: $tType,F: B > A,A6: set @ B,P2: A > $o] :
( ! [X: A] :
( ( member @ A @ X @ ( image2 @ B @ A @ F @ A6 ) )
=> ( P2 @ X ) )
=> ! [X5: B] :
( ( member @ B @ X5 @ A6 )
=> ( P2 @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_188_image__cong,axiom,
! [B: $tType,A: $tType,M: set @ A,N: set @ A,F: A > B,G: A > B] :
( ( M = N )
=> ( ! [X: A] :
( ( member @ A @ X @ N )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( image2 @ A @ B @ F @ M )
= ( image2 @ A @ B @ G @ N ) ) ) ) ).
% image_cong
thf(fact_189_bex__imageD,axiom,
! [A: $tType,B: $tType,F: B > A,A6: set @ B,P2: A > $o] :
( ? [X5: A] :
( ( member @ A @ X5 @ ( image2 @ B @ A @ F @ A6 ) )
& ( P2 @ X5 ) )
=> ? [X: B] :
( ( member @ B @ X @ A6 )
& ( P2 @ ( F @ X ) ) ) ) ).
% bex_imageD
thf(fact_190_image__iff,axiom,
! [A: $tType,B: $tType,Z3: A,F: B > A,A6: set @ B] :
( ( member @ A @ Z3 @ ( image2 @ B @ A @ F @ A6 ) )
= ( ? [X4: B] :
( ( member @ B @ X4 @ A6 )
& ( Z3
= ( F @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_191_imageI,axiom,
! [B: $tType,A: $tType,X2: A,A6: set @ A,F: A > B] :
( ( member @ A @ X2 @ A6 )
=> ( member @ B @ ( F @ X2 ) @ ( image2 @ A @ B @ F @ A6 ) ) ) ).
% imageI
thf(fact_192_Sup_OSUP__cong,axiom,
! [A: $tType,B: $tType,A6: set @ B,B6: set @ B,C5: B > A,D3: B > A,Sup: ( set @ A ) > A] :
( ( A6 = B6 )
=> ( ! [X: B] :
( ( member @ B @ X @ B6 )
=> ( ( C5 @ X )
= ( D3 @ X ) ) )
=> ( ( Sup @ ( image2 @ B @ A @ C5 @ A6 ) )
= ( Sup @ ( image2 @ B @ A @ D3 @ B6 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_193_Inf_OINF__cong,axiom,
! [A: $tType,B: $tType,A6: set @ B,B6: set @ B,C5: B > A,D3: B > A,Inf: ( set @ A ) > A] :
( ( A6 = B6 )
=> ( ! [X: B] :
( ( member @ B @ X @ B6 )
=> ( ( C5 @ X )
= ( D3 @ X ) ) )
=> ( ( Inf @ ( image2 @ B @ A @ C5 @ A6 ) )
= ( Inf @ ( image2 @ B @ A @ D3 @ B6 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_194_comp__eq__dest__lhs,axiom,
! [C: $tType,B: $tType,A: $tType,A3: C > B,B2: A > C,C3: A > B,V: A] :
( ( ( comp @ C @ B @ A @ A3 @ B2 )
= C3 )
=> ( ( A3 @ ( B2 @ V ) )
= ( C3 @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_195_comp__eq__elim,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,A3: C > B,B2: A > C,C3: D > B,D4: A > D] :
( ( ( comp @ C @ B @ A @ A3 @ B2 )
= ( comp @ D @ B @ A @ C3 @ D4 ) )
=> ! [V4: A] :
( ( A3 @ ( B2 @ V4 ) )
= ( C3 @ ( D4 @ V4 ) ) ) ) ).
% comp_eq_elim
thf(fact_196_comp__eq__dest,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,A3: C > B,B2: A > C,C3: D > B,D4: A > D,V: A] :
( ( ( comp @ C @ B @ A @ A3 @ B2 )
= ( comp @ D @ B @ A @ C3 @ D4 ) )
=> ( ( A3 @ ( B2 @ V ) )
= ( C3 @ ( D4 @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_197_comp__assoc,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType,F: D > B,G: C > D,H: A > C] :
( ( comp @ C @ B @ A @ ( comp @ D @ B @ C @ F @ G ) @ H )
= ( comp @ D @ B @ A @ F @ ( comp @ C @ D @ A @ G @ H ) ) ) ).
% comp_assoc
thf(fact_198_comp__def,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comp @ B @ C @ A )
= ( ^ [F4: B > C,G4: A > B,X4: A] : ( F4 @ ( G4 @ X4 ) ) ) ) ).
% comp_def
thf(fact_199_apfst__convE,axiom,
! [C: $tType,A: $tType,B: $tType,Q: product_prod @ A @ B,F: C > A,P: product_prod @ C @ B] :
( ( Q
= ( product_apfst @ C @ A @ B @ F @ P ) )
=> ~ ! [X: C,Y: B] :
( ( P
= ( product_Pair @ C @ B @ X @ Y ) )
=> ( Q
!= ( product_Pair @ A @ B @ ( F @ X ) @ Y ) ) ) ) ).
% apfst_convE
thf(fact_200_prod_Omap__comp,axiom,
! [D: $tType,F2: $tType,E: $tType,C: $tType,B: $tType,A: $tType,G1: C > E,G22: D > F2,F1: A > C,F23: B > D,V: product_prod @ A @ B] :
( ( product_map_prod @ C @ E @ D @ F2 @ G1 @ G22 @ ( product_map_prod @ A @ C @ B @ D @ F1 @ F23 @ V ) )
= ( product_map_prod @ A @ E @ B @ F2 @ ( comp @ C @ E @ A @ G1 @ F1 ) @ ( comp @ D @ F2 @ B @ G22 @ F23 ) @ V ) ) ).
% prod.map_comp
thf(fact_201_subrelI,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
( ! [X: A,Y: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ R )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ S2 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S2 ) ) ).
% subrelI
thf(fact_202_all__subset__image,axiom,
! [A: $tType,B: $tType,F: B > A,A6: set @ B,P2: ( set @ A ) > $o] :
( ( ! [B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B7 @ ( image2 @ B @ A @ F @ A6 ) )
=> ( P2 @ B7 ) ) )
= ( ! [B7: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ B7 @ A6 )
=> ( P2 @ ( image2 @ B @ A @ F @ B7 ) ) ) ) ) ).
% all_subset_image
thf(fact_203_rel__fun__Collect__case__prodD,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,A6: A > B > $o,B6: C > D > $o,F: A > C,G: B > D,X6: set @ ( product_prod @ A @ B ),X2: product_prod @ A @ B] :
( ( bNF_rel_fun @ A @ B @ C @ D @ A6 @ B6 @ F @ G )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ X6 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A6 ) ) )
=> ( ( member @ ( product_prod @ A @ B ) @ X2 @ X6 )
=> ( B6 @ ( comp @ A @ C @ ( product_prod @ A @ B ) @ F @ ( product_fst @ A @ B ) @ X2 ) @ ( comp @ B @ D @ ( product_prod @ A @ B ) @ G @ ( product_snd @ A @ B ) @ X2 ) ) ) ) ) ).
% rel_fun_Collect_case_prodD
thf(fact_204_convol__image__vimage2p,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,F: C > A,G: D > B,R3: A > B > $o] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( bNF_convol @ ( product_prod @ C @ D ) @ A @ B @ ( comp @ C @ A @ ( product_prod @ C @ D ) @ F @ ( product_fst @ C @ D ) ) @ ( comp @ D @ B @ ( product_prod @ C @ D ) @ G @ ( product_snd @ C @ D ) ) ) @ ( collect @ ( product_prod @ C @ D ) @ ( product_case_prod @ C @ D @ $o @ ( bNF_vimage2p @ C @ A @ D @ B @ $o @ F @ G @ R3 ) ) ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R3 ) ) ) ).
% convol_image_vimage2p
thf(fact_205_prod_Osize__gen__o__map,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F: C > nat,Fa: D > nat,G: A > C,Ga: B > D] :
( ( comp @ ( product_prod @ C @ D ) @ nat @ ( product_prod @ A @ B ) @ ( basic_BNF_size_prod @ C @ D @ F @ Fa ) @ ( product_map_prod @ A @ C @ B @ D @ G @ Ga ) )
= ( basic_BNF_size_prod @ A @ B @ ( comp @ C @ nat @ A @ F @ G ) @ ( comp @ D @ nat @ B @ Fa @ Ga ) ) ) ).
% prod.size_gen_o_map
thf(fact_206_relChain__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( bNF_Ca3754400796208372196lChain @ A @ B )
= ( ^ [R5: set @ ( product_prod @ A @ A ),As: A > B] :
! [I3: A,J: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I3 @ J ) @ R5 )
=> ( ord_less_eq @ B @ ( As @ I3 ) @ ( As @ J ) ) ) ) ) ) ).
% relChain_def
thf(fact_207_Collect__case__prod__in__rel__leI,axiom,
! [B: $tType,A: $tType,X6: set @ ( product_prod @ A @ B ),Y4: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ X6 @ Y4 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ X6 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( fun_in_rel @ A @ B @ Y4 ) ) ) ) ) ).
% Collect_case_prod_in_rel_leI
thf(fact_208_Collect__case__prod__in__rel__leE,axiom,
! [B: $tType,A: $tType,X6: set @ ( product_prod @ A @ B ),Y4: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ X6 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( fun_in_rel @ A @ B @ Y4 ) ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ X6 @ Y4 ) ) ).
% Collect_case_prod_in_rel_leE
thf(fact_209_relcompp__assoc,axiom,
! [A: $tType,D: $tType,B: $tType,C: $tType,R: A > D > $o,S2: D > C > $o,T3: C > B > $o] :
( ( relcompp @ A @ C @ B @ ( relcompp @ A @ D @ C @ R @ S2 ) @ T3 )
= ( relcompp @ A @ D @ B @ R @ ( relcompp @ D @ C @ B @ S2 @ T3 ) ) ) ).
% relcompp_assoc
thf(fact_210_relcompp__apply,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( relcompp @ A @ B @ C )
= ( ^ [R6: A > B > $o,S4: B > C > $o,A7: A,C6: C] :
? [B5: B] :
( ( R6 @ A7 @ B5 )
& ( S4 @ B5 @ C6 ) ) ) ) ).
% relcompp_apply
thf(fact_211_rel__funI,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A6: A > B > $o,B6: C > D > $o,F: A > C,G: B > D] :
( ! [X: A,Y: B] :
( ( A6 @ X @ Y )
=> ( B6 @ ( F @ X ) @ ( G @ Y ) ) )
=> ( bNF_rel_fun @ A @ B @ C @ D @ A6 @ B6 @ F @ G ) ) ).
% rel_funI
thf(fact_212_fun_Omap__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,G2: $tType,F2: $tType,Rb: A > F2 > $o,Sd: B > G2 > $o] :
( bNF_rel_fun @ ( A > B ) @ ( F2 > G2 ) @ ( ( D > A ) > D > B ) @ ( ( D > F2 ) > D > G2 ) @ ( bNF_rel_fun @ A @ F2 @ B @ G2 @ Rb @ Sd )
@ ( bNF_rel_fun @ ( D > A ) @ ( D > F2 ) @ ( D > B ) @ ( D > G2 )
@ ( bNF_rel_fun @ D @ D @ A @ F2
@ ^ [Y2: D,Z2: D] : Y2 = Z2
@ Rb )
@ ( bNF_rel_fun @ D @ D @ B @ G2
@ ^ [Y2: D,Z2: D] : Y2 = Z2
@ Sd ) )
@ ( comp @ A @ B @ D )
@ ( comp @ F2 @ G2 @ D ) ) ).
% fun.map_transfer
thf(fact_213_rel__fun__iff__leq__vimage2p,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType] :
( ( bNF_rel_fun @ A @ B @ C @ D )
= ( ^ [R6: A > B > $o,S4: C > D > $o,F4: A > C,G4: B > D] : ( ord_less_eq @ ( A > B > $o ) @ R6 @ ( bNF_vimage2p @ A @ C @ B @ D @ $o @ F4 @ G4 @ S4 ) ) ) ) ).
% rel_fun_iff_leq_vimage2p
thf(fact_214_predicate2D__vimage2p,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,R3: A > B > $o,F: A > C,G: B > D,S5: C > D > $o,X2: A,Y3: B] :
( ( ord_less_eq @ ( A > B > $o ) @ R3 @ ( bNF_vimage2p @ A @ C @ B @ D @ $o @ F @ G @ S5 ) )
=> ( ( R3 @ X2 @ Y3 )
=> ( S5 @ ( F @ X2 ) @ ( G @ Y3 ) ) ) ) ).
% predicate2D_vimage2p
thf(fact_215_rel__fun__mono_H,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,Y4: A > B > $o,X6: A > B > $o,A6: C > D > $o,B6: C > D > $o,F: A > C,G: B > D] :
( ! [X: A,Y: B] :
( ( Y4 @ X @ Y )
=> ( X6 @ X @ Y ) )
=> ( ! [X: C,Y: D] :
( ( A6 @ X @ Y )
=> ( B6 @ X @ Y ) )
=> ( ( bNF_rel_fun @ A @ B @ C @ D @ X6 @ A6 @ F @ G )
=> ( bNF_rel_fun @ A @ B @ C @ D @ Y4 @ B6 @ F @ G ) ) ) ) ).
% rel_fun_mono'
thf(fact_216_rel__fun__mono,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,X6: A > B > $o,A6: C > D > $o,F: A > C,G: B > D,Y4: A > B > $o,B6: C > D > $o] :
( ( bNF_rel_fun @ A @ B @ C @ D @ X6 @ A6 @ F @ G )
=> ( ! [X: A,Y: B] :
( ( Y4 @ X @ Y )
=> ( X6 @ X @ Y ) )
=> ( ! [X: C,Y: D] :
( ( A6 @ X @ Y )
=> ( B6 @ X @ Y ) )
=> ( bNF_rel_fun @ A @ B @ C @ D @ Y4 @ B6 @ F @ G ) ) ) ) ).
% rel_fun_mono
thf(fact_217_vimage2pI,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType,R3: A > B > $o,F: C > A,X2: C,G: D > B,Y3: D] :
( ( R3 @ ( F @ X2 ) @ ( G @ Y3 ) )
=> ( bNF_vimage2p @ C @ A @ D @ B @ $o @ F @ G @ R3 @ X2 @ Y3 ) ) ).
% vimage2pI
thf(fact_218_rel__funD,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A6: A > B > $o,B6: C > D > $o,F: A > C,G: B > D,X2: A,Y3: B] :
( ( bNF_rel_fun @ A @ B @ C @ D @ A6 @ B6 @ F @ G )
=> ( ( A6 @ X2 @ Y3 )
=> ( B6 @ ( F @ X2 ) @ ( G @ Y3 ) ) ) ) ).
% rel_funD
thf(fact_219_fun_Orel__compp,axiom,
! [A: $tType,D: $tType,C: $tType,B: $tType,R3: A > B > $o,S5: B > C > $o] :
( ( bNF_rel_fun @ D @ D @ A @ C
@ ^ [Y2: D,Z2: D] : Y2 = Z2
@ ( relcompp @ A @ B @ C @ R3 @ S5 ) )
= ( relcompp @ ( D > A ) @ ( D > B ) @ ( D > C )
@ ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y2: D,Z2: D] : Y2 = Z2
@ R3 )
@ ( bNF_rel_fun @ D @ D @ B @ C
@ ^ [Y2: D,Z2: D] : Y2 = Z2
@ S5 ) ) ) ).
% fun.rel_compp
thf(fact_220_relcompp__mono,axiom,
! [A: $tType,C: $tType,B: $tType,R2: A > B > $o,R: A > B > $o,S3: B > C > $o,S2: B > C > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ R2 @ R )
=> ( ( ord_less_eq @ ( B > C > $o ) @ S3 @ S2 )
=> ( ord_less_eq @ ( A > C > $o ) @ ( relcompp @ A @ B @ C @ R2 @ S3 ) @ ( relcompp @ A @ B @ C @ R @ S2 ) ) ) ) ).
% relcompp_mono
thf(fact_221_leq__OOI,axiom,
! [A: $tType,R3: A > A > $o] :
( ( R3
= ( ^ [Y2: A,Z2: A] : Y2 = Z2 ) )
=> ( ord_less_eq @ ( A > A > $o ) @ R3 @ ( relcompp @ A @ A @ A @ R3 @ R3 ) ) ) ).
% leq_OOI
thf(fact_222_convol__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( bNF_convol @ A @ B @ C )
= ( ^ [F4: A > B,G4: A > C,A7: A] : ( product_Pair @ B @ C @ ( F4 @ A7 ) @ ( G4 @ A7 ) ) ) ) ).
% convol_def
thf(fact_223_in__rel__def,axiom,
! [B: $tType,A: $tType] :
( ( fun_in_rel @ A @ B )
= ( ^ [R6: set @ ( product_prod @ A @ B ),X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ R6 ) ) ) ).
% in_rel_def
thf(fact_224_in__rel__Collect__case__prod__eq,axiom,
! [B: $tType,A: $tType,X6: A > B > $o] :
( ( fun_in_rel @ A @ B @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ X6 ) ) )
= X6 ) ).
% in_rel_Collect_case_prod_eq
thf(fact_225_Grp__mono,axiom,
! [B: $tType,A: $tType,A6: set @ A,B6: set @ A,F: A > B] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ( ord_less_eq @ ( A > B > $o ) @ ( bNF_Grp @ A @ B @ A6 @ F ) @ ( bNF_Grp @ A @ B @ B6 @ F ) ) ) ).
% Grp_mono
thf(fact_226_Collect__case__prod__mono,axiom,
! [B: $tType,A: $tType,A6: A > B > $o,B6: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ A6 @ B6 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A6 ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ B6 ) ) ) ) ).
% Collect_case_prod_mono
thf(fact_227_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_228_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_229_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_230_relcompp_Ocases,axiom,
! [A: $tType,B: $tType,C: $tType,R: A > B > $o,S2: B > C > $o,A1: A,A22: C] :
( ( relcompp @ A @ B @ C @ R @ S2 @ A1 @ A22 )
=> ~ ! [B4: B] :
( ( R @ A1 @ B4 )
=> ~ ( S2 @ B4 @ A22 ) ) ) ).
% relcompp.cases
thf(fact_231_relcompp_Osimps,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( relcompp @ A @ B @ C )
= ( ^ [R5: A > B > $o,S: B > C > $o,A12: A,A23: C] :
? [A7: A,B5: B,C6: C] :
( ( A12 = A7 )
& ( A23 = C6 )
& ( R5 @ A7 @ B5 )
& ( S @ B5 @ C6 ) ) ) ) ).
% relcompp.simps
thf(fact_232_OO__eq,axiom,
! [B: $tType,A: $tType,R3: A > B > $o] :
( ( relcompp @ A @ B @ B @ R3
@ ^ [Y2: B,Z2: B] : Y2 = Z2 )
= R3 ) ).
% OO_eq
thf(fact_233_eq__OO,axiom,
! [B: $tType,A: $tType,R3: A > B > $o] :
( ( relcompp @ A @ A @ B
@ ^ [Y2: A,Z2: A] : Y2 = Z2
@ R3 )
= R3 ) ).
% eq_OO
thf(fact_234_relcomppE,axiom,
! [A: $tType,B: $tType,C: $tType,R: A > B > $o,S2: B > C > $o,A3: A,C3: C] :
( ( relcompp @ A @ B @ C @ R @ S2 @ A3 @ C3 )
=> ~ ! [B4: B] :
( ( R @ A3 @ B4 )
=> ~ ( S2 @ B4 @ C3 ) ) ) ).
% relcomppE
thf(fact_235_relcomppI,axiom,
! [A: $tType,B: $tType,C: $tType,R: A > B > $o,A3: A,B2: B,S2: B > C > $o,C3: C] :
( ( R @ A3 @ B2 )
=> ( ( S2 @ B2 @ C3 )
=> ( relcompp @ A @ B @ C @ R @ S2 @ A3 @ C3 ) ) ) ).
% relcomppI
thf(fact_236_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_237_convol__expand__snd_H,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > ( product_prod @ B @ C ),G: A > B,H: A > C] :
( ( ( comp @ ( product_prod @ B @ C ) @ B @ A @ ( product_fst @ B @ C ) @ F )
= G )
=> ( ( H
= ( comp @ ( product_prod @ B @ C ) @ C @ A @ ( product_snd @ B @ C ) @ F ) )
= ( ( bNF_convol @ A @ B @ C @ G @ H )
= F ) ) ) ).
% convol_expand_snd'
thf(fact_238_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_239_convol__o,axiom,
! [B: $tType,C: $tType,D: $tType,A: $tType,F: D > B,G: D > C,H: A > D] :
( ( comp @ D @ ( product_prod @ B @ C ) @ A @ ( bNF_convol @ D @ B @ C @ F @ G ) @ H )
= ( bNF_convol @ A @ B @ C @ ( comp @ D @ B @ A @ F @ H ) @ ( comp @ D @ C @ A @ G @ H ) ) ) ).
% convol_o
thf(fact_240_vimage2p__relcompp__mono,axiom,
! [C: $tType,F2: $tType,E: $tType,D: $tType,B: $tType,A: $tType,R3: A > C > $o,S5: C > B > $o,T4: A > B > $o,F: D > A,G: F2 > C,H: E > B] :
( ( ord_less_eq @ ( A > B > $o ) @ ( relcompp @ A @ C @ B @ R3 @ S5 ) @ T4 )
=> ( ord_less_eq @ ( D > E > $o ) @ ( relcompp @ D @ F2 @ E @ ( bNF_vimage2p @ D @ A @ F2 @ C @ $o @ F @ G @ R3 ) @ ( bNF_vimage2p @ F2 @ C @ E @ B @ $o @ G @ H @ S5 ) ) @ ( bNF_vimage2p @ D @ A @ E @ B @ $o @ F @ H @ T4 ) ) ) ).
% vimage2p_relcompp_mono
thf(fact_241_snd__convol_H,axiom,
! [B: $tType,A: $tType,C: $tType,F: C > B,G: C > A,X2: C] :
( ( product_snd @ B @ A @ ( bNF_convol @ C @ B @ A @ F @ G @ X2 ) )
= ( G @ X2 ) ) ).
% snd_convol'
thf(fact_242_fst__convol_H,axiom,
! [B: $tType,A: $tType,C: $tType,F: C > A,G: C > B,X2: C] :
( ( product_fst @ A @ B @ ( bNF_convol @ C @ A @ B @ F @ G @ X2 ) )
= ( F @ X2 ) ) ).
% fst_convol'
thf(fact_243_vimage2p__comp,axiom,
! [E: $tType,D: $tType,F2: $tType,A: $tType,C: $tType,B: $tType,G2: $tType,F1: F2 > A,F23: D > F2,G1: G2 > B,G22: E > G2] :
( ( bNF_vimage2p @ D @ A @ E @ B @ C @ ( comp @ F2 @ A @ D @ F1 @ F23 ) @ ( comp @ G2 @ B @ E @ G1 @ G22 ) )
= ( comp @ ( F2 > G2 > C ) @ ( D > E > C ) @ ( A > B > C ) @ ( bNF_vimage2p @ D @ F2 @ E @ G2 @ C @ F23 @ G22 ) @ ( bNF_vimage2p @ F2 @ A @ G2 @ B @ C @ F1 @ G1 ) ) ) ).
% vimage2p_comp
thf(fact_244_pos__fun__distr,axiom,
! [E: $tType,C: $tType,A: $tType,B: $tType,D: $tType,F2: $tType,R3: A > E > $o,S5: B > F2 > $o,R7: E > C > $o,S6: F2 > D > $o] : ( ord_less_eq @ ( ( A > B ) > ( C > D ) > $o ) @ ( relcompp @ ( A > B ) @ ( E > F2 ) @ ( C > D ) @ ( bNF_rel_fun @ A @ E @ B @ F2 @ R3 @ S5 ) @ ( bNF_rel_fun @ E @ C @ F2 @ D @ R7 @ S6 ) ) @ ( bNF_rel_fun @ A @ C @ B @ D @ ( relcompp @ A @ E @ C @ R3 @ R7 ) @ ( relcompp @ B @ F2 @ D @ S5 @ S6 ) ) ) ).
% pos_fun_distr
thf(fact_245_o__rsp_I2_J,axiom,
! [E: $tType,F2: $tType,H5: $tType,G2: $tType,R1: E > F2 > $o] :
( bNF_rel_fun @ ( G2 > H5 ) @ ( G2 > H5 ) @ ( ( E > G2 ) > E > H5 ) @ ( ( F2 > G2 ) > F2 > H5 )
@ ^ [Y2: G2 > H5,Z2: G2 > H5] : Y2 = Z2
@ ( bNF_rel_fun @ ( E > G2 ) @ ( F2 > G2 ) @ ( E > H5 ) @ ( F2 > H5 )
@ ( bNF_rel_fun @ E @ F2 @ G2 @ G2 @ R1
@ ^ [Y2: G2,Z2: G2] : Y2 = Z2 )
@ ( bNF_rel_fun @ E @ F2 @ H5 @ H5 @ R1
@ ^ [Y2: H5,Z2: H5] : Y2 = Z2 ) )
@ ( comp @ G2 @ H5 @ E )
@ ( comp @ G2 @ H5 @ F2 ) ) ).
% o_rsp(2)
thf(fact_246_o__rsp_I1_J,axiom,
! [A: $tType,B: $tType,E: $tType,F2: $tType,D: $tType,C: $tType,R22: A > C > $o,R32: B > D > $o,R1: E > F2 > $o] : ( bNF_rel_fun @ ( A > B ) @ ( C > D ) @ ( ( E > A ) > E > B ) @ ( ( F2 > C ) > F2 > D ) @ ( bNF_rel_fun @ A @ C @ B @ D @ R22 @ R32 ) @ ( bNF_rel_fun @ ( E > A ) @ ( F2 > C ) @ ( E > B ) @ ( F2 > D ) @ ( bNF_rel_fun @ E @ F2 @ A @ C @ R1 @ R22 ) @ ( bNF_rel_fun @ E @ F2 @ B @ D @ R1 @ R32 ) ) @ ( comp @ A @ B @ E ) @ ( comp @ C @ D @ F2 ) ) ).
% o_rsp(1)
thf(fact_247_ge__eq__refl,axiom,
! [A: $tType,R3: A > A > $o,X2: A] :
( ( ord_less_eq @ ( A > A > $o )
@ ^ [Y2: A,Z2: A] : Y2 = Z2
@ R3 )
=> ( R3 @ X2 @ X2 ) ) ).
% ge_eq_refl
thf(fact_248_refl__ge__eq,axiom,
! [A: $tType,R3: A > A > $o] :
( ! [X: A] : ( R3 @ X @ X )
=> ( ord_less_eq @ ( A > A > $o )
@ ^ [Y2: A,Z2: A] : Y2 = Z2
@ R3 ) ) ).
% refl_ge_eq
thf(fact_249_rewriteR__comp__comp2,axiom,
! [C: $tType,B: $tType,E: $tType,D: $tType,A: $tType,G: C > B,H: A > C,R12: D > B,R23: A > D,F: B > E,L: D > E] :
( ( ( comp @ C @ B @ A @ G @ H )
= ( comp @ D @ B @ A @ R12 @ R23 ) )
=> ( ( ( comp @ B @ E @ D @ F @ R12 )
= L )
=> ( ( comp @ C @ E @ A @ ( comp @ B @ E @ C @ F @ G ) @ H )
= ( comp @ D @ E @ A @ L @ R23 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_250_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,H: E > A,R: E > D] :
( ( ( comp @ C @ B @ A @ F @ G )
= ( comp @ D @ B @ A @ L1 @ L2 ) )
=> ( ( ( comp @ A @ D @ E @ L2 @ H )
= R )
=> ( ( comp @ C @ B @ E @ F @ ( comp @ A @ C @ E @ G @ H ) )
= ( comp @ D @ B @ E @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_251_rewriteR__comp__comp,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,G: C > B,H: A > C,R: A > B,F: B > D] :
( ( ( comp @ C @ B @ A @ G @ H )
= R )
=> ( ( comp @ C @ D @ A @ ( comp @ B @ D @ C @ F @ G ) @ H )
= ( comp @ B @ D @ A @ F @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_252_rewriteL__comp__comp,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType,F: C > B,G: A > C,L: A > B,H: D > A] :
( ( ( comp @ C @ B @ A @ F @ G )
= L )
=> ( ( comp @ C @ B @ D @ F @ ( comp @ A @ C @ D @ G @ H ) )
= ( comp @ A @ B @ D @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_253_type__copy__map__cong0,axiom,
! [B: $tType,D: $tType,E: $tType,A: $tType,C: $tType,M: B > A,G: C > B,X2: C,N: D > A,H: C > D,F: A > E] :
( ( ( M @ ( G @ X2 ) )
= ( N @ ( H @ X2 ) ) )
=> ( ( comp @ B @ E @ C @ ( comp @ A @ E @ B @ F @ M ) @ G @ X2 )
= ( comp @ D @ E @ C @ ( comp @ A @ E @ D @ F @ N ) @ H @ X2 ) ) ) ).
% type_copy_map_cong0
thf(fact_254_eq__comp__r,axiom,
! [A: $tType,R3: A > A > $o] :
( ( relcompp @ A @ A @ A
@ ^ [Y2: A,Z2: A] : Y2 = Z2
@ ( relcompp @ A @ A @ A @ R3
@ ^ [Y2: A,Z2: A] : Y2 = Z2 ) )
= R3 ) ).
% eq_comp_r
thf(fact_255_comp__transfer,axiom,
! [A: $tType,B: $tType,E: $tType,F2: $tType,D: $tType,C: $tType,B6: A > C > $o,C5: B > D > $o,A6: E > F2 > $o] : ( bNF_rel_fun @ ( A > B ) @ ( C > D ) @ ( ( E > A ) > E > B ) @ ( ( F2 > C ) > F2 > D ) @ ( bNF_rel_fun @ A @ C @ B @ D @ B6 @ C5 ) @ ( bNF_rel_fun @ ( E > A ) @ ( F2 > C ) @ ( E > B ) @ ( F2 > D ) @ ( bNF_rel_fun @ E @ F2 @ A @ C @ A6 @ B6 ) @ ( bNF_rel_fun @ E @ F2 @ B @ D @ A6 @ C5 ) ) @ ( comp @ A @ B @ E ) @ ( comp @ C @ D @ F2 ) ) ).
% comp_transfer
thf(fact_256_rel__fun__iff__geq__image2p,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType] :
( ( bNF_rel_fun @ A @ B @ C @ D )
= ( ^ [R6: A > B > $o,S4: C > D > $o,F4: A > C,G4: B > D] : ( ord_less_eq @ ( C > D > $o ) @ ( bNF_Greatest_image2p @ A @ C @ B @ D @ F4 @ G4 @ R6 ) @ S4 ) ) ) ).
% rel_fun_iff_geq_image2p
thf(fact_257_map__prod__o__convol__id,axiom,
! [B: $tType,A: $tType,C: $tType,F: C > A,G: C > B,X2: 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 ) @ X2 )
= ( bNF_convol @ C @ A @ B @ ( comp @ A @ A @ C @ ( id @ A ) @ F ) @ G @ X2 ) ) ).
% map_prod_o_convol_id
thf(fact_258_in__rel__Gr,axiom,
! [B: $tType,A: $tType,A6: set @ A,F: A > B] :
( ( fun_in_rel @ A @ B @ ( bNF_Gr @ A @ B @ A6 @ F ) )
= ( bNF_Grp @ A @ B @ A6 @ F ) ) ).
% in_rel_Gr
thf(fact_259_image__Fpow__mono,axiom,
! [B: $tType,A: $tType,F: B > A,A6: set @ B,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F @ A6 ) @ B6 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( image2 @ ( set @ B ) @ ( set @ A ) @ ( image2 @ B @ A @ F ) @ ( finite_Fpow @ B @ A6 ) ) @ ( finite_Fpow @ A @ B6 ) ) ) ).
% image_Fpow_mono
thf(fact_260_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_261_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_262_effect__freezeI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [H2: heap_ext @ product_unit,H: heap_ext @ product_unit,R: list @ A,A3: array @ A] :
( ( H2 = H )
=> ( ( R
= ( get @ A @ H @ A3 ) )
=> ( heap_effect @ ( list @ A ) @ ( freeze @ A @ A3 ) @ H @ H2 @ R ) ) ) ) ).
% effect_freezeI
thf(fact_263_effect__freezeE,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [A3: array @ A,H: heap_ext @ product_unit,H2: heap_ext @ product_unit,R: list @ A] :
( ( heap_effect @ ( list @ A ) @ ( freeze @ A @ A3 ) @ H @ H2 @ R )
=> ~ ( ( H2 = H )
=> ( R
!= ( get @ A @ H @ A3 ) ) ) ) ) ).
% effect_freezeE
thf(fact_264_convol__mem__GrpI,axiom,
! [B: $tType,A: $tType,X2: A,A6: set @ A,G: A > B] :
( ( member @ A @ X2 @ A6 )
=> ( member @ ( product_prod @ A @ B ) @ ( bNF_convol @ A @ A @ B @ ( id @ A ) @ G @ X2 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( bNF_Grp @ A @ B @ A6 @ G ) ) ) ) ) ).
% convol_mem_GrpI
thf(fact_265_success__bind__effectI,axiom,
! [A: $tType,B: $tType,F: heap_Heap @ A,H: heap_ext @ product_unit,H2: heap_ext @ product_unit,X2: A,G: A > ( heap_Heap @ B )] :
( ( heap_effect @ A @ F @ H @ H2 @ X2 )
=> ( ( heap_success @ B @ ( G @ X2 ) @ H2 )
=> ( heap_success @ B @ ( heap_bind @ A @ B @ F @ G ) @ H ) ) ) ).
% success_bind_effectI
thf(fact_266_effect__lengthI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [H2: heap_ext @ product_unit,H: heap_ext @ product_unit,R: nat,A3: array @ A] :
( ( H2 = H )
=> ( ( R
= ( length @ A @ H @ A3 ) )
=> ( heap_effect @ nat @ ( len @ A @ A3 ) @ H @ H2 @ R ) ) ) ) ).
% effect_lengthI
thf(fact_267_image__id,axiom,
! [A: $tType] :
( ( image2 @ A @ A @ ( id @ A ) )
= ( id @ ( set @ A ) ) ) ).
% image_id
thf(fact_268_fun_Omap__id,axiom,
! [A: $tType,D: $tType,T3: D > A] :
( ( comp @ A @ A @ D @ ( id @ A ) @ T3 )
= T3 ) ).
% fun.map_id
thf(fact_269_comp__id,axiom,
! [B: $tType,A: $tType,F: A > B] :
( ( comp @ A @ B @ A @ F @ ( id @ A ) )
= F ) ).
% comp_id
thf(fact_270_id__comp,axiom,
! [B: $tType,A: $tType,G: A > B] :
( ( comp @ B @ B @ A @ ( id @ B ) @ G )
= G ) ).
% id_comp
thf(fact_271_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_272_apfst__id,axiom,
! [B: $tType,A: $tType] :
( ( product_apfst @ A @ A @ B @ ( id @ A ) )
= ( id @ ( product_prod @ A @ B ) ) ) ).
% apfst_id
thf(fact_273_apsnd__id,axiom,
! [B: $tType,A: $tType] :
( ( product_apsnd @ B @ B @ A @ ( id @ B ) )
= ( id @ ( product_prod @ A @ B ) ) ) ).
% apsnd_id
thf(fact_274_range__fst,axiom,
! [B: $tType,A: $tType] :
( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% range_fst
thf(fact_275_range__snd,axiom,
! [B: $tType,A: $tType] :
( ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% range_snd
thf(fact_276_success__ifI,axiom,
! [A: $tType,C3: $o,T3: heap_Heap @ A,H: heap_ext @ product_unit,E3: heap_Heap @ A] :
( ( C3
=> ( heap_success @ A @ T3 @ H ) )
=> ( ( ~ C3
=> ( heap_success @ A @ E3 @ H ) )
=> ( heap_success @ A @ ( if @ ( heap_Heap @ A ) @ C3 @ T3 @ E3 ) @ H ) ) ) ).
% success_ifI
thf(fact_277_success__freezeI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [A3: array @ A,H: heap_ext @ product_unit] : ( heap_success @ ( list @ A ) @ ( freeze @ A @ A3 ) @ H ) ) ).
% success_freezeI
thf(fact_278_success__lenI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [A3: array @ A,H: heap_ext @ product_unit] : ( heap_success @ nat @ ( len @ A @ A3 ) @ H ) ) ).
% success_lenI
thf(fact_279_fun_Omap__id0,axiom,
! [A: $tType,D: $tType] :
( ( comp @ A @ A @ D @ ( id @ A ) )
= ( id @ ( D > A ) ) ) ).
% fun.map_id0
thf(fact_280_surj__id,axiom,
! [A: $tType] :
( ( image2 @ A @ A @ ( id @ A ) @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% surj_id
thf(fact_281_Grp__UNIV__idI,axiom,
! [A: $tType,X2: A,Y3: A] :
( ( X2 = Y3 )
=> ( bNF_Grp @ A @ A @ ( top_top @ ( set @ A ) ) @ ( id @ A ) @ X2 @ Y3 ) ) ).
% Grp_UNIV_idI
thf(fact_282_eq__alt,axiom,
! [A: $tType] :
( ( ^ [Y2: A,Z2: A] : Y2 = Z2 )
= ( bNF_Grp @ A @ A @ ( top_top @ ( set @ A ) ) @ ( id @ A ) ) ) ).
% eq_alt
thf(fact_283_image2pE,axiom,
! [D: $tType,B: $tType,A: $tType,C: $tType,F: A > B,G: C > D,R3: A > C > $o,Fx: B,Gy: D] :
( ( bNF_Greatest_image2p @ A @ B @ C @ D @ F @ G @ R3 @ Fx @ Gy )
=> ~ ! [X: A] :
( ( Fx
= ( F @ X ) )
=> ! [Y: C] :
( ( Gy
= ( G @ Y ) )
=> ~ ( R3 @ X @ Y ) ) ) ) ).
% image2pE
thf(fact_284_image2pI,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,R3: A > B > $o,X2: A,Y3: B,F: A > C,G: B > D] :
( ( R3 @ X2 @ Y3 )
=> ( bNF_Greatest_image2p @ A @ C @ B @ D @ F @ G @ R3 @ ( F @ X2 ) @ ( G @ Y3 ) ) ) ).
% image2pI
thf(fact_285_Inf_OINF__id__eq,axiom,
! [A: $tType,Inf: ( set @ A ) > A,A6: set @ A] :
( ( Inf @ ( image2 @ A @ A @ ( id @ A ) @ A6 ) )
= ( Inf @ A6 ) ) ).
% Inf.INF_id_eq
thf(fact_286_Sup_OSUP__id__eq,axiom,
! [A: $tType,Sup: ( set @ A ) > A,A6: set @ A] :
( ( Sup @ ( image2 @ A @ A @ ( id @ A ) @ A6 ) )
= ( Sup @ A6 ) ) ).
% Sup.SUP_id_eq
thf(fact_287_comp__eq__id__dest,axiom,
! [C: $tType,B: $tType,A: $tType,A3: C > B,B2: A > C,C3: A > B,V: A] :
( ( ( comp @ C @ B @ A @ A3 @ B2 )
= ( comp @ B @ B @ A @ ( id @ B ) @ C3 ) )
=> ( ( A3 @ ( B2 @ V ) )
= ( C3 @ V ) ) ) ).
% comp_eq_id_dest
thf(fact_288_pointfree__idE,axiom,
! [B: $tType,A: $tType,F: B > A,G: A > B,X2: A] :
( ( ( comp @ B @ A @ A @ F @ G )
= ( id @ A ) )
=> ( ( F @ ( G @ X2 ) )
= X2 ) ) ).
% pointfree_idE
thf(fact_289_surjD,axiom,
! [A: $tType,B: $tType,F: B > A,Y3: A] :
( ( ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ? [X: B] :
( Y3
= ( F @ X ) ) ) ).
% surjD
thf(fact_290_surjE,axiom,
! [A: $tType,B: $tType,F: B > A,Y3: A] :
( ( ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ~ ! [X: B] :
( Y3
!= ( F @ X ) ) ) ).
% surjE
thf(fact_291_surjI,axiom,
! [B: $tType,A: $tType,G: B > A,F: A > B] :
( ! [X: A] :
( ( G @ ( F @ X ) )
= X )
=> ( ( image2 @ B @ A @ G @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surjI
thf(fact_292_rangeI,axiom,
! [A: $tType,B: $tType,F: B > A,X2: B] : ( member @ A @ ( F @ X2 ) @ ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ).
% rangeI
thf(fact_293_surj__def,axiom,
! [B: $tType,A: $tType,F: B > A] :
( ( ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
= ( ! [Y5: A] :
? [X4: B] :
( Y5
= ( F @ X4 ) ) ) ) ).
% surj_def
thf(fact_294_range__eqI,axiom,
! [A: $tType,B: $tType,B2: A,F: B > A,X2: B] :
( ( B2
= ( F @ X2 ) )
=> ( member @ A @ B2 @ ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ) ).
% range_eqI
thf(fact_295_effect__success,axiom,
! [A: $tType,C3: heap_Heap @ A,H: heap_ext @ product_unit,H2: heap_ext @ product_unit,R: A] :
( ( heap_effect @ A @ C3 @ H @ H2 @ R )
=> ( heap_success @ A @ C3 @ H ) ) ).
% effect_success
thf(fact_296_success__effectE,axiom,
! [A: $tType,C3: heap_Heap @ A,H: heap_ext @ product_unit] :
( ( heap_success @ A @ C3 @ H )
=> ~ ! [R4: A,H4: heap_ext @ product_unit] :
~ ( heap_effect @ A @ C3 @ H @ H4 @ R4 ) ) ).
% success_effectE
thf(fact_297_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_298_GrD1,axiom,
! [B: $tType,A: $tType,X2: A,Fx: B,A6: set @ A,F: A > B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Fx ) @ ( bNF_Gr @ A @ B @ A6 @ F ) )
=> ( member @ A @ X2 @ A6 ) ) ).
% GrD1
thf(fact_299_GrD2,axiom,
! [A: $tType,B: $tType,X2: A,Fx: B,A6: set @ A,F: A > B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Fx ) @ ( bNF_Gr @ A @ B @ A6 @ F ) )
=> ( ( F @ X2 )
= Fx ) ) ).
% GrD2
thf(fact_300_success__returnI,axiom,
! [A: $tType,X2: A,H: heap_ext @ product_unit] : ( heap_success @ A @ ( heap_return @ A @ X2 ) @ H ) ).
% success_returnI
thf(fact_301_success__heapI,axiom,
! [A: $tType,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( heap_ext @ product_unit ) ),H: heap_ext @ product_unit] : ( heap_success @ A @ ( heap_heap @ A @ F ) @ H ) ).
% success_heapI
thf(fact_302_success__guardI,axiom,
! [A: $tType,P2: ( heap_ext @ product_unit ) > $o,H: heap_ext @ product_unit,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( heap_ext @ product_unit ) )] :
( ( P2 @ H )
=> ( heap_success @ A @ ( heap_guard @ A @ P2 @ F ) @ H ) ) ).
% success_guardI
thf(fact_303_success__tapI,axiom,
! [A: $tType,F: ( heap_ext @ product_unit ) > A,H: heap_ext @ product_unit] : ( heap_success @ A @ ( heap_tap @ A @ F ) @ H ) ).
% success_tapI
thf(fact_304_success__refI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [V: A,H: heap_ext @ product_unit] : ( heap_success @ ( ref @ A ) @ ( ref3 @ A @ V ) @ H ) ) ).
% success_refI
thf(fact_305_success__updateI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R: ref @ A,V: A,H: heap_ext @ product_unit] : ( heap_success @ product_unit @ ( update2 @ A @ R @ V ) @ H ) ) ).
% success_updateI
thf(fact_306_success__lookupI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R: ref @ A,H: heap_ext @ product_unit] : ( heap_success @ A @ ( lookup @ A @ R ) @ H ) ) ).
% success_lookupI
thf(fact_307_fun_Orel__cong,axiom,
! [A: $tType,B: $tType,D: $tType,X2: D > A,Ya: D > A,Y3: D > B,Xa2: D > B,R3: A > B > $o,Ra: A > B > $o] :
( ( X2 = Ya )
=> ( ( Y3 = Xa2 )
=> ( ! [Z4: A,Yb: B] :
( ( member @ A @ Z4 @ ( image2 @ D @ A @ Ya @ ( top_top @ ( set @ D ) ) ) )
=> ( ( member @ B @ Yb @ ( image2 @ D @ B @ Xa2 @ ( top_top @ ( set @ D ) ) ) )
=> ( ( R3 @ Z4 @ Yb )
= ( Ra @ Z4 @ Yb ) ) ) )
=> ( ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y2: D,Z2: D] : Y2 = Z2
@ R3
@ X2
@ Y3 )
= ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y2: D,Z2: D] : Y2 = Z2
@ Ra
@ Ya
@ Xa2 ) ) ) ) ) ).
% fun.rel_cong
thf(fact_308_fun_Orel__mono__strong,axiom,
! [A: $tType,B: $tType,D: $tType,R3: A > B > $o,X2: D > A,Y3: D > B,Ra: A > B > $o] :
( ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y2: D,Z2: D] : Y2 = Z2
@ R3
@ X2
@ Y3 )
=> ( ! [Z4: A,Yb: B] :
( ( member @ A @ Z4 @ ( image2 @ D @ A @ X2 @ ( top_top @ ( set @ D ) ) ) )
=> ( ( member @ B @ Yb @ ( image2 @ D @ B @ Y3 @ ( top_top @ ( set @ D ) ) ) )
=> ( ( R3 @ Z4 @ Yb )
=> ( Ra @ Z4 @ Yb ) ) ) )
=> ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y2: D,Z2: D] : Y2 = Z2
@ Ra
@ X2
@ Y3 ) ) ) ).
% fun.rel_mono_strong
thf(fact_309_fun_Orel__refl__strong,axiom,
! [A: $tType,B: $tType,X2: B > A,Ra: A > A > $o] :
( ! [Z4: A] :
( ( member @ A @ Z4 @ ( image2 @ B @ A @ X2 @ ( top_top @ ( set @ B ) ) ) )
=> ( Ra @ Z4 @ Z4 ) )
=> ( bNF_rel_fun @ B @ B @ A @ A
@ ^ [Y2: B,Z2: B] : Y2 = Z2
@ Ra
@ X2
@ X2 ) ) ).
% fun.rel_refl_strong
thf(fact_310_range__subsetD,axiom,
! [B: $tType,A: $tType,F: B > A,B6: set @ A,I: B] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) @ B6 )
=> ( member @ A @ ( F @ I ) @ B6 ) ) ).
% range_subsetD
thf(fact_311_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_312_fun_Omap__cong,axiom,
! [B: $tType,A: $tType,D: $tType,X2: D > A,Ya: D > A,F: A > B,G: A > B] :
( ( X2 = Ya )
=> ( ! [Z4: A] :
( ( member @ A @ Z4 @ ( image2 @ D @ A @ Ya @ ( top_top @ ( set @ D ) ) ) )
=> ( ( F @ Z4 )
= ( G @ Z4 ) ) )
=> ( ( comp @ A @ B @ D @ F @ X2 )
= ( comp @ A @ B @ D @ G @ Ya ) ) ) ) ).
% fun.map_cong
thf(fact_313_fun_Omap__cong0,axiom,
! [B: $tType,A: $tType,D: $tType,X2: D > A,F: A > B,G: A > B] :
( ! [Z4: A] :
( ( member @ A @ Z4 @ ( image2 @ D @ A @ X2 @ ( top_top @ ( set @ D ) ) ) )
=> ( ( F @ Z4 )
= ( G @ Z4 ) ) )
=> ( ( comp @ A @ B @ D @ F @ X2 )
= ( comp @ A @ B @ D @ G @ X2 ) ) ) ).
% fun.map_cong0
thf(fact_314_fun_Oinj__map__strong,axiom,
! [B: $tType,A: $tType,D: $tType,X2: D > A,Xa2: D > A,F: A > B,Fa: A > B] :
( ! [Z4: A,Za: A] :
( ( member @ A @ Z4 @ ( image2 @ D @ A @ X2 @ ( top_top @ ( set @ D ) ) ) )
=> ( ( member @ A @ Za @ ( image2 @ D @ A @ Xa2 @ ( top_top @ ( set @ D ) ) ) )
=> ( ( ( F @ Z4 )
= ( Fa @ Za ) )
=> ( Z4 = Za ) ) ) )
=> ( ( ( comp @ A @ B @ D @ F @ X2 )
= ( comp @ A @ B @ D @ Fa @ Xa2 ) )
=> ( X2 = Xa2 ) ) ) ).
% fun.inj_map_strong
thf(fact_315_surj__fun__eq,axiom,
! [B: $tType,C: $tType,A: $tType,F: B > A,X6: set @ B,G1: A > C,G22: A > C] :
( ( ( image2 @ B @ A @ F @ X6 )
= ( top_top @ ( set @ A ) ) )
=> ( ! [X: B] :
( ( member @ B @ X @ X6 )
=> ( ( comp @ A @ C @ B @ G1 @ F @ X )
= ( comp @ A @ C @ B @ G22 @ F @ X ) ) )
=> ( G1 = G22 ) ) ) ).
% surj_fun_eq
thf(fact_316_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_317_success__changeI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [F: A > A,R: ref @ A,H: heap_ext @ product_unit] : ( heap_success @ A @ ( change @ A @ F @ R ) @ H ) ) ).
% success_changeI
thf(fact_318_apfst__def,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( product_apfst @ A @ C @ B )
= ( ^ [F4: A > C] : ( product_map_prod @ A @ C @ B @ B @ F4 @ ( id @ B ) ) ) ) ).
% apfst_def
thf(fact_319_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_320_rel__fun__image2p,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,R3: A > B > $o,F: A > C,G: B > D] : ( bNF_rel_fun @ A @ B @ C @ D @ R3 @ ( bNF_Greatest_image2p @ A @ C @ B @ D @ F @ G @ R3 ) @ F @ G ) ).
% rel_fun_image2p
thf(fact_321_effect__lengthE,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [A3: array @ A,H: heap_ext @ product_unit,H2: heap_ext @ product_unit,R: nat] :
( ( heap_effect @ nat @ ( len @ A @ A3 ) @ H @ H2 @ R )
=> ~ ( ( R
= ( length @ A @ H2 @ A3 ) )
=> ( H2 != H ) ) ) ) ).
% effect_lengthE
thf(fact_322_comp__fun__idem__on_Ocomp__comp__fun__idem__on,axiom,
! [B: $tType,A: $tType,C: $tType,S5: set @ A,F: A > B > B,G: C > A,R3: set @ C] :
( ( finite673082921795544331dem_on @ A @ B @ S5 @ F )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ C @ A @ G @ ( top_top @ ( set @ C ) ) ) @ S5 )
=> ( finite673082921795544331dem_on @ C @ B @ R3 @ ( comp @ A @ ( B > B ) @ C @ F @ G ) ) ) ) ).
% comp_fun_idem_on.comp_comp_fun_idem_on
thf(fact_323_comp__fun__commute__on_Ocomp__comp__fun__commute__on,axiom,
! [B: $tType,A: $tType,C: $tType,S5: set @ A,F: A > B > B,G: C > A,R3: set @ C] :
( ( finite4664212375090638736ute_on @ A @ B @ S5 @ F )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ C @ A @ G @ ( top_top @ ( set @ C ) ) ) @ S5 )
=> ( finite4664212375090638736ute_on @ C @ B @ R3 @ ( comp @ A @ ( B > B ) @ C @ F @ G ) ) ) ) ).
% comp_fun_commute_on.comp_comp_fun_commute_on
thf(fact_324_image__Pow__mono,axiom,
! [B: $tType,A: $tType,F: B > A,A6: set @ B,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F @ A6 ) @ B6 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( image2 @ ( set @ B ) @ ( set @ A ) @ ( image2 @ B @ A @ F ) @ ( pow @ B @ A6 ) ) @ ( pow @ A @ B6 ) ) ) ).
% image_Pow_mono
thf(fact_325_vimage2p__Grp,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType] :
( ( bNF_vimage2p @ A @ C @ B @ D @ $o )
= ( ^ [F4: A > C,G4: B > D,P4: C > D > $o] : ( relcompp @ A @ C @ B @ ( bNF_Grp @ A @ C @ ( top_top @ ( set @ A ) ) @ F4 ) @ ( relcompp @ C @ D @ B @ P4 @ ( conversep @ B @ D @ ( bNF_Grp @ B @ D @ ( top_top @ ( set @ B ) ) @ G4 ) ) ) ) ) ) ).
% vimage2p_Grp
thf(fact_326_fun_Oin__rel,axiom,
! [B: $tType,A: $tType,D: $tType,R3: A > B > $o,A3: D > A,B2: D > B] :
( ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y2: D,Z2: D] : Y2 = Z2
@ R3
@ A3
@ B2 )
= ( ? [Z5: D > ( product_prod @ A @ B )] :
( ( member @ ( D > ( product_prod @ A @ B ) ) @ Z5
@ ( collect @ ( D > ( product_prod @ A @ B ) )
@ ^ [X4: D > ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ D @ ( product_prod @ A @ B ) @ X4 @ ( top_top @ ( set @ D ) ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R3 ) ) ) ) )
& ( ( 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_327_Grp__UNIV__id,axiom,
! [A: $tType,F: A > A] :
( ( F
= ( id @ A ) )
=> ( ( relcompp @ A @ A @ A @ ( conversep @ A @ A @ ( bNF_Grp @ A @ A @ ( top_top @ ( set @ A ) ) @ F ) ) @ ( bNF_Grp @ A @ A @ ( top_top @ ( set @ A ) ) @ F ) )
= ( bNF_Grp @ A @ A @ ( top_top @ ( set @ A ) ) @ F ) ) ) ).
% Grp_UNIV_id
thf(fact_328_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 ) ) )
=> ( ! [X: A] :
( ( G @ ( F @ X ) )
= X )
=> ( order_strict_mono @ B @ A @ G ) ) ) ) ) ).
% strict_mono_inv
thf(fact_329_Id__on__Gr,axiom,
! [A: $tType] :
( ( id_on @ A )
= ( ^ [A8: set @ A] : ( bNF_Gr @ A @ A @ A8 @ ( id @ A ) ) ) ) ).
% Id_on_Gr
thf(fact_330_type__copy__map__id0,axiom,
! [B: $tType,A: $tType,Rep: A > B,Abs: B > A,M: B > B] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ( ( M
= ( id @ B ) )
=> ( ( comp @ B @ A @ A @ ( comp @ B @ A @ B @ Abs @ M ) @ Rep )
= ( id @ A ) ) ) ) ).
% type_copy_map_id0
thf(fact_331_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_332_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_333_case__prodI2_H,axiom,
! [A: $tType,B: $tType,C: $tType,P: product_prod @ A @ B,C3: A > B > C > $o,X2: C] :
( ! [A5: A,B4: B] :
( ( ( product_Pair @ A @ B @ A5 @ B4 )
= P )
=> ( C3 @ A5 @ B4 @ X2 ) )
=> ( product_case_prod @ A @ B @ ( C > $o ) @ C3 @ P @ X2 ) ) ).
% case_prodI2'
thf(fact_334_image__ident,axiom,
! [A: $tType,Y4: set @ A] :
( ( image2 @ A @ A
@ ^ [X4: A] : X4
@ Y4 )
= Y4 ) ).
% image_ident
thf(fact_335_split__part,axiom,
! [B: $tType,A: $tType,P2: $o,Q2: A > B > $o] :
( ( product_case_prod @ A @ B @ $o
@ ^ [A7: A,B5: B] :
( P2
& ( Q2 @ A7 @ B5 ) ) )
= ( ^ [Ab: product_prod @ A @ B] :
( P2
& ( product_case_prod @ A @ B @ $o @ Q2 @ Ab ) ) ) ) ).
% split_part
thf(fact_336_map__prod__ident,axiom,
! [B: $tType,A: $tType] :
( ( product_map_prod @ A @ A @ B @ B
@ ^ [X4: A] : X4
@ ^ [Y5: B] : Y5 )
= ( ^ [Z5: product_prod @ A @ B] : Z5 ) ) ).
% map_prod_ident
thf(fact_337_Heap__Monad_Obind__bind,axiom,
! [C: $tType,B: $tType,A: $tType,F: heap_Heap @ A,G: A > ( heap_Heap @ C ),K: C > ( heap_Heap @ B )] :
( ( heap_bind @ C @ B @ ( heap_bind @ A @ C @ F @ G ) @ K )
= ( heap_bind @ A @ B @ F
@ ^ [X4: A] : ( heap_bind @ C @ B @ ( G @ X4 ) @ K ) ) ) ).
% Heap_Monad.bind_bind
thf(fact_338_Id__onI,axiom,
! [A: $tType,A3: A,A6: set @ A] :
( ( member @ A @ A3 @ A6 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ ( id_on @ A @ A6 ) ) ) ).
% Id_onI
thf(fact_339_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_340_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_341_case__prodI2,axiom,
! [B: $tType,A: $tType,P: product_prod @ A @ B,C3: A > B > $o] :
( ! [A5: A,B4: B] :
( ( P
= ( product_Pair @ A @ B @ A5 @ B4 ) )
=> ( C3 @ A5 @ B4 ) )
=> ( product_case_prod @ A @ B @ $o @ C3 @ P ) ) ).
% case_prodI2
thf(fact_342_mem__case__prodI,axiom,
! [A: $tType,B: $tType,C: $tType,Z3: A,C3: B > C > ( set @ A ),A3: B,B2: C] :
( ( member @ A @ Z3 @ ( C3 @ A3 @ B2 ) )
=> ( member @ A @ Z3 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C3 @ ( product_Pair @ B @ C @ A3 @ B2 ) ) ) ) ).
% mem_case_prodI
thf(fact_343_mem__case__prodI2,axiom,
! [C: $tType,B: $tType,A: $tType,P: product_prod @ A @ B,Z3: C,C3: A > B > ( set @ C )] :
( ! [A5: A,B4: B] :
( ( P
= ( product_Pair @ A @ B @ A5 @ B4 ) )
=> ( member @ C @ Z3 @ ( C3 @ A5 @ B4 ) ) )
=> ( member @ C @ Z3 @ ( product_case_prod @ A @ B @ ( set @ C ) @ C3 @ P ) ) ) ).
% mem_case_prodI2
thf(fact_344_case__swap,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > B > A,P: product_prod @ C @ B] :
( ( product_case_prod @ B @ C @ A
@ ^ [Y5: B,X4: C] : ( F @ X4 @ Y5 )
@ ( product_swap @ C @ B @ P ) )
= ( product_case_prod @ C @ B @ A @ F @ P ) ) ).
% case_swap
thf(fact_345_pair__imageI,axiom,
! [C: $tType,B: $tType,A: $tType,A3: A,B2: B,A6: set @ ( product_prod @ A @ B ),F: A > B > C] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ A6 )
=> ( member @ C @ ( F @ A3 @ B2 ) @ ( image2 @ ( product_prod @ A @ B ) @ C @ ( product_case_prod @ A @ B @ C @ F ) @ A6 ) ) ) ).
% pair_imageI
thf(fact_346_Inf_OINF__identity__eq,axiom,
! [A: $tType,Inf: ( set @ A ) > A,A6: set @ A] :
( ( Inf
@ ( image2 @ A @ A
@ ^ [X4: A] : X4
@ A6 ) )
= ( Inf @ A6 ) ) ).
% Inf.INF_identity_eq
thf(fact_347_Sup_OSUP__identity__eq,axiom,
! [A: $tType,Sup: ( set @ A ) > A,A6: set @ A] :
( ( Sup
@ ( image2 @ A @ A
@ ^ [X4: A] : X4
@ A6 ) )
= ( Sup @ A6 ) ) ).
% Sup.SUP_identity_eq
thf(fact_348_imageE,axiom,
! [A: $tType,B: $tType,B2: A,F: B > A,A6: set @ B] :
( ( member @ A @ B2 @ ( image2 @ B @ A @ F @ A6 ) )
=> ~ ! [X: B] :
( ( B2
= ( F @ X ) )
=> ~ ( member @ B @ X @ A6 ) ) ) ).
% imageE
thf(fact_349_image__image,axiom,
! [A: $tType,B: $tType,C: $tType,F: B > A,G: C > B,A6: set @ C] :
( ( image2 @ B @ A @ F @ ( image2 @ C @ B @ G @ A6 ) )
= ( image2 @ C @ A
@ ^ [X4: C] : ( F @ ( G @ X4 ) )
@ A6 ) ) ).
% image_image
thf(fact_350_Compr__image__eq,axiom,
! [A: $tType,B: $tType,F: B > A,A6: set @ B,P2: A > $o] :
( ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ ( image2 @ B @ A @ F @ A6 ) )
& ( P2 @ X4 ) ) )
= ( image2 @ B @ A @ F
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ A6 )
& ( P2 @ ( F @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_351_Id__on__def_H,axiom,
! [A: $tType,A6: A > $o] :
( ( id_on @ A @ ( collect @ A @ A6 ) )
= ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X4: A,Y5: A] :
( ( X4 = Y5 )
& ( A6 @ X4 ) ) ) ) ) ).
% Id_on_def'
thf(fact_352_pred__equals__eq2,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S5: set @ ( product_prod @ A @ B )] :
( ( ( ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ R3 ) )
= ( ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ S5 ) ) )
= ( R3 = S5 ) ) ).
% pred_equals_eq2
thf(fact_353_case__prodE_H,axiom,
! [B: $tType,A: $tType,C: $tType,C3: A > B > C > $o,P: product_prod @ A @ B,Z3: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ C3 @ P @ Z3 )
=> ~ ! [X: A,Y: B] :
( ( P
= ( product_Pair @ A @ B @ X @ Y ) )
=> ~ ( C3 @ X @ Y @ Z3 ) ) ) ).
% case_prodE'
thf(fact_354_case__prodD_H,axiom,
! [B: $tType,A: $tType,C: $tType,R3: A > B > C > $o,A3: A,B2: B,C3: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ R3 @ ( product_Pair @ A @ B @ A3 @ B2 ) @ C3 )
=> ( R3 @ A3 @ B2 @ C3 ) ) ).
% case_prodD'
thf(fact_355_case__prod__Pair__iden,axiom,
! [B: $tType,A: $tType,P: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) @ P )
= P ) ).
% case_prod_Pair_iden
thf(fact_356_subset__CollectI,axiom,
! [A: $tType,B6: set @ A,A6: set @ A,Q2: A > $o,P2: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B6 @ A6 )
=> ( ! [X: A] :
( ( member @ A @ X @ B6 )
=> ( ( Q2 @ X )
=> ( P2 @ X ) ) )
=> ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ B6 )
& ( Q2 @ X4 ) ) )
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A6 )
& ( P2 @ X4 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_357_subset__Collect__iff,axiom,
! [A: $tType,B6: set @ A,A6: set @ A,P2: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B6 @ A6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A6 )
& ( P2 @ X4 ) ) ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ B6 )
=> ( P2 @ X4 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_358_prop__restrict,axiom,
! [A: $tType,X2: A,Z6: set @ A,X6: set @ A,P2: A > $o] :
( ( member @ A @ X2 @ Z6 )
=> ( ( ord_less_eq @ ( set @ A ) @ Z6
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ X6 )
& ( P2 @ X4 ) ) ) )
=> ( P2 @ X2 ) ) ) ).
% prop_restrict
thf(fact_359_Collect__restrict,axiom,
! [A: $tType,X6: set @ A,P2: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ X6 )
& ( P2 @ X4 ) ) )
@ X6 ) ).
% Collect_restrict
thf(fact_360_fun_Omap__ident,axiom,
! [A: $tType,D: $tType,T3: D > A] :
( ( comp @ A @ A @ D
@ ^ [X4: A] : X4
@ T3 )
= T3 ) ).
% fun.map_ident
thf(fact_361_case__prod__app,axiom,
! [A: $tType,D: $tType,C: $tType,B: $tType] :
( ( product_case_prod @ B @ C @ ( D > A ) )
= ( ^ [F4: B > C > D > A,X4: product_prod @ B @ C,Y5: D] :
( product_case_prod @ B @ C @ A
@ ^ [L3: B,R5: C] : ( F4 @ L3 @ R5 @ Y5 )
@ X4 ) ) ) ).
% case_prod_app
thf(fact_362_prod_Ocase__distrib,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,H: C > D,F: A > B > C,Prod: product_prod @ A @ B] :
( ( H @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
= ( product_case_prod @ A @ B @ D
@ ^ [X12: A,X23: B] : ( H @ ( F @ X12 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_363_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_364_Grp__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Grp @ A @ B )
= ( ^ [A8: set @ A,F4: A > B,A7: A,B5: B] :
( ( B5
= ( F4 @ A7 ) )
& ( member @ A @ A7 @ A8 ) ) ) ) ).
% Grp_def
thf(fact_365_effect__LetI,axiom,
! [A: $tType,B: $tType,X2: A,T3: A,F: A > ( heap_Heap @ B ),H: heap_ext @ product_unit,H2: heap_ext @ product_unit,R: B] :
( ( X2 = T3 )
=> ( ( heap_effect @ B @ ( F @ X2 ) @ H @ H2 @ R )
=> ( heap_effect @ B @ ( F @ T3 ) @ H @ H2 @ R ) ) ) ).
% effect_LetI
thf(fact_366_success__LetI,axiom,
! [A: $tType,B: $tType,X2: A,T3: A,F: A > ( heap_Heap @ B ),H: heap_ext @ product_unit] :
( ( X2 = T3 )
=> ( ( heap_success @ B @ ( F @ X2 ) @ H )
=> ( heap_success @ B @ ( F @ T3 ) @ H ) ) ) ).
% success_LetI
thf(fact_367_vimage2p__def,axiom,
! [B: $tType,E: $tType,C: $tType,D: $tType,A: $tType] :
( ( bNF_vimage2p @ A @ D @ B @ E @ C )
= ( ^ [F4: A > D,G4: B > E,R6: D > E > C,X4: A,Y5: B] : ( R6 @ ( F4 @ X4 ) @ ( G4 @ Y5 ) ) ) ) ).
% vimage2p_def
thf(fact_368_curry__K,axiom,
! [B: $tType,C: $tType,A: $tType,C3: C] :
( ( product_curry @ A @ B @ C
@ ^ [X4: product_prod @ A @ B] : C3 )
= ( ^ [X4: A,Y5: B] : C3 ) ) ).
% curry_K
thf(fact_369_scomp__unfold,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType] :
( ( product_scomp @ A @ B @ C @ D )
= ( ^ [F4: A > ( product_prod @ B @ C ),G4: B > C > D,X4: A] : ( G4 @ ( product_fst @ B @ C @ ( F4 @ X4 ) ) @ ( product_snd @ B @ C @ ( F4 @ X4 ) ) ) ) ) ).
% scomp_unfold
thf(fact_370_image__Collect__subsetI,axiom,
! [A: $tType,B: $tType,P2: A > $o,F: A > B,B6: set @ B] :
( ! [X: A] :
( ( P2 @ X )
=> ( member @ B @ ( F @ X ) @ B6 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F @ ( collect @ A @ P2 ) ) @ B6 ) ) ).
% image_Collect_subsetI
thf(fact_371_range__composition,axiom,
! [A: $tType,C: $tType,B: $tType,F: C > A,G: B > C] :
( ( image2 @ B @ A
@ ^ [X4: B] : ( F @ ( G @ X4 ) )
@ ( top_top @ ( set @ B ) ) )
= ( image2 @ C @ A @ F @ ( image2 @ B @ C @ G @ ( top_top @ ( set @ B ) ) ) ) ) ).
% range_composition
thf(fact_372_rangeE,axiom,
! [A: $tType,B: $tType,B2: A,F: B > A] :
( ( member @ A @ B2 @ ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) )
=> ~ ! [X: B] :
( B2
!= ( F @ X ) ) ) ).
% rangeE
thf(fact_373_fun_Orel__map_I1_J,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,Sb: C > B > $o,I: A > C,X2: D > A,Y3: D > B] :
( ( bNF_rel_fun @ D @ D @ C @ B
@ ^ [Y2: D,Z2: D] : Y2 = Z2
@ Sb
@ ( comp @ A @ C @ D @ I @ X2 )
@ Y3 )
= ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y2: D,Z2: D] : Y2 = Z2
@ ^ [X4: A] : ( Sb @ ( I @ X4 ) )
@ X2
@ Y3 ) ) ).
% fun.rel_map(1)
thf(fact_374_fun_Orel__map_I2_J,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,Sa: A > C > $o,X2: D > A,G: B > C,Y3: D > B] :
( ( bNF_rel_fun @ D @ D @ A @ C
@ ^ [Y2: D,Z2: D] : Y2 = Z2
@ Sa
@ X2
@ ( comp @ B @ C @ D @ G @ Y3 ) )
= ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y2: D,Z2: D] : Y2 = Z2
@ ^ [X4: A,Y5: B] : ( Sa @ X4 @ ( G @ Y5 ) )
@ X2
@ Y3 ) ) ).
% fun.rel_map(2)
thf(fact_375_case__prodE2,axiom,
! [B: $tType,A: $tType,C: $tType,Q2: A > $o,P2: B > C > A,Z3: product_prod @ B @ C] :
( ( Q2 @ ( product_case_prod @ B @ C @ A @ P2 @ Z3 ) )
=> ~ ! [X: B,Y: C] :
( ( Z3
= ( product_Pair @ B @ C @ X @ Y ) )
=> ~ ( Q2 @ ( P2 @ X @ Y ) ) ) ) ).
% case_prodE2
thf(fact_376_case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F: ( product_prod @ A @ B ) > C] :
( ( product_case_prod @ A @ B @ C
@ ^ [X4: A,Y5: B] : ( F @ ( product_Pair @ A @ B @ X4 @ Y5 ) ) )
= F ) ).
% case_prod_eta
thf(fact_377_cond__case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > B > C,G: ( product_prod @ A @ B ) > C] :
( ! [X: A,Y: B] :
( ( F @ X @ Y )
= ( G @ ( product_Pair @ A @ B @ X @ Y ) ) )
=> ( ( product_case_prod @ A @ B @ C @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_378_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_379_case__prodE,axiom,
! [A: $tType,B: $tType,C3: A > B > $o,P: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ $o @ C3 @ P )
=> ~ ! [X: A,Y: B] :
( ( P
= ( product_Pair @ A @ B @ X @ Y ) )
=> ~ ( C3 @ X @ Y ) ) ) ).
% case_prodE
thf(fact_380_fst__def,axiom,
! [B: $tType,A: $tType] :
( ( product_fst @ A @ B )
= ( product_case_prod @ A @ B @ A
@ ^ [X12: A,X23: B] : X12 ) ) ).
% fst_def
thf(fact_381_snd__def,axiom,
! [B: $tType,A: $tType] :
( ( product_snd @ A @ B )
= ( product_case_prod @ A @ B @ B
@ ^ [X12: A,X23: B] : X23 ) ) ).
% snd_def
thf(fact_382_map__prod__def,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType] :
( ( product_map_prod @ A @ C @ B @ D )
= ( ^ [F4: A > C,G4: B > D] :
( product_case_prod @ A @ B @ ( product_prod @ C @ D )
@ ^ [X4: A,Y5: B] : ( product_Pair @ C @ D @ ( F4 @ X4 ) @ ( G4 @ Y5 ) ) ) ) ) ).
% map_prod_def
thf(fact_383_case__prod__map__prod,axiom,
! [C: $tType,A: $tType,B: $tType,E: $tType,D: $tType,H: B > C > A,F: D > B,G: E > C,X2: product_prod @ D @ E] :
( ( product_case_prod @ B @ C @ A @ H @ ( product_map_prod @ D @ B @ E @ C @ F @ G @ X2 ) )
= ( product_case_prod @ D @ E @ A
@ ^ [L3: D,R5: E] : ( H @ ( F @ L3 ) @ ( G @ R5 ) )
@ X2 ) ) ).
% case_prod_map_prod
thf(fact_384_curry__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_curry @ A @ B @ C )
= ( ^ [C6: ( product_prod @ A @ B ) > C,X4: A,Y5: B] : ( C6 @ ( product_Pair @ A @ B @ X4 @ Y5 ) ) ) ) ).
% curry_def
thf(fact_385_converse__relcompp,axiom,
! [A: $tType,C: $tType,B: $tType,R: B > C > $o,S2: C > A > $o] :
( ( conversep @ B @ A @ ( relcompp @ B @ C @ A @ R @ S2 ) )
= ( relcompp @ A @ C @ B @ ( conversep @ C @ A @ S2 ) @ ( conversep @ B @ C @ R ) ) ) ).
% converse_relcompp
thf(fact_386_leq__conversepI,axiom,
! [A: $tType,R3: A > A > $o] :
( ( R3
= ( ^ [Y2: A,Z2: A] : Y2 = Z2 ) )
=> ( ord_less_eq @ ( A > A > $o ) @ R3 @ ( conversep @ A @ A @ R3 ) ) ) ).
% leq_conversepI
thf(fact_387_map__prod_Oidentity,axiom,
! [B: $tType,A: $tType] :
( ( product_map_prod @ A @ A @ B @ B
@ ^ [X4: A] : X4
@ ^ [X4: B] : X4 )
= ( id @ ( product_prod @ A @ B ) ) ) ).
% map_prod.identity
thf(fact_388_lookup__chain,axiom,
! [B: $tType,A: $tType] :
( ( heap @ B )
=> ! [R: ref @ B,F: heap_Heap @ A] :
( ( heap_bind @ B @ A @ ( lookup @ B @ R )
@ ^ [Uu: B] : F )
= F ) ) ).
% lookup_chain
thf(fact_389_scomp__scomp,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F2: $tType,E: $tType,F: A > ( product_prod @ E @ F2 ),G: E > F2 > ( product_prod @ C @ D ),H: C > D > B] :
( ( product_scomp @ A @ C @ D @ B @ ( product_scomp @ A @ E @ F2 @ ( product_prod @ C @ D ) @ F @ G ) @ H )
= ( product_scomp @ A @ E @ F2 @ B @ F
@ ^ [X4: E] : ( product_scomp @ F2 @ C @ D @ B @ ( G @ X4 ) @ H ) ) ) ).
% scomp_scomp
thf(fact_390_len__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( len @ A )
= ( ^ [A7: array @ A] :
( heap_tap @ nat
@ ^ [H6: heap_ext @ product_unit] : ( length @ A @ H6 @ A7 ) ) ) ) ) ).
% len_def
thf(fact_391_flip__pred,axiom,
! [A: $tType,B: $tType,A6: set @ ( product_prod @ A @ B ),R3: B > A > $o] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A6 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( conversep @ B @ A @ R3 ) ) ) )
=> ( 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 )
@ ^ [X4: A,Y5: B] : ( product_Pair @ B @ A @ Y5 @ X4 ) )
@ A6 )
@ ( collect @ ( product_prod @ B @ A ) @ ( product_case_prod @ B @ A @ $o @ R3 ) ) ) ) ).
% flip_pred
thf(fact_392_freeze__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( freeze @ A )
= ( ^ [A7: array @ A] :
( heap_tap @ ( list @ A )
@ ^ [H6: heap_ext @ product_unit] : ( get @ A @ H6 @ A7 ) ) ) ) ) ).
% freeze_def
thf(fact_393_type__copy__wit,axiom,
! [A: $tType,C: $tType,B: $tType,Rep: A > B,Abs: B > A,X2: C,S5: B > ( set @ C ),Y3: B] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ( ( member @ C @ X2 @ ( comp @ B @ ( set @ C ) @ A @ S5 @ Rep @ ( Abs @ Y3 ) ) )
=> ( member @ C @ X2 @ ( S5 @ Y3 ) ) ) ) ).
% type_copy_wit
thf(fact_394_Cantors__paradox,axiom,
! [A: $tType,A6: set @ A] :
~ ? [F6: A > ( set @ A )] :
( ( image2 @ A @ ( set @ A ) @ F6 @ A6 )
= ( pow @ A @ A6 ) ) ).
% Cantors_paradox
thf(fact_395_type__copy__vimage2p__Grp__Rep,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,Rep: A > B,Abs: B > A,F: C > D,P2: D > $o,H: 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 @ P2 ) @ H ) )
= ( bNF_Grp @ C @ A
@ ( collect @ C
@ ^ [X4: C] : ( P2 @ ( F @ X4 ) ) )
@ ( comp @ D @ A @ C @ ( comp @ B @ A @ D @ Abs @ H ) @ F ) ) ) ) ).
% type_copy_vimage2p_Grp_Rep
thf(fact_396_type__copy__vimage2p__Grp__Abs,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Rep: A > B,Abs: B > A,G: D > C,P2: C > $o,H: 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 @ P2 ) @ H ) )
= ( bNF_Grp @ D @ B
@ ( collect @ D
@ ^ [X4: D] : ( P2 @ ( G @ X4 ) ) )
@ ( comp @ C @ B @ D @ ( comp @ A @ B @ C @ Rep @ H ) @ G ) ) ) ) ).
% type_copy_vimage2p_Grp_Abs
thf(fact_397_exE__realizer,axiom,
! [C: $tType,A: $tType,B: $tType,P2: A > B > $o,P: product_prod @ B @ A,Q2: C > $o,F: B > A > C] :
( ( P2 @ ( product_snd @ B @ A @ P ) @ ( product_fst @ B @ A @ P ) )
=> ( ! [X: B,Y: A] :
( ( P2 @ Y @ X )
=> ( Q2 @ ( F @ X @ Y ) ) )
=> ( Q2 @ ( product_case_prod @ B @ A @ C @ F @ P ) ) ) ) ).
% exE_realizer
thf(fact_398_split__comp__eq,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,F: A > B > C,G: D > A] :
( ( ^ [U: product_prod @ D @ B] : ( F @ ( G @ ( product_fst @ D @ B @ U ) ) @ ( product_snd @ D @ B @ U ) ) )
= ( product_case_prod @ D @ B @ C
@ ^ [X4: D] : ( F @ ( G @ X4 ) ) ) ) ).
% split_comp_eq
thf(fact_399_case__prod__beta_H,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ C )
= ( ^ [F4: A > B > C,X4: product_prod @ A @ B] : ( F4 @ ( product_fst @ A @ B @ X4 ) @ ( product_snd @ A @ B @ X4 ) ) ) ) ).
% case_prod_beta'
thf(fact_400_case__prod__unfold,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ C )
= ( ^ [C6: A > B > C,P3: product_prod @ A @ B] : ( C6 @ ( product_fst @ A @ B @ P3 ) @ ( product_snd @ A @ B @ P3 ) ) ) ) ).
% case_prod_unfold
thf(fact_401_pred__subset__eq2,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S5: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( A > B > $o )
@ ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ R3 )
@ ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ S5 ) )
= ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ S5 ) ) ).
% pred_subset_eq2
thf(fact_402_comp__fun__commute__on__def,axiom,
! [B: $tType,A: $tType] :
( ( finite4664212375090638736ute_on @ A @ B )
= ( ^ [S4: set @ A,F4: A > B > B] :
! [X4: A,Y5: A] :
( ( member @ A @ X4 @ S4 )
=> ( ( member @ A @ Y5 @ S4 )
=> ( ( comp @ B @ B @ B @ ( F4 @ Y5 ) @ ( F4 @ X4 ) )
= ( comp @ B @ B @ B @ ( F4 @ X4 ) @ ( F4 @ Y5 ) ) ) ) ) ) ) ).
% comp_fun_commute_on_def
thf(fact_403_comp__fun__commute__on_Ocomp__fun__commute__on,axiom,
! [B: $tType,A: $tType,S5: set @ A,F: A > B > B,X2: A,Y3: A] :
( ( finite4664212375090638736ute_on @ A @ B @ S5 @ F )
=> ( ( member @ A @ X2 @ S5 )
=> ( ( member @ A @ Y3 @ S5 )
=> ( ( comp @ B @ B @ B @ ( F @ Y3 ) @ ( F @ X2 ) )
= ( comp @ B @ B @ B @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ) ) ) ).
% comp_fun_commute_on.comp_fun_commute_on
thf(fact_404_comp__fun__commute__on_Ocommute__left__comp,axiom,
! [A: $tType,B: $tType,C: $tType,S5: set @ A,F: A > B > B,X2: A,Y3: A,G: C > B] :
( ( finite4664212375090638736ute_on @ A @ B @ S5 @ F )
=> ( ( member @ A @ X2 @ S5 )
=> ( ( member @ A @ Y3 @ S5 )
=> ( ( comp @ B @ B @ C @ ( F @ Y3 ) @ ( comp @ B @ B @ C @ ( F @ X2 ) @ G ) )
= ( comp @ B @ B @ C @ ( F @ X2 ) @ ( comp @ B @ B @ C @ ( F @ Y3 ) @ G ) ) ) ) ) ) ).
% comp_fun_commute_on.commute_left_comp
thf(fact_405_comp__fun__commute__on_Ointro,axiom,
! [B: $tType,A: $tType,S5: set @ A,F: A > B > B] :
( ! [X: A,Y: A] :
( ( member @ A @ X @ S5 )
=> ( ( member @ A @ Y @ S5 )
=> ( ( comp @ B @ B @ B @ ( F @ Y ) @ ( F @ X ) )
= ( comp @ B @ B @ B @ ( F @ X ) @ ( F @ Y ) ) ) ) )
=> ( finite4664212375090638736ute_on @ A @ B @ S5 @ F ) ) ).
% comp_fun_commute_on.intro
thf(fact_406_case__prod__o__map__prod,axiom,
! [B: $tType,D: $tType,C: $tType,E: $tType,A: $tType,F: D > E > C,G1: A > D,G22: B > E] :
( ( comp @ ( product_prod @ D @ E ) @ C @ ( product_prod @ A @ B ) @ ( product_case_prod @ D @ E @ C @ F ) @ ( product_map_prod @ A @ D @ B @ E @ G1 @ G22 ) )
= ( product_case_prod @ A @ B @ C
@ ^ [L3: A,R5: B] : ( F @ ( G1 @ L3 ) @ ( G22 @ R5 ) ) ) ) ).
% case_prod_o_map_prod
thf(fact_407_vimage2p__relcompp__converse,axiom,
! [E: $tType,C: $tType,D: $tType,A: $tType,F2: $tType,B: $tType,Rep: A > B,Abs: B > A,F: C > E,G: D > F2,R3: B > E > $o,S5: B > F2 > $o] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ( ( bNF_vimage2p @ C @ E @ D @ F2 @ $o @ F @ G @ ( relcompp @ E @ B @ F2 @ ( conversep @ B @ E @ R3 ) @ S5 ) )
= ( relcompp @ C @ A @ D @ ( conversep @ A @ C @ ( bNF_vimage2p @ A @ B @ C @ E @ $o @ Rep @ F @ R3 ) ) @ ( bNF_vimage2p @ A @ B @ D @ F2 @ $o @ Rep @ G @ S5 ) ) ) ) ).
% vimage2p_relcompp_converse
thf(fact_408_Id__on__iff,axiom,
! [A: $tType,X2: A,Y3: A,A6: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ ( id_on @ A @ A6 ) )
= ( ( X2 = Y3 )
& ( member @ A @ X2 @ A6 ) ) ) ).
% Id_on_iff
thf(fact_409_Id__on__eqI,axiom,
! [A: $tType,A3: A,B2: A,A6: set @ A] :
( ( A3 = B2 )
=> ( ( member @ A @ A3 @ A6 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( id_on @ A @ A6 ) ) ) ) ).
% Id_on_eqI
thf(fact_410_Id__onE,axiom,
! [A: $tType,C3: product_prod @ A @ A,A6: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ C3 @ ( id_on @ A @ A6 ) )
=> ~ ! [X: A] :
( ( member @ A @ X @ A6 )
=> ( C3
!= ( product_Pair @ A @ A @ X @ X ) ) ) ) ).
% Id_onE
thf(fact_411_predicate2__transferD,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,R1: A > B > $o,R22: C > D > $o,P2: A > C > $o,Q2: B > D > $o,A3: product_prod @ A @ B,A6: set @ ( product_prod @ A @ B ),B2: product_prod @ C @ D,B6: set @ ( product_prod @ C @ D )] :
( ( bNF_rel_fun @ A @ B @ ( C > $o ) @ ( D > $o ) @ R1
@ ( bNF_rel_fun @ C @ D @ $o @ $o @ R22
@ ^ [Y2: $o,Z2: $o] : Y2 = Z2 )
@ P2
@ Q2 )
=> ( ( member @ ( product_prod @ A @ B ) @ A3 @ A6 )
=> ( ( member @ ( product_prod @ C @ D ) @ B2 @ B6 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A6 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R1 ) ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ C @ D ) ) @ B6 @ ( collect @ ( product_prod @ C @ D ) @ ( product_case_prod @ C @ D @ $o @ R22 ) ) )
=> ( ( P2 @ ( product_fst @ A @ B @ A3 ) @ ( product_fst @ C @ D @ B2 ) )
= ( Q2 @ ( product_snd @ A @ B @ A3 ) @ ( product_snd @ C @ D @ B2 ) ) ) ) ) ) ) ) ).
% predicate2_transferD
thf(fact_412_fst__diag__id,axiom,
! [A: $tType,Z3: A] :
( ( comp @ ( product_prod @ A @ A ) @ A @ A @ ( product_fst @ A @ A )
@ ^ [X4: A] : ( product_Pair @ A @ A @ X4 @ X4 )
@ Z3 )
= ( id @ A @ Z3 ) ) ).
% fst_diag_id
thf(fact_413_snd__diag__id,axiom,
! [A: $tType,Z3: A] :
( ( comp @ ( product_prod @ A @ A ) @ A @ A @ ( product_snd @ A @ A )
@ ^ [X4: A] : ( product_Pair @ A @ A @ X4 @ X4 )
@ Z3 )
= ( id @ A @ Z3 ) ) ).
% snd_diag_id
thf(fact_414_case__prod__comp,axiom,
! [D: $tType,A: $tType,C: $tType,B: $tType,F: D > C > A,G: B > D,X2: product_prod @ B @ C] :
( ( product_case_prod @ B @ C @ A @ ( comp @ D @ ( C > A ) @ B @ F @ G ) @ X2 )
= ( F @ ( G @ ( product_fst @ B @ C @ X2 ) ) @ ( product_snd @ B @ C @ X2 ) ) ) ).
% case_prod_comp
thf(fact_415_lookup__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( lookup @ A )
= ( ^ [R5: ref @ A] :
( heap_tap @ A
@ ^ [H6: heap_ext @ product_unit] : ( get2 @ A @ H6 @ R5 ) ) ) ) ) ).
% lookup_def
thf(fact_416_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 )
@ ^ [X4: A,Y5: B] : ( product_Pair @ B @ A @ Y5 @ X4 ) ) ) ) ).
% fst_snd_flip
thf(fact_417_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 )
@ ^ [X4: B,Y5: A] : ( product_Pair @ A @ B @ Y5 @ X4 ) ) ) ) ).
% snd_fst_flip
thf(fact_418_comp__fun__idem__on_Ocomp__fun__idem__on,axiom,
! [B: $tType,A: $tType,S5: set @ A,F: A > B > B,X2: A] :
( ( finite673082921795544331dem_on @ A @ B @ S5 @ F )
=> ( ( member @ A @ X2 @ S5 )
=> ( ( comp @ B @ B @ B @ ( F @ X2 ) @ ( F @ X2 ) )
= ( F @ X2 ) ) ) ) ).
% comp_fun_idem_on.comp_fun_idem_on
thf(fact_419_bind__lift,axiom,
! [A: $tType,B: $tType,F: heap_Heap @ B,G: B > A] :
( ( heap_bind @ B @ A @ F @ ( heap_lift @ B @ A @ G ) )
= ( heap_bind @ B @ A @ F
@ ^ [X4: B] : ( heap_return @ A @ ( G @ X4 ) ) ) ) ).
% bind_lift
thf(fact_420_OO__Grp__cong,axiom,
! [B: $tType,C: $tType,A: $tType,A6: set @ A,B6: set @ A,F: A > B,G: A > C] :
( ( A6 = B6 )
=> ( ( relcompp @ B @ A @ C @ ( conversep @ A @ B @ ( bNF_Grp @ A @ B @ A6 @ F ) ) @ ( bNF_Grp @ A @ C @ A6 @ G ) )
= ( relcompp @ B @ A @ C @ ( conversep @ A @ B @ ( bNF_Grp @ A @ B @ B6 @ F ) ) @ ( bNF_Grp @ A @ C @ B6 @ G ) ) ) ) ).
% OO_Grp_cong
thf(fact_421_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 )
@ ^ [X4: A] : ( product_Pair @ A @ A @ X4 @ X4 )
@ ( product_fst @ A @ B ) ) )
= ( product_fst @ A @ B ) ) ).
% fst_diag_fst
thf(fact_422_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 )
@ ^ [X4: B] : ( product_Pair @ B @ B @ X4 @ X4 )
@ ( product_snd @ A @ B ) ) )
= ( product_snd @ A @ B ) ) ).
% snd_diag_snd
thf(fact_423_type__copy__map__comp0,axiom,
! [F2: $tType,D: $tType,B: $tType,A: $tType,C: $tType,E: $tType,Rep: A > B,Abs: B > A,M: C > D,M1: B > D,M2: C > B,F: D > F2,G: E > C] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ( ( M
= ( comp @ B @ D @ C @ M1 @ M2 ) )
=> ( ( comp @ C @ F2 @ E @ ( comp @ D @ F2 @ C @ F @ M ) @ G )
= ( comp @ A @ F2 @ E @ ( comp @ B @ F2 @ A @ ( comp @ D @ F2 @ B @ F @ M1 ) @ Rep ) @ ( comp @ C @ A @ E @ ( comp @ B @ A @ C @ Abs @ M2 ) @ G ) ) ) ) ) ).
% type_copy_map_comp0
thf(fact_424_type__copy__map__comp0__undo,axiom,
! [E: $tType,A: $tType,C: $tType,B: $tType,D: $tType,F2: $tType,Rep: A > B,Abs: B > A,Rep2: C > D,Abs2: D > C,Rep3: E > F2,Abs3: F2 > E,M: F2 > D,M1: B > D,M2: F2 > B] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ( ( type_definition @ C @ D @ Rep2 @ Abs2 @ ( top_top @ ( set @ D ) ) )
=> ( ( type_definition @ E @ F2 @ Rep3 @ Abs3 @ ( top_top @ ( set @ F2 ) ) )
=> ( ( ( comp @ F2 @ C @ E @ ( comp @ D @ C @ F2 @ Abs2 @ M ) @ Rep3 )
= ( comp @ A @ C @ E @ ( comp @ B @ C @ A @ ( comp @ D @ C @ B @ Abs2 @ M1 ) @ Rep ) @ ( comp @ F2 @ A @ E @ ( comp @ B @ A @ F2 @ Abs @ M2 ) @ Rep3 ) ) )
=> ( ( comp @ B @ D @ F2 @ M1 @ M2 )
= M ) ) ) ) ) ).
% type_copy_map_comp0_undo
thf(fact_425_fun_Orel__compp__Grp,axiom,
! [D: $tType,B: $tType,A: $tType,R3: A > B > $o] :
( ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y2: D,Z2: D] : Y2 = Z2
@ R3 )
= ( 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 ) )
@ ^ [X4: D > ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ D @ ( product_prod @ A @ B ) @ X4 @ ( top_top @ ( set @ D ) ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R3 ) ) ) )
@ ( 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 ) )
@ ^ [X4: D > ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ D @ ( product_prod @ A @ B ) @ X4 @ ( top_top @ ( set @ D ) ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R3 ) ) ) )
@ ( comp @ ( product_prod @ A @ B ) @ B @ D @ ( product_snd @ A @ B ) ) ) ) ) ).
% fun.rel_compp_Grp
thf(fact_426_image__Pow__surj,axiom,
! [B: $tType,A: $tType,F: B > A,A6: set @ B,B6: set @ A] :
( ( ( image2 @ B @ A @ F @ A6 )
= B6 )
=> ( ( image2 @ ( set @ B ) @ ( set @ A ) @ ( image2 @ B @ A @ F ) @ ( pow @ B @ A6 ) )
= ( pow @ A @ B6 ) ) ) ).
% image_Pow_surj
thf(fact_427_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 )
@ ^ [X4: B] : ( product_Pair @ B @ B @ X4 @ X4 )
@ ( product_snd @ A @ B ) ) )
= ( product_snd @ A @ B ) ) ).
% fst_diag_snd
thf(fact_428_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 )
@ ^ [X4: A] : ( product_Pair @ A @ A @ X4 @ X4 )
@ ( product_fst @ A @ B ) ) )
= ( product_fst @ A @ B ) ) ).
% snd_diag_fst
thf(fact_429_fun_Orel__Grp,axiom,
! [D: $tType,B: $tType,A: $tType,A6: set @ A,F: A > B] :
( ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y2: D,Z2: D] : Y2 = Z2
@ ( bNF_Grp @ A @ B @ A6 @ F ) )
= ( bNF_Grp @ ( D > A ) @ ( D > B )
@ ( collect @ ( D > A )
@ ^ [X4: D > A] : ( ord_less_eq @ ( set @ A ) @ ( image2 @ D @ A @ X4 @ ( top_top @ ( set @ D ) ) ) @ A6 ) )
@ ( comp @ A @ B @ D @ F ) ) ) ).
% fun.rel_Grp
thf(fact_430_Grp__fst__snd,axiom,
! [B: $tType,A: $tType,R3: A > B > $o] :
( ( relcompp @ A @ ( product_prod @ A @ B ) @ B @ ( conversep @ ( product_prod @ A @ B ) @ A @ ( bNF_Grp @ ( product_prod @ A @ B ) @ A @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R3 ) ) @ ( product_fst @ A @ B ) ) ) @ ( bNF_Grp @ ( product_prod @ A @ B ) @ B @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R3 ) ) @ ( product_snd @ A @ B ) ) )
= R3 ) ).
% Grp_fst_snd
thf(fact_431_change__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( change @ A )
= ( ^ [F4: A > A,R5: ref @ A] :
( heap_bind @ A @ A @ ( lookup @ A @ R5 )
@ ^ [X4: A] :
( heap_bind @ product_unit @ A @ ( update2 @ A @ R5 @ ( F4 @ X4 ) )
@ ^ [Uu: product_unit] : ( heap_return @ A @ ( F4 @ X4 ) ) ) ) ) ) ) ).
% change_def
thf(fact_432_type__copy__set__map0,axiom,
! [A: $tType,B: $tType,D: $tType,E: $tType,C: $tType,F2: $tType,Rep: A > B,Abs: B > A,S5: B > ( set @ D ),M: C > B,F: E > D,S6: C > ( set @ E ),G: F2 > C] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ( ( ( comp @ B @ ( set @ D ) @ C @ S5 @ M )
= ( comp @ ( set @ E ) @ ( set @ D ) @ C @ ( image2 @ E @ D @ F ) @ S6 ) )
=> ( ( comp @ A @ ( set @ D ) @ F2 @ ( comp @ B @ ( set @ D ) @ A @ S5 @ Rep ) @ ( comp @ C @ A @ F2 @ ( comp @ B @ A @ C @ Abs @ M ) @ G ) )
= ( comp @ ( set @ E ) @ ( set @ D ) @ F2 @ ( image2 @ E @ D @ F ) @ ( comp @ C @ ( set @ E ) @ F2 @ S6 @ G ) ) ) ) ) ).
% type_copy_set_map0
thf(fact_433_type__definition_ORep__range,axiom,
! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A6: set @ A] :
( ( type_definition @ B @ A @ Rep @ Abs @ A6 )
=> ( ( image2 @ B @ A @ Rep @ ( top_top @ ( set @ B ) ) )
= A6 ) ) ).
% type_definition.Rep_range
thf(fact_434_type__definition_OAbs__image,axiom,
! [A: $tType,B: $tType,Rep: B > A,Abs: A > B,A6: set @ A] :
( ( type_definition @ B @ A @ Rep @ Abs @ A6 )
=> ( ( image2 @ A @ B @ Abs @ A6 )
= ( top_top @ ( set @ B ) ) ) ) ).
% type_definition.Abs_image
thf(fact_435_Quotient__alt__def5,axiom,
! [B: $tType,A: $tType] :
( ( quotient @ A @ B )
= ( ^ [R6: A > A > $o,Abs4: A > B,Rep4: B > A,T5: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ T5 @ ( bNF_Grp @ A @ B @ ( top_top @ ( set @ A ) ) @ Abs4 ) )
& ( ord_less_eq @ ( B > A > $o ) @ ( bNF_Grp @ B @ A @ ( top_top @ ( set @ B ) ) @ Rep4 ) @ ( conversep @ A @ B @ T5 ) )
& ( R6
= ( relcompp @ A @ B @ A @ T5 @ ( conversep @ A @ B @ T5 ) ) ) ) ) ) ).
% Quotient_alt_def5
thf(fact_436_K__record__comp,axiom,
! [C: $tType,B: $tType,A: $tType,C3: B,F: A > C] :
( ( comp @ C @ B @ A
@ ^ [X4: C] : C3
@ F )
= ( ^ [X4: A] : C3 ) ) ).
% K_record_comp
thf(fact_437_image__paired__Times,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,F: C > A,G: D > B,A6: set @ C,B6: set @ D] :
( ( image2 @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ C @ D @ ( product_prod @ A @ B )
@ ^ [X4: C,Y5: D] : ( product_Pair @ A @ B @ ( F @ X4 ) @ ( G @ Y5 ) ) )
@ ( product_Sigma @ C @ D @ A6
@ ^ [Uu: C] : B6 ) )
= ( product_Sigma @ A @ B @ ( image2 @ C @ A @ F @ A6 )
@ ^ [Uu: A] : ( image2 @ D @ B @ G @ B6 ) ) ) ).
% image_paired_Times
thf(fact_438_Gr__incl,axiom,
! [A: $tType,B: $tType,A6: set @ A,F: A > B,B6: set @ B] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( bNF_Gr @ A @ B @ A6 @ F )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu: A] : B6 ) )
= ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F @ A6 ) @ B6 ) ) ).
% Gr_incl
thf(fact_439_subset__fst__snd,axiom,
! [B: $tType,A: $tType,A6: set @ ( product_prod @ A @ B )] :
( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A6
@ ( product_Sigma @ A @ B @ ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A6 )
@ ^ [Uu: A] : ( image2 @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A6 ) ) ) ).
% subset_fst_snd
thf(fact_440_inv__image__def,axiom,
! [A: $tType,B: $tType] :
( ( inv_image @ B @ A )
= ( ^ [R5: set @ ( product_prod @ B @ B ),F4: A > B] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X4: A,Y5: A] : ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F4 @ X4 ) @ ( F4 @ Y5 ) ) @ R5 ) ) ) ) ) ).
% inv_image_def
thf(fact_441_The__case__prod,axiom,
! [B: $tType,A: $tType,P2: A > B > $o] :
( ( the @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P2 ) )
= ( the @ ( product_prod @ A @ B )
@ ^ [Xy: product_prod @ A @ B] : ( P2 @ ( product_fst @ A @ B @ Xy ) @ ( product_snd @ A @ B @ Xy ) ) ) ) ).
% The_case_prod
thf(fact_442_Quotient__composition__ge__eq,axiom,
! [B: $tType,A: $tType,T4: A > B > $o,R3: B > B > $o] :
( ( left_total @ A @ B @ T4 )
=> ( ( ord_less_eq @ ( B > B > $o )
@ ^ [Y2: B,Z2: B] : Y2 = Z2
@ R3 )
=> ( ord_less_eq @ ( A > A > $o )
@ ^ [Y2: A,Z2: A] : Y2 = Z2
@ ( relcompp @ A @ B @ A @ T4 @ ( relcompp @ B @ B @ A @ R3 @ ( conversep @ A @ B @ T4 ) ) ) ) ) ) ).
% Quotient_composition_ge_eq
thf(fact_443_Quotient__composition__le__eq,axiom,
! [B: $tType,A: $tType,T4: A > B > $o,R3: B > B > $o] :
( ( left_unique @ A @ B @ T4 )
=> ( ( ord_less_eq @ ( B > B > $o ) @ R3
@ ^ [Y2: B,Z2: B] : Y2 = Z2 )
=> ( ord_less_eq @ ( A > A > $o ) @ ( relcompp @ A @ B @ A @ T4 @ ( relcompp @ B @ B @ A @ R3 @ ( conversep @ A @ B @ T4 ) ) )
@ ^ [Y2: A,Z2: A] : Y2 = Z2 ) ) ) ).
% Quotient_composition_le_eq
thf(fact_444_mem__Sigma__iff,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,A6: set @ A,B6: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( product_Sigma @ A @ B @ A6 @ B6 ) )
= ( ( member @ A @ A3 @ A6 )
& ( member @ B @ B2 @ ( B6 @ A3 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_445_SigmaI,axiom,
! [B: $tType,A: $tType,A3: A,A6: set @ A,B2: B,B6: A > ( set @ B )] :
( ( member @ A @ A3 @ A6 )
=> ( ( member @ B @ B2 @ ( B6 @ A3 ) )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( product_Sigma @ A @ B @ A6 @ B6 ) ) ) ) ).
% SigmaI
thf(fact_446_in__inv__image,axiom,
! [A: $tType,B: $tType,X2: A,Y3: A,R: set @ ( product_prod @ B @ B ),F: A > B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ ( inv_image @ B @ A @ R @ F ) )
= ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F @ X2 ) @ ( F @ Y3 ) ) @ R ) ) ).
% in_inv_image
thf(fact_447_Collect__case__prod,axiom,
! [B: $tType,A: $tType,P2: A > $o,Q2: B > $o] :
( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A7: A,B5: B] :
( ( P2 @ A7 )
& ( Q2 @ B5 ) ) ) )
= ( product_Sigma @ A @ B @ ( collect @ A @ P2 )
@ ^ [Uu: A] : ( collect @ B @ Q2 ) ) ) ).
% Collect_case_prod
thf(fact_448_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_449_The__split__eq,axiom,
! [A: $tType,B: $tType,X2: A,Y3: B] :
( ( the @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X7: A,Y6: B] :
( ( X2 = X7 )
& ( Y3 = Y6 ) ) ) )
= ( product_Pair @ A @ B @ X2 @ Y3 ) ) ).
% The_split_eq
thf(fact_450_Times__eq__cancel2,axiom,
! [A: $tType,B: $tType,X2: A,C5: set @ A,A6: set @ B,B6: set @ B] :
( ( member @ A @ X2 @ C5 )
=> ( ( ( product_Sigma @ B @ A @ A6
@ ^ [Uu: B] : C5 )
= ( product_Sigma @ B @ A @ B6
@ ^ [Uu: B] : C5 ) )
= ( A6 = B6 ) ) ) ).
% Times_eq_cancel2
thf(fact_451_Sigma__cong,axiom,
! [B: $tType,A: $tType,A6: set @ A,B6: set @ A,C5: A > ( set @ B ),D3: A > ( set @ B )] :
( ( A6 = B6 )
=> ( ! [X: A] :
( ( member @ A @ X @ B6 )
=> ( ( C5 @ X )
= ( D3 @ X ) ) )
=> ( ( product_Sigma @ A @ B @ A6 @ C5 )
= ( product_Sigma @ A @ B @ B6 @ D3 ) ) ) ) ).
% Sigma_cong
thf(fact_452_SigmaE2,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,A6: set @ A,B6: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( product_Sigma @ A @ B @ A6 @ B6 ) )
=> ~ ( ( member @ A @ A3 @ A6 )
=> ~ ( member @ B @ B2 @ ( B6 @ A3 ) ) ) ) ).
% SigmaE2
thf(fact_453_SigmaD2,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,A6: set @ A,B6: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( product_Sigma @ A @ B @ A6 @ B6 ) )
=> ( member @ B @ B2 @ ( B6 @ A3 ) ) ) ).
% SigmaD2
thf(fact_454_SigmaD1,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,A6: set @ A,B6: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( product_Sigma @ A @ B @ A6 @ B6 ) )
=> ( member @ A @ A3 @ A6 ) ) ).
% SigmaD1
thf(fact_455_SigmaE,axiom,
! [A: $tType,B: $tType,C3: product_prod @ A @ B,A6: set @ A,B6: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ C3 @ ( product_Sigma @ A @ B @ A6 @ B6 ) )
=> ~ ! [X: A] :
( ( member @ A @ X @ A6 )
=> ! [Y: B] :
( ( member @ B @ Y @ ( B6 @ X ) )
=> ( C3
!= ( product_Pair @ A @ B @ X @ Y ) ) ) ) ) ).
% SigmaE
thf(fact_456_Collect__case__prod__Sigma,axiom,
! [B: $tType,A: $tType,P2: A > $o,Q2: A > B > $o] :
( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X4: A,Y5: B] :
( ( P2 @ X4 )
& ( Q2 @ X4 @ Y5 ) ) ) )
= ( product_Sigma @ A @ B @ ( collect @ A @ P2 )
@ ^ [X4: A] : ( collect @ B @ ( Q2 @ X4 ) ) ) ) ).
% Collect_case_prod_Sigma
thf(fact_457_Sigma__mono,axiom,
! [B: $tType,A: $tType,A6: set @ A,C5: set @ A,B6: A > ( set @ B ),D3: A > ( set @ B )] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ C5 )
=> ( ! [X: A] :
( ( member @ A @ X @ A6 )
=> ( ord_less_eq @ ( set @ B ) @ ( B6 @ X ) @ ( D3 @ X ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ A6 @ B6 ) @ ( product_Sigma @ A @ B @ C5 @ D3 ) ) ) ) ).
% Sigma_mono
thf(fact_458_Quotient__alt__def4,axiom,
! [B: $tType,A: $tType] :
( ( quotient @ A @ B )
= ( ^ [R6: A > A > $o,Abs4: A > B,Rep4: B > A,T5: A > B > $o] :
( ! [A7: A,B5: B] :
( ( T5 @ A7 @ B5 )
=> ( ( Abs4 @ A7 )
= B5 ) )
& ! [B5: B] : ( T5 @ ( Rep4 @ B5 ) @ B5 )
& ( R6
= ( relcompp @ A @ B @ A @ T5 @ ( conversep @ A @ B @ T5 ) ) ) ) ) ) ).
% Quotient_alt_def4
thf(fact_459_Times__subset__cancel2,axiom,
! [A: $tType,B: $tType,X2: A,C5: set @ A,A6: set @ B,B6: set @ B] :
( ( member @ A @ X2 @ C5 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) )
@ ( product_Sigma @ B @ A @ A6
@ ^ [Uu: B] : C5 )
@ ( product_Sigma @ B @ A @ B6
@ ^ [Uu: B] : C5 ) )
= ( ord_less_eq @ ( set @ B ) @ A6 @ B6 ) ) ) ).
% Times_subset_cancel2
thf(fact_460_mem__Times__iff,axiom,
! [A: $tType,B: $tType,X2: product_prod @ A @ B,A6: set @ A,B6: set @ B] :
( ( member @ ( product_prod @ A @ B ) @ X2
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu: A] : B6 ) )
= ( ( member @ A @ ( product_fst @ A @ B @ X2 ) @ A6 )
& ( member @ B @ ( product_snd @ A @ B @ X2 ) @ B6 ) ) ) ).
% mem_Times_iff
thf(fact_461_product__swap,axiom,
! [B: $tType,A: $tType,A6: set @ B,B6: set @ A] :
( ( image2 @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A )
@ ( product_Sigma @ B @ A @ A6
@ ^ [Uu: B] : B6 ) )
= ( product_Sigma @ A @ B @ B6
@ ^ [Uu: A] : A6 ) ) ).
% product_swap
thf(fact_462_Quotient__compose,axiom,
! [A: $tType,C: $tType,B: $tType,R1: A > A > $o,Abs1: A > B,Rep1: B > A,T1: A > B > $o,R22: B > B > $o,Abs22: B > C,Rep22: C > B,T22: B > C > $o] :
( ( quotient @ A @ B @ R1 @ Abs1 @ Rep1 @ T1 )
=> ( ( quotient @ B @ C @ R22 @ Abs22 @ Rep22 @ T22 )
=> ( quotient @ A @ C @ ( relcompp @ A @ B @ A @ T1 @ ( relcompp @ B @ B @ A @ R22 @ ( 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_463_swap__product,axiom,
! [B: $tType,A: $tType,A6: 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,J: A] : ( product_Pair @ A @ B @ J @ I3 ) )
@ ( product_Sigma @ B @ A @ A6
@ ^ [Uu: B] : B6 ) )
= ( product_Sigma @ A @ B @ B6
@ ^ [Uu: A] : A6 ) ) ).
% swap_product
thf(fact_464_map__prod__surj__on,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType,F: B > A,A6: set @ B,A9: set @ A,G: D > C,B6: set @ D,B8: set @ C] :
( ( ( image2 @ B @ A @ F @ A6 )
= A9 )
=> ( ( ( image2 @ D @ C @ G @ B6 )
= B8 )
=> ( ( image2 @ ( product_prod @ B @ D ) @ ( product_prod @ A @ C ) @ ( product_map_prod @ B @ A @ D @ C @ F @ G )
@ ( product_Sigma @ B @ D @ A6
@ ^ [Uu: B] : B6 ) )
= ( product_Sigma @ A @ C @ A9
@ ^ [Uu: A] : B8 ) ) ) ) ).
% map_prod_surj_on
thf(fact_465_old_Orec__prod__def,axiom,
! [T: $tType,B: $tType,A: $tType] :
( ( product_rec_prod @ A @ B @ T )
= ( ^ [F12: A > B > T,X4: product_prod @ A @ B] : ( the @ T @ ( product_rec_set_prod @ A @ B @ T @ F12 @ X4 ) ) ) ) ).
% old.rec_prod_def
thf(fact_466_left__total__alt__def,axiom,
! [B: $tType,A: $tType] :
( ( left_total @ A @ B )
= ( ^ [R6: A > B > $o] :
( ord_less_eq @ ( A > A > $o )
@ ^ [Y2: A,Z2: A] : Y2 = Z2
@ ( relcompp @ A @ B @ A @ R6 @ ( conversep @ A @ B @ R6 ) ) ) ) ) ).
% left_total_alt_def
thf(fact_467_left__unique__alt__def,axiom,
! [B: $tType,A: $tType] :
( ( left_unique @ A @ B )
= ( ^ [R6: A > B > $o] :
( ord_less_eq @ ( A > A > $o ) @ ( relcompp @ A @ B @ A @ R6 @ ( conversep @ A @ B @ R6 ) )
@ ^ [Y2: A,Z2: A] : Y2 = Z2 ) ) ) ).
% left_unique_alt_def
thf(fact_468_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 ),S4: set @ ( product_prod @ A @ A ),F4: B > A] : ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( inv_image @ A @ B @ R6 @ F4 ) @ ( inv_image @ A @ B @ S4 @ F4 ) ) ) ) ).
% rp_inv_image_def
thf(fact_469_left__total__OO,axiom,
! [A: $tType,C: $tType,B: $tType,R3: A > B > $o,S5: B > C > $o] :
( ( left_total @ A @ B @ R3 )
=> ( ( left_total @ B @ C @ S5 )
=> ( left_total @ A @ C @ ( relcompp @ A @ B @ C @ R3 @ S5 ) ) ) ) ).
% left_total_OO
thf(fact_470_left__unique__OO,axiom,
! [A: $tType,C: $tType,B: $tType,R3: A > B > $o,S5: B > C > $o] :
( ( left_unique @ A @ B @ R3 )
=> ( ( left_unique @ B @ C @ S5 )
=> ( left_unique @ A @ C @ ( relcompp @ A @ B @ C @ R3 @ S5 ) ) ) ) ).
% left_unique_OO
thf(fact_471_image__o__collect,axiom,
! [B: $tType,C: $tType,A: $tType,G: C > B,F7: set @ ( A > ( set @ C ) )] :
( ( bNF_collect @ A @ B @ ( image2 @ ( A > ( set @ C ) ) @ ( A > ( set @ B ) ) @ ( comp @ ( set @ C ) @ ( set @ B ) @ A @ ( image2 @ C @ B @ G ) ) @ F7 ) )
= ( comp @ ( set @ C ) @ ( set @ B ) @ A @ ( image2 @ C @ B @ G ) @ ( bNF_collect @ A @ C @ F7 ) ) ) ).
% image_o_collect
thf(fact_472_Product__Type_Oproduct__def,axiom,
! [B: $tType,A: $tType] :
( ( product_product @ A @ B )
= ( ^ [A8: set @ A,B7: set @ B] :
( product_Sigma @ A @ B @ A8
@ ^ [Uu: A] : B7 ) ) ) ).
% Product_Type.product_def
thf(fact_473_member__product,axiom,
! [B: $tType,A: $tType,X2: product_prod @ A @ B,A6: set @ A,B6: set @ B] :
( ( member @ ( product_prod @ A @ B ) @ X2 @ ( product_product @ A @ B @ A6 @ B6 ) )
= ( member @ ( product_prod @ A @ B ) @ X2
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu: A] : B6 ) ) ) ).
% member_product
thf(fact_474_snd__image__times,axiom,
! [B: $tType,A: $tType,A6: set @ B,B6: set @ A] :
( ( ( A6
= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A )
@ ( product_Sigma @ B @ A @ A6
@ ^ [Uu: B] : B6 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A )
@ ( product_Sigma @ B @ A @ A6
@ ^ [Uu: B] : B6 ) )
= B6 ) ) ) ).
% snd_image_times
thf(fact_475_fst__image__times,axiom,
! [B: $tType,A: $tType,B6: set @ B,A6: set @ A] :
( ( ( B6
= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
@ ( product_Sigma @ A @ B @ A6
@ ^ [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 @ A6
@ ^ [Uu: A] : B6 ) )
= A6 ) ) ) ).
% fst_image_times
thf(fact_476_image__is__empty,axiom,
! [A: $tType,B: $tType,F: B > A,A6: set @ B] :
( ( ( image2 @ B @ A @ F @ A6 )
= ( bot_bot @ ( set @ A ) ) )
= ( A6
= ( bot_bot @ ( set @ B ) ) ) ) ).
% image_is_empty
thf(fact_477_empty__is__image,axiom,
! [A: $tType,B: $tType,F: B > A,A6: set @ B] :
( ( ( bot_bot @ ( set @ A ) )
= ( image2 @ B @ A @ F @ A6 ) )
= ( A6
= ( bot_bot @ ( set @ B ) ) ) ) ).
% empty_is_image
thf(fact_478_image__empty,axiom,
! [B: $tType,A: $tType,F: B > A] :
( ( image2 @ B @ A @ F @ ( bot_bot @ ( set @ B ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% image_empty
thf(fact_479_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_480_Sigma__empty2,axiom,
! [B: $tType,A: $tType,A6: set @ A] :
( ( product_Sigma @ A @ B @ A6
@ ^ [Uu: A] : ( bot_bot @ ( set @ B ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% Sigma_empty2
thf(fact_481_Times__empty,axiom,
! [A: $tType,B: $tType,A6: set @ A,B6: set @ B] :
( ( ( product_Sigma @ A @ B @ A6
@ ^ [Uu: A] : B6 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( ( A6
= ( bot_bot @ ( set @ A ) ) )
| ( B6
= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% Times_empty
thf(fact_482_Collect__const__case__prod,axiom,
! [B: $tType,A: $tType,P2: $o] :
( ( P2
=> ( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A7: A,B5: B] : P2 ) )
= ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) )
& ( ~ P2
=> ( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A7: A,B5: B] : P2 ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% Collect_const_case_prod
thf(fact_483_Sigma__empty__iff,axiom,
! [B: $tType,A: $tType,I4: set @ A,X6: A > ( set @ B )] :
( ( ( product_Sigma @ A @ B @ I4 @ X6 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ I4 )
=> ( ( X6 @ X4 )
= ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% Sigma_empty_iff
thf(fact_484_subset__emptyI,axiom,
! [A: $tType,A6: set @ A] :
( ! [X: A] :
~ ( member @ A @ X @ A6 )
=> ( ord_less_eq @ ( set @ A ) @ A6 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_emptyI
thf(fact_485_less__by__empty,axiom,
! [A: $tType,A6: set @ ( product_prod @ A @ A ),B6: set @ ( product_prod @ A @ A )] :
( ( A6
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ A6 @ B6 ) ) ).
% less_by_empty
thf(fact_486_times__eq__iff,axiom,
! [A: $tType,B: $tType,A6: set @ A,B6: set @ B,C5: set @ A,D3: set @ B] :
( ( ( product_Sigma @ A @ B @ A6
@ ^ [Uu: A] : B6 )
= ( product_Sigma @ A @ B @ C5
@ ^ [Uu: A] : D3 ) )
= ( ( ( A6 = C5 )
& ( B6 = D3 ) )
| ( ( ( A6
= ( bot_bot @ ( set @ A ) ) )
| ( B6
= ( bot_bot @ ( set @ B ) ) ) )
& ( ( C5
= ( bot_bot @ ( set @ A ) ) )
| ( D3
= ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ).
% times_eq_iff
thf(fact_487_times__subset__iff,axiom,
! [A: $tType,B: $tType,A6: set @ A,C5: set @ B,B6: set @ A,D3: set @ B] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu: A] : C5 )
@ ( product_Sigma @ A @ B @ B6
@ ^ [Uu: A] : D3 ) )
= ( ( A6
= ( bot_bot @ ( set @ A ) ) )
| ( C5
= ( bot_bot @ ( set @ B ) ) )
| ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
& ( ord_less_eq @ ( set @ B ) @ C5 @ D3 ) ) ) ) ).
% times_subset_iff
thf(fact_488_fst__image__Sigma,axiom,
! [B: $tType,A: $tType,A6: set @ A,B6: A > ( set @ B )] :
( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( product_Sigma @ A @ B @ A6 @ B6 ) )
= ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A6 )
& ( ( B6 @ X4 )
!= ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% fst_image_Sigma
thf(fact_489_collect__comp,axiom,
! [A: $tType,B: $tType,C: $tType,F7: set @ ( C > ( set @ B ) ),G: A > C] :
( ( comp @ C @ ( set @ B ) @ A @ ( bNF_collect @ C @ B @ F7 ) @ G )
= ( bNF_collect @ A @ B
@ ( image2 @ ( C > ( set @ B ) ) @ ( A > ( set @ B ) )
@ ^ [F4: C > ( set @ B )] : ( comp @ C @ ( set @ B ) @ A @ F4 @ G )
@ F7 ) ) ) ).
% collect_comp
thf(fact_490_empty__natural,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,F: A > C,G: D > B] :
( ( comp @ C @ ( set @ B ) @ A
@ ^ [Uu: C] : ( bot_bot @ ( set @ B ) )
@ F )
= ( comp @ ( set @ D ) @ ( set @ B ) @ A @ ( image2 @ D @ B @ G )
@ ^ [Uu: A] : ( bot_bot @ ( set @ D ) ) ) ) ).
% empty_natural
thf(fact_491_rp__inv__image__rp,axiom,
! [A: $tType,B: $tType,P2: product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ),F: B > A] :
( ( fun_reduction_pair @ A @ P2 )
=> ( fun_reduction_pair @ B @ ( fun_rp_inv_image @ A @ B @ P2 @ F ) ) ) ).
% rp_inv_image_rp
thf(fact_492_the__elem__image__unique,axiom,
! [B: $tType,A: $tType,A6: set @ A,F: A > B,X2: A] :
( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [Y: A] :
( ( member @ A @ Y @ A6 )
=> ( ( F @ Y )
= ( F @ X2 ) ) )
=> ( ( the_elem @ B @ ( image2 @ A @ B @ F @ A6 ) )
= ( F @ X2 ) ) ) ) ).
% the_elem_image_unique
thf(fact_493_snd__image__Sigma,axiom,
! [A: $tType,B: $tType,A6: set @ B,B6: B > ( set @ A )] :
( ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( product_Sigma @ B @ A @ A6 @ B6 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B6 @ A6 ) ) ) ).
% snd_image_Sigma
thf(fact_494_old_Orec__unit__def,axiom,
! [T: $tType] :
( ( product_rec_unit @ T )
= ( ^ [F12: T,X4: product_unit] : ( the @ T @ ( product_rec_set_unit @ T @ F12 @ X4 ) ) ) ) ).
% old.rec_unit_def
thf(fact_495_range__constant,axiom,
! [B: $tType,A: $tType,X2: A] :
( ( image2 @ B @ A
@ ^ [Uu: B] : X2
@ ( top_top @ ( set @ B ) ) )
= ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% range_constant
thf(fact_496_INT__constant,axiom,
! [B: $tType,A: $tType,A6: set @ B,C3: set @ A] :
( ( ( A6
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y5: B] : C3
@ A6 ) )
= ( top_top @ ( set @ A ) ) ) )
& ( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y5: B] : C3
@ A6 ) )
= C3 ) ) ) ).
% INT_constant
thf(fact_497_neg__fun__distr1,axiom,
! [D: $tType,A: $tType,B: $tType,C: $tType,E: $tType,F2: $tType,R3: A > B > $o,R7: B > C > $o,S5: D > F2 > $o,S6: F2 > E > $o] :
( ( left_unique @ A @ B @ R3 )
=> ( ( right_total @ A @ B @ R3 )
=> ( ( right_unique @ B @ C @ R7 )
=> ( ( left_total @ B @ C @ R7 )
=> ( ord_less_eq @ ( ( A > D ) > ( C > E ) > $o ) @ ( bNF_rel_fun @ A @ C @ D @ E @ ( relcompp @ A @ B @ C @ R3 @ R7 ) @ ( relcompp @ D @ F2 @ E @ S5 @ S6 ) ) @ ( relcompp @ ( A > D ) @ ( B > F2 ) @ ( C > E ) @ ( bNF_rel_fun @ A @ B @ D @ F2 @ R3 @ S5 ) @ ( bNF_rel_fun @ B @ C @ F2 @ E @ R7 @ S6 ) ) ) ) ) ) ) ).
% neg_fun_distr1
thf(fact_498_neg__fun__distr2,axiom,
! [F2: $tType,E: $tType,A: $tType,B: $tType,D: $tType,C: $tType,R7: A > B > $o,S6: C > D > $o,R3: E > A > $o,S5: F2 > C > $o] :
( ( right_unique @ A @ B @ R7 )
=> ( ( left_total @ A @ B @ R7 )
=> ( ( left_unique @ C @ D @ S6 )
=> ( ( right_total @ C @ D @ S6 )
=> ( ord_less_eq @ ( ( E > F2 ) > ( B > D ) > $o ) @ ( bNF_rel_fun @ E @ B @ F2 @ D @ ( relcompp @ E @ A @ B @ R3 @ R7 ) @ ( relcompp @ F2 @ C @ D @ S5 @ S6 ) ) @ ( relcompp @ ( E > F2 ) @ ( A > C ) @ ( B > D ) @ ( bNF_rel_fun @ E @ A @ F2 @ C @ R3 @ S5 ) @ ( bNF_rel_fun @ A @ B @ C @ D @ R7 @ S6 ) ) ) ) ) ) ) ).
% neg_fun_distr2
thf(fact_499_refl__onI,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A6: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ A6
@ ^ [Uu: A] : A6 ) )
=> ( ! [X: A] :
( ( member @ A @ X @ A6 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ R ) )
=> ( refl_on @ A @ A6 @ R ) ) ) ).
% refl_onI
thf(fact_500_refl__on__def,axiom,
! [A: $tType] :
( ( refl_on @ A )
= ( ^ [A8: set @ A,R5: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R5
@ ( product_Sigma @ A @ A @ A8
@ ^ [Uu: A] : A8 ) )
& ! [X4: A] :
( ( member @ A @ X4 @ A8 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) @ R5 ) ) ) ) ) ).
% refl_on_def
thf(fact_501_right__unique__alt__def,axiom,
! [B: $tType,A: $tType] :
( ( right_unique @ A @ B )
= ( ^ [R6: A > B > $o] :
( ord_less_eq @ ( B > B > $o ) @ ( relcompp @ B @ A @ B @ ( conversep @ A @ B @ R6 ) @ R6 )
@ ^ [Y2: B,Z2: B] : Y2 = Z2 ) ) ) ).
% right_unique_alt_def
thf(fact_502_image__insert,axiom,
! [A: $tType,B: $tType,F: B > A,A3: B,B6: set @ B] :
( ( image2 @ B @ A @ F @ ( insert @ B @ A3 @ B6 ) )
= ( insert @ A @ ( F @ A3 ) @ ( image2 @ B @ A @ F @ B6 ) ) ) ).
% image_insert
thf(fact_503_insert__image,axiom,
! [B: $tType,A: $tType,X2: A,A6: set @ A,F: A > B] :
( ( member @ A @ X2 @ A6 )
=> ( ( insert @ B @ ( F @ X2 ) @ ( image2 @ A @ B @ F @ A6 ) )
= ( image2 @ A @ B @ F @ A6 ) ) ) ).
% insert_image
thf(fact_504_UN__ball__bex__simps_I4_J,axiom,
! [F2: $tType,E: $tType,B6: E > ( set @ F2 ),A6: set @ E,P2: F2 > $o] :
( ( ? [X4: F2] :
( ( member @ F2 @ X4 @ ( complete_Sup_Sup @ ( set @ F2 ) @ ( image2 @ E @ ( set @ F2 ) @ B6 @ A6 ) ) )
& ( P2 @ X4 ) ) )
= ( ? [X4: E] :
( ( member @ E @ X4 @ A6 )
& ? [Y5: F2] :
( ( member @ F2 @ Y5 @ ( B6 @ X4 ) )
& ( P2 @ Y5 ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_505_UN__ball__bex__simps_I2_J,axiom,
! [C: $tType,B: $tType,B6: B > ( set @ C ),A6: set @ B,P2: C > $o] :
( ( ! [X4: C] :
( ( member @ C @ X4 @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ B6 @ A6 ) ) )
=> ( P2 @ X4 ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A6 )
=> ! [Y5: C] :
( ( member @ C @ Y5 @ ( B6 @ X4 ) )
=> ( P2 @ Y5 ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_506_bex__UN,axiom,
! [A: $tType,B: $tType,B6: B > ( set @ A ),A6: set @ B,P2: A > $o] :
( ( ? [X4: A] :
( ( member @ A @ X4 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B6 @ A6 ) ) )
& ( P2 @ X4 ) ) )
= ( ? [X4: B] :
( ( member @ B @ X4 @ A6 )
& ? [Y5: A] :
( ( member @ A @ Y5 @ ( B6 @ X4 ) )
& ( P2 @ Y5 ) ) ) ) ) ).
% bex_UN
thf(fact_507_ball__UN,axiom,
! [A: $tType,B: $tType,B6: B > ( set @ A ),A6: set @ B,P2: A > $o] :
( ( ! [X4: A] :
( ( member @ A @ X4 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B6 @ A6 ) ) )
=> ( P2 @ X4 ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A6 )
=> ! [Y5: A] :
( ( member @ A @ Y5 @ ( B6 @ X4 ) )
=> ( P2 @ Y5 ) ) ) ) ) ).
% ball_UN
thf(fact_508_SUP__identity__eq,axiom,
! [A: $tType] :
( ( complete_Sup @ A )
=> ! [A6: set @ A] :
( ( complete_Sup_Sup @ A
@ ( image2 @ A @ A
@ ^ [X4: A] : X4
@ A6 ) )
= ( complete_Sup_Sup @ A @ A6 ) ) ) ).
% SUP_identity_eq
thf(fact_509_INF__identity__eq,axiom,
! [A: $tType] :
( ( complete_Inf @ A )
=> ! [A6: set @ A] :
( ( complete_Inf_Inf @ A
@ ( image2 @ A @ A
@ ^ [X4: A] : X4
@ A6 ) )
= ( complete_Inf_Inf @ A @ A6 ) ) ) ).
% INF_identity_eq
thf(fact_510_UN__iff,axiom,
! [A: $tType,B: $tType,B2: A,B6: B > ( set @ A ),A6: set @ B] :
( ( member @ A @ B2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B6 @ A6 ) ) )
= ( ? [X4: B] :
( ( member @ B @ X4 @ A6 )
& ( member @ A @ B2 @ ( B6 @ X4 ) ) ) ) ) ).
% UN_iff
thf(fact_511_UN__I,axiom,
! [B: $tType,A: $tType,A3: A,A6: set @ A,B2: B,B6: A > ( set @ B )] :
( ( member @ A @ A3 @ A6 )
=> ( ( member @ B @ B2 @ ( B6 @ A3 ) )
=> ( member @ B @ B2 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B6 @ A6 ) ) ) ) ) ).
% UN_I
thf(fact_512_INT__iff,axiom,
! [A: $tType,B: $tType,B2: A,B6: B > ( set @ A ),A6: set @ B] :
( ( member @ A @ B2 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B6 @ A6 ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A6 )
=> ( member @ A @ B2 @ ( B6 @ X4 ) ) ) ) ) ).
% INT_iff
thf(fact_513_INT__I,axiom,
! [B: $tType,A: $tType,A6: set @ A,B2: B,B6: A > ( set @ B )] :
( ! [X: A] :
( ( member @ A @ X @ A6 )
=> ( member @ B @ B2 @ ( B6 @ X ) ) )
=> ( member @ B @ B2 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B6 @ A6 ) ) ) ) ).
% INT_I
thf(fact_514_Sup__apply,axiom,
! [B: $tType,A: $tType] :
( ( complete_Sup @ B )
=> ( ( complete_Sup_Sup @ ( A > B ) )
= ( ^ [A8: set @ ( A > B ),X4: A] :
( complete_Sup_Sup @ B
@ ( image2 @ ( A > B ) @ B
@ ^ [F4: A > B] : ( F4 @ X4 )
@ A8 ) ) ) ) ) ).
% Sup_apply
thf(fact_515_Inf__apply,axiom,
! [B: $tType,A: $tType] :
( ( complete_Inf @ B )
=> ( ( complete_Inf_Inf @ ( A > B ) )
= ( ^ [A8: set @ ( A > B ),X4: A] :
( complete_Inf_Inf @ B
@ ( image2 @ ( A > B ) @ B
@ ^ [F4: A > B] : ( F4 @ X4 )
@ A8 ) ) ) ) ) ).
% Inf_apply
thf(fact_516_SUP__id__eq,axiom,
! [A: $tType] :
( ( complete_Sup @ A )
=> ! [A6: set @ A] :
( ( complete_Sup_Sup @ A @ ( image2 @ A @ A @ ( id @ A ) @ A6 ) )
= ( complete_Sup_Sup @ A @ A6 ) ) ) ).
% SUP_id_eq
thf(fact_517_INF__id__eq,axiom,
! [A: $tType] :
( ( complete_Inf @ A )
=> ! [A6: set @ A] :
( ( complete_Inf_Inf @ A @ ( image2 @ A @ A @ ( id @ A ) @ A6 ) )
= ( complete_Inf_Inf @ A @ A6 ) ) ) ).
% INF_id_eq
thf(fact_518_SUP__bot,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ B] :
( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [X4: B] : ( bot_bot @ A )
@ A6 ) )
= ( bot_bot @ A ) ) ) ).
% SUP_bot
thf(fact_519_SUP__bot__conv_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B6: B > A,A6: set @ B] :
( ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ B6 @ A6 ) )
= ( bot_bot @ A ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A6 )
=> ( ( B6 @ X4 )
= ( bot_bot @ A ) ) ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_520_SUP__bot__conv_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B6: B > A,A6: set @ B] :
( ( ( bot_bot @ A )
= ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ B6 @ A6 ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A6 )
=> ( ( B6 @ X4 )
= ( bot_bot @ A ) ) ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_521_SUP__const,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ B,F: A] :
( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [I3: B] : F
@ A6 ) )
= F ) ) ) ).
% SUP_const
thf(fact_522_INF__const,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ B,F: A] :
( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [I3: B] : F
@ A6 ) )
= F ) ) ) ).
% INF_const
thf(fact_523_INF__top__conv_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B6: B > A,A6: set @ B] :
( ( ( top_top @ A )
= ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ B6 @ A6 ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A6 )
=> ( ( B6 @ X4 )
= ( top_top @ A ) ) ) ) ) ) ).
% INF_top_conv(2)
thf(fact_524_INF__top__conv_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B6: B > A,A6: set @ B] :
( ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ B6 @ A6 ) )
= ( top_top @ A ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A6 )
=> ( ( B6 @ X4 )
= ( top_top @ A ) ) ) ) ) ) ).
% INF_top_conv(1)
thf(fact_525_INF__top,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ B] :
( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [X4: B] : ( top_top @ A )
@ A6 ) )
= ( top_top @ A ) ) ) ).
% INF_top
thf(fact_526_UN__constant,axiom,
! [B: $tType,A: $tType,A6: set @ B,C3: set @ A] :
( ( ( A6
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y5: B] : C3
@ A6 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y5: B] : C3
@ A6 ) )
= C3 ) ) ) ).
% UN_constant
thf(fact_527_SUP__apply,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( complete_Sup @ A )
=> ! [F: C > B > A,A6: set @ C,X2: B] :
( ( complete_Sup_Sup @ ( B > A ) @ ( image2 @ C @ ( B > A ) @ F @ A6 ) @ X2 )
= ( complete_Sup_Sup @ A
@ ( image2 @ C @ A
@ ^ [Y5: C] : ( F @ Y5 @ X2 )
@ A6 ) ) ) ) ).
% SUP_apply
thf(fact_528_INF__apply,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( complete_Inf @ A )
=> ! [F: C > B > A,A6: set @ C,X2: B] :
( ( complete_Inf_Inf @ ( B > A ) @ ( image2 @ C @ ( B > A ) @ F @ A6 ) @ X2 )
= ( complete_Inf_Inf @ A
@ ( image2 @ C @ A
@ ^ [Y5: C] : ( F @ Y5 @ X2 )
@ A6 ) ) ) ) ).
% INF_apply
thf(fact_529_UN__simps_I1_J,axiom,
! [A: $tType,B: $tType,C5: set @ B,A3: A,B6: B > ( set @ A )] :
( ( ( C5
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X4: B] : ( insert @ A @ A3 @ ( B6 @ X4 ) )
@ C5 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( C5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X4: B] : ( insert @ A @ A3 @ ( B6 @ X4 ) )
@ C5 ) )
= ( insert @ A @ A3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B6 @ C5 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_530_UN__singleton,axiom,
! [A: $tType,A6: set @ A] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ A @ ( set @ A )
@ ^ [X4: A] : ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) )
@ A6 ) )
= A6 ) ).
% UN_singleton
thf(fact_531_refl__on__UNION,axiom,
! [B: $tType,A: $tType,S5: set @ A,A6: A > ( set @ B ),R: A > ( set @ ( product_prod @ B @ B ) )] :
( ! [X: A] :
( ( member @ A @ X @ S5 )
=> ( refl_on @ B @ ( A6 @ X ) @ ( R @ X ) ) )
=> ( refl_on @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A6 @ S5 ) ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ B @ B ) ) @ ( image2 @ A @ ( set @ ( product_prod @ B @ B ) ) @ R @ S5 ) ) ) ) ).
% refl_on_UNION
thf(fact_532_refl__on__INTER,axiom,
! [B: $tType,A: $tType,S5: set @ A,A6: A > ( set @ B ),R: A > ( set @ ( product_prod @ B @ B ) )] :
( ! [X: A] :
( ( member @ A @ X @ S5 )
=> ( refl_on @ B @ ( A6 @ X ) @ ( R @ X ) ) )
=> ( refl_on @ B @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A6 @ S5 ) ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ B @ B ) ) @ ( image2 @ A @ ( set @ ( product_prod @ B @ B ) ) @ R @ S5 ) ) ) ) ).
% refl_on_INTER
thf(fact_533_refl__on__singleton,axiom,
! [A: $tType,X2: A] : ( refl_on @ A @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% refl_on_singleton
thf(fact_534_INT__extend__simps_I9_J,axiom,
! [S7: $tType,R8: $tType,Q4: $tType,C5: R8 > ( set @ S7 ),B6: Q4 > ( set @ R8 ),A6: set @ Q4] :
( ( complete_Inf_Inf @ ( set @ S7 )
@ ( image2 @ Q4 @ ( set @ S7 )
@ ^ [X4: Q4] : ( complete_Inf_Inf @ ( set @ S7 ) @ ( image2 @ R8 @ ( set @ S7 ) @ C5 @ ( B6 @ X4 ) ) )
@ A6 ) )
= ( complete_Inf_Inf @ ( set @ S7 ) @ ( image2 @ R8 @ ( set @ S7 ) @ C5 @ ( complete_Sup_Sup @ ( set @ R8 ) @ ( image2 @ Q4 @ ( set @ R8 ) @ B6 @ A6 ) ) ) ) ) ).
% INT_extend_simps(9)
thf(fact_535_INT__insert__distrib,axiom,
! [B: $tType,A: $tType,U2: A,A6: set @ A,A3: B,B6: A > ( set @ B )] :
( ( member @ A @ U2 @ A6 )
=> ( ( complete_Inf_Inf @ ( set @ B )
@ ( image2 @ A @ ( set @ B )
@ ^ [X4: A] : ( insert @ B @ A3 @ ( B6 @ X4 ) )
@ A6 ) )
= ( insert @ B @ A3 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B6 @ A6 ) ) ) ) ) ).
% INT_insert_distrib
thf(fact_536_INT__extend__simps_I5_J,axiom,
! [I5: $tType,J2: $tType,A3: I5,B6: J2 > ( set @ I5 ),C5: set @ J2] :
( ( insert @ I5 @ A3 @ ( complete_Inf_Inf @ ( set @ I5 ) @ ( image2 @ J2 @ ( set @ I5 ) @ B6 @ C5 ) ) )
= ( complete_Inf_Inf @ ( set @ I5 )
@ ( image2 @ J2 @ ( set @ I5 )
@ ^ [X4: J2] : ( insert @ I5 @ A3 @ ( B6 @ X4 ) )
@ C5 ) ) ) ).
% INT_extend_simps(5)
thf(fact_537_UN__insert__distrib,axiom,
! [B: $tType,A: $tType,U2: A,A6: set @ A,A3: B,B6: A > ( set @ B )] :
( ( member @ A @ U2 @ A6 )
=> ( ( complete_Sup_Sup @ ( set @ B )
@ ( image2 @ A @ ( set @ B )
@ ^ [X4: A] : ( insert @ B @ A3 @ ( B6 @ X4 ) )
@ A6 ) )
= ( insert @ B @ A3 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B6 @ A6 ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_538_Sigma__Union,axiom,
! [B: $tType,A: $tType,X6: set @ ( set @ A ),B6: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ ( complete_Sup_Sup @ ( set @ A ) @ X6 ) @ B6 )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ ( set @ A ) @ ( set @ ( product_prod @ A @ B ) )
@ ^ [A8: set @ A] : ( product_Sigma @ A @ B @ A8 @ B6 )
@ X6 ) ) ) ).
% Sigma_Union
thf(fact_539_Inf__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( complete_Inf @ B )
=> ( ( complete_Inf_Inf @ ( A > B ) )
= ( ^ [A8: set @ ( A > B ),X4: A] :
( complete_Inf_Inf @ B
@ ( image2 @ ( A > B ) @ B
@ ^ [F4: A > B] : ( F4 @ X4 )
@ A8 ) ) ) ) ) ).
% Inf_fun_def
thf(fact_540_Sup__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( complete_Sup @ B )
=> ( ( complete_Sup_Sup @ ( A > B ) )
= ( ^ [A8: set @ ( A > B ),X4: A] :
( complete_Sup_Sup @ B
@ ( image2 @ ( A > B ) @ B
@ ^ [F4: A > B] : ( F4 @ X4 )
@ A8 ) ) ) ) ) ).
% Sup_fun_def
thf(fact_541_INF__cong,axiom,
! [A: $tType,B: $tType] :
( ( complete_Inf @ A )
=> ! [A6: set @ B,B6: set @ B,C5: B > A,D3: B > A] :
( ( A6 = B6 )
=> ( ! [X: B] :
( ( member @ B @ X @ B6 )
=> ( ( C5 @ X )
= ( D3 @ X ) ) )
=> ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ C5 @ A6 ) )
= ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ D3 @ B6 ) ) ) ) ) ) ).
% INF_cong
thf(fact_542_SUP__cong,axiom,
! [A: $tType,B: $tType] :
( ( complete_Sup @ A )
=> ! [A6: set @ B,B6: set @ B,C5: B > A,D3: B > A] :
( ( A6 = B6 )
=> ( ! [X: B] :
( ( member @ B @ X @ B6 )
=> ( ( C5 @ X )
= ( D3 @ X ) ) )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ C5 @ A6 ) )
= ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ D3 @ B6 ) ) ) ) ) ) ).
% SUP_cong
thf(fact_543_SUP__UNION,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: B > A,G: C > ( set @ B ),A6: set @ C] :
( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ C @ ( set @ B ) @ G @ A6 ) ) ) )
= ( complete_Sup_Sup @ A
@ ( image2 @ C @ A
@ ^ [Y5: C] : ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ ( G @ Y5 ) ) )
@ A6 ) ) ) ) ).
% SUP_UNION
thf(fact_544_insert__subsetI,axiom,
! [A: $tType,X2: A,A6: set @ A,X6: set @ A] :
( ( member @ A @ X2 @ A6 )
=> ( ( ord_less_eq @ ( set @ A ) @ X6 @ A6 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X2 @ X6 ) @ A6 ) ) ) ).
% insert_subsetI
thf(fact_545_UN__extend__simps_I1_J,axiom,
! [A: $tType,B: $tType,C5: set @ B,A3: A,B6: B > ( set @ A )] :
( ( ( C5
= ( bot_bot @ ( set @ B ) ) )
=> ( ( insert @ A @ A3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B6 @ C5 ) ) )
= ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ( C5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( insert @ A @ A3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B6 @ C5 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X4: B] : ( insert @ A @ A3 @ ( B6 @ X4 ) )
@ C5 ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_546_Sigma__def,axiom,
! [B: $tType,A: $tType] :
( ( product_Sigma @ A @ B )
= ( ^ [A8: set @ A,B7: A > ( set @ B )] :
( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ A @ ( set @ ( product_prod @ A @ B ) )
@ ^ [X4: A] :
( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ B @ ( set @ ( product_prod @ A @ B ) )
@ ^ [Y5: B] : ( insert @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
@ ( B7 @ X4 ) ) )
@ A8 ) ) ) ) ).
% Sigma_def
thf(fact_547_UN__Times__distrib,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,E4: C > ( set @ A ),F7: D > ( set @ B ),A6: 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] : ( F7 @ B5 ) ) )
@ ( product_Sigma @ C @ D @ A6
@ ^ [Uu: C] : B6 ) ) )
= ( product_Sigma @ A @ B @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ C @ ( set @ A ) @ E4 @ A6 ) )
@ ^ [Uu: A] : ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ D @ ( set @ B ) @ F7 @ B6 ) ) ) ) ).
% UN_Times_distrib
thf(fact_548_Pow__INT__eq,axiom,
! [A: $tType,B: $tType,B6: B > ( set @ A ),A6: set @ B] :
( ( pow @ A @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B6 @ A6 ) ) )
= ( complete_Inf_Inf @ ( set @ ( set @ A ) )
@ ( image2 @ B @ ( set @ ( set @ A ) )
@ ^ [X4: B] : ( pow @ A @ ( B6 @ X4 ) )
@ A6 ) ) ) ).
% Pow_INT_eq
thf(fact_549_right__total__OO,axiom,
! [A: $tType,C: $tType,B: $tType,A6: A > B > $o,B6: B > C > $o] :
( ( right_total @ A @ B @ A6 )
=> ( ( right_total @ B @ C @ B6 )
=> ( right_total @ A @ C @ ( relcompp @ A @ B @ C @ A6 @ B6 ) ) ) ) ).
% right_total_OO
thf(fact_550_refl__onD,axiom,
! [A: $tType,A6: set @ A,R: set @ ( product_prod @ A @ A ),A3: A] :
( ( refl_on @ A @ A6 @ R )
=> ( ( member @ A @ A3 @ A6 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ R ) ) ) ).
% refl_onD
thf(fact_551_refl__onD1,axiom,
! [A: $tType,A6: set @ A,R: set @ ( product_prod @ A @ A ),X2: A,Y3: A] :
( ( refl_on @ A @ A6 @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R )
=> ( member @ A @ X2 @ A6 ) ) ) ).
% refl_onD1
thf(fact_552_refl__onD2,axiom,
! [A: $tType,A6: set @ A,R: set @ ( product_prod @ A @ A ),X2: A,Y3: A] :
( ( refl_on @ A @ A6 @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R )
=> ( member @ A @ Y3 @ A6 ) ) ) ).
% refl_onD2
thf(fact_553_INT__extend__simps_I8_J,axiom,
! [P5: $tType,O: $tType,B6: O > ( set @ P5 ),A6: set @ ( set @ O )] :
( ( complete_Inf_Inf @ ( set @ P5 )
@ ( image2 @ ( set @ O ) @ ( set @ P5 )
@ ^ [Y5: set @ O] : ( complete_Inf_Inf @ ( set @ P5 ) @ ( image2 @ O @ ( set @ P5 ) @ B6 @ Y5 ) )
@ A6 ) )
= ( complete_Inf_Inf @ ( set @ P5 ) @ ( image2 @ O @ ( set @ P5 ) @ B6 @ ( complete_Sup_Sup @ ( set @ O ) @ A6 ) ) ) ) ).
% INT_extend_simps(8)
thf(fact_554_Id__on__def,axiom,
! [A: $tType] :
( ( id_on @ A )
= ( ^ [A8: set @ A] :
( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ A @ ( set @ ( product_prod @ A @ A ) )
@ ^ [X4: A] : ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
@ A8 ) ) ) ) ).
% Id_on_def
thf(fact_555_INF__le__SUP,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ B,F: B > A] :
( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A6 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A6 ) ) ) ) ) ).
% INF_le_SUP
thf(fact_556_right__unique__OO,axiom,
! [A: $tType,C: $tType,B: $tType,A6: A > B > $o,B6: B > C > $o] :
( ( right_unique @ A @ B @ A6 )
=> ( ( right_unique @ B @ C @ B6 )
=> ( right_unique @ A @ C @ ( relcompp @ A @ B @ C @ A6 @ B6 ) ) ) ) ).
% right_unique_OO
thf(fact_557_INF__commute,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: B > C > A,B6: set @ C,A6: set @ B] :
( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [I3: B] : ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ ( F @ I3 ) @ B6 ) )
@ A6 ) )
= ( complete_Inf_Inf @ A
@ ( image2 @ C @ A
@ ^ [J: C] :
( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [I3: B] : ( F @ I3 @ J )
@ A6 ) )
@ B6 ) ) ) ) ).
% INF_commute
thf(fact_558_SUP__commute,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: B > C > A,B6: set @ C,A6: set @ B] :
( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [I3: B] : ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ ( F @ I3 ) @ B6 ) )
@ A6 ) )
= ( complete_Sup_Sup @ A
@ ( image2 @ C @ A
@ ^ [J: C] :
( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [I3: B] : ( F @ I3 @ J )
@ A6 ) )
@ B6 ) ) ) ) ).
% SUP_commute
thf(fact_559_INT__E,axiom,
! [A: $tType,B: $tType,B2: A,B6: B > ( set @ A ),A6: set @ B,A3: B] :
( ( member @ A @ B2 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B6 @ A6 ) ) )
=> ( ~ ( member @ A @ B2 @ ( B6 @ A3 ) )
=> ~ ( member @ B @ A3 @ A6 ) ) ) ).
% INT_E
thf(fact_560_INT__D,axiom,
! [A: $tType,B: $tType,B2: A,B6: B > ( set @ A ),A6: set @ B,A3: B] :
( ( member @ A @ B2 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B6 @ A6 ) ) )
=> ( ( member @ B @ A3 @ A6 )
=> ( member @ A @ B2 @ ( B6 @ A3 ) ) ) ) ).
% INT_D
thf(fact_561_UN__UN__flatten,axiom,
! [A: $tType,B: $tType,C: $tType,C5: B > ( set @ A ),B6: C > ( set @ B ),A6: set @ C] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ C5 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ C @ ( set @ B ) @ B6 @ A6 ) ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [Y5: C] : ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ C5 @ ( B6 @ Y5 ) ) )
@ A6 ) ) ) ).
% UN_UN_flatten
thf(fact_562_UN__E,axiom,
! [A: $tType,B: $tType,B2: A,B6: B > ( set @ A ),A6: set @ B] :
( ( member @ A @ B2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B6 @ A6 ) ) )
=> ~ ! [X: B] :
( ( member @ B @ X @ A6 )
=> ~ ( member @ A @ B2 @ ( B6 @ X ) ) ) ) ).
% UN_E
thf(fact_563_UN__extend__simps_I9_J,axiom,
! [S7: $tType,R8: $tType,Q4: $tType,C5: R8 > ( set @ S7 ),B6: Q4 > ( set @ R8 ),A6: set @ Q4] :
( ( complete_Sup_Sup @ ( set @ S7 )
@ ( image2 @ Q4 @ ( set @ S7 )
@ ^ [X4: Q4] : ( complete_Sup_Sup @ ( set @ S7 ) @ ( image2 @ R8 @ ( set @ S7 ) @ C5 @ ( B6 @ X4 ) ) )
@ A6 ) )
= ( complete_Sup_Sup @ ( set @ S7 ) @ ( image2 @ R8 @ ( set @ S7 ) @ C5 @ ( complete_Sup_Sup @ ( set @ R8 ) @ ( image2 @ Q4 @ ( set @ R8 ) @ B6 @ A6 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_564_UNION__singleton__eq__range,axiom,
! [A: $tType,B: $tType,F: B > A,A6: set @ B] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X4: B] : ( insert @ A @ ( F @ X4 ) @ ( bot_bot @ ( set @ A ) ) )
@ A6 ) )
= ( image2 @ B @ A @ F @ A6 ) ) ).
% UNION_singleton_eq_range
thf(fact_565_in__Union__o__assoc,axiom,
! [B: $tType,A: $tType,C: $tType,X2: A,Gset: B > ( set @ ( set @ A ) ),Gmap: C > B,A6: C] :
( ( member @ A @ X2 @ ( comp @ B @ ( set @ A ) @ C @ ( comp @ ( set @ ( set @ A ) ) @ ( set @ A ) @ B @ ( complete_Sup_Sup @ ( set @ A ) ) @ Gset ) @ Gmap @ A6 ) )
=> ( member @ A @ X2 @ ( comp @ ( set @ ( set @ A ) ) @ ( set @ A ) @ C @ ( complete_Sup_Sup @ ( set @ A ) ) @ ( comp @ B @ ( set @ ( set @ A ) ) @ C @ Gset @ Gmap ) @ A6 ) ) ) ).
% in_Union_o_assoc
thf(fact_566_UN__Pow__subset,axiom,
! [A: $tType,B: $tType,B6: B > ( set @ A ),A6: set @ B] :
( ord_less_eq @ ( set @ ( set @ A ) )
@ ( complete_Sup_Sup @ ( set @ ( set @ A ) )
@ ( image2 @ B @ ( set @ ( set @ A ) )
@ ^ [X4: B] : ( pow @ A @ ( B6 @ X4 ) )
@ A6 ) )
@ ( pow @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B6 @ A6 ) ) ) ) ).
% UN_Pow_subset
thf(fact_567_INF__eq,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ B,B6: set @ C,G: C > A,F: B > A] :
( ! [I6: B] :
( ( member @ B @ I6 @ A6 )
=> ? [X5: C] :
( ( member @ C @ X5 @ B6 )
& ( ord_less_eq @ A @ ( G @ X5 ) @ ( F @ I6 ) ) ) )
=> ( ! [J3: C] :
( ( member @ C @ J3 @ B6 )
=> ? [X5: B] :
( ( member @ B @ X5 @ A6 )
& ( ord_less_eq @ A @ ( F @ X5 ) @ ( G @ J3 ) ) ) )
=> ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A6 ) )
= ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ G @ B6 ) ) ) ) ) ) ).
% INF_eq
thf(fact_568_SUP__eq,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ B,B6: set @ C,F: B > A,G: C > A] :
( ! [I6: B] :
( ( member @ B @ I6 @ A6 )
=> ? [X5: C] :
( ( member @ C @ X5 @ B6 )
& ( ord_less_eq @ A @ ( F @ I6 ) @ ( G @ X5 ) ) ) )
=> ( ! [J3: C] :
( ( member @ C @ J3 @ B6 )
=> ? [X5: B] :
( ( member @ B @ X5 @ A6 )
& ( ord_less_eq @ A @ ( G @ J3 ) @ ( F @ X5 ) ) ) )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A6 ) )
= ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ G @ B6 ) ) ) ) ) ) ).
% SUP_eq
thf(fact_569_INF__eq__const,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I4: set @ B,F: B > A,X2: A] :
( ( I4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I6: B] :
( ( member @ B @ I6 @ I4 )
=> ( ( F @ I6 )
= X2 ) )
=> ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ I4 ) )
= X2 ) ) ) ) ).
% INF_eq_const
thf(fact_570_SUP__eq__const,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I4: set @ B,F: B > A,X2: A] :
( ( I4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I6: B] :
( ( member @ B @ I6 @ I4 )
=> ( ( F @ I6 )
= X2 ) )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ I4 ) )
= X2 ) ) ) ) ).
% SUP_eq_const
thf(fact_571_INF__image,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( complete_Inf @ A )
=> ! [G: B > A,F: C > B,A6: set @ C] :
( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G @ ( image2 @ C @ B @ F @ A6 ) ) )
= ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ ( comp @ B @ A @ C @ G @ F ) @ A6 ) ) ) ) ).
% INF_image
thf(fact_572_SUP__image,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( complete_Sup @ A )
=> ! [G: B > A,F: C > B,A6: set @ C] :
( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G @ ( image2 @ C @ B @ F @ A6 ) ) )
= ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ ( comp @ B @ A @ C @ G @ F ) @ A6 ) ) ) ) ).
% SUP_image
thf(fact_573_collect__def,axiom,
! [A: $tType,B: $tType] :
( ( bNF_collect @ B @ A )
= ( ^ [F8: set @ ( B > ( set @ A ) ),X4: B] :
( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ ( B > ( set @ A ) ) @ ( set @ A )
@ ^ [F4: B > ( set @ A )] : ( F4 @ X4 )
@ F8 ) ) ) ) ).
% collect_def
thf(fact_574_INF__greatest,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ B,U2: A,F: B > A] :
( ! [I6: B] :
( ( member @ B @ I6 @ A6 )
=> ( ord_less_eq @ A @ U2 @ ( F @ I6 ) ) )
=> ( ord_less_eq @ A @ U2 @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A6 ) ) ) ) ) ).
% INF_greatest
thf(fact_575_le__INF__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [U2: A,F: B > A,A6: set @ B] :
( ( ord_less_eq @ A @ U2 @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A6 ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A6 )
=> ( ord_less_eq @ A @ U2 @ ( F @ X4 ) ) ) ) ) ) ).
% le_INF_iff
thf(fact_576_INF__lower2,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I: B,A6: set @ B,F: B > A,U2: A] :
( ( member @ B @ I @ A6 )
=> ( ( ord_less_eq @ A @ ( F @ I ) @ U2 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A6 ) ) @ U2 ) ) ) ) ).
% INF_lower2
thf(fact_577_INF__mono_H,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: B > A,G: B > A,A6: set @ B] :
( ! [X: B] : ( ord_less_eq @ A @ ( F @ X ) @ ( G @ X ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A6 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G @ A6 ) ) ) ) ) ).
% INF_mono'
thf(fact_578_INF__lower,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I: B,A6: set @ B,F: B > A] :
( ( member @ B @ I @ A6 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A6 ) ) @ ( F @ I ) ) ) ) ).
% INF_lower
thf(fact_579_INF__mono,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B6: set @ B,A6: set @ C,F: C > A,G: B > A] :
( ! [M3: B] :
( ( member @ B @ M3 @ B6 )
=> ? [X5: C] :
( ( member @ C @ X5 @ A6 )
& ( ord_less_eq @ A @ ( F @ X5 ) @ ( G @ M3 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ F @ A6 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G @ B6 ) ) ) ) ) ).
% INF_mono
thf(fact_580_INF__eqI,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ B,X2: A,F: B > A] :
( ! [I6: B] :
( ( member @ B @ I6 @ A6 )
=> ( ord_less_eq @ A @ X2 @ ( F @ I6 ) ) )
=> ( ! [Y: A] :
( ! [I7: B] :
( ( member @ B @ I7 @ A6 )
=> ( ord_less_eq @ A @ Y @ ( F @ I7 ) ) )
=> ( ord_less_eq @ A @ Y @ X2 ) )
=> ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A6 ) )
= X2 ) ) ) ) ).
% INF_eqI
thf(fact_581_SUP__upper2,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I: B,A6: set @ B,U2: A,F: B > A] :
( ( member @ B @ I @ A6 )
=> ( ( ord_less_eq @ A @ U2 @ ( F @ I ) )
=> ( ord_less_eq @ A @ U2 @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A6 ) ) ) ) ) ) ).
% SUP_upper2
thf(fact_582_SUP__le__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: B > A,A6: set @ B,U2: A] :
( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A6 ) ) @ U2 )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A6 )
=> ( ord_less_eq @ A @ ( F @ X4 ) @ U2 ) ) ) ) ) ).
% SUP_le_iff
thf(fact_583_SUP__upper,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I: B,A6: set @ B,F: B > A] :
( ( member @ B @ I @ A6 )
=> ( ord_less_eq @ A @ ( F @ I ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A6 ) ) ) ) ) ).
% SUP_upper
thf(fact_584_SUP__mono_H,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: B > A,G: B > A,A6: set @ B] :
( ! [X: B] : ( ord_less_eq @ A @ ( F @ X ) @ ( G @ X ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A6 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G @ A6 ) ) ) ) ) ).
% SUP_mono'
thf(fact_585_SUP__least,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ B,F: B > A,U2: A] :
( ! [I6: B] :
( ( member @ B @ I6 @ A6 )
=> ( ord_less_eq @ A @ ( F @ I6 ) @ U2 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A6 ) ) @ U2 ) ) ) ).
% SUP_least
thf(fact_586_SUP__mono,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ B,B6: set @ C,F: B > A,G: C > A] :
( ! [N2: B] :
( ( member @ B @ N2 @ A6 )
=> ? [X5: C] :
( ( member @ C @ X5 @ B6 )
& ( ord_less_eq @ A @ ( F @ N2 ) @ ( G @ X5 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A6 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ G @ B6 ) ) ) ) ) ).
% SUP_mono
thf(fact_587_SUP__eqI,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ B,F: B > A,X2: A] :
( ! [I6: B] :
( ( member @ B @ I6 @ A6 )
=> ( ord_less_eq @ A @ ( F @ I6 ) @ X2 ) )
=> ( ! [Y: A] :
( ! [I7: B] :
( ( member @ B @ I7 @ A6 )
=> ( ord_less_eq @ A @ ( F @ I7 ) @ Y ) )
=> ( ord_less_eq @ A @ X2 @ Y ) )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A6 ) )
= X2 ) ) ) ) ).
% SUP_eqI
thf(fact_588_INT__extend__simps_I10_J,axiom,
! [V5: $tType,U3: $tType,T: $tType,B6: U3 > ( set @ V5 ),F: T > U3,A6: set @ T] :
( ( complete_Inf_Inf @ ( set @ V5 )
@ ( image2 @ T @ ( set @ V5 )
@ ^ [A7: T] : ( B6 @ ( F @ A7 ) )
@ A6 ) )
= ( complete_Inf_Inf @ ( set @ V5 ) @ ( image2 @ U3 @ ( set @ V5 ) @ B6 @ ( image2 @ T @ U3 @ F @ A6 ) ) ) ) ).
% INT_extend_simps(10)
thf(fact_589_INT__lower,axiom,
! [B: $tType,A: $tType,A3: A,A6: set @ A,B6: A > ( set @ B )] :
( ( member @ A @ A3 @ A6 )
=> ( ord_less_eq @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B6 @ A6 ) ) @ ( B6 @ A3 ) ) ) ).
% INT_lower
thf(fact_590_INT__greatest,axiom,
! [B: $tType,A: $tType,A6: set @ A,C5: set @ B,B6: A > ( set @ B )] :
( ! [X: A] :
( ( member @ A @ X @ A6 )
=> ( ord_less_eq @ ( set @ B ) @ C5 @ ( B6 @ X ) ) )
=> ( ord_less_eq @ ( set @ B ) @ C5 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B6 @ A6 ) ) ) ) ).
% INT_greatest
thf(fact_591_INT__anti__mono,axiom,
! [B: $tType,A: $tType,A6: set @ A,B6: set @ A,F: A > ( set @ B ),G: A > ( set @ B )] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ( ! [X: A] :
( ( member @ A @ X @ A6 )
=> ( ord_less_eq @ ( set @ B ) @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F @ B6 ) ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ G @ A6 ) ) ) ) ) ).
% INT_anti_mono
thf(fact_592_INT__subset__iff,axiom,
! [A: $tType,B: $tType,B6: set @ A,A6: B > ( set @ A ),I4: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B6 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A6 @ I4 ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ I4 )
=> ( ord_less_eq @ ( set @ A ) @ B6 @ ( A6 @ X4 ) ) ) ) ) ).
% INT_subset_iff
thf(fact_593_image__constant__conv,axiom,
! [B: $tType,A: $tType,A6: set @ B,C3: A] :
( ( ( A6
= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ B @ A
@ ^ [X4: B] : C3
@ A6 )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ B @ A
@ ^ [X4: B] : C3
@ A6 )
= ( insert @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% image_constant_conv
thf(fact_594_image__constant,axiom,
! [A: $tType,B: $tType,X2: A,A6: set @ A,C3: B] :
( ( member @ A @ X2 @ A6 )
=> ( ( image2 @ A @ B
@ ^ [X4: A] : C3
@ A6 )
= ( insert @ B @ C3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ).
% image_constant
thf(fact_595_INTER__UNIV__conv_I2_J,axiom,
! [A: $tType,B: $tType,B6: B > ( set @ A ),A6: set @ B] :
( ( ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B6 @ A6 ) )
= ( top_top @ ( set @ A ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A6 )
=> ( ( B6 @ X4 )
= ( top_top @ ( set @ A ) ) ) ) ) ) ).
% INTER_UNIV_conv(2)
thf(fact_596_INTER__UNIV__conv_I1_J,axiom,
! [A: $tType,B: $tType,B6: B > ( set @ A ),A6: set @ B] :
( ( ( top_top @ ( set @ A ) )
= ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B6 @ A6 ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A6 )
=> ( ( B6 @ X4 )
= ( top_top @ ( set @ A ) ) ) ) ) ) ).
% INTER_UNIV_conv(1)
thf(fact_597_UN__extend__simps_I10_J,axiom,
! [V5: $tType,U3: $tType,T: $tType,B6: U3 > ( set @ V5 ),F: T > U3,A6: set @ T] :
( ( complete_Sup_Sup @ ( set @ V5 )
@ ( image2 @ T @ ( set @ V5 )
@ ^ [A7: T] : ( B6 @ ( F @ A7 ) )
@ A6 ) )
= ( complete_Sup_Sup @ ( set @ V5 ) @ ( image2 @ U3 @ ( set @ V5 ) @ B6 @ ( image2 @ T @ U3 @ F @ A6 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_598_image__UN,axiom,
! [A: $tType,B: $tType,C: $tType,F: B > A,B6: C > ( set @ B ),A6: set @ C] :
( ( image2 @ B @ A @ F @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ C @ ( set @ B ) @ B6 @ A6 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [X4: C] : ( image2 @ B @ A @ F @ ( B6 @ X4 ) )
@ A6 ) ) ) ).
% image_UN
thf(fact_599_UN__empty2,axiom,
! [B: $tType,A: $tType,A6: set @ B] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X4: B] : ( bot_bot @ ( set @ A ) )
@ A6 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% UN_empty2
thf(fact_600_UN__empty,axiom,
! [B: $tType,A: $tType,B6: B > ( set @ A )] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B6 @ ( bot_bot @ ( set @ B ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% UN_empty
thf(fact_601_UNION__empty__conv_I1_J,axiom,
! [A: $tType,B: $tType,B6: B > ( set @ A ),A6: set @ B] :
( ( ( bot_bot @ ( set @ A ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B6 @ A6 ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A6 )
=> ( ( B6 @ X4 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% UNION_empty_conv(1)
thf(fact_602_UNION__empty__conv_I2_J,axiom,
! [A: $tType,B: $tType,B6: B > ( set @ A ),A6: set @ B] :
( ( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B6 @ A6 ) )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A6 )
=> ( ( B6 @ X4 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% UNION_empty_conv(2)
thf(fact_603_UN__mono,axiom,
! [B: $tType,A: $tType,A6: set @ A,B6: set @ A,F: A > ( set @ B ),G: A > ( set @ B )] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ( ! [X: A] :
( ( member @ A @ X @ A6 )
=> ( ord_less_eq @ ( set @ B ) @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq @ ( set @ B ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F @ A6 ) ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ G @ B6 ) ) ) ) ) ).
% UN_mono
thf(fact_604_UN__least,axiom,
! [A: $tType,B: $tType,A6: set @ A,B6: A > ( set @ B ),C5: set @ B] :
( ! [X: A] :
( ( member @ A @ X @ A6 )
=> ( ord_less_eq @ ( set @ B ) @ ( B6 @ X ) @ C5 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B6 @ A6 ) ) @ C5 ) ) ).
% UN_least
thf(fact_605_UN__upper,axiom,
! [B: $tType,A: $tType,A3: A,A6: set @ A,B6: A > ( set @ B )] :
( ( member @ A @ A3 @ A6 )
=> ( ord_less_eq @ ( set @ B ) @ ( B6 @ A3 ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B6 @ A6 ) ) ) ) ).
% UN_upper
thf(fact_606_UN__image__subset,axiom,
! [C: $tType,A: $tType,B: $tType,F: B > ( set @ A ),G: C > ( set @ B ),X2: C,X6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ F @ ( G @ X2 ) ) ) @ X6 )
= ( ord_less_eq @ ( set @ B ) @ ( G @ X2 )
@ ( collect @ B
@ ^ [X4: B] : ( ord_less_eq @ ( set @ A ) @ ( F @ X4 ) @ X6 ) ) ) ) ).
% UN_image_subset
thf(fact_607_UN__subset__iff,axiom,
! [A: $tType,B: $tType,A6: B > ( set @ A ),I4: set @ B,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A6 @ I4 ) ) @ B6 )
= ( ! [X4: B] :
( ( member @ B @ X4 @ I4 )
=> ( ord_less_eq @ ( set @ A ) @ ( A6 @ X4 ) @ B6 ) ) ) ) ).
% UN_subset_iff
thf(fact_608_Union__natural,axiom,
! [B: $tType,A: $tType,F: A > B] :
( ( comp @ ( set @ ( set @ B ) ) @ ( set @ B ) @ ( set @ ( set @ A ) ) @ ( complete_Sup_Sup @ ( set @ B ) ) @ ( image2 @ ( set @ A ) @ ( set @ B ) @ ( image2 @ A @ B @ F ) ) )
= ( comp @ ( set @ A ) @ ( set @ B ) @ ( set @ ( set @ A ) ) @ ( image2 @ A @ B @ F ) @ ( complete_Sup_Sup @ ( set @ A ) ) ) ) ).
% Union_natural
thf(fact_609_image__Union,axiom,
! [A: $tType,B: $tType,F: B > A,S5: set @ ( set @ B )] :
( ( image2 @ B @ A @ F @ ( complete_Sup_Sup @ ( set @ B ) @ S5 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( set @ B ) @ ( set @ A ) @ ( image2 @ B @ A @ F ) @ S5 ) ) ) ).
% image_Union
thf(fact_610_UN__extend__simps_I8_J,axiom,
! [P5: $tType,O: $tType,B6: O > ( set @ P5 ),A6: set @ ( set @ O )] :
( ( complete_Sup_Sup @ ( set @ P5 )
@ ( image2 @ ( set @ O ) @ ( set @ P5 )
@ ^ [Y5: set @ O] : ( complete_Sup_Sup @ ( set @ P5 ) @ ( image2 @ O @ ( set @ P5 ) @ B6 @ Y5 ) )
@ A6 ) )
= ( complete_Sup_Sup @ ( set @ P5 ) @ ( image2 @ O @ ( set @ P5 ) @ B6 @ ( complete_Sup_Sup @ ( set @ O ) @ A6 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_611_INF__eq__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I4: set @ B,F: B > A,C3: A] :
( ( I4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I6: B] :
( ( member @ B @ I6 @ I4 )
=> ( ord_less_eq @ A @ ( F @ I6 ) @ C3 ) )
=> ( ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ I4 ) )
= C3 )
= ( ! [X4: B] :
( ( member @ B @ X4 @ I4 )
=> ( ( F @ X4 )
= C3 ) ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_612_SUP__eq__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I4: set @ B,C3: A,F: B > A] :
( ( I4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I6: B] :
( ( member @ B @ I6 @ I4 )
=> ( ord_less_eq @ A @ C3 @ ( F @ I6 ) ) )
=> ( ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ I4 ) )
= C3 )
= ( ! [X4: B] :
( ( member @ B @ X4 @ I4 )
=> ( ( F @ X4 )
= C3 ) ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_613_range__eq__singletonD,axiom,
! [B: $tType,A: $tType,F: B > A,A3: A,X2: B] :
( ( ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( F @ X2 )
= A3 ) ) ).
% range_eq_singletonD
thf(fact_614_INF__superset__mono,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B6: set @ B,A6: set @ B,F: B > A,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ B6 @ A6 )
=> ( ! [X: B] :
( ( member @ B @ X @ B6 )
=> ( ord_less_eq @ A @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A6 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G @ B6 ) ) ) ) ) ) ).
% INF_superset_mono
thf(fact_615_SUP__subset__mono,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ B,B6: set @ B,F: B > A,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ A6 @ B6 )
=> ( ! [X: B] :
( ( member @ B @ X @ A6 )
=> ( ord_less_eq @ A @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A6 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G @ B6 ) ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_616_INF__constant,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ B,C3: A] :
( ( ( A6
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [Y5: B] : C3
@ A6 ) )
= ( top_top @ A ) ) )
& ( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [Y5: B] : C3
@ A6 ) )
= C3 ) ) ) ) ).
% INF_constant
thf(fact_617_INF__empty,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: B > A] :
( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ ( bot_bot @ ( set @ B ) ) ) )
= ( top_top @ A ) ) ) ).
% INF_empty
thf(fact_618_SUP__constant,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ B,C3: A] :
( ( ( A6
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [Y5: B] : C3
@ A6 ) )
= ( bot_bot @ A ) ) )
& ( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [Y5: B] : C3
@ A6 ) )
= C3 ) ) ) ) ).
% SUP_constant
thf(fact_619_SUP__empty,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: B > A] :
( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ ( bot_bot @ ( set @ B ) ) ) )
= ( bot_bot @ A ) ) ) ).
% SUP_empty
thf(fact_620_right__total__alt__def,axiom,
! [B: $tType,A: $tType] :
( ( right_total @ A @ B )
= ( ^ [R6: A > B > $o] :
( ord_less_eq @ ( B > B > $o )
@ ^ [Y2: B,Z2: B] : Y2 = Z2
@ ( relcompp @ B @ A @ B @ ( conversep @ A @ B @ R6 ) @ R6 ) ) ) ) ).
% right_total_alt_def
thf(fact_621_INT__empty,axiom,
! [B: $tType,A: $tType,B6: B > ( set @ A )] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B6 @ ( bot_bot @ ( set @ B ) ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% INT_empty
thf(fact_622_cINF__const,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A6: set @ B,C3: A] :
( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [X4: B] : C3
@ A6 ) )
= C3 ) ) ) ).
% cINF_const
thf(fact_623_cSUP__const,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A6: set @ B,C3: A] :
( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [X4: B] : C3
@ A6 ) )
= C3 ) ) ) ).
% cSUP_const
thf(fact_624_SUP__INF,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [P2: C > B > A] :
( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [Y5: B] :
( complete_Inf_Inf @ A
@ ( image2 @ C @ A
@ ^ [X4: C] : ( P2 @ X4 @ Y5 )
@ ( top_top @ ( set @ C ) ) ) )
@ ( top_top @ ( set @ B ) ) ) )
= ( complete_Inf_Inf @ A
@ ( image2 @ ( B > C ) @ A
@ ^ [X4: B > C] :
( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [Y5: B] : ( P2 @ ( X4 @ Y5 ) @ Y5 )
@ ( top_top @ ( set @ B ) ) ) )
@ ( top_top @ ( set @ ( B > C ) ) ) ) ) ) ) ).
% SUP_INF
thf(fact_625_INF__SUP,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [P2: C > B > A] :
( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [Y5: B] :
( complete_Sup_Sup @ A
@ ( image2 @ C @ A
@ ^ [X4: C] : ( P2 @ X4 @ Y5 )
@ ( top_top @ ( set @ C ) ) ) )
@ ( top_top @ ( set @ B ) ) ) )
= ( complete_Sup_Sup @ A
@ ( image2 @ ( B > C ) @ A
@ ^ [F4: B > C] :
( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [X4: B] : ( P2 @ ( F4 @ X4 ) @ X4 )
@ ( top_top @ ( set @ B ) ) ) )
@ ( top_top @ ( set @ ( B > C ) ) ) ) ) ) ) ).
% INF_SUP
thf(fact_626_cINF__greatest,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A6: set @ B,M4: A,F: B > A] :
( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X: B] :
( ( member @ B @ X @ A6 )
=> ( ord_less_eq @ A @ M4 @ ( F @ X ) ) )
=> ( ord_less_eq @ A @ M4 @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A6 ) ) ) ) ) ) ).
% cINF_greatest
thf(fact_627_cSUP__least,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A6: set @ B,F: B > A,M: A] :
( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X: B] :
( ( member @ B @ X @ A6 )
=> ( ord_less_eq @ A @ ( F @ X ) @ M ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A6 ) ) @ M ) ) ) ) ).
% cSUP_least
thf(fact_628_UN__UN__split__split__eq,axiom,
! [A: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A6: B > C > D > E > ( set @ A ),Y4: set @ ( product_prod @ D @ E ),X6: set @ ( product_prod @ B @ C )] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ ( product_prod @ B @ C ) @ ( set @ A )
@ ( product_case_prod @ B @ C @ ( set @ A )
@ ^ [X12: B,X23: C] : ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( product_prod @ D @ E ) @ ( set @ A ) @ ( product_case_prod @ D @ E @ ( set @ A ) @ ( A6 @ X12 @ X23 ) ) @ Y4 ) ) )
@ X6 ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ ( product_prod @ B @ C ) @ ( set @ A )
@ ^ [X4: product_prod @ B @ C] :
( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ ( product_prod @ D @ E ) @ ( set @ A )
@ ^ [Y5: product_prod @ D @ E] :
( product_case_prod @ B @ C @ ( set @ A )
@ ^ [X12: B,X23: C] : ( product_case_prod @ D @ E @ ( set @ A ) @ ( A6 @ X12 @ X23 ) @ Y5 )
@ X4 )
@ Y4 ) )
@ X6 ) ) ) ).
% UN_UN_split_split_eq
thf(fact_629_UN__constant__eq,axiom,
! [A: $tType,B: $tType,A3: A,A6: set @ A,F: A > ( set @ B ),C3: set @ B] :
( ( member @ A @ A3 @ A6 )
=> ( ! [X: A] :
( ( member @ A @ X @ A6 )
=> ( ( F @ X )
= C3 ) )
=> ( ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F @ A6 ) )
= C3 ) ) ) ).
% UN_constant_eq
thf(fact_630_refl__on__domain,axiom,
! [A: $tType,A6: set @ A,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( refl_on @ A @ A6 @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
=> ( ( member @ A @ A3 @ A6 )
& ( member @ A @ B2 @ A6 ) ) ) ) ).
% refl_on_domain
thf(fact_631_INT__simps_I4_J,axiom,
! [G2: $tType,H5: $tType,C5: set @ H5,A6: set @ G2,B6: H5 > ( set @ G2 )] :
( ( ( C5
= ( bot_bot @ ( set @ H5 ) ) )
=> ( ( complete_Inf_Inf @ ( set @ G2 )
@ ( image2 @ H5 @ ( set @ G2 )
@ ^ [X4: H5] : ( minus_minus @ ( set @ G2 ) @ A6 @ ( B6 @ X4 ) )
@ C5 ) )
= ( top_top @ ( set @ G2 ) ) ) )
& ( ( C5
!= ( bot_bot @ ( set @ H5 ) ) )
=> ( ( complete_Inf_Inf @ ( set @ G2 )
@ ( image2 @ H5 @ ( set @ G2 )
@ ^ [X4: H5] : ( minus_minus @ ( set @ G2 ) @ A6 @ ( B6 @ X4 ) )
@ C5 ) )
= ( minus_minus @ ( set @ G2 ) @ A6 @ ( complete_Sup_Sup @ ( set @ G2 ) @ ( image2 @ H5 @ ( set @ G2 ) @ B6 @ C5 ) ) ) ) ) ) ).
% INT_simps(4)
thf(fact_632_relcompp__bot2,axiom,
! [C: $tType,B: $tType,A: $tType,R3: A > C > $o] :
( ( relcompp @ A @ C @ B @ R3 @ ( bot_bot @ ( C > B > $o ) ) )
= ( bot_bot @ ( A > B > $o ) ) ) ).
% relcompp_bot2
thf(fact_633_relcompp__bot1,axiom,
! [C: $tType,B: $tType,A: $tType,R3: C > B > $o] :
( ( relcompp @ A @ C @ B @ ( bot_bot @ ( A > C > $o ) ) @ R3 )
= ( bot_bot @ ( A > B > $o ) ) ) ).
% relcompp_bot1
thf(fact_634_surj__diff__right,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A] :
( ( image2 @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ X4 @ A3 )
@ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surj_diff_right
thf(fact_635_INT__simps_I3_J,axiom,
! [E: $tType,F2: $tType,C5: set @ E,A6: E > ( set @ F2 ),B6: set @ F2] :
( ( ( C5
= ( bot_bot @ ( set @ E ) ) )
=> ( ( complete_Inf_Inf @ ( set @ F2 )
@ ( image2 @ E @ ( set @ F2 )
@ ^ [X4: E] : ( minus_minus @ ( set @ F2 ) @ ( A6 @ X4 ) @ B6 )
@ C5 ) )
= ( top_top @ ( set @ F2 ) ) ) )
& ( ( C5
!= ( bot_bot @ ( set @ E ) ) )
=> ( ( complete_Inf_Inf @ ( set @ F2 )
@ ( image2 @ E @ ( set @ F2 )
@ ^ [X4: E] : ( minus_minus @ ( set @ F2 ) @ ( A6 @ X4 ) @ B6 )
@ C5 ) )
= ( minus_minus @ ( set @ F2 ) @ ( complete_Inf_Inf @ ( set @ F2 ) @ ( image2 @ E @ ( set @ F2 ) @ A6 @ C5 ) ) @ B6 ) ) ) ) ).
% INT_simps(3)
thf(fact_636_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_637_translation__subtract__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,S2: set @ A,T3: set @ A] :
( ( image2 @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ X4 @ A3 )
@ ( minus_minus @ ( set @ A ) @ S2 @ T3 ) )
= ( minus_minus @ ( set @ A )
@ ( image2 @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ X4 @ A3 )
@ S2 )
@ ( image2 @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ X4 @ A3 )
@ T3 ) ) ) ) ).
% translation_subtract_diff
thf(fact_638_Times__Diff__distrib1,axiom,
! [B: $tType,A: $tType,A6: set @ A,B6: set @ A,C5: set @ B] :
( ( product_Sigma @ A @ B @ ( minus_minus @ ( set @ A ) @ A6 @ B6 )
@ ^ [Uu: A] : C5 )
= ( minus_minus @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu: A] : C5 )
@ ( product_Sigma @ A @ B @ B6
@ ^ [Uu: A] : C5 ) ) ) ).
% Times_Diff_distrib1
thf(fact_639_Sigma__Diff__distrib2,axiom,
! [B: $tType,A: $tType,I4: set @ A,A6: A > ( set @ B ),B6: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ I4
@ ^ [I3: A] : ( minus_minus @ ( set @ B ) @ ( A6 @ I3 ) @ ( B6 @ I3 ) ) )
= ( minus_minus @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I4 @ A6 ) @ ( product_Sigma @ A @ B @ I4 @ B6 ) ) ) ).
% Sigma_Diff_distrib2
thf(fact_640_Sup__SUP__eq,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( A > $o ) )
= ( ^ [S4: set @ ( A > $o ),X4: A] : ( member @ A @ X4 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( A > $o ) @ ( set @ A ) @ ( collect @ A ) @ S4 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_641_Inf__INT__eq,axiom,
! [A: $tType] :
( ( complete_Inf_Inf @ ( A > $o ) )
= ( ^ [S4: set @ ( A > $o ),X4: A] : ( member @ A @ X4 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ ( A > $o ) @ ( set @ A ) @ ( collect @ A ) @ S4 ) ) ) ) ) ).
% Inf_INT_eq
thf(fact_642_INF__INT__eq2,axiom,
! [B: $tType,C: $tType,A: $tType,R: C > ( set @ ( product_prod @ A @ B ) ),S5: set @ C] :
( ( complete_Inf_Inf @ ( A > B > $o )
@ ( image2 @ C @ ( A > B > $o )
@ ^ [I3: C,X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ ( R @ I3 ) )
@ S5 ) )
= ( ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ C @ ( set @ ( product_prod @ A @ B ) ) @ R @ S5 ) ) ) ) ) ).
% INF_INT_eq2
thf(fact_643_INF__Int__eq2,axiom,
! [B: $tType,A: $tType,S5: 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 ),X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ I3 )
@ S5 ) )
= ( ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) ) @ S5 ) ) ) ) ).
% INF_Int_eq2
thf(fact_644_SUP__UN__eq2,axiom,
! [B: $tType,C: $tType,A: $tType,R: C > ( set @ ( product_prod @ A @ B ) ),S5: set @ C] :
( ( complete_Sup_Sup @ ( A > B > $o )
@ ( image2 @ C @ ( A > B > $o )
@ ^ [I3: C,X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ ( R @ I3 ) )
@ S5 ) )
= ( ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ C @ ( set @ ( product_prod @ A @ B ) ) @ R @ S5 ) ) ) ) ) ).
% SUP_UN_eq2
thf(fact_645_SUP__Sup__eq2,axiom,
! [B: $tType,A: $tType,S5: 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 ),X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ I3 )
@ S5 ) )
= ( ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ S5 ) ) ) ) ).
% SUP_Sup_eq2
thf(fact_646_SUP__Sup__eq,axiom,
! [A: $tType,S5: set @ ( set @ A )] :
( ( complete_Sup_Sup @ ( A > $o )
@ ( image2 @ ( set @ A ) @ ( A > $o )
@ ^ [I3: set @ A,X4: A] : ( member @ A @ X4 @ I3 )
@ S5 ) )
= ( ^ [X4: A] : ( member @ A @ X4 @ ( complete_Sup_Sup @ ( set @ A ) @ S5 ) ) ) ) ).
% SUP_Sup_eq
thf(fact_647_INF__Int__eq,axiom,
! [A: $tType,S5: set @ ( set @ A )] :
( ( complete_Inf_Inf @ ( A > $o )
@ ( image2 @ ( set @ A ) @ ( A > $o )
@ ^ [I3: set @ A,X4: A] : ( member @ A @ X4 @ I3 )
@ S5 ) )
= ( ^ [X4: A] : ( member @ A @ X4 @ ( complete_Inf_Inf @ ( set @ A ) @ S5 ) ) ) ) ).
% INF_Int_eq
thf(fact_648_relcompp__SUP__distrib,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,S2: A > C > $o,R: D > C > B > $o,I4: set @ D] :
( ( relcompp @ A @ C @ B @ S2 @ ( complete_Sup_Sup @ ( C > B > $o ) @ ( image2 @ D @ ( C > B > $o ) @ R @ I4 ) ) )
= ( complete_Sup_Sup @ ( A > B > $o )
@ ( image2 @ D @ ( A > B > $o )
@ ^ [I3: D] : ( relcompp @ A @ C @ B @ S2 @ ( R @ I3 ) )
@ I4 ) ) ) ).
% relcompp_SUP_distrib
thf(fact_649_relcompp__SUP__distrib2,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,R: D > A > C > $o,I4: set @ D,S2: C > B > $o] :
( ( relcompp @ A @ C @ B @ ( complete_Sup_Sup @ ( A > C > $o ) @ ( image2 @ D @ ( A > C > $o ) @ R @ I4 ) ) @ S2 )
= ( complete_Sup_Sup @ ( A > B > $o )
@ ( image2 @ D @ ( A > B > $o )
@ ^ [I3: D] : ( relcompp @ A @ C @ B @ ( R @ I3 ) @ S2 )
@ I4 ) ) ) ).
% relcompp_SUP_distrib2
thf(fact_650_Sup__set__def,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( set @ A ) )
= ( ^ [A8: set @ ( set @ A )] :
( collect @ A
@ ^ [X4: A] : ( complete_Sup_Sup @ $o @ ( image2 @ ( set @ A ) @ $o @ ( member @ A @ X4 ) @ A8 ) ) ) ) ) ).
% Sup_set_def
thf(fact_651_Inf__set__def,axiom,
! [A: $tType] :
( ( complete_Inf_Inf @ ( set @ A ) )
= ( ^ [A8: set @ ( set @ A )] :
( collect @ A
@ ^ [X4: A] : ( complete_Inf_Inf @ $o @ ( image2 @ ( set @ A ) @ $o @ ( member @ A @ X4 ) @ A8 ) ) ) ) ) ).
% Inf_set_def
thf(fact_652_image__diff__subset,axiom,
! [A: $tType,B: $tType,F: B > A,A6: set @ B,B6: set @ B] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ ( image2 @ B @ A @ F @ A6 ) @ ( image2 @ B @ A @ F @ B6 ) ) @ ( image2 @ B @ A @ F @ ( minus_minus @ ( set @ B ) @ A6 @ B6 ) ) ) ).
% image_diff_subset
thf(fact_653_Inf__INT__eq2,axiom,
! [B: $tType,A: $tType] :
( ( complete_Inf_Inf @ ( A > B > $o ) )
= ( ^ [S4: set @ ( A > B > $o ),X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ ( 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 ) @ S4 ) ) ) ) ) ) ).
% Inf_INT_eq2
thf(fact_654_Sup__SUP__eq2,axiom,
! [B: $tType,A: $tType] :
( ( complete_Sup_Sup @ ( A > B > $o ) )
= ( ^ [S4: set @ ( A > B > $o ),X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ ( 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 ) @ S4 ) ) ) ) ) ) ).
% Sup_SUP_eq2
thf(fact_655_SUP__UN__eq,axiom,
! [B: $tType,A: $tType,R: B > ( set @ A ),S5: set @ B] :
( ( complete_Sup_Sup @ ( A > $o )
@ ( image2 @ B @ ( A > $o )
@ ^ [I3: B,X4: A] : ( member @ A @ X4 @ ( R @ I3 ) )
@ S5 ) )
= ( ^ [X4: A] : ( member @ A @ X4 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ R @ S5 ) ) ) ) ) ).
% SUP_UN_eq
thf(fact_656_bot__empty__eq2,axiom,
! [B: $tType,A: $tType] :
( ( bot_bot @ ( A > B > $o ) )
= ( ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% bot_empty_eq2
thf(fact_657_INF__INT__eq,axiom,
! [B: $tType,A: $tType,R: B > ( set @ A ),S5: set @ B] :
( ( complete_Inf_Inf @ ( A > $o )
@ ( image2 @ B @ ( A > $o )
@ ^ [I3: B,X4: A] : ( member @ A @ X4 @ ( R @ I3 ) )
@ S5 ) )
= ( ^ [X4: A] : ( member @ A @ X4 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ R @ S5 ) ) ) ) ) ).
% INF_INT_eq
thf(fact_658_top__empty__eq2,axiom,
! [B: $tType,A: $tType] :
( ( top_top @ ( A > B > $o ) )
= ( ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% top_empty_eq2
thf(fact_659_UN__extend__simps_I6_J,axiom,
! [L4: $tType,K2: $tType,A6: K2 > ( set @ L4 ),C5: set @ K2,B6: set @ L4] :
( ( minus_minus @ ( set @ L4 ) @ ( complete_Sup_Sup @ ( set @ L4 ) @ ( image2 @ K2 @ ( set @ L4 ) @ A6 @ C5 ) ) @ B6 )
= ( complete_Sup_Sup @ ( set @ L4 )
@ ( image2 @ K2 @ ( set @ L4 )
@ ^ [X4: K2] : ( minus_minus @ ( set @ L4 ) @ ( A6 @ X4 ) @ B6 )
@ C5 ) ) ) ).
% UN_extend_simps(6)
thf(fact_660_in__image__insert__iff,axiom,
! [A: $tType,B6: set @ ( set @ A ),X2: A,A6: set @ A] :
( ! [C4: set @ A] :
( ( member @ ( set @ A ) @ C4 @ B6 )
=> ~ ( member @ A @ X2 @ C4 ) )
=> ( ( member @ ( set @ A ) @ A6 @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( insert @ A @ X2 ) @ B6 ) )
= ( ( member @ A @ X2 @ A6 )
& ( member @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) @ B6 ) ) ) ) ).
% in_image_insert_iff
thf(fact_661_UN__extend__simps_I7_J,axiom,
! [M5: $tType,N3: $tType,A6: set @ M5,B6: N3 > ( set @ M5 ),C5: set @ N3] :
( ( minus_minus @ ( set @ M5 ) @ A6 @ ( complete_Inf_Inf @ ( set @ M5 ) @ ( image2 @ N3 @ ( set @ M5 ) @ B6 @ C5 ) ) )
= ( complete_Sup_Sup @ ( set @ M5 )
@ ( image2 @ N3 @ ( set @ M5 )
@ ^ [X4: N3] : ( minus_minus @ ( set @ M5 ) @ A6 @ ( B6 @ X4 ) )
@ C5 ) ) ) ).
% UN_extend_simps(7)
thf(fact_662_INT__extend__simps_I3_J,axiom,
! [F2: $tType,E: $tType,C5: set @ E,A6: E > ( set @ F2 ),B6: set @ F2] :
( ( ( C5
= ( bot_bot @ ( set @ E ) ) )
=> ( ( minus_minus @ ( set @ F2 ) @ ( complete_Inf_Inf @ ( set @ F2 ) @ ( image2 @ E @ ( set @ F2 ) @ A6 @ C5 ) ) @ B6 )
= ( minus_minus @ ( set @ F2 ) @ ( top_top @ ( set @ F2 ) ) @ B6 ) ) )
& ( ( C5
!= ( bot_bot @ ( set @ E ) ) )
=> ( ( minus_minus @ ( set @ F2 ) @ ( complete_Inf_Inf @ ( set @ F2 ) @ ( image2 @ E @ ( set @ F2 ) @ A6 @ C5 ) ) @ B6 )
= ( complete_Inf_Inf @ ( set @ F2 )
@ ( image2 @ E @ ( set @ F2 )
@ ^ [X4: E] : ( minus_minus @ ( set @ F2 ) @ ( A6 @ X4 ) @ B6 )
@ C5 ) ) ) ) ) ).
% INT_extend_simps(3)
thf(fact_663_INT__extend__simps_I4_J,axiom,
! [G2: $tType,H5: $tType,C5: set @ H5,A6: set @ G2,B6: H5 > ( set @ G2 )] :
( ( ( C5
= ( bot_bot @ ( set @ H5 ) ) )
=> ( ( minus_minus @ ( set @ G2 ) @ A6 @ ( complete_Sup_Sup @ ( set @ G2 ) @ ( image2 @ H5 @ ( set @ G2 ) @ B6 @ C5 ) ) )
= A6 ) )
& ( ( C5
!= ( bot_bot @ ( set @ H5 ) ) )
=> ( ( minus_minus @ ( set @ G2 ) @ A6 @ ( complete_Sup_Sup @ ( set @ G2 ) @ ( image2 @ H5 @ ( set @ G2 ) @ B6 @ C5 ) ) )
= ( complete_Inf_Inf @ ( set @ G2 )
@ ( image2 @ H5 @ ( set @ G2 )
@ ^ [X4: H5] : ( minus_minus @ ( set @ G2 ) @ A6 @ ( B6 @ X4 ) )
@ C5 ) ) ) ) ) ).
% INT_extend_simps(4)
thf(fact_664_linear__order__on__singleton,axiom,
! [A: $tType,X2: A] : ( order_679001287576687338der_on @ A @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% linear_order_on_singleton
thf(fact_665_total__on__singleton,axiom,
! [A: $tType,X2: A] : ( total_on @ A @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% total_on_singleton
thf(fact_666_relation__of__def,axiom,
! [A: $tType] :
( ( order_relation_of @ A )
= ( ^ [P4: A > A > $o,A8: 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 @ A8
@ ^ [Uu: A] : A8 ) )
& ( P4 @ A7 @ B5 ) ) ) ) ) ) ).
% relation_of_def
thf(fact_667_insert__Times__insert,axiom,
! [B: $tType,A: $tType,A3: A,A6: set @ A,B2: B,B6: set @ B] :
( ( product_Sigma @ A @ B @ ( insert @ A @ A3 @ A6 )
@ ^ [Uu: A] : ( insert @ B @ B2 @ B6 ) )
= ( insert @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 )
@ ( sup_sup @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu: A] : ( insert @ B @ B2 @ B6 ) )
@ ( product_Sigma @ A @ B @ ( insert @ A @ A3 @ A6 )
@ ^ [Uu: A] : B6 ) ) ) ) ).
% insert_Times_insert
thf(fact_668_INT__simps_I2_J,axiom,
! [C: $tType,D: $tType,C5: set @ D,A6: set @ C,B6: D > ( set @ C )] :
( ( ( C5
= ( bot_bot @ ( set @ D ) ) )
=> ( ( complete_Inf_Inf @ ( set @ C )
@ ( image2 @ D @ ( set @ C )
@ ^ [X4: D] : ( inf_inf @ ( set @ C ) @ A6 @ ( B6 @ X4 ) )
@ C5 ) )
= ( top_top @ ( set @ C ) ) ) )
& ( ( C5
!= ( bot_bot @ ( set @ D ) ) )
=> ( ( complete_Inf_Inf @ ( set @ C )
@ ( image2 @ D @ ( set @ C )
@ ^ [X4: D] : ( inf_inf @ ( set @ C ) @ A6 @ ( B6 @ X4 ) )
@ C5 ) )
= ( inf_inf @ ( set @ C ) @ A6 @ ( complete_Inf_Inf @ ( set @ C ) @ ( image2 @ D @ ( set @ C ) @ B6 @ C5 ) ) ) ) ) ) ).
% INT_simps(2)
thf(fact_669_INT__simps_I1_J,axiom,
! [A: $tType,B: $tType,C5: set @ A,A6: A > ( set @ B ),B6: set @ B] :
( ( ( C5
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ ( set @ B )
@ ( image2 @ A @ ( set @ B )
@ ^ [X4: A] : ( inf_inf @ ( set @ B ) @ ( A6 @ X4 ) @ B6 )
@ C5 ) )
= ( top_top @ ( set @ B ) ) ) )
& ( ( C5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ ( set @ B )
@ ( image2 @ A @ ( set @ B )
@ ^ [X4: A] : ( inf_inf @ ( set @ B ) @ ( A6 @ X4 ) @ B6 )
@ C5 ) )
= ( inf_inf @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A6 @ C5 ) ) @ B6 ) ) ) ) ).
% INT_simps(1)
thf(fact_670_cSUP__UNION,axiom,
! [B: $tType,D: $tType,C: $tType] :
( ( condit1219197933456340205attice @ B )
=> ! [A6: set @ C,B6: C > ( set @ D ),F: D > B] :
( ( A6
!= ( bot_bot @ ( set @ C ) ) )
=> ( ! [X: C] :
( ( member @ C @ X @ A6 )
=> ( ( B6 @ X )
!= ( bot_bot @ ( set @ D ) ) ) )
=> ( ( condit941137186595557371_above @ B
@ ( complete_Sup_Sup @ ( set @ B )
@ ( image2 @ C @ ( set @ B )
@ ^ [X4: C] : ( image2 @ D @ B @ F @ ( B6 @ X4 ) )
@ A6 ) ) )
=> ( ( complete_Sup_Sup @ B @ ( image2 @ D @ B @ F @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ B6 @ A6 ) ) ) )
= ( complete_Sup_Sup @ B
@ ( image2 @ C @ B
@ ^ [X4: C] : ( complete_Sup_Sup @ B @ ( image2 @ D @ B @ F @ ( B6 @ X4 ) ) )
@ A6 ) ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_671_cINF__UNION,axiom,
! [B: $tType,D: $tType,C: $tType] :
( ( condit1219197933456340205attice @ B )
=> ! [A6: set @ C,B6: C > ( set @ D ),F: D > B] :
( ( A6
!= ( bot_bot @ ( set @ C ) ) )
=> ( ! [X: C] :
( ( member @ C @ X @ A6 )
=> ( ( B6 @ X )
!= ( bot_bot @ ( set @ D ) ) ) )
=> ( ( condit1013018076250108175_below @ B
@ ( complete_Sup_Sup @ ( set @ B )
@ ( image2 @ C @ ( set @ B )
@ ^ [X4: C] : ( image2 @ D @ B @ F @ ( B6 @ X4 ) )
@ A6 ) ) )
=> ( ( complete_Inf_Inf @ B @ ( image2 @ D @ B @ F @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ B6 @ A6 ) ) ) )
= ( complete_Inf_Inf @ B
@ ( image2 @ C @ B
@ ^ [X4: C] : ( complete_Inf_Inf @ B @ ( image2 @ D @ B @ F @ ( B6 @ X4 ) ) )
@ A6 ) ) ) ) ) ) ) ).
% cINF_UNION
thf(fact_672_Ex__inj__on__UNION__Sigma,axiom,
! [A: $tType,B: $tType,A6: B > ( set @ A ),I4: set @ B] :
? [F3: A > ( product_prod @ B @ A )] :
( ( inj_on @ A @ ( product_prod @ B @ A ) @ F3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A6 @ I4 ) ) )
& ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) ) @ ( image2 @ A @ ( product_prod @ B @ A ) @ F3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A6 @ I4 ) ) ) @ ( product_Sigma @ B @ A @ I4 @ A6 ) ) ) ).
% Ex_inj_on_UNION_Sigma
thf(fact_673_prod__set__simps_I1_J,axiom,
! [B: $tType,A: $tType,X2: A,Y3: B] :
( ( basic_fsts @ A @ B @ ( product_Pair @ A @ B @ X2 @ Y3 ) )
= ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% prod_set_simps(1)
thf(fact_674_SUP2__I,axiom,
! [B: $tType,A: $tType,C: $tType,A3: A,A6: set @ A,B6: A > B > C > $o,B2: B,C3: C] :
( ( member @ A @ A3 @ A6 )
=> ( ( B6 @ A3 @ B2 @ C3 )
=> ( complete_Sup_Sup @ ( B > C > $o ) @ ( image2 @ A @ ( B > C > $o ) @ B6 @ A6 ) @ B2 @ C3 ) ) ) ).
% SUP2_I
thf(fact_675_INF1__I,axiom,
! [A: $tType,B: $tType,A6: set @ A,B6: A > B > $o,B2: B] :
( ! [X: A] :
( ( member @ A @ X @ A6 )
=> ( B6 @ X @ B2 ) )
=> ( complete_Inf_Inf @ ( B > $o ) @ ( image2 @ A @ ( B > $o ) @ B6 @ A6 ) @ B2 ) ) ).
% INF1_I
thf(fact_676_INF2__I,axiom,
! [B: $tType,A: $tType,C: $tType,A6: set @ A,B6: A > B > C > $o,B2: B,C3: C] :
( ! [X: A] :
( ( member @ A @ X @ A6 )
=> ( B6 @ X @ B2 @ C3 ) )
=> ( complete_Inf_Inf @ ( B > C > $o ) @ ( image2 @ A @ ( B > C > $o ) @ B6 @ A6 ) @ B2 @ C3 ) ) ).
% INF2_I
thf(fact_677_SUP1__I,axiom,
! [A: $tType,B: $tType,A3: A,A6: set @ A,B6: A > B > $o,B2: B] :
( ( member @ A @ A3 @ A6 )
=> ( ( B6 @ A3 @ B2 )
=> ( complete_Sup_Sup @ ( B > $o ) @ ( image2 @ A @ ( B > $o ) @ B6 @ A6 ) @ B2 ) ) ) ).
% SUP1_I
thf(fact_678_if__image__distrib,axiom,
! [A: $tType,B: $tType,P2: B > $o,F: B > A,G: B > A,S5: set @ B] :
( ( image2 @ B @ A
@ ^ [X4: B] : ( if @ A @ ( P2 @ X4 ) @ ( F @ X4 ) @ ( G @ X4 ) )
@ S5 )
= ( sup_sup @ ( set @ A ) @ ( image2 @ B @ A @ F @ ( inf_inf @ ( set @ B ) @ S5 @ ( collect @ B @ P2 ) ) )
@ ( image2 @ B @ A @ G
@ ( inf_inf @ ( set @ B ) @ S5
@ ( collect @ B
@ ^ [X4: B] :
~ ( P2 @ X4 ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_679_UN__Un,axiom,
! [A: $tType,B: $tType,M: B > ( set @ A ),A6: set @ B,B6: set @ B] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ M @ ( sup_sup @ ( set @ B ) @ A6 @ B6 ) ) )
= ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ M @ A6 ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ M @ B6 ) ) ) ) ).
% UN_Un
thf(fact_680_bdd__below__image__inf,axiom,
! [A: $tType,B: $tType] :
( ( lattice @ A )
=> ! [F: B > A,G: B > A,A6: set @ B] :
( ( condit1013018076250108175_below @ A
@ ( image2 @ B @ A
@ ^ [X4: B] : ( inf_inf @ A @ ( F @ X4 ) @ ( G @ X4 ) )
@ A6 ) )
= ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F @ A6 ) )
& ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ G @ A6 ) ) ) ) ) ).
% bdd_below_image_inf
thf(fact_681_bdd__above__image__sup,axiom,
! [A: $tType,B: $tType] :
( ( lattice @ A )
=> ! [F: B > A,G: B > A,A6: set @ B] :
( ( condit941137186595557371_above @ A
@ ( image2 @ B @ A
@ ^ [X4: B] : ( sup_sup @ A @ ( F @ X4 ) @ ( G @ X4 ) )
@ A6 ) )
= ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F @ A6 ) )
& ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ G @ A6 ) ) ) ) ) ).
% bdd_above_image_sup
thf(fact_682_UN__simps_I3_J,axiom,
! [E: $tType,F2: $tType,C5: set @ F2,A6: set @ E,B6: F2 > ( set @ E )] :
( ( ( C5
= ( bot_bot @ ( set @ F2 ) ) )
=> ( ( complete_Sup_Sup @ ( set @ E )
@ ( image2 @ F2 @ ( set @ E )
@ ^ [X4: F2] : ( sup_sup @ ( set @ E ) @ A6 @ ( B6 @ X4 ) )
@ C5 ) )
= ( bot_bot @ ( set @ E ) ) ) )
& ( ( C5
!= ( bot_bot @ ( set @ F2 ) ) )
=> ( ( complete_Sup_Sup @ ( set @ E )
@ ( image2 @ F2 @ ( set @ E )
@ ^ [X4: F2] : ( sup_sup @ ( set @ E ) @ A6 @ ( B6 @ X4 ) )
@ C5 ) )
= ( sup_sup @ ( set @ E ) @ A6 @ ( complete_Sup_Sup @ ( set @ E ) @ ( image2 @ F2 @ ( set @ E ) @ B6 @ C5 ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_683_UN__simps_I2_J,axiom,
! [C: $tType,D: $tType,C5: set @ C,A6: C > ( set @ D ),B6: set @ D] :
( ( ( C5
= ( bot_bot @ ( set @ C ) ) )
=> ( ( complete_Sup_Sup @ ( set @ D )
@ ( image2 @ C @ ( set @ D )
@ ^ [X4: C] : ( sup_sup @ ( set @ D ) @ ( A6 @ X4 ) @ B6 )
@ C5 ) )
= ( bot_bot @ ( set @ D ) ) ) )
& ( ( C5
!= ( bot_bot @ ( set @ C ) ) )
=> ( ( complete_Sup_Sup @ ( set @ D )
@ ( image2 @ C @ ( set @ D )
@ ^ [X4: C] : ( sup_sup @ ( set @ D ) @ ( A6 @ X4 ) @ B6 )
@ C5 ) )
= ( sup_sup @ ( set @ D ) @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ A6 @ C5 ) ) @ B6 ) ) ) ) ).
% UN_simps(2)
thf(fact_684_UN__insert,axiom,
! [A: $tType,B: $tType,B6: B > ( set @ A ),A3: B,A6: set @ B] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B6 @ ( insert @ B @ A3 @ A6 ) ) )
= ( sup_sup @ ( set @ A ) @ ( B6 @ A3 ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B6 @ A6 ) ) ) ) ).
% UN_insert
thf(fact_685_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_686_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_687_INT__insert,axiom,
! [A: $tType,B: $tType,B6: B > ( set @ A ),A3: B,A6: set @ B] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B6 @ ( insert @ B @ A3 @ A6 ) ) )
= ( inf_inf @ ( set @ A ) @ ( B6 @ A3 ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B6 @ A6 ) ) ) ) ).
% INT_insert
thf(fact_688_inj__on__apfst,axiom,
! [B: $tType,C: $tType,A: $tType,F: A > C,A6: set @ A] :
( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ C @ B ) @ ( product_apfst @ A @ C @ B @ F )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu: A] : ( top_top @ ( set @ B ) ) ) )
= ( inj_on @ A @ C @ F @ A6 ) ) ).
% inj_on_apfst
thf(fact_689_inj__on__apsnd,axiom,
! [A: $tType,C: $tType,B: $tType,F: B > C,A6: 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] : A6 ) )
= ( inj_on @ B @ C @ F @ A6 ) ) ).
% inj_on_apsnd
thf(fact_690_inj__on__insert,axiom,
! [B: $tType,A: $tType,F: A > B,A3: A,A6: set @ A] :
( ( inj_on @ A @ B @ F @ ( insert @ A @ A3 @ A6 ) )
= ( ( inj_on @ A @ B @ F @ A6 )
& ~ ( member @ B @ ( F @ A3 ) @ ( image2 @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_691_inj__on__Un__image__eq__iff,axiom,
! [B: $tType,A: $tType,F: A > B,A6: set @ A,B6: set @ A] :
( ( inj_on @ A @ B @ F @ ( sup_sup @ ( set @ A ) @ A6 @ B6 ) )
=> ( ( ( image2 @ A @ B @ F @ A6 )
= ( image2 @ A @ B @ F @ B6 ) )
= ( A6 = B6 ) ) ) ).
% inj_on_Un_image_eq_iff
thf(fact_692_INF2__D,axiom,
! [A: $tType,C: $tType,B: $tType,B6: C > A > B > $o,A6: set @ C,B2: A,C3: B,A3: C] :
( ( complete_Inf_Inf @ ( A > B > $o ) @ ( image2 @ C @ ( A > B > $o ) @ B6 @ A6 ) @ B2 @ C3 )
=> ( ( member @ C @ A3 @ A6 )
=> ( B6 @ A3 @ B2 @ C3 ) ) ) ).
% INF2_D
thf(fact_693_INF2__E,axiom,
! [B: $tType,A: $tType,C: $tType,B6: C > A > B > $o,A6: set @ C,B2: A,C3: B,A3: C] :
( ( complete_Inf_Inf @ ( A > B > $o ) @ ( image2 @ C @ ( A > B > $o ) @ B6 @ A6 ) @ B2 @ C3 )
=> ( ~ ( B6 @ A3 @ B2 @ C3 )
=> ~ ( member @ C @ A3 @ A6 ) ) ) ).
% INF2_E
thf(fact_694_INF1__D,axiom,
! [B: $tType,A: $tType,B6: B > A > $o,A6: set @ B,B2: A,A3: B] :
( ( complete_Inf_Inf @ ( A > $o ) @ ( image2 @ B @ ( A > $o ) @ B6 @ A6 ) @ B2 )
=> ( ( member @ B @ A3 @ A6 )
=> ( B6 @ A3 @ B2 ) ) ) ).
% INF1_D
thf(fact_695_INF1__E,axiom,
! [A: $tType,B: $tType,B6: B > A > $o,A6: set @ B,B2: A,A3: B] :
( ( complete_Inf_Inf @ ( A > $o ) @ ( image2 @ B @ ( A > $o ) @ B6 @ A6 ) @ B2 )
=> ( ~ ( B6 @ A3 @ B2 )
=> ~ ( member @ B @ A3 @ A6 ) ) ) ).
% INF1_E
thf(fact_696_SUP1__E,axiom,
! [B: $tType,A: $tType,B6: B > A > $o,A6: set @ B,B2: A] :
( ( complete_Sup_Sup @ ( A > $o ) @ ( image2 @ B @ ( A > $o ) @ B6 @ A6 ) @ B2 )
=> ~ ! [X: B] :
( ( member @ B @ X @ A6 )
=> ~ ( B6 @ X @ B2 ) ) ) ).
% SUP1_E
thf(fact_697_SUP2__E,axiom,
! [A: $tType,C: $tType,B: $tType,B6: C > A > B > $o,A6: set @ C,B2: A,C3: B] :
( ( complete_Sup_Sup @ ( A > B > $o ) @ ( image2 @ C @ ( A > B > $o ) @ B6 @ A6 ) @ B2 @ C3 )
=> ~ ! [X: C] :
( ( member @ C @ X @ A6 )
=> ~ ( B6 @ X @ B2 @ C3 ) ) ) ).
% SUP2_E
thf(fact_698_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_699_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_700_inj__on__disjoint__Un,axiom,
! [B: $tType,A: $tType,F: A > B,A6: set @ A,G: A > B,B6: set @ A] :
( ( inj_on @ A @ B @ F @ A6 )
=> ( ( inj_on @ A @ B @ G @ B6 )
=> ( ( ( inf_inf @ ( set @ B ) @ ( image2 @ A @ B @ F @ A6 ) @ ( image2 @ A @ B @ G @ B6 ) )
= ( bot_bot @ ( set @ B ) ) )
=> ( inj_on @ A @ B
@ ^ [X4: A] : ( if @ B @ ( member @ A @ X4 @ A6 ) @ ( F @ X4 ) @ ( G @ X4 ) )
@ ( sup_sup @ ( set @ A ) @ A6 @ B6 ) ) ) ) ) ).
% inj_on_disjoint_Un
thf(fact_701_Sigma__Un__distrib2,axiom,
! [B: $tType,A: $tType,I4: set @ A,A6: A > ( set @ B ),B6: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ I4
@ ^ [I3: A] : ( sup_sup @ ( set @ B ) @ ( A6 @ I3 ) @ ( B6 @ I3 ) ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I4 @ A6 ) @ ( product_Sigma @ A @ B @ I4 @ B6 ) ) ) ).
% Sigma_Un_distrib2
thf(fact_702_Times__Un__distrib1,axiom,
! [B: $tType,A: $tType,A6: set @ A,B6: set @ A,C5: set @ B] :
( ( product_Sigma @ A @ B @ ( sup_sup @ ( set @ A ) @ A6 @ B6 )
@ ^ [Uu: A] : C5 )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu: A] : C5 )
@ ( product_Sigma @ A @ B @ B6
@ ^ [Uu: A] : C5 ) ) ) ).
% Times_Un_distrib1
thf(fact_703_Times__Int__Times,axiom,
! [B: $tType,A: $tType,A6: set @ A,B6: set @ B,C5: set @ A,D3: set @ B] :
( ( inf_inf @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu: A] : B6 )
@ ( product_Sigma @ A @ B @ C5
@ ^ [Uu: A] : D3 ) )
= ( product_Sigma @ A @ B @ ( inf_inf @ ( set @ A ) @ A6 @ C5 )
@ ^ [Uu: A] : ( inf_inf @ ( set @ B ) @ B6 @ D3 ) ) ) ).
% Times_Int_Times
thf(fact_704_Sigma__Int__distrib2,axiom,
! [B: $tType,A: $tType,I4: set @ A,A6: A > ( set @ B ),B6: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ I4
@ ^ [I3: A] : ( inf_inf @ ( set @ B ) @ ( A6 @ I3 ) @ ( B6 @ I3 ) ) )
= ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I4 @ A6 ) @ ( product_Sigma @ A @ B @ I4 @ B6 ) ) ) ).
% Sigma_Int_distrib2
thf(fact_705_Times__Int__distrib1,axiom,
! [B: $tType,A: $tType,A6: set @ A,B6: set @ A,C5: set @ B] :
( ( product_Sigma @ A @ B @ ( inf_inf @ ( set @ A ) @ A6 @ B6 )
@ ^ [Uu: A] : C5 )
= ( inf_inf @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu: A] : C5 )
@ ( product_Sigma @ A @ B @ B6
@ ^ [Uu: A] : C5 ) ) ) ).
% Times_Int_distrib1
thf(fact_706_inj__on__Un,axiom,
! [A: $tType,B: $tType,F: A > B,A6: set @ A,B6: set @ A] :
( ( inj_on @ A @ B @ F @ ( sup_sup @ ( set @ A ) @ A6 @ B6 ) )
= ( ( inj_on @ A @ B @ F @ A6 )
& ( inj_on @ A @ B @ F @ B6 )
& ( ( inf_inf @ ( set @ B ) @ ( image2 @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ A6 @ B6 ) ) @ ( image2 @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ B6 @ A6 ) ) )
= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% inj_on_Un
thf(fact_707_inj__on__image,axiom,
! [B: $tType,A: $tType,F: A > B,A6: set @ ( set @ A )] :
( ( inj_on @ A @ B @ F @ ( complete_Sup_Sup @ ( set @ A ) @ A6 ) )
=> ( inj_on @ ( set @ A ) @ ( set @ B ) @ ( image2 @ A @ B @ F ) @ A6 ) ) ).
% inj_on_image
thf(fact_708_cSUP__union,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A6: set @ B,F: B > A,B6: set @ B] :
( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F @ A6 ) )
=> ( ( B6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F @ B6 ) )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ ( sup_sup @ ( set @ B ) @ A6 @ B6 ) ) )
= ( sup_sup @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A6 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ B6 ) ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_709_cINF__union,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A6: set @ B,F: B > A,B6: set @ B] :
( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F @ A6 ) )
=> ( ( B6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F @ B6 ) )
=> ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ ( sup_sup @ ( set @ B ) @ A6 @ B6 ) ) )
= ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A6 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ B6 ) ) ) ) ) ) ) ) ) ).
% cINF_union
thf(fact_710_inj__on__image__Int,axiom,
! [B: $tType,A: $tType,F: A > B,C5: set @ A,A6: set @ A,B6: set @ A] :
( ( inj_on @ A @ B @ F @ C5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ C5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ C5 )
=> ( ( image2 @ A @ B @ F @ ( inf_inf @ ( set @ A ) @ A6 @ B6 ) )
= ( inf_inf @ ( set @ B ) @ ( image2 @ A @ B @ F @ A6 ) @ ( image2 @ A @ B @ F @ B6 ) ) ) ) ) ) ).
% inj_on_image_Int
thf(fact_711_image__Int,axiom,
! [B: $tType,A: $tType,F: A > B,A6: set @ A,B6: set @ A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( image2 @ A @ B @ F @ ( inf_inf @ ( set @ A ) @ A6 @ B6 ) )
= ( inf_inf @ ( set @ B ) @ ( image2 @ A @ B @ F @ A6 ) @ ( image2 @ A @ B @ F @ B6 ) ) ) ) ).
% image_Int
thf(fact_712_inj__on__image__Pow,axiom,
! [B: $tType,A: $tType,F: A > B,A6: set @ A] :
( ( inj_on @ A @ B @ F @ A6 )
=> ( inj_on @ ( set @ A ) @ ( set @ B ) @ ( image2 @ A @ B @ F ) @ ( pow @ A @ A6 ) ) ) ).
% inj_on_image_Pow
thf(fact_713_inj__on__image__iff,axiom,
! [B: $tType,A: $tType,A6: set @ A,G: A > B,F: A > A] :
( ! [X: A] :
( ( member @ A @ X @ A6 )
=> ! [Xa: A] :
( ( member @ A @ Xa @ A6 )
=> ( ( ( G @ ( F @ X ) )
= ( G @ ( F @ Xa ) ) )
= ( ( G @ X )
= ( G @ Xa ) ) ) ) )
=> ( ( inj_on @ A @ A @ F @ A6 )
=> ( ( inj_on @ A @ B @ G @ ( image2 @ A @ A @ F @ A6 ) )
= ( inj_on @ A @ B @ G @ A6 ) ) ) ) ).
% inj_on_image_iff
thf(fact_714_inj__on__image__Fpow,axiom,
! [B: $tType,A: $tType,F: A > B,A6: set @ A] :
( ( inj_on @ A @ B @ F @ A6 )
=> ( inj_on @ ( set @ A ) @ ( set @ B ) @ ( image2 @ A @ B @ F ) @ ( finite_Fpow @ A @ A6 ) ) ) ).
% inj_on_image_Fpow
thf(fact_715_inj__on__imageI2,axiom,
! [B: $tType,C: $tType,A: $tType,F5: C > B,F: A > C,A6: set @ A] :
( ( inj_on @ A @ B @ ( comp @ C @ B @ A @ F5 @ F ) @ A6 )
=> ( inj_on @ A @ C @ F @ A6 ) ) ).
% inj_on_imageI2
thf(fact_716_SUP__union,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [M: B > A,A6: set @ B,B6: set @ B] :
( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ M @ ( sup_sup @ ( set @ B ) @ A6 @ B6 ) ) )
= ( sup_sup @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ M @ A6 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ M @ B6 ) ) ) ) ) ).
% SUP_union
thf(fact_717_map__prod__inj__on,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,F: A > B,A6: set @ A,G: C > D,B6: set @ C] :
( ( inj_on @ A @ B @ F @ A6 )
=> ( ( 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 @ A6
@ ^ [Uu: A] : B6 ) ) ) ) ).
% map_prod_inj_on
thf(fact_718_INF__union,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [M: B > A,A6: set @ B,B6: set @ B] :
( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ M @ ( sup_sup @ ( set @ B ) @ A6 @ B6 ) ) )
= ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ M @ A6 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ M @ B6 ) ) ) ) ) ).
% INF_union
thf(fact_719_image__Un,axiom,
! [A: $tType,B: $tType,F: B > A,A6: set @ B,B6: set @ B] :
( ( image2 @ B @ A @ F @ ( sup_sup @ ( set @ B ) @ A6 @ B6 ) )
= ( sup_sup @ ( set @ A ) @ ( image2 @ B @ A @ F @ A6 ) @ ( image2 @ B @ A @ F @ B6 ) ) ) ).
% image_Un
thf(fact_720_INT__Un,axiom,
! [A: $tType,B: $tType,M: B > ( set @ A ),A6: set @ B,B6: set @ B] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ M @ ( sup_sup @ ( set @ B ) @ A6 @ B6 ) ) )
= ( inf_inf @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ M @ A6 ) ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ M @ B6 ) ) ) ) ).
% INT_Un
thf(fact_721_inj__swap,axiom,
! [B: $tType,A: $tType,A6: set @ ( product_prod @ A @ B )] : ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A ) @ ( product_swap @ A @ B ) @ A6 ) ).
% inj_swap
thf(fact_722_inj__on__convol__ident,axiom,
! [B: $tType,A: $tType,F: A > B,X6: set @ A] :
( inj_on @ A @ ( product_prod @ A @ B )
@ ^ [X4: A] : ( product_Pair @ A @ B @ X4 @ ( F @ X4 ) )
@ X6 ) ).
% inj_on_convol_ident
thf(fact_723_total__onI,axiom,
! [A: $tType,A6: set @ A,R: set @ ( product_prod @ A @ A )] :
( ! [X: A,Y: A] :
( ( member @ A @ X @ A6 )
=> ( ( member @ A @ Y @ A6 )
=> ( ( X != Y )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ R ) ) ) ) )
=> ( total_on @ A @ A6 @ R ) ) ).
% total_onI
thf(fact_724_total__on__def,axiom,
! [A: $tType] :
( ( total_on @ A )
= ( ^ [A8: set @ A,R5: set @ ( product_prod @ A @ A )] :
! [X4: A] :
( ( member @ A @ X4 @ A8 )
=> ! [Y5: A] :
( ( member @ A @ Y5 @ A8 )
=> ( ( X4 != Y5 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R5 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X4 ) @ R5 ) ) ) ) ) ) ) ).
% total_on_def
thf(fact_725_SUP__UNIV__bool__expand,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: $o > A] :
( ( complete_Sup_Sup @ A @ ( image2 @ $o @ A @ A6 @ ( top_top @ ( set @ $o ) ) ) )
= ( sup_sup @ A @ ( A6 @ $true ) @ ( A6 @ $false ) ) ) ) ).
% SUP_UNIV_bool_expand
thf(fact_726_INF__UNIV__bool__expand,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: $o > A] :
( ( complete_Inf_Inf @ A @ ( image2 @ $o @ A @ A6 @ ( top_top @ ( set @ $o ) ) ) )
= ( inf_inf @ A @ ( A6 @ $true ) @ ( A6 @ $false ) ) ) ) ).
% INF_UNIV_bool_expand
thf(fact_727_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_728_Un__eq__UN,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A8: set @ A,B7: set @ A] :
( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ $o @ ( set @ A )
@ ^ [B5: $o] : ( if @ ( set @ A ) @ B5 @ A8 @ B7 )
@ ( top_top @ ( set @ $o ) ) ) ) ) ) ).
% Un_eq_UN
thf(fact_729_UN__bool__eq,axiom,
! [A: $tType,A6: $o > ( set @ A )] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ $o @ ( set @ A ) @ A6 @ ( top_top @ ( set @ $o ) ) ) )
= ( sup_sup @ ( set @ A ) @ ( A6 @ $true ) @ ( A6 @ $false ) ) ) ).
% UN_bool_eq
thf(fact_730_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A6: set @ B,F: B > A,G: B > A] :
( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F @ A6 ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ G @ A6 ) )
=> ( ( sup_sup @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A6 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G @ A6 ) ) )
= ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [A7: B] : ( sup_sup @ A @ ( F @ A7 ) @ ( G @ A7 ) )
@ A6 ) ) ) ) ) ) ) ).
% conditionally_complete_lattice_class.SUP_sup_distrib
thf(fact_731_cINF__inf__distrib,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A6: set @ B,F: B > A,G: B > A] :
( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F @ A6 ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ G @ A6 ) )
=> ( ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A6 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G @ A6 ) ) )
= ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [A7: B] : ( inf_inf @ A @ ( F @ A7 ) @ ( G @ A7 ) )
@ A6 ) ) ) ) ) ) ) ).
% cINF_inf_distrib
thf(fact_732_fsts_Ointros,axiom,
! [B: $tType,A: $tType,P: product_prod @ A @ B] : ( member @ A @ ( product_fst @ A @ B @ P ) @ ( basic_fsts @ A @ B @ P ) ) ).
% fsts.intros
thf(fact_733_fsts_Osimps,axiom,
! [B: $tType,A: $tType,A3: A,P: product_prod @ A @ B] :
( ( member @ A @ A3 @ ( basic_fsts @ A @ B @ P ) )
= ( A3
= ( product_fst @ A @ B @ P ) ) ) ).
% fsts.simps
thf(fact_734_fsts_Ocases,axiom,
! [B: $tType,A: $tType,A3: A,P: product_prod @ A @ B] :
( ( member @ A @ A3 @ ( basic_fsts @ A @ B @ P ) )
=> ( A3
= ( product_fst @ A @ B @ P ) ) ) ).
% fsts.cases
thf(fact_735_INT__bool__eq,axiom,
! [A: $tType,A6: $o > ( set @ A )] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ $o @ ( set @ A ) @ A6 @ ( top_top @ ( set @ $o ) ) ) )
= ( inf_inf @ ( set @ A ) @ ( A6 @ $true ) @ ( A6 @ $false ) ) ) ).
% INT_bool_eq
thf(fact_736_cSUP__eq__cINF__D,axiom,
! [B: $tType,C: $tType] :
( ( condit1219197933456340205attice @ B )
=> ! [F: C > B,A6: set @ C,A3: C] :
( ( ( complete_Sup_Sup @ B @ ( image2 @ C @ B @ F @ A6 ) )
= ( complete_Inf_Inf @ B @ ( image2 @ C @ B @ F @ A6 ) ) )
=> ( ( condit941137186595557371_above @ B @ ( image2 @ C @ B @ F @ A6 ) )
=> ( ( condit1013018076250108175_below @ B @ ( image2 @ C @ B @ F @ A6 ) )
=> ( ( member @ C @ A3 @ A6 )
=> ( ( F @ A3 )
= ( complete_Inf_Inf @ B @ ( image2 @ C @ B @ F @ A6 ) ) ) ) ) ) ) ) ).
% cSUP_eq_cINF_D
thf(fact_737_cSUP__insert,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A6: set @ B,F: B > A,A3: B] :
( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F @ A6 ) )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ ( insert @ B @ A3 @ A6 ) ) )
= ( sup_sup @ A @ ( F @ A3 ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A6 ) ) ) ) ) ) ) ).
% cSUP_insert
thf(fact_738_cINF__insert,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A6: set @ B,F: B > A,A3: B] :
( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F @ A6 ) )
=> ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ ( insert @ B @ A3 @ A6 ) ) )
= ( inf_inf @ A @ ( F @ A3 ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A6 ) ) ) ) ) ) ) ).
% cINF_insert
thf(fact_739_inj__on__image__mem__iff,axiom,
! [B: $tType,A: $tType,F: A > B,B6: set @ A,A3: A,A6: set @ A] :
( ( inj_on @ A @ B @ F @ B6 )
=> ( ( member @ A @ A3 @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ( ( member @ B @ ( F @ A3 ) @ ( image2 @ A @ B @ F @ A6 ) )
= ( member @ A @ A3 @ A6 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_740_inj__on__image__eq__iff,axiom,
! [B: $tType,A: $tType,F: A > B,C5: set @ A,A6: set @ A,B6: set @ A] :
( ( inj_on @ A @ B @ F @ C5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ C5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ C5 )
=> ( ( ( image2 @ A @ B @ F @ A6 )
= ( image2 @ A @ B @ F @ B6 ) )
= ( A6 = B6 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_741_subset__image__inj,axiom,
! [A: $tType,B: $tType,S5: set @ A,F: B > A,T4: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ S5 @ ( image2 @ B @ A @ F @ T4 ) )
= ( ? [U4: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ U4 @ T4 )
& ( inj_on @ B @ A @ F @ U4 )
& ( S5
= ( image2 @ B @ A @ F @ U4 ) ) ) ) ) ).
% subset_image_inj
thf(fact_742_inj__img__insertE,axiom,
! [B: $tType,A: $tType,F: A > B,A6: set @ A,X2: B,B6: set @ B] :
( ( inj_on @ A @ B @ F @ A6 )
=> ( ~ ( member @ B @ X2 @ B6 )
=> ( ( ( insert @ B @ X2 @ B6 )
= ( image2 @ A @ B @ F @ A6 ) )
=> ~ ! [X8: A,A10: set @ A] :
( ~ ( member @ A @ X8 @ A10 )
=> ( ( A6
= ( insert @ A @ X8 @ A10 ) )
=> ( ( X2
= ( F @ X8 ) )
=> ( B6
!= ( image2 @ A @ B @ F @ A10 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_743_inj__image__mem__iff,axiom,
! [B: $tType,A: $tType,F: A > B,A3: A,A6: set @ A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( member @ B @ ( F @ A3 ) @ ( image2 @ A @ B @ F @ A6 ) )
= ( member @ A @ A3 @ A6 ) ) ) ).
% inj_image_mem_iff
thf(fact_744_inj__image__eq__iff,axiom,
! [B: $tType,A: $tType,F: A > B,A6: set @ A,B6: set @ A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( ( image2 @ A @ B @ F @ A6 )
= ( image2 @ A @ B @ F @ B6 ) )
= ( A6 = B6 ) ) ) ).
% inj_image_eq_iff
thf(fact_745_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 ) ) ) )
= ( ? [X4: A] :
( ( B2
= ( F @ X4 ) )
& ! [Y5: A] :
( ( B2
= ( F @ Y5 ) )
=> ( Y5 = X4 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_746_comp__inj__on,axiom,
! [B: $tType,C: $tType,A: $tType,F: A > B,A6: set @ A,G: B > C] :
( ( inj_on @ A @ B @ F @ A6 )
=> ( ( inj_on @ B @ C @ G @ ( image2 @ A @ B @ F @ A6 ) )
=> ( inj_on @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ A6 ) ) ) ).
% comp_inj_on
thf(fact_747_inj__on__imageI,axiom,
! [B: $tType,C: $tType,A: $tType,G: C > B,F: A > C,A6: set @ A] :
( ( inj_on @ A @ B @ ( comp @ C @ B @ A @ G @ F ) @ A6 )
=> ( inj_on @ C @ B @ G @ ( image2 @ A @ C @ F @ A6 ) ) ) ).
% inj_on_imageI
thf(fact_748_comp__inj__on__iff,axiom,
! [B: $tType,C: $tType,A: $tType,F: A > B,A6: set @ A,F5: B > C] :
( ( inj_on @ A @ B @ F @ A6 )
=> ( ( inj_on @ B @ C @ F5 @ ( image2 @ A @ B @ F @ A6 ) )
= ( inj_on @ A @ C @ ( comp @ B @ C @ A @ F5 @ F ) @ A6 ) ) ) ).
% comp_inj_on_iff
thf(fact_749_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_750_inf__Sup,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A3: A,B6: set @ A] :
( ( inf_inf @ A @ A3 @ ( complete_Sup_Sup @ A @ B6 ) )
= ( complete_Sup_Sup @ A @ ( image2 @ A @ A @ ( inf_inf @ A @ A3 ) @ B6 ) ) ) ) ).
% inf_Sup
thf(fact_751_sup__Inf,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A3: A,B6: set @ A] :
( ( sup_sup @ A @ A3 @ ( complete_Inf_Inf @ A @ B6 ) )
= ( complete_Inf_Inf @ A @ ( image2 @ A @ A @ ( sup_sup @ A @ A3 ) @ B6 ) ) ) ) ).
% sup_Inf
thf(fact_752_bdd__below_OI2,axiom,
! [A: $tType,B: $tType] :
( ( preorder @ A )
=> ! [A6: set @ B,M: A,F: B > A] :
( ! [X: B] :
( ( member @ B @ X @ A6 )
=> ( ord_less_eq @ A @ M @ ( F @ X ) ) )
=> ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F @ A6 ) ) ) ) ).
% bdd_below.I2
thf(fact_753_bdd__belowI2,axiom,
! [A: $tType,B: $tType] :
( ( preorder @ A )
=> ! [A6: set @ B,M4: A,F: B > A] :
( ! [X: B] :
( ( member @ B @ X @ A6 )
=> ( ord_less_eq @ A @ M4 @ ( F @ X ) ) )
=> ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F @ A6 ) ) ) ) ).
% bdd_belowI2
thf(fact_754_bdd__above_OI2,axiom,
! [A: $tType,B: $tType] :
( ( preorder @ A )
=> ! [A6: set @ B,F: B > A,M: A] :
( ! [X: B] :
( ( member @ B @ X @ A6 )
=> ( ord_less_eq @ A @ ( F @ X ) @ M ) )
=> ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F @ A6 ) ) ) ) ).
% bdd_above.I2
thf(fact_755_image__Int__subset,axiom,
! [A: $tType,B: $tType,F: B > A,A6: set @ B,B6: set @ B] : ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F @ ( inf_inf @ ( set @ B ) @ A6 @ B6 ) ) @ ( inf_inf @ ( set @ A ) @ ( image2 @ B @ A @ F @ A6 ) @ ( image2 @ B @ A @ F @ B6 ) ) ) ).
% image_Int_subset
thf(fact_756_swap__inj__on,axiom,
! [B: $tType,A: $tType,A6: 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,J: B] : ( product_Pair @ B @ A @ J @ I3 ) )
@ A6 ) ).
% swap_inj_on
thf(fact_757_complete__lattice__class_OSUP__sup__distrib,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: B > A,A6: set @ B,G: B > A] :
( ( sup_sup @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A6 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G @ A6 ) ) )
= ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [A7: B] : ( sup_sup @ A @ ( F @ A7 ) @ ( G @ A7 ) )
@ A6 ) ) ) ) ).
% complete_lattice_class.SUP_sup_distrib
thf(fact_758_SUP__absorb,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [K: B,I4: set @ B,A6: B > A] :
( ( member @ B @ K @ I4 )
=> ( ( sup_sup @ A @ ( A6 @ K ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ A6 @ I4 ) ) )
= ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ A6 @ I4 ) ) ) ) ) ).
% SUP_absorb
thf(fact_759_SUP__inf,axiom,
! [A: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [F: B > A,B6: set @ B,A3: A] :
( ( inf_inf @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ B6 ) ) @ A3 )
= ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [B5: B] : ( inf_inf @ A @ ( F @ B5 ) @ A3 )
@ B6 ) ) ) ) ).
% SUP_inf
thf(fact_760_Sup__inf,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [B6: set @ A,A3: A] :
( ( inf_inf @ A @ ( complete_Sup_Sup @ A @ B6 ) @ A3 )
= ( complete_Sup_Sup @ A
@ ( image2 @ A @ A
@ ^ [B5: A] : ( inf_inf @ A @ B5 @ A3 )
@ B6 ) ) ) ) ).
% Sup_inf
thf(fact_761_inf__SUP,axiom,
! [A: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A3: A,F: B > A,B6: set @ B] :
( ( inf_inf @ A @ A3 @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ B6 ) ) )
= ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [B5: B] : ( inf_inf @ A @ A3 @ ( F @ B5 ) )
@ B6 ) ) ) ) ).
% inf_SUP
thf(fact_762_SUP__inf__distrib2,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [F: B > A,A6: set @ B,G: C > A,B6: set @ C] :
( ( inf_inf @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A6 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ G @ B6 ) ) )
= ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [A7: B] :
( complete_Sup_Sup @ A
@ ( image2 @ C @ A
@ ^ [B5: C] : ( inf_inf @ A @ ( F @ A7 ) @ ( G @ B5 ) )
@ B6 ) )
@ A6 ) ) ) ) ).
% SUP_inf_distrib2
thf(fact_763_INF__sup,axiom,
! [A: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [F: B > A,B6: set @ B,A3: A] :
( ( sup_sup @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ B6 ) ) @ A3 )
= ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [B5: B] : ( sup_sup @ A @ ( F @ B5 ) @ A3 )
@ B6 ) ) ) ) ).
% INF_sup
thf(fact_764_Inf__sup,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [B6: set @ A,A3: A] :
( ( sup_sup @ A @ ( complete_Inf_Inf @ A @ B6 ) @ A3 )
= ( complete_Inf_Inf @ A
@ ( image2 @ A @ A
@ ^ [B5: A] : ( sup_sup @ A @ B5 @ A3 )
@ B6 ) ) ) ) ).
% Inf_sup
thf(fact_765_sup__INF,axiom,
! [A: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A3: A,F: B > A,B6: set @ B] :
( ( sup_sup @ A @ A3 @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ B6 ) ) )
= ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [B5: B] : ( sup_sup @ A @ A3 @ ( F @ B5 ) )
@ B6 ) ) ) ) ).
% sup_INF
thf(fact_766_INF__sup__distrib2,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [F: B > A,A6: set @ B,G: C > A,B6: set @ C] :
( ( sup_sup @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A6 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ G @ B6 ) ) )
= ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [A7: B] :
( complete_Inf_Inf @ A
@ ( image2 @ C @ A
@ ^ [B5: C] : ( sup_sup @ A @ ( F @ A7 ) @ ( G @ B5 ) )
@ B6 ) )
@ A6 ) ) ) ) ).
% INF_sup_distrib2
thf(fact_767_INF__inf__distrib,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: B > A,A6: set @ B,G: B > A] :
( ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A6 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G @ A6 ) ) )
= ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [A7: B] : ( inf_inf @ A @ ( F @ A7 ) @ ( G @ A7 ) )
@ A6 ) ) ) ) ).
% INF_inf_distrib
thf(fact_768_INF__absorb,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [K: B,I4: set @ B,A6: B > A] :
( ( member @ B @ K @ I4 )
=> ( ( inf_inf @ A @ ( A6 @ K ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ A6 @ I4 ) ) )
= ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ A6 @ I4 ) ) ) ) ) ).
% INF_absorb
thf(fact_769_translation__subtract__Int,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,S2: set @ A,T3: set @ A] :
( ( image2 @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ X4 @ A3 )
@ ( inf_inf @ ( set @ A ) @ S2 @ T3 ) )
= ( inf_inf @ ( set @ A )
@ ( image2 @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ X4 @ A3 )
@ S2 )
@ ( image2 @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ X4 @ A3 )
@ T3 ) ) ) ) ).
% translation_subtract_Int
thf(fact_770_Un__Union__image,axiom,
! [A: $tType,B: $tType,A6: B > ( set @ A ),B6: B > ( set @ A ),C5: set @ B] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X4: B] : ( sup_sup @ ( set @ A ) @ ( A6 @ X4 ) @ ( B6 @ X4 ) )
@ C5 ) )
= ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A6 @ C5 ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B6 @ C5 ) ) ) ) ).
% Un_Union_image
thf(fact_771_UN__Un__distrib,axiom,
! [A: $tType,B: $tType,A6: B > ( set @ A ),B6: B > ( set @ A ),I4: set @ B] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [I3: B] : ( sup_sup @ ( set @ A ) @ ( A6 @ I3 ) @ ( B6 @ I3 ) )
@ I4 ) )
= ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A6 @ I4 ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B6 @ I4 ) ) ) ) ).
% UN_Un_distrib
thf(fact_772_UN__absorb,axiom,
! [B: $tType,A: $tType,K: A,I4: set @ A,A6: A > ( set @ B )] :
( ( member @ A @ K @ I4 )
=> ( ( sup_sup @ ( set @ B ) @ ( A6 @ K ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A6 @ I4 ) ) )
= ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A6 @ I4 ) ) ) ) ).
% UN_absorb
thf(fact_773_Int__UN__distrib2,axiom,
! [A: $tType,C: $tType,B: $tType,A6: B > ( set @ A ),I4: set @ B,B6: C > ( set @ A ),J4: set @ C] :
( ( inf_inf @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A6 @ I4 ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ C @ ( set @ A ) @ B6 @ J4 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [I3: B] :
( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [J: C] : ( inf_inf @ ( set @ A ) @ ( A6 @ I3 ) @ ( B6 @ J ) )
@ J4 ) )
@ I4 ) ) ) ).
% Int_UN_distrib2
thf(fact_774_Int__UN__distrib,axiom,
! [A: $tType,B: $tType,B6: set @ A,A6: B > ( set @ A ),I4: set @ B] :
( ( inf_inf @ ( set @ A ) @ B6 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A6 @ I4 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [I3: B] : ( inf_inf @ ( set @ A ) @ B6 @ ( A6 @ I3 ) )
@ I4 ) ) ) ).
% Int_UN_distrib
thf(fact_775_UN__extend__simps_I4_J,axiom,
! [H5: $tType,G2: $tType,A6: G2 > ( set @ H5 ),C5: set @ G2,B6: set @ H5] :
( ( inf_inf @ ( set @ H5 ) @ ( complete_Sup_Sup @ ( set @ H5 ) @ ( image2 @ G2 @ ( set @ H5 ) @ A6 @ C5 ) ) @ B6 )
= ( complete_Sup_Sup @ ( set @ H5 )
@ ( image2 @ G2 @ ( set @ H5 )
@ ^ [X4: G2] : ( inf_inf @ ( set @ H5 ) @ ( A6 @ X4 ) @ B6 )
@ C5 ) ) ) ).
% UN_extend_simps(4)
thf(fact_776_UN__extend__simps_I5_J,axiom,
! [I5: $tType,J2: $tType,A6: set @ I5,B6: J2 > ( set @ I5 ),C5: set @ J2] :
( ( inf_inf @ ( set @ I5 ) @ A6 @ ( complete_Sup_Sup @ ( set @ I5 ) @ ( image2 @ J2 @ ( set @ I5 ) @ B6 @ C5 ) ) )
= ( complete_Sup_Sup @ ( set @ I5 )
@ ( image2 @ J2 @ ( set @ I5 )
@ ^ [X4: J2] : ( inf_inf @ ( set @ I5 ) @ A6 @ ( B6 @ X4 ) )
@ C5 ) ) ) ).
% UN_extend_simps(5)
thf(fact_777_Un__INT__distrib2,axiom,
! [A: $tType,C: $tType,B: $tType,A6: B > ( set @ A ),I4: set @ B,B6: C > ( set @ A ),J4: set @ C] :
( ( sup_sup @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A6 @ I4 ) ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ C @ ( set @ A ) @ B6 @ J4 ) ) )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [I3: B] :
( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [J: C] : ( sup_sup @ ( set @ A ) @ ( A6 @ I3 ) @ ( B6 @ J ) )
@ J4 ) )
@ I4 ) ) ) ).
% Un_INT_distrib2
thf(fact_778_Un__INT__distrib,axiom,
! [A: $tType,B: $tType,B6: set @ A,A6: B > ( set @ A ),I4: set @ B] :
( ( sup_sup @ ( set @ A ) @ B6 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A6 @ I4 ) ) )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [I3: B] : ( sup_sup @ ( set @ A ) @ B6 @ ( A6 @ I3 ) )
@ I4 ) ) ) ).
% Un_INT_distrib
thf(fact_779_INT__extend__simps_I6_J,axiom,
! [L4: $tType,K2: $tType,A6: K2 > ( set @ L4 ),C5: set @ K2,B6: set @ L4] :
( ( sup_sup @ ( set @ L4 ) @ ( complete_Inf_Inf @ ( set @ L4 ) @ ( image2 @ K2 @ ( set @ L4 ) @ A6 @ C5 ) ) @ B6 )
= ( complete_Inf_Inf @ ( set @ L4 )
@ ( image2 @ K2 @ ( set @ L4 )
@ ^ [X4: K2] : ( sup_sup @ ( set @ L4 ) @ ( A6 @ X4 ) @ B6 )
@ C5 ) ) ) ).
% INT_extend_simps(6)
thf(fact_780_INT__extend__simps_I7_J,axiom,
! [M5: $tType,N3: $tType,A6: set @ M5,B6: N3 > ( set @ M5 ),C5: set @ N3] :
( ( sup_sup @ ( set @ M5 ) @ A6 @ ( complete_Inf_Inf @ ( set @ M5 ) @ ( image2 @ N3 @ ( set @ M5 ) @ B6 @ C5 ) ) )
= ( complete_Inf_Inf @ ( set @ M5 )
@ ( image2 @ N3 @ ( set @ M5 )
@ ^ [X4: N3] : ( sup_sup @ ( set @ M5 ) @ A6 @ ( B6 @ X4 ) )
@ C5 ) ) ) ).
% INT_extend_simps(7)
thf(fact_781_Int__Inter__image,axiom,
! [A: $tType,B: $tType,A6: B > ( set @ A ),B6: B > ( set @ A ),C5: set @ B] :
( ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X4: B] : ( inf_inf @ ( set @ A ) @ ( A6 @ X4 ) @ ( B6 @ X4 ) )
@ C5 ) )
= ( inf_inf @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A6 @ C5 ) ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B6 @ C5 ) ) ) ) ).
% Int_Inter_image
thf(fact_782_INT__Int__distrib,axiom,
! [A: $tType,B: $tType,A6: B > ( set @ A ),B6: B > ( set @ A ),I4: set @ B] :
( ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [I3: B] : ( inf_inf @ ( set @ A ) @ ( A6 @ I3 ) @ ( B6 @ I3 ) )
@ I4 ) )
= ( inf_inf @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A6 @ I4 ) ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B6 @ I4 ) ) ) ) ).
% INT_Int_distrib
thf(fact_783_INT__absorb,axiom,
! [B: $tType,A: $tType,K: A,I4: set @ A,A6: A > ( set @ B )] :
( ( member @ A @ K @ I4 )
=> ( ( inf_inf @ ( set @ B ) @ ( A6 @ K ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A6 @ I4 ) ) )
= ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A6 @ I4 ) ) ) ) ).
% INT_absorb
thf(fact_784_Int__Union2,axiom,
! [A: $tType,B6: set @ ( set @ A ),A6: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ B6 ) @ A6 )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ ( set @ A ) @ ( set @ A )
@ ^ [C7: set @ A] : ( inf_inf @ ( set @ A ) @ C7 @ A6 )
@ B6 ) ) ) ).
% Int_Union2
thf(fact_785_Int__Union,axiom,
! [A: $tType,A6: set @ A,B6: set @ ( set @ A )] :
( ( inf_inf @ ( set @ A ) @ A6 @ ( complete_Sup_Sup @ ( set @ A ) @ B6 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A6 ) @ B6 ) ) ) ).
% Int_Union
thf(fact_786_Un__Inter,axiom,
! [A: $tType,A6: set @ A,B6: set @ ( set @ A )] :
( ( sup_sup @ ( set @ A ) @ A6 @ ( complete_Inf_Inf @ ( set @ A ) @ B6 ) )
= ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A6 ) @ B6 ) ) ) ).
% Un_Inter
thf(fact_787_inj__on__iff__surj,axiom,
! [A: $tType,B: $tType,A6: set @ A,A9: set @ B] :
( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ? [F4: A > B] :
( ( inj_on @ A @ B @ F4 @ A6 )
& ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F4 @ A6 ) @ A9 ) ) )
= ( ? [G4: B > A] :
( ( image2 @ B @ A @ G4 @ A9 )
= A6 ) ) ) ) ).
% inj_on_iff_surj
thf(fact_788_Int__Inter__eq_I1_J,axiom,
! [A: $tType,B9: set @ ( set @ A ),A6: set @ A] :
( ( ( B9
= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ A6 @ ( complete_Inf_Inf @ ( set @ A ) @ B9 ) )
= A6 ) )
& ( ( B9
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ A6 @ ( complete_Inf_Inf @ ( set @ A ) @ B9 ) )
= ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A6 ) @ B9 ) ) ) ) ) ).
% Int_Inter_eq(1)
thf(fact_789_Int__Inter__eq_I2_J,axiom,
! [A: $tType,B9: set @ ( set @ A ),A6: set @ A] :
( ( ( B9
= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ B9 ) @ A6 )
= A6 ) )
& ( ( B9
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ B9 ) @ A6 )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ ( set @ A ) @ ( set @ A )
@ ^ [B7: set @ A] : ( inf_inf @ ( set @ A ) @ B7 @ A6 )
@ B9 ) ) ) ) ) ).
% Int_Inter_eq(2)
thf(fact_790_inj__image__subset__iff,axiom,
! [B: $tType,A: $tType,F: A > B,A6: set @ A,B6: set @ A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F @ A6 ) @ ( image2 @ A @ B @ F @ B6 ) )
= ( ord_less_eq @ ( set @ A ) @ A6 @ B6 ) ) ) ).
% inj_image_subset_iff
thf(fact_791_inj__on__image__set__diff,axiom,
! [B: $tType,A: $tType,F: A > B,C5: set @ A,A6: set @ A,B6: set @ A] :
( ( inj_on @ A @ B @ F @ C5 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A6 @ B6 ) @ C5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ C5 )
=> ( ( image2 @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ A6 @ B6 ) )
= ( minus_minus @ ( set @ B ) @ ( image2 @ A @ B @ F @ A6 ) @ ( image2 @ A @ B @ F @ B6 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_792_image__set__diff,axiom,
! [B: $tType,A: $tType,F: A > B,A6: set @ A,B6: set @ A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( image2 @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ A6 @ B6 ) )
= ( minus_minus @ ( set @ B ) @ ( image2 @ A @ B @ F @ A6 ) @ ( image2 @ A @ B @ F @ B6 ) ) ) ) ).
% image_set_diff
thf(fact_793_prod_Oset__map_I1_J,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,F1: A > C,F23: B > D,V: product_prod @ A @ B] :
( ( basic_fsts @ C @ D @ ( product_map_prod @ A @ C @ B @ D @ F1 @ F23 @ V ) )
= ( image2 @ A @ C @ F1 @ ( basic_fsts @ A @ B @ V ) ) ) ).
% prod.set_map(1)
thf(fact_794_cINF__lower2,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [F: B > A,A6: set @ B,X2: B,U2: A] :
( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F @ A6 ) )
=> ( ( member @ B @ X2 @ A6 )
=> ( ( ord_less_eq @ A @ ( F @ X2 ) @ U2 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A6 ) ) @ U2 ) ) ) ) ) ).
% cINF_lower2
thf(fact_795_cINF__lower,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [F: B > A,A6: set @ B,X2: B] :
( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F @ A6 ) )
=> ( ( member @ B @ X2 @ A6 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A6 ) ) @ ( F @ X2 ) ) ) ) ) ).
% cINF_lower
thf(fact_796_cSUP__upper2,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [F: B > A,A6: set @ B,X2: B,U2: A] :
( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F @ A6 ) )
=> ( ( member @ B @ X2 @ A6 )
=> ( ( ord_less_eq @ A @ U2 @ ( F @ X2 ) )
=> ( ord_less_eq @ A @ U2 @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A6 ) ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_797_cSUP__upper,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X2: B,A6: set @ B,F: B > A] :
( ( member @ B @ X2 @ A6 )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F @ A6 ) )
=> ( ord_less_eq @ A @ ( F @ X2 ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A6 ) ) ) ) ) ) ).
% cSUP_upper
thf(fact_798_Union__image__empty,axiom,
! [B: $tType,A: $tType,A6: set @ A,F: B > ( set @ A )] :
( ( sup_sup @ ( set @ A ) @ A6 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ F @ ( bot_bot @ ( set @ B ) ) ) ) )
= A6 ) ).
% Union_image_empty
thf(fact_799_Union__image__insert,axiom,
! [A: $tType,B: $tType,F: B > ( set @ A ),A3: B,B6: set @ B] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ F @ ( insert @ B @ A3 @ B6 ) ) )
= ( sup_sup @ ( set @ A ) @ ( F @ A3 ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ F @ B6 ) ) ) ) ).
% Union_image_insert
thf(fact_800_inj__on__UNION__chain,axiom,
! [C: $tType,B: $tType,A: $tType,I4: set @ A,A6: A > ( set @ B ),F: B > C] :
( ! [I6: A,J3: A] :
( ( member @ A @ I6 @ I4 )
=> ( ( member @ A @ J3 @ I4 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( A6 @ I6 ) @ ( A6 @ J3 ) )
| ( ord_less_eq @ ( set @ B ) @ ( A6 @ J3 ) @ ( A6 @ I6 ) ) ) ) )
=> ( ! [I6: A] :
( ( member @ A @ I6 @ I4 )
=> ( inj_on @ B @ C @ F @ ( A6 @ I6 ) ) )
=> ( inj_on @ B @ C @ F @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A6 @ I4 ) ) ) ) ) ).
% inj_on_UNION_chain
thf(fact_801_inj__on__INTER,axiom,
! [C: $tType,B: $tType,A: $tType,I4: set @ A,F: B > C,A6: A > ( set @ B )] :
( ( I4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [I6: A] :
( ( member @ A @ I6 @ I4 )
=> ( inj_on @ B @ C @ F @ ( A6 @ I6 ) ) )
=> ( inj_on @ B @ C @ F @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A6 @ I4 ) ) ) ) ) ).
% inj_on_INTER
thf(fact_802_SUP__insert,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: B > A,A3: B,A6: set @ B] :
( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ ( insert @ B @ A3 @ A6 ) ) )
= ( sup_sup @ A @ ( F @ A3 ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A6 ) ) ) ) ) ).
% SUP_insert
thf(fact_803_INF__inf__const2,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I4: set @ B,F: B > A,X2: A] :
( ( I4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [I3: B] : ( inf_inf @ A @ ( F @ I3 ) @ X2 )
@ I4 ) )
= ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ I4 ) ) @ X2 ) ) ) ) ).
% INF_inf_const2
thf(fact_804_INF__inf__const1,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I4: set @ B,X2: A,F: B > A] :
( ( I4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [I3: B] : ( inf_inf @ A @ X2 @ ( F @ I3 ) )
@ I4 ) )
= ( inf_inf @ A @ X2 @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ I4 ) ) ) ) ) ) ).
% INF_inf_const1
thf(fact_805_INF__insert,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: B > A,A3: B,A6: set @ B] :
( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ ( insert @ B @ A3 @ A6 ) ) )
= ( inf_inf @ A @ ( F @ A3 ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A6 ) ) ) ) ) ).
% INF_insert
thf(fact_806_UN__extend__simps_I2_J,axiom,
! [D: $tType,C: $tType,C5: set @ C,A6: C > ( set @ D ),B6: set @ D] :
( ( ( C5
= ( bot_bot @ ( set @ C ) ) )
=> ( ( sup_sup @ ( set @ D ) @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ A6 @ C5 ) ) @ B6 )
= B6 ) )
& ( ( C5
!= ( bot_bot @ ( set @ C ) ) )
=> ( ( sup_sup @ ( set @ D ) @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ A6 @ C5 ) ) @ B6 )
= ( complete_Sup_Sup @ ( set @ D )
@ ( image2 @ C @ ( set @ D )
@ ^ [X4: C] : ( sup_sup @ ( set @ D ) @ ( A6 @ X4 ) @ B6 )
@ C5 ) ) ) ) ) ).
% UN_extend_simps(2)
thf(fact_807_UN__extend__simps_I3_J,axiom,
! [E: $tType,F2: $tType,C5: set @ F2,A6: set @ E,B6: F2 > ( set @ E )] :
( ( ( C5
= ( bot_bot @ ( set @ F2 ) ) )
=> ( ( sup_sup @ ( set @ E ) @ A6 @ ( complete_Sup_Sup @ ( set @ E ) @ ( image2 @ F2 @ ( set @ E ) @ B6 @ C5 ) ) )
= A6 ) )
& ( ( C5
!= ( bot_bot @ ( set @ F2 ) ) )
=> ( ( sup_sup @ ( set @ E ) @ A6 @ ( complete_Sup_Sup @ ( set @ E ) @ ( image2 @ F2 @ ( set @ E ) @ B6 @ C5 ) ) )
= ( complete_Sup_Sup @ ( set @ E )
@ ( image2 @ F2 @ ( set @ E )
@ ^ [X4: F2] : ( sup_sup @ ( set @ E ) @ A6 @ ( B6 @ X4 ) )
@ C5 ) ) ) ) ) ).
% UN_extend_simps(3)
thf(fact_808_INT__extend__simps_I1_J,axiom,
! [B: $tType,A: $tType,C5: set @ A,A6: A > ( set @ B ),B6: set @ B] :
( ( ( C5
= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A6 @ C5 ) ) @ B6 )
= B6 ) )
& ( ( C5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A6 @ C5 ) ) @ B6 )
= ( complete_Inf_Inf @ ( set @ B )
@ ( image2 @ A @ ( set @ B )
@ ^ [X4: A] : ( inf_inf @ ( set @ B ) @ ( A6 @ X4 ) @ B6 )
@ C5 ) ) ) ) ) ).
% INT_extend_simps(1)
thf(fact_809_INT__extend__simps_I2_J,axiom,
! [C: $tType,D: $tType,C5: set @ D,A6: set @ C,B6: D > ( set @ C )] :
( ( ( C5
= ( bot_bot @ ( set @ D ) ) )
=> ( ( inf_inf @ ( set @ C ) @ A6 @ ( complete_Inf_Inf @ ( set @ C ) @ ( image2 @ D @ ( set @ C ) @ B6 @ C5 ) ) )
= A6 ) )
& ( ( C5
!= ( bot_bot @ ( set @ D ) ) )
=> ( ( inf_inf @ ( set @ C ) @ A6 @ ( complete_Inf_Inf @ ( set @ C ) @ ( image2 @ D @ ( set @ C ) @ B6 @ C5 ) ) )
= ( complete_Inf_Inf @ ( set @ C )
@ ( image2 @ D @ ( set @ C )
@ ^ [X4: D] : ( inf_inf @ ( set @ C ) @ A6 @ ( B6 @ X4 ) )
@ C5 ) ) ) ) ) ).
% INT_extend_simps(2)
thf(fact_810_le__cINF__iff,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A6: set @ B,F: B > A,U2: A] :
( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F @ A6 ) )
=> ( ( ord_less_eq @ A @ U2 @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A6 ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A6 )
=> ( ord_less_eq @ A @ U2 @ ( F @ X4 ) ) ) ) ) ) ) ) ).
% le_cINF_iff
thf(fact_811_cINF__mono,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [B6: set @ B,F: C > A,A6: set @ C,G: B > A] :
( ( B6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ C @ A @ F @ A6 ) )
=> ( ! [M3: B] :
( ( member @ B @ M3 @ B6 )
=> ? [X5: C] :
( ( member @ C @ X5 @ A6 )
& ( ord_less_eq @ A @ ( F @ X5 ) @ ( G @ M3 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ F @ A6 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G @ B6 ) ) ) ) ) ) ) ).
% cINF_mono
thf(fact_812_cSUP__le__iff,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A6: set @ B,F: B > A,U2: A] :
( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F @ A6 ) )
=> ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A6 ) ) @ U2 )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A6 )
=> ( ord_less_eq @ A @ ( F @ X4 ) @ U2 ) ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_813_cSUP__mono,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A6: set @ B,G: C > A,B6: set @ C,F: B > A] :
( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ C @ A @ G @ B6 ) )
=> ( ! [N2: B] :
( ( member @ B @ N2 @ A6 )
=> ? [X5: C] :
( ( member @ C @ X5 @ B6 )
& ( ord_less_eq @ A @ ( F @ N2 ) @ ( G @ X5 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A6 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ G @ B6 ) ) ) ) ) ) ) ).
% cSUP_mono
thf(fact_814_prod__set__defs_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( basic_fsts @ A @ B )
= ( ^ [P3: product_prod @ A @ B] : ( insert @ A @ ( product_fst @ A @ B @ P3 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% prod_set_defs(1)
thf(fact_815_image__INT,axiom,
! [B: $tType,A: $tType,C: $tType,F: A > B,C5: set @ A,A6: set @ C,B6: C > ( set @ A ),J5: C] :
( ( inj_on @ A @ B @ F @ C5 )
=> ( ! [X: C] :
( ( member @ C @ X @ A6 )
=> ( ord_less_eq @ ( set @ A ) @ ( B6 @ X ) @ C5 ) )
=> ( ( member @ C @ J5 @ A6 )
=> ( ( image2 @ A @ B @ F @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ C @ ( set @ A ) @ B6 @ A6 ) ) )
= ( complete_Inf_Inf @ ( set @ B )
@ ( image2 @ C @ ( set @ B )
@ ^ [X4: C] : ( image2 @ A @ B @ F @ ( B6 @ X4 ) )
@ A6 ) ) ) ) ) ) ).
% image_INT
thf(fact_816_cINF__superset__mono,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A6: set @ B,G: B > A,B6: set @ B,F: B > A] :
( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ G @ B6 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ A6 @ B6 )
=> ( ! [X: B] :
( ( member @ B @ X @ B6 )
=> ( ord_less_eq @ A @ ( G @ X ) @ ( F @ X ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G @ B6 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A6 ) ) ) ) ) ) ) ) ).
% cINF_superset_mono
thf(fact_817_cSUP__subset__mono,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A6: set @ B,G: B > A,B6: set @ B,F: B > A] :
( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ G @ B6 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ A6 @ B6 )
=> ( ! [X: B] :
( ( member @ B @ X @ A6 )
=> ( ord_less_eq @ A @ ( F @ X ) @ ( G @ X ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A6 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G @ B6 ) ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_818_UNION__fun__upd,axiom,
! [B: $tType,A: $tType,A6: B > ( set @ A ),I: B,B6: set @ A,J4: set @ B] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ ( fun_upd @ B @ ( set @ A ) @ A6 @ I @ B6 ) @ J4 ) )
= ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A6 @ ( minus_minus @ ( set @ B ) @ J4 @ ( insert @ B @ I @ ( bot_bot @ ( set @ B ) ) ) ) ) ) @ ( if @ ( set @ A ) @ ( member @ B @ I @ J4 ) @ B6 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% UNION_fun_upd
thf(fact_819_image__split__eq__Sigma,axiom,
! [C: $tType,B: $tType,A: $tType,F: C > A,G: C > B,A6: set @ C] :
( ( image2 @ C @ ( product_prod @ A @ B )
@ ^ [X4: C] : ( product_Pair @ A @ B @ ( F @ X4 ) @ ( G @ X4 ) )
@ A6 )
= ( product_Sigma @ A @ B @ ( image2 @ C @ A @ F @ A6 )
@ ^ [X4: A] : ( image2 @ C @ B @ G @ ( inf_inf @ ( set @ C ) @ ( vimage @ C @ A @ F @ ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) @ A6 ) ) ) ) ).
% image_split_eq_Sigma
thf(fact_820_If__the__inv__into__f__f,axiom,
! [B: $tType,A: $tType,I: A,C5: set @ A,G: A > B,X2: 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 ) @ X2 )
@ G
@ I )
= ( id @ A @ I ) ) ) ) ).
% If_the_inv_into_f_f
thf(fact_821_Field__insert,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A )] :
( ( field2 @ A @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R ) )
= ( sup_sup @ ( set @ A ) @ ( insert @ A @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( field2 @ A @ R ) ) ) ).
% Field_insert
thf(fact_822_bdd__below__image__antimono,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F: A > B,A6: set @ A] :
( ( order_antimono @ A @ B @ F )
=> ( ( condit941137186595557371_above @ A @ A6 )
=> ( condit1013018076250108175_below @ B @ ( image2 @ A @ B @ F @ A6 ) ) ) ) ) ).
% bdd_below_image_antimono
thf(fact_823_bdd__above__image__antimono,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F: A > B,A6: set @ A] :
( ( order_antimono @ A @ B @ F )
=> ( ( condit1013018076250108175_below @ A @ A6 )
=> ( condit941137186595557371_above @ B @ ( image2 @ A @ B @ F @ A6 ) ) ) ) ) ).
% bdd_above_image_antimono
thf(fact_824_the__inv__f__o__f__id,axiom,
! [B: $tType,A: $tType,F: A > B,Z3: 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 @ Z3 )
= ( id @ A @ Z3 ) ) ) ).
% the_inv_f_o_f_id
thf(fact_825_If__the__inv__into__in__Func,axiom,
! [B: $tType,A: $tType,G: A > B,C5: set @ A,B6: set @ A,X2: A] :
( ( inj_on @ A @ B @ G @ C5 )
=> ( ( ord_less_eq @ ( set @ A ) @ C5 @ ( sup_sup @ ( set @ A ) @ B6 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ ( B > A )
@ ^ [I3: B] : ( if @ A @ ( member @ B @ I3 @ ( image2 @ A @ B @ G @ C5 ) ) @ ( the_inv_into @ A @ B @ C5 @ G @ I3 ) @ X2 )
@ ( bNF_Wellorder_Func @ B @ A @ ( top_top @ ( set @ B ) ) @ ( sup_sup @ ( set @ A ) @ B6 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% If_the_inv_into_in_Func
thf(fact_826_mono__cInf,axiom,
! [B: $tType,A: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( condit1219197933456340205attice @ B ) )
=> ! [F: A > B,A6: set @ A] :
( ( order_mono @ A @ B @ F )
=> ( ( condit1013018076250108175_below @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ B @ ( F @ ( complete_Inf_Inf @ A @ A6 ) ) @ ( complete_Inf_Inf @ B @ ( image2 @ A @ B @ F @ A6 ) ) ) ) ) ) ) ).
% mono_cInf
thf(fact_827_mono__cINF,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( condit1219197933456340205attice @ B ) )
=> ! [F: A > B,A6: C > A,I4: set @ C] :
( ( order_mono @ A @ B @ F )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ C @ A @ A6 @ I4 ) )
=> ( ( I4
!= ( bot_bot @ ( set @ C ) ) )
=> ( ord_less_eq @ B @ ( F @ ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ A6 @ I4 ) ) )
@ ( complete_Inf_Inf @ B
@ ( image2 @ C @ B
@ ^ [X4: C] : ( F @ ( A6 @ X4 ) )
@ I4 ) ) ) ) ) ) ) ).
% mono_cINF
thf(fact_828_relcompp__distrib2,axiom,
! [A: $tType,B: $tType,C: $tType,S5: A > C > $o,T4: A > C > $o,R3: C > B > $o] :
( ( relcompp @ A @ C @ B @ ( sup_sup @ ( A > C > $o ) @ S5 @ T4 ) @ R3 )
= ( sup_sup @ ( A > B > $o ) @ ( relcompp @ A @ C @ B @ S5 @ R3 ) @ ( relcompp @ A @ C @ B @ T4 @ R3 ) ) ) ).
% relcompp_distrib2
thf(fact_829_relcompp__distrib,axiom,
! [A: $tType,B: $tType,C: $tType,R3: A > C > $o,S5: C > B > $o,T4: C > B > $o] :
( ( relcompp @ A @ C @ B @ R3 @ ( sup_sup @ ( C > B > $o ) @ S5 @ T4 ) )
= ( sup_sup @ ( A > B > $o ) @ ( relcompp @ A @ C @ B @ R3 @ S5 ) @ ( relcompp @ A @ C @ B @ R3 @ T4 ) ) ) ).
% relcompp_distrib
thf(fact_830_Field__Union,axiom,
! [A: $tType,R3: set @ ( set @ ( product_prod @ A @ A ) )] :
( ( field2 @ A @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) ) @ R3 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ A ) @ ( field2 @ A ) @ R3 ) ) ) ).
% Field_Union
thf(fact_831_the__inv__into__onto,axiom,
! [B: $tType,A: $tType,F: A > B,A6: set @ A] :
( ( inj_on @ A @ B @ F @ A6 )
=> ( ( image2 @ B @ A @ ( the_inv_into @ A @ B @ A6 @ F ) @ ( image2 @ A @ B @ F @ A6 ) )
= A6 ) ) ).
% the_inv_into_onto
thf(fact_832_image__vimage__eq,axiom,
! [A: $tType,B: $tType,F: B > A,A6: set @ A] :
( ( image2 @ B @ A @ F @ ( vimage @ B @ A @ F @ A6 ) )
= ( inf_inf @ ( set @ A ) @ A6 @ ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ) ).
% image_vimage_eq
thf(fact_833_vimage__comp,axiom,
! [A: $tType,B: $tType,C: $tType,F: A > B,G: B > C,X2: set @ C] :
( ( vimage @ A @ B @ F @ ( vimage @ B @ C @ G @ X2 ) )
= ( vimage @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ X2 ) ) ).
% vimage_comp
thf(fact_834_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_835_FieldI2,axiom,
! [A: $tType,I: A,J5: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J5 ) @ R3 )
=> ( member @ A @ J5 @ ( field2 @ A @ R3 ) ) ) ).
% FieldI2
thf(fact_836_FieldI1,axiom,
! [A: $tType,I: A,J5: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J5 ) @ R3 )
=> ( member @ A @ I @ ( field2 @ A @ R3 ) ) ) ).
% FieldI1
thf(fact_837_inf__Int__eq2,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S5: set @ ( product_prod @ A @ B )] :
( ( inf_inf @ ( A > B > $o )
@ ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ R3 )
@ ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ S5 ) )
= ( ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ S5 ) ) ) ) ).
% inf_Int_eq2
thf(fact_838_fun__upd__comp,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > B,G: A > C,X2: A,Y3: C] :
( ( comp @ C @ B @ A @ F @ ( fun_upd @ A @ C @ G @ X2 @ Y3 ) )
= ( fun_upd @ A @ B @ ( comp @ C @ B @ A @ F @ G ) @ X2 @ ( F @ Y3 ) ) ) ).
% fun_upd_comp
thf(fact_839_image__vimage__subset,axiom,
! [B: $tType,A: $tType,F: B > A,A6: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F @ ( vimage @ B @ A @ F @ A6 ) ) @ A6 ) ).
% image_vimage_subset
thf(fact_840_image__subset__iff__subset__vimage,axiom,
! [B: $tType,A: $tType,F: B > A,A6: set @ B,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F @ A6 ) @ B6 )
= ( ord_less_eq @ ( set @ B ) @ A6 @ ( vimage @ B @ A @ F @ B6 ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_841_surj__image__vimage__eq,axiom,
! [B: $tType,A: $tType,F: B > A,A6: set @ A] :
( ( ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( image2 @ B @ A @ F @ ( vimage @ B @ A @ F @ A6 ) )
= A6 ) ) ).
% surj_image_vimage_eq
thf(fact_842_bdd__below__image__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F: A > B,A6: set @ A] :
( ( order_mono @ A @ B @ F )
=> ( ( condit1013018076250108175_below @ A @ A6 )
=> ( condit1013018076250108175_below @ B @ ( image2 @ A @ B @ F @ A6 ) ) ) ) ) ).
% bdd_below_image_mono
thf(fact_843_bdd__above__image__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F: A > B,A6: set @ A] :
( ( order_mono @ A @ B @ F )
=> ( ( condit941137186595557371_above @ A @ A6 )
=> ( condit941137186595557371_above @ B @ ( image2 @ A @ B @ F @ A6 ) ) ) ) ) ).
% bdd_above_image_mono
thf(fact_844_vimage__UN,axiom,
! [A: $tType,B: $tType,C: $tType,F: A > B,B6: C > ( set @ B ),A6: set @ C] :
( ( vimage @ A @ B @ F @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ C @ ( set @ B ) @ B6 @ A6 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [X4: C] : ( vimage @ A @ B @ F @ ( B6 @ X4 ) )
@ A6 ) ) ) ).
% vimage_UN
thf(fact_845_vimage__INT,axiom,
! [A: $tType,B: $tType,C: $tType,F: A > B,B6: C > ( set @ B ),A6: set @ C] :
( ( vimage @ A @ B @ F @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ C @ ( set @ B ) @ B6 @ A6 ) ) )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [X4: C] : ( vimage @ A @ B @ F @ ( B6 @ X4 ) )
@ A6 ) ) ) ).
% vimage_INT
thf(fact_846_sup__Un__eq2,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S5: set @ ( product_prod @ A @ B )] :
( ( sup_sup @ ( A > B > $o )
@ ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ R3 )
@ ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ S5 ) )
= ( ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ S5 ) ) ) ) ).
% sup_Un_eq2
thf(fact_847_vimage__Union,axiom,
! [A: $tType,B: $tType,F: A > B,A6: set @ ( set @ B )] :
( ( vimage @ A @ B @ F @ ( complete_Sup_Sup @ ( set @ B ) @ A6 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( set @ B ) @ ( set @ A ) @ ( vimage @ A @ B @ F ) @ A6 ) ) ) ).
% vimage_Union
thf(fact_848_vimage__Times,axiom,
! [A: $tType,B: $tType,C: $tType,F: A > ( product_prod @ B @ C ),A6: set @ B,B6: set @ C] :
( ( vimage @ A @ ( product_prod @ B @ C ) @ F
@ ( product_Sigma @ B @ C @ A6
@ ^ [Uu: B] : B6 ) )
= ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ ( comp @ ( product_prod @ B @ C ) @ B @ A @ ( product_fst @ B @ C ) @ F ) @ A6 ) @ ( vimage @ A @ C @ ( comp @ ( product_prod @ B @ C ) @ C @ A @ ( product_snd @ B @ C ) @ F ) @ B6 ) ) ) ).
% vimage_Times
thf(fact_849_inj__on__fun__updI,axiom,
! [B: $tType,A: $tType,F: A > B,A6: set @ A,Y3: B,X2: A] :
( ( inj_on @ A @ B @ F @ A6 )
=> ( ~ ( member @ B @ Y3 @ ( image2 @ A @ B @ F @ A6 ) )
=> ( inj_on @ A @ B @ ( fun_upd @ A @ B @ F @ X2 @ Y3 ) @ A6 ) ) ) ).
% inj_on_fun_updI
thf(fact_850_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_851_surj__vimage__empty,axiom,
! [B: $tType,A: $tType,F: B > A,A6: set @ A] :
( ( ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( ( vimage @ B @ A @ F @ A6 )
= ( bot_bot @ ( set @ B ) ) )
= ( A6
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% surj_vimage_empty
thf(fact_852_vimage__subsetD,axiom,
! [A: $tType,B: $tType,F: B > A,B6: set @ A,A6: set @ B] :
( ( ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ B ) @ ( vimage @ B @ A @ F @ B6 ) @ A6 )
=> ( ord_less_eq @ ( set @ A ) @ B6 @ ( image2 @ B @ A @ F @ A6 ) ) ) ) ).
% vimage_subsetD
thf(fact_853_Pair__vimage__Sigma,axiom,
! [B: $tType,A: $tType,X2: B,A6: set @ B,F: B > ( set @ A )] :
( ( ( member @ B @ X2 @ A6 )
=> ( ( vimage @ A @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X2 ) @ ( product_Sigma @ B @ A @ A6 @ F ) )
= ( F @ X2 ) ) )
& ( ~ ( member @ B @ X2 @ A6 )
=> ( ( vimage @ A @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X2 ) @ ( product_Sigma @ B @ A @ A6 @ F ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Pair_vimage_Sigma
thf(fact_854_inj__vimage__image__eq,axiom,
! [B: $tType,A: $tType,F: A > B,A6: set @ A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( vimage @ A @ B @ F @ ( image2 @ A @ B @ F @ A6 ) )
= A6 ) ) ).
% inj_vimage_image_eq
thf(fact_855_inj__graph,axiom,
! [B: $tType,A: $tType] :
( inj_on @ ( A > B ) @ ( set @ ( product_prod @ A @ B ) )
@ ^ [F4: A > B] :
( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X4: A,Y5: B] :
( Y5
= ( F4 @ X4 ) ) ) )
@ ( top_top @ ( set @ ( A > B ) ) ) ) ).
% inj_graph
thf(fact_856_f__the__inv__into__f,axiom,
! [A: $tType,B: $tType,F: A > B,A6: set @ A,Y3: B] :
( ( inj_on @ A @ B @ F @ A6 )
=> ( ( member @ B @ Y3 @ ( image2 @ A @ B @ F @ A6 ) )
=> ( ( F @ ( the_inv_into @ A @ B @ A6 @ F @ Y3 ) )
= Y3 ) ) ) ).
% f_the_inv_into_f
thf(fact_857_inj__on__the__inv__into,axiom,
! [B: $tType,A: $tType,F: A > B,A6: set @ A] :
( ( inj_on @ A @ B @ F @ A6 )
=> ( inj_on @ B @ A @ ( the_inv_into @ A @ B @ A6 @ F ) @ ( image2 @ A @ B @ F @ A6 ) ) ) ).
% inj_on_the_inv_into
thf(fact_858_vimage__fst,axiom,
! [B: $tType,A: $tType,A6: set @ A] :
( ( vimage @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A6 )
= ( product_Sigma @ A @ B @ A6
@ ^ [Uu: A] : ( top_top @ ( set @ B ) ) ) ) ).
% vimage_fst
thf(fact_859_vimage__snd,axiom,
! [B: $tType,A: $tType,A6: set @ B] :
( ( vimage @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A6 )
= ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
@ ^ [Uu: A] : A6 ) ) ).
% vimage_snd
thf(fact_860_mono__SUP,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( comple6319245703460814977attice @ B ) )
=> ! [F: A > B,A6: C > A,I4: set @ C] :
( ( order_mono @ A @ B @ F )
=> ( ord_less_eq @ B
@ ( complete_Sup_Sup @ B
@ ( image2 @ C @ B
@ ^ [X4: C] : ( F @ ( A6 @ X4 ) )
@ I4 ) )
@ ( F @ ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ A6 @ I4 ) ) ) ) ) ) ).
% mono_SUP
thf(fact_861_mono__Sup,axiom,
! [B: $tType,A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( comple6319245703460814977attice @ B ) )
=> ! [F: A > B,A6: set @ A] :
( ( order_mono @ A @ B @ F )
=> ( ord_less_eq @ B @ ( complete_Sup_Sup @ B @ ( image2 @ A @ B @ F @ A6 ) ) @ ( F @ ( complete_Sup_Sup @ A @ A6 ) ) ) ) ) ).
% mono_Sup
thf(fact_862_mono__INF,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( comple6319245703460814977attice @ B ) )
=> ! [F: A > B,A6: C > A,I4: set @ C] :
( ( order_mono @ A @ B @ F )
=> ( ord_less_eq @ B @ ( F @ ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ A6 @ I4 ) ) )
@ ( complete_Inf_Inf @ B
@ ( image2 @ C @ B
@ ^ [X4: C] : ( F @ ( A6 @ X4 ) )
@ I4 ) ) ) ) ) ).
% mono_INF
thf(fact_863_mono__Inf,axiom,
! [B: $tType,A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( comple6319245703460814977attice @ B ) )
=> ! [F: A > B,A6: set @ A] :
( ( order_mono @ A @ B @ F )
=> ( ord_less_eq @ B @ ( F @ ( complete_Inf_Inf @ A @ A6 ) ) @ ( complete_Inf_Inf @ B @ ( image2 @ A @ B @ F @ A6 ) ) ) ) ) ).
% mono_Inf
thf(fact_864_Pow__insert,axiom,
! [A: $tType,A3: A,A6: set @ A] :
( ( pow @ A @ ( insert @ A @ A3 @ A6 ) )
= ( sup_sup @ ( set @ ( set @ A ) ) @ ( pow @ A @ A6 ) @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( insert @ A @ A3 ) @ ( pow @ A @ A6 ) ) ) ) ).
% Pow_insert
thf(fact_865_vimage__subsetI,axiom,
! [B: $tType,A: $tType,F: A > B,B6: set @ B,A6: set @ A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ B ) @ B6 @ ( image2 @ A @ B @ F @ A6 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( vimage @ A @ B @ F @ B6 ) @ A6 ) ) ) ).
% vimage_subsetI
thf(fact_866_vimage__eq__UN,axiom,
! [B: $tType,A: $tType] :
( ( vimage @ A @ B )
= ( ^ [F4: A > B,B7: set @ B] :
( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y5: B] : ( vimage @ A @ B @ F4 @ ( insert @ B @ Y5 @ ( bot_bot @ ( set @ B ) ) ) )
@ B7 ) ) ) ) ).
% vimage_eq_UN
thf(fact_867_the__inv__into__into,axiom,
! [B: $tType,A: $tType,F: A > B,A6: set @ A,X2: B,B6: set @ A] :
( ( inj_on @ A @ B @ F @ A6 )
=> ( ( member @ B @ X2 @ ( image2 @ A @ B @ F @ A6 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
=> ( member @ A @ ( the_inv_into @ A @ B @ A6 @ F @ X2 ) @ B6 ) ) ) ) ).
% the_inv_into_into
thf(fact_868_the__inv__into__comp,axiom,
! [A: $tType,C: $tType,B: $tType,F: A > B,G: C > A,A6: set @ C,X2: B] :
( ( inj_on @ A @ B @ F @ ( image2 @ C @ A @ G @ A6 ) )
=> ( ( inj_on @ C @ A @ G @ A6 )
=> ( ( member @ B @ X2 @ ( image2 @ A @ B @ F @ ( image2 @ C @ A @ G @ A6 ) ) )
=> ( ( the_inv_into @ C @ B @ A6 @ ( comp @ A @ B @ C @ F @ G ) @ X2 )
= ( comp @ A @ C @ B @ ( the_inv_into @ C @ A @ A6 @ G ) @ ( the_inv_into @ A @ B @ ( image2 @ C @ A @ G @ A6 ) @ F ) @ X2 ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_869_fun__upd__image,axiom,
! [A: $tType,B: $tType,X2: B,A6: set @ B,F: B > A,Y3: A] :
( ( ( member @ B @ X2 @ A6 )
=> ( ( image2 @ B @ A @ ( fun_upd @ B @ A @ F @ X2 @ Y3 ) @ A6 )
= ( insert @ A @ Y3 @ ( image2 @ B @ A @ F @ ( minus_minus @ ( set @ B ) @ A6 @ ( insert @ B @ X2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member @ B @ X2 @ A6 )
=> ( ( image2 @ B @ A @ ( fun_upd @ B @ A @ F @ X2 @ Y3 ) @ A6 )
= ( image2 @ B @ A @ F @ A6 ) ) ) ) ).
% fun_upd_image
thf(fact_870_mono__cSUP,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( condit1219197933456340205attice @ B ) )
=> ! [F: A > B,A6: C > A,I4: set @ C] :
( ( order_mono @ A @ B @ F )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ C @ A @ A6 @ I4 ) )
=> ( ( I4
!= ( bot_bot @ ( set @ C ) ) )
=> ( ord_less_eq @ B
@ ( complete_Sup_Sup @ B
@ ( image2 @ C @ B
@ ^ [X4: C] : ( F @ ( A6 @ X4 ) )
@ I4 ) )
@ ( F @ ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ A6 @ I4 ) ) ) ) ) ) ) ) ).
% mono_cSUP
thf(fact_871_mono__cSup,axiom,
! [B: $tType,A: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( condit1219197933456340205attice @ B ) )
=> ! [F: A > B,A6: set @ A] :
( ( order_mono @ A @ B @ F )
=> ( ( condit941137186595557371_above @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ B @ ( complete_Sup_Sup @ B @ ( image2 @ A @ B @ F @ A6 ) ) @ ( F @ ( complete_Sup_Sup @ A @ A6 ) ) ) ) ) ) ) ).
% mono_cSup
thf(fact_872_Func__map__surj,axiom,
! [C: $tType,A: $tType,D: $tType,B: $tType,F1: B > A,A13: set @ B,B1: set @ A,F23: C > D,B22: set @ C,A24: set @ D] :
( ( ( image2 @ B @ A @ F1 @ A13 )
= B1 )
=> ( ( inj_on @ C @ D @ F23 @ B22 )
=> ( ( ord_less_eq @ ( set @ D ) @ ( image2 @ C @ D @ F23 @ B22 ) @ A24 )
=> ( ( ( B22
= ( bot_bot @ ( set @ C ) ) )
=> ( A24
= ( bot_bot @ ( set @ D ) ) ) )
=> ( ( bNF_Wellorder_Func @ C @ A @ B22 @ B1 )
= ( image2 @ ( D > B ) @ ( C > A ) @ ( bNF_We4925052301507509544nc_map @ C @ B @ A @ D @ B22 @ F1 @ F23 ) @ ( bNF_Wellorder_Func @ D @ B @ A24 @ A13 ) ) ) ) ) ) ) ).
% Func_map_surj
thf(fact_873_cofinal__def,axiom,
! [A: $tType] :
( ( bNF_Ca7293521722713021262ofinal @ A )
= ( ^ [A8: set @ A,R5: set @ ( product_prod @ A @ A )] :
! [X4: A] :
( ( member @ A @ X4 @ ( field2 @ A @ R5 ) )
=> ? [Y5: A] :
( ( member @ A @ Y5 @ A8 )
& ( X4 != Y5 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R5 ) ) ) ) ) ).
% cofinal_def
thf(fact_874_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
= ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ A5 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ) ) ) ).
% Total_subset_Id
thf(fact_875_inf__img__fin__dom_H,axiom,
! [A: $tType,B: $tType,F: B > A,A6: set @ B] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ F @ A6 ) )
=> ( ~ ( finite_finite2 @ B @ A6 )
=> ? [X: A] :
( ( member @ A @ X @ ( image2 @ B @ A @ F @ A6 ) )
& ~ ( finite_finite2 @ B @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ F @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ A6 ) ) ) ) ) ).
% inf_img_fin_dom'
thf(fact_876_inf__img__fin__domE_H,axiom,
! [A: $tType,B: $tType,F: B > A,A6: set @ B] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ F @ A6 ) )
=> ( ~ ( finite_finite2 @ B @ A6 )
=> ~ ! [Y: A] :
( ( member @ A @ Y @ ( image2 @ B @ A @ F @ A6 ) )
=> ( finite_finite2 @ B @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ F @ ( insert @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) @ A6 ) ) ) ) ) ).
% inf_img_fin_domE'
thf(fact_877_prod__set__simps_I2_J,axiom,
! [A: $tType,B: $tType,X2: A,Y3: B] :
( ( basic_snds @ A @ B @ ( product_Pair @ A @ B @ X2 @ Y3 ) )
= ( insert @ B @ Y3 @ ( bot_bot @ ( set @ B ) ) ) ) ).
% prod_set_simps(2)
thf(fact_878_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_879_aboveS__def,axiom,
! [A: $tType] :
( ( order_aboveS @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A ),A7: A] :
( collect @ A
@ ^ [B5: A] :
( ( B5 != A7 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B5 ) @ R5 ) ) ) ) ) ).
% aboveS_def
thf(fact_880_card__vimage__inj,axiom,
! [A: $tType,B: $tType,F: A > B,A6: set @ B] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ B ) @ A6 @ ( image2 @ A @ B @ F @ ( top_top @ ( set @ A ) ) ) )
=> ( ( finite_card @ A @ ( vimage @ A @ B @ F @ A6 ) )
= ( finite_card @ B @ A6 ) ) ) ) ).
% card_vimage_inj
thf(fact_881_Linear__order__Well__order__iff,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
= ( ! [A8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A8 @ ( field2 @ A @ R ) )
=> ( ( A8
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X4: A] :
( ( member @ A @ X4 @ A8 )
& ! [Y5: A] :
( ( member @ A @ Y5 @ A8 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R ) ) ) ) ) ) ) ) ).
% Linear_order_Well_order_iff
thf(fact_882_finite__imageI,axiom,
! [B: $tType,A: $tType,F7: set @ A,H: A > B] :
( ( finite_finite2 @ A @ F7 )
=> ( finite_finite2 @ B @ ( image2 @ A @ B @ H @ F7 ) ) ) ).
% finite_imageI
thf(fact_883_IdI,axiom,
! [A: $tType,A3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ ( id2 @ A ) ) ).
% IdI
thf(fact_884_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_885_finite__UN,axiom,
! [B: $tType,A: $tType,A6: set @ A,B6: A > ( set @ B )] :
( ( finite_finite2 @ A @ A6 )
=> ( ( finite_finite2 @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B6 @ A6 ) ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A6 )
=> ( finite_finite2 @ B @ ( B6 @ X4 ) ) ) ) ) ) ).
% finite_UN
thf(fact_886_finite__UN__I,axiom,
! [B: $tType,A: $tType,A6: set @ A,B6: A > ( set @ B )] :
( ( finite_finite2 @ A @ A6 )
=> ( ! [A5: A] :
( ( member @ A @ A5 @ A6 )
=> ( finite_finite2 @ B @ ( B6 @ A5 ) ) )
=> ( finite_finite2 @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B6 @ A6 ) ) ) ) ) ).
% finite_UN_I
thf(fact_887_finite__INT,axiom,
! [B: $tType,A: $tType,I4: set @ A,A6: A > ( set @ B )] :
( ? [X5: A] :
( ( member @ A @ X5 @ I4 )
& ( finite_finite2 @ B @ ( A6 @ X5 ) ) )
=> ( finite_finite2 @ B @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A6 @ I4 ) ) ) ) ).
% finite_INT
thf(fact_888_bdd__below__UN,axiom,
! [A: $tType,B: $tType] :
( ( lattice @ A )
=> ! [I4: set @ B,A6: B > ( set @ A )] :
( ( finite_finite2 @ B @ I4 )
=> ( ( condit1013018076250108175_below @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A6 @ I4 ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ I4 )
=> ( condit1013018076250108175_below @ A @ ( A6 @ X4 ) ) ) ) ) ) ) ).
% bdd_below_UN
thf(fact_889_bdd__above__UN,axiom,
! [A: $tType,B: $tType] :
( ( lattice @ A )
=> ! [I4: set @ B,A6: B > ( set @ A )] :
( ( finite_finite2 @ B @ I4 )
=> ( ( condit941137186595557371_above @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A6 @ I4 ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ I4 )
=> ( condit941137186595557371_above @ A @ ( A6 @ X4 ) ) ) ) ) ) ) ).
% bdd_above_UN
thf(fact_890_card__image__le,axiom,
! [B: $tType,A: $tType,A6: set @ A,F: A > B] :
( ( finite_finite2 @ A @ A6 )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ ( image2 @ A @ B @ F @ A6 ) ) @ ( finite_card @ A @ A6 ) ) ) ).
% card_image_le
thf(fact_891_surj__card__le,axiom,
! [B: $tType,A: $tType,A6: set @ A,B6: set @ B,F: A > B] :
( ( finite_finite2 @ A @ A6 )
=> ( ( ord_less_eq @ ( set @ B ) @ B6 @ ( image2 @ A @ B @ F @ A6 ) )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ B6 ) @ ( finite_card @ A @ A6 ) ) ) ) ).
% surj_card_le
thf(fact_892_eq__card__imp__inj__on,axiom,
! [B: $tType,A: $tType,A6: set @ A,F: A > B] :
( ( finite_finite2 @ A @ A6 )
=> ( ( ( finite_card @ B @ ( image2 @ A @ B @ F @ A6 ) )
= ( finite_card @ A @ A6 ) )
=> ( inj_on @ A @ B @ F @ A6 ) ) ) ).
% eq_card_imp_inj_on
thf(fact_893_inj__on__iff__eq__card,axiom,
! [B: $tType,A: $tType,A6: set @ A,F: A > B] :
( ( finite_finite2 @ A @ A6 )
=> ( ( inj_on @ A @ B @ F @ A6 )
= ( ( finite_card @ B @ ( image2 @ A @ B @ F @ A6 ) )
= ( finite_card @ A @ A6 ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_894_well__order__on__domain,axiom,
! [A: $tType,A6: set @ A,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( order_well_order_on @ A @ A6 @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
=> ( ( member @ A @ A3 @ A6 )
& ( member @ A @ B2 @ A6 ) ) ) ) ).
% well_order_on_domain
thf(fact_895_infinite__countable__subset,axiom,
! [A: $tType,S5: set @ A] :
( ~ ( finite_finite2 @ A @ S5 )
=> ? [F3: nat > A] :
( ( inj_on @ nat @ A @ F3 @ ( top_top @ ( set @ nat ) ) )
& ( ord_less_eq @ ( set @ A ) @ ( image2 @ nat @ A @ F3 @ ( top_top @ ( set @ nat ) ) ) @ S5 ) ) ) ).
% infinite_countable_subset
thf(fact_896_infinite__iff__countable__subset,axiom,
! [A: $tType,S5: set @ A] :
( ( ~ ( finite_finite2 @ A @ S5 ) )
= ( ? [F4: nat > A] :
( ( inj_on @ nat @ A @ F4 @ ( top_top @ ( set @ nat ) ) )
& ( ord_less_eq @ ( set @ A ) @ ( image2 @ nat @ A @ F4 @ ( top_top @ ( set @ nat ) ) ) @ S5 ) ) ) ) ).
% infinite_iff_countable_subset
thf(fact_897_IdE,axiom,
! [A: $tType,P: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ P @ ( id2 @ A ) )
=> ~ ! [X: A] :
( P
!= ( product_Pair @ A @ A @ X @ X ) ) ) ).
% IdE
thf(fact_898_IdD,axiom,
! [A: $tType,A3: A,B2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( id2 @ A ) )
=> ( A3 = B2 ) ) ).
% IdD
thf(fact_899_card__bij__eq,axiom,
! [A: $tType,B: $tType,F: A > B,A6: set @ A,B6: set @ B,G: B > A] :
( ( inj_on @ A @ B @ F @ A6 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F @ A6 ) @ B6 )
=> ( ( inj_on @ B @ A @ G @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ G @ B6 ) @ A6 )
=> ( ( finite_finite2 @ A @ A6 )
=> ( ( finite_finite2 @ B @ B6 )
=> ( ( finite_card @ A @ A6 )
= ( finite_card @ B @ B6 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_900_card__le__inj,axiom,
! [B: $tType,A: $tType,A6: set @ A,B6: set @ B] :
( ( finite_finite2 @ A @ A6 )
=> ( ( finite_finite2 @ B @ B6 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ B @ B6 ) )
=> ? [F3: A > B] :
( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ A6 ) @ B6 )
& ( inj_on @ A @ B @ F3 @ A6 ) ) ) ) ) ).
% card_le_inj
thf(fact_901_card__inj__on__le,axiom,
! [A: $tType,B: $tType,F: A > B,A6: set @ A,B6: set @ B] :
( ( inj_on @ A @ B @ F @ A6 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F @ A6 ) @ B6 )
=> ( ( finite_finite2 @ B @ B6 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ B @ B6 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_902_inj__on__iff__card__le,axiom,
! [A: $tType,B: $tType,A6: set @ A,B6: set @ B] :
( ( finite_finite2 @ A @ A6 )
=> ( ( finite_finite2 @ B @ B6 )
=> ( ( ? [F4: A > B] :
( ( inj_on @ A @ B @ F4 @ A6 )
& ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F4 @ A6 ) @ B6 ) ) )
= ( ord_less_eq @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ B @ B6 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_903_surjective__iff__injective__gen,axiom,
! [B: $tType,A: $tType,S5: set @ A,T4: set @ B,F: A > B] :
( ( finite_finite2 @ A @ S5 )
=> ( ( finite_finite2 @ B @ T4 )
=> ( ( ( finite_card @ A @ S5 )
= ( finite_card @ B @ T4 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F @ S5 ) @ T4 )
=> ( ( ! [X4: B] :
( ( member @ B @ X4 @ T4 )
=> ? [Y5: A] :
( ( member @ A @ Y5 @ S5 )
& ( ( F @ Y5 )
= X4 ) ) ) )
= ( inj_on @ A @ B @ F @ S5 ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_904_pigeonhole__infinite,axiom,
! [B: $tType,A: $tType,A6: set @ A,F: A > B] :
( ~ ( finite_finite2 @ A @ A6 )
=> ( ( finite_finite2 @ B @ ( image2 @ A @ B @ F @ A6 ) )
=> ? [X: A] :
( ( member @ A @ X @ A6 )
& ~ ( finite_finite2 @ A
@ ( collect @ A
@ ^ [A7: A] :
( ( member @ A @ A7 @ A6 )
& ( ( F @ A7 )
= ( F @ X ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_905_snds_Ointros,axiom,
! [B: $tType,A: $tType,P: product_prod @ A @ B] : ( member @ B @ ( product_snd @ A @ B @ P ) @ ( basic_snds @ A @ B @ P ) ) ).
% snds.intros
thf(fact_906_snds_Osimps,axiom,
! [B: $tType,A: $tType,A3: B,P: product_prod @ A @ B] :
( ( member @ B @ A3 @ ( basic_snds @ A @ B @ P ) )
= ( A3
= ( product_snd @ A @ B @ P ) ) ) ).
% snds.simps
thf(fact_907_snds_Ocases,axiom,
! [B: $tType,A: $tType,A3: B,P: product_prod @ A @ B] :
( ( member @ B @ A3 @ ( basic_snds @ A @ B @ P ) )
=> ( A3
= ( product_snd @ A @ B @ P ) ) ) ).
% snds.cases
thf(fact_908_card__image,axiom,
! [B: $tType,A: $tType,F: A > B,A6: set @ A] :
( ( inj_on @ A @ B @ F @ A6 )
=> ( ( finite_card @ B @ ( image2 @ A @ B @ F @ A6 ) )
= ( finite_card @ A @ A6 ) ) ) ).
% card_image
thf(fact_909_all__finite__subset__image,axiom,
! [A: $tType,B: $tType,F: B > A,A6: set @ B,P2: ( set @ A ) > $o] :
( ( ! [B7: set @ A] :
( ( ( finite_finite2 @ A @ B7 )
& ( ord_less_eq @ ( set @ A ) @ B7 @ ( image2 @ B @ A @ F @ A6 ) ) )
=> ( P2 @ B7 ) ) )
= ( ! [B7: set @ B] :
( ( ( finite_finite2 @ B @ B7 )
& ( ord_less_eq @ ( set @ B ) @ B7 @ A6 ) )
=> ( P2 @ ( image2 @ B @ A @ F @ B7 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_910_ex__finite__subset__image,axiom,
! [A: $tType,B: $tType,F: B > A,A6: set @ B,P2: ( set @ A ) > $o] :
( ( ? [B7: set @ A] :
( ( finite_finite2 @ A @ B7 )
& ( ord_less_eq @ ( set @ A ) @ B7 @ ( image2 @ B @ A @ F @ A6 ) )
& ( P2 @ B7 ) ) )
= ( ? [B7: set @ B] :
( ( finite_finite2 @ B @ B7 )
& ( ord_less_eq @ ( set @ B ) @ B7 @ A6 )
& ( P2 @ ( image2 @ B @ A @ F @ B7 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_911_finite__subset__image,axiom,
! [A: $tType,B: $tType,B6: set @ A,F: B > A,A6: set @ B] :
( ( finite_finite2 @ A @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ ( image2 @ B @ A @ F @ A6 ) )
=> ? [C4: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ C4 @ A6 )
& ( finite_finite2 @ B @ C4 )
& ( B6
= ( image2 @ B @ A @ F @ C4 ) ) ) ) ) ).
% finite_subset_image
thf(fact_912_finite__surj,axiom,
! [A: $tType,B: $tType,A6: set @ A,B6: set @ B,F: A > B] :
( ( finite_finite2 @ A @ A6 )
=> ( ( ord_less_eq @ ( set @ B ) @ B6 @ ( image2 @ A @ B @ F @ A6 ) )
=> ( finite_finite2 @ B @ B6 ) ) ) ).
% finite_surj
thf(fact_913_finite__image__iff,axiom,
! [B: $tType,A: $tType,F: A > B,A6: set @ A] :
( ( inj_on @ A @ B @ F @ A6 )
=> ( ( finite_finite2 @ B @ ( image2 @ A @ B @ F @ A6 ) )
= ( finite_finite2 @ A @ A6 ) ) ) ).
% finite_image_iff
thf(fact_914_finite__imageD,axiom,
! [A: $tType,B: $tType,F: B > A,A6: set @ B] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ F @ A6 ) )
=> ( ( inj_on @ B @ A @ F @ A6 )
=> ( finite_finite2 @ B @ A6 ) ) ) ).
% finite_imageD
thf(fact_915_finite__range__imageI,axiom,
! [C: $tType,A: $tType,B: $tType,G: B > A,F: A > C] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ G @ ( top_top @ ( set @ B ) ) ) )
=> ( finite_finite2 @ C
@ ( image2 @ B @ C
@ ^ [X4: B] : ( F @ ( G @ X4 ) )
@ ( top_top @ ( set @ B ) ) ) ) ) ).
% finite_range_imageI
thf(fact_916_Id__fstsnd__eq,axiom,
! [A: $tType] :
( ( id2 @ A )
= ( collect @ ( product_prod @ A @ A )
@ ^ [X4: product_prod @ A @ A] :
( ( product_fst @ A @ A @ X4 )
= ( product_snd @ A @ A @ X4 ) ) ) ) ).
% Id_fstsnd_eq
thf(fact_917_prod_Oset__map_I2_J,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,F1: A > C,F23: B > D,V: product_prod @ A @ B] :
( ( basic_snds @ C @ D @ ( product_map_prod @ A @ C @ B @ D @ F1 @ F23 @ V ) )
= ( image2 @ B @ D @ F23 @ ( basic_snds @ A @ B @ V ) ) ) ).
% prod.set_map(2)
thf(fact_918_finite__surj__inj,axiom,
! [A: $tType,A6: set @ A,F: A > A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ ( image2 @ A @ A @ F @ A6 ) )
=> ( inj_on @ A @ A @ F @ A6 ) ) ) ).
% finite_surj_inj
thf(fact_919_inj__on__finite,axiom,
! [B: $tType,A: $tType,F: A > B,A6: set @ A,B6: set @ B] :
( ( inj_on @ A @ B @ F @ A6 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F @ A6 ) @ B6 )
=> ( ( finite_finite2 @ B @ B6 )
=> ( finite_finite2 @ A @ A6 ) ) ) ) ).
% inj_on_finite
thf(fact_920_endo__inj__surj,axiom,
! [A: $tType,A6: set @ A,F: A > A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ A @ A @ F @ A6 ) @ A6 )
=> ( ( inj_on @ A @ A @ F @ A6 )
=> ( ( image2 @ A @ A @ F @ A6 )
= A6 ) ) ) ) ).
% endo_inj_surj
thf(fact_921_finite__UNIV__surj__inj,axiom,
! [A: $tType,F: A > A] :
( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
=> ( ( ( image2 @ A @ A @ F @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( inj_on @ A @ A @ F @ ( top_top @ ( set @ A ) ) ) ) ) ).
% finite_UNIV_surj_inj
thf(fact_922_finite__UNIV__inj__surj,axiom,
! [A: $tType,F: A > A] :
( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
=> ( ( inj_on @ A @ A @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( image2 @ A @ A @ F @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ) ).
% finite_UNIV_inj_surj
thf(fact_923_finite__vimageD,axiom,
! [A: $tType,B: $tType,H: A > B,F7: set @ B] :
( ( finite_finite2 @ A @ ( vimage @ A @ B @ H @ F7 ) )
=> ( ( ( image2 @ A @ B @ H @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ B ) ) )
=> ( finite_finite2 @ B @ F7 ) ) ) ).
% finite_vimageD
thf(fact_924_prod__set__defs_I2_J,axiom,
! [D: $tType,C: $tType] :
( ( basic_snds @ C @ D )
= ( ^ [P3: product_prod @ C @ D] : ( insert @ D @ ( product_snd @ C @ D @ P3 ) @ ( bot_bot @ ( set @ D ) ) ) ) ) ).
% prod_set_defs(2)
thf(fact_925_inf__img__fin__dom,axiom,
! [B: $tType,A: $tType,F: B > A,A6: set @ B] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ F @ A6 ) )
=> ( ~ ( finite_finite2 @ B @ A6 )
=> ? [X: A] :
( ( member @ A @ X @ ( image2 @ B @ A @ F @ A6 ) )
& ~ ( finite_finite2 @ B @ ( vimage @ B @ A @ F @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% inf_img_fin_dom
thf(fact_926_inf__img__fin__domE,axiom,
! [B: $tType,A: $tType,F: B > A,A6: set @ B] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ F @ A6 ) )
=> ( ~ ( finite_finite2 @ B @ A6 )
=> ~ ! [Y: A] :
( ( member @ A @ Y @ ( image2 @ B @ A @ F @ A6 ) )
=> ( finite_finite2 @ B @ ( vimage @ B @ A @ F @ ( insert @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% inf_img_fin_domE
thf(fact_927_finite__vimageD_H,axiom,
! [A: $tType,B: $tType,F: A > B,A6: set @ B] :
( ( finite_finite2 @ A @ ( vimage @ A @ B @ F @ A6 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ A6 @ ( image2 @ A @ B @ F @ ( top_top @ ( set @ A ) ) ) )
=> ( finite_finite2 @ B @ A6 ) ) ) ).
% finite_vimageD'
thf(fact_928_Func__map,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,G: A > B,A24: set @ A,A13: set @ B,F1: B > C,B1: set @ C,F23: D > A,B22: set @ D] :
( ( member @ ( A > B ) @ G @ ( bNF_Wellorder_Func @ A @ B @ A24 @ A13 ) )
=> ( ( ord_less_eq @ ( set @ C ) @ ( image2 @ B @ C @ F1 @ A13 ) @ B1 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ D @ A @ F23 @ B22 ) @ A24 )
=> ( member @ ( D > C ) @ ( bNF_We4925052301507509544nc_map @ D @ B @ C @ A @ B22 @ F1 @ F23 @ G ) @ ( bNF_Wellorder_Func @ D @ C @ B22 @ B1 ) ) ) ) ) ).
% Func_map
thf(fact_929_reflcl__set__eq,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( sup_sup @ ( A > A > $o )
@ ^ [X4: A,Y5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R )
@ ^ [Y2: A,Z2: A] : Y2 = Z2 )
= ( ^ [X4: A,Y5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( id2 @ A ) ) ) ) ) ).
% reflcl_set_eq
thf(fact_930_INF__filter__not__bot,axiom,
! [I5: $tType,A: $tType,B6: set @ I5,F7: I5 > ( filter @ A )] :
( ! [X9: set @ I5] :
( ( ord_less_eq @ ( set @ I5 ) @ X9 @ B6 )
=> ( ( finite_finite2 @ I5 @ X9 )
=> ( ( complete_Inf_Inf @ ( filter @ A ) @ ( image2 @ I5 @ ( filter @ A ) @ F7 @ X9 ) )
!= ( bot_bot @ ( filter @ A ) ) ) ) )
=> ( ( complete_Inf_Inf @ ( filter @ A ) @ ( image2 @ I5 @ ( filter @ A ) @ F7 @ B6 ) )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% INF_filter_not_bot
thf(fact_931_Pow__fold,axiom,
! [A: $tType,A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( pow @ A @ A6 )
= ( finite_fold @ A @ ( set @ ( set @ A ) )
@ ^ [X4: A,A8: set @ ( set @ A )] : ( sup_sup @ ( set @ ( set @ A ) ) @ A8 @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( insert @ A @ X4 ) @ A8 ) )
@ ( insert @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) )
@ A6 ) ) ) ).
% Pow_fold
thf(fact_932_fold__union__pair,axiom,
! [B: $tType,A: $tType,B6: set @ A,X2: B,A6: 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 ) )
@ ^ [Y5: A] : ( insert @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X2 @ Y5 ) @ ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) )
@ B6 ) )
@ A6 )
= ( finite_fold @ A @ ( set @ ( product_prod @ B @ A ) )
@ ^ [Y5: A] : ( insert @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X2 @ Y5 ) )
@ A6
@ B6 ) ) ) ).
% fold_union_pair
thf(fact_933_relInvImage__Id__on,axiom,
! [B: $tType,A: $tType,F: A > B,A6: set @ A,B6: set @ B] :
( ! [A14: A,A25: A] :
( ( ( F @ A14 )
= ( F @ A25 ) )
= ( A14 = A25 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Gr7122648621184425601vImage @ A @ B @ A6 @ ( id_on @ B @ B6 ) @ F ) @ ( id2 @ A ) ) ) ).
% relInvImage_Id_on
thf(fact_934_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 ) ) )
= ( ! [A8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A8 @ ( field2 @ A @ R ) )
=> ( ( A8
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X4: A] :
( ( member @ A @ X4 @ A8 )
& ! [Y5: A] :
( ( member @ A @ Y5 @ A8 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R ) ) ) ) ) ) ) ) ).
% Linear_order_wf_diff_Id
thf(fact_935_bsqr__def,axiom,
! [A: $tType] :
( ( bNF_Wellorder_bsqr @ A )
= ( ^ [R5: 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 ) @ ( insert @ A @ A12 @ ( insert @ A @ A23 @ ( insert @ A @ B12 @ ( insert @ A @ B23 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) @ ( field2 @ A @ R5 ) )
& ( ( ( A12 = B12 )
& ( A23 = B23 ) )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( bNF_We1388413361240627857o_max2 @ A @ R5 @ A12 @ A23 ) @ ( bNF_We1388413361240627857o_max2 @ A @ R5 @ B12 @ B23 ) ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R5 @ ( id2 @ A ) ) )
| ( ( ( bNF_We1388413361240627857o_max2 @ A @ R5 @ A12 @ A23 )
= ( bNF_We1388413361240627857o_max2 @ A @ R5 @ B12 @ B23 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A12 @ B12 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R5 @ ( id2 @ A ) ) ) )
| ( ( ( bNF_We1388413361240627857o_max2 @ A @ R5 @ A12 @ A23 )
= ( bNF_We1388413361240627857o_max2 @ A @ R5 @ B12 @ B23 ) )
& ( A12 = B12 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A23 @ B23 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R5 @ ( id2 @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% bsqr_def
thf(fact_936_INF__fold__inf,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ B,F: B > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A6 ) )
= ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( inf_inf @ A ) @ F ) @ ( top_top @ A ) @ A6 ) ) ) ) ).
% INF_fold_inf
thf(fact_937_SUP__fold__sup,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ B,F: B > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A6 ) )
= ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( sup_sup @ A ) @ F ) @ ( bot_bot @ A ) @ A6 ) ) ) ) ).
% SUP_fold_sup
thf(fact_938_INF__filter__bot__base,axiom,
! [B: $tType,A: $tType,I4: set @ A,F7: A > ( filter @ B )] :
( ! [I6: A] :
( ( member @ A @ I6 @ I4 )
=> ! [J3: A] :
( ( member @ A @ J3 @ I4 )
=> ? [X5: A] :
( ( member @ A @ X5 @ I4 )
& ( ord_less_eq @ ( filter @ B ) @ ( F7 @ X5 ) @ ( inf_inf @ ( filter @ B ) @ ( F7 @ I6 ) @ ( F7 @ J3 ) ) ) ) ) )
=> ( ( ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ F7 @ I4 ) )
= ( bot_bot @ ( filter @ B ) ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ I4 )
& ( ( F7 @ X4 )
= ( bot_bot @ ( filter @ B ) ) ) ) ) ) ) ).
% INF_filter_bot_base
thf(fact_939_bsqr__max2,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A1: A,A22: A,B13: A,B24: 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 @ B13 @ B24 ) ) @ ( 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 @ B13 @ B24 ) ) @ R ) ) ) ).
% bsqr_max2
thf(fact_940_relInvImage__mono,axiom,
! [A: $tType,B: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ A @ A ),A6: set @ B,F: B > A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R1 @ R22 )
=> ( ord_less_eq @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Gr7122648621184425601vImage @ B @ A @ A6 @ R1 @ F ) @ ( bNF_Gr7122648621184425601vImage @ B @ A @ A6 @ R22 @ F ) ) ) ).
% relInvImage_mono
thf(fact_941_fold__image,axiom,
! [C: $tType,B: $tType,A: $tType,G: A > B,A6: set @ A,F: B > C > C,Z3: C] :
( ( inj_on @ A @ B @ G @ A6 )
=> ( ( finite_fold @ B @ C @ F @ Z3 @ ( image2 @ A @ B @ G @ A6 ) )
= ( finite_fold @ A @ C @ ( comp @ B @ ( C > C ) @ A @ F @ G ) @ Z3 @ A6 ) ) ) ).
% fold_image
thf(fact_942_reduction__pair__lemma,axiom,
! [A: $tType,P2: product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ),R3: set @ ( product_prod @ A @ A ),S5: set @ ( product_prod @ A @ A )] :
( ( fun_reduction_pair @ A @ P2 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( product_fst @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P2 ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ S5 @ ( product_snd @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P2 ) )
=> ( ( wf @ A @ S5 )
=> ( wf @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ S5 ) ) ) ) ) ) ).
% reduction_pair_lemma
thf(fact_943_wf__eq__minimal2,axiom,
! [A: $tType] :
( ( wf @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
! [A8: set @ A] :
( ( ( ord_less_eq @ ( set @ A ) @ A8 @ ( field2 @ A @ R5 ) )
& ( A8
!= ( bot_bot @ ( set @ A ) ) ) )
=> ? [X4: A] :
( ( member @ A @ X4 @ A8 )
& ! [Y5: A] :
( ( member @ A @ Y5 @ A8 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X4 ) @ R5 ) ) ) ) ) ) ).
% wf_eq_minimal2
thf(fact_944_image__fold__insert,axiom,
! [B: $tType,A: $tType,A6: set @ A,F: A > B] :
( ( finite_finite2 @ A @ A6 )
=> ( ( image2 @ A @ B @ F @ A6 )
= ( finite_fold @ A @ ( set @ B )
@ ^ [K3: A] : ( insert @ B @ ( F @ K3 ) )
@ ( bot_bot @ ( set @ B ) )
@ A6 ) ) ) ).
% image_fold_insert
thf(fact_945_sup__SUP__fold__sup,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ B,B6: A,F: B > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ( sup_sup @ A @ B6 @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A6 ) ) )
= ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( sup_sup @ A ) @ F ) @ B6 @ A6 ) ) ) ) ).
% sup_SUP_fold_sup
thf(fact_946_inf__INF__fold__inf,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ B,B6: A,F: B > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ( inf_inf @ A @ B6 @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A6 ) ) )
= ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( inf_inf @ A ) @ F ) @ B6 @ A6 ) ) ) ) ).
% inf_INF_fold_inf
thf(fact_947_product__fold,axiom,
! [B: $tType,A: $tType,A6: set @ A,B6: set @ B] :
( ( finite_finite2 @ A @ A6 )
=> ( ( finite_finite2 @ B @ B6 )
=> ( ( product_Sigma @ A @ B @ A6
@ ^ [Uu: A] : B6 )
= ( finite_fold @ A @ ( set @ ( product_prod @ A @ B ) )
@ ^ [X4: A,Z5: set @ ( product_prod @ A @ B )] :
( finite_fold @ B @ ( set @ ( product_prod @ A @ B ) )
@ ^ [Y5: B] : ( insert @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) )
@ Z5
@ B6 )
@ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) )
@ A6 ) ) ) ) ).
% product_fold
thf(fact_948_Id__on__fold,axiom,
! [A: $tType,A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( id_on @ A @ A6 )
= ( finite_fold @ A @ ( set @ ( product_prod @ A @ A ) )
@ ^ [X4: A] : ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) )
@ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) )
@ A6 ) ) ) ).
% Id_on_fold
thf(fact_949_wf__map__prod__image,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),F: A > B] :
( ( wf @ A @ R )
=> ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( wf @ B @ ( image2 @ ( product_prod @ A @ A ) @ ( product_prod @ B @ B ) @ ( product_map_prod @ A @ B @ A @ B @ F @ F ) @ R ) ) ) ) ).
% wf_map_prod_image
thf(fact_950_wfI,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A6: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ A6
@ ^ [Uu: A] : B6 ) )
=> ( ! [X: A,P6: A > $o] :
( ! [Xa3: A] :
( ! [Y: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Xa3 ) @ R )
=> ( P6 @ Y ) )
=> ( P6 @ Xa3 ) )
=> ( ( member @ A @ X @ A6 )
=> ( ( member @ A @ X @ B6 )
=> ( P6 @ X ) ) ) )
=> ( wf @ A @ R ) ) ) ).
% wfI
thf(fact_951_wfE__min_H,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),Q2: set @ A] :
( ( wf @ A @ R3 )
=> ( ( Q2
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [Z4: A] :
( ( member @ A @ Z4 @ Q2 )
=> ~ ! [Y7: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y7 @ Z4 ) @ R3 )
=> ~ ( member @ A @ Y7 @ Q2 ) ) ) ) ) ).
% wfE_min'
thf(fact_952_relImage__relInvImage,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),F: B > A,A6: set @ B] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ ( image2 @ B @ A @ F @ A6 )
@ ^ [Uu: A] : ( image2 @ B @ A @ F @ A6 ) ) )
=> ( ( bNF_Gr4221423524335903396lImage @ B @ A @ ( bNF_Gr7122648621184425601vImage @ B @ A @ A6 @ R3 @ F ) @ F )
= R3 ) ) ).
% relImage_relInvImage
thf(fact_953_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 )
@ ^ [X7: A,Y6: B] :
( product_case_prod @ A @ B @ $o
@ ^ [X4: A,Y5: B] :
( ( X7 = X4 )
& ( P4 @ X4 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y6 @ Y5 ) @ ( R6 @ X4 ) ) ) ) ) ) ) ) ) ).
% same_fst_def
thf(fact_954_relInvImage__UNIV__relImage,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),F: A > B] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( bNF_Gr7122648621184425601vImage @ A @ B @ ( top_top @ ( set @ A ) ) @ ( bNF_Gr4221423524335903396lImage @ A @ B @ R3 @ F ) @ F ) ) ).
% relInvImage_UNIV_relImage
thf(fact_955_comp__fun__commute__product__fold,axiom,
! [A: $tType,B: $tType,B6: set @ A] :
( ( finite_finite2 @ A @ B6 )
=> ( finite6289374366891150609ommute @ B @ ( set @ ( product_prod @ B @ A ) )
@ ^ [X4: B,Z5: set @ ( product_prod @ B @ A )] :
( finite_fold @ A @ ( set @ ( product_prod @ B @ A ) )
@ ^ [Y5: A] : ( insert @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X4 @ Y5 ) )
@ Z5
@ B6 ) ) ) ).
% comp_fun_commute_product_fold
thf(fact_956_wf__def,axiom,
! [A: $tType] :
( ( wf @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
! [P4: A > $o] :
( ! [X4: A] :
( ! [Y5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X4 ) @ R5 )
=> ( P4 @ Y5 ) )
=> ( P4 @ X4 ) )
=> ! [X10: A] : ( P4 @ X10 ) ) ) ) ).
% wf_def
thf(fact_957_wfE__min,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),X2: A,Q2: set @ A] :
( ( wf @ A @ R3 )
=> ( ( member @ A @ X2 @ Q2 )
=> ~ ! [Z4: A] :
( ( member @ A @ Z4 @ Q2 )
=> ~ ! [Y7: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y7 @ Z4 ) @ R3 )
=> ~ ( member @ A @ Y7 @ Q2 ) ) ) ) ) ).
% wfE_min
thf(fact_958_same__fstI,axiom,
! [B: $tType,A: $tType,P2: A > $o,X2: A,Y8: B,Y3: B,R3: A > ( set @ ( product_prod @ B @ B ) )] :
( ( P2 @ X2 )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y8 @ Y3 ) @ ( R3 @ X2 ) )
=> ( 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 @ X2 @ Y8 ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) ) @ ( same_fst @ A @ B @ P2 @ R3 ) ) ) ) ).
% same_fstI
thf(fact_959_comp__fun__commute_Ointro,axiom,
! [B: $tType,A: $tType,F: A > B > B] :
( ! [Y: A,X: A] :
( ( comp @ B @ B @ B @ ( F @ Y ) @ ( F @ X ) )
= ( comp @ B @ B @ B @ ( F @ X ) @ ( F @ Y ) ) )
=> ( finite6289374366891150609ommute @ A @ B @ F ) ) ).
% comp_fun_commute.intro
thf(fact_960_comp__fun__commute_Ocomp__fun__commute,axiom,
! [B: $tType,A: $tType,F: A > B > B,Y3: A,X2: A] :
( ( finite6289374366891150609ommute @ A @ B @ F )
=> ( ( comp @ B @ B @ B @ ( F @ Y3 ) @ ( F @ X2 ) )
= ( comp @ B @ B @ B @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ) ).
% comp_fun_commute.comp_fun_commute
thf(fact_961_comp__fun__commute__def,axiom,
! [B: $tType,A: $tType] :
( ( finite6289374366891150609ommute @ A @ B )
= ( ^ [F4: A > B > B] :
! [Y5: A,X4: A] :
( ( comp @ B @ B @ B @ ( F4 @ Y5 ) @ ( F4 @ X4 ) )
= ( comp @ B @ B @ B @ ( F4 @ X4 ) @ ( F4 @ Y5 ) ) ) ) ) ).
% comp_fun_commute_def
thf(fact_962_comp__fun__commute_Ocomp__comp__fun__commute,axiom,
! [B: $tType,A: $tType,C: $tType,F: A > B > B,G: C > A] :
( ( finite6289374366891150609ommute @ A @ B @ F )
=> ( finite6289374366891150609ommute @ C @ B @ ( comp @ A @ ( B > B ) @ C @ F @ G ) ) ) ).
% comp_fun_commute.comp_comp_fun_commute
thf(fact_963_relImage__mono,axiom,
! [B: $tType,A: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ A @ A ),F: A > B] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R1 @ R22 )
=> ( ord_less_eq @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Gr4221423524335903396lImage @ A @ B @ R1 @ F ) @ ( bNF_Gr4221423524335903396lImage @ A @ B @ R22 @ F ) ) ) ).
% relImage_mono
thf(fact_964_comp__fun__commute__relcomp__fold,axiom,
! [A: $tType,B: $tType,C: $tType,S5: set @ ( product_prod @ A @ B )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ S5 )
=> ( 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 ) ) )
@ ^ [X4: C,Y5: A,A8: 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 ) ) )
@ ^ [W: A,Z5: B,A11: set @ ( product_prod @ C @ B )] : ( if @ ( set @ ( product_prod @ C @ B ) ) @ ( Y5 = W ) @ ( insert @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ X4 @ Z5 ) @ A11 ) @ A11 ) )
@ A8
@ S5 ) ) ) ) ).
% comp_fun_commute_relcomp_fold
thf(fact_965_wf__induct__rule,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),P2: A > $o,A3: A] :
( ( wf @ A @ R )
=> ( ! [X: A] :
( ! [Y7: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y7 @ X ) @ R )
=> ( P2 @ Y7 ) )
=> ( P2 @ X ) )
=> ( P2 @ A3 ) ) ) ).
% wf_induct_rule
thf(fact_966_wf__eq__minimal,axiom,
! [A: $tType] :
( ( wf @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
! [Q3: set @ A] :
( ? [X4: A] : ( member @ A @ X4 @ Q3 )
=> ? [X4: A] :
( ( member @ A @ X4 @ Q3 )
& ! [Y5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X4 ) @ R5 )
=> ~ ( member @ A @ Y5 @ Q3 ) ) ) ) ) ) ).
% wf_eq_minimal
thf(fact_967_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_968_wf__not__sym,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,X2: A] :
( ( wf @ A @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ X2 ) @ R )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ A3 ) @ R ) ) ) ).
% wf_not_sym
thf(fact_969_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_970_wf__induct,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),P2: A > $o,A3: A] :
( ( wf @ A @ R )
=> ( ! [X: A] :
( ! [Y7: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y7 @ X ) @ R )
=> ( P2 @ Y7 ) )
=> ( P2 @ X ) )
=> ( P2 @ A3 ) ) ) ).
% wf_induct
thf(fact_971_wf__asym,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,X2: A] :
( ( wf @ A @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ X2 ) @ R )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ A3 ) @ R ) ) ) ).
% wf_asym
thf(fact_972_wfUNIVI,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ! [P6: A > $o,X: A] :
( ! [Xa3: A] :
( ! [Y: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Xa3 ) @ R )
=> ( P6 @ Y ) )
=> ( P6 @ Xa3 ) )
=> ( P6 @ X ) )
=> ( wf @ A @ R ) ) ).
% wfUNIVI
thf(fact_973_wfI__min,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ! [X: A,Q5: set @ A] :
( ( member @ A @ X @ Q5 )
=> ? [Xa3: A] :
( ( member @ A @ Xa3 @ Q5 )
& ! [Y: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Xa3 ) @ R3 )
=> ~ ( member @ A @ Y @ Q5 ) ) ) )
=> ( wf @ A @ R3 ) ) ).
% wfI_min
thf(fact_974_lex__prod__def,axiom,
! [B: $tType,A: $tType] :
( ( lex_prod @ A @ B )
= ( ^ [Ra2: set @ ( product_prod @ A @ A ),Rb2: 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
@ ^ [A15: A,B10: B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ A15 ) @ Ra2 )
| ( ( A7 = A15 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B5 @ B10 ) @ Rb2 ) ) ) ) ) ) ) ) ) ).
% lex_prod_def
thf(fact_975_in__lex__prod,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,A4: A,B3: B,R: set @ ( product_prod @ A @ A ),S2: 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 @ A4 @ B3 ) ) @ ( lex_prod @ A @ B @ R @ S2 ) )
= ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A4 ) @ R )
| ( ( A3 = A4 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B2 @ B3 ) @ S2 ) ) ) ) ).
% in_lex_prod
thf(fact_976_dependent__wf__choice,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),P2: ( A > B ) > A > B > $o] :
( ( wf @ A @ R3 )
=> ( ! [F3: A > B,G3: A > B,X: A,R4: B] :
( ! [Z7: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z7 @ X ) @ R3 )
=> ( ( F3 @ Z7 )
= ( G3 @ Z7 ) ) )
=> ( ( P2 @ F3 @ X @ R4 )
= ( P2 @ G3 @ X @ R4 ) ) )
=> ( ! [X: A,F3: A > B] :
( ! [Y7: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y7 @ X ) @ R3 )
=> ( P2 @ F3 @ Y7 @ ( F3 @ Y7 ) ) )
=> ? [X_1: B] : ( P2 @ F3 @ X @ X_1 ) )
=> ? [F3: A > B] :
! [X5: A] : ( P2 @ F3 @ X5 @ ( F3 @ X5 ) ) ) ) ) ).
% dependent_wf_choice
thf(fact_977_reduction__pair__def,axiom,
! [A: $tType] :
( ( fun_reduction_pair @ A )
= ( ^ [P4: product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) )] :
( ( wf @ A @ ( product_fst @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P4 ) )
& ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ ( product_fst @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P4 ) @ ( product_snd @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P4 ) ) @ ( product_fst @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P4 ) ) ) ) ) ).
% reduction_pair_def
thf(fact_978_INF__principal__finite,axiom,
! [B: $tType,A: $tType,X6: set @ A,F: A > ( set @ B )] :
( ( finite_finite2 @ A @ X6 )
=> ( ( complete_Inf_Inf @ ( filter @ B )
@ ( image2 @ A @ ( filter @ B )
@ ^ [X4: A] : ( principal @ B @ ( F @ X4 ) )
@ X6 ) )
= ( principal @ B @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F @ X6 ) ) ) ) ) ).
% INF_principal_finite
thf(fact_979_insert__relcomp__union__fold,axiom,
! [C: $tType,B: $tType,A: $tType,S5: set @ ( product_prod @ A @ B ),X2: product_prod @ C @ A,X6: set @ ( product_prod @ C @ B )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ S5 )
=> ( ( sup_sup @ ( set @ ( product_prod @ C @ B ) ) @ ( relcomp @ C @ A @ B @ ( insert @ ( product_prod @ C @ A ) @ X2 @ ( bot_bot @ ( set @ ( product_prod @ C @ A ) ) ) ) @ S5 ) @ X6 )
= ( 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 ) ) )
@ ^ [W: A,Z5: B,A11: set @ ( product_prod @ C @ B )] :
( if @ ( set @ ( product_prod @ C @ B ) )
@ ( ( product_snd @ C @ A @ X2 )
= W )
@ ( insert @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ ( product_fst @ C @ A @ X2 ) @ Z5 ) @ A11 )
@ A11 ) )
@ X6
@ S5 ) ) ) ).
% insert_relcomp_union_fold
thf(fact_980_max__ext_Osimps,axiom,
! [A: $tType,A1: set @ A,A22: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A1 @ A22 ) @ ( max_ext @ A @ R3 ) )
= ( ( finite_finite2 @ A @ A1 )
& ( finite_finite2 @ A @ A22 )
& ( A22
!= ( bot_bot @ ( set @ A ) ) )
& ! [X4: A] :
( ( member @ A @ X4 @ A1 )
=> ? [Y5: A] :
( ( member @ A @ Y5 @ A22 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R3 ) ) ) ) ) ).
% max_ext.simps
thf(fact_981_max__ext_Ocases,axiom,
! [A: $tType,A1: set @ A,A22: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A1 @ A22 ) @ ( max_ext @ A @ R3 ) )
=> ~ ( ( finite_finite2 @ A @ A1 )
=> ( ( finite_finite2 @ A @ A22 )
=> ( ( A22
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [X5: A] :
( ( member @ A @ X5 @ A1 )
=> ? [Xa: A] :
( ( member @ A @ Xa @ A22 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Xa ) @ R3 ) ) ) ) ) ) ) ).
% max_ext.cases
thf(fact_982_SUP__principal,axiom,
! [A: $tType,B: $tType,A6: B > ( set @ A ),I4: set @ B] :
( ( complete_Sup_Sup @ ( filter @ A )
@ ( image2 @ B @ ( filter @ A )
@ ^ [I3: B] : ( principal @ A @ ( A6 @ I3 ) )
@ I4 ) )
= ( principal @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A6 @ I4 ) ) ) ) ).
% SUP_principal
thf(fact_983_reduction__pairI,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),S5: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ R3 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R3 @ S5 ) @ R3 )
=> ( fun_reduction_pair @ A @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ S5 ) ) ) ) ).
% reduction_pairI
thf(fact_984_max__ext__compat,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),S5: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R3 @ S5 ) @ R3 )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( relcomp @ ( set @ A ) @ ( set @ A ) @ ( set @ A ) @ ( max_ext @ A @ R3 ) @ ( sup_sup @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( max_ext @ A @ S5 ) @ ( insert @ ( 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 @ R3 ) ) ) ).
% max_ext_compat
thf(fact_985_relcompEpair,axiom,
! [A: $tType,B: $tType,C: $tType,A3: A,C3: B,R: set @ ( product_prod @ A @ C ),S2: set @ ( product_prod @ C @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ C3 ) @ ( relcomp @ A @ C @ B @ R @ S2 ) )
=> ~ ! [B4: C] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A3 @ B4 ) @ R )
=> ~ ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ B4 @ C3 ) @ S2 ) ) ) ).
% relcompEpair
thf(fact_986_relcompE,axiom,
! [A: $tType,B: $tType,C: $tType,Xz: product_prod @ A @ B,R: set @ ( product_prod @ A @ C ),S2: set @ ( product_prod @ C @ B )] :
( ( member @ ( product_prod @ A @ B ) @ Xz @ ( relcomp @ A @ C @ B @ R @ S2 ) )
=> ~ ! [X: A,Y: C,Z4: B] :
( ( Xz
= ( product_Pair @ A @ B @ X @ Z4 ) )
=> ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X @ Y ) @ R )
=> ~ ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ Y @ Z4 ) @ S2 ) ) ) ) ).
% relcompE
thf(fact_987_relcomp_OrelcompI,axiom,
! [A: $tType,C: $tType,B: $tType,A3: A,B2: B,R: set @ ( product_prod @ A @ B ),C3: C,S2: 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 @ C3 ) @ S2 )
=> ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A3 @ C3 ) @ ( relcomp @ A @ B @ C @ R @ S2 ) ) ) ) ).
% relcomp.relcompI
thf(fact_988_relcomp_Osimps,axiom,
! [B: $tType,C: $tType,A: $tType,A1: A,A22: C,R: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ B @ C )] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A1 @ A22 ) @ ( relcomp @ A @ B @ C @ R @ S2 ) )
= ( ? [A7: A,B5: B,C6: C] :
( ( A1 = A7 )
& ( A22 = C6 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A7 @ B5 ) @ R )
& ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B5 @ C6 ) @ S2 ) ) ) ) ).
% relcomp.simps
thf(fact_989_relcomp_Ocases,axiom,
! [A: $tType,C: $tType,B: $tType,A1: A,A22: C,R: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ B @ C )] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A1 @ A22 ) @ ( relcomp @ A @ B @ C @ R @ S2 ) )
=> ~ ! [B4: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A1 @ B4 ) @ R )
=> ~ ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B4 @ A22 ) @ S2 ) ) ) ).
% relcomp.cases
thf(fact_990_wf__no__loop,axiom,
! [B: $tType,R3: set @ ( product_prod @ B @ B )] :
( ( ( relcomp @ B @ B @ B @ R3 @ R3 )
= ( bot_bot @ ( set @ ( product_prod @ B @ B ) ) ) )
=> ( wf @ B @ R3 ) ) ).
% wf_no_loop
thf(fact_991_union__comp__emptyR,axiom,
! [A: $tType,A6: set @ ( product_prod @ A @ A ),B6: set @ ( product_prod @ A @ A ),C5: set @ ( product_prod @ A @ A )] :
( ( ( relcomp @ A @ A @ A @ A6 @ B6 )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ( ( relcomp @ A @ A @ A @ A6 @ C5 )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ( relcomp @ A @ A @ A @ A6 @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ B6 @ C5 ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ) ).
% union_comp_emptyR
thf(fact_992_union__comp__emptyL,axiom,
! [A: $tType,A6: set @ ( product_prod @ A @ A ),C5: set @ ( product_prod @ A @ A ),B6: set @ ( product_prod @ A @ A )] :
( ( ( relcomp @ A @ A @ A @ A6 @ C5 )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ( ( relcomp @ A @ A @ A @ B6 @ C5 )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ( relcomp @ A @ A @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ A6 @ B6 ) @ C5 )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ) ).
% union_comp_emptyL
thf(fact_993_comp__fun__commute__Image__fold,axiom,
! [B: $tType,A: $tType,S5: set @ A] :
( finite6289374366891150609ommute @ ( product_prod @ A @ B ) @ ( set @ B )
@ ( product_case_prod @ A @ B @ ( ( set @ B ) > ( set @ B ) )
@ ^ [X4: A,Y5: B,A8: set @ B] : ( if @ ( set @ B ) @ ( member @ A @ X4 @ S5 ) @ ( insert @ B @ Y5 @ A8 ) @ A8 ) ) ) ).
% comp_fun_commute_Image_fold
thf(fact_994_relcompp__relcomp__eq,axiom,
! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ B @ C )] :
( ( relcompp @ A @ B @ C
@ ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ R )
@ ^ [X4: B,Y5: C] : ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ X4 @ Y5 ) @ S2 ) )
= ( ^ [X4: A,Y5: C] : ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X4 @ Y5 ) @ ( relcomp @ A @ B @ C @ R @ S2 ) ) ) ) ).
% relcompp_relcomp_eq
thf(fact_995_relcompp__in__rel,axiom,
! [A: $tType,B: $tType,C: $tType,R3: set @ ( product_prod @ A @ C ),S5: set @ ( product_prod @ C @ B )] :
( ( relcompp @ A @ C @ B @ ( fun_in_rel @ A @ C @ R3 ) @ ( fun_in_rel @ C @ B @ S5 ) )
= ( fun_in_rel @ A @ B @ ( relcomp @ A @ C @ B @ R3 @ S5 ) ) ) ).
% relcompp_in_rel
thf(fact_996_relcomp__UNION__distrib,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,S2: set @ ( product_prod @ A @ C ),R: D > ( set @ ( product_prod @ C @ B ) ),I4: set @ D] :
( ( relcomp @ A @ C @ B @ S2 @ ( complete_Sup_Sup @ ( set @ ( product_prod @ C @ B ) ) @ ( image2 @ D @ ( set @ ( product_prod @ C @ B ) ) @ R @ I4 ) ) )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ D @ ( set @ ( product_prod @ A @ B ) )
@ ^ [I3: D] : ( relcomp @ A @ C @ B @ S2 @ ( R @ I3 ) )
@ I4 ) ) ) ).
% relcomp_UNION_distrib
thf(fact_997_relcomp__UNION__distrib2,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,R: D > ( set @ ( product_prod @ A @ C ) ),I4: set @ D,S2: set @ ( product_prod @ C @ B )] :
( ( relcomp @ A @ C @ B @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ C ) ) @ ( image2 @ D @ ( set @ ( product_prod @ A @ C ) ) @ R @ I4 ) ) @ S2 )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ D @ ( set @ ( product_prod @ A @ B ) )
@ ^ [I3: D] : ( relcomp @ A @ C @ B @ ( R @ I3 ) @ S2 )
@ I4 ) ) ) ).
% relcomp_UNION_distrib2
thf(fact_998_relcomp__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( relcomp @ A @ B @ C )
= ( ^ [R5: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ B @ C )] :
( collect @ ( product_prod @ A @ C )
@ ( product_case_prod @ A @ C @ $o
@ ( relcompp @ A @ B @ C
@ ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ R5 )
@ ^ [X4: B,Y5: C] : ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ X4 @ Y5 ) @ S ) ) ) ) ) ) ).
% relcomp_def
thf(fact_999_relcomp__fold,axiom,
! [C: $tType,B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S5: set @ ( product_prod @ B @ C )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ R3 )
=> ( ( finite_finite2 @ ( product_prod @ B @ C ) @ S5 )
=> ( ( relcomp @ A @ B @ C @ R3 @ S5 )
= ( 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 ) ) )
@ ^ [X4: A,Y5: B,A8: 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 ) ) )
@ ^ [W: B,Z5: C,A11: set @ ( product_prod @ A @ C )] : ( if @ ( set @ ( product_prod @ A @ C ) ) @ ( Y5 = W ) @ ( insert @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X4 @ Z5 ) @ A11 ) @ A11 ) )
@ A8
@ S5 ) )
@ ( bot_bot @ ( set @ ( product_prod @ A @ C ) ) )
@ R3 ) ) ) ) ).
% relcomp_fold
thf(fact_1000_insert__relcomp__fold,axiom,
! [C: $tType,B: $tType,A: $tType,S5: set @ ( product_prod @ A @ B ),X2: product_prod @ C @ A,R3: set @ ( product_prod @ C @ A )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ S5 )
=> ( ( relcomp @ C @ A @ B @ ( insert @ ( product_prod @ C @ A ) @ X2 @ R3 ) @ S5 )
= ( 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 ) ) )
@ ^ [W: A,Z5: B,A11: set @ ( product_prod @ C @ B )] :
( if @ ( set @ ( product_prod @ C @ B ) )
@ ( ( product_snd @ C @ A @ X2 )
= W )
@ ( insert @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ ( product_fst @ C @ A @ X2 ) @ Z5 ) @ A11 )
@ A11 ) )
@ ( relcomp @ C @ A @ B @ R3 @ S5 )
@ S5 ) ) ) ).
% insert_relcomp_fold
thf(fact_1001_comp__fun__commute__Pow__fold,axiom,
! [A: $tType] :
( finite6289374366891150609ommute @ A @ ( set @ ( set @ A ) )
@ ^ [X4: A,A8: set @ ( set @ A )] : ( sup_sup @ ( set @ ( set @ A ) ) @ A8 @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( insert @ A @ X4 ) @ A8 ) ) ) ).
% comp_fun_commute_Pow_fold
thf(fact_1002_max__ext_Omax__extI,axiom,
! [A: $tType,X6: set @ A,Y4: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ A @ X6 )
=> ( ( finite_finite2 @ A @ Y4 )
=> ( ( Y4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X: A] :
( ( member @ A @ X @ X6 )
=> ? [Xa3: A] :
( ( member @ A @ Xa3 @ Y4 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Xa3 ) @ R3 ) ) )
=> ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X6 @ Y4 ) @ ( max_ext @ A @ R3 ) ) ) ) ) ) ).
% max_ext.max_extI
thf(fact_1003_min__ext__compat,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),S5: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R3 @ S5 ) @ R3 )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( relcomp @ ( set @ A ) @ ( set @ A ) @ ( set @ A ) @ ( min_ext @ A @ R3 ) @ ( sup_sup @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( min_ext @ A @ S5 ) @ ( insert @ ( 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 @ R3 ) ) ) ).
% min_ext_compat
thf(fact_1004_max__extp__max__ext__eq,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( max_extp @ A
@ ^ [X4: A,Y5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R3 ) )
= ( ^ [X4: set @ A,Y5: set @ A] : ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X4 @ Y5 ) @ ( max_ext @ A @ R3 ) ) ) ) ).
% max_extp_max_ext_eq
thf(fact_1005_relInvImage__Gr,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),B6: set @ A,A6: set @ B,F: B > A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ B6
@ ^ [Uu: A] : B6 ) )
=> ( ( bNF_Gr7122648621184425601vImage @ B @ A @ A6 @ R3 @ F )
= ( relcomp @ B @ A @ B @ ( bNF_Gr @ B @ A @ A6 @ F ) @ ( relcomp @ A @ A @ B @ R3 @ ( converse @ B @ A @ ( bNF_Gr @ B @ A @ A6 @ F ) ) ) ) ) ) ).
% relInvImage_Gr
thf(fact_1006_relImage__Gr,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),A6: set @ A,F: A > B] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A6
@ ^ [Uu: A] : A6 ) )
=> ( ( bNF_Gr4221423524335903396lImage @ A @ B @ R3 @ F )
= ( relcomp @ B @ A @ B @ ( converse @ A @ B @ ( bNF_Gr @ A @ B @ A6 @ F ) ) @ ( relcomp @ A @ A @ B @ R3 @ ( bNF_Gr @ A @ B @ A6 @ F ) ) ) ) ) ).
% relImage_Gr
thf(fact_1007_Image__fold,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S5: set @ A] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ R3 )
=> ( ( image @ A @ B @ R3 @ S5 )
= ( finite_fold @ ( product_prod @ A @ B ) @ ( set @ B )
@ ( product_case_prod @ A @ B @ ( ( set @ B ) > ( set @ B ) )
@ ^ [X4: A,Y5: B,A8: set @ B] : ( if @ ( set @ B ) @ ( member @ A @ X4 @ S5 ) @ ( insert @ B @ Y5 @ A8 ) @ A8 ) )
@ ( bot_bot @ ( set @ B ) )
@ R3 ) ) ) ).
% Image_fold
thf(fact_1008_filterlim__base__iff,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,I4: set @ A,F7: A > ( set @ B ),F: B > C,G6: D > ( set @ C ),J4: set @ D] :
( ( I4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [I6: A] :
( ( member @ A @ I6 @ I4 )
=> ! [J3: A] :
( ( member @ A @ J3 @ I4 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( F7 @ I6 ) @ ( F7 @ J3 ) )
| ( ord_less_eq @ ( set @ B ) @ ( F7 @ J3 ) @ ( F7 @ I6 ) ) ) ) )
=> ( ( filterlim @ B @ C @ F
@ ( complete_Inf_Inf @ ( filter @ C )
@ ( image2 @ D @ ( filter @ C )
@ ^ [J: D] : ( principal @ C @ ( G6 @ J ) )
@ J4 ) )
@ ( complete_Inf_Inf @ ( filter @ B )
@ ( image2 @ A @ ( filter @ B )
@ ^ [I3: A] : ( principal @ B @ ( F7 @ I3 ) )
@ I4 ) ) )
= ( ! [X4: D] :
( ( member @ D @ X4 @ J4 )
=> ? [Y5: A] :
( ( member @ A @ Y5 @ I4 )
& ! [Z5: B] :
( ( member @ B @ Z5 @ ( F7 @ Y5 ) )
=> ( member @ C @ ( F @ Z5 ) @ ( G6 @ X4 ) ) ) ) ) ) ) ) ) ).
% filterlim_base_iff
thf(fact_1009_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 ) @ ( insert @ A @ A3 @ ( insert @ 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_1010_eventually__INF__base,axiom,
! [B: $tType,A: $tType,B6: set @ A,F7: A > ( filter @ B ),P2: B > $o] :
( ( B6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A5: A] :
( ( member @ A @ A5 @ B6 )
=> ! [B4: A] :
( ( member @ A @ B4 @ B6 )
=> ? [X5: A] :
( ( member @ A @ X5 @ B6 )
& ( ord_less_eq @ ( filter @ B ) @ ( F7 @ X5 ) @ ( inf_inf @ ( filter @ B ) @ ( F7 @ A5 ) @ ( F7 @ B4 ) ) ) ) ) )
=> ( ( eventually @ B @ P2 @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ F7 @ B6 ) ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ B6 )
& ( eventually @ B @ P2 @ ( F7 @ X4 ) ) ) ) ) ) ) ).
% eventually_INF_base
thf(fact_1011_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
@ ^ [X4: A,Y5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R6 ) ) ) ) ) ) ).
% max_ext_def
thf(fact_1012_ImageI,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,R: set @ ( product_prod @ A @ B ),A6: set @ A] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R )
=> ( ( member @ A @ A3 @ A6 )
=> ( member @ B @ B2 @ ( image @ A @ B @ R @ A6 ) ) ) ) ).
% ImageI
thf(fact_1013_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_1014_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 @ ( insert @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A3 @ B2 ) @ R ) ) ).
% Image_singleton_iff
thf(fact_1015_wo__rel_Owell__order__induct,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),P2: A > $o,A3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R )
=> ( ! [X: A] :
( ! [Y7: A] :
( ( ( Y7 != X )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y7 @ X ) @ R ) )
=> ( P2 @ Y7 ) )
=> ( P2 @ X ) )
=> ( P2 @ A3 ) ) ) ).
% wo_rel.well_order_induct
thf(fact_1016_ImageE,axiom,
! [A: $tType,B: $tType,B2: A,R: set @ ( product_prod @ B @ A ),A6: set @ B] :
( ( member @ A @ B2 @ ( image @ B @ A @ R @ A6 ) )
=> ~ ! [X: B] :
( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ B2 ) @ R )
=> ~ ( member @ B @ X @ A6 ) ) ) ).
% ImageE
thf(fact_1017_Image__iff,axiom,
! [A: $tType,B: $tType,B2: A,R: set @ ( product_prod @ B @ A ),A6: set @ B] :
( ( member @ A @ B2 @ ( image @ B @ A @ R @ A6 ) )
= ( ? [X4: B] :
( ( member @ B @ X4 @ A6 )
& ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X4 @ B2 ) @ R ) ) ) ) ).
% Image_iff
thf(fact_1018_rev__ImageI,axiom,
! [B: $tType,A: $tType,A3: A,A6: set @ A,B2: B,R: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A3 @ A6 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R )
=> ( member @ B @ B2 @ ( image @ A @ B @ R @ A6 ) ) ) ) ).
% rev_ImageI
thf(fact_1019_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_1020_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_1021_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_1022_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 ) )
=> ~ ! [X: B,Y: A] :
( ( Yx
= ( product_Pair @ A @ B @ Y @ X ) )
=> ~ ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y ) @ R ) ) ) ).
% converseE
thf(fact_1023_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_1024_wo__rel_OTOTALS,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( bNF_Wellorder_wo_rel @ A @ R )
=> ! [X5: A] :
( ( member @ A @ X5 @ ( field2 @ A @ R ) )
=> ! [Xa3: A] :
( ( member @ A @ Xa3 @ ( field2 @ A @ R ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Xa3 ) @ R )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Xa3 @ X5 ) @ R ) ) ) ) ) ).
% wo_rel.TOTALS
thf(fact_1025_converse__Times,axiom,
! [B: $tType,A: $tType,A6: set @ B,B6: set @ A] :
( ( converse @ B @ A
@ ( product_Sigma @ B @ A @ A6
@ ^ [Uu: B] : B6 ) )
= ( product_Sigma @ A @ B @ B6
@ ^ [Uu: A] : A6 ) ) ).
% converse_Times
thf(fact_1026_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_1027_well__order__induct__imp,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),P2: A > $o,A3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R )
=> ( ! [X: A] :
( ! [Y7: A] :
( ( ( Y7 != X )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y7 @ X ) @ R ) )
=> ( ( member @ A @ Y7 @ ( field2 @ A @ R ) )
=> ( P2 @ Y7 ) ) )
=> ( ( member @ A @ X @ ( field2 @ A @ R ) )
=> ( P2 @ X ) ) )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R ) )
=> ( P2 @ A3 ) ) ) ) ).
% well_order_induct_imp
thf(fact_1028_Image__UN,axiom,
! [A: $tType,B: $tType,C: $tType,R: set @ ( product_prod @ B @ A ),B6: C > ( set @ B ),A6: set @ C] :
( ( image @ B @ A @ R @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ C @ ( set @ B ) @ B6 @ A6 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [X4: C] : ( image @ B @ A @ R @ ( B6 @ X4 ) )
@ A6 ) ) ) ).
% Image_UN
thf(fact_1029_converse__unfold,axiom,
! [A: $tType,B: $tType] :
( ( converse @ B @ A )
= ( ^ [R5: set @ ( product_prod @ B @ A )] :
( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [Y5: A,X4: B] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X4 @ Y5 ) @ R5 ) ) ) ) ) ).
% converse_unfold
thf(fact_1030_conversep__converse__eq,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B )] :
( ( conversep @ A @ B
@ ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ R ) )
= ( ^ [X4: B,Y5: A] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X4 @ Y5 ) @ ( converse @ A @ B @ R ) ) ) ) ).
% conversep_converse_eq
thf(fact_1031_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_1032_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_1033_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_1034_conversep__in__rel,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ B @ A )] :
( ( conversep @ B @ A @ ( fun_in_rel @ B @ A @ R3 ) )
= ( fun_in_rel @ A @ B @ ( converse @ B @ A @ R3 ) ) ) ).
% conversep_in_rel
thf(fact_1035_eventually__INF1,axiom,
! [B: $tType,A: $tType,I: A,I4: set @ A,P2: B > $o,F7: A > ( filter @ B )] :
( ( member @ A @ I @ I4 )
=> ( ( eventually @ B @ P2 @ ( F7 @ I ) )
=> ( eventually @ B @ P2 @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ F7 @ I4 ) ) ) ) ) ).
% eventually_INF1
thf(fact_1036_converse__UNION,axiom,
! [B: $tType,A: $tType,C: $tType,R: C > ( set @ ( product_prod @ B @ A ) ),S5: set @ C] :
( ( converse @ B @ A @ ( complete_Sup_Sup @ ( set @ ( product_prod @ B @ A ) ) @ ( image2 @ C @ ( set @ ( product_prod @ B @ A ) ) @ R @ S5 ) ) )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ C @ ( set @ ( product_prod @ A @ B ) )
@ ^ [X4: C] : ( converse @ B @ A @ ( R @ X4 ) )
@ S5 ) ) ) ).
% converse_UNION
thf(fact_1037_filterlim__INF_H,axiom,
! [C: $tType,B: $tType,A: $tType,X2: A,A6: set @ A,F: B > C,F7: filter @ C,G6: A > ( filter @ B )] :
( ( member @ A @ X2 @ A6 )
=> ( ( filterlim @ B @ C @ F @ F7 @ ( G6 @ X2 ) )
=> ( filterlim @ B @ C @ F @ F7 @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ G6 @ A6 ) ) ) ) ) ).
% filterlim_INF'
thf(fact_1038_filterlim__INF,axiom,
! [A: $tType,B: $tType,C: $tType,F: A > B,G6: C > ( filter @ B ),B6: set @ C,F7: filter @ A] :
( ( filterlim @ A @ B @ F @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ C @ ( filter @ B ) @ G6 @ B6 ) ) @ F7 )
= ( ! [X4: C] :
( ( member @ C @ X4 @ B6 )
=> ( filterlim @ A @ B @ F @ ( G6 @ X4 ) @ F7 ) ) ) ) ).
% filterlim_INF
thf(fact_1039_converse__INTER,axiom,
! [B: $tType,A: $tType,C: $tType,R: C > ( set @ ( product_prod @ B @ A ) ),S5: set @ C] :
( ( converse @ B @ A @ ( complete_Inf_Inf @ ( set @ ( product_prod @ B @ A ) ) @ ( image2 @ C @ ( set @ ( product_prod @ B @ A ) ) @ R @ S5 ) ) )
= ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ C @ ( set @ ( product_prod @ A @ B ) )
@ ^ [X4: C] : ( converse @ B @ A @ ( R @ X4 ) )
@ S5 ) ) ) ).
% converse_INTER
thf(fact_1040_Image__singleton,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ B @ A ),A3: B] :
( ( image @ B @ A @ R @ ( insert @ 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_1041_Image__INT__subset,axiom,
! [A: $tType,B: $tType,C: $tType,R: set @ ( product_prod @ B @ A ),B6: C > ( set @ B ),A6: set @ C] :
( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ R @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ C @ ( set @ B ) @ B6 @ A6 ) ) )
@ ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [X4: C] : ( image @ B @ A @ R @ ( B6 @ X4 ) )
@ A6 ) ) ) ).
% Image_INT_subset
thf(fact_1042_converse__def,axiom,
! [B: $tType,A: $tType] :
( ( converse @ A @ B )
= ( ^ [R5: set @ ( product_prod @ A @ B )] :
( collect @ ( product_prod @ B @ A )
@ ( product_case_prod @ B @ A @ $o
@ ( conversep @ A @ B
@ ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ R5 ) ) ) ) ) ) ).
% converse_def
thf(fact_1043_eventually__INF__finite,axiom,
! [B: $tType,A: $tType,A6: set @ A,P2: B > $o,F7: A > ( filter @ B )] :
( ( finite_finite2 @ A @ A6 )
=> ( ( eventually @ B @ P2 @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ F7 @ A6 ) ) )
= ( ? [Q3: A > B > $o] :
( ! [X4: A] :
( ( member @ A @ X4 @ A6 )
=> ( eventually @ B @ ( Q3 @ X4 ) @ ( F7 @ X4 ) ) )
& ! [Y5: B] :
( ! [X4: A] :
( ( member @ A @ X4 @ A6 )
=> ( Q3 @ X4 @ Y5 ) )
=> ( P2 @ Y5 ) ) ) ) ) ) ).
% eventually_INF_finite
thf(fact_1044_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 ) @ ( insert @ A @ A3 @ ( insert @ 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_1045_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 ) @ ( insert @ A @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% wo_rel.max2_greater_among
thf(fact_1046_filterlim__base,axiom,
! [B: $tType,A: $tType,E: $tType,D: $tType,C: $tType,J4: set @ A,I: A > C,I4: set @ C,F7: C > ( set @ D ),F: D > E,G6: A > ( set @ E )] :
( ! [M3: A,X: B] :
( ( member @ A @ M3 @ J4 )
=> ( member @ C @ ( I @ M3 ) @ I4 ) )
=> ( ! [M3: A,X: D] :
( ( member @ A @ M3 @ J4 )
=> ( ( member @ D @ X @ ( F7 @ ( I @ M3 ) ) )
=> ( member @ E @ ( F @ X ) @ ( G6 @ M3 ) ) ) )
=> ( filterlim @ D @ E @ F
@ ( complete_Inf_Inf @ ( filter @ E )
@ ( image2 @ A @ ( filter @ E )
@ ^ [J: A] : ( principal @ E @ ( G6 @ J ) )
@ J4 ) )
@ ( complete_Inf_Inf @ ( filter @ D )
@ ( image2 @ C @ ( filter @ D )
@ ^ [I3: C] : ( principal @ D @ ( F7 @ I3 ) )
@ I4 ) ) ) ) ) ).
% filterlim_base
thf(fact_1047_Image__eq__UN,axiom,
! [A: $tType,B: $tType] :
( ( image @ B @ A )
= ( ^ [R5: set @ ( product_prod @ B @ A ),B7: set @ B] :
( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y5: B] : ( image @ B @ A @ R5 @ ( insert @ B @ Y5 @ ( bot_bot @ ( set @ B ) ) ) )
@ B7 ) ) ) ) ).
% Image_eq_UN
thf(fact_1048_image2__Gr,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( bNF_Greatest_image2 @ C @ A @ B )
= ( ^ [A8: set @ C,F4: C > A,G4: C > B] : ( relcomp @ A @ C @ B @ ( converse @ C @ A @ ( bNF_Gr @ C @ A @ A8 @ F4 ) ) @ ( bNF_Gr @ C @ B @ A8 @ G4 ) ) ) ) ).
% image2_Gr
thf(fact_1049_Sigma__Image,axiom,
! [A: $tType,B: $tType,A6: set @ B,B6: B > ( set @ A ),X6: set @ B] :
( ( image @ B @ A @ ( product_Sigma @ B @ A @ A6 @ B6 ) @ X6 )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B6 @ ( inf_inf @ ( set @ B ) @ X6 @ A6 ) ) ) ) ).
% Sigma_Image
thf(fact_1050_eventually__INF,axiom,
! [A: $tType,B: $tType,P2: A > $o,F7: B > ( filter @ A ),B6: set @ B] :
( ( eventually @ A @ P2 @ ( complete_Inf_Inf @ ( filter @ A ) @ ( image2 @ B @ ( filter @ A ) @ F7 @ B6 ) ) )
= ( ? [X10: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ X10 @ B6 )
& ( finite_finite2 @ B @ X10 )
& ( eventually @ A @ P2 @ ( complete_Inf_Inf @ ( filter @ A ) @ ( image2 @ B @ ( filter @ A ) @ F7 @ X10 ) ) ) ) ) ) ).
% eventually_INF
thf(fact_1051_UN__Image,axiom,
! [A: $tType,B: $tType,C: $tType,X6: C > ( set @ ( product_prod @ B @ A ) ),I4: set @ C,S5: set @ B] :
( ( image @ B @ A @ ( complete_Sup_Sup @ ( set @ ( product_prod @ B @ A ) ) @ ( image2 @ C @ ( set @ ( product_prod @ B @ A ) ) @ X6 @ I4 ) ) @ S5 )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [I3: C] : ( image @ B @ A @ ( X6 @ I3 ) @ S5 )
@ I4 ) ) ) ).
% UN_Image
thf(fact_1052_max__extp__eq,axiom,
! [A: $tType] :
( ( max_extp @ A )
= ( ^ [R5: A > A > $o,X4: set @ A,Y5: set @ A] : ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X4 @ Y5 ) @ ( max_ext @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R5 ) ) ) ) ) ) ).
% max_extp_eq
thf(fact_1053_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 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ R ) ) ) ) ) ).
% subset_Image1_Image1_iff
thf(fact_1054_map__filter__on__comp,axiom,
! [A: $tType,C: $tType,B: $tType,G: B > A,Y4: set @ B,X6: set @ A,F7: filter @ B,F: A > C] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ G @ Y4 ) @ X6 )
=> ( ( eventually @ B
@ ^ [X4: B] : ( member @ B @ X4 @ Y4 )
@ F7 )
=> ( ( map_filter_on @ A @ C @ X6 @ F @ ( map_filter_on @ B @ A @ Y4 @ G @ F7 ) )
= ( map_filter_on @ B @ C @ Y4 @ ( comp @ A @ C @ B @ F @ G ) @ F7 ) ) ) ) ).
% map_filter_on_comp
thf(fact_1055_quotient__def,axiom,
! [A: $tType] :
( ( equiv_quotient @ A )
= ( ^ [A8: set @ A,R5: set @ ( product_prod @ A @ A )] :
( complete_Sup_Sup @ ( set @ ( set @ A ) )
@ ( image2 @ A @ ( set @ ( set @ A ) )
@ ^ [X4: A] : ( insert @ ( set @ A ) @ ( image @ A @ A @ R5 @ ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) )
@ A8 ) ) ) ) ).
% quotient_def
thf(fact_1056_Image__INT__eq,axiom,
! [A: $tType,B: $tType,C: $tType,R: set @ ( product_prod @ B @ A ),A6: set @ C,B6: C > ( set @ B )] :
( ( single_valued @ A @ B @ ( converse @ B @ A @ R ) )
=> ( ( A6
!= ( bot_bot @ ( set @ C ) ) )
=> ( ( image @ B @ A @ R @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ C @ ( set @ B ) @ B6 @ A6 ) ) )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [X4: C] : ( image @ B @ A @ R @ ( B6 @ X4 ) )
@ A6 ) ) ) ) ) ).
% Image_INT_eq
thf(fact_1057_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_1058_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 )
=> ( ! [B4: A] :
( ( member @ A @ B4 @ B6 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B4 ) @ R ) )
=> ( A3
= ( bNF_We6954850376910717587_minim @ A @ R @ B6 ) ) ) ) ) ) ).
% wo_rel.equals_minim
thf(fact_1059_subset__Image__Image__iff,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A6: set @ A,B6: set @ A] :
( ( order_preorder_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ ( field2 @ A @ R ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ ( field2 @ A @ R ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R @ A6 ) @ ( image @ A @ A @ R @ B6 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A6 )
=> ? [Y5: A] :
( ( member @ A @ Y5 @ B6 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X4 ) @ R ) ) ) ) ) ) ) ) ).
% subset_Image_Image_iff
thf(fact_1060_filterlim__INF__INF,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,J4: set @ A,I4: set @ B,F: D > C,F7: B > ( filter @ D ),G6: A > ( filter @ C )] :
( ! [M3: A] :
( ( member @ A @ M3 @ J4 )
=> ? [X5: B] :
( ( member @ B @ X5 @ I4 )
& ( ord_less_eq @ ( filter @ C ) @ ( filtermap @ D @ C @ F @ ( F7 @ X5 ) ) @ ( G6 @ M3 ) ) ) )
=> ( filterlim @ D @ C @ F @ ( complete_Inf_Inf @ ( filter @ C ) @ ( image2 @ A @ ( filter @ C ) @ G6 @ J4 ) ) @ ( complete_Inf_Inf @ ( filter @ D ) @ ( image2 @ B @ ( filter @ D ) @ F7 @ I4 ) ) ) ) ).
% filterlim_INF_INF
thf(fact_1061_filtermap__principal,axiom,
! [A: $tType,B: $tType,F: B > A,A6: set @ B] :
( ( filtermap @ B @ A @ F @ ( principal @ B @ A6 ) )
= ( principal @ A @ ( image2 @ B @ A @ F @ A6 ) ) ) ).
% filtermap_principal
thf(fact_1062_single__valued__def,axiom,
! [B: $tType,A: $tType] :
( ( single_valued @ A @ B )
= ( ^ [R5: set @ ( product_prod @ A @ B )] :
! [X4: A,Y5: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ R5 )
=> ! [Z5: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Z5 ) @ R5 )
=> ( Y5 = Z5 ) ) ) ) ) ).
% single_valued_def
thf(fact_1063_single__valuedI,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B )] :
( ! [X: A,Y: B,Z4: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ R )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Z4 ) @ R )
=> ( Y = Z4 ) ) )
=> ( single_valued @ A @ B @ R ) ) ).
% single_valuedI
thf(fact_1064_single__valuedD,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B ),X2: A,Y3: B,Z3: B] :
( ( single_valued @ A @ B @ R )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ R )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Z3 ) @ R )
=> ( Y3 = Z3 ) ) ) ) ).
% single_valuedD
thf(fact_1065_filtermap__SUP,axiom,
! [A: $tType,B: $tType,C: $tType,F: B > A,F7: C > ( filter @ B ),B6: set @ C] :
( ( filtermap @ B @ A @ F @ ( complete_Sup_Sup @ ( filter @ B ) @ ( image2 @ C @ ( filter @ B ) @ F7 @ B6 ) ) )
= ( complete_Sup_Sup @ ( filter @ A )
@ ( image2 @ C @ ( filter @ A )
@ ^ [B5: C] : ( filtermap @ B @ A @ F @ ( F7 @ B5 ) )
@ B6 ) ) ) ).
% filtermap_SUP
thf(fact_1066_filtermap__INF,axiom,
! [A: $tType,B: $tType,C: $tType,F: B > A,F7: C > ( filter @ B ),B6: set @ C] :
( ord_less_eq @ ( filter @ A ) @ ( filtermap @ B @ A @ F @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ C @ ( filter @ B ) @ F7 @ B6 ) ) )
@ ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ C @ ( filter @ A )
@ ^ [B5: C] : ( filtermap @ B @ A @ F @ ( F7 @ B5 ) )
@ B6 ) ) ) ).
% filtermap_INF
thf(fact_1067_prod__filter__principal__singleton2,axiom,
! [B: $tType,A: $tType,F7: filter @ A,X2: B] :
( ( prod_filter @ A @ B @ F7 @ ( principal @ B @ ( insert @ B @ X2 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( filtermap @ A @ ( product_prod @ A @ B )
@ ^ [A7: A] : ( product_Pair @ A @ B @ A7 @ X2 )
@ F7 ) ) ).
% prod_filter_principal_singleton2
thf(fact_1068_in__quotient__imp__in__rel,axiom,
! [A: $tType,A6: set @ A,R: set @ ( product_prod @ A @ A ),X6: set @ A,X2: A,Y3: A] :
( ( equiv_equiv @ A @ A6 @ R )
=> ( ( member @ ( set @ A ) @ X6 @ ( equiv_quotient @ A @ A6 @ R ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X2 @ ( insert @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) @ X6 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R ) ) ) ) ).
% in_quotient_imp_in_rel
thf(fact_1069_relpow__finite__bounded,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),K: nat] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R3 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ K @ R3 )
@ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N4: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N4 @ R3 )
@ ( collect @ nat
@ ^ [N4: nat] : ( ord_less_eq @ nat @ N4 @ ( finite_card @ ( product_prod @ A @ A ) @ R3 ) ) ) ) ) ) ) ).
% relpow_finite_bounded
thf(fact_1070_equiv__class__nondisjoint,axiom,
! [A: $tType,A6: set @ A,R: set @ ( product_prod @ A @ A ),X2: A,A3: A,B2: A] :
( ( equiv_equiv @ A @ A6 @ R )
=> ( ( member @ A @ X2 @ ( inf_inf @ ( set @ A ) @ ( image @ A @ A @ R @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R ) ) ) ).
% equiv_class_nondisjoint
thf(fact_1071_equiv__class__subset,axiom,
! [A: $tType,A6: set @ A,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( equiv_equiv @ A @ A6 @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
=> ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% equiv_class_subset
thf(fact_1072_subset__equiv__class,axiom,
! [A: $tType,A6: set @ A,R: set @ ( product_prod @ A @ A ),B2: A,A3: A] :
( ( equiv_equiv @ A @ A6 @ R )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( member @ A @ B2 @ A6 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R ) ) ) ) ).
% subset_equiv_class
thf(fact_1073_single__valuedp__single__valued__eq,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B )] :
( ( single_valuedp @ A @ B
@ ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ R ) )
= ( single_valued @ A @ B @ R ) ) ).
% single_valuedp_single_valued_eq
thf(fact_1074_Range__Union,axiom,
! [A: $tType,B: $tType,S5: set @ ( set @ ( product_prod @ B @ A ) )] :
( ( range2 @ B @ A @ ( complete_Sup_Sup @ ( set @ ( product_prod @ B @ A ) ) @ S5 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( set @ ( product_prod @ B @ A ) ) @ ( set @ A ) @ ( range2 @ B @ A ) @ S5 ) ) ) ).
% Range_Union
thf(fact_1075_refines__equiv__image__eq,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),S5: set @ ( product_prod @ A @ A ),A6: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ S5 )
=> ( ( equiv_equiv @ A @ A6 @ R3 )
=> ( ( equiv_equiv @ A @ A6 @ S5 )
=> ( ( image2 @ ( set @ A ) @ ( set @ A ) @ ( image @ A @ A @ S5 ) @ ( equiv_quotient @ A @ A6 @ R3 ) )
= ( equiv_quotient @ A @ A6 @ S5 ) ) ) ) ) ).
% refines_equiv_image_eq
thf(fact_1076_Range__insert,axiom,
! [A: $tType,B: $tType,A3: B,B2: A,R: set @ ( product_prod @ B @ A )] :
( ( range2 @ B @ A @ ( insert @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A3 @ B2 ) @ R ) )
= ( insert @ A @ B2 @ ( range2 @ B @ A @ R ) ) ) ).
% Range_insert
thf(fact_1077_prod__filter__assoc,axiom,
! [A: $tType,B: $tType,C: $tType,F7: filter @ A,G6: filter @ B,H7: filter @ C] :
( ( prod_filter @ ( product_prod @ A @ B ) @ C @ ( prod_filter @ A @ B @ F7 @ G6 ) @ H7 )
= ( filtermap @ ( product_prod @ A @ ( product_prod @ B @ C ) ) @ ( product_prod @ ( product_prod @ A @ B ) @ C )
@ ( product_case_prod @ A @ ( product_prod @ B @ C ) @ ( product_prod @ ( product_prod @ A @ B ) @ C )
@ ^ [X4: A] :
( product_case_prod @ B @ C @ ( product_prod @ ( product_prod @ A @ B ) @ C )
@ ^ [Y5: B] : ( product_Pair @ ( product_prod @ A @ B ) @ C @ ( product_Pair @ A @ B @ X4 @ Y5 ) ) ) )
@ ( prod_filter @ A @ ( product_prod @ B @ C ) @ F7 @ ( prod_filter @ B @ C @ G6 @ H7 ) ) ) ) ).
% prod_filter_assoc
thf(fact_1078_relpow__Suc__D2_H,axiom,
! [A: $tType,N5: nat,R3: set @ ( product_prod @ A @ A ),X5: A,Y7: A,Z7: A] :
( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y7 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N5 @ R3 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y7 @ Z7 ) @ R3 ) )
=> ? [W2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ W2 ) @ R3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ W2 @ Z7 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N5 @ R3 ) ) ) ) ).
% relpow_Suc_D2'
thf(fact_1079_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_1080_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_1081_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_1082_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_1083_Range__iff,axiom,
! [A: $tType,B: $tType,A3: A,R: set @ ( product_prod @ B @ A )] :
( ( member @ A @ A3 @ ( range2 @ B @ A @ R ) )
= ( ? [Y5: B] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ Y5 @ A3 ) @ R ) ) ) ).
% Range_iff
thf(fact_1084_prod__filter__commute,axiom,
! [B: $tType,A: $tType] :
( ( prod_filter @ A @ B )
= ( ^ [F8: filter @ A,G7: filter @ B] : ( filtermap @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ ( prod_filter @ B @ A @ G7 @ F8 ) ) ) ) ).
% prod_filter_commute
thf(fact_1085_in__quotient__imp__closed,axiom,
! [A: $tType,A6: set @ A,R: set @ ( product_prod @ A @ A ),X6: set @ A,X2: A,Y3: A] :
( ( equiv_equiv @ A @ A6 @ R )
=> ( ( member @ ( set @ A ) @ X6 @ ( equiv_quotient @ A @ A6 @ R ) )
=> ( ( member @ A @ X2 @ X6 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R )
=> ( member @ A @ Y3 @ X6 ) ) ) ) ) ).
% in_quotient_imp_closed
thf(fact_1086_quotient__eq__iff,axiom,
! [A: $tType,A6: set @ A,R: set @ ( product_prod @ A @ A ),X6: set @ A,Y4: set @ A,X2: A,Y3: A] :
( ( equiv_equiv @ A @ A6 @ R )
=> ( ( member @ ( set @ A ) @ X6 @ ( equiv_quotient @ A @ A6 @ R ) )
=> ( ( member @ ( set @ A ) @ Y4 @ ( equiv_quotient @ A @ A6 @ R ) )
=> ( ( member @ A @ X2 @ X6 )
=> ( ( member @ A @ Y3 @ Y4 )
=> ( ( X6 = Y4 )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R ) ) ) ) ) ) ) ).
% quotient_eq_iff
thf(fact_1087_quotient__eqI,axiom,
! [A: $tType,A6: set @ A,R: set @ ( product_prod @ A @ A ),X6: set @ A,Y4: set @ A,X2: A,Y3: A] :
( ( equiv_equiv @ A @ A6 @ R )
=> ( ( member @ ( set @ A ) @ X6 @ ( equiv_quotient @ A @ A6 @ R ) )
=> ( ( member @ ( set @ A ) @ Y4 @ ( equiv_quotient @ A @ A6 @ R ) )
=> ( ( member @ A @ X2 @ X6 )
=> ( ( member @ A @ Y3 @ Y4 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R )
=> ( X6 = Y4 ) ) ) ) ) ) ) ).
% quotient_eqI
thf(fact_1088_eventually__prod__filter,axiom,
! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,F7: filter @ A,G6: filter @ B] :
( ( eventually @ ( product_prod @ A @ B ) @ P2 @ ( prod_filter @ A @ B @ F7 @ G6 ) )
= ( ? [Pf: A > $o,Pg: B > $o] :
( ( eventually @ A @ Pf @ F7 )
& ( eventually @ B @ Pg @ G6 )
& ! [X4: A,Y5: B] :
( ( Pf @ X4 )
=> ( ( Pg @ Y5 )
=> ( P2 @ ( product_Pair @ A @ B @ X4 @ Y5 ) ) ) ) ) ) ) ).
% eventually_prod_filter
thf(fact_1089_eventually__prod__same,axiom,
! [A: $tType,P2: ( product_prod @ A @ A ) > $o,F7: filter @ A] :
( ( eventually @ ( product_prod @ A @ A ) @ P2 @ ( prod_filter @ A @ A @ F7 @ F7 ) )
= ( ? [Q3: A > $o] :
( ( eventually @ A @ Q3 @ F7 )
& ! [X4: A,Y5: A] :
( ( Q3 @ X4 )
=> ( ( Q3 @ Y5 )
=> ( P2 @ ( product_Pair @ A @ A @ X4 @ Y5 ) ) ) ) ) ) ) ).
% eventually_prod_same
thf(fact_1090_Range__snd,axiom,
! [A: $tType,B: $tType] :
( ( range2 @ B @ A )
= ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) ) ) ).
% Range_snd
thf(fact_1091_snd__eq__Range,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ B @ A )] :
( ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ R3 )
= ( range2 @ B @ A @ R3 ) ) ).
% snd_eq_Range
thf(fact_1092_filtermap__fst__prod__filter,axiom,
! [B: $tType,A: $tType,A6: filter @ A,B6: filter @ B] : ( ord_less_eq @ ( filter @ A ) @ ( filtermap @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( prod_filter @ A @ B @ A6 @ B6 ) ) @ A6 ) ).
% filtermap_fst_prod_filter
thf(fact_1093_filtermap__snd__prod__filter,axiom,
! [B: $tType,A: $tType,A6: filter @ B,B6: filter @ A] : ( ord_less_eq @ ( filter @ A ) @ ( filtermap @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( prod_filter @ B @ A @ A6 @ B6 ) ) @ B6 ) ).
% filtermap_snd_prod_filter
thf(fact_1094_filterlim__Pair,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > B,G6: filter @ B,F7: filter @ A,G: A > C,H7: filter @ C] :
( ( filterlim @ A @ B @ F @ G6 @ F7 )
=> ( ( filterlim @ A @ C @ G @ H7 @ F7 )
=> ( filterlim @ A @ ( product_prod @ B @ C )
@ ^ [X4: A] : ( product_Pair @ B @ C @ ( F @ X4 ) @ ( G @ X4 ) )
@ ( prod_filter @ B @ C @ G6 @ H7 )
@ F7 ) ) ) ).
% filterlim_Pair
thf(fact_1095_filtermap__Pair,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > A,G: C > B,F7: filter @ C] :
( ord_less_eq @ ( filter @ ( product_prod @ A @ B ) )
@ ( filtermap @ C @ ( product_prod @ A @ B )
@ ^ [X4: C] : ( product_Pair @ A @ B @ ( F @ X4 ) @ ( G @ X4 ) )
@ F7 )
@ ( prod_filter @ A @ B @ ( filtermap @ C @ A @ F @ F7 ) @ ( filtermap @ C @ B @ G @ F7 ) ) ) ).
% filtermap_Pair
thf(fact_1096_prod__filter__def,axiom,
! [B: $tType,A: $tType] :
( ( prod_filter @ A @ B )
= ( ^ [F8: filter @ A,G7: filter @ B] :
( complete_Inf_Inf @ ( filter @ ( product_prod @ A @ B ) )
@ ( image2 @ ( product_prod @ ( A > $o ) @ ( B > $o ) ) @ ( filter @ ( product_prod @ A @ B ) )
@ ( product_case_prod @ ( A > $o ) @ ( B > $o ) @ ( filter @ ( product_prod @ A @ B ) )
@ ^ [P4: A > $o,Q3: B > $o] :
( principal @ ( product_prod @ A @ B )
@ ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X4: A,Y5: B] :
( ( P4 @ X4 )
& ( Q3 @ Y5 ) ) ) ) ) )
@ ( collect @ ( product_prod @ ( A > $o ) @ ( B > $o ) )
@ ( product_case_prod @ ( A > $o ) @ ( B > $o ) @ $o
@ ^ [P4: A > $o,Q3: B > $o] :
( ( eventually @ A @ P4 @ F8 )
& ( eventually @ B @ Q3 @ G7 ) ) ) ) ) ) ) ) ).
% prod_filter_def
thf(fact_1097_le__prod__filterI,axiom,
! [A: $tType,B: $tType,F7: filter @ ( product_prod @ A @ B ),A6: filter @ A,B6: filter @ B] :
( ( ord_less_eq @ ( filter @ A ) @ ( filtermap @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ F7 ) @ A6 )
=> ( ( ord_less_eq @ ( filter @ B ) @ ( filtermap @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ F7 ) @ B6 )
=> ( ord_less_eq @ ( filter @ ( product_prod @ A @ B ) ) @ F7 @ ( prod_filter @ A @ B @ A6 @ B6 ) ) ) ) ).
% le_prod_filterI
thf(fact_1098_eq__equiv__class,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B2: A,A6: set @ A] :
( ( ( image @ A @ A @ R @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( equiv_equiv @ A @ A6 @ R )
=> ( ( member @ A @ B2 @ A6 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R ) ) ) ) ).
% eq_equiv_class
thf(fact_1099_equiv__class__eq,axiom,
! [A: $tType,A6: set @ A,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( equiv_equiv @ A @ A6 @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
=> ( ( image @ A @ A @ R @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% equiv_class_eq
thf(fact_1100_eq__equiv__class__iff,axiom,
! [A: $tType,A6: set @ A,R: set @ ( product_prod @ A @ A ),X2: A,Y3: A] :
( ( equiv_equiv @ A @ A6 @ R )
=> ( ( member @ A @ X2 @ A6 )
=> ( ( member @ A @ Y3 @ A6 )
=> ( ( ( image @ A @ A @ R @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R @ ( insert @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R ) ) ) ) ) ).
% eq_equiv_class_iff
thf(fact_1101_equiv__class__eq__iff,axiom,
! [A: $tType,A6: set @ A,R: set @ ( product_prod @ A @ A ),X2: A,Y3: A] :
( ( equiv_equiv @ A @ A6 @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R )
= ( ( ( image @ A @ A @ R @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R @ ( insert @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( member @ A @ X2 @ A6 )
& ( member @ A @ Y3 @ A6 ) ) ) ) ).
% equiv_class_eq_iff
thf(fact_1102_eventually__prodI,axiom,
! [A: $tType,B: $tType,P2: A > $o,F7: filter @ A,Q2: B > $o,G6: filter @ B] :
( ( eventually @ A @ P2 @ F7 )
=> ( ( eventually @ B @ Q2 @ G6 )
=> ( eventually @ ( product_prod @ A @ B )
@ ^ [X4: product_prod @ A @ B] :
( ( P2 @ ( product_fst @ A @ B @ X4 ) )
& ( Q2 @ ( product_snd @ A @ B @ X4 ) ) )
@ ( prod_filter @ A @ B @ F7 @ G6 ) ) ) ) ).
% eventually_prodI
thf(fact_1103_eq__equiv__class__iff2,axiom,
! [A: $tType,A6: set @ A,R: set @ ( product_prod @ A @ A ),X2: A,Y3: A] :
( ( equiv_equiv @ A @ A6 @ R )
=> ( ( member @ A @ X2 @ A6 )
=> ( ( member @ A @ Y3 @ A6 )
=> ( ( ( equiv_quotient @ A @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) @ R )
= ( equiv_quotient @ A @ ( insert @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) @ R ) )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R ) ) ) ) ) ).
% eq_equiv_class_iff2
thf(fact_1104_eventually__prod2,axiom,
! [A: $tType,B: $tType,A6: filter @ A,P2: B > $o,B6: filter @ B] :
( ( A6
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( eventually @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X4: A] : P2 )
@ ( prod_filter @ A @ B @ A6 @ B6 ) )
= ( eventually @ B @ P2 @ B6 ) ) ) ).
% eventually_prod2
thf(fact_1105_eventually__prod1,axiom,
! [A: $tType,B: $tType,B6: filter @ A,P2: B > $o,A6: filter @ B] :
( ( B6
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( eventually @ ( product_prod @ B @ A )
@ ( product_case_prod @ B @ A @ $o
@ ^ [X4: B,Y5: A] : ( P2 @ X4 ) )
@ ( prod_filter @ B @ A @ A6 @ B6 ) )
= ( eventually @ B @ P2 @ A6 ) ) ) ).
% eventually_prod1
thf(fact_1106_prod__filter__INF2,axiom,
! [B: $tType,C: $tType,A: $tType,J4: set @ A,A6: filter @ B,B6: A > ( filter @ C )] :
( ( J4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( prod_filter @ B @ C @ A6 @ ( complete_Inf_Inf @ ( filter @ C ) @ ( image2 @ A @ ( filter @ C ) @ B6 @ J4 ) ) )
= ( complete_Inf_Inf @ ( filter @ ( product_prod @ B @ C ) )
@ ( image2 @ A @ ( filter @ ( product_prod @ B @ C ) )
@ ^ [I3: A] : ( prod_filter @ B @ C @ A6 @ ( B6 @ I3 ) )
@ J4 ) ) ) ) ).
% prod_filter_INF2
thf(fact_1107_prod__filter__INF1,axiom,
! [B: $tType,C: $tType,A: $tType,I4: set @ A,A6: A > ( filter @ B ),B6: filter @ C] :
( ( I4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( prod_filter @ B @ C @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ A6 @ I4 ) ) @ B6 )
= ( complete_Inf_Inf @ ( filter @ ( product_prod @ B @ C ) )
@ ( image2 @ A @ ( filter @ ( product_prod @ B @ C ) )
@ ^ [I3: A] : ( prod_filter @ B @ C @ ( A6 @ I3 ) @ B6 )
@ I4 ) ) ) ) ).
% prod_filter_INF1
thf(fact_1108_prod__filter__INF,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,I4: set @ A,J4: set @ B,A6: A > ( filter @ C ),B6: B > ( filter @ D )] :
( ( I4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( J4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( prod_filter @ C @ D @ ( complete_Inf_Inf @ ( filter @ C ) @ ( image2 @ A @ ( filter @ C ) @ A6 @ I4 ) ) @ ( complete_Inf_Inf @ ( filter @ D ) @ ( image2 @ B @ ( filter @ D ) @ B6 @ J4 ) ) )
= ( complete_Inf_Inf @ ( filter @ ( product_prod @ C @ D ) )
@ ( image2 @ A @ ( filter @ ( product_prod @ C @ D ) )
@ ^ [I3: A] :
( complete_Inf_Inf @ ( filter @ ( product_prod @ C @ D ) )
@ ( image2 @ B @ ( filter @ ( product_prod @ C @ D ) )
@ ^ [J: B] : ( prod_filter @ C @ D @ ( A6 @ I3 ) @ ( B6 @ J ) )
@ J4 ) )
@ I4 ) ) ) ) ) ).
% prod_filter_INF
thf(fact_1109_prod__filter__principal__singleton,axiom,
! [A: $tType,B: $tType,X2: A,F7: filter @ B] :
( ( prod_filter @ A @ B @ ( principal @ A @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) @ F7 )
= ( filtermap @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 ) @ F7 ) ) ).
% prod_filter_principal_singleton
thf(fact_1110_UN__equiv__class2,axiom,
! [A: $tType,C: $tType,B: $tType,A13: set @ A,R12: set @ ( product_prod @ A @ A ),A24: set @ B,R23: set @ ( product_prod @ B @ B ),F: A > B > ( set @ C ),A1: A,A22: B] :
( ( equiv_equiv @ A @ A13 @ R12 )
=> ( ( equiv_equiv @ B @ A24 @ R23 )
=> ( ( equiv_congruent2 @ A @ B @ ( set @ C ) @ R12 @ R23 @ F )
=> ( ( member @ A @ A1 @ A13 )
=> ( ( member @ B @ A22 @ A24 )
=> ( ( complete_Sup_Sup @ ( set @ C )
@ ( image2 @ A @ ( set @ C )
@ ^ [X12: A] : ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ ( F @ X12 ) @ ( image @ B @ B @ R23 @ ( insert @ B @ A22 @ ( bot_bot @ ( set @ B ) ) ) ) ) )
@ ( image @ A @ A @ R12 @ ( insert @ A @ A1 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
= ( F @ A1 @ A22 ) ) ) ) ) ) ) ).
% UN_equiv_class2
thf(fact_1111_UN__equiv__class,axiom,
! [B: $tType,A: $tType,A6: set @ A,R: set @ ( product_prod @ A @ A ),F: A > ( set @ B ),A3: A] :
( ( equiv_equiv @ A @ A6 @ R )
=> ( ( equiv_congruent @ A @ ( set @ B ) @ R @ F )
=> ( ( member @ A @ A3 @ A6 )
=> ( ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F @ ( image @ A @ A @ R @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
= ( F @ A3 ) ) ) ) ) ).
% UN_equiv_class
thf(fact_1112_UN__equiv__class__inject,axiom,
! [B: $tType,A: $tType,A6: set @ A,R: set @ ( product_prod @ A @ A ),F: A > ( set @ B ),X6: set @ A,Y4: set @ A] :
( ( equiv_equiv @ A @ A6 @ R )
=> ( ( equiv_congruent @ A @ ( set @ B ) @ R @ F )
=> ( ( ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F @ X6 ) )
= ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F @ Y4 ) ) )
=> ( ( member @ ( set @ A ) @ X6 @ ( equiv_quotient @ A @ A6 @ R ) )
=> ( ( member @ ( set @ A ) @ Y4 @ ( equiv_quotient @ A @ A6 @ R ) )
=> ( ! [X: A,Y: A] :
( ( member @ A @ X @ A6 )
=> ( ( member @ A @ Y @ A6 )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R ) ) ) )
=> ( X6 = Y4 ) ) ) ) ) ) ) ).
% UN_equiv_class_inject
thf(fact_1113_proj__iff,axiom,
! [A: $tType,A6: set @ A,R: set @ ( product_prod @ A @ A ),X2: A,Y3: A] :
( ( equiv_equiv @ A @ A6 @ R )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X2 @ ( insert @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) @ A6 )
=> ( ( ( equiv_proj @ A @ A @ R @ X2 )
= ( equiv_proj @ A @ A @ R @ Y3 ) )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R ) ) ) ) ).
% proj_iff
thf(fact_1114_UN__equiv__class__type2,axiom,
! [A: $tType,B: $tType,C: $tType,A13: set @ A,R12: set @ ( product_prod @ A @ A ),A24: set @ B,R23: set @ ( product_prod @ B @ B ),F: A > B > ( set @ C ),X13: set @ A,X24: set @ B,B6: set @ ( set @ C )] :
( ( equiv_equiv @ A @ A13 @ R12 )
=> ( ( equiv_equiv @ B @ A24 @ R23 )
=> ( ( equiv_congruent2 @ A @ B @ ( set @ C ) @ R12 @ R23 @ F )
=> ( ( member @ ( set @ A ) @ X13 @ ( equiv_quotient @ A @ A13 @ R12 ) )
=> ( ( member @ ( set @ B ) @ X24 @ ( equiv_quotient @ B @ A24 @ R23 ) )
=> ( ! [X14: A,X25: B] :
( ( member @ A @ X14 @ A13 )
=> ( ( member @ B @ X25 @ A24 )
=> ( member @ ( set @ C ) @ ( F @ X14 @ X25 ) @ B6 ) ) )
=> ( member @ ( set @ C )
@ ( complete_Sup_Sup @ ( set @ C )
@ ( image2 @ A @ ( set @ C )
@ ^ [X12: A] : ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ ( F @ X12 ) @ X24 ) )
@ X13 ) )
@ B6 ) ) ) ) ) ) ) ).
% UN_equiv_class_type2
thf(fact_1115_UN__equiv__class__type,axiom,
! [A: $tType,B: $tType,A6: set @ A,R: set @ ( product_prod @ A @ A ),F: A > ( set @ B ),X6: set @ A,B6: set @ ( set @ B )] :
( ( equiv_equiv @ A @ A6 @ R )
=> ( ( equiv_congruent @ A @ ( set @ B ) @ R @ F )
=> ( ( member @ ( set @ A ) @ X6 @ ( equiv_quotient @ A @ A6 @ R ) )
=> ( ! [X: A] :
( ( member @ A @ X @ A6 )
=> ( member @ ( set @ B ) @ ( F @ X ) @ B6 ) )
=> ( member @ ( set @ B ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F @ X6 ) ) @ B6 ) ) ) ) ) ).
% UN_equiv_class_type
thf(fact_1116_wf__UN,axiom,
! [B: $tType,A: $tType,I4: set @ A,R: A > ( set @ ( product_prod @ B @ B ) )] :
( ! [I6: A] :
( ( member @ A @ I6 @ I4 )
=> ( wf @ B @ ( R @ I6 ) ) )
=> ( ! [I6: A,J3: A] :
( ( member @ A @ I6 @ I4 )
=> ( ( member @ A @ J3 @ I4 )
=> ( ( ( R @ I6 )
!= ( R @ J3 ) )
=> ( ( inf_inf @ ( set @ B ) @ ( domain @ B @ B @ ( R @ I6 ) ) @ ( range2 @ B @ B @ ( R @ J3 ) ) )
= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( wf @ B @ ( complete_Sup_Sup @ ( set @ ( product_prod @ B @ B ) ) @ ( image2 @ A @ ( set @ ( product_prod @ B @ B ) ) @ R @ I4 ) ) ) ) ) ).
% wf_UN
thf(fact_1117_rtrancl__finite__eq__relpow,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R3 )
=> ( ( transitive_rtrancl @ A @ R3 )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N4: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N4 @ R3 )
@ ( collect @ nat
@ ^ [N4: nat] : ( ord_less_eq @ nat @ N4 @ ( finite_card @ ( product_prod @ A @ A ) @ R3 ) ) ) ) ) ) ) ).
% rtrancl_finite_eq_relpow
thf(fact_1118_relImage__proj,axiom,
! [A: $tType,A6: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( equiv_equiv @ A @ A6 @ R3 )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( bNF_Gr4221423524335903396lImage @ A @ ( set @ A ) @ R3 @ ( equiv_proj @ A @ A @ R3 ) ) @ ( id_on @ ( set @ A ) @ ( equiv_quotient @ A @ A6 @ R3 ) ) ) ) ).
% relImage_proj
thf(fact_1119_wf__insert,axiom,
! [A: $tType,Y3: A,X2: A,R: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X2 ) @ R ) )
= ( ( wf @ A @ R )
& ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ).
% wf_insert
thf(fact_1120_Domain__insert,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,R: set @ ( product_prod @ A @ B )] :
( ( domain @ A @ B @ ( insert @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R ) )
= ( insert @ A @ A3 @ ( domain @ A @ B @ R ) ) ) ).
% Domain_insert
thf(fact_1121_Not__Domain__rtrancl,axiom,
! [A: $tType,X2: A,R3: set @ ( product_prod @ A @ A ),Y3: A] :
( ~ ( member @ A @ X2 @ ( domain @ A @ A @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ ( transitive_rtrancl @ A @ R3 ) )
= ( X2 = Y3 ) ) ) ).
% Not_Domain_rtrancl
thf(fact_1122_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 )
=> ~ ! [B4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ B4 ) @ ( transitive_rtrancl @ A @ R ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ A22 ) @ R ) ) ) ) ).
% rtrancl.cases
thf(fact_1123_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,C6: A] :
( ( A1 = A7 )
& ( A22 = C6 )
& ( 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 @ C6 ) @ R ) ) ) ) ).
% rtrancl.simps
thf(fact_1124_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_1125_rtrancl_Ortrancl__into__rtrancl,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),C3: 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 @ C3 ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C3 ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ).
% rtrancl.rtrancl_into_rtrancl
thf(fact_1126_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 )
=> ~ ! [Y: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ Y ) @ ( transitive_rtrancl @ A @ R ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ B2 ) @ R ) ) ) ) ).
% rtranclE
thf(fact_1127_rtrancl__trans,axiom,
! [A: $tType,X2: A,Y3: A,R: set @ ( product_prod @ A @ A ),Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Z3 ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ).
% rtrancl_trans
thf(fact_1128_rtrancl__induct,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),P2: A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( P2 @ A3 )
=> ( ! [Y: A,Z4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ Y ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z4 ) @ R )
=> ( ( P2 @ Y )
=> ( P2 @ Z4 ) ) ) )
=> ( P2 @ B2 ) ) ) ) ).
% rtrancl_induct
thf(fact_1129_converse__rtranclE,axiom,
! [A: $tType,X2: A,Z3: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Z3 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( X2 != Z3 )
=> ~ ! [Y: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z3 ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ) ).
% converse_rtranclE
thf(fact_1130_converse__rtrancl__induct,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),P2: A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( P2 @ B2 )
=> ( ! [Y: A,Z4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z4 ) @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z4 @ B2 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( P2 @ Z4 )
=> ( P2 @ Y ) ) ) )
=> ( P2 @ A3 ) ) ) ) ).
% converse_rtrancl_induct
thf(fact_1131_converse__rtrancl__into__rtrancl,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),C3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C3 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C3 ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ).
% converse_rtrancl_into_rtrancl
thf(fact_1132_Domain__iff,axiom,
! [A: $tType,B: $tType,A3: A,R: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A3 @ ( domain @ A @ B @ R ) )
= ( ? [Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ Y5 ) @ R ) ) ) ).
% Domain_iff
thf(fact_1133_DomainE,axiom,
! [B: $tType,A: $tType,A3: A,R: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A3 @ ( domain @ A @ B @ R ) )
=> ~ ! [B4: B] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B4 ) @ R ) ) ).
% DomainE
thf(fact_1134_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_1135_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_1136_Domain_Ocases,axiom,
! [B: $tType,A: $tType,A3: A,R: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A3 @ ( domain @ A @ B @ R ) )
=> ~ ! [B4: B] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B4 ) @ R ) ) ).
% Domain.cases
thf(fact_1137_rtrancl__converseD,axiom,
! [A: $tType,X2: A,Y3: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ ( transitive_rtrancl @ A @ ( converse @ A @ A @ R ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X2 ) @ ( transitive_rtrancl @ A @ R ) ) ) ).
% rtrancl_converseD
thf(fact_1138_rtrancl__converseI,axiom,
! [A: $tType,Y3: A,X2: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X2 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ ( transitive_rtrancl @ A @ ( converse @ A @ A @ R ) ) ) ) ).
% rtrancl_converseI
thf(fact_1139_rtrancl__Un__separator__converseE,axiom,
! [A: $tType,A3: A,B2: A,P2: set @ ( product_prod @ A @ A ),Q2: 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 ) ) @ P2 @ Q2 ) ) )
=> ( ! [X: A,Y: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ B2 ) @ ( transitive_rtrancl @ A @ P2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ Q2 )
=> ( Y = X ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ P2 ) ) ) ) ).
% rtrancl_Un_separator_converseE
thf(fact_1140_rtrancl__Un__separatorE,axiom,
! [A: $tType,A3: A,B2: A,P2: set @ ( product_prod @ A @ A ),Q2: 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 ) ) @ P2 @ Q2 ) ) )
=> ( ! [X: A,Y: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ X ) @ ( transitive_rtrancl @ A @ P2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ Q2 )
=> ( X = Y ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ P2 ) ) ) ) ).
% rtrancl_Un_separatorE
thf(fact_1141_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 ) ),P2: 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 ) )
=> ( ( P2 @ Ax @ Ay )
=> ( ! [A5: A,B4: B,Aa2: 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 @ B4 ) ) @ ( 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 @ B4 ) @ ( product_Pair @ A @ B @ Aa2 @ Ba ) ) @ R )
=> ( ( P2 @ A5 @ B4 )
=> ( P2 @ Aa2 @ Ba ) ) ) )
=> ( P2 @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_1142_converse__rtranclE2,axiom,
! [B: $tType,A: $tType,Xa2: 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 @ Xa2 @ Xb ) @ ( product_Pair @ A @ B @ Za2 @ Zb ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R ) )
=> ( ( ( product_Pair @ A @ B @ Xa2 @ Xb )
!= ( product_Pair @ A @ B @ Za2 @ Zb ) )
=> ~ ! [A5: A,B4: 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 @ Xa2 @ Xb ) @ ( product_Pair @ A @ B @ A5 @ B4 ) ) @ 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 @ B4 ) @ ( product_Pair @ A @ B @ Za2 @ Zb ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_1143_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 ) ),P2: 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 ) )
=> ( ( P2 @ Bx @ By )
=> ( ! [A5: A,B4: B,Aa2: 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 @ B4 ) @ ( product_Pair @ A @ B @ Aa2 @ 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 @ Aa2 @ Ba ) @ ( product_Pair @ A @ B @ Bx @ By ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R ) )
=> ( ( P2 @ Aa2 @ Ba )
=> ( P2 @ A5 @ B4 ) ) ) )
=> ( P2 @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_1144_single__valued__confluent,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),X2: A,Y3: A,Z3: A] :
( ( single_valued @ A @ A @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Z3 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ ( transitive_rtrancl @ A @ R ) )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z3 @ Y3 ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ) ) ).
% single_valued_confluent
thf(fact_1145_congruent2I_H,axiom,
! [C: $tType,B: $tType,A: $tType,R12: set @ ( product_prod @ A @ A ),R23: set @ ( product_prod @ B @ B ),F: A > B > C] :
( ! [Y12: A,Z1: A,Y23: B,Z22: B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y12 @ Z1 ) @ R12 )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y23 @ Z22 ) @ R23 )
=> ( ( F @ Y12 @ Y23 )
= ( F @ Z1 @ Z22 ) ) ) )
=> ( equiv_congruent2 @ A @ B @ C @ R12 @ R23 @ F ) ) ).
% congruent2I'
thf(fact_1146_congruent2D,axiom,
! [A: $tType,C: $tType,B: $tType,R12: set @ ( product_prod @ A @ A ),R23: set @ ( product_prod @ B @ B ),F: A > B > C,Y1: A,Z12: A,Y22: B,Z23: B] :
( ( equiv_congruent2 @ A @ B @ C @ R12 @ R23 @ F )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y1 @ Z12 ) @ R12 )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y22 @ Z23 ) @ R23 )
=> ( ( F @ Y1 @ Y22 )
= ( F @ Z12 @ Z23 ) ) ) ) ) ).
% congruent2D
thf(fact_1147_congruentI,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),F: A > B] :
( ! [Y: A,Z4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z4 ) @ R )
=> ( ( F @ Y )
= ( F @ Z4 ) ) )
=> ( equiv_congruent @ A @ B @ R @ F ) ) ).
% congruentI
thf(fact_1148_congruentD,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),F: A > B,Y3: A,Z3: A] :
( ( equiv_congruent @ A @ B @ R @ F )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ R )
=> ( ( F @ Y3 )
= ( F @ Z3 ) ) ) ) ).
% congruentD
thf(fact_1149_fst__eq__Domain,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B )] :
( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ R3 )
= ( domain @ A @ B @ R3 ) ) ).
% fst_eq_Domain
thf(fact_1150_Domain__fst,axiom,
! [B: $tType,A: $tType] :
( ( domain @ A @ B )
= ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) ) ) ).
% Domain_fst
thf(fact_1151_trancl__subset__Sigma__aux,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),A6: 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 @ A6
@ ^ [Uu: A] : A6 ) )
=> ( ( A3 = B2 )
| ( member @ A @ A3 @ A6 ) ) ) ) ).
% trancl_subset_Sigma_aux
thf(fact_1152_proj__image,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A6: set @ A] :
( ( image2 @ A @ ( set @ A ) @ ( equiv_proj @ A @ A @ R ) @ A6 )
= ( equiv_quotient @ A @ A6 @ R ) ) ).
% proj_image
thf(fact_1153_rtrancl__insert,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A )] :
( ( transitive_rtrancl @ A @ ( insert @ ( 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
@ ^ [X4: A,Y5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ A3 ) @ ( transitive_rtrancl @ A @ R ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ Y5 ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ) ) ) ).
% rtrancl_insert
thf(fact_1154_congruent2__commuteI,axiom,
! [B: $tType,A: $tType,A6: set @ A,R: set @ ( product_prod @ A @ A ),F: A > A > B] :
( ( equiv_equiv @ A @ A6 @ R )
=> ( ! [Y: A,Z4: A] :
( ( member @ A @ Y @ A6 )
=> ( ( member @ A @ Z4 @ A6 )
=> ( ( F @ Y @ Z4 )
= ( F @ Z4 @ Y ) ) ) )
=> ( ! [Y: A,Z4: A,W2: A] :
( ( member @ A @ W2 @ A6 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z4 ) @ R )
=> ( ( F @ W2 @ Y )
= ( F @ W2 @ Z4 ) ) ) )
=> ( equiv_congruent2 @ A @ A @ B @ R @ R @ F ) ) ) ) ).
% congruent2_commuteI
thf(fact_1155_congruent2I,axiom,
! [C: $tType,B: $tType,A: $tType,A13: set @ A,R12: set @ ( product_prod @ A @ A ),A24: set @ B,R23: set @ ( product_prod @ B @ B ),F: A > B > C] :
( ( equiv_equiv @ A @ A13 @ R12 )
=> ( ( equiv_equiv @ B @ A24 @ R23 )
=> ( ! [Y: A,Z4: A,W2: B] :
( ( member @ B @ W2 @ A24 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z4 ) @ R12 )
=> ( ( F @ Y @ W2 )
= ( F @ Z4 @ W2 ) ) ) )
=> ( ! [Y: B,Z4: B,W2: A] :
( ( member @ A @ W2 @ A13 )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y @ Z4 ) @ R23 )
=> ( ( F @ W2 @ Y )
= ( F @ W2 @ Z4 ) ) ) )
=> ( equiv_congruent2 @ A @ B @ C @ R12 @ R23 @ F ) ) ) ) ) ).
% congruent2I
thf(fact_1156_rtrancl__is__UN__relpow,axiom,
! [A: $tType] :
( ( transitive_rtrancl @ A )
= ( ^ [R6: set @ ( product_prod @ A @ A )] :
( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N4: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N4 @ R6 )
@ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% rtrancl_is_UN_relpow
thf(fact_1157_rtrancl__imp__UN__relpow,axiom,
! [A: $tType,P: product_prod @ A @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ P @ ( transitive_rtrancl @ A @ R3 ) )
=> ( member @ ( product_prod @ A @ A ) @ P
@ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N4: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N4 @ R3 )
@ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% rtrancl_imp_UN_relpow
thf(fact_1158_Domain__Union,axiom,
! [B: $tType,A: $tType,S5: set @ ( set @ ( product_prod @ A @ B ) )] :
( ( domain @ A @ B @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ S5 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( set @ ( product_prod @ A @ B ) ) @ ( set @ A ) @ ( domain @ A @ B ) @ S5 ) ) ) ).
% Domain_Union
thf(fact_1159_wf__map__prod__image__Dom__Ran,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),F: A > B] :
( ( wf @ A @ R )
=> ( ! [A5: A,A16: A] :
( ( member @ A @ A5 @ ( domain @ A @ A @ R ) )
=> ( ( member @ A @ A16 @ ( range2 @ A @ A @ R ) )
=> ( ( ( F @ A5 )
= ( F @ A16 ) )
=> ( A5 = A16 ) ) ) )
=> ( wf @ B @ ( image2 @ ( product_prod @ A @ A ) @ ( product_prod @ B @ B ) @ ( product_map_prod @ A @ B @ A @ B @ F @ F ) @ R ) ) ) ) ).
% wf_map_prod_image_Dom_Ran
thf(fact_1160_congruent2__implies__congruent__UN,axiom,
! [B: $tType,C: $tType,A: $tType,A13: set @ A,R12: set @ ( product_prod @ A @ A ),A24: set @ B,R23: set @ ( product_prod @ B @ B ),F: A > B > ( set @ C ),A3: B] :
( ( equiv_equiv @ A @ A13 @ R12 )
=> ( ( equiv_equiv @ B @ A24 @ R23 )
=> ( ( equiv_congruent2 @ A @ B @ ( set @ C ) @ R12 @ R23 @ F )
=> ( ( member @ B @ A3 @ A24 )
=> ( equiv_congruent @ A @ ( set @ C ) @ R12
@ ^ [X12: A] : ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ ( F @ X12 ) @ ( image @ B @ B @ R23 @ ( insert @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ) ).
% congruent2_implies_congruent_UN
thf(fact_1161_equiv__proj,axiom,
! [A: $tType,A6: set @ A,R3: set @ ( product_prod @ A @ A ),Z3: product_prod @ A @ A] :
( ( equiv_equiv @ A @ A6 @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ Z3 @ R3 )
=> ( ( comp @ A @ ( set @ A ) @ ( product_prod @ A @ A ) @ ( equiv_proj @ A @ A @ R3 ) @ ( product_fst @ A @ A ) @ Z3 )
= ( comp @ A @ ( set @ A ) @ ( product_prod @ A @ A ) @ ( equiv_proj @ A @ A @ R3 ) @ ( product_snd @ A @ A ) @ Z3 ) ) ) ) ).
% equiv_proj
thf(fact_1162_univ__commute,axiom,
! [B: $tType,A: $tType,A6: set @ A,R: set @ ( product_prod @ A @ A ),F: A > B,X2: A] :
( ( equiv_equiv @ A @ A6 @ R )
=> ( ( equiv_congruent @ A @ B @ R @ F )
=> ( ( member @ A @ X2 @ A6 )
=> ( ( bNF_Greatest_univ @ A @ B @ F @ ( equiv_proj @ A @ A @ R @ X2 ) )
= ( F @ X2 ) ) ) ) ) ).
% univ_commute
thf(fact_1163_univ__preserves,axiom,
! [A: $tType,B: $tType,A6: set @ A,R: set @ ( product_prod @ A @ A ),F: A > B,B6: set @ B] :
( ( equiv_equiv @ A @ A6 @ R )
=> ( ( equiv_congruent @ A @ B @ R @ F )
=> ( ! [X: A] :
( ( member @ A @ X @ A6 )
=> ( member @ B @ ( F @ X ) @ B6 ) )
=> ! [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ ( equiv_quotient @ A @ A6 @ R ) )
=> ( member @ B @ ( bNF_Greatest_univ @ A @ B @ F @ X5 ) @ B6 ) ) ) ) ) ).
% univ_preserves
thf(fact_1164_acyclic__insert,axiom,
! [A: $tType,Y3: A,X2: A,R: set @ ( product_prod @ A @ A )] :
( ( transitive_acyclic @ A @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X2 ) @ R ) )
= ( ( transitive_acyclic @ A @ R )
& ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ).
% acyclic_insert
thf(fact_1165_disjnt__equiv__class,axiom,
! [A: $tType,A6: set @ A,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( equiv_equiv @ A @ A6 @ R )
=> ( ( disjnt @ A @ ( image @ A @ A @ R @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R ) ) ) ) ).
% disjnt_equiv_class
thf(fact_1166_trancl__insert,axiom,
! [A: $tType,Y3: A,X2: A,R: set @ ( product_prod @ A @ A )] :
( ( transitive_trancl @ A @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X2 ) @ 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 @ Y3 ) @ ( transitive_rtrancl @ A @ R ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ B5 ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ) ) ) ).
% trancl_insert
thf(fact_1167_rel__filter_Ointros,axiom,
! [A: $tType,B: $tType,R3: A > B > $o,Z6: filter @ ( product_prod @ A @ B ),F7: filter @ A,G6: filter @ B] :
( ( eventually @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R3 ) @ Z6 )
=> ( ( ( map_filter_on @ ( product_prod @ A @ B ) @ A @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R3 ) ) @ ( product_fst @ A @ B ) @ Z6 )
= F7 )
=> ( ( ( map_filter_on @ ( product_prod @ A @ B ) @ B @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R3 ) ) @ ( product_snd @ A @ B ) @ Z6 )
= G6 )
=> ( rel_filter @ A @ B @ R3 @ F7 @ G6 ) ) ) ) ).
% rel_filter.intros
thf(fact_1168_rel__filter_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( rel_filter @ A @ B )
= ( ^ [R6: A > B > $o,F8: filter @ A,G7: filter @ B] :
? [Z8: filter @ ( product_prod @ A @ B )] :
( ( eventually @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) @ Z8 )
& ( ( map_filter_on @ ( product_prod @ A @ B ) @ A @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) @ ( product_fst @ A @ B ) @ Z8 )
= F8 )
& ( ( map_filter_on @ ( product_prod @ A @ B ) @ B @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) @ ( product_snd @ A @ B ) @ Z8 )
= G7 ) ) ) ) ).
% rel_filter.simps
thf(fact_1169_rel__filter_Ocases,axiom,
! [A: $tType,B: $tType,R3: A > B > $o,F7: filter @ A,G6: filter @ B] :
( ( rel_filter @ A @ B @ R3 @ F7 @ G6 )
=> ~ ! [Z9: filter @ ( product_prod @ A @ B )] :
( ( eventually @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R3 ) @ Z9 )
=> ( ( ( map_filter_on @ ( product_prod @ A @ B ) @ A @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R3 ) ) @ ( product_fst @ A @ B ) @ Z9 )
= F7 )
=> ( ( map_filter_on @ ( product_prod @ A @ B ) @ B @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R3 ) ) @ ( product_snd @ A @ B ) @ Z9 )
!= G6 ) ) ) ) ).
% rel_filter.cases
thf(fact_1170_disjnt__Times1__iff,axiom,
! [A: $tType,B: $tType,C5: set @ A,A6: set @ B,B6: set @ B] :
( ( disjnt @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ C5
@ ^ [Uu: A] : A6 )
@ ( product_Sigma @ A @ B @ C5
@ ^ [Uu: A] : B6 ) )
= ( ( C5
= ( bot_bot @ ( set @ A ) ) )
| ( disjnt @ B @ A6 @ B6 ) ) ) ).
% disjnt_Times1_iff
thf(fact_1171_disjnt__Times2__iff,axiom,
! [B: $tType,A: $tType,A6: set @ A,C5: set @ B,B6: set @ A] :
( ( disjnt @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu: A] : C5 )
@ ( product_Sigma @ A @ B @ B6
@ ^ [Uu: A] : C5 ) )
= ( ( C5
= ( bot_bot @ ( set @ B ) ) )
| ( disjnt @ A @ A6 @ B6 ) ) ) ).
% disjnt_Times2_iff
thf(fact_1172_acyclicP__converse,axiom,
! [A: $tType,R: A > A > $o] :
( ( transitive_acyclic @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ ( conversep @ A @ A @ R ) ) ) )
= ( transitive_acyclic @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R ) ) ) ) ).
% acyclicP_converse
thf(fact_1173_acyclicI,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ! [X: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ ( transitive_trancl @ A @ R ) )
=> ( transitive_acyclic @ A @ R ) ) ).
% acyclicI
thf(fact_1174_acyclic__def,axiom,
! [A: $tType] :
( ( transitive_acyclic @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
! [X4: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) @ ( transitive_trancl @ A @ R5 ) ) ) ) ).
% acyclic_def
thf(fact_1175_converse__trancl__induct,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),P2: A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R ) )
=> ( ! [Y: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ B2 ) @ R )
=> ( P2 @ Y ) )
=> ( ! [Y: A,Z4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z4 ) @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z4 @ B2 ) @ ( transitive_trancl @ A @ R ) )
=> ( ( P2 @ Z4 )
=> ( P2 @ Y ) ) ) )
=> ( P2 @ A3 ) ) ) ) ).
% converse_trancl_induct
thf(fact_1176_trancl__trans__induct,axiom,
! [A: $tType,X2: A,Y3: A,R: set @ ( product_prod @ A @ A ),P2: A > A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ ( transitive_trancl @ A @ R ) )
=> ( ! [X: A,Y: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
=> ( P2 @ X @ Y ) )
=> ( ! [X: A,Y: A,Z4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ R ) )
=> ( ( P2 @ X @ Y )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z4 ) @ ( transitive_trancl @ A @ R ) )
=> ( ( P2 @ Y @ Z4 )
=> ( P2 @ X @ Z4 ) ) ) ) )
=> ( P2 @ X2 @ Y3 ) ) ) ) ).
% trancl_trans_induct
thf(fact_1177_trancl__into__trancl2,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),C3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C3 ) @ ( transitive_trancl @ A @ R ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C3 ) @ ( transitive_trancl @ A @ R ) ) ) ) ).
% trancl_into_trancl2
thf(fact_1178_Transitive__Closure_Otrancl__into__trancl,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),C3: 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 @ C3 ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C3 ) @ ( transitive_trancl @ A @ R ) ) ) ) ).
% Transitive_Closure.trancl_into_trancl
thf(fact_1179_irrefl__trancl__rD,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),X2: A,Y3: A] :
( ! [X: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ ( transitive_trancl @ A @ R ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R )
=> ( X2 != Y3 ) ) ) ).
% irrefl_trancl_rD
thf(fact_1180_converse__tranclE,axiom,
! [A: $tType,X2: A,Z3: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Z3 ) @ ( transitive_trancl @ A @ R ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Z3 ) @ R )
=> ~ ! [Y: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z3 ) @ ( transitive_trancl @ A @ R ) ) ) ) ) ).
% converse_tranclE
thf(fact_1181_r__r__into__trancl,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A ),C3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C3 ) @ R3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C3 ) @ ( transitive_trancl @ A @ R3 ) ) ) ) ).
% r_r_into_trancl
thf(fact_1182_trancl__induct,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),P2: A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R ) )
=> ( ! [Y: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ Y ) @ R )
=> ( P2 @ Y ) )
=> ( ! [Y: A,Z4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ Y ) @ ( transitive_trancl @ A @ R ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z4 ) @ R )
=> ( ( P2 @ Y )
=> ( P2 @ Z4 ) ) ) )
=> ( P2 @ B2 ) ) ) ) ).
% trancl_induct
thf(fact_1183_trancl__trans,axiom,
! [A: $tType,X2: A,Y3: A,R: set @ ( product_prod @ A @ A ),Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ ( transitive_trancl @ A @ R ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ ( transitive_trancl @ A @ R ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Z3 ) @ ( transitive_trancl @ A @ R ) ) ) ) ).
% trancl_trans
thf(fact_1184_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 )
=> ~ ! [C2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C2 ) @ ( transitive_trancl @ A @ R ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ C2 @ B2 ) @ R ) ) ) ) ).
% tranclE
thf(fact_1185_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_1186_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,C6: A] :
( ( A1 = A7 )
& ( A22 = C6 )
& ( 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 @ C6 ) @ R ) ) ) ) ).
% trancl.simps
thf(fact_1187_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 )
=> ~ ! [B4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ B4 ) @ ( transitive_trancl @ A @ R ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ A22 ) @ R ) ) ) ) ).
% trancl.cases
thf(fact_1188_rel__filter__distr,axiom,
! [A: $tType,B: $tType,C: $tType,A6: A > C > $o,B6: C > B > $o] :
( ( relcompp @ ( filter @ A ) @ ( filter @ C ) @ ( filter @ B ) @ ( rel_filter @ A @ C @ A6 ) @ ( rel_filter @ C @ B @ B6 ) )
= ( rel_filter @ A @ B @ ( relcompp @ A @ C @ B @ A6 @ B6 ) ) ) ).
% rel_filter_distr
thf(fact_1189_disjnt__Sigma__iff,axiom,
! [B: $tType,A: $tType,A6: set @ A,C5: A > ( set @ B ),B6: set @ A] :
( ( disjnt @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ A6 @ C5 ) @ ( product_Sigma @ A @ B @ B6 @ C5 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( inf_inf @ ( set @ A ) @ A6 @ B6 ) )
=> ( ( C5 @ X4 )
= ( bot_bot @ ( set @ B ) ) ) )
| ( disjnt @ A @ A6 @ B6 ) ) ) ).
% disjnt_Sigma_iff
thf(fact_1190_trancl__rtrancl__trancl,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),C3: 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 @ C3 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C3 ) @ ( transitive_trancl @ A @ R ) ) ) ) ).
% trancl_rtrancl_trancl
thf(fact_1191_rtrancl__trancl__trancl,axiom,
! [A: $tType,X2: A,Y3: A,R: set @ ( product_prod @ A @ A ),Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ ( transitive_trancl @ A @ R ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Z3 ) @ ( transitive_trancl @ A @ R ) ) ) ) ).
% rtrancl_trancl_trancl
thf(fact_1192_rtrancl__into__trancl2,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),C3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C3 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C3 ) @ ( transitive_trancl @ A @ R ) ) ) ) ).
% rtrancl_into_trancl2
thf(fact_1193_rtrancl__into__trancl1,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),C3: 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 @ C3 ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C3 ) @ ( transitive_trancl @ A @ R ) ) ) ) ).
% rtrancl_into_trancl1
thf(fact_1194_rtrancl__eq__or__trancl,axiom,
! [A: $tType,X2: A,Y3: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ ( transitive_rtrancl @ A @ R3 ) )
= ( ( X2 = Y3 )
| ( ( X2 != Y3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ ( transitive_trancl @ A @ R3 ) ) ) ) ) ).
% rtrancl_eq_or_trancl
thf(fact_1195_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_1196_tranclD2,axiom,
! [A: $tType,X2: A,Y3: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ ( transitive_trancl @ A @ R3 ) )
=> ? [Z4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Z4 ) @ ( transitive_rtrancl @ A @ R3 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z4 @ Y3 ) @ R3 ) ) ) ).
% tranclD2
thf(fact_1197_rtranclD,axiom,
! [A: $tType,A3: A,B2: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( ( A3 = B2 )
| ( ( A3 != B2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R3 ) ) ) ) ) ).
% rtranclD
thf(fact_1198_tranclD,axiom,
! [A: $tType,X2: A,Y3: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ ( transitive_trancl @ A @ R3 ) )
=> ? [Z4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Z4 ) @ R3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z4 @ Y3 ) @ ( transitive_rtrancl @ A @ R3 ) ) ) ) ).
% tranclD
thf(fact_1199_trancl__converseI,axiom,
! [A: $tType,X2: A,Y3: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ ( converse @ A @ A @ ( transitive_trancl @ A @ R ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ ( transitive_trancl @ A @ ( converse @ A @ A @ R ) ) ) ) ).
% trancl_converseI
thf(fact_1200_trancl__converseD,axiom,
! [A: $tType,X2: A,Y3: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ ( transitive_trancl @ A @ ( converse @ A @ A @ R ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ ( converse @ A @ A @ ( transitive_trancl @ A @ R ) ) ) ) ).
% trancl_converseD
thf(fact_1201_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 ) ),P2: 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,B4: 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 @ B4 ) ) @ R )
=> ( P2 @ A5 @ B4 ) )
=> ( ! [A5: A,B4: B,Aa2: 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 @ B4 ) ) @ ( 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 @ B4 ) @ ( product_Pair @ A @ B @ Aa2 @ Ba ) ) @ R )
=> ( ( P2 @ A5 @ B4 )
=> ( P2 @ Aa2 @ Ba ) ) ) )
=> ( P2 @ Bx @ By ) ) ) ) ).
% trancl_induct2
thf(fact_1202_disjoint__UN__iff,axiom,
! [A: $tType,B: $tType,A6: set @ A,B6: B > ( set @ A ),I4: set @ B] :
( ( disjnt @ A @ A6 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B6 @ I4 ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ I4 )
=> ( disjnt @ A @ A6 @ ( B6 @ X4 ) ) ) ) ) ).
% disjoint_UN_iff
thf(fact_1203_disjnt__inj__on__iff,axiom,
! [B: $tType,A: $tType,F: A > B,A17: set @ ( set @ A ),X6: set @ A,Y4: set @ A] :
( ( inj_on @ A @ B @ F @ ( complete_Sup_Sup @ ( set @ A ) @ A17 ) )
=> ( ( member @ ( set @ A ) @ X6 @ A17 )
=> ( ( member @ ( set @ A ) @ Y4 @ A17 )
=> ( ( disjnt @ B @ ( image2 @ A @ B @ F @ X6 ) @ ( image2 @ A @ B @ F @ Y4 ) )
= ( disjnt @ A @ X6 @ Y4 ) ) ) ) ) ).
% disjnt_inj_on_iff
thf(fact_1204_trancl__insert2,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A )] :
( ( transitive_trancl @ A @ ( insert @ ( 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
@ ^ [X4: A,Y5: A] :
( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ A3 ) @ ( transitive_trancl @ A @ R ) )
| ( X4 = A3 ) )
& ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ Y5 ) @ ( transitive_trancl @ A @ R ) )
| ( Y5 = B2 ) ) ) ) ) ) ) ).
% trancl_insert2
thf(fact_1205_irrefl__tranclI,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),X2: 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 @ X2 @ X2 ) @ ( transitive_trancl @ A @ R ) ) ) ).
% irrefl_tranclI
thf(fact_1206_wf__linord__ex__has__least,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),P2: B > $o,K: B,M4: B > A] :
( ( wf @ A @ R )
=> ( ! [X: A,Y: 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 @ X ) @ ( transitive_rtrancl @ A @ R ) ) ) )
=> ( ( P2 @ K )
=> ? [X: B] :
( ( P2 @ X )
& ! [Y7: B] :
( ( P2 @ Y7 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( M4 @ X ) @ ( M4 @ Y7 ) ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ) ) ) ).
% wf_linord_ex_has_least
thf(fact_1207_Range__def,axiom,
! [B: $tType,A: $tType] :
( ( range2 @ A @ B )
= ( ^ [R5: set @ ( product_prod @ A @ B )] :
( collect @ B
@ ( rangep @ A @ B
@ ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ R5 ) ) ) ) ) ).
% Range_def
thf(fact_1208_Rangep__Range__eq,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B )] :
( ( rangep @ A @ B
@ ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ R ) )
= ( ^ [X4: B] : ( member @ B @ X4 @ ( range2 @ A @ B @ R ) ) ) ) ).
% Rangep_Range_eq
thf(fact_1209_Least__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F: A > B,S5: set @ A] :
( ( order_mono @ A @ B @ F )
=> ( ? [X5: A] :
( ( member @ A @ X5 @ S5 )
& ! [Xa: A] :
( ( member @ A @ Xa @ S5 )
=> ( ord_less_eq @ A @ X5 @ Xa ) ) )
=> ( ( ord_Least @ B
@ ^ [Y5: B] : ( member @ B @ Y5 @ ( image2 @ A @ B @ F @ S5 ) ) )
= ( F
@ ( ord_Least @ A
@ ^ [X4: A] : ( member @ A @ X4 @ S5 ) ) ) ) ) ) ) ).
% Least_mono
thf(fact_1210_filtercomap__SUP,axiom,
! [A: $tType,C: $tType,B: $tType,F: A > C,F7: B > ( filter @ C ),B6: set @ B] :
( ord_less_eq @ ( filter @ A )
@ ( complete_Sup_Sup @ ( filter @ A )
@ ( image2 @ B @ ( filter @ A )
@ ^ [B5: B] : ( filtercomap @ A @ C @ F @ ( F7 @ B5 ) )
@ B6 ) )
@ ( filtercomap @ A @ C @ F @ ( complete_Sup_Sup @ ( filter @ C ) @ ( image2 @ B @ ( filter @ C ) @ F7 @ B6 ) ) ) ) ).
% filtercomap_SUP
thf(fact_1211_bind__singleton__conv__image,axiom,
! [A: $tType,B: $tType,A6: set @ B,F: B > A] :
( ( bind4 @ B @ A @ A6
@ ^ [X4: B] : ( insert @ A @ ( F @ X4 ) @ ( bot_bot @ ( set @ A ) ) ) )
= ( image2 @ B @ A @ F @ A6 ) ) ).
% bind_singleton_conv_image
thf(fact_1212_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_1213_finite__subsets__at__top__def,axiom,
! [A: $tType] :
( ( finite5375528669736107172at_top @ A )
= ( ^ [A8: set @ A] :
( complete_Inf_Inf @ ( filter @ ( set @ A ) )
@ ( image2 @ ( set @ A ) @ ( filter @ ( set @ A ) )
@ ^ [X10: set @ A] :
( principal @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [Y9: set @ A] :
( ( finite_finite2 @ A @ Y9 )
& ( ord_less_eq @ ( set @ A ) @ X10 @ Y9 )
& ( ord_less_eq @ ( set @ A ) @ Y9 @ A8 ) ) ) )
@ ( collect @ ( set @ A )
@ ^ [X10: set @ A] :
( ( finite_finite2 @ A @ X10 )
& ( ord_less_eq @ ( set @ A ) @ X10 @ A8 ) ) ) ) ) ) ) ).
% finite_subsets_at_top_def
thf(fact_1214_cINF__less__iff,axiom,
! [A: $tType,B: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [A6: set @ B,F: B > A,A3: A] :
( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F @ A6 ) )
=> ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A6 ) ) @ A3 )
= ( ? [X4: B] :
( ( member @ B @ X4 @ A6 )
& ( ord_less @ A @ ( F @ X4 ) @ A3 ) ) ) ) ) ) ) ).
% cINF_less_iff
thf(fact_1215_member__bind,axiom,
! [A: $tType,B: $tType,X2: A,A6: set @ B,F: B > ( set @ A )] :
( ( member @ A @ X2 @ ( bind4 @ B @ A @ A6 @ F ) )
= ( member @ A @ X2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ F @ A6 ) ) ) ) ).
% member_bind
thf(fact_1216_SUP__eq__top__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F: B > A,A6: set @ B] :
( ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A6 ) )
= ( top_top @ A ) )
= ( ! [X4: A] :
( ( ord_less @ A @ X4 @ ( top_top @ A ) )
=> ? [Y5: B] :
( ( member @ B @ Y5 @ A6 )
& ( ord_less @ A @ X4 @ ( F @ Y5 ) ) ) ) ) ) ) ).
% SUP_eq_top_iff
thf(fact_1217_INF__eq__bot__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F: B > A,A6: set @ B] :
( ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A6 ) )
= ( bot_bot @ A ) )
= ( ! [X4: A] :
( ( ord_less @ A @ ( bot_bot @ A ) @ X4 )
=> ? [Y5: B] :
( ( member @ B @ Y5 @ A6 )
& ( ord_less @ A @ ( F @ Y5 ) @ X4 ) ) ) ) ) ) ).
% INF_eq_bot_iff
thf(fact_1218_nat__descend__induct,axiom,
! [N5: nat,P2: nat > $o,M4: nat] :
( ! [K4: nat] :
( ( ord_less @ nat @ N5 @ K4 )
=> ( P2 @ K4 ) )
=> ( ! [K4: nat] :
( ( ord_less_eq @ nat @ K4 @ N5 )
=> ( ! [I7: nat] :
( ( ord_less @ nat @ K4 @ I7 )
=> ( P2 @ I7 ) )
=> ( P2 @ K4 ) ) )
=> ( P2 @ M4 ) ) ) ).
% nat_descend_induct
thf(fact_1219_underS__I,axiom,
! [A: $tType,I: A,J5: A,R3: set @ ( product_prod @ A @ A )] :
( ( I != J5 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J5 ) @ R3 )
=> ( member @ A @ I @ ( order_underS @ A @ R3 @ J5 ) ) ) ) ).
% underS_I
thf(fact_1220_underS__E,axiom,
! [A: $tType,I: A,R3: set @ ( product_prod @ A @ A ),J5: A] :
( ( member @ A @ I @ ( order_underS @ A @ R3 @ J5 ) )
=> ( ( I != J5 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J5 ) @ R3 ) ) ) ).
% underS_E
thf(fact_1221_BNF__Least__Fixpoint_OunderS__Field,axiom,
! [A: $tType,I: A,R3: set @ ( product_prod @ A @ A ),J5: A] :
( ( member @ A @ I @ ( order_underS @ A @ R3 @ J5 ) )
=> ( member @ A @ I @ ( field2 @ A @ R3 ) ) ) ).
% BNF_Least_Fixpoint.underS_Field
thf(fact_1222_finite__imp__inj__to__nat__seg,axiom,
! [A: $tType,A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ? [F3: A > nat,N2: nat] :
( ( ( image2 @ A @ nat @ F3 @ A6 )
= ( collect @ nat
@ ^ [I3: nat] : ( ord_less @ nat @ I3 @ N2 ) ) )
& ( inj_on @ A @ nat @ F3 @ A6 ) ) ) ).
% finite_imp_inj_to_nat_seg
thf(fact_1223_Field__natLeq__on,axiom,
! [N5: nat] :
( ( field2 @ nat
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X4: nat,Y5: nat] :
( ( ord_less @ nat @ X4 @ N5 )
& ( ord_less @ nat @ Y5 @ N5 )
& ( ord_less_eq @ nat @ X4 @ Y5 ) ) ) ) )
= ( collect @ nat
@ ^ [X4: nat] : ( ord_less @ nat @ X4 @ N5 ) ) ) ).
% Field_natLeq_on
thf(fact_1224_natLeq__on__well__order__on,axiom,
! [N5: nat] :
( order_well_order_on @ nat
@ ( collect @ nat
@ ^ [X4: nat] : ( ord_less @ nat @ X4 @ N5 ) )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X4: nat,Y5: nat] :
( ( ord_less @ nat @ X4 @ N5 )
& ( ord_less @ nat @ Y5 @ N5 )
& ( ord_less_eq @ nat @ X4 @ Y5 ) ) ) ) ) ).
% natLeq_on_well_order_on
thf(fact_1225_natLeq__on__Well__order,axiom,
! [N5: nat] :
( order_well_order_on @ nat
@ ( field2 @ nat
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X4: nat,Y5: nat] :
( ( ord_less @ nat @ X4 @ N5 )
& ( ord_less @ nat @ Y5 @ N5 )
& ( ord_less_eq @ nat @ X4 @ Y5 ) ) ) ) )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X4: nat,Y5: nat] :
( ( ord_less @ nat @ X4 @ N5 )
& ( ord_less @ nat @ Y5 @ N5 )
& ( ord_less_eq @ nat @ X4 @ Y5 ) ) ) ) ) ).
% natLeq_on_Well_order
thf(fact_1226_natLeq__on__wo__rel,axiom,
! [N5: nat] :
( bNF_Wellorder_wo_rel @ nat
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X4: nat,Y5: nat] :
( ( ord_less @ nat @ X4 @ N5 )
& ( ord_less @ nat @ Y5 @ N5 )
& ( ord_less_eq @ nat @ X4 @ Y5 ) ) ) ) ) ).
% natLeq_on_wo_rel
thf(fact_1227_underS__def,axiom,
! [A: $tType] :
( ( order_underS @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A ),A7: A] :
( collect @ A
@ ^ [B5: A] :
( ( B5 != A7 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ A7 ) @ R5 ) ) ) ) ) ).
% underS_def
thf(fact_1228_inj__on__strict__subset,axiom,
! [B: $tType,A: $tType,F: A > B,B6: set @ A,A6: set @ A] :
( ( inj_on @ A @ B @ F @ B6 )
=> ( ( ord_less @ ( set @ A ) @ A6 @ B6 )
=> ( ord_less @ ( set @ B ) @ ( image2 @ A @ B @ F @ A6 ) @ ( image2 @ A @ B @ F @ B6 ) ) ) ) ).
% inj_on_strict_subset
thf(fact_1229_acyclicI__order,axiom,
! [A: $tType,B: $tType] :
( ( preorder @ A )
=> ! [R: set @ ( product_prod @ B @ B ),F: B > A] :
( ! [A5: B,B4: B] :
( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A5 @ B4 ) @ R )
=> ( ord_less @ A @ ( F @ B4 ) @ ( F @ A5 ) ) )
=> ( transitive_acyclic @ B @ R ) ) ) ).
% acyclicI_order
thf(fact_1230_SUP__lessD,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: B > A,A6: set @ B,Y3: A,I: B] :
( ( ord_less @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A6 ) ) @ Y3 )
=> ( ( member @ B @ I @ A6 )
=> ( ord_less @ A @ ( F @ I ) @ Y3 ) ) ) ) ).
% SUP_lessD
thf(fact_1231_less__SUP__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A3: A,F: B > A,A6: set @ B] :
( ( ord_less @ A @ A3 @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A6 ) ) )
= ( ? [X4: B] :
( ( member @ B @ X4 @ A6 )
& ( ord_less @ A @ A3 @ ( F @ X4 ) ) ) ) ) ) ).
% less_SUP_iff
thf(fact_1232_less__INF__D,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [Y3: A,F: B > A,A6: set @ B,I: B] :
( ( ord_less @ A @ Y3 @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A6 ) ) )
=> ( ( member @ B @ I @ A6 )
=> ( ord_less @ A @ Y3 @ ( F @ I ) ) ) ) ) ).
% less_INF_D
thf(fact_1233_INF__less__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F: B > A,A6: set @ B,A3: A] :
( ( ord_less @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A6 ) ) @ A3 )
= ( ? [X4: B] :
( ( member @ B @ X4 @ A6 )
& ( ord_less @ A @ ( F @ X4 ) @ A3 ) ) ) ) ) ).
% INF_less_iff
thf(fact_1234_filterlim__filtercomap__iff,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > B,G: B > C,G6: filter @ C,F7: filter @ A] :
( ( filterlim @ A @ B @ F @ ( filtercomap @ B @ C @ G @ G6 ) @ F7 )
= ( filterlim @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ G6 @ F7 ) ) ).
% filterlim_filtercomap_iff
thf(fact_1235_wf,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ( wf @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ ( ord_less @ A ) ) ) ) ) ).
% wf
thf(fact_1236_finite__conv__nat__seg__image,axiom,
! [A: $tType] :
( ( finite_finite2 @ A )
= ( ^ [A8: set @ A] :
? [N4: nat,F4: nat > A] :
( A8
= ( image2 @ nat @ A @ F4
@ ( collect @ nat
@ ^ [I3: nat] : ( ord_less @ nat @ I3 @ N4 ) ) ) ) ) ) ).
% finite_conv_nat_seg_image
thf(fact_1237_nat__seg__image__imp__finite,axiom,
! [A: $tType,A6: set @ A,F: nat > A,N5: nat] :
( ( A6
= ( image2 @ nat @ A @ F
@ ( collect @ nat
@ ^ [I3: nat] : ( ord_less @ nat @ I3 @ N5 ) ) ) )
=> ( finite_finite2 @ A @ A6 ) ) ).
% nat_seg_image_imp_finite
thf(fact_1238_wf__if__measure,axiom,
! [A: $tType,P2: A > $o,F: A > nat,G: A > A] :
( ! [X: A] :
( ( P2 @ X )
=> ( ord_less @ nat @ ( F @ ( G @ X ) ) @ ( F @ X ) ) )
=> ( wf @ A
@ ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [Y5: A,X4: A] :
( ( P2 @ X4 )
& ( Y5
= ( G @ X4 ) ) ) ) ) ) ) ).
% wf_if_measure
thf(fact_1239_bind__UNION,axiom,
! [A: $tType,B: $tType] :
( ( bind4 @ B @ A )
= ( ^ [A8: set @ B,F4: B > ( set @ A )] : ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ F4 @ A8 ) ) ) ) ).
% bind_UNION
thf(fact_1240_le__SUP__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [X2: A,F: B > A,A6: set @ B] :
( ( ord_less_eq @ A @ X2 @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A6 ) ) )
= ( ! [Y5: A] :
( ( ord_less @ A @ Y5 @ X2 )
=> ? [X4: B] :
( ( member @ B @ X4 @ A6 )
& ( ord_less @ A @ Y5 @ ( F @ X4 ) ) ) ) ) ) ) ).
% le_SUP_iff
thf(fact_1241_INF__le__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F: B > A,A6: set @ B,X2: A] :
( ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A6 ) ) @ X2 )
= ( ! [Y5: A] :
( ( ord_less @ A @ X2 @ Y5 )
=> ? [X4: B] :
( ( member @ B @ X4 @ A6 )
& ( ord_less @ A @ ( F @ X4 ) @ Y5 ) ) ) ) ) ) ).
% INF_le_iff
thf(fact_1242_bind__image,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > B,A6: set @ C,G: B > ( set @ A )] :
( ( bind4 @ B @ A @ ( image2 @ C @ B @ F @ A6 ) @ G )
= ( bind4 @ C @ A @ A6 @ ( comp @ B @ ( set @ A ) @ C @ G @ F ) ) ) ).
% bind_image
thf(fact_1243_filtercomap__INF,axiom,
! [A: $tType,B: $tType,C: $tType,F: A > B,F7: C > ( filter @ B ),B6: set @ C] :
( ( filtercomap @ A @ B @ F @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ C @ ( filter @ B ) @ F7 @ B6 ) ) )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ C @ ( filter @ A )
@ ^ [B5: C] : ( filtercomap @ A @ B @ F @ ( F7 @ B5 ) )
@ B6 ) ) ) ).
% filtercomap_INF
thf(fact_1244_pigeonhole,axiom,
! [A: $tType,B: $tType,F: B > A,A6: set @ B] :
( ( ord_less @ nat @ ( finite_card @ A @ ( image2 @ B @ A @ F @ A6 ) ) @ ( finite_card @ B @ A6 ) )
=> ~ ( inj_on @ B @ A @ F @ A6 ) ) ).
% pigeonhole
thf(fact_1245_wf__bounded__measure,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),Ub: A > nat,F: A > nat] :
( ! [A5: A,B4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ A5 ) @ R )
=> ( ( ord_less_eq @ nat @ ( Ub @ B4 ) @ ( Ub @ A5 ) )
& ( ord_less_eq @ nat @ ( F @ B4 ) @ ( Ub @ A5 ) )
& ( ord_less @ nat @ ( F @ A5 ) @ ( F @ B4 ) ) ) )
=> ( wf @ A @ R ) ) ).
% wf_bounded_measure
thf(fact_1246_image__bind,axiom,
! [A: $tType,B: $tType,C: $tType,F: B > A,A6: set @ C,G: C > ( set @ B )] :
( ( image2 @ B @ A @ F @ ( bind4 @ C @ B @ A6 @ G ) )
= ( bind4 @ C @ A @ A6 @ ( comp @ ( set @ B ) @ ( set @ A ) @ C @ ( image2 @ B @ A @ F ) @ G ) ) ) ).
% image_bind
thf(fact_1247_cSUP__lessD,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [F: B > A,A6: set @ B,Y3: A,I: B] :
( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F @ A6 ) )
=> ( ( ord_less @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A6 ) ) @ Y3 )
=> ( ( member @ B @ I @ A6 )
=> ( ord_less @ A @ ( F @ I ) @ Y3 ) ) ) ) ) ).
% cSUP_lessD
thf(fact_1248_filtercomap__neq__bot__surj,axiom,
! [A: $tType,B: $tType,F7: filter @ A,F: B > A] :
( ( F7
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( filtercomap @ B @ A @ F @ F7 )
!= ( bot_bot @ ( filter @ B ) ) ) ) ) ).
% filtercomap_neq_bot_surj
thf(fact_1249_less__cINF__D,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [F: B > A,A6: set @ B,Y3: A,I: B] :
( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F @ A6 ) )
=> ( ( ord_less @ A @ Y3 @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A6 ) ) )
=> ( ( member @ B @ I @ A6 )
=> ( ord_less @ A @ Y3 @ ( F @ I ) ) ) ) ) ) ).
% less_cINF_D
thf(fact_1250_finite__mono__strict__prefix__implies__finite__fixpoint,axiom,
! [A: $tType,F: nat > ( set @ A ),S5: set @ A] :
( ! [I6: nat] : ( ord_less_eq @ ( set @ A ) @ ( F @ I6 ) @ S5 )
=> ( ( finite_finite2 @ A @ S5 )
=> ( ? [N6: nat] :
( ! [N2: nat] :
( ( ord_less_eq @ nat @ N2 @ N6 )
=> ! [M3: nat] :
( ( ord_less_eq @ nat @ M3 @ N6 )
=> ( ( ord_less @ nat @ M3 @ N2 )
=> ( ord_less @ ( set @ A ) @ ( F @ M3 ) @ ( F @ N2 ) ) ) ) )
& ! [N2: nat] :
( ( ord_less_eq @ nat @ N6 @ N2 )
=> ( ( F @ N6 )
= ( F @ N2 ) ) ) )
=> ( ( F @ ( finite_card @ A @ S5 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ F @ ( top_top @ ( set @ nat ) ) ) ) ) ) ) ) ).
% finite_mono_strict_prefix_implies_finite_fixpoint
thf(fact_1251_finite__subset__wf,axiom,
! [A: $tType,A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( wf @ ( set @ A )
@ ( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
@ ^ [X10: set @ A,Y9: set @ A] :
( ( ord_less @ ( set @ A ) @ X10 @ Y9 )
& ( ord_less_eq @ ( set @ A ) @ Y9 @ A6 ) ) ) ) ) ) ).
% finite_subset_wf
thf(fact_1252_finite__imp__nat__seg__image__inj__on,axiom,
! [A: $tType,A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ? [N2: nat,F3: nat > A] :
( ( A6
= ( image2 @ nat @ A @ F3
@ ( collect @ nat
@ ^ [I3: nat] : ( ord_less @ nat @ I3 @ N2 ) ) ) )
& ( inj_on @ nat @ A @ F3
@ ( collect @ nat
@ ^ [I3: nat] : ( ord_less @ nat @ I3 @ N2 ) ) ) ) ) ).
% finite_imp_nat_seg_image_inj_on
thf(fact_1253_filtermap__image__finite__subsets__at__top,axiom,
! [B: $tType,A: $tType,F: A > B,A6: set @ A] :
( ( inj_on @ A @ B @ F @ A6 )
=> ( ( filtermap @ ( set @ A ) @ ( set @ B ) @ ( image2 @ A @ B @ F ) @ ( finite5375528669736107172at_top @ A @ A6 ) )
= ( finite5375528669736107172at_top @ B @ ( image2 @ A @ B @ F @ A6 ) ) ) ) ).
% filtermap_image_finite_subsets_at_top
thf(fact_1254_wf__bounded__set,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),Ub: A > ( set @ B ),F: A > ( set @ B )] :
( ! [A5: A,B4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ A5 ) @ R )
=> ( ( finite_finite2 @ B @ ( Ub @ A5 ) )
& ( ord_less_eq @ ( set @ B ) @ ( Ub @ B4 ) @ ( Ub @ A5 ) )
& ( ord_less_eq @ ( set @ B ) @ ( F @ B4 ) @ ( Ub @ A5 ) )
& ( ord_less @ ( set @ B ) @ ( F @ A5 ) @ ( F @ B4 ) ) ) )
=> ( wf @ A @ R ) ) ).
% wf_bounded_set
thf(fact_1255_less__cSUP__iff,axiom,
! [A: $tType,B: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [A6: set @ B,F: B > A,A3: A] :
( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F @ A6 ) )
=> ( ( ord_less @ A @ A3 @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A6 ) ) )
= ( ? [X4: B] :
( ( member @ B @ X4 @ A6 )
& ( ord_less @ A @ A3 @ ( F @ X4 ) ) ) ) ) ) ) ) ).
% less_cSUP_iff
thf(fact_1256_trancl__finite__eq__relpow,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R3 )
=> ( ( transitive_trancl @ A @ R3 )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N4: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N4 @ R3 )
@ ( collect @ nat
@ ^ [N4: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 )
& ( ord_less_eq @ nat @ N4 @ ( finite_card @ ( product_prod @ A @ A ) @ R3 ) ) ) ) ) ) ) ) ).
% trancl_finite_eq_relpow
thf(fact_1257_relpow__finite__bounded1,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),K: nat] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ K @ R3 )
@ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N4: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N4 @ R3 )
@ ( collect @ nat
@ ^ [N4: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 )
& ( ord_less_eq @ nat @ N4 @ ( finite_card @ ( product_prod @ A @ A ) @ R3 ) ) ) ) ) ) ) ) ) ).
% relpow_finite_bounded1
thf(fact_1258_finite__psubset__def,axiom,
! [A: $tType] :
( ( finite_psubset @ A )
= ( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
@ ^ [A8: set @ A,B7: set @ A] :
( ( ord_less @ ( set @ A ) @ A8 @ B7 )
& ( finite_finite2 @ A @ B7 ) ) ) ) ) ).
% finite_psubset_def
thf(fact_1259_mlex__eq,axiom,
! [A: $tType] :
( ( mlex_prod @ A )
= ( ^ [F4: A > nat,R6: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X4: A,Y5: A] :
( ( ord_less @ nat @ ( F4 @ X4 ) @ ( F4 @ Y5 ) )
| ( ( ord_less_eq @ nat @ ( F4 @ X4 ) @ ( F4 @ Y5 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R6 ) ) ) ) ) ) ) ).
% mlex_eq
thf(fact_1260_in__measure,axiom,
! [A: $tType,X2: A,Y3: A,F: A > nat] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ ( measure @ A @ F ) )
= ( ord_less @ nat @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ).
% in_measure
thf(fact_1261_Ref_Opresent__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( present2 @ A )
= ( ^ [H6: heap_ext @ product_unit,R5: ref @ A] : ( ord_less @ nat @ ( addr_of_ref @ A @ R5 ) @ ( lim @ product_unit @ H6 ) ) ) ) ) ).
% Ref.present_def
thf(fact_1262_Array_Opresent__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( present @ A )
= ( ^ [H6: heap_ext @ product_unit,A7: array @ A] : ( ord_less @ nat @ ( addr_of_array @ A @ A7 ) @ ( lim @ product_unit @ H6 ) ) ) ) ) ).
% Array.present_def
thf(fact_1263_success__swapI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,H: heap_ext @ product_unit,A3: array @ A,X2: A] :
( ( ord_less @ nat @ I @ ( length @ A @ H @ A3 ) )
=> ( heap_success @ A @ ( swap @ A @ I @ X2 @ A3 ) @ H ) ) ) ).
% success_swapI
thf(fact_1264_upd__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( upd @ A )
= ( ^ [I3: nat,X4: A,A7: array @ A] :
( heap_guard @ ( array @ A )
@ ^ [H6: heap_ext @ product_unit] : ( ord_less @ nat @ I3 @ ( length @ A @ H6 @ A7 ) )
@ ^ [H6: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( heap_ext @ product_unit ) @ A7 @ ( update @ A @ A7 @ I3 @ X4 @ H6 ) ) ) ) ) ) ).
% upd_def
thf(fact_1265_wf__less,axiom,
wf @ nat @ ( collect @ ( product_prod @ nat @ nat ) @ ( product_case_prod @ nat @ nat @ $o @ ( ord_less @ nat ) ) ) ).
% wf_less
thf(fact_1266_upd__return,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,X2: A,A3: array @ A] :
( ( heap_bind @ ( array @ A ) @ ( array @ A ) @ ( upd @ A @ I @ X2 @ A3 )
@ ^ [Uu: array @ A] : ( heap_return @ ( array @ A ) @ A3 ) )
= ( upd @ A @ I @ X2 @ A3 ) ) ) ).
% upd_return
thf(fact_1267_relpow__0__I,axiom,
! [A: $tType,X2: A,R3: set @ ( product_prod @ A @ A )] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( zero_zero @ nat ) @ R3 ) ) ).
% relpow_0_I
thf(fact_1268_relpow__0__E,axiom,
! [A: $tType,X2: A,Y3: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( zero_zero @ nat ) @ R3 ) )
=> ( X2 = Y3 ) ) ).
% relpow_0_E
thf(fact_1269_mlex__iff,axiom,
! [A: $tType,X2: A,Y3: A,F: A > nat,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ ( mlex_prod @ A @ F @ R3 ) )
= ( ( ord_less @ nat @ ( F @ X2 ) @ ( F @ Y3 ) )
| ( ( ( F @ X2 )
= ( F @ Y3 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R3 ) ) ) ) ).
% mlex_iff
thf(fact_1270_mlex__less,axiom,
! [A: $tType,F: A > nat,X2: A,Y3: A,R3: set @ ( product_prod @ A @ A )] :
( ( ord_less @ nat @ ( F @ X2 ) @ ( F @ Y3 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ ( mlex_prod @ A @ F @ R3 ) ) ) ).
% mlex_less
thf(fact_1271_success__updI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,H: heap_ext @ product_unit,A3: array @ A,X2: A] :
( ( ord_less @ nat @ I @ ( length @ A @ H @ A3 ) )
=> ( heap_success @ ( array @ A ) @ ( upd @ A @ I @ X2 @ A3 ) @ H ) ) ) ).
% success_updI
thf(fact_1272_card__range__greater__zero,axiom,
! [A: $tType,B: $tType,F: B > A] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ) ) ).
% card_range_greater_zero
thf(fact_1273_mlex__leq,axiom,
! [A: $tType,F: A > nat,X2: A,Y3: A,R3: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ nat @ ( F @ X2 ) @ ( F @ Y3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ ( mlex_prod @ A @ F @ R3 ) ) ) ) ).
% mlex_leq
thf(fact_1274_effect__updE,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,V: A,A3: array @ A,H: heap_ext @ product_unit,H2: heap_ext @ product_unit,R: array @ A] :
( ( heap_effect @ ( array @ A ) @ ( upd @ A @ I @ V @ A3 ) @ H @ H2 @ R )
=> ~ ( ( R = A3 )
=> ( ( H2
= ( update @ A @ A3 @ I @ V @ H ) )
=> ~ ( ord_less @ nat @ I @ ( length @ A @ H @ A3 ) ) ) ) ) ) ).
% effect_updE
thf(fact_1275_effect__updI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,H: heap_ext @ product_unit,A3: array @ A,H2: heap_ext @ product_unit,V: A] :
( ( ord_less @ nat @ I @ ( length @ A @ H @ A3 ) )
=> ( ( H2
= ( update @ A @ A3 @ I @ V @ H ) )
=> ( heap_effect @ ( array @ A ) @ ( upd @ A @ I @ V @ A3 ) @ H @ H2 @ A3 ) ) ) ) ).
% effect_updI
thf(fact_1276_natLess__def,axiom,
( bNF_Ca8459412986667044542atLess
= ( collect @ ( product_prod @ nat @ nat ) @ ( product_case_prod @ nat @ nat @ $o @ ( ord_less @ nat ) ) ) ) ).
% natLess_def
thf(fact_1277_swap__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( swap @ A )
= ( ^ [I3: nat,X4: A,A7: array @ A] :
( heap_guard @ A
@ ^ [H6: heap_ext @ product_unit] : ( ord_less @ nat @ I3 @ ( length @ A @ H6 @ A7 ) )
@ ^ [H6: heap_ext @ product_unit] : ( product_Pair @ A @ ( heap_ext @ product_unit ) @ ( nth3 @ A @ ( get @ A @ H6 @ A7 ) @ I3 ) @ ( update @ A @ A7 @ I3 @ X4 @ H6 ) ) ) ) ) ) ).
% swap_def
thf(fact_1278_success__nthI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,H: heap_ext @ product_unit,A3: array @ A] :
( ( ord_less @ nat @ I @ ( length @ A @ H @ A3 ) )
=> ( heap_success @ A @ ( nth @ A @ A3 @ I ) @ H ) ) ) ).
% success_nthI
thf(fact_1279_effect__swapE,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,X2: A,A3: array @ A,H: heap_ext @ product_unit,H2: heap_ext @ product_unit,R: A] :
( ( heap_effect @ A @ ( swap @ A @ I @ X2 @ A3 ) @ H @ H2 @ R )
=> ~ ( ( R
= ( nth3 @ A @ ( get @ A @ H @ A3 ) @ I ) )
=> ( ( H2
= ( update @ A @ A3 @ I @ X2 @ H ) )
=> ~ ( ord_less @ nat @ I @ ( length @ A @ H @ A3 ) ) ) ) ) ) ).
% effect_swapE
thf(fact_1280_effect__swapI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,H: heap_ext @ product_unit,A3: array @ A,H2: heap_ext @ product_unit,X2: A,R: A] :
( ( ord_less @ nat @ I @ ( length @ A @ H @ A3 ) )
=> ( ( H2
= ( update @ A @ A3 @ I @ X2 @ H ) )
=> ( ( R
= ( nth3 @ A @ ( get @ A @ H @ A3 ) @ I ) )
=> ( heap_effect @ A @ ( swap @ A @ I @ X2 @ A3 ) @ H @ H2 @ R ) ) ) ) ) ).
% effect_swapI
thf(fact_1281_finite__mono__remains__stable__implies__strict__prefix,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F: nat > A] :
( ( finite_finite2 @ A @ ( image2 @ nat @ A @ F @ ( top_top @ ( set @ nat ) ) ) )
=> ( ( order_mono @ nat @ A @ F )
=> ( ! [N2: nat] :
( ( ( F @ N2 )
= ( F @ ( suc @ N2 ) ) )
=> ( ( F @ ( suc @ N2 ) )
= ( F @ ( suc @ ( suc @ N2 ) ) ) ) )
=> ? [N7: nat] :
( ! [N8: nat] :
( ( ord_less_eq @ nat @ N8 @ N7 )
=> ! [M6: nat] :
( ( ord_less_eq @ nat @ M6 @ N7 )
=> ( ( ord_less @ nat @ M6 @ N8 )
=> ( ord_less @ A @ ( F @ M6 ) @ ( F @ N8 ) ) ) ) )
& ! [N8: nat] :
( ( ord_less_eq @ nat @ N7 @ N8 )
=> ( ( F @ N7 )
= ( F @ N8 ) ) ) ) ) ) ) ) ).
% finite_mono_remains_stable_implies_strict_prefix
thf(fact_1282_map__entry__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( map_entry @ A )
= ( ^ [I3: nat,F4: A > A,A7: array @ A] :
( heap_guard @ ( array @ A )
@ ^ [H6: heap_ext @ product_unit] : ( ord_less @ nat @ I3 @ ( length @ A @ H6 @ A7 ) )
@ ^ [H6: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( heap_ext @ product_unit ) @ A7 @ ( update @ A @ A7 @ I3 @ ( F4 @ ( nth3 @ A @ ( get @ A @ H6 @ A7 ) @ I3 ) ) @ H6 ) ) ) ) ) ) ).
% map_entry_def
thf(fact_1283_success__map__entryI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,H: heap_ext @ product_unit,A3: array @ A,F: A > A] :
( ( ord_less @ nat @ I @ ( length @ A @ H @ A3 ) )
=> ( heap_success @ ( array @ A ) @ ( map_entry @ A @ I @ F @ A3 ) @ H ) ) ) ).
% success_map_entryI
thf(fact_1284_get__update__elem__neqIndex,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,J5: nat,A3: array @ A,V: A,H: heap_ext @ product_unit] :
( ( I != J5 )
=> ( ( nth3 @ A @ ( get @ A @ ( update @ A @ A3 @ J5 @ V @ H ) @ A3 ) @ I )
= ( nth3 @ A @ ( get @ A @ H @ A3 ) @ I ) ) ) ) ).
% get_update_elem_neqIndex
thf(fact_1285_card_Ocomp__fun__commute__on,axiom,
( ( comp @ nat @ nat @ nat @ suc @ suc )
= ( comp @ nat @ nat @ nat @ suc @ suc ) ) ).
% card.comp_fun_commute_on
thf(fact_1286_wf__no__infinite__down__chainE,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),F: nat > A] :
( ( wf @ A @ R )
=> ~ ! [K4: nat] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F @ ( suc @ K4 ) ) @ ( F @ K4 ) ) @ R ) ) ).
% wf_no_infinite_down_chainE
thf(fact_1287_wf__iff__no__infinite__down__chain,axiom,
! [A: $tType] :
( ( wf @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
~ ? [F4: nat > A] :
! [I3: nat] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F4 @ ( suc @ I3 ) ) @ ( F4 @ I3 ) ) @ R5 ) ) ) ).
% wf_iff_no_infinite_down_chain
thf(fact_1288_relpow__Suc__E,axiom,
! [A: $tType,X2: A,Z3: A,N5: nat,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N5 ) @ R3 ) )
=> ~ ! [Y: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N5 @ R3 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z3 ) @ R3 ) ) ) ).
% relpow_Suc_E
thf(fact_1289_relpow__Suc__I,axiom,
! [A: $tType,X2: A,Y3: A,N5: nat,R3: set @ ( product_prod @ A @ A ),Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N5 @ R3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ R3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N5 ) @ R3 ) ) ) ) ).
% relpow_Suc_I
thf(fact_1290_relpow__Suc__D2,axiom,
! [A: $tType,X2: A,Z3: A,N5: nat,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N5 ) @ R3 ) )
=> ? [Y: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N5 @ R3 ) ) ) ) ).
% relpow_Suc_D2
thf(fact_1291_relpow__Suc__E2,axiom,
! [A: $tType,X2: A,Z3: A,N5: nat,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N5 ) @ R3 ) )
=> ~ ! [Y: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R3 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N5 @ R3 ) ) ) ) ).
% relpow_Suc_E2
thf(fact_1292_relpow__Suc__I2,axiom,
! [A: $tType,X2: A,Y3: A,R3: set @ ( product_prod @ A @ A ),Z3: A,N5: nat] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N5 @ R3 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N5 ) @ R3 ) ) ) ) ).
% relpow_Suc_I2
thf(fact_1293_effect__nthI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,H: heap_ext @ product_unit,A3: array @ A,H2: heap_ext @ product_unit,R: A] :
( ( ord_less @ nat @ I @ ( length @ A @ H @ A3 ) )
=> ( ( H2 = H )
=> ( ( R
= ( nth3 @ A @ ( get @ A @ H @ A3 ) @ I ) )
=> ( heap_effect @ A @ ( nth @ A @ A3 @ I ) @ H @ H2 @ R ) ) ) ) ) ).
% effect_nthI
thf(fact_1294_effect__nthE,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [A3: array @ A,I: nat,H: heap_ext @ product_unit,H2: heap_ext @ product_unit,R: A] :
( ( heap_effect @ A @ ( nth @ A @ A3 @ I ) @ H @ H2 @ R )
=> ~ ( ( ord_less @ nat @ I @ ( length @ A @ H @ A3 ) )
=> ( ( R
= ( nth3 @ A @ ( get @ A @ H @ A3 ) @ I ) )
=> ( H2 != H ) ) ) ) ) ).
% effect_nthE
thf(fact_1295_effect__map__entryE,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,F: A > A,A3: array @ A,H: heap_ext @ product_unit,H2: heap_ext @ product_unit,R: array @ A] :
( ( heap_effect @ ( array @ A ) @ ( map_entry @ A @ I @ F @ A3 ) @ H @ H2 @ R )
=> ~ ( ( R = A3 )
=> ( ( H2
= ( update @ A @ A3 @ I @ ( F @ ( nth3 @ A @ ( get @ A @ H @ A3 ) @ I ) ) @ H ) )
=> ~ ( ord_less @ nat @ I @ ( length @ A @ H @ A3 ) ) ) ) ) ) ).
% effect_map_entryE
thf(fact_1296_effect__map__entryI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,H: heap_ext @ product_unit,A3: array @ A,H2: heap_ext @ product_unit,F: A > A,R: array @ A] :
( ( ord_less @ nat @ I @ ( length @ A @ H @ A3 ) )
=> ( ( H2
= ( update @ A @ A3 @ I @ ( F @ ( nth3 @ A @ ( get @ A @ H @ A3 ) @ I ) ) @ H ) )
=> ( ( R = A3 )
=> ( heap_effect @ ( array @ A ) @ ( map_entry @ A @ I @ F @ A3 ) @ H @ H2 @ R ) ) ) ) ) ).
% effect_map_entryI
thf(fact_1297_relpow__E,axiom,
! [A: $tType,X2: A,Z3: A,N5: nat,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N5 @ R3 ) )
=> ( ( ( N5
= ( zero_zero @ nat ) )
=> ( X2 != Z3 ) )
=> ~ ! [Y: A,M3: nat] :
( ( N5
= ( suc @ M3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ M3 @ R3 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z3 ) @ R3 ) ) ) ) ) ).
% relpow_E
thf(fact_1298_relpow__E2,axiom,
! [A: $tType,X2: A,Z3: A,N5: nat,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N5 @ R3 ) )
=> ( ( ( N5
= ( zero_zero @ nat ) )
=> ( X2 != Z3 ) )
=> ~ ! [Y: A,M3: nat] :
( ( N5
= ( suc @ M3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R3 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ M3 @ R3 ) ) ) ) ) ) ).
% relpow_E2
thf(fact_1299_Array_Onth__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( nth @ A )
= ( ^ [A7: array @ A,I3: nat] :
( heap_guard @ A
@ ^ [H6: heap_ext @ product_unit] : ( ord_less @ nat @ I3 @ ( length @ A @ H6 @ A7 ) )
@ ^ [H6: heap_ext @ product_unit] : ( product_Pair @ A @ ( heap_ext @ product_unit ) @ ( nth3 @ A @ ( get @ A @ H6 @ A7 ) @ I3 ) @ H6 ) ) ) ) ) ).
% Array.nth_def
thf(fact_1300_relpow__fun__conv,axiom,
! [A: $tType,A3: A,B2: A,N5: nat,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N5 @ R3 ) )
= ( ? [F4: nat > A] :
( ( ( F4 @ ( zero_zero @ nat ) )
= A3 )
& ( ( F4 @ N5 )
= B2 )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N5 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F4 @ I3 ) @ ( F4 @ ( suc @ I3 ) ) ) @ R3 ) ) ) ) ) ).
% relpow_fun_conv
thf(fact_1301_ntrancl__def,axiom,
! [A: $tType] :
( ( transitive_ntrancl @ A )
= ( ^ [N4: nat,R6: set @ ( product_prod @ A @ A )] :
( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [I3: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ I3 @ R6 )
@ ( collect @ nat
@ ^ [I3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ I3 )
& ( ord_less_eq @ nat @ I3 @ ( suc @ N4 ) ) ) ) ) ) ) ) ).
% ntrancl_def
thf(fact_1302_nth__update__neq,axiom,
! [B: $tType,A: $tType] :
( ( ( heap @ A )
& ( heap @ B ) )
=> ! [A3: array @ A,B2: array @ B,J5: nat,V: B,H: heap_ext @ product_unit,I: nat] :
( ( noteq @ A @ B @ A3 @ B2 )
=> ( ( nth3 @ A @ ( get @ A @ ( update @ B @ B2 @ J5 @ V @ H ) @ A3 ) @ I )
= ( nth3 @ A @ ( get @ A @ H @ A3 ) @ I ) ) ) ) ).
% nth_update_neq
thf(fact_1303_pred__nat__def,axiom,
( pred_nat
= ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [M7: nat,N4: nat] :
( N4
= ( suc @ M7 ) ) ) ) ) ).
% pred_nat_def
thf(fact_1304_zero__notin__Suc__image,axiom,
! [A6: set @ nat] :
~ ( member @ nat @ ( zero_zero @ nat ) @ ( image2 @ nat @ nat @ suc @ A6 ) ) ).
% zero_notin_Suc_image
thf(fact_1305_size__prod__simp,axiom,
! [B: $tType,A: $tType] :
( ( basic_BNF_size_prod @ A @ B )
= ( ^ [F4: A > nat,G4: B > nat,P3: product_prod @ A @ B] : ( plus_plus @ nat @ ( plus_plus @ nat @ ( F4 @ ( product_fst @ A @ B @ P3 ) ) @ ( G4 @ ( product_snd @ A @ B @ P3 ) ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% size_prod_simp
thf(fact_1306_Restr__natLeq2,axiom,
! [N5: nat] :
( ( inf_inf @ ( set @ ( product_prod @ nat @ nat ) ) @ bNF_Ca8665028551170535155natLeq
@ ( product_Sigma @ nat @ nat @ ( order_underS @ nat @ bNF_Ca8665028551170535155natLeq @ N5 )
@ ^ [Uu: nat] : ( order_underS @ nat @ bNF_Ca8665028551170535155natLeq @ N5 ) ) )
= ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X4: nat,Y5: nat] :
( ( ord_less @ nat @ X4 @ N5 )
& ( ord_less @ nat @ Y5 @ N5 )
& ( ord_less_eq @ nat @ X4 @ Y5 ) ) ) ) ) ).
% Restr_natLeq2
thf(fact_1307_Restr__natLeq,axiom,
! [N5: nat] :
( ( inf_inf @ ( set @ ( product_prod @ nat @ nat ) ) @ bNF_Ca8665028551170535155natLeq
@ ( product_Sigma @ nat @ nat
@ ( collect @ nat
@ ^ [X4: nat] : ( ord_less @ nat @ X4 @ N5 ) )
@ ^ [Uu: nat] :
( collect @ nat
@ ^ [X4: nat] : ( ord_less @ nat @ X4 @ N5 ) ) ) )
= ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X4: nat,Y5: nat] :
( ( ord_less @ nat @ X4 @ N5 )
& ( ord_less @ nat @ Y5 @ N5 )
& ( ord_less_eq @ nat @ X4 @ Y5 ) ) ) ) ) ).
% Restr_natLeq
thf(fact_1308_surj__from__nat,axiom,
! [A: $tType] :
( ( countable @ A )
=> ( ( image2 @ nat @ A @ ( from_nat @ A ) @ ( top_top @ ( set @ nat ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surj_from_nat
thf(fact_1309_Array_Ounequal,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [A3: array @ A,A4: array @ A] :
( ( A3 != A4 )
= ( noteq @ A @ A @ A3 @ A4 ) ) ) ).
% Array.unequal
thf(fact_1310_image__add__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [S5: set @ A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ ( zero_zero @ A ) ) @ S5 )
= S5 ) ) ).
% image_add_0
thf(fact_1311_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_1312_Array_Onoteq__irrefl,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R: array @ A] :
~ ( noteq @ A @ A @ R @ R ) ) ).
% Array.noteq_irrefl
thf(fact_1313_Array_Onoteq__sym,axiom,
! [B: $tType,A: $tType] :
( ( ( heap @ A )
& ( heap @ B ) )
=> ! [A3: array @ A,B2: array @ B] :
( ( noteq @ A @ B @ A3 @ B2 )
=> ( noteq @ B @ A @ B2 @ A3 ) ) ) ).
% Array.noteq_sym
thf(fact_1314_translation__Int,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,S2: set @ A,T3: set @ A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ ( inf_inf @ ( set @ A ) @ S2 @ T3 ) )
= ( inf_inf @ ( set @ A ) @ ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ S2 ) @ ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ T3 ) ) ) ) ).
% translation_Int
thf(fact_1315_translation__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,S2: set @ A,T3: set @ A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ ( minus_minus @ ( set @ A ) @ S2 @ T3 ) )
= ( minus_minus @ ( set @ A ) @ ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ S2 ) @ ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ T3 ) ) ) ) ).
% translation_diff
thf(fact_1316_update__swap__neq,axiom,
! [B: $tType,A: $tType] :
( ( ( heap @ A )
& ( heap @ B ) )
=> ! [A3: array @ A,A4: array @ B,I: nat,V: A,I2: nat,V2: B,H: heap_ext @ product_unit] :
( ( noteq @ A @ B @ A3 @ A4 )
=> ( ( update @ A @ A3 @ I @ V @ ( update @ B @ A4 @ I2 @ V2 @ H ) )
= ( update @ B @ A4 @ I2 @ V2 @ ( update @ A @ A3 @ I @ V @ H ) ) ) ) ) ).
% update_swap_neq
thf(fact_1317_natLeq__def,axiom,
( bNF_Ca8665028551170535155natLeq
= ( collect @ ( product_prod @ nat @ nat ) @ ( product_case_prod @ nat @ nat @ $o @ ( ord_less_eq @ nat ) ) ) ) ).
% natLeq_def
thf(fact_1318_rat__denum,axiom,
? [F3: nat > rat] :
( ( image2 @ nat @ rat @ F3 @ ( top_top @ ( set @ nat ) ) )
= ( top_top @ ( set @ rat ) ) ) ).
% rat_denum
thf(fact_1319_UN__finite2__subset,axiom,
! [A: $tType,A6: nat > ( set @ A ),B6: nat > ( set @ A ),K: nat] :
( ! [N2: nat] : ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ B6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ N2 @ K ) ) ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A6 @ ( top_top @ ( set @ nat ) ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ B6 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% UN_finite2_subset
thf(fact_1320_Array_Oget__set__neq,axiom,
! [B: $tType,A: $tType] :
( ( ( heap @ A )
& ( heap @ B ) )
=> ! [R: array @ A,S2: array @ B,X2: list @ B,H: heap_ext @ product_unit] :
( ( noteq @ A @ B @ R @ S2 )
=> ( ( get @ A @ ( set2 @ B @ S2 @ X2 @ H ) @ R )
= ( get @ A @ H @ R ) ) ) ) ).
% Array.get_set_neq
thf(fact_1321_UN__finite2__eq,axiom,
! [A: $tType,A6: nat > ( set @ A ),B6: nat > ( set @ A ),K: nat] :
( ! [N2: nat] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ B6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ N2 @ K ) ) ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A6 @ ( top_top @ ( set @ nat ) ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ B6 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% UN_finite2_eq
thf(fact_1322_UN__finite__subset,axiom,
! [A: $tType,A6: nat > ( set @ A ),C5: set @ A] :
( ! [N2: nat] : ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) @ C5 )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A6 @ ( top_top @ ( set @ nat ) ) ) ) @ C5 ) ) ).
% UN_finite_subset
thf(fact_1323_greaterThan__0,axiom,
( ( set_ord_greaterThan @ nat @ ( zero_zero @ nat ) )
= ( image2 @ nat @ nat @ suc @ ( top_top @ ( set @ nat ) ) ) ) ).
% greaterThan_0
thf(fact_1324_UN__UN__finite__eq,axiom,
! [A: $tType,A6: nat > ( set @ A )] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ nat @ ( set @ A )
@ ^ [N4: nat] : ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N4 ) ) )
@ ( top_top @ ( set @ nat ) ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A6 @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% UN_UN_finite_eq
thf(fact_1325_INF__bool__eq,axiom,
! [A: $tType] :
( ( ^ [A8: set @ A,F4: A > $o] : ( complete_Inf_Inf @ $o @ ( image2 @ A @ $o @ F4 @ A8 ) ) )
= ( ball @ A ) ) ).
% INF_bool_eq
thf(fact_1326_Max__add__commute,axiom,
! [B: $tType,A: $tType] :
( ( linord4140545234300271783up_add @ A )
=> ! [S5: set @ B,F: B > A,K: A] :
( ( finite_finite2 @ B @ S5 )
=> ( ( S5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798349783984er_Max @ A
@ ( image2 @ B @ A
@ ^ [X4: B] : ( plus_plus @ A @ ( F @ X4 ) @ K )
@ S5 ) )
= ( plus_plus @ A @ ( lattic643756798349783984er_Max @ A @ ( image2 @ B @ A @ F @ S5 ) ) @ K ) ) ) ) ) ).
% Max_add_commute
thf(fact_1327_Array_Oset__same,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R: array @ A,X2: list @ A,Y3: list @ A,H: heap_ext @ product_unit] :
( ( set2 @ A @ R @ X2 @ ( set2 @ A @ R @ Y3 @ H ) )
= ( set2 @ A @ R @ X2 @ H ) ) ) ).
% Array.set_same
thf(fact_1328_not__False__in__image__Ball,axiom,
! [A: $tType,P2: A > $o,A6: set @ A] :
( ( ~ ( member @ $o @ $false @ ( image2 @ A @ $o @ P2 @ A6 ) ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A6 )
=> ( P2 @ X4 ) ) ) ) ).
% not_False_in_image_Ball
thf(fact_1329_Array_Oget__set__eq,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R: array @ A,X2: list @ A,H: heap_ext @ product_unit] :
( ( get @ A @ ( set2 @ A @ R @ X2 @ H ) @ R )
= X2 ) ) ).
% Array.get_set_eq
thf(fact_1330_image__add__atLeastLessThan,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: A,I: A,J5: A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ K ) @ ( set_or7035219750837199246ssThan @ A @ I @ J5 ) )
= ( set_or7035219750837199246ssThan @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J5 @ K ) ) ) ) ).
% image_add_atLeastLessThan
thf(fact_1331_image__Suc__atLeastLessThan,axiom,
! [I: nat,J5: nat] :
( ( image2 @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ I @ J5 ) )
= ( set_or7035219750837199246ssThan @ nat @ ( suc @ I ) @ ( suc @ J5 ) ) ) ).
% image_Suc_atLeastLessThan
thf(fact_1332_image__add__atLeastLessThan_H,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: A,I: A,J5: A] :
( ( image2 @ A @ A
@ ^ [N4: A] : ( plus_plus @ A @ N4 @ K )
@ ( set_or7035219750837199246ssThan @ A @ I @ J5 ) )
= ( set_or7035219750837199246ssThan @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J5 @ K ) ) ) ) ).
% image_add_atLeastLessThan'
thf(fact_1333_Max__const,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ B,C3: A] :
( ( finite_finite2 @ B @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798349783984er_Max @ A
@ ( image2 @ B @ A
@ ^ [Uu: B] : C3
@ A6 ) )
= C3 ) ) ) ) ).
% Max_const
thf(fact_1334_Ball__image__comp,axiom,
! [A: $tType,B: $tType,F: B > A,A6: set @ B,G: A > $o] :
( ( ! [X4: A] :
( ( member @ A @ X4 @ ( image2 @ B @ A @ F @ A6 ) )
=> ( G @ X4 ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A6 )
=> ( comp @ A @ $o @ B @ G @ F @ X4 ) ) ) ) ).
% Ball_image_comp
thf(fact_1335_Ball__Collect,axiom,
! [A: $tType] :
( ( ball @ A )
= ( ^ [A8: set @ A,P4: A > $o] : ( ord_less_eq @ ( set @ A ) @ A8 @ ( collect @ A @ P4 ) ) ) ) ).
% Ball_Collect
thf(fact_1336_Ball__comp__iff,axiom,
! [C: $tType,B: $tType,A: $tType,A6: B > ( set @ C ),F: C > $o,G: A > B] :
( ( comp @ B @ $o @ A
@ ^ [X4: B] :
! [Y5: C] :
( ( member @ C @ Y5 @ ( A6 @ X4 ) )
=> ( F @ Y5 ) )
@ G )
= ( ^ [X4: A] :
! [Y5: C] :
( ( member @ C @ Y5 @ ( comp @ B @ ( set @ C ) @ A @ A6 @ G @ X4 ) )
=> ( F @ Y5 ) ) ) ) ).
% Ball_comp_iff
thf(fact_1337_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_1338_set__array__set__swap,axiom,
! [B: $tType,A: $tType] :
( ( ( heap @ A )
& ( heap @ B ) )
=> ! [A3: array @ A,Xs: list @ A,R: ref @ B,X3: B,H: heap_ext @ product_unit] :
( ( set2 @ A @ A3 @ Xs @ ( set4 @ B @ R @ X3 @ H ) )
= ( set4 @ B @ R @ X3 @ ( set2 @ A @ A3 @ Xs @ H ) ) ) ) ).
% set_array_set_swap
thf(fact_1339_INTER__eq,axiom,
! [B: $tType,A: $tType,B6: B > ( set @ A ),A6: set @ B] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B6 @ A6 ) )
= ( collect @ A
@ ^ [Y5: A] :
! [X4: B] :
( ( member @ B @ X4 @ A6 )
=> ( member @ A @ Y5 @ ( B6 @ X4 ) ) ) ) ) ).
% INTER_eq
thf(fact_1340_Collect__ball__eq,axiom,
! [A: $tType,B: $tType,A6: set @ B,P2: A > B > $o] :
( ( collect @ A
@ ^ [X4: A] :
! [Y5: B] :
( ( member @ B @ Y5 @ A6 )
=> ( P2 @ X4 @ Y5 ) ) )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y5: B] :
( collect @ A
@ ^ [X4: A] : ( P2 @ X4 @ Y5 ) )
@ A6 ) ) ) ).
% Collect_ball_eq
thf(fact_1341_Array_Oset__set__swap,axiom,
! [B: $tType,A: $tType] :
( ( ( heap @ A )
& ( heap @ B ) )
=> ! [R: array @ A,R2: array @ B,X2: list @ A,X3: list @ B,H: heap_ext @ product_unit] :
( ( noteq @ A @ B @ R @ R2 )
=> ( ( set2 @ A @ R @ X2 @ ( set2 @ B @ R2 @ X3 @ H ) )
= ( set2 @ B @ R2 @ X3 @ ( set2 @ A @ R @ X2 @ H ) ) ) ) ) ).
% Array.set_set_swap
thf(fact_1342_congruent__def,axiom,
! [B: $tType,A: $tType] :
( ( equiv_congruent @ A @ B )
= ( ^ [R5: set @ ( product_prod @ A @ A ),F4: A > B] :
! [X4: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X4 @ R5 )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y5: A,Z5: A] :
( ( F4 @ Y5 )
= ( F4 @ Z5 ) )
@ X4 ) ) ) ) ).
% congruent_def
thf(fact_1343_congruent2__def,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( equiv_congruent2 @ A @ B @ C )
= ( ^ [R13: set @ ( product_prod @ A @ A ),R24: set @ ( product_prod @ B @ B ),F4: A > B > C] :
! [X4: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X4 @ R13 )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y13: A,Z13: A] :
! [Y5: product_prod @ B @ B] :
( ( member @ ( product_prod @ B @ B ) @ Y5 @ R24 )
=> ( product_case_prod @ B @ B @ $o
@ ^ [Y24: B,Z24: B] :
( ( F4 @ Y13 @ Y24 )
= ( F4 @ Z13 @ Z24 ) )
@ Y5 ) )
@ X4 ) ) ) ) ).
% congruent2_def
thf(fact_1344_mono__image__least,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [F: A > B,M4: A,N5: A,M8: B,N9: B] :
( ( order_mono @ A @ B @ F )
=> ( ( ( image2 @ A @ B @ F @ ( set_or7035219750837199246ssThan @ A @ M4 @ N5 ) )
= ( set_or7035219750837199246ssThan @ B @ M8 @ N9 ) )
=> ( ( ord_less @ A @ M4 @ N5 )
=> ( ( F @ M4 )
= M8 ) ) ) ) ) ).
% mono_image_least
thf(fact_1345_atLeast0__lessThan__Suc__eq__insert__0,axiom,
! [N5: nat] :
( ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N5 ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( image2 @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N5 ) ) ) ) ).
% atLeast0_lessThan_Suc_eq_insert_0
thf(fact_1346_refl__on__def_H,axiom,
! [A: $tType] :
( ( refl_on @ A )
= ( ^ [A8: set @ A,R5: set @ ( product_prod @ A @ A )] :
( ! [X4: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X4 @ R5 )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y5: A,Z5: A] :
( ( member @ A @ Y5 @ A8 )
& ( member @ A @ Z5 @ A8 ) )
@ X4 ) )
& ! [X4: A] :
( ( member @ A @ X4 @ A8 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) @ R5 ) ) ) ) ) ).
% refl_on_def'
thf(fact_1347_image__minus__const__atLeastLessThan__nat,axiom,
! [C3: nat,Y3: nat,X2: nat] :
( ( ( ord_less @ nat @ C3 @ Y3 )
=> ( ( image2 @ nat @ nat
@ ^ [I3: nat] : ( minus_minus @ nat @ I3 @ C3 )
@ ( set_or7035219750837199246ssThan @ nat @ X2 @ Y3 ) )
= ( set_or7035219750837199246ssThan @ nat @ ( minus_minus @ nat @ X2 @ C3 ) @ ( minus_minus @ nat @ Y3 @ C3 ) ) ) )
& ( ~ ( ord_less @ nat @ C3 @ Y3 )
=> ( ( ( ord_less @ nat @ X2 @ Y3 )
=> ( ( image2 @ nat @ nat
@ ^ [I3: nat] : ( minus_minus @ nat @ I3 @ C3 )
@ ( set_or7035219750837199246ssThan @ nat @ X2 @ Y3 ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) )
& ( ~ ( ord_less @ nat @ X2 @ Y3 )
=> ( ( image2 @ nat @ nat
@ ^ [I3: nat] : ( minus_minus @ nat @ I3 @ C3 )
@ ( set_or7035219750837199246ssThan @ nat @ X2 @ Y3 ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% image_minus_const_atLeastLessThan_nat
thf(fact_1348_mono__Max__commute,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F: A > B,A6: set @ A] :
( ( order_mono @ A @ B @ F )
=> ( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( F @ ( lattic643756798349783984er_Max @ A @ A6 ) )
= ( lattic643756798349783984er_Max @ B @ ( image2 @ A @ B @ F @ A6 ) ) ) ) ) ) ) ).
% mono_Max_commute
thf(fact_1349_UnderS__def,axiom,
! [A: $tType] :
( ( order_UnderS @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A ),A8: set @ A] :
( collect @ A
@ ^ [B5: A] :
( ( member @ A @ B5 @ ( field2 @ A @ R5 ) )
& ! [X4: A] :
( ( member @ A @ X4 @ A8 )
=> ( ( B5 != X4 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ X4 ) @ R5 ) ) ) ) ) ) ) ).
% UnderS_def
thf(fact_1350_Under__def,axiom,
! [A: $tType] :
( ( order_Under @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A ),A8: set @ A] :
( collect @ A
@ ^ [B5: A] :
( ( member @ A @ B5 @ ( field2 @ A @ R5 ) )
& ! [X4: A] :
( ( member @ A @ X4 @ A8 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ X4 ) @ R5 ) ) ) ) ) ) ).
% Under_def
thf(fact_1351_Above__def,axiom,
! [A: $tType] :
( ( order_Above @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A ),A8: set @ A] :
( collect @ A
@ ^ [B5: A] :
( ( member @ A @ B5 @ ( field2 @ A @ R5 ) )
& ! [X4: A] :
( ( member @ A @ X4 @ A8 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ B5 ) @ R5 ) ) ) ) ) ) ).
% Above_def
thf(fact_1352_surj__nat__to__rat__surj,axiom,
( ( image2 @ nat @ rat @ nat_to_rat_surj @ ( top_top @ ( set @ nat ) ) )
= ( top_top @ ( set @ rat ) ) ) ).
% surj_nat_to_rat_surj
thf(fact_1353_range__mod,axiom,
! [N5: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N5 )
=> ( ( image2 @ nat @ nat
@ ^ [M7: nat] : ( modulo_modulo @ nat @ M7 @ N5 )
@ ( top_top @ ( set @ nat ) ) )
= ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N5 ) ) ) ).
% range_mod
thf(fact_1354_AboveS__def,axiom,
! [A: $tType] :
( ( order_AboveS @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A ),A8: set @ A] :
( collect @ A
@ ^ [B5: A] :
( ( member @ A @ B5 @ ( field2 @ A @ R5 ) )
& ! [X4: A] :
( ( member @ A @ X4 @ A8 )
=> ( ( B5 != X4 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ B5 ) @ R5 ) ) ) ) ) ) ) ).
% AboveS_def
thf(fact_1355_sum_OatLeast0__lessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N5: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N5 ) ) )
= ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N5 ) ) ) ) ) ).
% sum.atLeast0_lessThan_Suc_shift
thf(fact_1356_Chains__def,axiom,
! [A: $tType] :
( ( chains @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
( collect @ ( set @ A )
@ ^ [C7: set @ A] :
! [X4: A] :
( ( member @ A @ X4 @ C7 )
=> ! [Y5: A] :
( ( member @ A @ Y5 @ C7 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R5 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X4 ) @ R5 ) ) ) ) ) ) ) ).
% Chains_def
thf(fact_1357_conj__comp__iff,axiom,
! [B: $tType,A: $tType,P2: B > $o,Q2: B > $o,G: A > B] :
( ( comp @ B @ $o @ A
@ ^ [X4: B] :
( ( P2 @ X4 )
& ( Q2 @ X4 ) )
@ G )
= ( ^ [X4: A] :
( ( comp @ B @ $o @ A @ P2 @ G @ X4 )
& ( comp @ B @ $o @ A @ Q2 @ G @ X4 ) ) ) ) ).
% conj_comp_iff
thf(fact_1358_sum_OatLeast__Suc__lessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M4: nat,N5: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M4 ) @ ( suc @ N5 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ M4 @ N5 ) ) ) ) ).
% sum.atLeast_Suc_lessThan_Suc_shift
thf(fact_1359_sum_OatLeastLessThan__shift__bounds,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M4: nat,K: nat,N5: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M4 @ K ) @ ( plus_plus @ nat @ N5 @ K ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ K ) ) @ ( set_or7035219750837199246ssThan @ nat @ M4 @ N5 ) ) ) ) ).
% sum.atLeastLessThan_shift_bounds
thf(fact_1360_sum_OatLeastLessThan__shift__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M4: nat,N5: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M4 @ N5 ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M4 ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N5 @ M4 ) ) ) ) ) ).
% sum.atLeastLessThan_shift_0
thf(fact_1361_sum_OatLeast__lessThan__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M4: nat,N5: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ( comp @ nat @ A @ nat @ G
@ ^ [N4: nat] : ( minus_minus @ nat @ N4 @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M4 ) @ ( suc @ N5 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M4 @ N5 ) ) ) ) ).
% sum.atLeast_lessThan_pred_shift
thf(fact_1362_sum_OUNION__disjoint,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I4: set @ B,A6: B > ( set @ C ),G: C > A] :
( ( finite_finite2 @ B @ I4 )
=> ( ! [X: B] :
( ( member @ B @ X @ I4 )
=> ( finite_finite2 @ C @ ( A6 @ X ) ) )
=> ( ! [X: B] :
( ( member @ B @ X @ I4 )
=> ! [Xa: B] :
( ( member @ B @ Xa @ I4 )
=> ( ( X != Xa )
=> ( ( inf_inf @ ( set @ C ) @ ( A6 @ X ) @ ( A6 @ Xa ) )
= ( bot_bot @ ( set @ C ) ) ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ C @ A @ G @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ A6 @ I4 ) ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( groups7311177749621191930dd_sum @ C @ A @ G @ ( A6 @ X4 ) )
@ I4 ) ) ) ) ) ) ).
% sum.UNION_disjoint
thf(fact_1363_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 )
=> ( ! [I6: C] :
( ( member @ C @ I6 @ I4 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( G @ ( F @ I6 ) ) ) )
=> ( 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_1364_sum_Oeq__fold,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ( ( groups7311177749621191930dd_sum @ B @ A )
= ( ^ [G4: B > A] : ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( plus_plus @ A ) @ G4 ) @ ( zero_zero @ A ) ) ) ) ) ).
% sum.eq_fold
thf(fact_1365_sum_OUnion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [C5: set @ ( set @ B ),G: B > A] :
( ! [X: set @ B] :
( ( member @ ( set @ B ) @ X @ C5 )
=> ( finite_finite2 @ B @ X ) )
=> ( ! [X: set @ B] :
( ( member @ ( set @ B ) @ X @ C5 )
=> ! [Xa: set @ B] :
( ( member @ ( set @ B ) @ Xa @ C5 )
=> ( ( X != Xa )
=> ( ( inf_inf @ ( set @ B ) @ X @ Xa )
= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( complete_Sup_Sup @ ( set @ B ) @ C5 ) )
= ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7311177749621191930dd_sum @ ( set @ B ) @ A ) @ ( groups7311177749621191930dd_sum @ B @ A ) @ G @ C5 ) ) ) ) ) ).
% sum.Union_disjoint
thf(fact_1366_sum_Oreindex__nontrivial,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A6: set @ B,H: B > C,G: C > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ! [X: B,Y: B] :
( ( member @ B @ X @ A6 )
=> ( ( member @ B @ Y @ A6 )
=> ( ( X != Y )
=> ( ( ( H @ X )
= ( H @ Y ) )
=> ( ( G @ ( H @ X ) )
= ( zero_zero @ A ) ) ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ C @ A @ G @ ( image2 @ B @ C @ H @ A6 ) )
= ( groups7311177749621191930dd_sum @ B @ A @ ( comp @ C @ A @ B @ G @ H ) @ A6 ) ) ) ) ) ).
% sum.reindex_nontrivial
thf(fact_1367_sum_OSigma,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A6: set @ B,B6: B > ( set @ C ),G: B > C > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ! [X: B] :
( ( member @ B @ X @ A6 )
=> ( finite_finite2 @ C @ ( B6 @ X ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( groups7311177749621191930dd_sum @ C @ A @ ( G @ X4 ) @ ( B6 @ X4 ) )
@ A6 )
= ( groups7311177749621191930dd_sum @ ( product_prod @ B @ C ) @ A @ ( product_case_prod @ B @ C @ A @ G ) @ ( product_Sigma @ B @ C @ A6 @ B6 ) ) ) ) ) ) ).
% sum.Sigma
thf(fact_1368_sum_Ogroup,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S5: set @ B,T4: set @ C,G: B > C,H: B > A] :
( ( finite_finite2 @ B @ S5 )
=> ( ( finite_finite2 @ C @ T4 )
=> ( ( ord_less_eq @ ( set @ C ) @ ( image2 @ B @ C @ G @ S5 ) @ T4 )
=> ( ( groups7311177749621191930dd_sum @ C @ A
@ ^ [Y5: C] :
( groups7311177749621191930dd_sum @ B @ A @ H
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ S5 )
& ( ( G @ X4 )
= Y5 ) ) ) )
@ T4 )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ S5 ) ) ) ) ) ) ).
% sum.group
thf(fact_1369_sum_Oreindex,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [H: B > C,A6: set @ B,G: C > A] :
( ( inj_on @ B @ C @ H @ A6 )
=> ( ( groups7311177749621191930dd_sum @ C @ A @ G @ ( image2 @ B @ C @ H @ A6 ) )
= ( groups7311177749621191930dd_sum @ B @ A @ ( comp @ C @ A @ B @ G @ H ) @ A6 ) ) ) ) ).
% sum.reindex
thf(fact_1370_card__UN__le,axiom,
! [B: $tType,A: $tType,I4: set @ A,A6: A > ( set @ B )] :
( ( finite_finite2 @ A @ I4 )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A6 @ I4 ) ) )
@ ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [I3: A] : ( finite_card @ B @ ( A6 @ I3 ) )
@ I4 ) ) ) ).
% card_UN_le
thf(fact_1371_card__UN__disjoint,axiom,
! [B: $tType,A: $tType,I4: set @ A,A6: A > ( set @ B )] :
( ( finite_finite2 @ A @ I4 )
=> ( ! [X: A] :
( ( member @ A @ X @ I4 )
=> ( finite_finite2 @ B @ ( A6 @ X ) ) )
=> ( ! [X: A] :
( ( member @ A @ X @ I4 )
=> ! [Xa: A] :
( ( member @ A @ Xa @ I4 )
=> ( ( X != Xa )
=> ( ( inf_inf @ ( set @ B ) @ ( A6 @ X ) @ ( A6 @ Xa ) )
= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ( finite_card @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A6 @ I4 ) ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [I3: A] : ( finite_card @ B @ ( A6 @ I3 ) )
@ I4 ) ) ) ) ) ).
% card_UN_disjoint
thf(fact_1372_sum_Oreindex__cong,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [L: B > C,B6: set @ B,A6: set @ C,G: C > A,H: B > A] :
( ( inj_on @ B @ C @ L @ B6 )
=> ( ( A6
= ( image2 @ B @ C @ L @ B6 ) )
=> ( ! [X: B] :
( ( member @ B @ X @ B6 )
=> ( ( G @ ( L @ X ) )
= ( H @ X ) ) )
=> ( ( groups7311177749621191930dd_sum @ C @ A @ G @ A6 )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ B6 ) ) ) ) ) ) ).
% sum.reindex_cong
thf(fact_1373_sum_Oimage__gen,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S5: set @ B,H: B > A,G: B > C] :
( ( finite_finite2 @ B @ S5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ H @ S5 )
= ( groups7311177749621191930dd_sum @ C @ A
@ ^ [Y5: C] :
( groups7311177749621191930dd_sum @ B @ A @ H
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ S5 )
& ( ( G @ X4 )
= Y5 ) ) ) )
@ ( image2 @ B @ C @ G @ S5 ) ) ) ) ) ).
% sum.image_gen
thf(fact_1374_sum_Oimage__eq,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: B > A,A6: set @ B] :
( ( inj_on @ B @ A @ G @ A6 )
=> ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [X4: A] : X4
@ ( image2 @ B @ A @ G @ A6 ) )
= ( groups7311177749621191930dd_sum @ B @ A @ G @ A6 ) ) ) ) ).
% sum.image_eq
thf(fact_1375_sum_Ocartesian__product,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: B > C > A,B6: set @ C,A6: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( groups7311177749621191930dd_sum @ C @ A @ ( G @ X4 ) @ B6 )
@ A6 )
= ( groups7311177749621191930dd_sum @ ( product_prod @ B @ C ) @ A @ ( product_case_prod @ B @ C @ A @ G )
@ ( product_Sigma @ B @ C @ A6
@ ^ [Uu: B] : B6 ) ) ) ) ).
% sum.cartesian_product
thf(fact_1376_sum__comp__morphism,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( comm_monoid_add @ B )
& ( comm_monoid_add @ A ) )
=> ! [H: B > A,G: C > B,A6: set @ C] :
( ( ( H @ ( zero_zero @ B ) )
= ( zero_zero @ A ) )
=> ( ! [X: B,Y: B] :
( ( H @ ( plus_plus @ B @ X @ Y ) )
= ( plus_plus @ A @ ( H @ X ) @ ( H @ Y ) ) )
=> ( ( groups7311177749621191930dd_sum @ C @ A @ ( comp @ B @ A @ C @ H @ G ) @ A6 )
= ( H @ ( groups7311177749621191930dd_sum @ C @ B @ G @ A6 ) ) ) ) ) ) ).
% sum_comp_morphism
thf(fact_1377_sum_OUnion__comp,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [B6: set @ ( set @ B ),G: B > A] :
( ! [X: set @ B] :
( ( member @ ( set @ B ) @ X @ B6 )
=> ( finite_finite2 @ B @ X ) )
=> ( ! [A18: set @ B] :
( ( member @ ( set @ B ) @ A18 @ B6 )
=> ! [A26: set @ B] :
( ( member @ ( set @ B ) @ A26 @ B6 )
=> ( ( A18 != A26 )
=> ! [X: B] :
( ( member @ B @ X @ A18 )
=> ( ( member @ B @ X @ A26 )
=> ( ( G @ X )
= ( zero_zero @ A ) ) ) ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( complete_Sup_Sup @ ( set @ B ) @ B6 ) )
= ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7311177749621191930dd_sum @ ( set @ B ) @ A ) @ ( groups7311177749621191930dd_sum @ B @ A ) @ G @ B6 ) ) ) ) ) ).
% sum.Union_comp
thf(fact_1378_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_1379_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_1380_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 ) )
@ ^ [X4: A,Y5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R ) ) )
@ ( chains @ A @ R ) ) ) ).
% Chains_subset'
thf(fact_1381_sum_Otriangle__reindex__eq,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > nat > A,N5: 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,J: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I3 @ J ) @ N5 ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G @ I3 @ ( minus_minus @ nat @ K3 @ I3 ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_atMost @ nat @ N5 ) ) ) ) ).
% sum.triangle_reindex_eq
thf(fact_1382_sum__card__image,axiom,
! [B: $tType,A: $tType,A6: set @ A,F: A > ( set @ B )] :
( ( finite_finite2 @ A @ A6 )
=> ( ( pairwise @ A
@ ^ [S: A,T2: A] : ( disjnt @ B @ ( F @ S ) @ ( F @ T2 ) )
@ A6 )
=> ( ( groups7311177749621191930dd_sum @ ( set @ B ) @ nat @ ( finite_card @ B ) @ ( image2 @ A @ ( set @ B ) @ F @ A6 ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [A7: A] : ( finite_card @ B @ ( F @ A7 ) )
@ A6 ) ) ) ) ).
% sum_card_image
thf(fact_1383_sum_OatLeastAtMost__shift__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M4: nat,N5: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M4 @ N5 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M4 @ N5 ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M4 ) ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N5 @ M4 ) ) ) ) ) ) ).
% sum.atLeastAtMost_shift_0
thf(fact_1384_divmod__nat__if,axiom,
( divmod_nat
= ( ^ [M7: nat,N4: nat] :
( if @ ( product_prod @ nat @ nat )
@ ( ( N4
= ( zero_zero @ nat ) )
| ( ord_less @ nat @ M7 @ N4 ) )
@ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ M7 )
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [Q6: nat] : ( product_Pair @ nat @ nat @ ( suc @ Q6 ) )
@ ( divmod_nat @ ( minus_minus @ nat @ M7 @ N4 ) @ N4 ) ) ) ) ) ).
% divmod_nat_if
thf(fact_1385_Rats__eq__range__nat__to__rat__surj,axiom,
( ( field_char_0_Rats @ rat )
= ( image2 @ nat @ rat @ nat_to_rat_surj @ ( top_top @ ( set @ nat ) ) ) ) ).
% Rats_eq_range_nat_to_rat_surj
thf(fact_1386_image__add__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: A,I: A,J5: A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ K ) @ ( set_or1337092689740270186AtMost @ A @ I @ J5 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J5 @ K ) ) ) ) ).
% image_add_atLeastAtMost
thf(fact_1387_image__diff__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [D4: A,A3: A,B2: A] :
( ( image2 @ A @ A @ ( minus_minus @ A @ D4 ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ D4 @ B2 ) @ ( minus_minus @ A @ D4 @ A3 ) ) ) ) ).
% image_diff_atLeastAtMost
thf(fact_1388_image__add__atMost,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [C3: A,A3: A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ C3 ) @ ( set_ord_atMost @ A @ A3 ) )
= ( set_ord_atMost @ A @ ( plus_plus @ A @ C3 @ A3 ) ) ) ) ).
% image_add_atMost
thf(fact_1389_image__add__atLeastAtMost_H,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: A,I: A,J5: A] :
( ( image2 @ A @ A
@ ^ [N4: A] : ( plus_plus @ A @ N4 @ K )
@ ( set_or1337092689740270186AtMost @ A @ I @ J5 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J5 @ K ) ) ) ) ).
% image_add_atLeastAtMost'
thf(fact_1390_image__Suc__atLeastAtMost,axiom,
! [I: nat,J5: nat] :
( ( image2 @ nat @ nat @ suc @ ( set_or1337092689740270186AtMost @ nat @ I @ J5 ) )
= ( set_or1337092689740270186AtMost @ nat @ ( suc @ I ) @ ( suc @ J5 ) ) ) ).
% image_Suc_atLeastAtMost
thf(fact_1391_image__minus__const__atLeastAtMost_H,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [D4: A,A3: A,B2: A] :
( ( image2 @ A @ A
@ ^ [T2: A] : ( minus_minus @ A @ T2 @ D4 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ A3 @ D4 ) @ ( minus_minus @ A @ B2 @ D4 ) ) ) ) ).
% image_minus_const_atLeastAtMost'
thf(fact_1392_pairwise__imageI,axiom,
! [B: $tType,A: $tType,A6: set @ A,F: A > B,P2: B > B > $o] :
( ! [X: A,Y: A] :
( ( member @ A @ X @ A6 )
=> ( ( member @ A @ Y @ A6 )
=> ( ( X != Y )
=> ( ( ( F @ X )
!= ( F @ Y ) )
=> ( P2 @ ( F @ X ) @ ( F @ Y ) ) ) ) ) )
=> ( pairwise @ B @ P2 @ ( image2 @ A @ B @ F @ A6 ) ) ) ).
% pairwise_imageI
thf(fact_1393_UN__le__add__shift,axiom,
! [A: $tType,M: nat > ( set @ A ),K: nat,N5: nat] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ nat @ ( set @ A )
@ ^ [I3: nat] : ( M @ ( plus_plus @ nat @ I3 @ K ) )
@ ( set_ord_atMost @ nat @ N5 ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ M @ ( set_or1337092689740270186AtMost @ nat @ K @ ( plus_plus @ nat @ N5 @ K ) ) ) ) ) ).
% UN_le_add_shift
thf(fact_1394_pairwise__image,axiom,
! [A: $tType,B: $tType,R: A > A > $o,F: B > A,S2: set @ B] :
( ( pairwise @ A @ R @ ( image2 @ B @ A @ F @ S2 ) )
= ( pairwise @ B
@ ^ [X4: B,Y5: B] :
( ( ( F @ X4 )
!= ( F @ Y5 ) )
=> ( R @ ( F @ X4 ) @ ( F @ Y5 ) ) )
@ S2 ) ) ).
% pairwise_image
thf(fact_1395_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_1396_sum_OatLeast__Suc__atMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M4: nat,N5: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M4 ) @ ( suc @ N5 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ M4 @ N5 ) ) ) ) ).
% sum.atLeast_Suc_atMost_Suc_shift
thf(fact_1397_sum_OatLeastAtMost__shift__bounds,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M4: nat,K: nat,N5: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M4 @ K ) @ ( plus_plus @ nat @ N5 @ K ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ K ) ) @ ( set_or1337092689740270186AtMost @ nat @ M4 @ N5 ) ) ) ) ).
% sum.atLeastAtMost_shift_bounds
thf(fact_1398_atMost__Suc__eq__insert__0,axiom,
! [N5: nat] :
( ( set_ord_atMost @ nat @ ( suc @ N5 ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( image2 @ nat @ nat @ suc @ ( set_ord_atMost @ nat @ N5 ) ) ) ) ).
% atMost_Suc_eq_insert_0
thf(fact_1399_disjoint__image__subset,axiom,
! [A: $tType,A17: set @ ( set @ A ),F: ( set @ A ) > ( set @ A )] :
( ( pairwise @ ( set @ A ) @ ( disjnt @ A ) @ A17 )
=> ( ! [X9: set @ A] :
( ( member @ ( set @ A ) @ X9 @ A17 )
=> ( ord_less_eq @ ( set @ A ) @ ( F @ X9 ) @ X9 ) )
=> ( pairwise @ ( set @ A ) @ ( disjnt @ A ) @ ( image2 @ ( set @ A ) @ ( set @ A ) @ F @ A17 ) ) ) ) ).
% disjoint_image_subset
thf(fact_1400_atLeast0__atMost__Suc__eq__insert__0,axiom,
! [N5: nat] :
( ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N5 ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( image2 @ nat @ nat @ suc @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N5 ) ) ) ) ).
% atLeast0_atMost_Suc_eq_insert_0
thf(fact_1401_sum_OatLeast0__atMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N5: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N5 ) ) )
= ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N5 ) ) ) ) ) ).
% sum.atLeast0_atMost_Suc_shift
thf(fact_1402_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_1403_sum_OatLeast__atMost__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M4: nat,N5: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ( comp @ nat @ A @ nat @ G
@ ^ [N4: nat] : ( minus_minus @ nat @ N4 @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M4 ) @ ( suc @ N5 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M4 @ N5 ) ) ) ) ).
% sum.atLeast_atMost_pred_shift
thf(fact_1404_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 ) )
@ ^ [X4: A,Y5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R ) ) ) ) ) ).
% Chains_alt_def
thf(fact_1405_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 ) )
@ ^ [X4: A,Y5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R ) ) ) ) ).
% Chains_subset
thf(fact_1406_divides__aux__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [Q: A,R: A] :
( ( unique5940410009612947441es_aux @ A @ ( product_Pair @ A @ A @ Q @ R ) )
= ( R
= ( zero_zero @ A ) ) ) ) ).
% divides_aux_eq
thf(fact_1407_at__bot__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( at_bot @ A )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ A @ ( filter @ A )
@ ^ [K3: A] : ( principal @ A @ ( set_ord_atMost @ A @ K3 ) )
@ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% at_bot_def
thf(fact_1408_sum_Otriangle__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > nat > A,N5: 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,J: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I3 @ J ) @ N5 ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I3: nat] : ( G @ I3 @ ( minus_minus @ nat @ K3 @ I3 ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_lessThan @ nat @ N5 ) ) ) ) ).
% sum.triangle_reindex
thf(fact_1409_at__bot__sub,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C3: A] :
( ( at_bot @ A )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ A @ ( filter @ A )
@ ^ [K3: A] : ( principal @ A @ ( set_ord_atMost @ A @ K3 ) )
@ ( set_ord_atMost @ A @ C3 ) ) ) ) ) ).
% at_bot_sub
thf(fact_1410_UN__le__eq__Un0,axiom,
! [A: $tType,M: nat > ( set @ A ),N5: nat] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ M @ ( set_ord_atMost @ nat @ N5 ) ) )
= ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ M @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N5 ) ) ) @ ( M @ ( zero_zero @ nat ) ) ) ) ).
% UN_le_eq_Un0
thf(fact_1411_divides__aux__def,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique5940410009612947441es_aux @ A )
= ( ^ [Qr: product_prod @ A @ A] :
( ( product_snd @ A @ A @ Qr )
= ( zero_zero @ A ) ) ) ) ) ).
% divides_aux_def
thf(fact_1412_wo__rel_Osuc__ofilter__in,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A6: set @ A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R )
=> ( ( order_ofilter @ A @ R @ A6 )
=> ( ( ( order_AboveS @ A @ R @ A6 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ ( bNF_Wellorder_wo_suc @ A @ R @ A6 ) ) @ R )
=> ( ( B2
!= ( bNF_Wellorder_wo_suc @ A @ R @ A6 ) )
=> ( member @ A @ B2 @ A6 ) ) ) ) ) ) ).
% wo_rel.suc_ofilter_in
thf(fact_1413_image__affinity__atLeastAtMost__diff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,M4: A,C3: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ ( times_times @ A @ M4 @ X4 ) @ C3 )
@ ( 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 ) @ M4 )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ ( times_times @ A @ M4 @ X4 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( times_times @ A @ M4 @ A3 ) @ C3 ) @ ( minus_minus @ A @ ( times_times @ A @ M4 @ B2 ) @ C3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M4 )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ ( times_times @ A @ M4 @ X4 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( times_times @ A @ M4 @ B2 ) @ C3 ) @ ( minus_minus @ A @ ( times_times @ A @ M4 @ A3 ) @ C3 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_diff
thf(fact_1414_image__mult__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [D4: A,A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ D4 )
=> ( ( image2 @ A @ A @ ( times_times @ A @ D4 ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ D4 @ A3 ) @ ( times_times @ A @ D4 @ B2 ) ) ) ) ) ).
% image_mult_atLeastAtMost
thf(fact_1415_image__Suc__lessThan,axiom,
! [N5: nat] :
( ( image2 @ nat @ nat @ suc @ ( set_ord_lessThan @ nat @ N5 ) )
= ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N5 ) ) ).
% image_Suc_lessThan
thf(fact_1416_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_1417_wo__rel_Oofilter__UNION,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),I4: set @ B,A6: B > ( set @ A )] :
( ( bNF_Wellorder_wo_rel @ A @ R )
=> ( ! [I6: B] :
( ( member @ B @ I6 @ I4 )
=> ( order_ofilter @ A @ R @ ( A6 @ I6 ) ) )
=> ( order_ofilter @ A @ R @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A6 @ I4 ) ) ) ) ) ).
% wo_rel.ofilter_UNION
thf(fact_1418_lessThan__Suc__eq__insert__0,axiom,
! [N5: nat] :
( ( set_ord_lessThan @ nat @ ( suc @ N5 ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( image2 @ nat @ nat @ suc @ ( set_ord_lessThan @ nat @ N5 ) ) ) ) ).
% lessThan_Suc_eq_insert_0
thf(fact_1419_UN__le__add__shift__strict,axiom,
! [A: $tType,M: nat > ( set @ A ),K: nat,N5: nat] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ nat @ ( set @ A )
@ ^ [I3: nat] : ( M @ ( plus_plus @ nat @ I3 @ K ) )
@ ( set_ord_lessThan @ nat @ N5 ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ M @ ( set_or7035219750837199246ssThan @ nat @ K @ ( plus_plus @ nat @ N5 @ K ) ) ) ) ) ).
% UN_le_add_shift_strict
thf(fact_1420_image__Suc__atMost,axiom,
! [N5: nat] :
( ( image2 @ nat @ nat @ suc @ ( set_ord_atMost @ nat @ N5 ) )
= ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ ( suc @ N5 ) ) ) ).
% image_Suc_atMost
thf(fact_1421_sum__le__card__Max,axiom,
! [A: $tType,A6: set @ A,F: A > nat] :
( ( finite_finite2 @ A @ A6 )
=> ( ord_less_eq @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F @ A6 ) @ ( times_times @ nat @ ( finite_card @ A @ A6 ) @ ( lattic643756798349783984er_Max @ nat @ ( image2 @ A @ nat @ F @ A6 ) ) ) ) ) ).
% sum_le_card_Max
thf(fact_1422_image__mult__atLeastAtMost__if,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,X2: A,Y3: A] :
( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( image2 @ A @ A @ ( times_times @ A @ C3 ) @ ( set_or1337092689740270186AtMost @ A @ X2 @ Y3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ C3 @ X2 ) @ ( times_times @ A @ C3 @ Y3 ) ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less_eq @ A @ X2 @ Y3 )
=> ( ( image2 @ A @ A @ ( times_times @ A @ C3 ) @ ( set_or1337092689740270186AtMost @ A @ X2 @ Y3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ C3 @ Y3 ) @ ( times_times @ A @ C3 @ X2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ X2 @ Y3 )
=> ( ( image2 @ A @ A @ ( times_times @ A @ C3 ) @ ( set_or1337092689740270186AtMost @ A @ X2 @ Y3 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% image_mult_atLeastAtMost_if
thf(fact_1423_image__mult__atLeastAtMost__if_H,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,Y3: A,C3: A] :
( ( ( ord_less_eq @ A @ X2 @ Y3 )
=> ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( times_times @ A @ X4 @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ X2 @ Y3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ X2 @ C3 ) @ ( times_times @ A @ Y3 @ C3 ) ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( times_times @ A @ X4 @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ X2 @ Y3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ Y3 @ C3 ) @ ( times_times @ A @ X2 @ C3 ) ) ) ) ) )
& ( ~ ( ord_less_eq @ A @ X2 @ Y3 )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( times_times @ A @ X4 @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ X2 @ Y3 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% image_mult_atLeastAtMost_if'
thf(fact_1424_image__affinity__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,M4: A,C3: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( plus_plus @ A @ ( times_times @ A @ M4 @ X4 ) @ C3 )
@ ( 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 ) @ M4 )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( plus_plus @ A @ ( times_times @ A @ M4 @ X4 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( times_times @ A @ M4 @ A3 ) @ C3 ) @ ( plus_plus @ A @ ( times_times @ A @ M4 @ B2 ) @ C3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M4 )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( plus_plus @ A @ ( times_times @ A @ M4 @ X4 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( times_times @ A @ M4 @ B2 ) @ C3 ) @ ( plus_plus @ A @ ( times_times @ A @ M4 @ A3 ) @ C3 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost
thf(fact_1425_UNION__inj__on__ofilter,axiom,
! [C: $tType,A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),I4: set @ B,A6: B > ( set @ A ),F: A > C] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ! [I6: B] :
( ( member @ B @ I6 @ I4 )
=> ( order_ofilter @ A @ R @ ( A6 @ I6 ) ) )
=> ( ! [I6: B] :
( ( member @ B @ I6 @ I4 )
=> ( inj_on @ A @ C @ F @ ( A6 @ I6 ) ) )
=> ( inj_on @ A @ C @ F @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A6 @ I4 ) ) ) ) ) ) ).
% UNION_inj_on_ofilter
thf(fact_1426_ofilterIncl__def,axiom,
! [A: $tType] :
( ( bNF_We413866401316099525erIncl @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
@ ^ [A8: set @ A,B7: set @ A] :
( ( order_ofilter @ A @ R5 @ A8 )
& ( A8
!= ( field2 @ A @ R5 ) )
& ( order_ofilter @ A @ R5 @ B7 )
& ( B7
!= ( field2 @ A @ R5 ) )
& ( ord_less @ ( set @ A ) @ A8 @ B7 ) ) ) ) ) ) ).
% ofilterIncl_def
thf(fact_1427_sum__mult__sum__if__inj,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( semiring_0 @ B )
=> ! [F: A > B,G: C > B,A6: set @ A,B6: set @ C] :
( ( inj_on @ ( product_prod @ A @ C ) @ B
@ ( product_case_prod @ A @ C @ B
@ ^ [A7: A,B5: C] : ( times_times @ B @ ( F @ A7 ) @ ( G @ B5 ) ) )
@ ( product_Sigma @ A @ C @ A6
@ ^ [Uu: A] : B6 ) )
=> ( ( times_times @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F @ A6 ) @ ( groups7311177749621191930dd_sum @ C @ B @ G @ B6 ) )
= ( groups7311177749621191930dd_sum @ B @ B @ ( id @ B )
@ ( collect @ B
@ ^ [Uu: B] :
? [A7: A,B5: C] :
( ( Uu
= ( times_times @ B @ ( F @ A7 ) @ ( G @ B5 ) ) )
& ( member @ A @ A7 @ A6 )
& ( member @ C @ B5 @ B6 ) ) ) ) ) ) ) ).
% sum_mult_sum_if_inj
thf(fact_1428_Ref_Oalloc__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( alloc2 @ A )
= ( ^ [X4: A,H6: heap_ext @ product_unit] :
( product_Pair @ ( ref @ A ) @ ( heap_ext @ product_unit ) @ ( ref2 @ A @ ( lim @ product_unit @ H6 ) )
@ ( set4 @ A @ ( ref2 @ A @ ( lim @ product_unit @ H6 ) ) @ X4
@ ( lim_update @ product_unit
@ ^ [Uu: nat] : ( plus_plus @ nat @ ( lim @ product_unit @ H6 ) @ ( one_one @ nat ) )
@ H6 ) ) ) ) ) ) ).
% Ref.alloc_def
thf(fact_1429_image__affinity__atLeastAtMost__div__diff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,M4: A,C3: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ ( divide_divide @ A @ X4 @ M4 ) @ C3 )
@ ( 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 ) @ M4 )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ ( divide_divide @ A @ X4 @ M4 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( divide_divide @ A @ A3 @ M4 ) @ C3 ) @ ( minus_minus @ A @ ( divide_divide @ A @ B2 @ M4 ) @ C3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M4 )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ ( divide_divide @ A @ X4 @ M4 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( divide_divide @ A @ B2 @ M4 ) @ C3 ) @ ( minus_minus @ A @ ( divide_divide @ A @ A3 @ M4 ) @ C3 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_div_diff
thf(fact_1430_image__affinity__atLeastAtMost__div,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,M4: A,C3: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( plus_plus @ A @ ( divide_divide @ A @ X4 @ M4 ) @ C3 )
@ ( 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 ) @ M4 )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( plus_plus @ A @ ( divide_divide @ A @ X4 @ M4 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( divide_divide @ A @ A3 @ M4 ) @ C3 ) @ ( plus_plus @ A @ ( divide_divide @ A @ B2 @ M4 ) @ C3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M4 )
=> ( ( image2 @ A @ A
@ ^ [X4: A] : ( plus_plus @ A @ ( divide_divide @ A @ X4 @ M4 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( divide_divide @ A @ B2 @ M4 ) @ C3 ) @ ( plus_plus @ A @ ( divide_divide @ A @ A3 @ M4 ) @ C3 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_div
thf(fact_1431_sum__fun__comp,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( semiring_1 @ C )
=> ! [S5: set @ A,R3: set @ B,G: A > B,F: B > C] :
( ( finite_finite2 @ A @ S5 )
=> ( ( finite_finite2 @ B @ R3 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ G @ S5 ) @ R3 )
=> ( ( groups7311177749621191930dd_sum @ A @ C
@ ^ [X4: A] : ( F @ ( G @ X4 ) )
@ S5 )
= ( groups7311177749621191930dd_sum @ B @ C
@ ^ [Y5: B] :
( times_times @ C
@ ( semiring_1_of_nat @ C
@ ( finite_card @ A
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ S5 )
& ( ( G @ X4 )
= Y5 ) ) ) ) )
@ ( F @ Y5 ) )
@ R3 ) ) ) ) ) ) ).
% sum_fun_comp
thf(fact_1432_prod_OatLeastAtMost__shift__0,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M4: nat,N5: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M4 @ N5 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M4 @ N5 ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M4 ) ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N5 @ M4 ) ) ) ) ) ) ).
% prod.atLeastAtMost_shift_0
thf(fact_1433_Domain__Collect__case__prod,axiom,
! [B: $tType,A: $tType,P2: A > B > $o] :
( ( domain @ A @ B @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P2 ) ) )
= ( collect @ A
@ ^ [X4: A] :
? [X10: B] : ( P2 @ X4 @ X10 ) ) ) ).
% Domain_Collect_case_prod
thf(fact_1434_Range__Collect__case__prod,axiom,
! [B: $tType,A: $tType,P2: B > A > $o] :
( ( range2 @ B @ A @ ( collect @ ( product_prod @ B @ A ) @ ( product_case_prod @ B @ A @ $o @ P2 ) ) )
= ( collect @ A
@ ^ [Y5: A] :
? [X4: B] : ( P2 @ X4 @ Y5 ) ) ) ).
% Range_Collect_case_prod
thf(fact_1435_image__divide__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [D4: A,A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ D4 )
=> ( ( image2 @ A @ A
@ ^ [C6: A] : ( divide_divide @ A @ C6 @ D4 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( divide_divide @ A @ A3 @ D4 ) @ ( divide_divide @ A @ B2 @ D4 ) ) ) ) ) ).
% image_divide_atLeastAtMost
thf(fact_1436_prod_Ocartesian__product,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: B > C > A,B6: set @ C,A6: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X4: B] : ( groups7121269368397514597t_prod @ C @ A @ ( G @ X4 ) @ B6 )
@ A6 )
= ( groups7121269368397514597t_prod @ ( product_prod @ B @ C ) @ A @ ( product_case_prod @ B @ C @ A @ G )
@ ( product_Sigma @ B @ C @ A6
@ ^ [Uu: B] : B6 ) ) ) ) ).
% prod.cartesian_product
thf(fact_1437_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_1438_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_1439_setcompr__eq__image,axiom,
! [A: $tType,B: $tType,F: B > A,P2: B > $o] :
( ( collect @ A
@ ^ [Uu: A] :
? [X4: B] :
( ( Uu
= ( F @ X4 ) )
& ( P2 @ X4 ) ) )
= ( image2 @ B @ A @ F @ ( collect @ B @ P2 ) ) ) ).
% setcompr_eq_image
thf(fact_1440_Setcompr__eq__image,axiom,
! [A: $tType,B: $tType,F: B > A,A6: set @ B] :
( ( collect @ A
@ ^ [Uu: A] :
? [X4: B] :
( ( Uu
= ( F @ X4 ) )
& ( member @ B @ X4 @ A6 ) ) )
= ( image2 @ B @ A @ F @ A6 ) ) ).
% Setcompr_eq_image
thf(fact_1441_prod_OSigma,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A6: set @ B,B6: B > ( set @ C ),G: B > C > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ! [X: B] :
( ( member @ B @ X @ A6 )
=> ( finite_finite2 @ C @ ( B6 @ X ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X4: B] : ( groups7121269368397514597t_prod @ C @ A @ ( G @ X4 ) @ ( B6 @ X4 ) )
@ A6 )
= ( groups7121269368397514597t_prod @ ( product_prod @ B @ C ) @ A @ ( product_case_prod @ B @ C @ A @ G ) @ ( product_Sigma @ B @ C @ A6 @ B6 ) ) ) ) ) ) ).
% prod.Sigma
thf(fact_1442_OO__def,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( relcompp @ A @ C @ B )
= ( ^ [R6: A > C > $o,S4: C > B > $o,X4: A,Z5: B] :
? [Y5: C] :
( ( R6 @ X4 @ Y5 )
& ( S4 @ Y5 @ Z5 ) ) ) ) ).
% OO_def
thf(fact_1443_prod_Oreindex__cong,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [L: B > C,B6: set @ B,A6: set @ C,G: C > A,H: B > A] :
( ( inj_on @ B @ C @ L @ B6 )
=> ( ( A6
= ( image2 @ B @ C @ L @ B6 ) )
=> ( ! [X: B] :
( ( member @ B @ X @ B6 )
=> ( ( G @ ( L @ X ) )
= ( H @ X ) ) )
=> ( ( groups7121269368397514597t_prod @ C @ A @ G @ A6 )
= ( groups7121269368397514597t_prod @ B @ A @ H @ B6 ) ) ) ) ) ) ).
% prod.reindex_cong
thf(fact_1444_prod_OUnion__comp,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [B6: set @ ( set @ B ),G: B > A] :
( ! [X: set @ B] :
( ( member @ ( set @ B ) @ X @ B6 )
=> ( finite_finite2 @ B @ X ) )
=> ( ! [A18: set @ B] :
( ( member @ ( set @ B ) @ A18 @ B6 )
=> ! [A26: set @ B] :
( ( member @ ( set @ B ) @ A26 @ B6 )
=> ( ( A18 != A26 )
=> ! [X: B] :
( ( member @ B @ X @ A18 )
=> ( ( member @ B @ X @ A26 )
=> ( ( G @ X )
= ( 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_1445_image2p__def,axiom,
! [D: $tType,B: $tType,A: $tType,C: $tType] :
( ( bNF_Greatest_image2p @ C @ A @ D @ B )
= ( ^ [F4: C > A,G4: D > B,R6: C > D > $o,X4: A,Y5: B] :
? [X7: C,Y6: D] :
( ( R6 @ X7 @ Y6 )
& ( ( F4 @ X7 )
= X4 )
& ( ( G4 @ Y6 )
= Y5 ) ) ) ) ).
% image2p_def
thf(fact_1446_prod_Oimage__gen,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S5: set @ B,H: B > A,G: B > C] :
( ( finite_finite2 @ B @ S5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ H @ S5 )
= ( groups7121269368397514597t_prod @ C @ A
@ ^ [Y5: C] :
( groups7121269368397514597t_prod @ B @ A @ H
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ S5 )
& ( ( G @ X4 )
= Y5 ) ) ) )
@ ( image2 @ B @ C @ G @ S5 ) ) ) ) ) ).
% prod.image_gen
thf(fact_1447_sum_OatLeast__int__atMost__int__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: int > A,M4: nat,N5: nat] :
( ( groups7311177749621191930dd_sum @ int @ A @ G @ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ M4 ) @ ( semiring_1_of_nat @ int @ N5 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ int @ A @ nat @ G @ ( semiring_1_of_nat @ int ) ) @ ( set_or1337092689740270186AtMost @ nat @ M4 @ N5 ) ) ) ) ).
% sum.atLeast_int_atMost_int_shift
thf(fact_1448_sum_OatLeast__int__lessThan__int__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: int > A,M4: nat,N5: nat] :
( ( groups7311177749621191930dd_sum @ int @ A @ G @ ( set_or7035219750837199246ssThan @ int @ ( semiring_1_of_nat @ int @ M4 ) @ ( semiring_1_of_nat @ int @ N5 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ int @ A @ nat @ G @ ( semiring_1_of_nat @ int ) ) @ ( set_or7035219750837199246ssThan @ nat @ M4 @ N5 ) ) ) ) ).
% sum.atLeast_int_lessThan_int_shift
thf(fact_1449_prod_Oimage__eq,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: B > A,A6: set @ B] :
( ( inj_on @ B @ A @ G @ A6 )
=> ( ( groups7121269368397514597t_prod @ A @ A
@ ^ [X4: A] : X4
@ ( image2 @ B @ A @ G @ A6 ) )
= ( groups7121269368397514597t_prod @ B @ A @ G @ A6 ) ) ) ) ).
% prod.image_eq
thf(fact_1450_full__SetCompr__eq,axiom,
! [A: $tType,B: $tType,F: B > A] :
( ( collect @ A
@ ^ [U: A] :
? [X4: B] :
( U
= ( F @ X4 ) ) )
= ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ).
% full_SetCompr_eq
thf(fact_1451_prod_OUnion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C5: set @ ( set @ B ),G: B > A] :
( ! [X: set @ B] :
( ( member @ ( set @ B ) @ X @ C5 )
=> ( finite_finite2 @ B @ X ) )
=> ( ! [X: set @ B] :
( ( member @ ( set @ B ) @ X @ C5 )
=> ! [Xa: set @ B] :
( ( member @ ( set @ B ) @ Xa @ C5 )
=> ( ( X != Xa )
=> ( ( inf_inf @ ( set @ B ) @ X @ Xa )
= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( complete_Sup_Sup @ ( set @ B ) @ C5 ) )
= ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7121269368397514597t_prod @ ( set @ B ) @ A ) @ ( groups7121269368397514597t_prod @ B @ A ) @ G @ C5 ) ) ) ) ) ).
% prod.Union_disjoint
thf(fact_1452_Id__def,axiom,
! [A: $tType] :
( ( id2 @ A )
= ( collect @ ( product_prod @ A @ A )
@ ^ [P3: product_prod @ A @ A] :
? [X4: A] :
( P3
= ( product_Pair @ A @ A @ X4 @ X4 ) ) ) ) ).
% Id_def
thf(fact_1453_Domain__unfold,axiom,
! [B: $tType,A: $tType] :
( ( domain @ A @ B )
= ( ^ [R5: set @ ( product_prod @ A @ B )] :
( collect @ A
@ ^ [X4: A] :
? [Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ R5 ) ) ) ) ).
% Domain_unfold
thf(fact_1454_Gr__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Gr @ A @ B )
= ( ^ [A8: set @ A,F4: A > B] :
( collect @ ( product_prod @ A @ B )
@ ^ [Uu: product_prod @ A @ B] :
? [A7: A] :
( ( Uu
= ( product_Pair @ A @ B @ A7 @ ( F4 @ A7 ) ) )
& ( member @ A @ A7 @ A8 ) ) ) ) ) ).
% Gr_def
thf(fact_1455_relImage__def,axiom,
! [A: $tType,B: $tType] :
( ( bNF_Gr4221423524335903396lImage @ B @ A )
= ( ^ [R6: set @ ( product_prod @ B @ B ),F4: B > A] :
( collect @ ( product_prod @ A @ A )
@ ^ [Uu: product_prod @ A @ A] :
? [A12: B,A23: B] :
( ( Uu
= ( product_Pair @ A @ A @ ( F4 @ A12 ) @ ( F4 @ A23 ) ) )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A12 @ A23 ) @ R6 ) ) ) ) ) ).
% relImage_def
thf(fact_1456_relInvImage__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Gr7122648621184425601vImage @ A @ B )
= ( ^ [A8: set @ A,R6: set @ ( product_prod @ B @ B ),F4: 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 @ A8 )
& ( member @ A @ A23 @ A8 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F4 @ A12 ) @ ( F4 @ A23 ) ) @ R6 ) ) ) ) ) ).
% relInvImage_def
thf(fact_1457_prod_Oreindex,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [H: B > C,A6: set @ B,G: C > A] :
( ( inj_on @ B @ C @ H @ A6 )
=> ( ( groups7121269368397514597t_prod @ C @ A @ G @ ( image2 @ B @ C @ H @ A6 ) )
= ( groups7121269368397514597t_prod @ B @ A @ ( comp @ C @ A @ B @ G @ H ) @ A6 ) ) ) ) ).
% prod.reindex
thf(fact_1458_prod_OatLeast__Suc__atMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M4: nat,N5: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M4 ) @ ( suc @ N5 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ M4 @ N5 ) ) ) ) ).
% prod.atLeast_Suc_atMost_Suc_shift
thf(fact_1459_prod_OatLeast__Suc__lessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M4: nat,N5: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M4 ) @ ( suc @ N5 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ M4 @ N5 ) ) ) ) ).
% prod.atLeast_Suc_lessThan_Suc_shift
thf(fact_1460_prod_Ogroup,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S5: set @ B,T4: set @ C,G: B > C,H: B > A] :
( ( finite_finite2 @ B @ S5 )
=> ( ( finite_finite2 @ C @ T4 )
=> ( ( ord_less_eq @ ( set @ C ) @ ( image2 @ B @ C @ G @ S5 ) @ T4 )
=> ( ( groups7121269368397514597t_prod @ C @ A
@ ^ [Y5: C] :
( groups7121269368397514597t_prod @ B @ A @ H
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ S5 )
& ( ( G @ X4 )
= Y5 ) ) ) )
@ T4 )
= ( groups7121269368397514597t_prod @ B @ A @ H @ S5 ) ) ) ) ) ) ).
% prod.group
thf(fact_1461_prod_OatLeastAtMost__shift__bounds,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M4: nat,K: nat,N5: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M4 @ K ) @ ( plus_plus @ nat @ N5 @ K ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ K ) ) @ ( set_or1337092689740270186AtMost @ nat @ M4 @ N5 ) ) ) ) ).
% prod.atLeastAtMost_shift_bounds
thf(fact_1462_prod_OatLeastLessThan__shift__bounds,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M4: nat,K: nat,N5: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M4 @ K ) @ ( plus_plus @ nat @ N5 @ K ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ K ) ) @ ( set_or7035219750837199246ssThan @ nat @ M4 @ N5 ) ) ) ) ).
% prod.atLeastLessThan_shift_bounds
thf(fact_1463_image2__def,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( bNF_Greatest_image2 @ C @ A @ B )
= ( ^ [A8: set @ C,F4: C > A,G4: C > B] :
( collect @ ( product_prod @ A @ B )
@ ^ [Uu: product_prod @ A @ B] :
? [A7: C] :
( ( Uu
= ( product_Pair @ A @ B @ ( F4 @ A7 ) @ ( G4 @ A7 ) ) )
& ( member @ C @ A7 @ A8 ) ) ) ) ) ).
% image2_def
thf(fact_1464_Collect__ex__eq,axiom,
! [A: $tType,B: $tType,P2: A > B > $o] :
( ( collect @ A
@ ^ [X4: A] :
? [X10: B] : ( P2 @ X4 @ X10 ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y5: B] :
( collect @ A
@ ^ [X4: A] : ( P2 @ X4 @ Y5 ) )
@ ( top_top @ ( set @ B ) ) ) ) ) ).
% Collect_ex_eq
thf(fact_1465_relcomp__unfold,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( relcomp @ A @ C @ B )
= ( ^ [R5: set @ ( product_prod @ A @ C ),S: set @ ( product_prod @ C @ B )] :
( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X4: A,Z5: B] :
? [Y5: C] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X4 @ Y5 ) @ R5 )
& ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ Y5 @ Z5 ) @ S ) ) ) ) ) ) ).
% relcomp_unfold
thf(fact_1466_OO__Grp__alt,axiom,
! [B: $tType,C: $tType,A: $tType,A6: set @ C,F: C > A,G: C > B] :
( ( relcompp @ A @ C @ B @ ( conversep @ C @ A @ ( bNF_Grp @ C @ A @ A6 @ F ) ) @ ( bNF_Grp @ C @ B @ A6 @ G ) )
= ( ^ [X4: A,Y5: B] :
? [Z5: C] :
( ( member @ C @ Z5 @ A6 )
& ( ( F @ Z5 )
= X4 )
& ( ( G @ Z5 )
= Y5 ) ) ) ) ).
% OO_Grp_alt
thf(fact_1467_Inf__Sup,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A6: set @ ( set @ A )] :
( ( complete_Inf_Inf @ A @ ( image2 @ ( set @ A ) @ A @ ( complete_Sup_Sup @ A ) @ A6 ) )
= ( complete_Sup_Sup @ A
@ ( image2 @ ( set @ A ) @ A @ ( complete_Inf_Inf @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu: set @ A] :
? [F4: ( set @ A ) > A] :
( ( Uu
= ( image2 @ ( set @ A ) @ A @ F4 @ A6 ) )
& ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ A6 )
=> ( member @ A @ ( F4 @ X4 ) @ X4 ) ) ) ) ) ) ) ) ).
% Inf_Sup
thf(fact_1468_Sup__Inf,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A6: set @ ( set @ A )] :
( ( complete_Sup_Sup @ A @ ( image2 @ ( set @ A ) @ A @ ( complete_Inf_Inf @ A ) @ A6 ) )
= ( complete_Inf_Inf @ A
@ ( image2 @ ( set @ A ) @ A @ ( complete_Sup_Sup @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu: set @ A] :
? [F4: ( set @ A ) > A] :
( ( Uu
= ( image2 @ ( set @ A ) @ A @ F4 @ A6 ) )
& ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ A6 )
=> ( member @ A @ ( F4 @ X4 ) @ X4 ) ) ) ) ) ) ) ) ).
% Sup_Inf
thf(fact_1469_prod_Oreindex__nontrivial,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A6: set @ B,H: B > C,G: C > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ! [X: B,Y: B] :
( ( member @ B @ X @ A6 )
=> ( ( member @ B @ Y @ A6 )
=> ( ( X != Y )
=> ( ( ( H @ X )
= ( H @ Y ) )
=> ( ( G @ ( H @ X ) )
= ( one_one @ A ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ C @ A @ G @ ( image2 @ B @ C @ H @ A6 ) )
= ( groups7121269368397514597t_prod @ B @ A @ ( comp @ C @ A @ B @ G @ H ) @ A6 ) ) ) ) ) ).
% prod.reindex_nontrivial
thf(fact_1470_prod_Oeq__fold,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ( ( groups7121269368397514597t_prod @ B @ A )
= ( ^ [G4: B > A] : ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( times_times @ A ) @ G4 ) @ ( one_one @ A ) ) ) ) ) ).
% prod.eq_fold
thf(fact_1471_Inf__Sup__le,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A6: set @ ( set @ A )] :
( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ ( set @ A ) @ A @ ( complete_Sup_Sup @ A ) @ A6 ) )
@ ( complete_Sup_Sup @ A
@ ( image2 @ ( set @ A ) @ A @ ( complete_Inf_Inf @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu: set @ A] :
? [F4: ( set @ A ) > A] :
( ( Uu
= ( image2 @ ( set @ A ) @ A @ F4 @ A6 ) )
& ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ A6 )
=> ( member @ A @ ( F4 @ X4 ) @ X4 ) ) ) ) ) ) ) ) ).
% Inf_Sup_le
thf(fact_1472_Sup__Inf__le,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ ( set @ A )] :
( ord_less_eq @ A
@ ( complete_Sup_Sup @ A
@ ( image2 @ ( set @ A ) @ A @ ( complete_Inf_Inf @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu: set @ A] :
? [F4: ( set @ A ) > A] :
( ( Uu
= ( image2 @ ( set @ A ) @ A @ F4 @ A6 ) )
& ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ A6 )
=> ( member @ A @ ( F4 @ X4 ) @ X4 ) ) ) ) ) )
@ ( complete_Inf_Inf @ A @ ( image2 @ ( set @ A ) @ A @ ( complete_Sup_Sup @ A ) @ A6 ) ) ) ) ).
% Sup_Inf_le
thf(fact_1473_finite__Inf__Sup,axiom,
! [A: $tType] :
( ( finite8700451911770168679attice @ A )
=> ! [A6: set @ ( set @ A )] :
( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ ( set @ A ) @ A @ ( complete_Sup_Sup @ A ) @ A6 ) )
@ ( complete_Sup_Sup @ A
@ ( image2 @ ( set @ A ) @ A @ ( complete_Inf_Inf @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu: set @ A] :
? [F4: ( set @ A ) > A] :
( ( Uu
= ( image2 @ ( set @ A ) @ A @ F4 @ A6 ) )
& ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ A6 )
=> ( member @ A @ ( F4 @ X4 ) @ X4 ) ) ) ) ) ) ) ) ).
% finite_Inf_Sup
thf(fact_1474_INF__SUP__set,axiom,
! [A: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [G: B > A,A6: set @ ( set @ B )] :
( ( complete_Inf_Inf @ A
@ ( image2 @ ( set @ B ) @ A
@ ^ [B7: set @ B] : ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G @ B7 ) )
@ A6 ) )
= ( complete_Sup_Sup @ A
@ ( image2 @ ( set @ B ) @ A
@ ^ [B7: set @ B] : ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G @ B7 ) )
@ ( collect @ ( set @ B )
@ ^ [Uu: set @ B] :
? [F4: ( set @ B ) > B] :
( ( Uu
= ( image2 @ ( set @ B ) @ B @ F4 @ A6 ) )
& ! [X4: set @ B] :
( ( member @ ( set @ B ) @ X4 @ A6 )
=> ( member @ B @ ( F4 @ X4 ) @ X4 ) ) ) ) ) ) ) ) ).
% INF_SUP_set
thf(fact_1475_SUP__INF__set,axiom,
! [A: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [G: B > A,A6: set @ ( set @ B )] :
( ( complete_Sup_Sup @ A
@ ( image2 @ ( set @ B ) @ A
@ ^ [X4: set @ B] : ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G @ X4 ) )
@ A6 ) )
= ( complete_Inf_Inf @ A
@ ( image2 @ ( set @ B ) @ A
@ ^ [X4: set @ B] : ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G @ X4 ) )
@ ( collect @ ( set @ B )
@ ^ [Uu: set @ B] :
? [F4: ( set @ B ) > B] :
( ( Uu
= ( image2 @ ( set @ B ) @ B @ F4 @ A6 ) )
& ! [X4: set @ B] :
( ( member @ ( set @ B ) @ X4 @ A6 )
=> ( member @ B @ ( F4 @ X4 ) @ X4 ) ) ) ) ) ) ) ) ).
% SUP_INF_set
thf(fact_1476_prod_OatLeast0__atMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N5: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N5 ) ) )
= ( times_times @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N5 ) ) ) ) ) ).
% prod.atLeast0_atMost_Suc_shift
thf(fact_1477_prod_OatLeast0__lessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N5: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N5 ) ) )
= ( times_times @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N5 ) ) ) ) ) ).
% prod.atLeast0_lessThan_Suc_shift
thf(fact_1478_prod_OatLeastLessThan__shift__0,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M4: nat,N5: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M4 @ N5 ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M4 ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N5 @ M4 ) ) ) ) ) ).
% prod.atLeastLessThan_shift_0
thf(fact_1479_prod_OUNION__disjoint,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I4: set @ B,A6: B > ( set @ C ),G: C > A] :
( ( finite_finite2 @ B @ I4 )
=> ( ! [X: B] :
( ( member @ B @ X @ I4 )
=> ( finite_finite2 @ C @ ( A6 @ X ) ) )
=> ( ! [X: B] :
( ( member @ B @ X @ I4 )
=> ! [Xa: B] :
( ( member @ B @ Xa @ I4 )
=> ( ( X != Xa )
=> ( ( inf_inf @ ( set @ C ) @ ( A6 @ X ) @ ( A6 @ Xa ) )
= ( bot_bot @ ( set @ C ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ C @ A @ G @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ A6 @ I4 ) ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X4: B] : ( groups7121269368397514597t_prod @ C @ A @ G @ ( A6 @ X4 ) )
@ I4 ) ) ) ) ) ) ).
% prod.UNION_disjoint
thf(fact_1480_prod_OatLeast__atMost__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M4: nat,N5: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ( comp @ nat @ A @ nat @ G
@ ^ [N4: nat] : ( minus_minus @ nat @ N4 @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M4 ) @ ( suc @ N5 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M4 @ N5 ) ) ) ) ).
% prod.atLeast_atMost_pred_shift
thf(fact_1481_prod_OatLeast__lessThan__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M4: nat,N5: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ( comp @ nat @ A @ nat @ G
@ ^ [N4: nat] : ( minus_minus @ nat @ N4 @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M4 ) @ ( suc @ N5 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M4 @ N5 ) ) ) ) ).
% prod.atLeast_lessThan_pred_shift
thf(fact_1482_len_H__def,axiom,
! [A: $tType,B: $tType] :
( ( ( heap @ B )
& ( semiring_1 @ A ) )
=> ( ( len2 @ B @ A )
= ( ^ [A7: array @ B] :
( heap_bind @ nat @ A @ ( len @ B @ A7 )
@ ^ [N4: nat] : ( heap_return @ A @ ( semiring_1_of_nat @ A @ N4 ) ) ) ) ) ) ).
% len'_def
thf(fact_1483_fold__congs_I3_J,axiom,
! [Z10: $tType,R: heap_ext @ Z10,R2: heap_ext @ Z10,V2: nat,F: nat > nat,F5: nat > nat] :
( ( R = R2 )
=> ( ( ( lim @ Z10 @ R2 )
= V2 )
=> ( ! [V6: nat] :
( ( V2 = V6 )
=> ( ( F @ V6 )
= ( F5 @ V6 ) ) )
=> ( ( lim_update @ Z10 @ F @ R )
= ( lim_update @ Z10 @ F5 @ R2 ) ) ) ) ) ).
% fold_congs(3)
thf(fact_1484_unfold__congs_I3_J,axiom,
! [Z10: $tType,R: heap_ext @ Z10,R2: heap_ext @ Z10,V2: nat,F: nat > nat,F5: nat > nat] :
( ( R = R2 )
=> ( ( ( lim @ Z10 @ R2 )
= V2 )
=> ( ! [V6: nat] :
( ( V6 = V2 )
=> ( ( F @ V6 )
= ( F5 @ V6 ) ) )
=> ( ( lim_update @ Z10 @ F @ R )
= ( lim_update @ Z10 @ F5 @ R2 ) ) ) ) ) ).
% unfold_congs(3)
thf(fact_1485_prod_Otriangle__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > nat > A,N5: 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,J: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I3 @ J ) @ N5 ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G @ I3 @ ( minus_minus @ nat @ K3 @ I3 ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_lessThan @ nat @ N5 ) ) ) ) ).
% prod.triangle_reindex
thf(fact_1486_prod_Otriangle__reindex__eq,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > nat > A,N5: 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,J: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I3 @ J ) @ N5 ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I3: nat] : ( G @ I3 @ ( minus_minus @ nat @ K3 @ I3 ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_atMost @ nat @ N5 ) ) ) ) ).
% prod.triangle_reindex_eq
thf(fact_1487_card__Min__le__sum,axiom,
! [A: $tType,A6: set @ A,F: A > nat] :
( ( finite_finite2 @ A @ A6 )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ ( finite_card @ A @ A6 ) @ ( lattic643756798350308766er_Min @ nat @ ( image2 @ A @ nat @ F @ A6 ) ) ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F @ A6 ) ) ) ).
% card_Min_le_sum
thf(fact_1488_embedS__Field,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),R2: set @ ( product_prod @ B @ B ),F: A > B] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( bNF_Wellorder_embedS @ A @ B @ R @ R2 @ F )
=> ( ord_less @ ( set @ B ) @ ( image2 @ A @ B @ F @ ( field2 @ A @ R ) ) @ ( field2 @ B @ R2 ) ) ) ) ).
% embedS_Field
thf(fact_1489_wo__rel_Oofilter__under__UNION,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A6: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R )
=> ( ( order_ofilter @ A @ R @ A6 )
=> ( A6
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ A @ ( set @ A ) @ ( order_under @ A @ R ) @ A6 ) ) ) ) ) ).
% wo_rel.ofilter_under_UNION
thf(fact_1490_Min__const,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ B,C3: A] :
( ( finite_finite2 @ B @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798350308766er_Min @ A
@ ( image2 @ B @ A
@ ^ [Uu: B] : C3
@ A6 ) )
= C3 ) ) ) ) ).
% Min_const
thf(fact_1491_image__add__int__atLeastLessThan,axiom,
! [L: int,U2: int] :
( ( image2 @ int @ int
@ ^ [X4: int] : ( plus_plus @ int @ X4 @ L )
@ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ ( minus_minus @ int @ U2 @ L ) ) )
= ( set_or7035219750837199246ssThan @ int @ L @ U2 ) ) ).
% image_add_int_atLeastLessThan
thf(fact_1492_under__def,axiom,
! [A: $tType] :
( ( order_under @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A ),A7: A] :
( collect @ A
@ ^ [B5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ A7 ) @ R5 ) ) ) ) ).
% under_def
thf(fact_1493_prod_OatLeast__int__atMost__int__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: int > A,M4: nat,N5: nat] :
( ( groups7121269368397514597t_prod @ int @ A @ G @ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ M4 ) @ ( semiring_1_of_nat @ int @ N5 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ int @ A @ nat @ G @ ( semiring_1_of_nat @ int ) ) @ ( set_or1337092689740270186AtMost @ nat @ M4 @ N5 ) ) ) ) ).
% prod.atLeast_int_atMost_int_shift
thf(fact_1494_prod_OatLeast__int__lessThan__int__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: int > A,M4: nat,N5: nat] :
( ( groups7121269368397514597t_prod @ int @ A @ G @ ( set_or7035219750837199246ssThan @ int @ ( semiring_1_of_nat @ int @ M4 ) @ ( semiring_1_of_nat @ int @ N5 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ int @ A @ nat @ G @ ( semiring_1_of_nat @ int ) ) @ ( set_or7035219750837199246ssThan @ nat @ M4 @ N5 ) ) ) ) ).
% prod.atLeast_int_lessThan_int_shift
thf(fact_1495_mono__Min__commute,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F: A > B,A6: set @ A] :
( ( order_mono @ A @ B @ F )
=> ( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( F @ ( lattic643756798350308766er_Min @ A @ A6 ) )
= ( lattic643756798350308766er_Min @ B @ ( image2 @ A @ B @ F @ A6 ) ) ) ) ) ) ) ).
% mono_Min_commute
thf(fact_1496_Min__add__commute,axiom,
! [B: $tType,A: $tType] :
( ( linord4140545234300271783up_add @ A )
=> ! [S5: set @ B,F: B > A,K: A] :
( ( finite_finite2 @ B @ S5 )
=> ( ( S5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798350308766er_Min @ A
@ ( image2 @ B @ A
@ ^ [X4: B] : ( plus_plus @ A @ ( F @ X4 ) @ K )
@ S5 ) )
= ( plus_plus @ A @ ( lattic643756798350308766er_Min @ A @ ( image2 @ B @ A @ F @ S5 ) ) @ K ) ) ) ) ) ).
% Min_add_commute
thf(fact_1497_minus__Min__eq__Max,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [S5: set @ A] :
( ( finite_finite2 @ A @ S5 )
=> ( ( S5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( uminus_uminus @ A @ ( lattic643756798350308766er_Min @ A @ S5 ) )
= ( lattic643756798349783984er_Max @ A @ ( image2 @ A @ A @ ( uminus_uminus @ A ) @ S5 ) ) ) ) ) ) ).
% minus_Min_eq_Max
thf(fact_1498_minus__Max__eq__Min,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [S5: set @ A] :
( ( finite_finite2 @ A @ S5 )
=> ( ( S5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( uminus_uminus @ A @ ( lattic643756798349783984er_Max @ A @ S5 ) )
= ( lattic643756798350308766er_Min @ A @ ( image2 @ A @ A @ ( uminus_uminus @ A ) @ S5 ) ) ) ) ) ) ).
% minus_Max_eq_Min
thf(fact_1499_wellorders__totally__ordered__aux,axiom,
! [A2: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R2: set @ ( product_prod @ A2 @ A2 ),A3: A,F: A > A2] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( order_well_order_on @ A2 @ ( field2 @ A2 @ R2 ) @ R2 )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R ) )
=> ( ! [X: A] :
( ( member @ A @ X @ ( order_underS @ A @ R @ A3 ) )
=> ( bij_betw @ A @ A2 @ F @ ( order_under @ A @ R @ X ) @ ( order_under @ A2 @ R2 @ ( F @ X ) ) ) )
=> ( ( ( image2 @ A @ A2 @ F @ ( order_underS @ A @ R @ A3 ) )
!= ( field2 @ A2 @ R2 ) )
=> ( ( ( F @ A3 )
= ( bNF_Wellorder_wo_suc @ A2 @ R2 @ ( image2 @ A @ A2 @ F @ ( order_underS @ A @ R @ A3 ) ) ) )
=> ( bij_betw @ A @ A2 @ F @ ( order_under @ A @ R @ A3 ) @ ( order_under @ A2 @ R2 @ ( F @ A3 ) ) ) ) ) ) ) ) ) ).
% wellorders_totally_ordered_aux
thf(fact_1500_ordLess__def,axiom,
! [A2: $tType,A: $tType] :
( ( bNF_We4044943003108391690rdLess @ A @ A2 )
= ( collect @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A2 @ A2 ) ) )
@ ( product_case_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A2 @ A2 ) ) @ $o
@ ^ [R5: set @ ( product_prod @ A @ A ),R9: set @ ( product_prod @ A2 @ A2 )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R5 ) @ R5 )
& ( order_well_order_on @ A2 @ ( field2 @ A2 @ R9 ) @ R9 )
& ? [X10: A > A2] : ( bNF_Wellorder_embedS @ A @ A2 @ R5 @ R9 @ X10 ) ) ) ) ) ).
% ordLess_def
thf(fact_1501_pair__lessI2,axiom,
! [A3: nat,B2: nat,S2: nat,T3: nat] :
( ( ord_less_eq @ nat @ A3 @ B2 )
=> ( ( ord_less @ nat @ S2 @ T3 )
=> ( 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 @ S2 ) @ ( product_Pair @ nat @ nat @ B2 @ T3 ) ) @ fun_pair_less ) ) ) ).
% pair_lessI2
thf(fact_1502_pair__less__iff1,axiom,
! [X2: nat,Y3: nat,Z3: 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 @ X2 @ Y3 ) @ ( product_Pair @ nat @ nat @ X2 @ Z3 ) ) @ fun_pair_less )
= ( ord_less @ nat @ Y3 @ Z3 ) ) ).
% pair_less_iff1
thf(fact_1503_Array_Oalloc__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( alloc @ A )
= ( ^ [Xs2: list @ A,H6: heap_ext @ product_unit] :
( product_Pair @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array2 @ A @ ( lim @ product_unit @ H6 ) )
@ ( set2 @ A @ ( array2 @ A @ ( lim @ product_unit @ H6 ) ) @ Xs2
@ ( lim_update @ product_unit
@ ^ [Uu: nat] : ( plus_plus @ nat @ ( lim @ product_unit @ H6 ) @ ( one_one @ nat ) )
@ H6 ) ) ) ) ) ) ).
% Array.alloc_def
thf(fact_1504_embed__determined,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R2: set @ ( product_prod @ B @ B ),F: A > B,A3: A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( order_well_order_on @ B @ ( field2 @ B @ R2 ) @ R2 )
=> ( ( bNF_Wellorder_embed @ A @ B @ R @ R2 @ F )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R ) )
=> ( ( F @ A3 )
= ( bNF_Wellorder_wo_suc @ B @ R2 @ ( image2 @ A @ B @ F @ ( order_underS @ A @ R @ A3 ) ) ) ) ) ) ) ) ).
% embed_determined
thf(fact_1505_total__pair__less,axiom,
! [A6: set @ ( product_prod @ nat @ nat )] : ( total_on @ ( product_prod @ nat @ nat ) @ A6 @ fun_pair_less ) ).
% total_pair_less
thf(fact_1506_bij__betw__of__nat,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: set @ nat,A6: set @ A] :
( ( bij_betw @ nat @ A @ ( semiring_1_of_nat @ A ) @ N @ A6 )
= ( ( image2 @ nat @ A @ ( semiring_1_of_nat @ A ) @ N )
= A6 ) ) ) ).
% bij_betw_of_nat
thf(fact_1507_Compl__Times__UNIV2,axiom,
! [B: $tType,A: $tType,A6: set @ A] :
( ( uminus_uminus @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu: A] : ( top_top @ ( set @ B ) ) ) )
= ( product_Sigma @ A @ B @ ( uminus_uminus @ ( set @ A ) @ A6 )
@ ^ [Uu: A] : ( top_top @ ( set @ B ) ) ) ) ).
% Compl_Times_UNIV2
thf(fact_1508_Compl__Times__UNIV1,axiom,
! [B: $tType,A: $tType,A6: set @ B] :
( ( uminus_uminus @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
@ ^ [Uu: A] : A6 ) )
= ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
@ ^ [Uu: A] : ( uminus_uminus @ ( set @ B ) @ A6 ) ) ) ).
% Compl_Times_UNIV1
thf(fact_1509_image__uminus__atLeastAtMost,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X2: A,Y3: A] :
( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_or1337092689740270186AtMost @ A @ X2 @ Y3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( uminus_uminus @ A @ Y3 ) @ ( uminus_uminus @ A @ X2 ) ) ) ) ).
% image_uminus_atLeastAtMost
thf(fact_1510_bij__betw__add,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A3: A,A6: set @ A,B6: set @ A] :
( ( bij_betw @ A @ A @ ( plus_plus @ A @ A3 ) @ A6 @ B6 )
= ( ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ A6 )
= B6 ) ) ) ).
% bij_betw_add
thf(fact_1511_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_1512_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_1513_Compl__UN,axiom,
! [A: $tType,B: $tType,B6: B > ( set @ A ),A6: set @ B] :
( ( uminus_uminus @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B6 @ A6 ) ) )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X4: B] : ( uminus_uminus @ ( set @ A ) @ ( B6 @ X4 ) )
@ A6 ) ) ) ).
% Compl_UN
thf(fact_1514_Compl__INT,axiom,
! [A: $tType,B: $tType,B6: B > ( set @ A ),A6: set @ B] :
( ( uminus_uminus @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B6 @ A6 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X4: B] : ( uminus_uminus @ ( set @ A ) @ ( B6 @ X4 ) )
@ A6 ) ) ) ).
% Compl_INT
thf(fact_1515_bdd__below__uminus,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X6: set @ A] :
( ( condit1013018076250108175_below @ A @ ( image2 @ A @ A @ ( uminus_uminus @ A ) @ X6 ) )
= ( condit941137186595557371_above @ A @ X6 ) ) ) ).
% bdd_below_uminus
thf(fact_1516_bdd__above__uminus,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X6: set @ A] :
( ( condit941137186595557371_above @ A @ ( image2 @ A @ A @ ( uminus_uminus @ A ) @ X6 ) )
= ( condit1013018076250108175_below @ A @ X6 ) ) ) ).
% bdd_above_uminus
thf(fact_1517_image__uminus__lessThan,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X2: A] :
( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_ord_lessThan @ A @ X2 ) )
= ( set_ord_greaterThan @ A @ ( uminus_uminus @ A @ X2 ) ) ) ) ).
% image_uminus_lessThan
thf(fact_1518_image__uminus__greaterThan,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X2: A] :
( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_ord_greaterThan @ A @ X2 ) )
= ( set_ord_lessThan @ A @ ( uminus_uminus @ A @ X2 ) ) ) ) ).
% image_uminus_greaterThan
thf(fact_1519_Inf__int__def,axiom,
( ( complete_Inf_Inf @ int )
= ( ^ [X10: set @ int] : ( uminus_uminus @ int @ ( complete_Sup_Sup @ int @ ( image2 @ int @ int @ ( uminus_uminus @ int ) @ X10 ) ) ) ) ) ).
% Inf_int_def
thf(fact_1520_wf__pair__less,axiom,
wf @ ( product_prod @ nat @ nat ) @ fun_pair_less ).
% wf_pair_less
thf(fact_1521_bij__betwI_H,axiom,
! [A: $tType,B: $tType,X6: set @ A,F: A > B,Y4: set @ B] :
( ! [X: A] :
( ( member @ A @ X @ X6 )
=> ! [Y: A] :
( ( member @ A @ Y @ X6 )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) )
=> ( ! [X: A] :
( ( member @ A @ X @ X6 )
=> ( member @ B @ ( F @ X ) @ Y4 ) )
=> ( ! [Y: B] :
( ( member @ B @ Y @ Y4 )
=> ? [X5: A] :
( ( member @ A @ X5 @ X6 )
& ( Y
= ( F @ X5 ) ) ) )
=> ( bij_betw @ A @ B @ F @ X6 @ Y4 ) ) ) ) ).
% bij_betwI'
thf(fact_1522_bij__betw__Pow,axiom,
! [A: $tType,B: $tType,F: A > B,A6: set @ A,B6: set @ B] :
( ( bij_betw @ A @ B @ F @ A6 @ B6 )
=> ( bij_betw @ ( set @ A ) @ ( set @ B ) @ ( image2 @ A @ B @ F ) @ ( pow @ A @ A6 ) @ ( pow @ B @ B6 ) ) ) ).
% bij_betw_Pow
thf(fact_1523_bij__image__Compl__eq,axiom,
! [B: $tType,A: $tType,F: A > B,A6: set @ A] :
( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( image2 @ A @ B @ F @ ( uminus_uminus @ ( set @ A ) @ A6 ) )
= ( uminus_uminus @ ( set @ B ) @ ( image2 @ A @ B @ F @ A6 ) ) ) ) ).
% bij_image_Compl_eq
thf(fact_1524_bij__betw__imp__surj__on,axiom,
! [A: $tType,B: $tType,F: A > B,A6: set @ A,B6: set @ B] :
( ( bij_betw @ A @ B @ F @ A6 @ B6 )
=> ( ( image2 @ A @ B @ F @ A6 )
= B6 ) ) ).
% bij_betw_imp_surj_on
thf(fact_1525_bijI_H,axiom,
! [A: $tType,B: $tType,F: A > B] :
( ! [X: A,Y: A] :
( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) )
=> ( ! [Y: B] :
? [X5: A] :
( Y
= ( F @ X5 ) )
=> ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) ) ) ) ).
% bijI'
thf(fact_1526_bij__betw__trans,axiom,
! [A: $tType,B: $tType,C: $tType,F: A > B,A6: set @ A,B6: set @ B,G: B > C,C5: set @ C] :
( ( bij_betw @ A @ B @ F @ A6 @ B6 )
=> ( ( bij_betw @ B @ C @ G @ B6 @ C5 )
=> ( bij_betw @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ A6 @ C5 ) ) ) ).
% bij_betw_trans
thf(fact_1527_bij__betw__comp__iff,axiom,
! [A: $tType,B: $tType,C: $tType,F: A > B,A6: set @ A,A9: set @ B,F5: B > C,A19: set @ C] :
( ( bij_betw @ A @ B @ F @ A6 @ A9 )
=> ( ( bij_betw @ B @ C @ F5 @ A9 @ A19 )
= ( bij_betw @ A @ C @ ( comp @ B @ C @ A @ F5 @ F ) @ A6 @ A19 ) ) ) ).
% bij_betw_comp_iff
thf(fact_1528_comp__embed,axiom,
! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R2: set @ ( product_prod @ B @ B ),F: A > B,R10: set @ ( product_prod @ C @ C ),F5: B > C] :
( ( bNF_Wellorder_embed @ A @ B @ R @ R2 @ F )
=> ( ( bNF_Wellorder_embed @ B @ C @ R2 @ R10 @ F5 )
=> ( bNF_Wellorder_embed @ A @ C @ R @ R10 @ ( comp @ B @ C @ A @ F5 @ F ) ) ) ) ).
% comp_embed
thf(fact_1529_BNF__Wellorder__Constructions_OordLess__Field,axiom,
! [A: $tType,B: $tType,R12: set @ ( product_prod @ A @ A ),R23: 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 ) ) @ R12 @ R23 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
=> ( ( bNF_Wellorder_embed @ A @ B @ R12 @ R23 @ F )
=> ( ( image2 @ A @ B @ F @ ( field2 @ A @ R12 ) )
!= ( field2 @ B @ R23 ) ) ) ) ).
% BNF_Wellorder_Constructions.ordLess_Field
thf(fact_1530_Array_Oget__alloc,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [Xs: list @ A,H: heap_ext @ product_unit,Ys: list @ A] :
( ( get @ A @ ( product_snd @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( alloc @ A @ Xs @ H ) ) @ ( product_fst @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( alloc @ A @ Ys @ H ) ) )
= Xs ) ) ).
% Array.get_alloc
thf(fact_1531_bij__betw__byWitness,axiom,
! [A: $tType,B: $tType,A6: set @ A,F5: B > A,F: A > B,A9: set @ B] :
( ! [X: A] :
( ( member @ A @ X @ A6 )
=> ( ( F5 @ ( F @ X ) )
= X ) )
=> ( ! [X: B] :
( ( member @ B @ X @ A9 )
=> ( ( F @ ( F5 @ X ) )
= X ) )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F @ A6 ) @ A9 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F5 @ A9 ) @ A6 )
=> ( bij_betw @ A @ B @ F @ A6 @ A9 ) ) ) ) ) ).
% bij_betw_byWitness
thf(fact_1532_bij__betw__subset,axiom,
! [A: $tType,B: $tType,F: A > B,A6: set @ A,A9: set @ B,B6: set @ A,B8: set @ B] :
( ( bij_betw @ A @ B @ F @ A6 @ A9 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ A6 )
=> ( ( ( image2 @ A @ B @ F @ B6 )
= B8 )
=> ( bij_betw @ A @ B @ F @ B6 @ B8 ) ) ) ) ).
% bij_betw_subset
thf(fact_1533_bij__betw__imp__surj,axiom,
! [A: $tType,B: $tType,F: A > B,A6: set @ A] :
( ( bij_betw @ A @ B @ F @ A6 @ ( top_top @ ( set @ B ) ) )
=> ( ( image2 @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ B ) ) ) ) ).
% bij_betw_imp_surj
thf(fact_1534_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_1535_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_1536_inj__on__imp__bij__betw,axiom,
! [B: $tType,A: $tType,F: A > B,A6: set @ A] :
( ( inj_on @ A @ B @ F @ A6 )
=> ( bij_betw @ A @ B @ F @ A6 @ ( image2 @ A @ B @ F @ A6 ) ) ) ).
% inj_on_imp_bij_betw
thf(fact_1537_bij__betw__imageI,axiom,
! [A: $tType,B: $tType,F: A > B,A6: set @ A,B6: set @ B] :
( ( inj_on @ A @ B @ F @ A6 )
=> ( ( ( image2 @ A @ B @ F @ A6 )
= B6 )
=> ( bij_betw @ A @ B @ F @ A6 @ B6 ) ) ) ).
% bij_betw_imageI
thf(fact_1538_bij__betw__def,axiom,
! [B: $tType,A: $tType] :
( ( bij_betw @ A @ B )
= ( ^ [F4: A > B,A8: set @ A,B7: set @ B] :
( ( inj_on @ A @ B @ F4 @ A8 )
& ( ( image2 @ A @ B @ F4 @ A8 )
= B7 ) ) ) ) ).
% bij_betw_def
thf(fact_1539_Array_Opresent__alloc,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [Xs: list @ A,H: heap_ext @ product_unit] : ( present @ A @ ( product_snd @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( alloc @ A @ Xs @ H ) ) @ ( product_fst @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( alloc @ A @ Xs @ H ) ) ) ) ).
% Array.present_alloc
thf(fact_1540_Array_Onot__present__alloc,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [H: heap_ext @ product_unit,Xs: list @ A] :
~ ( present @ A @ H @ ( product_fst @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( alloc @ A @ Xs @ H ) ) ) ) ).
% Array.not_present_alloc
thf(fact_1541_set,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [Ls: list @ A,H: heap_ext @ product_unit,New_ls: list @ A] :
( ( set2 @ A @ ( product_fst @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( alloc @ A @ Ls @ H ) ) @ New_ls @ ( product_snd @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( alloc @ A @ Ls @ H ) ) )
= ( product_snd @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( alloc @ A @ New_ls @ H ) ) ) ) ).
% set
thf(fact_1542_translation__Compl,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,T3: set @ A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ ( uminus_uminus @ ( set @ A ) @ T3 ) )
= ( uminus_uminus @ ( set @ A ) @ ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ T3 ) ) ) ) ).
% translation_Compl
thf(fact_1543_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_1544_translation__subtract__Compl,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,T3: set @ A] :
( ( image2 @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ X4 @ A3 )
@ ( uminus_uminus @ ( set @ A ) @ T3 ) )
= ( uminus_uminus @ ( set @ A )
@ ( image2 @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ X4 @ A3 )
@ T3 ) ) ) ) ).
% translation_subtract_Compl
thf(fact_1545_bij__betw__comp__iff2,axiom,
! [C: $tType,A: $tType,B: $tType,F5: A > B,A9: set @ A,A19: set @ B,F: C > A,A6: set @ C] :
( ( bij_betw @ A @ B @ F5 @ A9 @ A19 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ C @ A @ F @ A6 ) @ A9 )
=> ( ( bij_betw @ C @ A @ F @ A6 @ A9 )
= ( bij_betw @ C @ B @ ( comp @ A @ B @ C @ F5 @ F ) @ A6 @ A19 ) ) ) ) ).
% bij_betw_comp_iff2
thf(fact_1546_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_1547_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_1548_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_1549_embed__ordLess__ofilterIncl,axiom,
! [B: $tType,A: $tType,C: $tType,R12: set @ ( product_prod @ A @ A ),R23: 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 ) ) @ R12 @ R23 ) @ ( 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 ) ) @ R23 @ R33 ) @ ( bNF_We4044943003108391690rdLess @ B @ C ) )
=> ( ( bNF_Wellorder_embed @ A @ C @ R12 @ R33 @ F13 )
=> ( ( bNF_Wellorder_embed @ B @ C @ R23 @ R33 @ F232 )
=> ( member @ ( product_prod @ ( set @ C ) @ ( set @ C ) ) @ ( product_Pair @ ( set @ C ) @ ( set @ C ) @ ( image2 @ A @ C @ F13 @ ( field2 @ A @ R12 ) ) @ ( image2 @ B @ C @ F232 @ ( field2 @ B @ R23 ) ) ) @ ( bNF_We413866401316099525erIncl @ C @ R33 ) ) ) ) ) ) ).
% embed_ordLess_ofilterIncl
thf(fact_1550_embed__Field,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),R2: set @ ( product_prod @ B @ B ),F: A > B] :
( ( bNF_Wellorder_embed @ A @ B @ R @ R2 @ F )
=> ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F @ ( field2 @ A @ R ) ) @ ( field2 @ B @ R2 ) ) ) ).
% embed_Field
thf(fact_1551_surj__Compl__image__subset,axiom,
! [A: $tType,B: $tType,F: B > A,A6: 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 @ A6 ) ) @ ( image2 @ B @ A @ F @ ( uminus_uminus @ ( set @ B ) @ A6 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_1552_bij__betw__UNION__chain,axiom,
! [B: $tType,C: $tType,A: $tType,I4: set @ A,A6: A > ( set @ B ),F: B > C,A9: A > ( set @ C )] :
( ! [I6: A,J3: A] :
( ( member @ A @ I6 @ I4 )
=> ( ( member @ A @ J3 @ I4 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( A6 @ I6 ) @ ( A6 @ J3 ) )
| ( ord_less_eq @ ( set @ B ) @ ( A6 @ J3 ) @ ( A6 @ I6 ) ) ) ) )
=> ( ! [I6: A] :
( ( member @ A @ I6 @ I4 )
=> ( bij_betw @ B @ C @ F @ ( A6 @ I6 ) @ ( A9 @ I6 ) ) )
=> ( bij_betw @ B @ C @ F @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A6 @ I4 ) ) @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ A @ ( set @ C ) @ A9 @ I4 ) ) ) ) ) ).
% bij_betw_UNION_chain
thf(fact_1553_uminus__Inf,axiom,
! [A: $tType] :
( ( comple489889107523837845lgebra @ A )
=> ! [A6: set @ A] :
( ( uminus_uminus @ A @ ( complete_Inf_Inf @ A @ A6 ) )
= ( complete_Sup_Sup @ A @ ( image2 @ A @ A @ ( uminus_uminus @ A ) @ A6 ) ) ) ) ).
% uminus_Inf
thf(fact_1554_uminus__Sup,axiom,
! [A: $tType] :
( ( comple489889107523837845lgebra @ A )
=> ! [A6: set @ A] :
( ( uminus_uminus @ A @ ( complete_Sup_Sup @ A @ A6 ) )
= ( complete_Inf_Inf @ A @ ( image2 @ A @ A @ ( uminus_uminus @ A ) @ A6 ) ) ) ) ).
% uminus_Sup
thf(fact_1555_uminus__INF,axiom,
! [A: $tType,B: $tType] :
( ( comple489889107523837845lgebra @ A )
=> ! [B6: B > A,A6: set @ B] :
( ( uminus_uminus @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ B6 @ A6 ) ) )
= ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [X4: B] : ( uminus_uminus @ A @ ( B6 @ X4 ) )
@ A6 ) ) ) ) ).
% uminus_INF
thf(fact_1556_uminus__SUP,axiom,
! [A: $tType,B: $tType] :
( ( comple489889107523837845lgebra @ A )
=> ! [B6: B > A,A6: set @ B] :
( ( uminus_uminus @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ B6 @ A6 ) ) )
= ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [X4: B] : ( uminus_uminus @ A @ ( B6 @ X4 ) )
@ A6 ) ) ) ) ).
% uminus_SUP
thf(fact_1557_present__alloc__noteq,axiom,
! [A: $tType,B: $tType] :
( ( ( heap @ B )
& ( heap @ A ) )
=> ! [H: heap_ext @ product_unit,A3: array @ A,Xs: list @ B] :
( ( present @ A @ H @ A3 )
=> ( noteq @ A @ B @ A3 @ ( product_fst @ ( array @ B ) @ ( heap_ext @ product_unit ) @ ( alloc @ B @ Xs @ H ) ) ) ) ) ).
% present_alloc_noteq
thf(fact_1558_vimage__subset__eq,axiom,
! [B: $tType,A: $tType,F: A > B,B6: set @ B,A6: 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 ) @ A6 )
= ( ord_less_eq @ ( set @ B ) @ B6 @ ( image2 @ A @ B @ F @ A6 ) ) ) ) ).
% vimage_subset_eq
thf(fact_1559_mono__bij__Inf,axiom,
! [B: $tType,A: $tType] :
( ( ( comple5582772986160207858norder @ A )
& ( comple5582772986160207858norder @ B ) )
=> ! [F: A > B,A6: set @ A] :
( ( order_mono @ A @ B @ F )
=> ( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( F @ ( complete_Inf_Inf @ A @ A6 ) )
= ( complete_Inf_Inf @ B @ ( image2 @ A @ B @ F @ A6 ) ) ) ) ) ) ).
% mono_bij_Inf
thf(fact_1560_wellorders__totally__ordered__aux2,axiom,
! [A: $tType,A2: $tType,R: set @ ( product_prod @ A @ A ),R2: set @ ( product_prod @ A2 @ A2 ),G: A > $o,F: A > A2,A3: A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( order_well_order_on @ A2 @ ( field2 @ A2 @ R2 ) @ R2 )
=> ( ! [A5: A] :
( ( ( ~ ( member @ $o @ $false @ ( image2 @ A @ $o @ G @ ( order_underS @ A @ R @ A5 ) ) )
& ( ( image2 @ A @ A2 @ F @ ( order_underS @ A @ R @ A5 ) )
!= ( field2 @ A2 @ R2 ) ) )
=> ( ( ( F @ A5 )
= ( bNF_Wellorder_wo_suc @ A2 @ R2 @ ( image2 @ A @ A2 @ F @ ( order_underS @ A @ R @ A5 ) ) ) )
& ( G @ A5 ) ) )
& ( ~ ( ~ ( member @ $o @ $false @ ( image2 @ A @ $o @ G @ ( order_underS @ A @ R @ A5 ) ) )
& ( ( image2 @ A @ A2 @ F @ ( order_underS @ A @ R @ A5 ) )
!= ( field2 @ A2 @ R2 ) ) )
=> ~ ( G @ A5 ) ) )
=> ( ! [A5: A] :
( ( ( member @ A @ A5 @ ( field2 @ A @ R ) )
& ~ ( member @ $o @ $false @ ( image2 @ A @ $o @ G @ ( order_under @ A @ R @ A5 ) ) ) )
=> ( bij_betw @ A @ A2 @ F @ ( order_under @ A @ R @ A5 ) @ ( order_under @ A2 @ R2 @ ( F @ A5 ) ) ) )
=> ( ( ( member @ A @ A3 @ ( field2 @ A @ R ) )
& ( member @ $o @ $false @ ( image2 @ A @ $o @ G @ ( order_under @ A @ R @ A3 ) ) ) )
=> ? [X_12: A2 > A] : ( bNF_Wellorder_embed @ A2 @ A @ R2 @ R @ X_12 ) ) ) ) ) ) ).
% wellorders_totally_ordered_aux2
thf(fact_1561_bij__image__INT,axiom,
! [B: $tType,A: $tType,C: $tType,F: A > B,B6: C > ( set @ A ),A6: set @ C] :
( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( image2 @ A @ B @ F @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ C @ ( set @ A ) @ B6 @ A6 ) ) )
= ( complete_Inf_Inf @ ( set @ B )
@ ( image2 @ C @ ( set @ B )
@ ^ [X4: C] : ( image2 @ A @ B @ F @ ( B6 @ X4 ) )
@ A6 ) ) ) ) ).
% bij_image_INT
thf(fact_1562_pair__lessI1,axiom,
! [A3: nat,B2: nat,S2: nat,T3: 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 @ S2 ) @ ( product_Pair @ nat @ nat @ B2 @ T3 ) ) @ fun_pair_less ) ) ).
% pair_lessI1
thf(fact_1563_inj__image__Compl__subset,axiom,
! [B: $tType,A: $tType,F: A > B,A6: set @ A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F @ ( uminus_uminus @ ( set @ A ) @ A6 ) ) @ ( uminus_uminus @ ( set @ B ) @ ( image2 @ A @ B @ F @ A6 ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_1564_embed__preserves__ofilter,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R2: set @ ( product_prod @ B @ B ),F: A > B,A6: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( order_well_order_on @ B @ ( field2 @ B @ R2 ) @ R2 )
=> ( ( bNF_Wellorder_embed @ A @ B @ R @ R2 @ F )
=> ( ( order_ofilter @ A @ R @ A6 )
=> ( order_ofilter @ B @ R2 @ ( image2 @ A @ B @ F @ A6 ) ) ) ) ) ) ).
% embed_preserves_ofilter
thf(fact_1565_embed__Field__ofilter,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R2: set @ ( product_prod @ B @ B ),F: A > B] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( order_well_order_on @ B @ ( field2 @ B @ R2 ) @ R2 )
=> ( ( bNF_Wellorder_embed @ A @ B @ R @ R2 @ F )
=> ( order_ofilter @ B @ R2 @ ( image2 @ A @ B @ F @ ( field2 @ A @ R ) ) ) ) ) ) ).
% embed_Field_ofilter
thf(fact_1566_embedS__comp__embed,axiom,
! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R2: set @ ( product_prod @ B @ B ),F: A > B,R10: set @ ( product_prod @ C @ C ),F5: B > C] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( order_well_order_on @ B @ ( field2 @ B @ R2 ) @ R2 )
=> ( ( bNF_Wellorder_embedS @ A @ B @ R @ R2 @ F )
=> ( ( bNF_Wellorder_embed @ B @ C @ R2 @ R10 @ F5 )
=> ( bNF_Wellorder_embedS @ A @ C @ R @ R10 @ ( comp @ B @ C @ A @ F5 @ F ) ) ) ) ) ) ).
% embedS_comp_embed
thf(fact_1567_embed__comp__embedS,axiom,
! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R2: set @ ( product_prod @ B @ B ),F: A > B,R10: set @ ( product_prod @ C @ C ),F5: B > C] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( order_well_order_on @ B @ ( field2 @ B @ R2 ) @ R2 )
=> ( ( bNF_Wellorder_embed @ A @ B @ R @ R2 @ F )
=> ( ( bNF_Wellorder_embedS @ B @ C @ R2 @ R10 @ F5 )
=> ( bNF_Wellorder_embedS @ A @ C @ R @ R10 @ ( comp @ B @ C @ A @ F5 @ F ) ) ) ) ) ) ).
% embed_comp_embedS
thf(fact_1568_sum_OatLeastAtMost__reindex,axiom,
! [B: $tType,A: $tType] :
( ( ( comm_monoid_add @ A )
& ( ord @ B ) )
=> ! [H: nat > B,M4: nat,N5: nat,G: B > A] :
( ( bij_betw @ nat @ B @ H @ ( set_or1337092689740270186AtMost @ nat @ M4 @ N5 ) @ ( set_or1337092689740270186AtMost @ B @ ( H @ M4 ) @ ( H @ N5 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( set_or1337092689740270186AtMost @ B @ ( H @ M4 ) @ ( H @ N5 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ B @ A @ nat @ G @ H ) @ ( set_or1337092689740270186AtMost @ nat @ M4 @ N5 ) ) ) ) ) ).
% sum.atLeastAtMost_reindex
thf(fact_1569_sum_OatLeastLessThan__reindex,axiom,
! [B: $tType,A: $tType] :
( ( ( comm_monoid_add @ A )
& ( ord @ B ) )
=> ! [H: nat > B,M4: nat,N5: nat,G: B > A] :
( ( bij_betw @ nat @ B @ H @ ( set_or7035219750837199246ssThan @ nat @ M4 @ N5 ) @ ( set_or7035219750837199246ssThan @ B @ ( H @ M4 ) @ ( H @ N5 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( set_or7035219750837199246ssThan @ B @ ( H @ M4 ) @ ( H @ N5 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ B @ A @ nat @ G @ H ) @ ( set_or7035219750837199246ssThan @ nat @ M4 @ N5 ) ) ) ) ) ).
% sum.atLeastLessThan_reindex
thf(fact_1570_prod_OatLeastAtMost__reindex,axiom,
! [B: $tType,A: $tType] :
( ( ( comm_monoid_mult @ A )
& ( ord @ B ) )
=> ! [H: nat > B,M4: nat,N5: nat,G: B > A] :
( ( bij_betw @ nat @ B @ H @ ( set_or1337092689740270186AtMost @ nat @ M4 @ N5 ) @ ( set_or1337092689740270186AtMost @ B @ ( H @ M4 ) @ ( H @ N5 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( set_or1337092689740270186AtMost @ B @ ( H @ M4 ) @ ( H @ N5 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ B @ A @ nat @ G @ H ) @ ( set_or1337092689740270186AtMost @ nat @ M4 @ N5 ) ) ) ) ) ).
% prod.atLeastAtMost_reindex
thf(fact_1571_prod_OatLeastLessThan__reindex,axiom,
! [B: $tType,A: $tType] :
( ( ( comm_monoid_mult @ A )
& ( ord @ B ) )
=> ! [H: nat > B,M4: nat,N5: nat,G: B > A] :
( ( bij_betw @ nat @ B @ H @ ( set_or7035219750837199246ssThan @ nat @ M4 @ N5 ) @ ( set_or7035219750837199246ssThan @ B @ ( H @ M4 ) @ ( H @ N5 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( set_or7035219750837199246ssThan @ B @ ( H @ M4 ) @ ( H @ N5 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ B @ A @ nat @ G @ H ) @ ( set_or7035219750837199246ssThan @ nat @ M4 @ N5 ) ) ) ) ) ).
% prod.atLeastLessThan_reindex
thf(fact_1572_embedS__iff,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),R2: set @ ( product_prod @ B @ B ),F: A > B] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( bNF_Wellorder_embed @ A @ B @ R @ R2 @ F )
=> ( ( bNF_Wellorder_embedS @ A @ B @ R @ R2 @ F )
= ( ord_less @ ( set @ B ) @ ( image2 @ A @ B @ F @ ( field2 @ A @ R ) ) @ ( field2 @ B @ R2 ) ) ) ) ) ).
% embedS_iff
thf(fact_1573_bij__betw__Suc,axiom,
! [M: set @ nat,N: set @ nat] :
( ( bij_betw @ nat @ nat @ suc @ M @ N )
= ( ( image2 @ nat @ nat @ suc @ M )
= N ) ) ).
% bij_betw_Suc
thf(fact_1574_Schroeder__Bernstein,axiom,
! [A: $tType,B: $tType,F: A > B,A6: set @ A,B6: set @ B,G: B > A] :
( ( inj_on @ A @ B @ F @ A6 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F @ A6 ) @ B6 )
=> ( ( inj_on @ B @ A @ G @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ G @ B6 ) @ A6 )
=> ? [H8: A > B] : ( bij_betw @ A @ B @ H8 @ A6 @ B6 ) ) ) ) ) ).
% Schroeder_Bernstein
thf(fact_1575_embed__iff__compat__inj__on__ofilter,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R2: set @ ( product_prod @ B @ B ),F: A > B] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( order_well_order_on @ B @ ( field2 @ B @ R2 ) @ R2 )
=> ( ( bNF_Wellorder_embed @ A @ B @ R @ R2 @ F )
= ( ( bNF_Wellorder_compat @ A @ B @ R @ R2 @ F )
& ( inj_on @ A @ B @ F @ ( field2 @ A @ R ) )
& ( order_ofilter @ B @ R2 @ ( image2 @ A @ B @ F @ ( field2 @ A @ R ) ) ) ) ) ) ) ).
% embed_iff_compat_inj_on_ofilter
thf(fact_1576_effect__of__listE,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [Xs: list @ A,H: heap_ext @ product_unit,H2: heap_ext @ product_unit,R: array @ A] :
( ( heap_effect @ ( array @ A ) @ ( of_list @ A @ Xs ) @ H @ H2 @ R )
=> ~ ( ( R
= ( product_fst @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( alloc @ A @ Xs @ H ) ) )
=> ( ( H2
= ( product_snd @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( alloc @ A @ Xs @ H ) ) )
=> ( ( ( get @ A @ H2 @ R )
= Xs )
=> ( ( present @ A @ H2 @ R )
=> ( present @ A @ H @ R ) ) ) ) ) ) ) ).
% effect_of_listE
thf(fact_1577_effect__of__listI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [A3: array @ A,H2: heap_ext @ product_unit,Xs: list @ A,H: heap_ext @ product_unit] :
( ( ( product_Pair @ ( array @ A ) @ ( heap_ext @ product_unit ) @ A3 @ H2 )
= ( alloc @ A @ Xs @ H ) )
=> ( heap_effect @ ( array @ A ) @ ( of_list @ A @ Xs ) @ H @ H2 @ A3 ) ) ) ).
% effect_of_listI
thf(fact_1578_embed__implies__iso__Restr,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),R2: set @ ( product_prod @ B @ B ),F: B > A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( order_well_order_on @ B @ ( field2 @ B @ R2 ) @ R2 )
=> ( ( bNF_Wellorder_embed @ B @ A @ R2 @ R @ F )
=> ( bNF_Wellorder_iso @ B @ A @ R2
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ ( image2 @ B @ A @ F @ ( field2 @ B @ R2 ) )
@ ^ [Uu: A] : ( image2 @ B @ A @ F @ ( field2 @ B @ R2 ) ) ) )
@ F ) ) ) ) ).
% embed_implies_iso_Restr
thf(fact_1579_pair__leqI2,axiom,
! [A3: nat,B2: nat,S2: nat,T3: nat] :
( ( ord_less_eq @ nat @ A3 @ B2 )
=> ( ( ord_less_eq @ nat @ S2 @ T3 )
=> ( 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 @ S2 ) @ ( product_Pair @ nat @ nat @ B2 @ T3 ) ) @ fun_pair_leq ) ) ) ).
% pair_leqI2
thf(fact_1580_pair__leqI1,axiom,
! [A3: nat,B2: nat,S2: nat,T3: 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 @ S2 ) @ ( product_Pair @ nat @ nat @ B2 @ T3 ) ) @ fun_pair_leq ) ) ).
% pair_leqI1
thf(fact_1581_iso__forward,axiom,
! [A: $tType,B: $tType,X2: A,Y3: A,R: set @ ( product_prod @ A @ A ),R2: set @ ( product_prod @ B @ B ),F: A > B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R )
=> ( ( bNF_Wellorder_iso @ A @ B @ R @ R2 @ F )
=> ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F @ X2 ) @ ( F @ Y3 ) ) @ R2 ) ) ) ).
% iso_forward
thf(fact_1582_comp__iso,axiom,
! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R2: set @ ( product_prod @ B @ B ),F: A > B,R10: set @ ( product_prod @ C @ C ),F5: B > C] :
( ( bNF_Wellorder_iso @ A @ B @ R @ R2 @ F )
=> ( ( bNF_Wellorder_iso @ B @ C @ R2 @ R10 @ F5 )
=> ( bNF_Wellorder_iso @ A @ C @ R @ R10 @ ( comp @ B @ C @ A @ F5 @ F ) ) ) ) ).
% comp_iso
thf(fact_1583_compat__def,axiom,
! [A2: $tType,A: $tType] :
( ( bNF_Wellorder_compat @ A @ A2 )
= ( ^ [R5: set @ ( product_prod @ A @ A ),R9: set @ ( product_prod @ A2 @ A2 ),F4: A > A2] :
! [A7: A,B5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B5 ) @ R5 )
=> ( member @ ( product_prod @ A2 @ A2 ) @ ( product_Pair @ A2 @ A2 @ ( F4 @ A7 ) @ ( F4 @ B5 ) ) @ R9 ) ) ) ) ).
% compat_def
thf(fact_1584_iso__Field,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),R2: set @ ( product_prod @ B @ B ),F: A > B] :
( ( bNF_Wellorder_iso @ A @ B @ R @ R2 @ F )
=> ( ( image2 @ A @ B @ F @ ( field2 @ A @ R ) )
= ( field2 @ B @ R2 ) ) ) ).
% iso_Field
thf(fact_1585_iso__comp__embed,axiom,
! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R2: set @ ( product_prod @ B @ B ),F: A > B,R10: set @ ( product_prod @ C @ C ),F5: B > C] :
( ( bNF_Wellorder_iso @ A @ B @ R @ R2 @ F )
=> ( ( bNF_Wellorder_embed @ B @ C @ R2 @ R10 @ F5 )
=> ( bNF_Wellorder_embed @ A @ C @ R @ R10 @ ( comp @ B @ C @ A @ F5 @ F ) ) ) ) ).
% iso_comp_embed
thf(fact_1586_embed__comp__iso,axiom,
! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R2: set @ ( product_prod @ B @ B ),F: A > B,R10: set @ ( product_prod @ C @ C ),F5: B > C] :
( ( bNF_Wellorder_embed @ A @ B @ R @ R2 @ F )
=> ( ( bNF_Wellorder_iso @ B @ C @ R2 @ R10 @ F5 )
=> ( bNF_Wellorder_embed @ A @ C @ R @ R10 @ ( comp @ B @ C @ A @ F5 @ F ) ) ) ) ).
% embed_comp_iso
thf(fact_1587_embedS__comp__iso,axiom,
! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R2: set @ ( product_prod @ B @ B ),F: A > B,R10: set @ ( product_prod @ C @ C ),F5: B > C] :
( ( bNF_Wellorder_embedS @ A @ B @ R @ R2 @ F )
=> ( ( bNF_Wellorder_iso @ B @ C @ R2 @ R10 @ F5 )
=> ( bNF_Wellorder_embedS @ A @ C @ R @ R10 @ ( comp @ B @ C @ A @ F5 @ F ) ) ) ) ).
% embedS_comp_iso
thf(fact_1588_of__list__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( of_list @ A )
= ( ^ [Xs2: list @ A] : ( heap_heap @ ( array @ A ) @ ( alloc @ A @ Xs2 ) ) ) ) ) ).
% of_list_def
thf(fact_1589_pair__leq__def,axiom,
( fun_pair_leq
= ( sup_sup @ ( set @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) ) @ fun_pair_less @ ( id2 @ ( product_prod @ nat @ nat ) ) ) ) ).
% pair_leq_def
thf(fact_1590_iso__iff2,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Wellorder_iso @ A @ B )
= ( ^ [R5: set @ ( product_prod @ A @ A ),R9: set @ ( product_prod @ B @ B ),F4: A > B] :
( ( bij_betw @ A @ B @ F4 @ ( field2 @ A @ R5 ) @ ( field2 @ B @ R9 ) )
& ! [X4: A] :
( ( member @ A @ X4 @ ( field2 @ A @ R5 ) )
=> ! [Y5: A] :
( ( member @ A @ Y5 @ ( field2 @ A @ R5 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R5 )
= ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F4 @ X4 ) @ ( F4 @ Y5 ) ) @ R9 ) ) ) ) ) ) ) ).
% iso_iff2
thf(fact_1591_success__of__listI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [Xs: list @ A,H: heap_ext @ product_unit] : ( heap_success @ ( array @ A ) @ ( of_list @ A @ Xs ) @ H ) ) ).
% success_of_listI
thf(fact_1592_iso__iff,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),R2: set @ ( product_prod @ B @ B ),F: A > B] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( bNF_Wellorder_iso @ A @ B @ R @ R2 @ F )
= ( ( bNF_Wellorder_embed @ A @ B @ R @ R2 @ F )
& ( ( image2 @ A @ B @ F @ ( field2 @ A @ R ) )
= ( field2 @ B @ R2 ) ) ) ) ) ).
% iso_iff
thf(fact_1593_iso__comp__embedS,axiom,
! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R2: set @ ( product_prod @ B @ B ),F: A > B,R10: set @ ( product_prod @ C @ C ),F5: B > C] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( order_well_order_on @ B @ ( field2 @ B @ R2 ) @ R2 )
=> ( ( bNF_Wellorder_iso @ A @ B @ R @ R2 @ F )
=> ( ( bNF_Wellorder_embedS @ B @ C @ R2 @ R10 @ F5 )
=> ( bNF_Wellorder_embedS @ A @ C @ R @ R10 @ ( comp @ B @ C @ A @ F5 @ F ) ) ) ) ) ) ).
% iso_comp_embedS
thf(fact_1594_wmax__insertI,axiom,
! [Y3: product_prod @ nat @ nat,YS: set @ ( product_prod @ nat @ nat ),X2: product_prod @ nat @ nat,XS: set @ ( product_prod @ nat @ nat )] :
( ( member @ ( product_prod @ nat @ nat ) @ Y3 @ YS )
=> ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X2 @ Y3 ) @ 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 ) ) @ ( insert @ ( product_prod @ nat @ nat ) @ X2 @ XS ) @ YS ) @ fun_max_weak ) ) ) ) ).
% wmax_insertI
thf(fact_1595_wmin__insertI,axiom,
! [X2: product_prod @ nat @ nat,XS: set @ ( product_prod @ nat @ nat ),Y3: product_prod @ nat @ nat,YS: set @ ( product_prod @ nat @ nat )] :
( ( member @ ( product_prod @ nat @ nat ) @ X2 @ XS )
=> ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X2 @ Y3 ) @ 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 @ ( insert @ ( product_prod @ nat @ nat ) @ Y3 @ YS ) ) @ fun_min_weak ) ) ) ) ).
% wmin_insertI
thf(fact_1596_smax__insertI,axiom,
! [Y3: product_prod @ nat @ nat,Y4: set @ ( product_prod @ nat @ nat ),X2: product_prod @ nat @ nat,X6: set @ ( product_prod @ nat @ nat )] :
( ( member @ ( product_prod @ nat @ nat ) @ Y3 @ Y4 )
=> ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X2 @ Y3 ) @ 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 ) ) @ X6 @ Y4 ) @ 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 ) ) @ ( insert @ ( product_prod @ nat @ nat ) @ X2 @ X6 ) @ Y4 ) @ fun_max_strict ) ) ) ) ).
% smax_insertI
thf(fact_1597_smin__insertI,axiom,
! [X2: product_prod @ nat @ nat,XS: set @ ( product_prod @ nat @ nat ),Y3: product_prod @ nat @ nat,YS: set @ ( product_prod @ nat @ nat )] :
( ( member @ ( product_prod @ nat @ nat ) @ X2 @ XS )
=> ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X2 @ Y3 ) @ 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 @ ( insert @ ( product_prod @ nat @ nat ) @ Y3 @ YS ) ) @ fun_min_strict ) ) ) ) ).
% smin_insertI
thf(fact_1598_inv__into__ofilter__embed,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),A6: set @ A,F: A > B,R2: set @ ( product_prod @ B @ B )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( order_ofilter @ A @ R @ A6 )
=> ( ! [X: A] :
( ( member @ A @ X @ A6 )
=> ( bij_betw @ A @ B @ F @ ( order_under @ A @ R @ X ) @ ( order_under @ B @ R2 @ ( F @ X ) ) ) )
=> ( ( ( image2 @ A @ B @ F @ A6 )
= ( field2 @ B @ R2 ) )
=> ( bNF_Wellorder_embed @ B @ A @ R2 @ R @ ( hilbert_inv_into @ A @ B @ A6 @ F ) ) ) ) ) ) ).
% inv_into_ofilter_embed
thf(fact_1599_inv__into__underS__embed,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,F: A > B,R2: set @ ( product_prod @ B @ B )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ! [X: A] :
( ( member @ A @ X @ ( order_underS @ A @ R @ A3 ) )
=> ( bij_betw @ A @ B @ F @ ( order_under @ A @ R @ X ) @ ( order_under @ B @ R2 @ ( F @ X ) ) ) )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R ) )
=> ( ( ( image2 @ A @ B @ F @ ( order_underS @ A @ R @ A3 ) )
= ( field2 @ B @ R2 ) )
=> ( bNF_Wellorder_embed @ B @ A @ R2 @ R @ ( hilbert_inv_into @ A @ B @ ( order_underS @ A @ R @ A3 ) @ F ) ) ) ) ) ) ).
% inv_into_underS_embed
thf(fact_1600_ordIso__def,axiom,
! [A2: $tType,A: $tType] :
( ( bNF_Wellorder_ordIso @ A @ A2 )
= ( collect @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A2 @ A2 ) ) )
@ ( product_case_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A2 @ A2 ) ) @ $o
@ ^ [R5: set @ ( product_prod @ A @ A ),R9: set @ ( product_prod @ A2 @ A2 )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R5 ) @ R5 )
& ( order_well_order_on @ A2 @ ( field2 @ A2 @ R9 ) @ R9 )
& ? [X10: A > A2] : ( bNF_Wellorder_iso @ A @ A2 @ R5 @ R9 @ X10 ) ) ) ) ) ).
% ordIso_def
thf(fact_1601_ordLeq__def,axiom,
! [A2: $tType,A: $tType] :
( ( bNF_Wellorder_ordLeq @ A @ A2 )
= ( collect @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A2 @ A2 ) ) )
@ ( product_case_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A2 @ A2 ) ) @ $o
@ ^ [R5: set @ ( product_prod @ A @ A ),R9: set @ ( product_prod @ A2 @ A2 )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R5 ) @ R5 )
& ( order_well_order_on @ A2 @ ( field2 @ A2 @ R9 ) @ R9 )
& ? [X10: A > A2] : ( bNF_Wellorder_embed @ A @ A2 @ R5 @ R9 @ X10 ) ) ) ) ) ).
% ordLeq_def
thf(fact_1602_inv__into__image__cancel,axiom,
! [B: $tType,A: $tType,F: A > B,A6: set @ A,S5: set @ A] :
( ( inj_on @ A @ B @ F @ A6 )
=> ( ( ord_less_eq @ ( set @ A ) @ S5 @ A6 )
=> ( ( image2 @ B @ A @ ( hilbert_inv_into @ A @ B @ A6 @ F ) @ ( image2 @ A @ B @ F @ S5 ) )
= S5 ) ) ) ).
% inv_into_image_cancel
thf(fact_1603_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_1604_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_1605_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_1606_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_1607_wmax__emptyI,axiom,
! [X6: set @ ( product_prod @ nat @ nat )] :
( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ X6 )
=> ( 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 ) ) ) @ X6 ) @ fun_max_weak ) ) ).
% wmax_emptyI
thf(fact_1608_smax__emptyI,axiom,
! [Y4: set @ ( product_prod @ nat @ nat )] :
( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ Y4 )
=> ( ( Y4
!= ( 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 ) ) ) @ Y4 ) @ fun_max_strict ) ) ) ).
% smax_emptyI
thf(fact_1609_smin__emptyI,axiom,
! [X6: set @ ( product_prod @ nat @ nat )] :
( ( X6
!= ( 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 ) ) @ X6 @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ fun_min_strict ) ) ).
% smin_emptyI
thf(fact_1610_wmin__emptyI,axiom,
! [X6: 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 ) ) @ X6 @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ fun_min_weak ) ).
% wmin_emptyI
thf(fact_1611_f__inv__into__f,axiom,
! [B: $tType,A: $tType,Y3: A,F: B > A,A6: set @ B] :
( ( member @ A @ Y3 @ ( image2 @ B @ A @ F @ A6 ) )
=> ( ( F @ ( hilbert_inv_into @ B @ A @ A6 @ F @ Y3 ) )
= Y3 ) ) ).
% f_inv_into_f
thf(fact_1612_inv__into__into,axiom,
! [A: $tType,B: $tType,X2: A,F: B > A,A6: set @ B] :
( ( member @ A @ X2 @ ( image2 @ B @ A @ F @ A6 ) )
=> ( member @ B @ ( hilbert_inv_into @ B @ A @ A6 @ F @ X2 ) @ A6 ) ) ).
% inv_into_into
thf(fact_1613_inv__into__injective,axiom,
! [A: $tType,B: $tType,A6: set @ A,F: A > B,X2: B,Y3: B] :
( ( ( hilbert_inv_into @ A @ B @ A6 @ F @ X2 )
= ( hilbert_inv_into @ A @ B @ A6 @ F @ Y3 ) )
=> ( ( member @ B @ X2 @ ( image2 @ A @ B @ F @ A6 ) )
=> ( ( member @ B @ Y3 @ ( image2 @ A @ B @ F @ A6 ) )
=> ( X2 = Y3 ) ) ) ) ).
% inv_into_injective
thf(fact_1614_image__inv__into__cancel,axiom,
! [B: $tType,A: $tType,F: B > A,A6: set @ B,A9: set @ A,B8: set @ A] :
( ( ( image2 @ B @ A @ F @ A6 )
= A9 )
=> ( ( ord_less_eq @ ( set @ A ) @ B8 @ A9 )
=> ( ( image2 @ B @ A @ F @ ( image2 @ A @ B @ ( hilbert_inv_into @ B @ A @ A6 @ F ) @ B8 ) )
= B8 ) ) ) ).
% image_inv_into_cancel
thf(fact_1615_surj__f__inv__f,axiom,
! [B: $tType,A: $tType,F: B > A,Y3: A] :
( ( ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( F @ ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F @ Y3 ) )
= Y3 ) ) ).
% surj_f_inv_f
thf(fact_1616_surj__iff__all,axiom,
! [B: $tType,A: $tType,F: B > A] :
( ( ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
= ( ! [X4: A] :
( ( F @ ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F @ X4 ) )
= X4 ) ) ) ).
% surj_iff_all
thf(fact_1617_image__f__inv__f,axiom,
! [B: $tType,A: $tType,F: B > A,A6: set @ A] :
( ( ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( image2 @ B @ A @ F @ ( image2 @ A @ B @ ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F ) @ A6 ) )
= A6 ) ) ).
% image_f_inv_f
thf(fact_1618_surj__imp__inv__eq,axiom,
! [B: $tType,A: $tType,F: B > A,G: A > B] :
( ( ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ! [X: B] :
( ( G @ ( F @ X ) )
= X )
=> ( ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F )
= G ) ) ) ).
% surj_imp_inv_eq
thf(fact_1619_inj__on__inv__into,axiom,
! [B: $tType,A: $tType,B6: set @ A,F: B > A,A6: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B6 @ ( image2 @ B @ A @ F @ A6 ) )
=> ( inj_on @ A @ B @ ( hilbert_inv_into @ B @ A @ A6 @ F ) @ B6 ) ) ).
% inj_on_inv_into
thf(fact_1620_inj__transfer,axiom,
! [B: $tType,A: $tType,F: A > B,P2: A > $o,X2: A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ! [Y: B] :
( ( member @ B @ Y @ ( image2 @ A @ B @ F @ ( top_top @ ( set @ A ) ) ) )
=> ( P2 @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F @ Y ) ) )
=> ( P2 @ X2 ) ) ) ).
% inj_transfer
thf(fact_1621_image__inv__f__f,axiom,
! [B: $tType,A: $tType,F: A > B,A6: set @ A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( image2 @ B @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F ) @ ( image2 @ A @ B @ F @ A6 ) )
= A6 ) ) ).
% image_inv_f_f
thf(fact_1622_inj__imp__surj__inv,axiom,
! [B: $tType,A: $tType,F: A > B] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( image2 @ B @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F ) @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% inj_imp_surj_inv
thf(fact_1623_surj__imp__inj__inv,axiom,
! [B: $tType,A: $tType,F: B > A] :
( ( ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( inj_on @ A @ B @ ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F ) @ ( top_top @ ( set @ A ) ) ) ) ).
% surj_imp_inj_inv
thf(fact_1624_inv__into__comp,axiom,
! [A: $tType,C: $tType,B: $tType,F: A > B,G: C > A,A6: set @ C,X2: B] :
( ( inj_on @ A @ B @ F @ ( image2 @ C @ A @ G @ A6 ) )
=> ( ( inj_on @ C @ A @ G @ A6 )
=> ( ( member @ B @ X2 @ ( image2 @ A @ B @ F @ ( image2 @ C @ A @ G @ A6 ) ) )
=> ( ( hilbert_inv_into @ C @ B @ A6 @ ( comp @ A @ B @ C @ F @ G ) @ X2 )
= ( comp @ A @ C @ B @ ( hilbert_inv_into @ C @ A @ A6 @ G ) @ ( hilbert_inv_into @ A @ B @ ( image2 @ C @ A @ G @ A6 ) @ F ) @ X2 ) ) ) ) ) ).
% inv_into_comp
thf(fact_1625_bij__betw__inv__into__subset,axiom,
! [B: $tType,A: $tType,F: A > B,A6: set @ A,A9: set @ B,B6: set @ A,B8: set @ B] :
( ( bij_betw @ A @ B @ F @ A6 @ A9 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ A6 )
=> ( ( ( image2 @ A @ B @ F @ B6 )
= B8 )
=> ( bij_betw @ B @ A @ ( hilbert_inv_into @ A @ B @ A6 @ F ) @ B8 @ B6 ) ) ) ) ).
% bij_betw_inv_into_subset
thf(fact_1626_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_1627_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_1628_iso__backward,axiom,
! [A: $tType,B: $tType,X2: A,Y3: A,R2: set @ ( product_prod @ A @ A ),R: set @ ( product_prod @ B @ B ),F: B > A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R2 )
=> ( ( bNF_Wellorder_iso @ B @ A @ R @ R2 @ F )
=> ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( hilbert_inv_into @ B @ A @ ( field2 @ B @ R ) @ F @ X2 ) @ ( hilbert_inv_into @ B @ A @ ( field2 @ B @ R ) @ F @ Y3 ) ) @ R ) ) ) ).
% iso_backward
thf(fact_1629_bij__image__Collect__eq,axiom,
! [A: $tType,B: $tType,F: A > B,P2: A > $o] :
( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( image2 @ A @ B @ F @ ( collect @ A @ P2 ) )
= ( collect @ B
@ ^ [Y5: B] : ( P2 @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F @ Y5 ) ) ) ) ) ).
% bij_image_Collect_eq
thf(fact_1630_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_1631_inj__imp__bij__betw__inv,axiom,
! [B: $tType,A: $tType,F: A > B,M: set @ A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( bij_betw @ B @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F ) @ ( image2 @ A @ B @ F @ M ) @ M ) ) ).
% inj_imp_bij_betw_inv
thf(fact_1632_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_1633_bij__vimage__eq__inv__image,axiom,
! [A: $tType,B: $tType,F: A > B,A6: set @ B] :
( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( vimage @ A @ B @ F @ A6 )
= ( image2 @ B @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F ) @ A6 ) ) ) ).
% bij_vimage_eq_inv_image
thf(fact_1634_max__strict__def,axiom,
( fun_max_strict
= ( max_ext @ ( product_prod @ nat @ nat ) @ fun_pair_less ) ) ).
% max_strict_def
thf(fact_1635_min__strict__def,axiom,
( fun_min_strict
= ( min_ext @ ( product_prod @ nat @ nat ) @ fun_pair_less ) ) ).
% min_strict_def
thf(fact_1636_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 ) @ ( insert @ ( 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_1637_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 ) @ ( insert @ ( 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_1638_inv__into__Field__embed,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R2: set @ ( product_prod @ B @ B ),F: A > B] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( bNF_Wellorder_embed @ A @ B @ R @ R2 @ F )
=> ( ( ord_less_eq @ ( set @ B ) @ ( field2 @ B @ R2 ) @ ( image2 @ A @ B @ F @ ( field2 @ A @ R ) ) )
=> ( bNF_Wellorder_embed @ B @ A @ R2 @ R @ ( hilbert_inv_into @ A @ B @ ( field2 @ A @ R ) @ F ) ) ) ) ) ).
% inv_into_Field_embed
thf(fact_1639_strict__mono__inv__on__range,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F: A > B] :
( ( order_strict_mono @ A @ B @ F )
=> ( strict_mono_on @ B @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F ) @ ( image2 @ A @ B @ F @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% strict_mono_inv_on_range
thf(fact_1640_comp__set__bd__Union__o__collect,axiom,
! [C: $tType,B: $tType,A: $tType,X2: C,X6: 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 ) )
@ ^ [F4: C > ( set @ ( set @ A ) )] : ( F4 @ X2 )
@ X6 ) ) ) )
@ 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 ) @ X6 ) @ X2 ) ) @ Hbd ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ).
% comp_set_bd_Union_o_collect
thf(fact_1641_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_1642_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_1643_image__atLeastZeroLessThan__int,axiom,
! [U2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ U2 )
=> ( ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ U2 )
= ( image2 @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_ord_lessThan @ nat @ ( nat2 @ U2 ) ) ) ) ) ).
% image_atLeastZeroLessThan_int
thf(fact_1644_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_1645_Array_Olength__alloc,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [Xs: list @ A,H: heap_ext @ product_unit,Ys: list @ A] :
( ( length @ A @ ( product_snd @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( alloc @ A @ Xs @ H ) ) @ ( product_fst @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( alloc @ A @ Ys @ H ) ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% Array.length_alloc
thf(fact_1646_min__ext__def,axiom,
! [A: $tType] :
( ( min_ext @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ^ [Uu: product_prod @ ( set @ A ) @ ( set @ A )] :
? [X10: set @ A,Y9: set @ A] :
( ( Uu
= ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X10 @ Y9 ) )
& ( X10
!= ( bot_bot @ ( set @ A ) ) )
& ! [X4: A] :
( ( member @ A @ X4 @ Y9 )
=> ? [Y5: A] :
( ( member @ A @ Y5 @ X10 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X4 ) @ R5 ) ) ) ) ) ) ) ).
% min_ext_def
thf(fact_1647_True__in__image__Bex,axiom,
! [A: $tType,P2: A > $o,A6: set @ A] :
( ( member @ $o @ $true @ ( image2 @ A @ $o @ P2 @ A6 ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ A6 )
& ( P2 @ X4 ) ) ) ) ).
% True_in_image_Bex
thf(fact_1648_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_1649_Image__Collect__case__prod,axiom,
! [B: $tType,A: $tType,P2: B > A > $o,A6: set @ B] :
( ( image @ B @ A @ ( collect @ ( product_prod @ B @ A ) @ ( product_case_prod @ B @ A @ $o @ P2 ) ) @ A6 )
= ( collect @ A
@ ^ [Y5: A] :
? [X4: B] :
( ( member @ B @ X4 @ A6 )
& ( P2 @ X4 @ Y5 ) ) ) ) ).
% Image_Collect_case_prod
thf(fact_1650_image__minus__const__atLeast,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C3: A,A3: A] :
( ( image2 @ A @ A @ ( minus_minus @ A @ C3 ) @ ( set_ord_atLeast @ A @ A3 ) )
= ( set_ord_atMost @ A @ ( minus_minus @ A @ C3 @ A3 ) ) ) ) ).
% image_minus_const_atLeast
thf(fact_1651_image__minus__const__AtMost,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C3: A,B2: A] :
( ( image2 @ A @ A @ ( minus_minus @ A @ C3 ) @ ( set_ord_atMost @ A @ B2 ) )
= ( set_ord_atLeast @ A @ ( minus_minus @ A @ C3 @ B2 ) ) ) ) ).
% image_minus_const_AtMost
thf(fact_1652_image__uminus__atLeast,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X2: A] :
( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_ord_atLeast @ A @ X2 ) )
= ( set_ord_atMost @ A @ ( uminus_uminus @ A @ X2 ) ) ) ) ).
% image_uminus_atLeast
thf(fact_1653_image__uminus__atMost,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X2: A] :
( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_ord_atMost @ A @ X2 ) )
= ( set_ord_atLeast @ A @ ( uminus_uminus @ A @ X2 ) ) ) ) ).
% image_uminus_atMost
thf(fact_1654_SUP__bool__eq,axiom,
! [A: $tType] :
( ( ^ [A8: set @ A,F4: A > $o] : ( complete_Sup_Sup @ $o @ ( image2 @ A @ $o @ F4 @ A8 ) ) )
= ( bex @ A ) ) ).
% SUP_bool_eq
thf(fact_1655_image__def,axiom,
! [B: $tType,A: $tType] :
( ( image2 @ A @ B )
= ( ^ [F4: A > B,A8: set @ A] :
( collect @ B
@ ^ [Y5: B] :
? [X4: A] :
( ( member @ A @ X4 @ A8 )
& ( Y5
= ( F4 @ X4 ) ) ) ) ) ) ).
% image_def
thf(fact_1656_card__of__UNION__Sigma,axiom,
! [B: $tType,A: $tType,A6: 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 ) @ A6 @ I4 ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A ) @ ( product_Sigma @ B @ A @ I4 @ A6 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( product_prod @ B @ A ) ) ) ).
% card_of_UNION_Sigma
thf(fact_1657_UNION__eq,axiom,
! [B: $tType,A: $tType,B6: B > ( set @ A ),A6: set @ B] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B6 @ A6 ) )
= ( collect @ A
@ ^ [Y5: A] :
? [X4: B] :
( ( member @ B @ X4 @ A6 )
& ( member @ A @ Y5 @ ( B6 @ X4 ) ) ) ) ) ).
% UNION_eq
thf(fact_1658_Collect__bex__eq,axiom,
! [A: $tType,B: $tType,A6: set @ B,P2: A > B > $o] :
( ( collect @ A
@ ^ [X4: A] :
? [Y5: B] :
( ( member @ B @ Y5 @ A6 )
& ( P2 @ X4 @ Y5 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y5: B] :
( collect @ A
@ ^ [X4: A] : ( P2 @ X4 @ Y5 ) )
@ A6 ) ) ) ).
% Collect_bex_eq
thf(fact_1659_Image__def,axiom,
! [B: $tType,A: $tType] :
( ( image @ A @ B )
= ( ^ [R5: set @ ( product_prod @ A @ B ),S: set @ A] :
( collect @ B
@ ^ [Y5: B] :
? [X4: A] :
( ( member @ A @ X4 @ S )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ R5 ) ) ) ) ) ).
% Image_def
thf(fact_1660_vimage__image__eq,axiom,
! [B: $tType,A: $tType,F: A > B,A6: set @ A] :
( ( vimage @ A @ B @ F @ ( image2 @ A @ B @ F @ A6 ) )
= ( collect @ A
@ ^ [Y5: A] :
? [X4: A] :
( ( member @ A @ X4 @ A6 )
& ( ( F @ X4 )
= ( F @ Y5 ) ) ) ) ) ).
% vimage_image_eq
thf(fact_1661_length__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( length @ A )
= ( ^ [H6: heap_ext @ product_unit,A7: array @ A] : ( size_size @ ( list @ A ) @ ( get @ A @ H6 @ A7 ) ) ) ) ) ).
% length_def
thf(fact_1662_Set_Obind__def,axiom,
! [B: $tType,A: $tType] :
( ( bind4 @ A @ B )
= ( ^ [A8: set @ A,F4: A > ( set @ B )] :
( collect @ B
@ ^ [X4: B] :
? [Y5: set @ B] :
( ( member @ ( set @ B ) @ Y5 @ ( image2 @ A @ ( set @ B ) @ F4 @ A8 ) )
& ( member @ B @ X4 @ Y5 ) ) ) ) ) ).
% Set.bind_def
thf(fact_1663_bijection_Osurj,axiom,
! [A: $tType,F: A > A] :
( ( hilbert_bijection @ A @ F )
=> ( ( image2 @ A @ A @ F @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% bijection.surj
thf(fact_1664_card__of__image,axiom,
! [B: $tType,A: $tType,F: B > A,A6: 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 @ A6 ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A6 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ).
% card_of_image
thf(fact_1665_ex__bij__betw,axiom,
! [B: $tType,A: $tType,A6: 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 @ A6 ) @ R ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ? [F3: B > A,B11: set @ B] : ( bij_betw @ B @ A @ F3 @ B11 @ A6 ) ) ).
% ex_bij_betw
thf(fact_1666_type__copy__set__bd,axiom,
! [A: $tType,D: $tType,C: $tType,B: $tType,S5: A > ( set @ B ),Bd: set @ ( product_prod @ C @ C ),Rep: D > A,X2: D] :
( ! [Y: 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 @ ( S5 @ Y ) ) @ 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 @ S5 @ Rep @ X2 ) ) @ Bd ) @ ( bNF_Wellorder_ordLeq @ B @ C ) ) ) ).
% type_copy_set_bd
thf(fact_1667_card__of__ordLeq2,axiom,
! [B: $tType,A: $tType,A6: set @ A,B6: set @ B] :
( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ? [G4: B > A] :
( ( image2 @ B @ A @ G4 @ B6 )
= A6 ) )
= ( 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 @ A6 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ) ).
% card_of_ordLeq2
thf(fact_1668_surj__imp__ordLeq,axiom,
! [B: $tType,A: $tType,B6: set @ A,F: B > A,A6: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B6 @ ( image2 @ B @ A @ F @ A6 ) )
=> ( 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 @ A6 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ).
% surj_imp_ordLeq
thf(fact_1669_card__of__ordLess2,axiom,
! [A: $tType,B: $tType,B6: set @ A,A6: set @ B] :
( ( B6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ~ ? [F4: B > A] :
( ( image2 @ B @ A @ F4 @ A6 )
= 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 @ A6 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B6 ) ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) ) ) ) ).
% card_of_ordLess2
thf(fact_1670_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 )
@ ^ [F4: A > B] : ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F4 @ ( top_top @ ( set @ A ) ) ) @ B6 ) ) ) )
@ ( bNF_Wellorder_ordIso @ ( A > B ) @ ( A > B ) ) ) ).
% card_of_Func_UNIV
thf(fact_1671_bijection_Osurj__inv,axiom,
! [A: $tType,F: A > A] :
( ( hilbert_bijection @ A @ F )
=> ( ( image2 @ A @ A @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F ) @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% bijection.surj_inv
thf(fact_1672_card__of__ordLeq,axiom,
! [B: $tType,A: $tType,A6: set @ A,B6: set @ B] :
( ( ? [F4: A > B] :
( ( inj_on @ A @ B @ F4 @ A6 )
& ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F4 @ A6 ) @ 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 @ A6 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ).
% card_of_ordLeq
thf(fact_1673_card__of__ordLess,axiom,
! [A: $tType,B: $tType,A6: set @ A,B6: set @ B] :
( ( ~ ? [F4: A > B] :
( ( inj_on @ A @ B @ F4 @ A6 )
& ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F4 @ A6 ) @ 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 @ A6 ) ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) ) ) ).
% card_of_ordLess
thf(fact_1674_card__of__UNION__ordLeq__infinite,axiom,
! [B: $tType,A: $tType,C: $tType,B6: set @ A,I4: set @ B,A6: 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 ) )
=> ( ! [X: B] :
( ( member @ B @ X @ 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 @ ( A6 @ X ) ) @ ( 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 ) @ A6 @ I4 ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) ) ) ) ).
% card_of_UNION_ordLeq_infinite
thf(fact_1675_at__top__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( at_top @ A )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ A @ ( filter @ A )
@ ^ [K3: A] : ( principal @ A @ ( set_ord_atLeast @ A @ K3 ) )
@ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% at_top_def
thf(fact_1676_array__of__list__make,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( of_list @ A )
= ( ^ [Xs2: list @ A] : ( make @ A @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( nth3 @ A @ Xs2 ) ) ) ) ) ).
% array_of_list_make
thf(fact_1677_at__top__sub,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C3: A] :
( ( at_top @ A )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ A @ ( filter @ A )
@ ^ [K3: A] : ( principal @ A @ ( set_ord_atLeast @ A @ K3 ) )
@ ( set_ord_atLeast @ A @ C3 ) ) ) ) ) ).
% at_top_sub
thf(fact_1678_card__of__UNION__ordLeq__infinite__Field,axiom,
! [B: $tType,A: $tType,C: $tType,R: set @ ( product_prod @ A @ A ),I4: set @ B,A6: 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 ) )
=> ( ! [X: B] :
( ( member @ B @ X @ 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 @ ( A6 @ X ) ) @ 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 ) @ A6 @ I4 ) ) ) @ R ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) ) ) ) ) ).
% card_of_UNION_ordLeq_infinite_Field
thf(fact_1679_product__nth,axiom,
! [A: $tType,B: $tType,N5: nat,Xs: list @ A,Ys: list @ B] :
( ( ord_less @ nat @ N5 @ ( times_times @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) ) )
=> ( ( nth3 @ ( product_prod @ A @ B ) @ ( product @ A @ B @ Xs @ Ys ) @ N5 )
= ( product_Pair @ A @ B @ ( nth3 @ A @ Xs @ ( divide_divide @ nat @ N5 @ ( size_size @ ( list @ B ) @ Ys ) ) ) @ ( nth3 @ B @ Ys @ ( modulo_modulo @ nat @ N5 @ ( size_size @ ( list @ B ) @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_1680_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
@ ^ [X10: set @ A,Y9: set @ A] :
( ( finite_finite2 @ A @ X10 )
& ( finite_finite2 @ A @ Y9 )
& ( Y9
!= ( bot_bot @ ( set @ A ) ) )
& ! [X4: A] :
( ( member @ A @ X4 @ X10 )
=> ? [Y5: A] :
( ( member @ A @ Y5 @ Y9 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R6 ) ) ) ) ) ) ) ) ).
% max_ext_eq
thf(fact_1681_remdups__adj__altdef,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( remdups_adj @ A @ Xs )
= Ys )
= ( ? [F4: nat > nat] :
( ( order_mono @ nat @ nat @ F4 )
& ( ( image2 @ nat @ nat @ F4 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) )
= ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Ys ) ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth3 @ A @ Xs @ I3 )
= ( nth3 @ A @ Ys @ ( F4 @ I3 ) ) ) )
& ! [I3: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ I3 @ ( one_one @ nat ) ) @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( nth3 @ A @ Xs @ I3 )
= ( nth3 @ A @ Xs @ ( plus_plus @ nat @ I3 @ ( one_one @ nat ) ) ) )
= ( ( F4 @ I3 )
= ( F4 @ ( plus_plus @ nat @ I3 @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ).
% remdups_adj_altdef
thf(fact_1682_right__total__relcompp__transfer,axiom,
! [C: $tType,A: $tType,E: $tType,F2: $tType,B: $tType,D: $tType,B6: A > B > $o,A6: C > D > $o,C5: E > F2 > $o] :
( ( right_total @ A @ B @ B6 )
=> ( bNF_rel_fun @ ( C > A > $o ) @ ( D > B > $o ) @ ( ( A > E > $o ) > C > E > $o ) @ ( ( B > F2 > $o ) > D > F2 > $o )
@ ( bNF_rel_fun @ C @ D @ ( A > $o ) @ ( B > $o ) @ A6
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ B6
@ ^ [Y2: $o,Z2: $o] : Y2 = Z2 ) )
@ ( bNF_rel_fun @ ( A > E > $o ) @ ( B > F2 > $o ) @ ( C > E > $o ) @ ( D > F2 > $o )
@ ( bNF_rel_fun @ A @ B @ ( E > $o ) @ ( F2 > $o ) @ B6
@ ( bNF_rel_fun @ E @ F2 @ $o @ $o @ C5
@ ^ [Y2: $o,Z2: $o] : Y2 = Z2 ) )
@ ( bNF_rel_fun @ C @ D @ ( E > $o ) @ ( F2 > $o ) @ A6
@ ( bNF_rel_fun @ E @ F2 @ $o @ $o @ C5
@ ^ [Y2: $o,Z2: $o] : Y2 = Z2 ) ) )
@ ^ [R6: C > A > $o,S4: A > E > $o,X4: C,Z5: E] :
? [Y5: A] :
( ( member @ A @ Y5 @ ( collect @ A @ ( domainp @ A @ B @ B6 ) ) )
& ( R6 @ X4 @ Y5 )
& ( S4 @ Y5 @ Z5 ) )
@ ( relcompp @ D @ B @ F2 ) ) ) ).
% right_total_relcompp_transfer
thf(fact_1683_rel__fun__def,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType] :
( ( bNF_rel_fun @ A @ C @ B @ D )
= ( ^ [A8: A > C > $o,B7: B > D > $o,F4: A > B,G4: C > D] :
! [X4: A,Y5: C] :
( ( A8 @ X4 @ Y5 )
=> ( B7 @ ( F4 @ X4 ) @ ( G4 @ Y5 ) ) ) ) ) ).
% rel_fun_def
thf(fact_1684_Card__order__trans,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),X2: A,Y3: A,Z3: A] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( X2 != Y3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R )
=> ( ( Y3 != Z3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ R )
=> ( ( X2 != Z3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Z3 ) @ R ) ) ) ) ) ) ) ).
% Card_order_trans
thf(fact_1685_Card__order__wo__rel,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( bNF_Wellorder_wo_rel @ A @ R ) ) ).
% Card_order_wo_rel
thf(fact_1686_pcr__Domainp__par,axiom,
! [A: $tType,B: $tType,C: $tType,B6: A > B > $o,P23: A > $o,A6: C > A > $o,P12: C > $o,P24: C > $o] :
( ( ( domainp @ A @ B @ B6 )
= P23 )
=> ( ( ( domainp @ C @ A @ A6 )
= P12 )
=> ( ( bNF_rel_fun @ C @ A @ $o @ $o @ A6
@ ^ [Y2: $o,Z2: $o] : Y2 = Z2
@ P24
@ P23 )
=> ( ( domainp @ C @ B @ ( relcompp @ C @ A @ B @ A6 @ B6 ) )
= ( inf_inf @ ( C > $o ) @ P12 @ P24 ) ) ) ) ) ).
% pcr_Domainp_par
thf(fact_1687_pcr__Domainp__total,axiom,
! [A: $tType,B: $tType,C: $tType,B6: A > B > $o,A6: C > A > $o,P2: C > $o] :
( ( left_total @ A @ B @ B6 )
=> ( ( ( domainp @ C @ A @ A6 )
= P2 )
=> ( ( domainp @ C @ B @ ( relcompp @ C @ A @ B @ A6 @ B6 ) )
= P2 ) ) ) ).
% pcr_Domainp_total
thf(fact_1688_pcr__Domainp,axiom,
! [B: $tType,A: $tType,C: $tType,B6: A > B > $o,P2: A > $o,A6: C > A > $o] :
( ( ( domainp @ A @ B @ B6 )
= P2 )
=> ( ( domainp @ C @ B @ ( relcompp @ C @ A @ B @ A6 @ B6 ) )
= ( ^ [X4: C] :
? [Y5: A] :
( ( A6 @ X4 @ Y5 )
& ( P2 @ Y5 ) ) ) ) ) ).
% pcr_Domainp
thf(fact_1689_Domain__def,axiom,
! [B: $tType,A: $tType] :
( ( domain @ A @ B )
= ( ^ [R5: set @ ( product_prod @ A @ B )] :
( collect @ A
@ ( domainp @ A @ B
@ ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ R5 ) ) ) ) ) ).
% Domain_def
thf(fact_1690_Domainp__Domain__eq,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B )] :
( ( domainp @ A @ B
@ ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ R ) )
= ( ^ [X4: A] : ( member @ A @ X4 @ ( domain @ A @ B @ R ) ) ) ) ).
% Domainp_Domain_eq
thf(fact_1691_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 ) )
=> ? [X: A] :
( ( member @ A @ X @ ( field2 @ A @ R ) )
& ( A3 != X )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ X ) @ R ) ) ) ) ) ).
% infinite_Card_order_limit
thf(fact_1692_Collect__all__eq,axiom,
! [A: $tType,B: $tType,P2: A > B > $o] :
( ( collect @ A
@ ^ [X4: A] :
! [X10: B] : ( P2 @ X4 @ X10 ) )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y5: B] :
( collect @ A
@ ^ [X4: A] : ( P2 @ X4 @ Y5 ) )
@ ( top_top @ ( set @ B ) ) ) ) ) ).
% Collect_all_eq
thf(fact_1693_success__makeI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [N5: nat,F: nat > A,H: heap_ext @ product_unit] : ( heap_success @ ( array @ A ) @ ( make @ A @ N5 @ F ) @ H ) ) ).
% success_makeI
thf(fact_1694_pcr__Domainp__par__left__total,axiom,
! [A: $tType,B: $tType,C: $tType,B6: A > B > $o,P2: A > $o,A6: C > A > $o,P7: C > $o] :
( ( ( domainp @ A @ B @ B6 )
= P2 )
=> ( ( left_total @ C @ A @ A6 )
=> ( ( bNF_rel_fun @ C @ A @ $o @ $o @ A6
@ ^ [Y2: $o,Z2: $o] : Y2 = Z2
@ P7
@ P2 )
=> ( ( domainp @ C @ B @ ( relcompp @ C @ A @ B @ A6 @ B6 ) )
= P7 ) ) ) ) ).
% pcr_Domainp_par_left_total
thf(fact_1695_regularCard__UNION,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),As2: A > ( set @ B ),B6: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( bNF_Ca7133664381575040944arCard @ A @ R )
=> ( ( bNF_Ca3754400796208372196lChain @ A @ ( set @ B ) @ R @ As2 )
=> ( ( ord_less_eq @ ( set @ B ) @ B6 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ As2 @ ( 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 ) )
=> ? [X: A] :
( ( member @ A @ X @ ( field2 @ A @ R ) )
& ( ord_less_eq @ ( set @ B ) @ B6 @ ( As2 @ X ) ) ) ) ) ) ) ) ).
% regularCard_UNION
thf(fact_1696_ex__toCard__pred,axiom,
! [B: $tType,A: $tType,A6: 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 @ A6 ) @ R ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R ) @ R )
=> ? [X_12: A > B] : ( bNF_Gr1419584066657907630d_pred @ A @ B @ A6 @ R @ X_12 ) ) ) ).
% ex_toCard_pred
thf(fact_1697_toCard__pred__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Gr1419584066657907630d_pred @ A @ B )
= ( ^ [A8: set @ A,R5: set @ ( product_prod @ B @ B ),F4: A > B] :
( ( inj_on @ A @ B @ F4 @ A8 )
& ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F4 @ A8 ) @ ( field2 @ B @ R5 ) )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R5 ) @ R5 ) ) ) ) ).
% toCard_pred_def
thf(fact_1698_cardSuc__UNION,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),As2: ( 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 ) @ As2 )
=> ( ( ord_less_eq @ ( set @ B ) @ B6 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ ( set @ A ) @ ( set @ B ) @ As2 @ ( 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 ) )
=> ? [X: set @ A] :
( ( member @ ( set @ A ) @ X @ ( field2 @ ( set @ A ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R ) ) )
& ( ord_less_eq @ ( set @ B ) @ B6 @ ( As2 @ X ) ) ) ) ) ) ) ) ).
% cardSuc_UNION
thf(fact_1699_toCard__inj,axiom,
! [B: $tType,A: $tType,A6: set @ A,R: set @ ( product_prod @ B @ B ),X2: A,Y3: 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 @ A6 ) @ R ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R ) @ R )
=> ( ( member @ A @ X2 @ A6 )
=> ( ( member @ A @ Y3 @ A6 )
=> ( ( ( bNF_Greatest_toCard @ A @ B @ A6 @ R @ X2 )
= ( bNF_Greatest_toCard @ A @ B @ A6 @ R @ Y3 ) )
= ( X2 = Y3 ) ) ) ) ) ) ).
% toCard_inj
thf(fact_1700_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 ),X2: D] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ Fbd ) @ Fbd )
=> ( ! [X: 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 @ X ) ) @ Fbd ) @ ( bNF_Wellorder_ordLeq @ C @ A ) )
=> ( ! [X: 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 @ X ) ) @ 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 @ X2 ) ) ) ) @ ( bNF_Cardinal_cprod @ E @ A @ Gbd @ Fbd ) ) @ ( bNF_Wellorder_ordLeq @ C @ ( product_prod @ E @ A ) ) ) ) ) ) ).
% comp_single_set_bd
thf(fact_1701_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
@ ^ [R5: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R5 @ S )
& ! [A7: A,B5: A,C6: A] :
( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B5 ) @ S )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ C6 ) @ R5 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B5 ) @ R5 ) ) ) ) ) ) ).
% init_seg_of_def
thf(fact_1702_wfP__SUP,axiom,
! [B: $tType,A: $tType,R: A > B > B > $o] :
( ! [I6: A] : ( wfP @ B @ ( R @ I6 ) )
=> ( ! [I6: A,J3: A] :
( ( ( R @ I6 )
!= ( R @ J3 ) )
=> ( ( inf_inf @ ( B > $o ) @ ( domainp @ B @ B @ ( R @ I6 ) ) @ ( rangep @ B @ B @ ( R @ J3 ) ) )
= ( bot_bot @ ( B > $o ) ) ) )
=> ( wfP @ B @ ( complete_Sup_Sup @ ( B > B > $o ) @ ( image2 @ A @ ( B > B > $o ) @ R @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% wfP_SUP
thf(fact_1703_wfP__wf__eq,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( wfP @ A
@ ^ [X4: A,Y5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R ) )
= ( wf @ A @ R ) ) ).
% wfP_wf_eq
thf(fact_1704_wf__in__rel,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ R3 )
=> ( wfP @ A @ ( fun_in_rel @ A @ A @ R3 ) ) ) ).
% wf_in_rel
thf(fact_1705_wfP__def,axiom,
! [A: $tType] :
( ( wfP @ A )
= ( ^ [R5: A > A > $o] : ( wf @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R5 ) ) ) ) ) ).
% wfP_def
thf(fact_1706_wfP__acyclicP,axiom,
! [A: $tType,R: A > A > $o] :
( ( wfP @ A @ R )
=> ( transitive_acyclic @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R ) ) ) ) ).
% wfP_acyclicP
thf(fact_1707_toCard__pred__toCard,axiom,
! [A: $tType,B: $tType,A6: 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 @ A6 ) @ R ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R ) @ R )
=> ( bNF_Gr1419584066657907630d_pred @ A @ B @ A6 @ R @ ( bNF_Greatest_toCard @ A @ B @ A6 @ R ) ) ) ) ).
% toCard_pred_toCard
thf(fact_1708_fromCard__toCard,axiom,
! [B: $tType,A: $tType,A6: 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 @ A6 ) @ R ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R ) @ R )
=> ( ( member @ A @ B2 @ A6 )
=> ( ( bNF_Gr5436034075474128252omCard @ A @ B @ A6 @ R @ ( bNF_Greatest_toCard @ A @ B @ A6 @ R @ B2 ) )
= B2 ) ) ) ) ).
% fromCard_toCard
thf(fact_1709_cardSuc__UNION__Cinfinite,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),As2: ( 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 ) @ As2 )
=> ( ( ord_less_eq @ ( set @ B ) @ B6 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ ( set @ A ) @ ( set @ B ) @ As2 @ ( 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 ) )
=> ? [X: set @ A] :
( ( member @ ( set @ A ) @ X @ ( field2 @ ( set @ A ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R ) ) )
& ( ord_less_eq @ ( set @ B ) @ B6 @ ( As2 @ X ) ) ) ) ) ) ) ).
% cardSuc_UNION_Cinfinite
thf(fact_1710_composed__equiv__rel__eq__onp,axiom,
! [B: $tType,A: $tType,R3: A > B > $o,P2: A > $o,P7: B > $o,P8: A > $o] :
( ( left_unique @ A @ B @ R3 )
=> ( ( bNF_rel_fun @ A @ B @ $o @ $o @ R3
@ ^ [Y2: $o,Z2: $o] : Y2 = Z2
@ P2
@ P7 )
=> ( ( ( domainp @ A @ B @ R3 )
= P8 )
=> ( ( relcompp @ A @ B @ A @ R3 @ ( relcompp @ B @ B @ A @ ( bNF_eq_onp @ B @ P7 ) @ ( conversep @ A @ B @ R3 ) ) )
= ( bNF_eq_onp @ A @ ( inf_inf @ ( A > $o ) @ P8 @ P2 ) ) ) ) ) ) ).
% composed_equiv_rel_eq_onp
thf(fact_1711_nth__zip,axiom,
! [A: $tType,B: $tType,I: nat,Xs: list @ A,Ys: list @ B] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( nth3 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) @ I )
= ( product_Pair @ A @ B @ ( nth3 @ A @ Xs @ I ) @ ( nth3 @ B @ Ys @ I ) ) ) ) ) ).
% nth_zip
thf(fact_1712_relcompp__transfer,axiom,
! [C: $tType,A: $tType,E: $tType,F2: $tType,B: $tType,D: $tType,B6: A > B > $o,A6: C > D > $o,C5: E > F2 > $o] :
( ( bi_total @ A @ B @ B6 )
=> ( bNF_rel_fun @ ( C > A > $o ) @ ( D > B > $o ) @ ( ( A > E > $o ) > C > E > $o ) @ ( ( B > F2 > $o ) > D > F2 > $o )
@ ( bNF_rel_fun @ C @ D @ ( A > $o ) @ ( B > $o ) @ A6
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ B6
@ ^ [Y2: $o,Z2: $o] : Y2 = Z2 ) )
@ ( bNF_rel_fun @ ( A > E > $o ) @ ( B > F2 > $o ) @ ( C > E > $o ) @ ( D > F2 > $o )
@ ( bNF_rel_fun @ A @ B @ ( E > $o ) @ ( F2 > $o ) @ B6
@ ( bNF_rel_fun @ E @ F2 @ $o @ $o @ C5
@ ^ [Y2: $o,Z2: $o] : Y2 = Z2 ) )
@ ( bNF_rel_fun @ C @ D @ ( E > $o ) @ ( F2 > $o ) @ A6
@ ( bNF_rel_fun @ E @ F2 @ $o @ $o @ C5
@ ^ [Y2: $o,Z2: $o] : Y2 = Z2 ) ) )
@ ( relcompp @ C @ A @ E )
@ ( relcompp @ D @ B @ F2 ) ) ) ).
% relcompp_transfer
thf(fact_1713_UNION__Cinfinite__bound,axiom,
! [A: $tType,B: $tType,C: $tType,I4: set @ A,R: set @ ( product_prod @ B @ B ),A6: 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 ) )
=> ( ! [X: A] :
( ( member @ A @ X @ 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 @ ( A6 @ X ) ) @ 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 ) @ A6 @ I4 ) ) ) @ R ) @ ( bNF_Wellorder_ordLeq @ C @ B ) ) ) ) ) ).
% UNION_Cinfinite_bound
thf(fact_1714_Cinfinite__limit__finite,axiom,
! [A: $tType,X6: set @ A,R: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ A @ X6 )
=> ( ( ord_less_eq @ ( set @ A ) @ X6 @ ( field2 @ A @ R ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ A @ R )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
=> ? [X: A] :
( ( member @ A @ X @ ( field2 @ A @ R ) )
& ! [Xa3: A] :
( ( member @ A @ Xa3 @ X6 )
=> ( ( Xa3 != X )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Xa3 @ X ) @ R ) ) ) ) ) ) ) ).
% Cinfinite_limit_finite
thf(fact_1715_eq__onp__top__eq__eq,axiom,
! [A: $tType] :
( ( bNF_eq_onp @ A @ ( top_top @ ( A > $o ) ) )
= ( ^ [Y2: A,Z2: A] : Y2 = Z2 ) ) ).
% eq_onp_top_eq_eq
thf(fact_1716_eq__onp__def,axiom,
! [A: $tType] :
( ( bNF_eq_onp @ A )
= ( ^ [R6: A > $o,X4: A,Y5: A] :
( ( R6 @ X4 )
& ( X4 = Y5 ) ) ) ) ).
% eq_onp_def
thf(fact_1717_eq__onp__True,axiom,
! [A: $tType] :
( ( bNF_eq_onp @ A
@ ^ [Uu: A] : $true )
= ( ^ [Y2: A,Z2: A] : Y2 = Z2 ) ) ).
% eq_onp_True
thf(fact_1718_eq__onp__same__args,axiom,
! [A: $tType,P2: A > $o,X2: A] :
( ( bNF_eq_onp @ A @ P2 @ X2 @ X2 )
= ( P2 @ X2 ) ) ).
% eq_onp_same_args
thf(fact_1719_eq__onp__to__eq,axiom,
! [A: $tType,P2: A > $o,X2: A,Y3: A] :
( ( bNF_eq_onp @ A @ P2 @ X2 @ Y3 )
=> ( X2 = Y3 ) ) ).
% eq_onp_to_eq
thf(fact_1720_eq__onp__mono0,axiom,
! [A: $tType,A6: set @ A,P2: A > $o,Q2: A > $o] :
( ! [X: A] :
( ( member @ A @ X @ A6 )
=> ( ( P2 @ X )
=> ( Q2 @ X ) ) )
=> ! [X5: A] :
( ( member @ A @ X5 @ A6 )
=> ! [Xa3: A] :
( ( member @ A @ Xa3 @ A6 )
=> ( ( bNF_eq_onp @ A @ P2 @ X5 @ Xa3 )
=> ( bNF_eq_onp @ A @ Q2 @ X5 @ Xa3 ) ) ) ) ) ).
% eq_onp_mono0
thf(fact_1721_eq__onp__eqD,axiom,
! [A: $tType,P2: A > $o,Q2: A > A > $o,X2: A] :
( ( ( bNF_eq_onp @ A @ P2 )
= Q2 )
=> ( ( P2 @ X2 )
= ( Q2 @ X2 @ X2 ) ) ) ).
% eq_onp_eqD
thf(fact_1722_bi__total__OO,axiom,
! [A: $tType,C: $tType,B: $tType,A6: A > B > $o,B6: B > C > $o] :
( ( bi_total @ A @ B @ A6 )
=> ( ( bi_total @ B @ C @ B6 )
=> ( bi_total @ A @ C @ ( relcompp @ A @ B @ C @ A6 @ B6 ) ) ) ) ).
% bi_total_OO
thf(fact_1723_eq__onp__mono__iff,axiom,
! [A: $tType,P2: A > $o,Q2: A > $o] :
( ( ord_less_eq @ ( A > A > $o ) @ ( bNF_eq_onp @ A @ P2 ) @ ( bNF_eq_onp @ A @ Q2 ) )
= ( ord_less_eq @ ( A > $o ) @ P2 @ Q2 ) ) ).
% eq_onp_mono_iff
thf(fact_1724_eq__onp__Grp,axiom,
! [A: $tType] :
( ( bNF_eq_onp @ A )
= ( ^ [P4: A > $o] : ( bNF_Grp @ A @ A @ ( collect @ A @ P4 ) @ ( id @ A ) ) ) ) ).
% eq_onp_Grp
thf(fact_1725_Cinfinite__limit,axiom,
! [A: $tType,X2: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ A @ X2 @ ( field2 @ A @ R ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ A @ R )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
=> ? [X: A] :
( ( member @ A @ X @ ( field2 @ A @ R ) )
& ( X2 != X )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X ) @ R ) ) ) ) ).
% Cinfinite_limit
thf(fact_1726_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 ) )
=> ? [X: A] :
( ( member @ A @ X @ ( field2 @ A @ R ) )
& ( X1 != X )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X1 @ X ) @ R )
& ( X22 != X )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X22 @ X ) @ R ) ) ) ) ) ).
% Cinfinite_limit2
thf(fact_1727_composed__equiv__rel__eq__eq__onp,axiom,
! [B: $tType,A: $tType,R3: A > B > $o,P2: A > $o] :
( ( left_unique @ A @ B @ R3 )
=> ( ( ( domainp @ A @ B @ R3 )
= P2 )
=> ( ( relcompp @ A @ B @ A @ R3
@ ( relcompp @ B @ B @ A
@ ^ [Y2: B,Z2: B] : Y2 = Z2
@ ( conversep @ A @ B @ R3 ) ) )
= ( bNF_eq_onp @ A @ P2 ) ) ) ) ).
% composed_equiv_rel_eq_eq_onp
thf(fact_1728_in__set__zip,axiom,
! [B: $tType,A: $tType,P: product_prod @ A @ B,Xs: list @ A,Ys: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ P @ ( set3 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) )
= ( ? [N4: nat] :
( ( ( nth3 @ A @ Xs @ N4 )
= ( product_fst @ A @ B @ P ) )
& ( ( nth3 @ B @ Ys @ N4 )
= ( product_snd @ A @ B @ P ) )
& ( ord_less @ nat @ N4 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ord_less @ nat @ N4 @ ( size_size @ ( list @ B ) @ Ys ) ) ) ) ) ).
% in_set_zip
thf(fact_1729_effect__newE,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [N5: nat,X2: A,H: heap_ext @ product_unit,H2: heap_ext @ product_unit,R: array @ A] :
( ( heap_effect @ ( array @ A ) @ ( new @ A @ N5 @ X2 ) @ H @ H2 @ R )
=> ~ ( ( R
= ( product_fst @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( alloc @ A @ ( replicate @ A @ N5 @ X2 ) @ H ) ) )
=> ( ( H2
= ( product_snd @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( alloc @ A @ ( replicate @ A @ N5 @ X2 ) @ H ) ) )
=> ( ( ( get @ A @ H2 @ R )
= ( replicate @ A @ N5 @ X2 ) )
=> ( ( present @ A @ H2 @ R )
=> ( present @ A @ H @ R ) ) ) ) ) ) ) ).
% effect_newE
thf(fact_1730_Rats__eq__range__of__rat__o__nat__to__rat__surj,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( field_char_0_Rats @ A )
= ( image2 @ nat @ A @ ( comp @ rat @ A @ nat @ ( field_char_0_of_rat @ A ) @ nat_to_rat_surj ) @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% Rats_eq_range_of_rat_o_nat_to_rat_surj
thf(fact_1731_list__eq__iff__zip__eq,axiom,
! [A: $tType] :
( ( ^ [Y2: list @ A,Z2: list @ A] : Y2 = Z2 )
= ( ^ [Xs2: list @ A,Ys2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ A ) @ Ys2 ) )
& ! [X4: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X4 @ ( set3 @ ( product_prod @ A @ A ) @ ( zip @ A @ A @ Xs2 @ Ys2 ) ) )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y2: A,Z2: A] : Y2 = Z2
@ X4 ) ) ) ) ) ).
% list_eq_iff_zip_eq
thf(fact_1732_Nats__def,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_Nats @ A )
= ( image2 @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% Nats_def
thf(fact_1733_effect__makeE,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [N5: nat,F: nat > A,H: heap_ext @ product_unit,H2: heap_ext @ product_unit,R: array @ A] :
( ( heap_effect @ ( array @ A ) @ ( make @ A @ N5 @ F ) @ H @ H2 @ R )
=> ~ ( ( R
= ( product_fst @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( alloc @ A @ ( map @ nat @ A @ F @ ( upt @ ( zero_zero @ nat ) @ N5 ) ) @ H ) ) )
=> ( ( H2
= ( product_snd @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( alloc @ A @ ( map @ nat @ A @ F @ ( upt @ ( zero_zero @ nat ) @ N5 ) ) @ H ) ) )
=> ( ( ( get @ A @ H2 @ R )
= ( map @ nat @ A @ F @ ( upt @ ( zero_zero @ nat ) @ N5 ) ) )
=> ( ( present @ A @ H2 @ R )
=> ( present @ A @ H @ R ) ) ) ) ) ) ) ).
% effect_makeE
thf(fact_1734_Zorns__po__lemma,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( order_7125193373082350890der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ! [C4: set @ A] :
( ( member @ ( set @ A ) @ C4 @ ( chains @ A @ R ) )
=> ? [X5: A] :
( ( member @ A @ X5 @ ( field2 @ A @ R ) )
& ! [Xa: A] :
( ( member @ A @ Xa @ C4 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Xa @ X5 ) @ R ) ) ) )
=> ? [X: A] :
( ( member @ A @ X @ ( field2 @ A @ R ) )
& ! [Xa3: A] :
( ( member @ A @ Xa3 @ ( field2 @ A @ R ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Xa3 ) @ R )
=> ( Xa3 = X ) ) ) ) ) ) ).
% Zorns_po_lemma
thf(fact_1735_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_1736_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_1737_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_1738_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_1739_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_1740_list_Oset__map,axiom,
! [B: $tType,A: $tType,F: A > B,V: list @ A] :
( ( set3 @ B @ ( map @ A @ B @ F @ V ) )
= ( image2 @ A @ B @ F @ ( set3 @ A @ V ) ) ) ).
% list.set_map
thf(fact_1741_image__set,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: list @ B] :
( ( image2 @ B @ A @ F @ ( set3 @ B @ Xs ) )
= ( set3 @ A @ ( map @ B @ A @ F @ Xs ) ) ) ).
% image_set
thf(fact_1742_inj__on__mapI,axiom,
! [B: $tType,A: $tType,F: A > B,A6: set @ ( list @ A )] :
( ( inj_on @ A @ B @ F @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( list @ A ) @ ( set @ A ) @ ( set3 @ A ) @ A6 ) ) )
=> ( inj_on @ ( list @ A ) @ ( list @ B ) @ ( map @ A @ B @ F ) @ A6 ) ) ).
% inj_on_mapI
thf(fact_1743_zip__same,axiom,
! [A: $tType,A3: A,B2: A,Xs: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( set3 @ ( product_prod @ A @ A ) @ ( zip @ A @ A @ Xs @ Xs ) ) )
= ( ( member @ A @ A3 @ ( set3 @ A @ Xs ) )
& ( A3 = B2 ) ) ) ).
% zip_same
thf(fact_1744_in__set__zipE,axiom,
! [A: $tType,B: $tType,X2: A,Y3: B,Xs: list @ A,Ys: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ ( set3 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) )
=> ~ ( ( member @ A @ X2 @ ( set3 @ A @ Xs ) )
=> ~ ( member @ B @ Y3 @ ( set3 @ B @ Ys ) ) ) ) ).
% in_set_zipE
thf(fact_1745_set__zip__leftD,axiom,
! [B: $tType,A: $tType,X2: A,Y3: B,Xs: list @ A,Ys: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ ( set3 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) )
=> ( member @ A @ X2 @ ( set3 @ A @ Xs ) ) ) ).
% set_zip_leftD
thf(fact_1746_set__zip__rightD,axiom,
! [A: $tType,B: $tType,X2: A,Y3: B,Xs: list @ A,Ys: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ ( set3 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) )
=> ( member @ B @ Y3 @ ( set3 @ B @ Ys ) ) ) ).
% set_zip_rightD
thf(fact_1747_map2__map__map,axiom,
! [B: $tType,A: $tType,C: $tType,D: $tType,H: B > C > A,F: D > B,Xs: list @ D,G: D > C] :
( ( map @ ( product_prod @ B @ C ) @ A @ ( product_case_prod @ B @ C @ A @ H ) @ ( zip @ B @ C @ ( map @ D @ B @ F @ Xs ) @ ( map @ D @ C @ G @ Xs ) ) )
= ( map @ D @ A
@ ^ [X4: D] : ( H @ ( F @ X4 ) @ ( G @ X4 ) )
@ Xs ) ) ).
% map2_map_map
thf(fact_1748_in__set__impl__in__set__zip1,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,X2: A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( member @ A @ X2 @ ( set3 @ A @ Xs ) )
=> ~ ! [Y: B] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ ( set3 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip1
thf(fact_1749_in__set__impl__in__set__zip2,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,Y3: B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( member @ B @ Y3 @ ( set3 @ B @ Ys ) )
=> ~ ! [X: A] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y3 ) @ ( set3 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip2
thf(fact_1750_map__zip__map,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F: ( product_prod @ B @ C ) > A,G: D > B,Xs: list @ D,Ys: list @ C] :
( ( map @ ( product_prod @ B @ C ) @ A @ F @ ( zip @ B @ C @ ( map @ D @ B @ G @ Xs ) @ Ys ) )
= ( map @ ( product_prod @ D @ C ) @ A
@ ( product_case_prod @ D @ C @ A
@ ^ [X4: D,Y5: C] : ( F @ ( product_Pair @ B @ C @ ( G @ X4 ) @ Y5 ) ) )
@ ( zip @ D @ C @ Xs @ Ys ) ) ) ).
% map_zip_map
thf(fact_1751_map__zip__map2,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F: ( product_prod @ B @ C ) > A,Xs: list @ B,G: D > C,Ys: list @ D] :
( ( map @ ( product_prod @ B @ C ) @ A @ F @ ( zip @ B @ C @ Xs @ ( map @ D @ C @ G @ Ys ) ) )
= ( map @ ( product_prod @ B @ D ) @ A
@ ( product_case_prod @ B @ D @ A
@ ^ [X4: B,Y5: D] : ( F @ ( product_Pair @ B @ C @ X4 @ ( G @ Y5 ) ) ) )
@ ( zip @ B @ D @ Xs @ Ys ) ) ) ).
% map_zip_map2
thf(fact_1752_zip__map__map,axiom,
! [B: $tType,A: $tType,C: $tType,D: $tType,F: C > A,Xs: list @ C,G: D > B,Ys: list @ D] :
( ( zip @ A @ B @ ( map @ C @ A @ F @ Xs ) @ ( map @ D @ B @ G @ Ys ) )
= ( map @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ C @ D @ ( product_prod @ A @ B )
@ ^ [X4: C,Y5: D] : ( product_Pair @ A @ B @ ( F @ X4 ) @ ( G @ Y5 ) ) )
@ ( zip @ C @ D @ Xs @ Ys ) ) ) ).
% zip_map_map
thf(fact_1753_zip__map2,axiom,
! [B: $tType,A: $tType,C: $tType,Xs: list @ A,F: C > B,Ys: list @ C] :
( ( zip @ A @ B @ Xs @ ( map @ C @ B @ F @ Ys ) )
= ( map @ ( product_prod @ A @ C ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ C @ ( product_prod @ A @ B )
@ ^ [X4: A,Y5: C] : ( product_Pair @ A @ B @ X4 @ ( F @ Y5 ) ) )
@ ( zip @ A @ C @ Xs @ Ys ) ) ) ).
% zip_map2
thf(fact_1754_zip__map1,axiom,
! [A: $tType,C: $tType,B: $tType,F: C > A,Xs: list @ C,Ys: list @ B] :
( ( zip @ A @ B @ ( map @ C @ A @ F @ Xs ) @ Ys )
= ( map @ ( product_prod @ C @ B ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ C @ B @ ( product_prod @ A @ B )
@ ^ [X4: C] : ( product_Pair @ A @ B @ ( F @ X4 ) ) )
@ ( zip @ C @ B @ Xs @ Ys ) ) ) ).
% zip_map1
thf(fact_1755_new__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( new @ A )
= ( ^ [N4: nat,X4: A] : ( heap_heap @ ( array @ A ) @ ( alloc @ A @ ( replicate @ A @ N4 @ X4 ) ) ) ) ) ) ).
% new_def
thf(fact_1756_Array_Omake__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( make @ A )
= ( ^ [N4: nat,F4: nat > A] : ( heap_heap @ ( array @ A ) @ ( alloc @ A @ ( map @ nat @ A @ F4 @ ( upt @ ( zero_zero @ nat ) @ N4 ) ) ) ) ) ) ) ).
% Array.make_def
thf(fact_1757_effect__newI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [A3: array @ A,H2: heap_ext @ product_unit,N5: nat,X2: A,H: heap_ext @ product_unit] :
( ( ( product_Pair @ ( array @ A ) @ ( heap_ext @ product_unit ) @ A3 @ H2 )
= ( alloc @ A @ ( replicate @ A @ N5 @ X2 ) @ H ) )
=> ( heap_effect @ ( array @ A ) @ ( new @ A @ N5 @ X2 ) @ H @ H2 @ A3 ) ) ) ).
% effect_newI
thf(fact_1758_effect__makeI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [A3: array @ A,H2: heap_ext @ product_unit,F: nat > A,N5: nat,H: heap_ext @ product_unit] :
( ( ( product_Pair @ ( array @ A ) @ ( heap_ext @ product_unit ) @ A3 @ H2 )
= ( alloc @ A @ ( map @ nat @ A @ F @ ( upt @ ( zero_zero @ nat ) @ N5 ) ) @ H ) )
=> ( heap_effect @ ( array @ A ) @ ( make @ A @ N5 @ F ) @ H @ H2 @ A3 ) ) ) ).
% effect_makeI
thf(fact_1759_success__newI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [N5: nat,X2: A,H: heap_ext @ product_unit] : ( heap_success @ ( array @ A ) @ ( new @ A @ N5 @ X2 ) @ H ) ) ).
% success_newI
thf(fact_1760_array__make,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( new @ A )
= ( ^ [N4: nat,X4: A] :
( make @ A @ N4
@ ^ [Uu: nat] : X4 ) ) ) ) ).
% array_make
thf(fact_1761_Rats__def,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( field_char_0_Rats @ A )
= ( image2 @ rat @ A @ ( field_char_0_of_rat @ A ) @ ( top_top @ ( set @ rat ) ) ) ) ) ).
% Rats_def
thf(fact_1762_listrel__iff__zip,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B,R: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys ) @ ( listrel @ A @ B @ R ) )
= ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
& ! [X4: product_prod @ A @ B] :
( ( member @ ( product_prod @ A @ B ) @ X4 @ ( set3 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) )
=> ( product_case_prod @ A @ B @ $o
@ ^ [Y5: A,Z5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y5 @ Z5 ) @ R )
@ X4 ) ) ) ) ).
% listrel_iff_zip
thf(fact_1763_set__list__bind,axiom,
! [A: $tType,B: $tType,Xs: list @ B,F: B > ( list @ A )] :
( ( set3 @ A @ ( bind @ B @ A @ Xs @ F ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X4: B] : ( set3 @ A @ ( F @ X4 ) )
@ ( set3 @ B @ Xs ) ) ) ) ).
% set_list_bind
thf(fact_1764_nth__image,axiom,
! [A: $tType,L: nat,Xs: list @ A] :
( ( ord_less_eq @ nat @ L @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( image2 @ nat @ A @ ( nth3 @ A @ Xs ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ L ) )
= ( set3 @ A @ ( take @ A @ L @ Xs ) ) ) ) ).
% nth_image
thf(fact_1765_INF__set__fold,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: B > A,Xs: list @ B] :
( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ ( set3 @ B @ Xs ) ) )
= ( fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( inf_inf @ A ) @ F ) @ Xs @ ( top_top @ A ) ) ) ) ).
% INF_set_fold
thf(fact_1766_SUP__set__fold,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: B > A,Xs: list @ B] :
( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ ( set3 @ B @ Xs ) ) )
= ( fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( sup_sup @ A ) @ F ) @ Xs @ ( bot_bot @ A ) ) ) ) ).
% SUP_set_fold
thf(fact_1767_set__zip,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B] :
( ( set3 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) )
= ( collect @ ( product_prod @ A @ B )
@ ^ [Uu: product_prod @ A @ B] :
? [I3: nat] :
( ( Uu
= ( product_Pair @ A @ B @ ( nth3 @ A @ Xs @ I3 ) @ ( nth3 @ B @ Ys @ I3 ) ) )
& ( ord_less @ nat @ I3 @ ( ord_min @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) ) ) ) ) ) ).
% set_zip
thf(fact_1768_map__fst__zip,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) )
= Xs ) ) ).
% map_fst_zip
thf(fact_1769_map__snd__zip,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) )
= Ys ) ) ).
% map_snd_zip
thf(fact_1770_zip__replicate,axiom,
! [A: $tType,B: $tType,I: nat,X2: A,J5: nat,Y3: B] :
( ( zip @ A @ B @ ( replicate @ A @ I @ X2 ) @ ( replicate @ B @ J5 @ Y3 ) )
= ( replicate @ ( product_prod @ A @ B ) @ ( ord_min @ nat @ I @ J5 ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) ) ) ).
% zip_replicate
thf(fact_1771_map__fst__zip__take,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B] :
( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) )
= ( take @ A @ ( ord_min @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) ) @ Xs ) ) ).
% map_fst_zip_take
thf(fact_1772_map__snd__zip__take,axiom,
! [B: $tType,A: $tType,Xs: list @ B,Ys: list @ A] :
( ( map @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( zip @ B @ A @ Xs @ Ys ) )
= ( take @ A @ ( ord_min @ nat @ ( size_size @ ( list @ B ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys ) ) @ Ys ) ) ).
% map_snd_zip_take
thf(fact_1773_zip__replicate1,axiom,
! [A: $tType,B: $tType,N5: nat,X2: A,Ys: list @ B] :
( ( zip @ A @ B @ ( replicate @ A @ N5 @ X2 ) @ Ys )
= ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 ) @ ( take @ B @ N5 @ Ys ) ) ) ).
% zip_replicate1
thf(fact_1774_zip__replicate2,axiom,
! [B: $tType,A: $tType,Xs: list @ A,N5: nat,Y3: B] :
( ( zip @ A @ B @ Xs @ ( replicate @ B @ N5 @ Y3 ) )
= ( map @ A @ ( product_prod @ A @ B )
@ ^ [X4: A] : ( product_Pair @ A @ B @ X4 @ Y3 )
@ ( take @ A @ N5 @ Xs ) ) ) ).
% zip_replicate2
thf(fact_1775_pair__list__eqI,axiom,
! [B: $tType,A: $tType,Xs: list @ ( product_prod @ A @ B ),Ys: list @ ( product_prod @ A @ B )] :
( ( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs )
= ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Ys ) )
=> ( ( ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Xs )
= ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Ys ) )
=> ( Xs = Ys ) ) ) ).
% pair_list_eqI
thf(fact_1776_fold__commute__apply,axiom,
! [A: $tType,C: $tType,B: $tType,Xs: list @ A,H: B > C,G: A > B > B,F: A > C > C,S2: B] :
( ! [X: A] :
( ( member @ A @ X @ ( set3 @ A @ Xs ) )
=> ( ( comp @ B @ C @ B @ H @ ( G @ X ) )
= ( comp @ C @ C @ B @ ( F @ X ) @ H ) ) )
=> ( ( H @ ( fold @ A @ B @ G @ Xs @ S2 ) )
= ( fold @ A @ C @ F @ Xs @ ( H @ S2 ) ) ) ) ).
% fold_commute_apply
thf(fact_1777_fold__commute,axiom,
! [A: $tType,C: $tType,B: $tType,Xs: list @ A,H: B > C,G: A > B > B,F: A > C > C] :
( ! [X: A] :
( ( member @ A @ X @ ( set3 @ A @ Xs ) )
=> ( ( comp @ B @ C @ B @ H @ ( G @ X ) )
= ( comp @ C @ C @ B @ ( F @ X ) @ H ) ) )
=> ( ( comp @ B @ C @ B @ H @ ( fold @ A @ B @ G @ Xs ) )
= ( comp @ C @ C @ B @ ( fold @ A @ C @ F @ Xs ) @ H ) ) ) ).
% fold_commute
thf(fact_1778_zip__same__conv__map,axiom,
! [A: $tType,Xs: list @ A] :
( ( zip @ A @ A @ Xs @ Xs )
= ( map @ A @ ( product_prod @ A @ A )
@ ^ [X4: A] : ( product_Pair @ A @ A @ X4 @ X4 )
@ Xs ) ) ).
% zip_same_conv_map
thf(fact_1779_fold__map,axiom,
! [B: $tType,A: $tType,C: $tType,G: B > A > A,F: C > B,Xs: list @ C] :
( ( fold @ B @ A @ G @ ( map @ C @ B @ F @ Xs ) )
= ( fold @ C @ A @ ( comp @ B @ ( A > A ) @ C @ G @ F ) @ Xs ) ) ).
% fold_map
thf(fact_1780_zip__map__fst__snd,axiom,
! [B: $tType,A: $tType,Zs: list @ ( product_prod @ A @ B )] :
( ( zip @ A @ B @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Zs ) @ ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Zs ) )
= Zs ) ).
% zip_map_fst_snd
thf(fact_1781_zip__left__commute,axiom,
! [B: $tType,A: $tType,C: $tType,Xs: list @ A,Ys: list @ B,Zs: list @ C] :
( ( zip @ A @ ( product_prod @ B @ C ) @ Xs @ ( zip @ B @ C @ Ys @ Zs ) )
= ( 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 ) )
@ ^ [Y5: B] :
( product_case_prod @ A @ C @ ( product_prod @ A @ ( product_prod @ B @ C ) )
@ ^ [X4: A,Z5: C] : ( product_Pair @ A @ ( product_prod @ B @ C ) @ X4 @ ( product_Pair @ B @ C @ Y5 @ Z5 ) ) ) )
@ ( zip @ B @ ( product_prod @ A @ C ) @ Ys @ ( zip @ A @ C @ Xs @ Zs ) ) ) ) ).
% zip_left_commute
thf(fact_1782_zip__assoc,axiom,
! [B: $tType,A: $tType,C: $tType,Xs: list @ A,Ys: list @ B,Zs: list @ C] :
( ( zip @ A @ ( product_prod @ B @ C ) @ Xs @ ( zip @ B @ C @ Ys @ Zs ) )
= ( 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 ) ) )
@ ^ [X4: A,Y5: B,Z5: C] : ( product_Pair @ A @ ( product_prod @ B @ C ) @ X4 @ ( product_Pair @ B @ C @ Y5 @ Z5 ) ) ) )
@ ( zip @ ( product_prod @ A @ B ) @ C @ ( zip @ A @ B @ Xs @ Ys ) @ Zs ) ) ) ).
% zip_assoc
thf(fact_1783_zip__commute,axiom,
! [B: $tType,A: $tType] :
( ( zip @ A @ B )
= ( ^ [Xs2: list @ A,Ys2: list @ B] :
( map @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ B @ A @ ( product_prod @ A @ B )
@ ^ [X4: B,Y5: A] : ( product_Pair @ A @ B @ Y5 @ X4 ) )
@ ( zip @ B @ A @ Ys2 @ Xs2 ) ) ) ) ).
% zip_commute
thf(fact_1784_hom__Min__commute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [H: A > A,N: set @ A] :
( ! [X: A,Y: A] :
( ( H @ ( ord_min @ A @ X @ Y ) )
= ( ord_min @ A @ ( H @ X ) @ ( H @ Y ) ) )
=> ( ( finite_finite2 @ A @ N )
=> ( ( N
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H @ ( lattic643756798350308766er_Min @ A @ N ) )
= ( lattic643756798350308766er_Min @ A @ ( image2 @ A @ A @ H @ N ) ) ) ) ) ) ) ).
% hom_Min_commute
thf(fact_1785_zip__eq__conv,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,Zs: list @ ( product_prod @ A @ B )] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( ( zip @ A @ B @ Xs @ Ys )
= Zs )
= ( ( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Zs )
= Xs )
& ( ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Zs )
= Ys ) ) ) ) ).
% zip_eq_conv
thf(fact_1786_Id__on__set,axiom,
! [A: $tType,Xs: list @ A] :
( ( id_on @ A @ ( set3 @ A @ Xs ) )
= ( set3 @ ( product_prod @ A @ A )
@ ( map @ A @ ( product_prod @ A @ A )
@ ^ [X4: A] : ( product_Pair @ A @ A @ X4 @ X4 )
@ Xs ) ) ) ).
% Id_on_set
thf(fact_1787_listrel__iff__nth,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,R: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys ) @ ( listrel @ A @ B @ R ) )
= ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
& ! [N4: nat] :
( ( ord_less @ nat @ N4 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ ( nth3 @ A @ Xs @ N4 ) @ ( nth3 @ B @ Ys @ N4 ) ) @ R ) ) ) ) ).
% listrel_iff_nth
thf(fact_1788_lexord__take__index__conv,axiom,
! [A: $tType,X2: list @ A,Y3: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X2 @ Y3 ) @ ( lexord @ A @ R ) )
= ( ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ X2 ) @ ( size_size @ ( list @ A ) @ Y3 ) )
& ( ( take @ A @ ( size_size @ ( list @ A ) @ X2 ) @ Y3 )
= X2 ) )
| ? [I3: nat] :
( ( ord_less @ nat @ I3 @ ( ord_min @ nat @ ( size_size @ ( list @ A ) @ X2 ) @ ( size_size @ ( list @ A ) @ Y3 ) ) )
& ( ( take @ A @ I3 @ X2 )
= ( take @ A @ I3 @ Y3 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth3 @ A @ X2 @ I3 ) @ ( nth3 @ A @ Y3 @ I3 ) ) @ R ) ) ) ) ).
% lexord_take_index_conv
thf(fact_1789_lex__take__index,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lex @ A @ R ) )
=> ~ ! [I6: nat] :
( ( ord_less @ nat @ I6 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ I6 @ ( size_size @ ( list @ A ) @ Ys ) )
=> ( ( ( take @ A @ I6 @ Xs )
= ( take @ A @ I6 @ Ys ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth3 @ A @ Xs @ I6 ) @ ( nth3 @ A @ Ys @ I6 ) ) @ R ) ) ) ) ) ).
% lex_take_index
thf(fact_1790_product__code,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B] :
( ( product_product @ A @ B @ ( set3 @ A @ Xs ) @ ( set3 @ B @ Ys ) )
= ( set3 @ ( product_prod @ A @ B )
@ ( concat @ ( product_prod @ A @ B )
@ ( map @ A @ ( list @ ( product_prod @ A @ B ) )
@ ^ [X4: A] : ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 ) @ Ys )
@ Xs ) ) ) ) ).
% product_code
thf(fact_1791_subseqs__powset,axiom,
! [A: $tType,Xs: list @ A] :
( ( image2 @ ( list @ A ) @ ( set @ A ) @ ( set3 @ A ) @ ( set3 @ ( list @ A ) @ ( subseqs @ A @ Xs ) ) )
= ( pow @ A @ ( set3 @ A @ Xs ) ) ) ).
% subseqs_powset
thf(fact_1792_list_Oin__rel,axiom,
! [B: $tType,A: $tType] :
( ( list_all2 @ A @ B )
= ( ^ [R6: A > B > $o,A7: list @ A,B5: list @ B] :
? [Z5: list @ ( product_prod @ A @ B )] :
( ( member @ ( list @ ( product_prod @ A @ B ) ) @ Z5
@ ( collect @ ( list @ ( product_prod @ A @ B ) )
@ ^ [X4: list @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set3 @ ( product_prod @ A @ B ) @ X4 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) ) ) )
& ( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Z5 )
= A7 )
& ( ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Z5 )
= B5 ) ) ) ) ).
% list.in_rel
thf(fact_1793_list_Orel__compp__Grp,axiom,
! [B: $tType,A: $tType] :
( ( list_all2 @ A @ B )
= ( ^ [R6: A > B > $o] :
( relcompp @ ( list @ A ) @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ B )
@ ( conversep @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ A )
@ ( bNF_Grp @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ A )
@ ( collect @ ( list @ ( product_prod @ A @ B ) )
@ ^ [X4: list @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set3 @ ( product_prod @ A @ B ) @ X4 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) ) )
@ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) ) ) )
@ ( bNF_Grp @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ B )
@ ( collect @ ( list @ ( product_prod @ A @ B ) )
@ ^ [X4: list @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set3 @ ( product_prod @ A @ B ) @ X4 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) ) )
@ ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) ) ) ) ) ) ).
% list.rel_compp_Grp
thf(fact_1794_Pow__set_I2_J,axiom,
! [B: $tType,X2: B,Xs: list @ B] :
( ( pow @ B @ ( set3 @ B @ ( cons @ B @ X2 @ Xs ) ) )
= ( sup_sup @ ( set @ ( set @ B ) ) @ ( pow @ B @ ( set3 @ B @ Xs ) ) @ ( image2 @ ( set @ B ) @ ( set @ B ) @ ( insert @ B @ X2 ) @ ( pow @ B @ ( set3 @ B @ Xs ) ) ) ) ) ).
% Pow_set(2)
thf(fact_1795_zip__Cons__Cons,axiom,
! [A: $tType,B: $tType,X2: A,Xs: list @ A,Y3: B,Ys: list @ B] :
( ( zip @ A @ B @ ( cons @ A @ X2 @ Xs ) @ ( cons @ B @ Y3 @ Ys ) )
= ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) ).
% zip_Cons_Cons
thf(fact_1796_lexord__cons__cons,axiom,
! [A: $tType,A3: A,X2: list @ A,B2: A,Y3: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ A3 @ X2 ) @ ( cons @ A @ B2 @ Y3 ) ) @ ( 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 ) @ X2 @ Y3 ) @ ( lexord @ A @ R ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_1797_Cons__in__lex,axiom,
! [A: $tType,X2: A,Xs: list @ A,Y3: A,Ys: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs ) @ ( cons @ A @ Y3 @ Ys ) ) @ ( lex @ A @ R ) )
= ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R )
& ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) ) )
| ( ( X2 = Y3 )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lex @ A @ R ) ) ) ) ) ).
% Cons_in_lex
thf(fact_1798_set__concat,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( set3 @ A @ ( concat @ A @ Xs ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( list @ A ) @ ( set @ A ) @ ( set3 @ A ) @ ( set3 @ ( list @ A ) @ Xs ) ) ) ) ).
% set_concat
thf(fact_1799_inj__split__Cons,axiom,
! [A: $tType,X6: set @ ( product_prod @ ( list @ A ) @ A )] :
( inj_on @ ( product_prod @ ( list @ A ) @ A ) @ ( list @ A )
@ ( product_case_prod @ ( list @ A ) @ A @ ( list @ A )
@ ^ [Xs2: list @ A,N4: A] : ( cons @ A @ N4 @ Xs2 ) )
@ X6 ) ).
% inj_split_Cons
thf(fact_1800_zip__eq__ConsE,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,Xy2: product_prod @ A @ B,Xys: list @ ( product_prod @ A @ B )] :
( ( ( zip @ A @ B @ Xs @ Ys )
= ( cons @ ( product_prod @ A @ B ) @ Xy2 @ Xys ) )
=> ~ ! [X: A,Xs3: list @ A] :
( ( Xs
= ( cons @ A @ X @ Xs3 ) )
=> ! [Y: B,Ys3: list @ B] :
( ( Ys
= ( cons @ B @ Y @ Ys3 ) )
=> ( ( Xy2
= ( product_Pair @ A @ B @ X @ Y ) )
=> ( Xys
!= ( zip @ A @ B @ Xs3 @ Ys3 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_1801_list_Orel__compp,axiom,
! [A: $tType,C: $tType,B: $tType,R3: A > B > $o,S5: B > C > $o] :
( ( list_all2 @ A @ C @ ( relcompp @ A @ B @ C @ R3 @ S5 ) )
= ( relcompp @ ( list @ A ) @ ( list @ B ) @ ( list @ C ) @ ( list_all2 @ A @ B @ R3 ) @ ( list_all2 @ B @ C @ S5 ) ) ) ).
% list.rel_compp
thf(fact_1802_concat__injective,axiom,
! [A: $tType,Xs: list @ ( list @ A ),Ys: list @ ( list @ A )] :
( ( ( concat @ A @ Xs )
= ( concat @ A @ Ys ) )
=> ( ( ( size_size @ ( list @ ( list @ A ) ) @ Xs )
= ( size_size @ ( list @ ( list @ A ) ) @ Ys ) )
=> ( ! [X: product_prod @ ( list @ A ) @ ( list @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X @ ( set3 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( zip @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) ) )
=> ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Y5: list @ A,Z5: list @ A] :
( ( size_size @ ( list @ A ) @ Y5 )
= ( size_size @ ( list @ A ) @ Z5 ) )
@ X ) )
=> ( Xs = Ys ) ) ) ) ).
% concat_injective
thf(fact_1803_concat__eq__concat__iff,axiom,
! [A: $tType,Xs: list @ ( list @ A ),Ys: list @ ( list @ A )] :
( ! [X: product_prod @ ( list @ A ) @ ( list @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X @ ( set3 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( zip @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) ) )
=> ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Y5: list @ A,Z5: list @ A] :
( ( size_size @ ( list @ A ) @ Y5 )
= ( size_size @ ( list @ A ) @ Z5 ) )
@ X ) )
=> ( ( ( size_size @ ( list @ ( list @ A ) ) @ Xs )
= ( size_size @ ( list @ ( list @ A ) ) @ Ys ) )
=> ( ( ( concat @ A @ Xs )
= ( concat @ A @ Ys ) )
= ( Xs = Ys ) ) ) ) ).
% concat_eq_concat_iff
thf(fact_1804_length__Cons,axiom,
! [A: $tType,X2: A,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( cons @ A @ X2 @ Xs ) )
= ( suc @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% length_Cons
thf(fact_1805_fold__Cons,axiom,
! [B: $tType,A: $tType,F: A > B > B,X2: A,Xs: list @ A] :
( ( fold @ A @ B @ F @ ( cons @ A @ X2 @ Xs ) )
= ( comp @ B @ B @ B @ ( fold @ A @ B @ F @ Xs ) @ ( F @ X2 ) ) ) ).
% fold_Cons
thf(fact_1806_lexord__linear,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),X2: list @ A,Y3: list @ A] :
( ! [A5: A,B4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B4 ) @ R )
| ( A5 = B4 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ A5 ) @ R ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X2 @ Y3 ) @ ( lexord @ A @ R ) )
| ( X2 = Y3 )
| ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Y3 @ X2 ) @ ( lexord @ A @ R ) ) ) ) ).
% lexord_linear
thf(fact_1807_lexord__irreflexive,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),Xs: list @ A] :
( ! [X: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ R )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Xs ) @ ( lexord @ A @ R ) ) ) ).
% lexord_irreflexive
thf(fact_1808_listrel_OCons,axiom,
! [B: $tType,A: $tType,X2: A,Y3: B,R: set @ ( product_prod @ A @ B ),Xs: list @ A,Ys: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ R )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys ) @ ( listrel @ A @ B @ R ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ X2 @ Xs ) @ ( cons @ B @ Y3 @ Ys ) ) @ ( listrel @ A @ B @ R ) ) ) ) ).
% listrel.Cons
thf(fact_1809_listrel__Cons1,axiom,
! [B: $tType,A: $tType,Y3: A,Ys: 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 @ Y3 @ Ys ) @ Xs ) @ ( listrel @ A @ B @ R ) )
=> ~ ! [Y: B,Ys4: list @ B] :
( ( Xs
= ( cons @ B @ Y @ Ys4 ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y3 @ Y ) @ R )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Ys @ Ys4 ) @ ( listrel @ A @ B @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_1810_listrel__Cons2,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Y3: B,Ys: list @ B,R: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ ( cons @ B @ Y3 @ Ys ) ) @ ( listrel @ A @ B @ R ) )
=> ~ ! [X: A,Xs4: list @ A] :
( ( Xs
= ( cons @ A @ X @ Xs4 ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y3 ) @ R )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs4 @ Ys ) @ ( listrel @ A @ B @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_1811_lexord__partial__trans,axiom,
! [A: $tType,Xs: list @ A,R: set @ ( product_prod @ A @ A ),Ys: list @ A,Zs: list @ A] :
( ! [X: A,Y: A,Z4: A] :
( ( member @ A @ X @ ( set3 @ A @ Xs ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z4 ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z4 ) @ R ) ) ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lexord @ A @ R ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ 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_1812_list__all2I,axiom,
! [A: $tType,B: $tType,A3: list @ A,B2: list @ B,P2: A > B > $o] :
( ! [X: product_prod @ A @ B] :
( ( member @ ( product_prod @ A @ B ) @ X @ ( set3 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ A3 @ B2 ) ) )
=> ( product_case_prod @ A @ B @ $o @ P2 @ X ) )
=> ( ( ( size_size @ ( list @ A ) @ A3 )
= ( size_size @ ( list @ B ) @ B2 ) )
=> ( list_all2 @ A @ B @ P2 @ A3 @ B2 ) ) ) ).
% list_all2I
thf(fact_1813_product__concat__map,axiom,
! [B: $tType,A: $tType] :
( ( product @ A @ B )
= ( ^ [Xs2: list @ A,Ys2: list @ B] :
( concat @ ( product_prod @ A @ B )
@ ( map @ A @ ( list @ ( product_prod @ A @ B ) )
@ ^ [X4: A] : ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 ) @ Ys2 )
@ Xs2 ) ) ) ) ).
% product_concat_map
thf(fact_1814_lists__length__Suc__eq,axiom,
! [A: $tType,A6: set @ A,N5: nat] :
( ( collect @ ( list @ A )
@ ^ [Xs2: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set3 @ A @ Xs2 ) @ A6 )
& ( ( size_size @ ( list @ A ) @ Xs2 )
= ( suc @ N5 ) ) ) )
= ( image2 @ ( product_prod @ ( list @ A ) @ A ) @ ( list @ A )
@ ( product_case_prod @ ( list @ A ) @ A @ ( list @ A )
@ ^ [Xs2: list @ A,N4: A] : ( cons @ A @ N4 @ Xs2 ) )
@ ( product_Sigma @ ( list @ A ) @ A
@ ( collect @ ( list @ A )
@ ^ [Xs2: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set3 @ A @ Xs2 ) @ A6 )
& ( ( size_size @ ( list @ A ) @ Xs2 )
= N5 ) ) )
@ ^ [Uu: list @ A] : A6 ) ) ) ).
% lists_length_Suc_eq
thf(fact_1815_list__all2__iff,axiom,
! [B: $tType,A: $tType] :
( ( list_all2 @ A @ B )
= ( ^ [P4: A > B > $o,Xs2: list @ A,Ys2: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys2 ) )
& ! [X4: product_prod @ A @ B] :
( ( member @ ( product_prod @ A @ B ) @ X4 @ ( set3 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs2 @ Ys2 ) ) )
=> ( product_case_prod @ A @ B @ $o @ P4 @ X4 ) ) ) ) ) ).
% list_all2_iff
thf(fact_1816_subset__subseqs,axiom,
! [A: $tType,X6: set @ A,Xs: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ X6 @ ( set3 @ A @ Xs ) )
=> ( member @ ( set @ A ) @ X6 @ ( image2 @ ( list @ A ) @ ( set @ A ) @ ( set3 @ A ) @ ( set3 @ ( list @ A ) @ ( subseqs @ A @ Xs ) ) ) ) ) ).
% subset_subseqs
thf(fact_1817_List_Olexordp__def,axiom,
! [A: $tType] :
( ( lexordp @ A )
= ( ^ [R5: A > A > $o,Xs2: list @ A,Ys2: list @ A] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) @ ( lexord @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R5 ) ) ) ) ) ) ).
% List.lexordp_def
thf(fact_1818_length__remdups__concat,axiom,
! [A: $tType,Xss: list @ ( list @ A )] :
( ( size_size @ ( list @ A ) @ ( remdups @ A @ ( concat @ A @ Xss ) ) )
= ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( list @ A ) @ ( set @ A ) @ ( set3 @ A ) @ ( set3 @ ( list @ A ) @ Xss ) ) ) ) ) ).
% length_remdups_concat
thf(fact_1819_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 @ ( set3 @ ( product_prod @ A @ C ) @ Xys ) @ ( set3 @ ( product_prod @ C @ B ) @ Yzs ) )
= ( set3 @ ( 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_1820_shift__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Greatest_shift @ A @ B )
= ( ^ [Lab: ( list @ A ) > B,K3: A,Kl: list @ A] : ( Lab @ ( cons @ A @ K3 @ Kl ) ) ) ) ).
% shift_def
thf(fact_1821_Cons__lenlex__iff,axiom,
! [A: $tType,M4: A,Ms: list @ A,N5: 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 @ M4 @ Ms ) @ ( cons @ A @ N5 @ 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 @ M4 @ N5 ) @ R ) )
| ( ( M4 = N5 )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ms @ Ns ) @ ( lenlex @ A @ R ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_1822_lexn_Osimps_I2_J,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),N5: nat] :
( ( lexn @ A @ R @ ( suc @ N5 ) )
= ( inf_inf @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( image2 @ ( product_prod @ ( product_prod @ A @ ( list @ A ) ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_map_prod @ ( product_prod @ A @ ( list @ A ) ) @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( list @ A ) @ ( product_case_prod @ A @ ( list @ A ) @ ( list @ A ) @ ( cons @ A ) ) @ ( product_case_prod @ A @ ( list @ A ) @ ( list @ A ) @ ( cons @ A ) ) ) @ ( lex_prod @ A @ ( list @ A ) @ R @ ( lexn @ A @ R @ N5 ) ) )
@ ( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs2: list @ A,Ys2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( suc @ N5 ) )
& ( ( size_size @ ( list @ A ) @ Ys2 )
= ( suc @ N5 ) ) ) ) ) ) ) ).
% lexn.simps(2)
thf(fact_1823_lenlex__conv,axiom,
! [A: $tType] :
( ( lenlex @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs2: list @ A,Ys2: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( size_size @ ( list @ A ) @ Ys2 ) )
| ( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ A ) @ Ys2 ) )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) @ ( lex @ A @ R5 ) ) ) ) ) ) ) ) ).
% lenlex_conv
thf(fact_1824_lenlex__irreflexive,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),Xs: list @ A] :
( ! [X: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ R )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Xs ) @ ( lenlex @ A @ R ) ) ) ).
% lenlex_irreflexive
thf(fact_1825_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 ) ) )
| ? [X4: A,Y5: B,Xs2: list @ A,Ys2: list @ B] :
( ( A1
= ( cons @ A @ X4 @ Xs2 ) )
& ( A22
= ( cons @ B @ Y5 @ Ys2 ) )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ R )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs2 @ Ys2 ) @ ( listrel @ A @ B @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_1826_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 ) ) )
=> ~ ! [X: A,Y: B,Xs4: list @ A] :
( ( A1
= ( cons @ A @ X @ Xs4 ) )
=> ! [Ys4: list @ B] :
( ( A22
= ( cons @ B @ Y @ Ys4 ) )
=> ( ( 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 ) @ Xs4 @ Ys4 ) @ ( listrel @ A @ B @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_1827_lex__def,axiom,
! [A: $tType] :
( ( lex @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] : ( complete_Sup_Sup @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( image2 @ nat @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( lexn @ A @ R5 ) @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% lex_def
thf(fact_1828_set__Cons__sing__Nil,axiom,
! [A: $tType,A6: set @ A] :
( ( set_Cons @ A @ A6 @ ( insert @ ( list @ A ) @ ( nil @ A ) @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) )
= ( image2 @ A @ ( list @ A )
@ ^ [X4: A] : ( cons @ A @ X4 @ ( nil @ A ) )
@ A6 ) ) ).
% set_Cons_sing_Nil
thf(fact_1829_zip__Cons1,axiom,
! [A: $tType,B: $tType,X2: A,Xs: list @ A,Ys: list @ B] :
( ( zip @ A @ B @ ( cons @ A @ X2 @ Xs ) @ Ys )
= ( case_list @ ( list @ ( product_prod @ A @ B ) ) @ B @ ( nil @ ( product_prod @ A @ B ) )
@ ^ [Y5: B,Ys2: list @ B] : ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y5 ) @ ( zip @ A @ B @ Xs @ Ys2 ) )
@ Ys ) ) ).
% zip_Cons1
thf(fact_1830_zip__Cons,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Y3: B,Ys: list @ B] :
( ( zip @ A @ B @ Xs @ ( cons @ B @ Y3 @ Ys ) )
= ( case_list @ ( list @ ( product_prod @ A @ B ) ) @ A @ ( nil @ ( product_prod @ A @ B ) )
@ ^ [Z5: A,Zs2: list @ A] : ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Z5 @ Y3 ) @ ( zip @ A @ B @ Zs2 @ Ys ) )
@ Xs ) ) ).
% zip_Cons
thf(fact_1831_f__arg__min__list__f,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [Xs: list @ A,F: A > B] :
( ( Xs
!= ( nil @ A ) )
=> ( ( F @ ( arg_min_list @ A @ B @ F @ Xs ) )
= ( lattic643756798350308766er_Min @ B @ ( image2 @ A @ B @ F @ ( set3 @ A @ Xs ) ) ) ) ) ) ).
% f_arg_min_list_f
thf(fact_1832_lexn__conv,axiom,
! [A: $tType] :
( ( lexn @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A ),N4: nat] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs2: list @ A,Ys2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= N4 )
& ( ( size_size @ ( list @ A ) @ Ys2 )
= N4 )
& ? [Xys2: list @ A,X4: A,Y5: A,Xs5: list @ A,Ys5: list @ A] :
( ( Xs2
= ( append @ A @ Xys2 @ ( cons @ A @ X4 @ Xs5 ) ) )
& ( Ys2
= ( append @ A @ Xys2 @ ( cons @ A @ Y5 @ Ys5 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R5 ) ) ) ) ) ) ) ).
% lexn_conv
thf(fact_1833_lex__conv,axiom,
! [A: $tType] :
( ( lex @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs2: list @ A,Ys2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ A ) @ Ys2 ) )
& ? [Xys2: list @ A,X4: A,Y5: A,Xs5: list @ A,Ys5: list @ A] :
( ( Xs2
= ( append @ A @ Xys2 @ ( cons @ A @ X4 @ Xs5 ) ) )
& ( Ys2
= ( append @ A @ Xys2 @ ( cons @ A @ Y5 @ Ys5 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R5 ) ) ) ) ) ) ) ).
% lex_conv
thf(fact_1834_listrel__def,axiom,
! [B: $tType,A: $tType] :
( ( listrel @ A @ B )
= ( ^ [R5: set @ ( product_prod @ A @ B )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ B ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ B ) @ $o
@ ( listrelp @ A @ B
@ ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ R5 ) ) ) ) ) ) ).
% listrel_def
thf(fact_1835_fold__append,axiom,
! [A: $tType,B: $tType,F: B > A > A,Xs: list @ B,Ys: list @ B] :
( ( fold @ B @ A @ F @ ( append @ B @ Xs @ Ys ) )
= ( comp @ A @ A @ A @ ( fold @ B @ A @ F @ Ys ) @ ( fold @ B @ A @ F @ Xs ) ) ) ).
% fold_append
thf(fact_1836_lexord__append__leftD,axiom,
! [A: $tType,X2: list @ A,U2: 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 @ X2 @ U2 ) @ ( append @ A @ X2 @ 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 ) @ U2 @ V ) @ ( lexord @ A @ R ) ) ) ) ).
% lexord_append_leftD
thf(fact_1837_lex__append__leftD,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ! [X: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ R )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) @ ( append @ A @ Xs @ Zs ) ) @ ( lex @ A @ R ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs ) @ ( lex @ A @ R ) ) ) ) ).
% lex_append_leftD
thf(fact_1838_lex__append__left__iff,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ! [X: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ R )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) @ ( append @ A @ Xs @ Zs ) ) @ ( lex @ A @ R ) )
= ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs ) @ ( lex @ A @ R ) ) ) ) ).
% lex_append_left_iff
thf(fact_1839_length__append__singleton,axiom,
! [A: $tType,Xs: list @ A,X2: A] :
( ( size_size @ ( list @ A ) @ ( append @ A @ Xs @ ( cons @ A @ X2 @ ( nil @ A ) ) ) )
= ( suc @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% length_append_singleton
thf(fact_1840_lexord__append__left__rightI,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),U2: list @ A,X2: list @ A,Y3: 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 @ U2 @ ( cons @ A @ A3 @ X2 ) ) @ ( append @ A @ U2 @ ( cons @ A @ B2 @ Y3 ) ) ) @ ( lexord @ A @ R ) ) ) ).
% lexord_append_left_rightI
thf(fact_1841_lexord__same__pref__iff,axiom,
! [A: $tType,Xs: list @ A,Ys: 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 @ Ys ) @ ( append @ A @ Xs @ Zs ) ) @ ( lexord @ A @ R ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ ( set3 @ A @ Xs ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) @ R ) )
| ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs ) @ ( lexord @ A @ R ) ) ) ) ).
% lexord_same_pref_iff
thf(fact_1842_product_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,X2: A,Xs: list @ A,Ys: list @ B] :
( ( product @ A @ B @ ( cons @ A @ X2 @ Xs ) @ Ys )
= ( append @ ( product_prod @ A @ B ) @ ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 ) @ Ys ) @ ( product @ A @ B @ Xs @ Ys ) ) ) ).
% product.simps(2)
thf(fact_1843_lexord__def,axiom,
! [A: $tType] :
( ( lexord @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [X4: list @ A,Y5: list @ A] :
? [A7: A,V3: list @ A] :
( ( Y5
= ( append @ A @ X4 @ ( cons @ A @ A7 @ V3 ) ) )
| ? [U: list @ A,B5: A,C6: A,W: list @ A,Z5: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ C6 ) @ R5 )
& ( X4
= ( append @ A @ U @ ( cons @ A @ B5 @ W ) ) )
& ( Y5
= ( append @ A @ U @ ( cons @ A @ C6 @ Z5 ) ) ) ) ) ) ) ) ) ).
% lexord_def
thf(fact_1844_listrelp__listrel__eq,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B )] :
( ( listrelp @ A @ B
@ ^ [X4: A,Y5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y5 ) @ R ) )
= ( ^ [X4: list @ A,Y5: list @ B] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ X4 @ Y5 ) @ ( listrel @ A @ B @ R ) ) ) ) ).
% listrelp_listrel_eq
thf(fact_1845_Succ__def,axiom,
! [A: $tType] :
( ( bNF_Greatest_Succ @ A )
= ( ^ [Kl2: set @ ( list @ A ),Kl: list @ A] :
( collect @ A
@ ^ [K3: A] : ( member @ ( list @ A ) @ ( append @ A @ Kl @ ( cons @ A @ K3 @ ( nil @ A ) ) ) @ Kl2 ) ) ) ) ).
% Succ_def
thf(fact_1846_listrel1__def,axiom,
! [A: $tType] :
( ( listrel1 @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs2: list @ A,Ys2: list @ A] :
? [Us: list @ A,Z5: A,Z11: A,Vs: list @ A] :
( ( Xs2
= ( append @ A @ Us @ ( cons @ A @ Z5 @ Vs ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z5 @ Z11 ) @ R5 )
& ( Ys2
= ( append @ A @ Us @ ( cons @ A @ Z11 @ Vs ) ) ) ) ) ) ) ) ).
% listrel1_def
thf(fact_1847_SuccI,axiom,
! [A: $tType,Kl3: list @ A,K: A,Kl4: set @ ( list @ A )] :
( ( member @ ( list @ A ) @ ( append @ A @ Kl3 @ ( cons @ A @ K @ ( nil @ A ) ) ) @ Kl4 )
=> ( member @ A @ K @ ( bNF_Greatest_Succ @ A @ Kl4 @ Kl3 ) ) ) ).
% SuccI
thf(fact_1848_SuccD,axiom,
! [A: $tType,K: A,Kl4: set @ ( list @ A ),Kl3: list @ A] :
( ( member @ A @ K @ ( bNF_Greatest_Succ @ A @ Kl4 @ Kl3 ) )
=> ( member @ ( list @ A ) @ ( append @ A @ Kl3 @ ( cons @ A @ K @ ( nil @ A ) ) ) @ Kl4 ) ) ).
% SuccD
thf(fact_1849_prod_Orel__compp__Grp,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType] :
( ( basic_rel_prod @ A @ C @ B @ D )
= ( ^ [R14: A > C > $o,R25: B > D > $o] :
( relcompp @ ( product_prod @ A @ B ) @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) @ ( product_prod @ C @ D )
@ ( conversep @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) @ ( product_prod @ A @ B )
@ ( bNF_Grp @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) @ ( product_prod @ A @ B )
@ ( collect @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) )
@ ^ [X4: product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ C ) ) @ ( basic_fsts @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ X4 ) @ ( collect @ ( product_prod @ A @ C ) @ ( product_case_prod @ A @ C @ $o @ R14 ) ) )
& ( ord_less_eq @ ( set @ ( product_prod @ B @ D ) ) @ ( basic_snds @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ X4 ) @ ( collect @ ( product_prod @ B @ D ) @ ( product_case_prod @ B @ D @ $o @ R25 ) ) ) ) )
@ ( product_map_prod @ ( product_prod @ A @ C ) @ A @ ( product_prod @ B @ D ) @ B @ ( product_fst @ A @ C ) @ ( product_fst @ B @ D ) ) ) )
@ ( bNF_Grp @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) @ ( product_prod @ C @ D )
@ ( collect @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) )
@ ^ [X4: product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ C ) ) @ ( basic_fsts @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ X4 ) @ ( collect @ ( product_prod @ A @ C ) @ ( product_case_prod @ A @ C @ $o @ R14 ) ) )
& ( ord_less_eq @ ( set @ ( product_prod @ B @ D ) ) @ ( basic_snds @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ X4 ) @ ( collect @ ( product_prod @ B @ D ) @ ( product_case_prod @ B @ D @ $o @ R25 ) ) ) ) )
@ ( product_map_prod @ ( product_prod @ A @ C ) @ C @ ( product_prod @ B @ D ) @ D @ ( product_snd @ A @ C ) @ ( product_snd @ B @ D ) ) ) ) ) ) ).
% prod.rel_compp_Grp
thf(fact_1850_prod_Oin__rel,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType] :
( ( basic_rel_prod @ A @ C @ B @ D )
= ( ^ [R14: A > C > $o,R25: B > D > $o,A7: product_prod @ A @ B,B5: product_prod @ C @ D] :
? [Z5: product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D )] :
( ( member @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) @ Z5
@ ( collect @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) )
@ ^ [X4: product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ C ) ) @ ( basic_fsts @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ X4 ) @ ( collect @ ( product_prod @ A @ C ) @ ( product_case_prod @ A @ C @ $o @ R14 ) ) )
& ( ord_less_eq @ ( set @ ( product_prod @ B @ D ) ) @ ( basic_snds @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ X4 ) @ ( collect @ ( product_prod @ B @ D ) @ ( product_case_prod @ B @ D @ $o @ R25 ) ) ) ) ) )
& ( ( product_map_prod @ ( product_prod @ A @ C ) @ A @ ( product_prod @ B @ D ) @ B @ ( product_fst @ A @ C ) @ ( product_fst @ B @ D ) @ Z5 )
= A7 )
& ( ( product_map_prod @ ( product_prod @ A @ C ) @ C @ ( product_prod @ B @ D ) @ D @ ( product_snd @ A @ C ) @ ( product_snd @ B @ D ) @ Z5 )
= B5 ) ) ) ) ).
% prod.in_rel
thf(fact_1851_in__measures_I2_J,axiom,
! [A: $tType,X2: A,Y3: A,F: A > nat,Fs: list @ ( A > nat )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ ( measures @ A @ ( cons @ ( A > nat ) @ F @ Fs ) ) )
= ( ( ord_less @ nat @ ( F @ X2 ) @ ( F @ Y3 ) )
| ( ( ( F @ X2 )
= ( F @ Y3 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ ( measures @ A @ Fs ) ) ) ) ) ).
% in_measures(2)
thf(fact_1852_rel__prod__inject,axiom,
! [B: $tType,A: $tType,C: $tType,D: $tType,R1: A > B > $o,R22: C > D > $o,A3: A,B2: C,C3: B,D4: D] :
( ( basic_rel_prod @ A @ B @ C @ D @ R1 @ R22 @ ( product_Pair @ A @ C @ A3 @ B2 ) @ ( product_Pair @ B @ D @ C3 @ D4 ) )
= ( ( R1 @ A3 @ C3 )
& ( R22 @ B2 @ D4 ) ) ) ).
% rel_prod_inject
thf(fact_1853_in__measures_I1_J,axiom,
! [A: $tType,X2: A,Y3: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ ( measures @ A @ ( nil @ ( A > nat ) ) ) ) ).
% in_measures(1)
thf(fact_1854_Cons__listrel1__Cons,axiom,
! [A: $tType,X2: A,Xs: list @ A,Y3: A,Ys: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs ) @ ( cons @ A @ Y3 @ Ys ) ) @ ( listrel1 @ A @ R ) )
= ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R )
& ( Xs = Ys ) )
| ( ( X2 = Y3 )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_1855_rel__prod_Ocases,axiom,
! [B: $tType,A: $tType,C: $tType,D: $tType,R1: A > B > $o,R22: C > D > $o,A1: product_prod @ A @ C,A22: product_prod @ B @ D] :
( ( basic_rel_prod @ A @ B @ C @ D @ R1 @ R22 @ A1 @ A22 )
=> ~ ! [A5: A,B4: B,C2: C] :
( ( A1
= ( product_Pair @ A @ C @ A5 @ C2 ) )
=> ! [D2: D] :
( ( A22
= ( product_Pair @ B @ D @ B4 @ D2 ) )
=> ( ( R1 @ A5 @ B4 )
=> ~ ( R22 @ C2 @ D2 ) ) ) ) ) ).
% rel_prod.cases
thf(fact_1856_rel__prod_Osimps,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType] :
( ( basic_rel_prod @ A @ B @ C @ D )
= ( ^ [R14: A > B > $o,R25: C > D > $o,A12: product_prod @ A @ C,A23: product_prod @ B @ D] :
? [A7: A,B5: B,C6: C,D5: D] :
( ( A12
= ( product_Pair @ A @ C @ A7 @ C6 ) )
& ( A23
= ( product_Pair @ B @ D @ B5 @ D5 ) )
& ( R14 @ A7 @ B5 )
& ( R25 @ C6 @ D5 ) ) ) ) ).
% rel_prod.simps
thf(fact_1857_rel__prod_Ointros,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,R1: A > B > $o,A3: A,B2: B,R22: C > D > $o,C3: C,D4: D] :
( ( R1 @ A3 @ B2 )
=> ( ( R22 @ C3 @ D4 )
=> ( basic_rel_prod @ A @ B @ C @ D @ R1 @ R22 @ ( product_Pair @ A @ C @ A3 @ C3 ) @ ( product_Pair @ B @ D @ B2 @ D4 ) ) ) ) ).
% rel_prod.intros
thf(fact_1858_Pair__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A6: A > B > $o,B6: C > D > $o] : ( bNF_rel_fun @ A @ B @ ( C > ( product_prod @ A @ C ) ) @ ( D > ( product_prod @ B @ D ) ) @ A6 @ ( bNF_rel_fun @ C @ D @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ B6 @ ( basic_rel_prod @ A @ B @ C @ D @ A6 @ B6 ) ) @ ( product_Pair @ A @ C ) @ ( product_Pair @ B @ D ) ) ).
% Pair_transfer
thf(fact_1859_fst__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,A6: A > C > $o,B6: B > D > $o] : ( bNF_rel_fun @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ A @ C @ ( basic_rel_prod @ A @ C @ B @ D @ A6 @ B6 ) @ A6 @ ( product_fst @ A @ B ) @ ( product_fst @ C @ D ) ) ).
% fst_transfer
thf(fact_1860_snd__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,A6: A > C > $o,B6: B > D > $o] : ( bNF_rel_fun @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ B @ D @ ( basic_rel_prod @ A @ C @ B @ D @ A6 @ B6 ) @ B6 @ ( product_snd @ A @ B ) @ ( product_snd @ C @ D ) ) ).
% snd_transfer
thf(fact_1861_prod_Orel__compp,axiom,
! [B: $tType,A: $tType,C: $tType,E: $tType,F2: $tType,D: $tType,R1: A > C > $o,S1: C > E > $o,R22: B > D > $o,S22: D > F2 > $o] :
( ( basic_rel_prod @ A @ E @ B @ F2 @ ( relcompp @ A @ C @ E @ R1 @ S1 ) @ ( relcompp @ B @ D @ F2 @ R22 @ S22 ) )
= ( relcompp @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( product_prod @ E @ F2 ) @ ( basic_rel_prod @ A @ C @ B @ D @ R1 @ R22 ) @ ( basic_rel_prod @ C @ E @ D @ F2 @ S1 @ S22 ) ) ) ).
% prod.rel_compp
thf(fact_1862_rel__prod__sel,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType] :
( ( basic_rel_prod @ A @ B @ C @ D )
= ( ^ [R14: A > B > $o,R25: C > D > $o,P3: product_prod @ A @ C,Q6: product_prod @ B @ D] :
( ( R14 @ ( product_fst @ A @ C @ P3 ) @ ( product_fst @ B @ D @ Q6 ) )
& ( R25 @ ( product_snd @ A @ C @ P3 ) @ ( product_snd @ B @ D @ Q6 ) ) ) ) ) ).
% rel_prod_sel
thf(fact_1863_rel__prod__conv,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType] :
( ( basic_rel_prod @ A @ C @ B @ D )
= ( ^ [R14: A > C > $o,R25: B > D > $o] :
( product_case_prod @ A @ B @ ( ( product_prod @ C @ D ) > $o )
@ ^ [A7: A,B5: B] :
( product_case_prod @ C @ D @ $o
@ ^ [C6: C,D5: D] :
( ( R14 @ A7 @ C6 )
& ( R25 @ B5 @ D5 ) ) ) ) ) ) ).
% rel_prod_conv
thf(fact_1864_case__prod__transfer,axiom,
! [A: $tType,B: $tType,C: $tType,F2: $tType,E: $tType,D: $tType,A6: A > D > $o,B6: B > E > $o,C5: C > F2 > $o] : ( bNF_rel_fun @ ( A > B > C ) @ ( D > E > F2 ) @ ( ( product_prod @ A @ B ) > C ) @ ( ( product_prod @ D @ E ) > F2 ) @ ( bNF_rel_fun @ A @ D @ ( B > C ) @ ( E > F2 ) @ A6 @ ( bNF_rel_fun @ B @ E @ C @ F2 @ B6 @ C5 ) ) @ ( bNF_rel_fun @ ( product_prod @ A @ B ) @ ( product_prod @ D @ E ) @ C @ F2 @ ( basic_rel_prod @ A @ D @ B @ E @ A6 @ B6 ) @ C5 ) @ ( product_case_prod @ A @ B @ C ) @ ( product_case_prod @ D @ E @ F2 ) ) ).
% case_prod_transfer
thf(fact_1865_curry__transfer,axiom,
! [A: $tType,B: $tType,C: $tType,F2: $tType,E: $tType,D: $tType,A6: A > D > $o,B6: B > E > $o,C5: C > F2 > $o] : ( bNF_rel_fun @ ( ( product_prod @ A @ B ) > C ) @ ( ( product_prod @ D @ E ) > F2 ) @ ( A > B > C ) @ ( D > E > F2 ) @ ( bNF_rel_fun @ ( product_prod @ A @ B ) @ ( product_prod @ D @ E ) @ C @ F2 @ ( basic_rel_prod @ A @ D @ B @ E @ A6 @ B6 ) @ C5 ) @ ( bNF_rel_fun @ A @ D @ ( B > C ) @ ( E > F2 ) @ A6 @ ( bNF_rel_fun @ B @ E @ C @ F2 @ B6 @ C5 ) ) @ ( product_curry @ A @ B @ C ) @ ( product_curry @ D @ E @ F2 ) ) ).
% curry_transfer
thf(fact_1866_Cons__listrel1E2,axiom,
! [A: $tType,Xs: list @ A,Y3: A,Ys: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ ( cons @ A @ Y3 @ Ys ) ) @ ( listrel1 @ A @ R ) )
=> ( ! [X: A] :
( ( Xs
= ( cons @ A @ X @ Ys ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R ) )
=> ~ ! [Zs3: list @ A] :
( ( Xs
= ( cons @ A @ Y3 @ Zs3 ) )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Zs3 @ Ys ) @ ( listrel1 @ A @ R ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_1867_Cons__listrel1E1,axiom,
! [A: $tType,X2: A,Xs: list @ A,Ys: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs ) @ Ys ) @ ( listrel1 @ A @ R ) )
=> ( ! [Y: A] :
( ( Ys
= ( cons @ A @ Y @ Xs ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R ) )
=> ~ ! [Zs3: list @ A] :
( ( Ys
= ( cons @ A @ X2 @ Zs3 ) )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Zs3 ) @ ( listrel1 @ A @ R ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_1868_listrel1I1,axiom,
! [A: $tType,X2: A,Y3: A,R: set @ ( product_prod @ A @ A ),Xs: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs ) @ ( cons @ A @ Y3 @ Xs ) ) @ ( listrel1 @ A @ R ) ) ) ).
% listrel1I1
thf(fact_1869_measures__less,axiom,
! [A: $tType,F: A > nat,X2: A,Y3: A,Fs: list @ ( A > nat )] :
( ( ord_less @ nat @ ( F @ X2 ) @ ( F @ Y3 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ ( measures @ A @ ( cons @ ( A > nat ) @ F @ Fs ) ) ) ) ).
% measures_less
thf(fact_1870_measures__lesseq,axiom,
! [A: $tType,F: A > nat,X2: A,Y3: A,Fs: list @ ( A > nat )] :
( ( ord_less_eq @ nat @ ( F @ X2 ) @ ( F @ Y3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ ( measures @ A @ Fs ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ ( measures @ A @ ( cons @ ( A > nat ) @ F @ Fs ) ) ) ) ) ).
% measures_lesseq
thf(fact_1871_listrel1E,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R ) )
=> ~ ! [X: A,Y: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
=> ! [Us2: list @ A,Vs2: list @ A] :
( ( Xs
= ( append @ A @ Us2 @ ( cons @ A @ X @ Vs2 ) ) )
=> ( Ys
!= ( append @ A @ Us2 @ ( cons @ A @ Y @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_1872_listrel1I,axiom,
! [A: $tType,X2: A,Y3: A,R: set @ ( product_prod @ A @ A ),Xs: list @ A,Us3: list @ A,Vs3: list @ A,Ys: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R )
=> ( ( Xs
= ( append @ A @ Us3 @ ( cons @ A @ X2 @ Vs3 ) ) )
=> ( ( Ys
= ( append @ A @ Us3 @ ( cons @ A @ Y3 @ Vs3 ) ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R ) ) ) ) ) ).
% listrel1I
thf(fact_1873_rtrancl__listrel1__ConsI2,axiom,
! [A: $tType,X2: A,Y3: A,R: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R ) ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs ) @ ( cons @ A @ Y3 @ Ys ) ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R ) ) ) ) ) ).
% rtrancl_listrel1_ConsI2
thf(fact_1874_snoc__listrel1__snoc__iff,axiom,
! [A: $tType,Xs: list @ A,X2: A,Ys: list @ A,Y3: 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 @ X2 @ ( nil @ A ) ) ) @ ( append @ A @ Ys @ ( cons @ A @ Y3 @ ( nil @ A ) ) ) ) @ ( listrel1 @ A @ R ) )
= ( ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R ) )
& ( X2 = Y3 ) )
| ( ( Xs = Ys )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_1875_listrel1p__def,axiom,
! [A: $tType] :
( ( listrel1p @ A )
= ( ^ [R5: A > A > $o,Xs2: list @ A,Ys2: list @ A] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) @ ( listrel1 @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R5 ) ) ) ) ) ) ).
% listrel1p_def
thf(fact_1876_empty__Shift,axiom,
! [A: $tType,Kl4: set @ ( list @ A ),K: A] :
( ( member @ ( list @ A ) @ ( nil @ A ) @ Kl4 )
=> ( ( member @ A @ K @ ( bNF_Greatest_Succ @ A @ Kl4 @ ( nil @ A ) ) )
=> ( member @ ( list @ A ) @ ( nil @ A ) @ ( bNF_Greatest_Shift @ A @ Kl4 @ K ) ) ) ) ).
% empty_Shift
thf(fact_1877_Succ__Shift,axiom,
! [A: $tType,Kl4: set @ ( list @ A ),K: A,Kl3: list @ A] :
( ( bNF_Greatest_Succ @ A @ ( bNF_Greatest_Shift @ A @ Kl4 @ K ) @ Kl3 )
= ( bNF_Greatest_Succ @ A @ Kl4 @ ( cons @ A @ K @ Kl3 ) ) ) ).
% Succ_Shift
thf(fact_1878_listrel1__iff__update,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R ) )
= ( ? [Y5: A,N4: nat] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth3 @ A @ Xs @ N4 ) @ Y5 ) @ R )
& ( ord_less @ nat @ N4 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( Ys
= ( list_update @ A @ Xs @ N4 @ Y5 ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_1879_lexordp__conv__lexord,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_lexordp @ A )
= ( ^ [Xs2: list @ A,Ys2: list @ A] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) @ ( lexord @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ ( ord_less @ A ) ) ) ) ) ) ) ) ).
% lexordp_conv_lexord
thf(fact_1880_list_Orec__o__map,axiom,
! [C: $tType,B: $tType,A: $tType,G: C,Ga: B > ( list @ B ) > C > C,F: A > B] :
( ( comp @ ( list @ B ) @ C @ ( list @ A ) @ ( rec_list @ C @ B @ G @ Ga ) @ ( map @ A @ B @ F ) )
= ( rec_list @ C @ A @ G
@ ^ [X4: A,Xa4: list @ A] : ( Ga @ ( F @ X4 ) @ ( map @ A @ B @ F @ Xa4 ) ) ) ) ).
% list.rec_o_map
thf(fact_1881_shuffles_Oelims,axiom,
! [A: $tType,X2: list @ A,Xa2: list @ A,Y3: set @ ( list @ A )] :
( ( ( shuffles @ A @ X2 @ Xa2 )
= Y3 )
=> ( ( ( X2
= ( nil @ A ) )
=> ( Y3
!= ( insert @ ( list @ A ) @ Xa2 @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) )
=> ( ( ( Xa2
= ( nil @ A ) )
=> ( Y3
!= ( insert @ ( list @ A ) @ X2 @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) )
=> ~ ! [X: A,Xs4: list @ A] :
( ( X2
= ( cons @ A @ X @ Xs4 ) )
=> ! [Y: A,Ys4: list @ A] :
( ( Xa2
= ( cons @ A @ Y @ Ys4 ) )
=> ( Y3
!= ( sup_sup @ ( set @ ( list @ A ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X ) @ ( shuffles @ A @ Xs4 @ ( cons @ A @ Y @ Ys4 ) ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y ) @ ( shuffles @ A @ ( cons @ A @ X @ Xs4 ) @ Ys4 ) ) ) ) ) ) ) ) ) ).
% shuffles.elims
thf(fact_1882_get__update__eq,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [A3: array @ A,I: nat,V: A,H: heap_ext @ product_unit] :
( ( get @ A @ ( update @ A @ A3 @ I @ V @ H ) @ A3 )
= ( list_update @ A @ ( get @ A @ H @ A3 ) @ I @ V ) ) ) ).
% get_update_eq
thf(fact_1883_zip__update,axiom,
! [A: $tType,B: $tType,Xs: list @ A,I: nat,X2: A,Ys: list @ B,Y3: B] :
( ( zip @ A @ B @ ( list_update @ A @ Xs @ I @ X2 ) @ ( list_update @ B @ Ys @ I @ Y3 ) )
= ( list_update @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) @ I @ ( product_Pair @ A @ B @ X2 @ Y3 ) ) ) ).
% zip_update
thf(fact_1884_Cons__shuffles__subset2,axiom,
! [A: $tType,Y3: A,Xs: list @ A,Ys: list @ A] : ( ord_less_eq @ ( set @ ( list @ A ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y3 ) @ ( shuffles @ A @ Xs @ Ys ) ) @ ( shuffles @ A @ Xs @ ( cons @ A @ Y3 @ Ys ) ) ) ).
% Cons_shuffles_subset2
thf(fact_1885_Cons__shuffles__subset1,axiom,
! [A: $tType,X2: A,Xs: list @ A,Ys: list @ A] : ( ord_less_eq @ ( set @ ( list @ A ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 ) @ ( shuffles @ A @ Xs @ Ys ) ) @ ( shuffles @ A @ ( cons @ A @ X2 @ Xs ) @ Ys ) ) ).
% Cons_shuffles_subset1
thf(fact_1886_shuffles_Osimps_I3_J,axiom,
! [A: $tType,X2: A,Xs: list @ A,Y3: A,Ys: list @ A] :
( ( shuffles @ A @ ( cons @ A @ X2 @ Xs ) @ ( cons @ A @ Y3 @ Ys ) )
= ( sup_sup @ ( set @ ( list @ A ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 ) @ ( shuffles @ A @ Xs @ ( cons @ A @ Y3 @ Ys ) ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y3 ) @ ( shuffles @ A @ ( cons @ A @ X2 @ Xs ) @ Ys ) ) ) ) ).
% shuffles.simps(3)
thf(fact_1887_update__zip,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B,I: nat,Xy2: product_prod @ A @ B] :
( ( list_update @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) @ I @ Xy2 )
= ( zip @ A @ B @ ( list_update @ A @ Xs @ I @ ( product_fst @ A @ B @ Xy2 ) ) @ ( list_update @ B @ Ys @ I @ ( product_snd @ A @ B @ Xy2 ) ) ) ) ).
% update_zip
thf(fact_1888_rec__list__Cons__imp,axiom,
! [B: $tType,A: $tType,F: ( list @ A ) > B,F1: B,F23: A > ( list @ A ) > B > B,X2: A,Xs: list @ A] :
( ( F
= ( rec_list @ B @ A @ F1 @ F23 ) )
=> ( ( F @ ( cons @ A @ X2 @ Xs ) )
= ( F23 @ X2 @ Xs @ ( F @ Xs ) ) ) ) ).
% rec_list_Cons_imp
thf(fact_1889_rec__list__Nil__imp,axiom,
! [A: $tType,B: $tType,F: ( list @ A ) > B,F1: B,F23: A > ( list @ A ) > B > B] :
( ( F
= ( rec_list @ B @ A @ F1 @ F23 ) )
=> ( ( F @ ( nil @ A ) )
= F1 ) ) ).
% rec_list_Nil_imp
thf(fact_1890_ShiftD,axiom,
! [A: $tType,Kl3: list @ A,Kl4: set @ ( list @ A ),K: A] :
( ( member @ ( list @ A ) @ Kl3 @ ( bNF_Greatest_Shift @ A @ Kl4 @ K ) )
=> ( member @ ( list @ A ) @ ( cons @ A @ K @ Kl3 ) @ Kl4 ) ) ).
% ShiftD
thf(fact_1891_Array_Oupdate__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( update @ A )
= ( ^ [A7: array @ A,I3: nat,X4: A,H6: heap_ext @ product_unit] : ( set2 @ A @ A7 @ ( list_update @ A @ ( get @ A @ H6 @ A7 ) @ I3 @ X4 ) @ H6 ) ) ) ) ).
% Array.update_def
thf(fact_1892_Shift__def,axiom,
! [A: $tType] :
( ( bNF_Greatest_Shift @ A )
= ( ^ [Kl2: set @ ( list @ A ),K3: A] :
( collect @ ( list @ A )
@ ^ [Kl: list @ A] : ( member @ ( list @ A ) @ ( cons @ A @ K3 @ Kl ) @ Kl2 ) ) ) ) ).
% Shift_def
thf(fact_1893_shuffles_Opelims,axiom,
! [A: $tType,X2: list @ A,Xa2: list @ A,Y3: set @ ( list @ A )] :
( ( ( shuffles @ A @ X2 @ Xa2 )
= Y3 )
=> ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X2 @ Xa2 ) )
=> ( ( ( X2
= ( nil @ A ) )
=> ( ( Y3
= ( insert @ ( list @ A ) @ Xa2 @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xa2 ) ) ) )
=> ( ( ( Xa2
= ( nil @ A ) )
=> ( ( Y3
= ( insert @ ( list @ A ) @ X2 @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X2 @ ( nil @ A ) ) ) ) )
=> ~ ! [X: A,Xs4: list @ A] :
( ( X2
= ( cons @ A @ X @ Xs4 ) )
=> ! [Y: A,Ys4: list @ A] :
( ( Xa2
= ( cons @ A @ Y @ Ys4 ) )
=> ( ( Y3
= ( sup_sup @ ( set @ ( list @ A ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X ) @ ( shuffles @ A @ Xs4 @ ( cons @ A @ Y @ Ys4 ) ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y ) @ ( shuffles @ A @ ( cons @ A @ X @ Xs4 ) @ Ys4 ) ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs4 ) @ ( cons @ A @ Y @ Ys4 ) ) ) ) ) ) ) ) ) ) ).
% shuffles.pelims
thf(fact_1894_fun_Opred__set,axiom,
! [A: $tType,D: $tType,P2: A > $o] :
( ( basic_pred_fun @ D @ A
@ ^ [Uu: D] : $true
@ P2 )
= ( ^ [X4: D > A] :
! [Y5: A] :
( ( member @ A @ Y5 @ ( image2 @ D @ A @ X4 @ ( top_top @ ( set @ D ) ) ) )
=> ( P2 @ Y5 ) ) ) ) ).
% fun.pred_set
thf(fact_1895_make_H__def,axiom,
! [B: $tType,A: $tType] :
( ( ( heap @ A )
& ( semiring_1 @ B ) )
=> ( ( make2 @ B @ A )
= ( ^ [I3: code_integer,F4: B > A] : ( make @ A @ ( code_nat_of_integer @ I3 ) @ ( comp @ B @ A @ nat @ F4 @ ( semiring_1_of_nat @ B ) ) ) ) ) ) ).
% make'_def
thf(fact_1896_Sup__fin_Ohom__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [H: A > A,N: set @ A] :
( ! [X: A,Y: A] :
( ( H @ ( sup_sup @ A @ X @ Y ) )
= ( sup_sup @ A @ ( H @ X ) @ ( H @ Y ) ) )
=> ( ( finite_finite2 @ A @ N )
=> ( ( N
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H @ ( lattic5882676163264333800up_fin @ A @ N ) )
= ( lattic5882676163264333800up_fin @ A @ ( image2 @ A @ A @ H @ N ) ) ) ) ) ) ) ).
% Sup_fin.hom_commute
thf(fact_1897_Inf__fin_Ohom__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [H: A > A,N: set @ A] :
( ! [X: A,Y: A] :
( ( H @ ( inf_inf @ A @ X @ Y ) )
= ( inf_inf @ A @ ( H @ X ) @ ( H @ Y ) ) )
=> ( ( finite_finite2 @ A @ N )
=> ( ( N
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H @ ( lattic7752659483105999362nf_fin @ A @ N ) )
= ( lattic7752659483105999362nf_fin @ A @ ( image2 @ A @ A @ H @ N ) ) ) ) ) ) ) ).
% Inf_fin.hom_commute
thf(fact_1898_irrefl__distinct,axiom,
! [A: $tType] :
( ( irrefl @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
! [X4: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X4 @ R5 )
=> ( product_case_prod @ A @ A @ $o
@ ^ [A7: A,B5: A] : A7 != B5
@ X4 ) ) ) ) ).
% irrefl_distinct
thf(fact_1899_wo__rel_OisMinim__def,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A6: set @ A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R )
=> ( ( bNF_We4791949203932849705sMinim @ A @ R @ A6 @ B2 )
= ( ( member @ A @ B2 @ A6 )
& ! [X4: A] :
( ( member @ A @ X4 @ A6 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ X4 ) @ R ) ) ) ) ) ).
% wo_rel.isMinim_def
thf(fact_1900_irreflI,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ! [A5: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ A5 ) @ R3 )
=> ( irrefl @ A @ R3 ) ) ).
% irreflI
thf(fact_1901_irrefl__def,axiom,
! [A: $tType] :
( ( irrefl @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
! [A7: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ A7 ) @ R5 ) ) ) ).
% irrefl_def
thf(fact_1902_fun_Omap__cong__pred,axiom,
! [B: $tType,A: $tType,D: $tType,X2: D > A,Ya: D > A,F: A > B,G: A > B] :
( ( X2 = Ya )
=> ( ( basic_pred_fun @ D @ A
@ ^ [Uu: D] : $true
@ ^ [Z5: A] :
( ( F @ Z5 )
= ( G @ Z5 ) )
@ Ya )
=> ( ( comp @ A @ B @ D @ F @ X2 )
= ( comp @ A @ B @ D @ G @ Ya ) ) ) ) ).
% fun.map_cong_pred
thf(fact_1903_fun_Opred__mono__strong,axiom,
! [A: $tType,D: $tType,P2: A > $o,X2: D > A,Pa: A > $o] :
( ( basic_pred_fun @ D @ A
@ ^ [Uu: D] : $true
@ P2
@ X2 )
=> ( ! [Z4: A] :
( ( member @ A @ Z4 @ ( image2 @ D @ A @ X2 @ ( top_top @ ( set @ D ) ) ) )
=> ( ( P2 @ Z4 )
=> ( Pa @ Z4 ) ) )
=> ( basic_pred_fun @ D @ A
@ ^ [Uu: D] : $true
@ Pa
@ X2 ) ) ) ).
% fun.pred_mono_strong
thf(fact_1904_fun_Opred__cong,axiom,
! [A: $tType,D: $tType,X2: D > A,Ya: D > A,P2: A > $o,Pa: A > $o] :
( ( X2 = Ya )
=> ( ! [Z4: A] :
( ( member @ A @ Z4 @ ( image2 @ D @ A @ Ya @ ( top_top @ ( set @ D ) ) ) )
=> ( ( P2 @ Z4 )
= ( Pa @ Z4 ) ) )
=> ( ( basic_pred_fun @ D @ A
@ ^ [Uu: D] : $true
@ P2
@ X2 )
= ( basic_pred_fun @ D @ A
@ ^ [Uu: D] : $true
@ Pa
@ Ya ) ) ) ) ).
% fun.pred_cong
thf(fact_1905_fun_Opred__map,axiom,
! [B: $tType,A: $tType,D: $tType,Q2: B > $o,F: A > B,X2: D > A] :
( ( basic_pred_fun @ D @ B
@ ^ [Uu: D] : $true
@ Q2
@ ( comp @ A @ B @ D @ F @ X2 ) )
= ( basic_pred_fun @ D @ A
@ ^ [Uu: D] : $true
@ ( comp @ B @ $o @ A @ Q2 @ F )
@ X2 ) ) ).
% fun.pred_map
thf(fact_1906_shuffles_Opsimps_I3_J,axiom,
! [A: $tType,X2: A,Xs: list @ A,Y3: A,Ys: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs ) @ ( cons @ A @ Y3 @ Ys ) ) )
=> ( ( shuffles @ A @ ( cons @ A @ X2 @ Xs ) @ ( cons @ A @ Y3 @ Ys ) )
= ( sup_sup @ ( set @ ( list @ A ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 ) @ ( shuffles @ A @ Xs @ ( cons @ A @ Y3 @ Ys ) ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y3 ) @ ( shuffles @ A @ ( cons @ A @ X2 @ Xs ) @ Ys ) ) ) ) ) ).
% shuffles.psimps(3)
thf(fact_1907_nth_H__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( nth2 @ A )
= ( ^ [A7: array @ A] : ( comp @ nat @ ( heap_Heap @ A ) @ code_integer @ ( nth @ A @ A7 ) @ code_nat_of_integer ) ) ) ) ).
% nth'_def
thf(fact_1908_new_H__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( new2 @ A )
= ( comp @ nat @ ( A > ( heap_Heap @ ( array @ A ) ) ) @ code_integer @ ( new @ A ) @ code_nat_of_integer ) ) ) ).
% new'_def
thf(fact_1909_pred__fun__True__id,axiom,
! [A: $tType,B: $tType,C: $tType,P: B > $o,F: C > B] :
( ( nO_MATCH @ ( A > A ) @ ( B > $o ) @ ( id @ A ) @ P )
=> ( ( basic_pred_fun @ C @ B
@ ^ [X4: C] : $true
@ P
@ F )
= ( basic_pred_fun @ C @ $o
@ ^ [X4: C] : $true
@ ( id @ $o )
@ ( comp @ B @ $o @ C @ P @ F ) ) ) ) ).
% pred_fun_True_id
thf(fact_1910_irreflp__irrefl__eq,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( irreflp @ A
@ ^ [A7: A,B5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B5 ) @ R3 ) )
= ( irrefl @ A @ R3 ) ) ).
% irreflp_irrefl_eq
thf(fact_1911_wf__finite__segments,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( irrefl @ A @ R )
=> ( ( trans @ A @ R )
=> ( ! [X: A] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [Y5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X ) @ R ) ) )
=> ( wf @ A @ R ) ) ) ) ).
% wf_finite_segments
thf(fact_1912_size__list__map,axiom,
! [A: $tType,B: $tType,F: A > nat,G: B > A,Xs: list @ B] :
( ( size_list @ A @ F @ ( map @ B @ A @ G @ Xs ) )
= ( size_list @ B @ ( comp @ A @ nat @ B @ F @ G ) @ Xs ) ) ).
% size_list_map
thf(fact_1913_fromCard__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Gr5436034075474128252omCard @ A @ B )
= ( ^ [A8: set @ A,R5: set @ ( product_prod @ B @ B ),K3: B] :
( fChoice @ A
@ ^ [B5: A] :
( ( member @ A @ B5 @ A8 )
& ( ( bNF_Greatest_toCard @ A @ B @ A8 @ R5 @ B5 )
= K3 ) ) ) ) ) ).
% fromCard_def
thf(fact_1914_trans__pair__less,axiom,
trans @ ( product_prod @ nat @ nat ) @ fun_pair_less ).
% trans_pair_less
thf(fact_1915_Eps__case__prod__eq,axiom,
! [A: $tType,B: $tType,X2: A,Y3: B] :
( ( fChoice @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X7: A,Y6: B] :
( ( X2 = X7 )
& ( Y3 = Y6 ) ) ) )
= ( product_Pair @ A @ B @ X2 @ Y3 ) ) ).
% Eps_case_prod_eq
thf(fact_1916_trans__def,axiom,
! [A: $tType] :
( ( trans @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
! [X4: A,Y5: A,Z5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R5 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ Z5 ) @ R5 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Z5 ) @ R5 ) ) ) ) ) ).
% trans_def
thf(fact_1917_transI,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ! [X: A,Y: A,Z4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z4 ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z4 ) @ R ) ) )
=> ( trans @ A @ R ) ) ).
% transI
thf(fact_1918_transE,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),X2: A,Y3: A,Z3: A] :
( ( trans @ A @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Z3 ) @ R ) ) ) ) ).
% transE
thf(fact_1919_transD,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),X2: A,Y3: A,Z3: A] :
( ( trans @ A @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Z3 ) @ R ) ) ) ) ).
% transD
thf(fact_1920_pick__middlep__def,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( bNF_pick_middlep @ B @ A @ C )
= ( ^ [P4: B > A > $o,Q3: A > C > $o,A7: B,C6: C] :
( fChoice @ A
@ ^ [B5: A] :
( ( P4 @ A7 @ B5 )
& ( Q3 @ B5 @ C6 ) ) ) ) ) ).
% pick_middlep_def
thf(fact_1921_univ__def,axiom,
! [A: $tType,B: $tType] :
( ( bNF_Greatest_univ @ B @ A )
= ( ^ [F4: B > A,X10: set @ B] :
( F4
@ ( fChoice @ B
@ ^ [X4: B] : ( member @ B @ X4 @ X10 ) ) ) ) ) ).
% univ_def
thf(fact_1922_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_1923_trans__INTER,axiom,
! [B: $tType,A: $tType,S5: set @ A,R: A > ( set @ ( product_prod @ B @ B ) )] :
( ! [X: A] :
( ( member @ A @ X @ S5 )
=> ( trans @ B @ ( R @ X ) ) )
=> ( trans @ B @ ( complete_Inf_Inf @ ( set @ ( product_prod @ B @ B ) ) @ ( image2 @ A @ ( set @ ( product_prod @ B @ B ) ) @ R @ S5 ) ) ) ) ).
% trans_INTER
thf(fact_1924_equiv__Eps__in,axiom,
! [A: $tType,A6: set @ A,R: set @ ( product_prod @ A @ A ),X6: set @ A] :
( ( equiv_equiv @ A @ A6 @ R )
=> ( ( member @ ( set @ A ) @ X6 @ ( equiv_quotient @ A @ A6 @ R ) )
=> ( member @ A
@ ( fChoice @ A
@ ^ [X4: A] : ( member @ A @ X4 @ X6 ) )
@ X6 ) ) ) ).
% equiv_Eps_in
thf(fact_1925_equiv__Eps__preserves,axiom,
! [A: $tType,A6: set @ A,R: set @ ( product_prod @ A @ A ),X6: set @ A] :
( ( equiv_equiv @ A @ A6 @ R )
=> ( ( member @ ( set @ A ) @ X6 @ ( equiv_quotient @ A @ A6 @ R ) )
=> ( member @ A
@ ( fChoice @ A
@ ^ [X4: A] : ( member @ A @ X4 @ X6 ) )
@ A6 ) ) ) ).
% equiv_Eps_preserves
thf(fact_1926_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_1927_list_Osize__gen__o__map,axiom,
! [B: $tType,A: $tType,F: B > nat,G: A > B] :
( ( comp @ ( list @ B ) @ nat @ ( list @ A ) @ ( size_list @ B @ F ) @ ( map @ A @ B @ G ) )
= ( size_list @ A @ ( comp @ B @ nat @ A @ F @ G ) ) ) ).
% list.size_gen_o_map
thf(fact_1928_trans__singleton,axiom,
! [A: $tType,A3: A] : ( trans @ A @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% trans_singleton
thf(fact_1929_Eps__case__prod,axiom,
! [B: $tType,A: $tType,P2: A > B > $o] :
( ( fChoice @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P2 ) )
= ( fChoice @ ( product_prod @ A @ B )
@ ^ [Xy: product_prod @ A @ B] : ( P2 @ ( product_fst @ A @ B @ Xy ) @ ( product_snd @ A @ B @ Xy ) ) ) ) ).
% Eps_case_prod
thf(fact_1930_toCard__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Greatest_toCard @ A @ B )
= ( ^ [A8: set @ A,R5: set @ ( product_prod @ B @ B )] : ( fChoice @ ( A > B ) @ ( bNF_Gr1419584066657907630d_pred @ A @ B @ A8 @ R5 ) ) ) ) ).
% toCard_def
thf(fact_1931_trans__join,axiom,
! [A: $tType] :
( ( trans @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
! [X4: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X4 @ R5 )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y5: A,Y13: A] :
! [Z5: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ Z5 @ R5 )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y24: A,Aa3: A] :
( ( Y13 = Y24 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ Aa3 ) @ R5 ) )
@ Z5 ) )
@ X4 ) ) ) ) ).
% trans_join
thf(fact_1932_proj__Eps,axiom,
! [A: $tType,A6: set @ A,R: set @ ( product_prod @ A @ A ),X6: set @ A] :
( ( equiv_equiv @ A @ A6 @ R )
=> ( ( member @ ( set @ A ) @ X6 @ ( equiv_quotient @ A @ A6 @ R ) )
=> ( ( equiv_proj @ A @ A @ R
@ ( fChoice @ A
@ ^ [X4: A] : ( member @ A @ X4 @ X6 ) ) )
= X6 ) ) ) ).
% proj_Eps
thf(fact_1933_ord__to__filter__def,axiom,
! [A: $tType] :
( ( bNF_We8469521843155493636filter @ A )
= ( ^ [R0: set @ ( product_prod @ A @ A ),R5: set @ ( product_prod @ A @ A )] : ( image2 @ A @ A @ ( fChoice @ ( A > A ) @ ( bNF_Wellorder_embed @ A @ A @ R5 @ R0 ) ) @ ( field2 @ A @ R5 ) ) ) ) ).
% ord_to_filter_def
thf(fact_1934_arg__min__SOME__Min,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [S5: set @ A,F: A > B] :
( ( finite_finite2 @ A @ S5 )
=> ( ( lattic7623131987881927897min_on @ A @ B @ F @ S5 )
= ( fChoice @ A
@ ^ [Y5: A] :
( ( member @ A @ Y5 @ S5 )
& ( ( F @ Y5 )
= ( lattic643756798350308766er_Min @ B @ ( image2 @ A @ B @ F @ S5 ) ) ) ) ) ) ) ) ).
% arg_min_SOME_Min
thf(fact_1935_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_1936_equivp__equiv,axiom,
! [A: $tType,A6: set @ ( product_prod @ A @ A )] :
( ( equiv_equiv @ A @ ( top_top @ ( set @ A ) ) @ A6 )
= ( equiv_equivp @ A
@ ^ [X4: A,Y5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ A6 ) ) ) ).
% equivp_equiv
thf(fact_1937_partition_Osimps_I2_J,axiom,
! [A: $tType,P2: A > $o,X2: A,Xs: list @ A] :
( ( partition @ A @ P2 @ ( cons @ A @ X2 @ Xs ) )
= ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ^ [Yes: list @ A,No: list @ A] : ( if @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( P2 @ X2 ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Yes ) @ No ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ ( cons @ A @ X2 @ No ) ) )
@ ( partition @ A @ P2 @ Xs ) ) ) ).
% partition.simps(2)
thf(fact_1938_less__eq__int_Orep__eq,axiom,
( ( ord_less_eq @ int )
= ( ^ [X4: int,Xa4: int] :
( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [Y5: nat,Z5: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U: nat,V3: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ Y5 @ V3 ) @ ( plus_plus @ nat @ U @ Z5 ) ) )
@ ( rep_Integ @ X4 )
@ ( rep_Integ @ Xa4 ) ) ) ) ).
% less_eq_int.rep_eq
thf(fact_1939_above__def,axiom,
! [A: $tType] :
( ( order_above @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A ),A7: A] :
( collect @ A
@ ^ [B5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B5 ) @ R5 ) ) ) ) ).
% above_def
thf(fact_1940_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_1941_antisymI,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ! [X: A,Y: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ R )
=> ( X = Y ) ) )
=> ( antisym @ A @ R ) ) ).
% antisymI
thf(fact_1942_antisym__def,axiom,
! [A: $tType] :
( ( antisym @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
! [X4: A,Y5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R5 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X4 ) @ R5 )
=> ( X4 = Y5 ) ) ) ) ) ).
% antisym_def
thf(fact_1943_nat_Orep__eq,axiom,
( nat2
= ( ^ [X4: int] : ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) @ ( rep_Integ @ X4 ) ) ) ) ).
% nat.rep_eq
thf(fact_1944_less__int_Orep__eq,axiom,
( ( ord_less @ int )
= ( ^ [X4: int,Xa4: int] :
( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [Y5: nat,Z5: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U: nat,V3: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ Y5 @ V3 ) @ ( plus_plus @ nat @ U @ Z5 ) ) )
@ ( rep_Integ @ X4 )
@ ( rep_Integ @ Xa4 ) ) ) ) ).
% less_int.rep_eq
thf(fact_1945_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
@ ^ [Y2: $o,Z2: $o] : Y2 = Z2 )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X4: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U: nat,V3: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X4 @ V3 ) @ ( plus_plus @ nat @ U @ Y5 ) ) ) )
@ ( ord_less_eq @ int ) ) ).
% less_eq_int.transfer
thf(fact_1946_less__eq__int_Oabs__eq,axiom,
! [Xa2: product_prod @ nat @ nat,X2: product_prod @ nat @ nat] :
( ( ord_less_eq @ int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X2 ) )
= ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X4: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U: nat,V3: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X4 @ V3 ) @ ( plus_plus @ nat @ U @ Y5 ) ) )
@ Xa2
@ X2 ) ) ).
% less_eq_int.abs_eq
thf(fact_1947_int__ge__less__than2__def,axiom,
( int_ge_less_than2
= ( ^ [D5: int] :
( collect @ ( product_prod @ int @ int )
@ ( product_case_prod @ int @ int @ $o
@ ^ [Z11: int,Z5: int] :
( ( ord_less_eq @ int @ D5 @ Z5 )
& ( ord_less @ int @ Z11 @ Z5 ) ) ) ) ) ) ).
% int_ge_less_than2_def
thf(fact_1948_int__ge__less__than__def,axiom,
( int_ge_less_than
= ( ^ [D5: int] :
( collect @ ( product_prod @ int @ int )
@ ( product_case_prod @ int @ int @ $o
@ ^ [Z11: int,Z5: int] :
( ( ord_less_eq @ int @ D5 @ Z11 )
& ( ord_less @ int @ Z11 @ Z5 ) ) ) ) ) ) ).
% int_ge_less_than_def
thf(fact_1949_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
@ ^ [Y2: $o,Z2: $o] : Y2 = Z2 )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X4: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U: nat,V3: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X4 @ V3 ) @ ( plus_plus @ nat @ U @ Y5 ) ) ) )
@ ( ord_less @ int ) ) ).
% less_int.transfer
thf(fact_1950_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 ) )
@ ^ [X4: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U: nat,V3: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ V3 ) @ ( plus_plus @ nat @ Y5 @ U ) ) ) )
@ ( minus_minus @ int ) ) ).
% minus_int.transfer
thf(fact_1951_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 ) )
@ ^ [X4: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U: nat,V3: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ U ) @ ( plus_plus @ nat @ Y5 @ V3 ) ) ) )
@ ( plus_plus @ int ) ) ).
% plus_int.transfer
thf(fact_1952_nat_Oabs__eq,axiom,
! [X2: product_prod @ nat @ nat] :
( ( nat2 @ ( abs_Integ @ X2 ) )
= ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) @ X2 ) ) ).
% nat.abs_eq
thf(fact_1953_nat_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ nat @ nat @ pcr_int
@ ^ [Y2: nat,Z2: nat] : Y2 = Z2
@ ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) )
@ nat2 ) ).
% nat.transfer
thf(fact_1954_uminus__int_Oabs__eq,axiom,
! [X2: product_prod @ nat @ nat] :
( ( uminus_uminus @ int @ ( abs_Integ @ X2 ) )
= ( abs_Integ
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [X4: nat,Y5: nat] : ( product_Pair @ nat @ nat @ Y5 @ X4 )
@ X2 ) ) ) ).
% uminus_int.abs_eq
thf(fact_1955_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 )
@ ^ [X4: nat,Y5: nat] : ( product_Pair @ nat @ nat @ Y5 @ X4 ) )
@ ( uminus_uminus @ int ) ) ).
% uminus_int.transfer
thf(fact_1956_times__int_Oabs__eq,axiom,
! [Xa2: product_prod @ nat @ nat,X2: product_prod @ nat @ nat] :
( ( times_times @ int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X2 ) )
= ( abs_Integ
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U: nat,V3: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ U ) @ ( times_times @ nat @ Y5 @ V3 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ V3 ) @ ( times_times @ nat @ Y5 @ U ) ) ) )
@ Xa2
@ X2 ) ) ) ).
% times_int.abs_eq
thf(fact_1957_plus__int_Oabs__eq,axiom,
! [Xa2: product_prod @ nat @ nat,X2: product_prod @ nat @ nat] :
( ( plus_plus @ int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X2 ) )
= ( abs_Integ
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U: nat,V3: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ U ) @ ( plus_plus @ nat @ Y5 @ V3 ) ) )
@ Xa2
@ X2 ) ) ) ).
% plus_int.abs_eq
thf(fact_1958_minus__int_Oabs__eq,axiom,
! [Xa2: product_prod @ nat @ nat,X2: product_prod @ nat @ nat] :
( ( minus_minus @ int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X2 ) )
= ( abs_Integ
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U: nat,V3: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ V3 ) @ ( plus_plus @ nat @ Y5 @ U ) ) )
@ Xa2
@ X2 ) ) ) ).
% minus_int.abs_eq
thf(fact_1959_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 ) )
@ ^ [X4: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U: nat,V3: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ U ) @ ( times_times @ nat @ Y5 @ V3 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ V3 ) @ ( times_times @ nat @ Y5 @ U ) ) ) ) )
@ ( times_times @ int ) ) ).
% times_int.transfer
thf(fact_1960_less__int_Oabs__eq,axiom,
! [Xa2: product_prod @ nat @ nat,X2: product_prod @ nat @ nat] :
( ( ord_less @ int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X2 ) )
= ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X4: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U: nat,V3: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X4 @ V3 ) @ ( plus_plus @ nat @ U @ Y5 ) ) )
@ Xa2
@ X2 ) ) ).
% less_int.abs_eq
thf(fact_1961_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 )
@ ^ [X4: nat,Y5: nat] : ( product_Pair @ nat @ nat @ Y5 @ X4 ) ) ) ) ).
% uminus_int_def
thf(fact_1962_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 ) )
@ ^ [X4: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U: nat,V3: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ U ) @ ( times_times @ nat @ Y5 @ V3 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ V3 ) @ ( times_times @ nat @ Y5 @ U ) ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U: nat,V3: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ U ) @ ( times_times @ nat @ Y5 @ V3 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ V3 ) @ ( times_times @ nat @ Y5 @ U ) ) ) ) ) ) ).
% times_int.rsp
thf(fact_1963_nat__def,axiom,
( nat2
= ( map_fun @ int @ ( product_prod @ nat @ nat ) @ nat @ nat @ rep_Integ @ ( id @ nat ) @ ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) ) ) ) ).
% nat_def
thf(fact_1964_of__int_Otransfer,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ A @ A @ pcr_int
@ ^ [Y2: A,Z2: A] : Y2 = Z2
@ ( product_case_prod @ nat @ nat @ A
@ ^ [I3: nat,J: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I3 ) @ ( semiring_1_of_nat @ A @ J ) ) )
@ ( ring_1_of_int @ A ) ) ) ).
% of_int.transfer
thf(fact_1965_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 ) )
@ ^ [X4: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U: nat,V3: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ V3 ) @ ( plus_plus @ nat @ Y5 @ U ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U: nat,V3: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ V3 ) @ ( plus_plus @ nat @ Y5 @ U ) ) ) ) ) ).
% minus_int.rsp
thf(fact_1966_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 ) )
@ ^ [X4: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U: nat,V3: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ U ) @ ( plus_plus @ nat @ Y5 @ V3 ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U: nat,V3: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ U ) @ ( plus_plus @ nat @ Y5 @ V3 ) ) ) ) ) ).
% plus_int.rsp
thf(fact_1967_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
@ ^ [Y2: $o,Z2: $o] : Y2 = Z2 )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X4: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U: nat,V3: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X4 @ V3 ) @ ( plus_plus @ nat @ U @ Y5 ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X4: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U: nat,V3: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X4 @ V3 ) @ ( plus_plus @ nat @ U @ Y5 ) ) ) ) ) ).
% less_eq_int.rsp
thf(fact_1968_map__fun_Ocompositionality,axiom,
! [D: $tType,F2: $tType,C: $tType,E: $tType,B: $tType,A: $tType,F: E > C,G: D > F2,H: C > A,I: B > D,Fun: A > B] :
( ( map_fun @ E @ C @ D @ F2 @ F @ G @ ( map_fun @ C @ A @ B @ D @ H @ I @ Fun ) )
= ( map_fun @ E @ A @ B @ F2 @ ( comp @ C @ A @ E @ H @ F ) @ ( comp @ D @ F2 @ B @ G @ I ) @ Fun ) ) ).
% map_fun.compositionality
thf(fact_1969_map__fun__def,axiom,
! [B: $tType,D: $tType,A: $tType,C: $tType] :
( ( map_fun @ C @ A @ B @ D )
= ( ^ [F4: C > A,G4: B > D,H6: A > B] : ( comp @ A @ D @ C @ ( comp @ B @ D @ A @ G4 @ H6 ) @ F4 ) ) ) ).
% map_fun_def
thf(fact_1970_map__fun_Ocomp,axiom,
! [E: $tType,C: $tType,A: $tType,F2: $tType,D: $tType,B: $tType,F: E > C,G: D > F2,H: C > A,I: B > D] :
( ( comp @ ( C > D ) @ ( E > F2 ) @ ( A > B ) @ ( map_fun @ E @ C @ D @ F2 @ F @ G ) @ ( map_fun @ C @ A @ B @ D @ H @ I ) )
= ( map_fun @ E @ A @ B @ F2 @ ( comp @ C @ A @ E @ H @ F ) @ ( comp @ D @ F2 @ B @ G @ I ) ) ) ).
% map_fun.comp
thf(fact_1971_ring__1__class_Oof__int__def,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_of_int @ A )
= ( map_fun @ int @ ( product_prod @ nat @ nat ) @ A @ A @ rep_Integ @ ( id @ A )
@ ( product_case_prod @ nat @ nat @ A
@ ^ [I3: nat,J: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I3 ) @ ( semiring_1_of_nat @ A @ J ) ) ) ) ) ) ).
% ring_1_class.of_int_def
thf(fact_1972_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 )
@ ^ [X4: nat,Y5: nat] : ( product_Pair @ nat @ nat @ Y5 @ X4 ) )
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [X4: nat,Y5: nat] : ( product_Pair @ nat @ nat @ Y5 @ X4 ) ) ) ).
% uminus_int.rsp
thf(fact_1973_nat_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ nat @ nat @ intrel
@ ^ [Y2: nat,Z2: nat] : Y2 = Z2
@ ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) )
@ ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) ) ) ).
% nat.rsp
thf(fact_1974_of__int_Orsp,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ A @ A @ intrel
@ ^ [Y2: A,Z2: A] : Y2 = Z2
@ ( product_case_prod @ nat @ nat @ A
@ ^ [I3: nat,J: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I3 ) @ ( semiring_1_of_nat @ A @ J ) ) )
@ ( product_case_prod @ nat @ nat @ A
@ ^ [I3: nat,J: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I3 ) @ ( semiring_1_of_nat @ A @ J ) ) ) ) ) ).
% of_int.rsp
thf(fact_1975_intrel__def,axiom,
( intrel
= ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X4: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U: nat,V3: nat] :
( ( plus_plus @ nat @ X4 @ V3 )
= ( plus_plus @ nat @ U @ Y5 ) ) ) ) ) ).
% intrel_def
thf(fact_1976_of__int_Oabs__eq,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X2: product_prod @ nat @ nat] :
( ( ring_1_of_int @ A @ ( abs_Integ @ X2 ) )
= ( product_case_prod @ nat @ nat @ A
@ ^ [I3: nat,J: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I3 ) @ ( semiring_1_of_nat @ A @ J ) )
@ X2 ) ) ) ).
% of_int.abs_eq
thf(fact_1977_of__int_Orep__eq,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_of_int @ A )
= ( ^ [X4: int] :
( product_case_prod @ nat @ nat @ A
@ ^ [I3: nat,J: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I3 ) @ ( semiring_1_of_nat @ A @ J ) )
@ ( rep_Integ @ X4 ) ) ) ) ) ).
% of_int.rep_eq
thf(fact_1978_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
@ ^ [Y2: $o,Z2: $o] : Y2 = Z2 )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X4: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U: nat,V3: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X4 @ V3 ) @ ( plus_plus @ nat @ U @ Y5 ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X4: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U: nat,V3: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X4 @ V3 ) @ ( plus_plus @ nat @ U @ Y5 ) ) ) ) ) ).
% less_int.rsp
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 ) )
@ ^ [X4: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U: nat,V3: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ U ) @ ( times_times @ nat @ Y5 @ V3 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ V3 ) @ ( times_times @ nat @ Y5 @ U ) ) ) ) ) ) ) ).
% 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 ) )
@ ^ [X4: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U: nat,V3: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ V3 ) @ ( plus_plus @ nat @ Y5 @ U ) ) ) ) ) ) ).
% 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 ) )
@ ^ [X4: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U: nat,V3: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ U ) @ ( plus_plus @ nat @ Y5 @ V3 ) ) ) ) ) ) ).
% plus_int_def
thf(fact_1982_antisymp__antisym__eq,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( antisymp @ A
@ ^ [X4: A,Y5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R ) )
= ( antisym @ A @ R ) ) ).
% antisymp_antisym_eq
thf(fact_1983_o__prs_I2_J,axiom,
! [F2: $tType,E: $tType,C: $tType,D: $tType,A: $tType,B: $tType,H5: $tType,G2: $tType,R1: A > A > $o,Abs1: A > B,Rep1: B > A,R22: C > C > $o,Abs22: C > D,Rep22: D > C,R32: E > E > $o,Abs32: E > F2,Rep32: F2 > E] :
( ( quotient3 @ A @ B @ R1 @ Abs1 @ Rep1 )
=> ( ( quotient3 @ C @ D @ R22 @ Abs22 @ Rep22 )
=> ( ( quotient3 @ E @ F2 @ R32 @ Abs32 @ Rep32 )
=> ( ( map_fun @ ( G2 > H5 ) @ ( G2 > H5 ) @ ( ( A > G2 ) > A > H5 ) @ ( ( B > G2 ) > B > H5 ) @ ( id @ ( G2 > H5 ) ) @ ( map_fun @ ( B > G2 ) @ ( A > G2 ) @ ( A > H5 ) @ ( B > H5 ) @ ( map_fun @ A @ B @ G2 @ G2 @ Abs1 @ ( id @ G2 ) ) @ ( map_fun @ B @ A @ H5 @ H5 @ Rep1 @ ( id @ H5 ) ) ) @ ( comp @ G2 @ H5 @ A ) )
= ( comp @ G2 @ H5 @ B ) ) ) ) ) ).
% o_prs(2)
thf(fact_1984_rat__less__code,axiom,
( ( ord_less @ rat )
= ( ^ [P3: rat,Q6: rat] :
( product_case_prod @ int @ int @ $o
@ ^ [A7: int,C6: int] :
( product_case_prod @ int @ int @ $o
@ ^ [B5: int,D5: int] : ( ord_less @ int @ ( times_times @ int @ A7 @ D5 ) @ ( times_times @ int @ C6 @ B5 ) )
@ ( quotient_of @ Q6 ) )
@ ( quotient_of @ P3 ) ) ) ) ).
% rat_less_code
thf(fact_1985_rat__less__eq__code,axiom,
( ( ord_less_eq @ rat )
= ( ^ [P3: rat,Q6: rat] :
( product_case_prod @ int @ int @ $o
@ ^ [A7: int,C6: int] :
( product_case_prod @ int @ int @ $o
@ ^ [B5: int,D5: int] : ( ord_less_eq @ int @ ( times_times @ int @ A7 @ D5 ) @ ( times_times @ int @ C6 @ B5 ) )
@ ( quotient_of @ Q6 ) )
@ ( quotient_of @ P3 ) ) ) ) ).
% rat_less_eq_code
thf(fact_1986_rat__uminus__code,axiom,
! [P: rat] :
( ( quotient_of @ ( uminus_uminus @ rat @ P ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A7: int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ A7 ) )
@ ( quotient_of @ P ) ) ) ).
% rat_uminus_code
thf(fact_1987_abs__o__rep,axiom,
! [A: $tType,B: $tType,R3: A > A > $o,Abs: A > B,Rep: B > A] :
( ( quotient3 @ A @ B @ R3 @ Abs @ Rep )
=> ( ( comp @ A @ B @ B @ Abs @ Rep )
= ( id @ B ) ) ) ).
% abs_o_rep
thf(fact_1988_o__prs_I1_J,axiom,
! [C: $tType,E: $tType,A: $tType,B: $tType,F2: $tType,D: $tType,R1: A > A > $o,Abs1: A > B,Rep1: B > A,R22: C > C > $o,Abs22: C > D,Rep22: D > C,R32: E > E > $o,Abs32: E > F2,Rep32: F2 > E] :
( ( quotient3 @ A @ B @ R1 @ Abs1 @ Rep1 )
=> ( ( quotient3 @ C @ D @ R22 @ Abs22 @ Rep22 )
=> ( ( quotient3 @ E @ F2 @ R32 @ Abs32 @ Rep32 )
=> ( ( map_fun @ ( D > F2 ) @ ( C > E ) @ ( ( A > C ) > A > E ) @ ( ( B > D ) > B > F2 ) @ ( map_fun @ C @ D @ F2 @ E @ Abs22 @ Rep32 ) @ ( map_fun @ ( B > D ) @ ( A > C ) @ ( A > E ) @ ( B > F2 ) @ ( map_fun @ A @ B @ D @ C @ Abs1 @ Rep22 ) @ ( map_fun @ B @ A @ E @ F2 @ Rep1 @ Abs32 ) ) @ ( comp @ C @ E @ A ) )
= ( comp @ D @ F2 @ B ) ) ) ) ) ).
% o_prs(1)
thf(fact_1989_OOO__eq__quotient3,axiom,
! [A: $tType,B: $tType,C: $tType,R1: A > A > $o,Abs1: A > B,Rep1: B > A,Abs22: B > C,Rep22: C > B] :
( ( quotient3 @ A @ B @ R1 @ Abs1 @ Rep1 )
=> ( ( quotient3 @ B @ C
@ ^ [Y2: B,Z2: B] : Y2 = Z2
@ Abs22
@ Rep22 )
=> ( quotient3 @ A @ C
@ ( relcompp @ A @ A @ A @ R1
@ ( relcompp @ A @ A @ A
@ ^ [Y2: A,Z2: A] : Y2 = Z2
@ R1 ) )
@ ( comp @ B @ C @ A @ Abs22 @ Abs1 )
@ ( comp @ B @ A @ C @ Rep1 @ Rep22 ) ) ) ) ).
% OOO_eq_quotient3
thf(fact_1990_OOO__quotient3,axiom,
! [A: $tType,B: $tType,C: $tType,R1: A > A > $o,Abs1: A > B,Rep1: B > A,R22: B > B > $o,Abs22: B > C,Rep22: C > B,R26: A > A > $o] :
( ( quotient3 @ A @ B @ R1 @ Abs1 @ Rep1 )
=> ( ( quotient3 @ B @ C @ R22 @ Abs22 @ Rep22 )
=> ( ! [X: A,Y: A] :
( ( R26 @ X @ Y )
=> ( ( R1 @ X @ X )
=> ( ( R1 @ Y @ Y )
=> ( R22 @ ( Abs1 @ X ) @ ( Abs1 @ Y ) ) ) ) )
=> ( ! [X: B,Y: B] :
( ( R22 @ X @ Y )
=> ( R26 @ ( Rep1 @ X ) @ ( Rep1 @ Y ) ) )
=> ( quotient3 @ A @ C @ ( relcompp @ A @ A @ A @ R1 @ ( relcompp @ A @ A @ A @ R26 @ R1 ) ) @ ( comp @ B @ C @ A @ Abs22 @ Abs1 ) @ ( comp @ B @ A @ C @ Rep1 @ Rep22 ) ) ) ) ) ) ).
% OOO_quotient3
thf(fact_1991_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 )
@ ^ [X4: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U: nat,V3: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X4 @ V3 ) @ ( plus_plus @ nat @ U @ Y5 ) ) ) ) ) ) ).
% less_int_def
thf(fact_1992_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 )
@ ^ [X4: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U: nat,V3: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X4 @ V3 ) @ ( plus_plus @ nat @ U @ Y5 ) ) ) ) ) ) ).
% less_eq_int_def
thf(fact_1993_rat__minus__code,axiom,
! [P: rat,Q: rat] :
( ( quotient_of @ ( minus_minus @ rat @ P @ Q ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A7: int,C6: 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 @ C6 ) ) @ ( times_times @ int @ C6 @ D5 ) ) )
@ ( quotient_of @ Q ) )
@ ( quotient_of @ P ) ) ) ).
% rat_minus_code
thf(fact_1994_rat__plus__code,axiom,
! [P: rat,Q: rat] :
( ( quotient_of @ ( plus_plus @ rat @ P @ Q ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A7: int,C6: 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 @ C6 ) ) @ ( times_times @ int @ C6 @ D5 ) ) )
@ ( quotient_of @ Q ) )
@ ( quotient_of @ P ) ) ) ).
% rat_plus_code
thf(fact_1995_rat__divide__code,axiom,
! [P: rat,Q: rat] :
( ( quotient_of @ ( divide_divide @ rat @ P @ Q ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A7: int,C6: 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 @ C6 @ B5 ) ) )
@ ( quotient_of @ Q ) )
@ ( quotient_of @ P ) ) ) ).
% rat_divide_code
thf(fact_1996_rat__times__code,axiom,
! [P: rat,Q: rat] :
( ( quotient_of @ ( times_times @ rat @ P @ Q ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A7: int,C6: 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 @ C6 @ D5 ) ) )
@ ( quotient_of @ Q ) )
@ ( quotient_of @ P ) ) ) ).
% rat_times_code
thf(fact_1997_ID_Oin__rel,axiom,
! [B: $tType,A: $tType] :
( ( bNF_id_bnf @ ( A > B > $o ) )
= ( ^ [R6: A > B > $o,A7: A,B5: B] :
? [Z5: product_prod @ A @ B] :
( ( member @ ( product_prod @ A @ B ) @ Z5
@ ( collect @ ( product_prod @ A @ B )
@ ^ [X4: product_prod @ A @ B] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( insert @ ( product_prod @ A @ B ) @ X4 @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) ) ) )
& ( ( bNF_id_bnf @ ( ( product_prod @ A @ B ) > A ) @ ( product_fst @ A @ B ) @ Z5 )
= A7 )
& ( ( bNF_id_bnf @ ( ( product_prod @ A @ B ) > B ) @ ( product_snd @ A @ B ) @ Z5 )
= B5 ) ) ) ) ).
% ID.in_rel
thf(fact_1998_execute__map__entry_I1_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,H: heap_ext @ product_unit,A3: array @ A,F: A > A] :
( ( ord_less @ nat @ I @ ( length @ A @ H @ A3 ) )
=> ( ( heap_execute @ ( array @ A ) @ ( map_entry @ A @ I @ F @ A3 ) @ H )
= ( some @ ( product_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) ) @ ( product_Pair @ ( array @ A ) @ ( heap_ext @ product_unit ) @ A3 @ ( update @ A @ A3 @ I @ ( F @ ( nth3 @ A @ ( get @ A @ H @ A3 ) @ I ) ) @ H ) ) ) ) ) ) ).
% execute_map_entry(1)
thf(fact_1999_foldr__fold,axiom,
! [B: $tType,A: $tType,Xs: list @ A,F: A > B > B] :
( ! [X: A,Y: A] :
( ( member @ A @ X @ ( set3 @ A @ Xs ) )
=> ( ( member @ A @ Y @ ( set3 @ A @ Xs ) )
=> ( ( comp @ B @ B @ B @ ( F @ Y ) @ ( F @ X ) )
= ( comp @ B @ B @ B @ ( F @ X ) @ ( F @ Y ) ) ) ) )
=> ( ( foldr @ A @ B @ F @ Xs )
= ( fold @ A @ B @ F @ Xs ) ) ) ).
% foldr_fold
thf(fact_2000_ID_Opred__map,axiom,
! [B: $tType,A: $tType,Q2: B > $o,F: A > B,X2: A] :
( ( bNF_id_bnf @ ( B > $o ) @ Q2 @ ( bNF_id_bnf @ ( A > B ) @ F @ X2 ) )
= ( bNF_id_bnf @ ( A > $o ) @ ( comp @ B @ $o @ A @ Q2 @ F ) @ X2 ) ) ).
% ID.pred_map
thf(fact_2001_execute__heap,axiom,
! [A: $tType,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( heap_ext @ product_unit ) )] :
( ( heap_execute @ A @ ( heap_heap @ A @ F ) )
= ( comp @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) @ ( option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) @ ( heap_ext @ product_unit ) @ ( some @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) @ F ) ) ).
% execute_heap
thf(fact_2002_execute__guard_I2_J,axiom,
! [A: $tType,P2: ( heap_ext @ product_unit ) > $o,H: heap_ext @ product_unit,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( heap_ext @ product_unit ) )] :
( ( P2 @ H )
=> ( ( heap_execute @ A @ ( heap_guard @ A @ P2 @ F ) @ H )
= ( some @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) @ ( F @ H ) ) ) ) ).
% execute_guard(2)
thf(fact_2003_Heap__eqI,axiom,
! [A: $tType,F: heap_Heap @ A,G: heap_Heap @ A] :
( ! [H8: heap_ext @ product_unit] :
( ( heap_execute @ A @ F @ H8 )
= ( heap_execute @ A @ G @ H8 ) )
=> ( F = G ) ) ).
% Heap_eqI
thf(fact_2004_effect__def,axiom,
! [A: $tType] :
( ( heap_effect @ A )
= ( ^ [C6: heap_Heap @ A,H6: heap_ext @ product_unit,H9: heap_ext @ product_unit,R5: A] :
( ( heap_execute @ A @ C6 @ H6 )
= ( some @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) @ ( product_Pair @ A @ ( heap_ext @ product_unit ) @ R5 @ H9 ) ) ) ) ) ).
% effect_def
thf(fact_2005_effectI,axiom,
! [A: $tType,C3: heap_Heap @ A,H: heap_ext @ product_unit,R: A,H2: heap_ext @ product_unit] :
( ( ( heap_execute @ A @ C3 @ H )
= ( some @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) @ ( product_Pair @ A @ ( heap_ext @ product_unit ) @ R @ H2 ) ) )
=> ( heap_effect @ A @ C3 @ H @ H2 @ R ) ) ).
% effectI
thf(fact_2006_successE,axiom,
! [A: $tType,F: heap_Heap @ A,H: heap_ext @ product_unit] :
( ( heap_success @ A @ F @ H )
=> ~ ! [R4: A,H4: heap_ext @ product_unit] :
( ( heap_execute @ A @ F @ H )
!= ( some @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) @ ( product_Pair @ A @ ( heap_ext @ product_unit ) @ R4 @ H4 ) ) ) ) ).
% successE
thf(fact_2007_execute__bind__eq__SomeI,axiom,
! [A: $tType,B: $tType,F: heap_Heap @ A,H: heap_ext @ product_unit,X2: A,H2: heap_ext @ product_unit,G: A > ( heap_Heap @ B ),Y3: B,H3: heap_ext @ product_unit] :
( ( ( heap_execute @ A @ F @ H )
= ( some @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) @ ( product_Pair @ A @ ( heap_ext @ product_unit ) @ X2 @ H2 ) ) )
=> ( ( ( heap_execute @ B @ ( G @ X2 ) @ H2 )
= ( some @ ( product_prod @ B @ ( heap_ext @ product_unit ) ) @ ( product_Pair @ B @ ( heap_ext @ product_unit ) @ Y3 @ H3 ) ) )
=> ( ( heap_execute @ B @ ( heap_bind @ A @ B @ F @ G ) @ H )
= ( some @ ( product_prod @ B @ ( heap_ext @ product_unit ) ) @ ( product_Pair @ B @ ( heap_ext @ product_unit ) @ Y3 @ H3 ) ) ) ) ) ).
% execute_bind_eq_SomeI
thf(fact_2008_execute__bind_I1_J,axiom,
! [A: $tType,B: $tType,F: heap_Heap @ A,H: heap_ext @ product_unit,X2: A,H2: heap_ext @ product_unit,G: A > ( heap_Heap @ B )] :
( ( ( heap_execute @ A @ F @ H )
= ( some @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) @ ( product_Pair @ A @ ( heap_ext @ product_unit ) @ X2 @ H2 ) ) )
=> ( ( heap_execute @ B @ ( heap_bind @ A @ B @ F @ G ) @ H )
= ( heap_execute @ B @ ( G @ X2 ) @ H2 ) ) ) ).
% execute_bind(1)
thf(fact_2009_execute__return,axiom,
! [A: $tType,X2: A] :
( ( heap_execute @ A @ ( heap_return @ A @ X2 ) )
= ( comp @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) @ ( option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) @ ( heap_ext @ product_unit ) @ ( some @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) @ ( product_Pair @ A @ ( heap_ext @ product_unit ) @ X2 ) ) ) ).
% execute_return
thf(fact_2010_execute__tap,axiom,
! [A: $tType,F: ( heap_ext @ product_unit ) > A,H: heap_ext @ product_unit] :
( ( heap_execute @ A @ ( heap_tap @ A @ F ) @ H )
= ( some @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) @ ( product_Pair @ A @ ( heap_ext @ product_unit ) @ ( F @ H ) @ H ) ) ) ).
% execute_tap
thf(fact_2011_execute__of__list,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [Xs: list @ A,H: heap_ext @ product_unit] :
( ( heap_execute @ ( array @ A ) @ ( of_list @ A @ Xs ) @ H )
= ( some @ ( product_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) ) @ ( alloc @ A @ Xs @ H ) ) ) ) ).
% execute_of_list
thf(fact_2012_success__bind__executeI,axiom,
! [A: $tType,B: $tType,F: heap_Heap @ A,H: heap_ext @ product_unit,X2: A,H2: heap_ext @ product_unit,G: A > ( heap_Heap @ B )] :
( ( ( heap_execute @ A @ F @ H )
= ( some @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) @ ( product_Pair @ A @ ( heap_ext @ product_unit ) @ X2 @ H2 ) ) )
=> ( ( heap_success @ B @ ( G @ X2 ) @ H2 )
=> ( heap_success @ B @ ( heap_bind @ A @ B @ F @ G ) @ H ) ) ) ).
% success_bind_executeI
thf(fact_2013_execute__new,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [N5: nat,X2: A,H: heap_ext @ product_unit] :
( ( heap_execute @ ( array @ A ) @ ( new @ A @ N5 @ X2 ) @ H )
= ( some @ ( product_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) ) @ ( alloc @ A @ ( replicate @ A @ N5 @ X2 ) @ H ) ) ) ) ).
% execute_new
thf(fact_2014_execute__lookup,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R: ref @ A,H: heap_ext @ product_unit] :
( ( heap_execute @ A @ ( lookup @ A @ R ) @ H )
= ( some @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) @ ( product_Pair @ A @ ( heap_ext @ product_unit ) @ ( get2 @ A @ H @ R ) @ H ) ) ) ) ).
% execute_lookup
thf(fact_2015_execute__change,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [F: A > A,R: ref @ A,H: heap_ext @ product_unit] :
( ( heap_execute @ A @ ( change @ A @ F @ R ) @ H )
= ( some @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) @ ( product_Pair @ A @ ( heap_ext @ product_unit ) @ ( F @ ( get2 @ A @ H @ R ) ) @ ( set4 @ A @ R @ ( F @ ( get2 @ A @ H @ R ) ) @ H ) ) ) ) ) ).
% execute_change
thf(fact_2016_foldr__Cons,axiom,
! [B: $tType,A: $tType,F: A > B > B,X2: A,Xs: list @ A] :
( ( foldr @ A @ B @ F @ ( cons @ A @ X2 @ Xs ) )
= ( comp @ B @ B @ B @ ( F @ X2 ) @ ( foldr @ A @ B @ F @ Xs ) ) ) ).
% foldr_Cons
thf(fact_2017_execute__make,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [N5: nat,F: nat > A,H: heap_ext @ product_unit] :
( ( heap_execute @ ( array @ A ) @ ( make @ A @ N5 @ F ) @ H )
= ( some @ ( product_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) ) @ ( alloc @ A @ ( map @ nat @ A @ F @ ( upt @ ( zero_zero @ nat ) @ N5 ) ) @ H ) ) ) ) ).
% execute_make
thf(fact_2018_foldr__map,axiom,
! [C: $tType,B: $tType,A: $tType,G: B > A > A,F: C > B,Xs: list @ C,A3: A] :
( ( foldr @ B @ A @ G @ ( map @ C @ B @ F @ Xs ) @ A3 )
= ( foldr @ C @ A @ ( comp @ B @ ( A > A ) @ C @ G @ F ) @ Xs @ A3 ) ) ).
% foldr_map
thf(fact_2019_execute__upd_I1_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,H: heap_ext @ product_unit,A3: array @ A,X2: A] :
( ( ord_less @ nat @ I @ ( length @ A @ H @ A3 ) )
=> ( ( heap_execute @ ( array @ A ) @ ( upd @ A @ I @ X2 @ A3 ) @ H )
= ( some @ ( product_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) ) @ ( product_Pair @ ( array @ A ) @ ( heap_ext @ product_unit ) @ A3 @ ( update @ A @ A3 @ I @ X2 @ H ) ) ) ) ) ) ).
% execute_upd(1)
thf(fact_2020_execute__nth_I1_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,H: heap_ext @ product_unit,A3: array @ A] :
( ( ord_less @ nat @ I @ ( length @ A @ H @ A3 ) )
=> ( ( heap_execute @ A @ ( nth @ A @ A3 @ I ) @ H )
= ( some @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) @ ( product_Pair @ A @ ( heap_ext @ product_unit ) @ ( nth3 @ A @ ( get @ A @ H @ A3 ) @ I ) @ H ) ) ) ) ) ).
% execute_nth(1)
thf(fact_2021_execute__swap_I1_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,H: heap_ext @ product_unit,A3: array @ A,X2: A] :
( ( ord_less @ nat @ I @ ( length @ A @ H @ A3 ) )
=> ( ( heap_execute @ A @ ( swap @ A @ I @ X2 @ A3 ) @ H )
= ( some @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) @ ( product_Pair @ A @ ( heap_ext @ product_unit ) @ ( nth3 @ A @ ( get @ A @ H @ A3 ) @ I ) @ ( update @ A @ A3 @ I @ X2 @ H ) ) ) ) ) ) ).
% execute_swap(1)
thf(fact_2022_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_2023_image__map__upd,axiom,
! [B: $tType,A: $tType,X2: A,A6: set @ A,M4: A > ( option @ B ),Y3: B] :
( ~ ( member @ A @ X2 @ A6 )
=> ( ( image2 @ A @ ( option @ B ) @ ( fun_upd @ A @ ( option @ B ) @ M4 @ X2 @ ( some @ B @ Y3 ) ) @ A6 )
= ( image2 @ A @ ( option @ B ) @ M4 @ A6 ) ) ) ).
% image_map_upd
thf(fact_2024_ctor__rec__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,R3: A > C > $o,S5: B > D > $o] : ( bNF_rel_fun @ ( A > B ) @ ( C > D ) @ ( A > B ) @ ( C > D ) @ ( bNF_rel_fun @ A @ C @ B @ D @ ( bNF_vimage2p @ A @ A @ C @ C @ $o @ ( bNF_id_bnf @ A ) @ ( bNF_id_bnf @ C ) @ R3 ) @ S5 ) @ ( bNF_rel_fun @ A @ C @ B @ D @ R3 @ S5 ) @ ( basic_BNF_ctor_rec @ ( A > B ) ) @ ( basic_BNF_ctor_rec @ ( C > D ) ) ) ).
% ctor_rec_transfer
thf(fact_2025_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_2026_execute__assert_I1_J,axiom,
! [A: $tType,P2: A > $o,X2: A,H: heap_ext @ product_unit] :
( ( P2 @ X2 )
=> ( ( heap_execute @ A @ ( heap_assert @ A @ P2 @ X2 ) @ H )
= ( some @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) @ ( product_Pair @ A @ ( heap_ext @ product_unit ) @ X2 @ H ) ) ) ) ).
% execute_assert(1)
thf(fact_2027_execute__bind__success,axiom,
! [B: $tType,A: $tType,F: heap_Heap @ A,H: heap_ext @ product_unit,G: A > ( heap_Heap @ B )] :
( ( heap_success @ A @ F @ H )
=> ( ( heap_execute @ B @ ( heap_bind @ A @ B @ F @ G ) @ H )
= ( heap_execute @ B @ ( G @ ( product_fst @ A @ ( heap_ext @ product_unit ) @ ( the2 @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) @ ( heap_execute @ A @ F @ H ) ) ) ) @ ( product_snd @ A @ ( heap_ext @ product_unit ) @ ( the2 @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) @ ( heap_execute @ A @ F @ H ) ) ) ) ) ) ).
% execute_bind_success
thf(fact_2028_effectE,axiom,
! [A: $tType,C3: heap_Heap @ A,H: heap_ext @ product_unit,H2: heap_ext @ product_unit,R: A] :
( ( heap_effect @ A @ C3 @ H @ H2 @ R )
=> ~ ( ( R
= ( product_fst @ A @ ( heap_ext @ product_unit ) @ ( the2 @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) @ ( heap_execute @ A @ C3 @ H ) ) ) )
=> ( ( H2
= ( product_snd @ A @ ( heap_ext @ product_unit ) @ ( the2 @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) @ ( heap_execute @ A @ C3 @ H ) ) ) )
=> ~ ( heap_success @ A @ C3 @ H ) ) ) ) ).
% effectE
thf(fact_2029_comp__the__Some,axiom,
! [A: $tType] :
( ( comp @ ( option @ A ) @ A @ A @ ( the2 @ A ) @ ( some @ A ) )
= ( id @ A ) ) ).
% comp_the_Some
thf(fact_2030_execute__ref,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [V: A,H: heap_ext @ product_unit] :
( ( heap_execute @ ( ref @ A ) @ ( ref3 @ A @ V ) @ H )
= ( some @ ( product_prod @ ( ref @ A ) @ ( heap_ext @ product_unit ) ) @ ( alloc2 @ A @ V @ H ) ) ) ) ).
% execute_ref
thf(fact_2031_execute__len,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [A3: array @ A,H: heap_ext @ product_unit] :
( ( heap_execute @ nat @ ( len @ A @ A3 ) @ H )
= ( some @ ( product_prod @ nat @ ( heap_ext @ product_unit ) ) @ ( product_Pair @ nat @ ( heap_ext @ product_unit ) @ ( length @ A @ H @ A3 ) @ H ) ) ) ) ).
% execute_len
thf(fact_2032_execute__freeze,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [A3: array @ A,H: heap_ext @ product_unit] :
( ( heap_execute @ ( list @ A ) @ ( freeze @ A @ A3 ) @ H )
= ( some @ ( product_prod @ ( list @ A ) @ ( heap_ext @ product_unit ) ) @ ( product_Pair @ ( list @ A ) @ ( heap_ext @ product_unit ) @ ( get @ A @ H @ A3 ) @ H ) ) ) ) ).
% execute_freeze
thf(fact_2033_Basic__BNF__LFPs_Octor__rec__def,axiom,
! [A: $tType] :
( ( basic_BNF_ctor_rec @ A )
= ( ^ [X4: A] : X4 ) ) ).
% Basic_BNF_LFPs.ctor_rec_def
thf(fact_2034_effect__assertI,axiom,
! [A: $tType,P2: A > $o,X2: A,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,R: A] :
( ( P2 @ X2 )
=> ( ( H2 = H )
=> ( ( R = X2 )
=> ( heap_effect @ A @ ( heap_assert @ A @ P2 @ X2 ) @ H @ H2 @ R ) ) ) ) ).
% effect_assertI
thf(fact_2035_effect__assertE,axiom,
! [A: $tType,P2: A > $o,X2: A,H: heap_ext @ product_unit,H2: heap_ext @ product_unit,R: A] :
( ( heap_effect @ A @ ( heap_assert @ A @ P2 @ X2 ) @ H @ H2 @ R )
=> ~ ( ( P2 @ X2 )
=> ( ( R = X2 )
=> ( H2 != H ) ) ) ) ).
% effect_assertE
thf(fact_2036_success__assertI,axiom,
! [A: $tType,P2: A > $o,X2: A,H: heap_ext @ product_unit] :
( ( P2 @ X2 )
=> ( heap_success @ A @ ( heap_assert @ A @ P2 @ X2 ) @ H ) ) ).
% success_assertI
thf(fact_2037_assert__cong,axiom,
! [B: $tType,A: $tType,P2: A > $o,P7: A > $o,F: A > ( heap_Heap @ B ),F5: A > ( heap_Heap @ B ),X2: A] :
( ( P2 = P7 )
=> ( ! [X: A] :
( ( P7 @ X )
=> ( ( F @ X )
= ( F5 @ X ) ) )
=> ( ( heap_bind @ A @ B @ ( heap_assert @ A @ P2 @ X2 ) @ F )
= ( heap_bind @ A @ B @ ( heap_assert @ A @ P7 @ X2 ) @ F5 ) ) ) ) ).
% assert_cong
thf(fact_2038_Basic__BNF__LFPs_Octor__rec__o__map,axiom,
! [B: $tType,C: $tType,A: $tType,F: C > B,G: A > C] :
( ( comp @ C @ B @ A @ ( basic_BNF_ctor_rec @ ( C > B ) @ F ) @ G )
= ( basic_BNF_ctor_rec @ ( A > B ) @ ( comp @ C @ B @ A @ F @ ( comp @ A @ C @ A @ ( comp @ C @ C @ A @ ( bNF_id_bnf @ C ) @ G ) @ ( bNF_id_bnf @ A ) ) ) ) ) ).
% Basic_BNF_LFPs.ctor_rec_o_map
thf(fact_2039_Basic__BNF__LFPs_Octor__rec__def__alt,axiom,
! [B: $tType,A: $tType,F: A > B] :
( F
= ( basic_BNF_ctor_rec @ ( A > B ) @ ( comp @ A @ B @ A @ F @ ( bNF_id_bnf @ A ) ) ) ) ).
% Basic_BNF_LFPs.ctor_rec_def_alt
thf(fact_2040_execute__fold__map__unchanged__heap,axiom,
! [A: $tType,B: $tType,Xs: list @ A,F: A > ( heap_Heap @ B ),H: heap_ext @ product_unit] :
( ! [X: A] :
( ( member @ A @ X @ ( set3 @ A @ Xs ) )
=> ? [Y7: B] :
( ( heap_execute @ B @ ( F @ X ) @ H )
= ( some @ ( product_prod @ B @ ( heap_ext @ product_unit ) ) @ ( product_Pair @ B @ ( heap_ext @ product_unit ) @ Y7 @ H ) ) ) )
=> ( ( heap_execute @ ( list @ B ) @ ( heap_fold_map @ A @ B @ F @ Xs ) @ H )
= ( some @ ( product_prod @ ( list @ B ) @ ( heap_ext @ product_unit ) )
@ ( product_Pair @ ( list @ B ) @ ( heap_ext @ product_unit )
@ ( map @ A @ B
@ ^ [X4: A] : ( product_fst @ B @ ( heap_ext @ product_unit ) @ ( the2 @ ( product_prod @ B @ ( heap_ext @ product_unit ) ) @ ( heap_execute @ B @ ( F @ X4 ) @ H ) ) )
@ Xs )
@ H ) ) ) ) ).
% execute_fold_map_unchanged_heap
thf(fact_2041_map__of__mapk__SomeI,axiom,
! [A: $tType,B: $tType,C: $tType,F: A > B,T3: list @ ( product_prod @ A @ C ),K: A,X2: C] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( ( map_of @ A @ C @ T3 @ K )
= ( some @ C @ X2 ) )
=> ( ( map_of @ B @ C
@ ( map @ ( product_prod @ A @ C ) @ ( product_prod @ B @ C )
@ ( product_case_prod @ A @ C @ ( product_prod @ B @ C )
@ ^ [K3: A] : ( product_Pair @ B @ C @ ( F @ K3 ) ) )
@ T3 )
@ ( F @ K ) )
= ( some @ C @ X2 ) ) ) ) ).
% map_of_mapk_SomeI
thf(fact_2042_Basic__BNF__LFPs_Octor__rec,axiom,
! [B: $tType,A: $tType,G: A > A,F: A > B,X2: A] :
( ( G
= ( id @ A ) )
=> ( ( basic_BNF_ctor_rec @ ( A > B ) @ F @ ( basic_BNF_xtor @ A @ X2 ) )
= ( F @ ( comp @ A @ A @ A @ ( comp @ A @ A @ A @ ( bNF_id_bnf @ A ) @ G ) @ ( bNF_id_bnf @ A ) @ X2 ) ) ) ) ).
% Basic_BNF_LFPs.ctor_rec
thf(fact_2043_ctor__rec__unique,axiom,
! [B: $tType,A: $tType,G: A > A,F: A > B,S2: A > B] :
( ( G
= ( id @ A ) )
=> ( ( ( comp @ A @ B @ A @ F @ ( basic_BNF_xtor @ A ) )
= ( comp @ A @ B @ A @ S2 @ ( comp @ A @ A @ A @ ( comp @ A @ A @ A @ ( bNF_id_bnf @ A ) @ G ) @ ( bNF_id_bnf @ A ) ) ) )
=> ( F
= ( basic_BNF_ctor_rec @ ( A > B ) @ S2 ) ) ) ) ).
% ctor_rec_unique
thf(fact_2044_these__image__Some__eq,axiom,
! [A: $tType,A6: set @ A] :
( ( these @ A @ ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ A6 ) )
= A6 ) ).
% these_image_Some_eq
thf(fact_2045_map__of_Osimps_I2_J,axiom,
! [B: $tType,A: $tType,P: product_prod @ A @ B,Ps: list @ ( product_prod @ A @ B )] :
( ( map_of @ A @ B @ ( cons @ ( product_prod @ A @ B ) @ P @ Ps ) )
= ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ Ps ) @ ( product_fst @ A @ B @ P ) @ ( some @ B @ ( product_snd @ A @ B @ P ) ) ) ) ).
% map_of.simps(2)
thf(fact_2046_execute__map__entry_I2_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [H: heap_ext @ product_unit,A3: array @ A,I: nat,F: A > A] :
( ( ord_less_eq @ nat @ ( length @ A @ H @ A3 ) @ I )
=> ( ( heap_execute @ ( array @ A ) @ ( map_entry @ A @ I @ F @ A3 ) @ H )
= ( none @ ( product_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) ) ) ) ) ) ).
% execute_map_entry(2)
thf(fact_2047_xtor__iff__xtor,axiom,
! [A: $tType,U2: A,W3: A] :
( ( U2
= ( basic_BNF_xtor @ A @ W3 ) )
= ( ( basic_BNF_xtor @ A @ U2 )
= W3 ) ) ).
% xtor_iff_xtor
thf(fact_2048_Basic__BNF__LFPs_Oxtor__inject,axiom,
! [A: $tType,X2: A,Y3: A] :
( ( ( basic_BNF_xtor @ A @ X2 )
= ( basic_BNF_xtor @ A @ Y3 ) )
= ( X2 = Y3 ) ) ).
% Basic_BNF_LFPs.xtor_inject
thf(fact_2049_Basic__BNF__LFPs_Oxtor__induct,axiom,
! [A: $tType,P2: A > $o,Z3: A] :
( ! [X: A] : ( P2 @ ( basic_BNF_xtor @ A @ X ) )
=> ( P2 @ Z3 ) ) ).
% Basic_BNF_LFPs.xtor_induct
thf(fact_2050_Basic__BNF__LFPs_Oxtor__xtor,axiom,
! [A: $tType,X2: A] :
( ( basic_BNF_xtor @ A @ ( basic_BNF_xtor @ A @ X2 ) )
= X2 ) ).
% Basic_BNF_LFPs.xtor_xtor
thf(fact_2051_Basic__BNF__LFPs_Oxtor__set,axiom,
! [A: $tType,B: $tType,F: B > A,X2: B] :
( ( F @ ( basic_BNF_xtor @ B @ X2 ) )
= ( F @ X2 ) ) ).
% Basic_BNF_LFPs.xtor_set
thf(fact_2052_Basic__BNF__LFPs_Oxtor__rel,axiom,
! [B: $tType,A: $tType,C: $tType,R3: B > C > A,X2: B,Y3: C] :
( ( R3 @ ( basic_BNF_xtor @ B @ X2 ) @ ( basic_BNF_xtor @ C @ Y3 ) )
= ( R3 @ X2 @ Y3 ) ) ).
% Basic_BNF_LFPs.xtor_rel
thf(fact_2053_Basic__BNF__LFPs_Oxtor__map,axiom,
! [A: $tType,B: $tType,F: B > A,X2: B] :
( ( F @ ( basic_BNF_xtor @ B @ X2 ) )
= ( basic_BNF_xtor @ A @ ( F @ X2 ) ) ) ).
% Basic_BNF_LFPs.xtor_map
thf(fact_2054_Basic__BNF__LFPs_Oxtor__def,axiom,
! [A: $tType] :
( ( basic_BNF_xtor @ A )
= ( ^ [X4: A] : X4 ) ) ).
% Basic_BNF_LFPs.xtor_def
thf(fact_2055_Option_Othese__def,axiom,
! [A: $tType] :
( ( these @ A )
= ( ^ [A8: set @ ( option @ A )] :
( image2 @ ( option @ A ) @ A @ ( the2 @ A )
@ ( collect @ ( option @ A )
@ ^ [X4: option @ A] :
( ( member @ ( option @ A ) @ X4 @ A8 )
& ( X4
!= ( none @ A ) ) ) ) ) ) ) ).
% Option.these_def
thf(fact_2056_map__of__eq__None__iff,axiom,
! [A: $tType,B: $tType,Xys: list @ ( product_prod @ B @ A ),X2: B] :
( ( ( map_of @ B @ A @ Xys @ X2 )
= ( none @ A ) )
= ( ~ ( member @ B @ X2 @ ( image2 @ ( product_prod @ B @ A ) @ B @ ( product_fst @ B @ A ) @ ( set3 @ ( product_prod @ B @ A ) @ Xys ) ) ) ) ) ).
% map_of_eq_None_iff
thf(fact_2057_xtor__map__unique,axiom,
! [B: $tType,A: $tType,U2: A > B,F: A > B] :
( ( ( comp @ A @ B @ A @ U2 @ ( basic_BNF_xtor @ A ) )
= ( comp @ B @ B @ A @ ( basic_BNF_xtor @ B ) @ F ) )
=> ( U2 = F ) ) ).
% xtor_map_unique
thf(fact_2058_Some__image__these__eq,axiom,
! [A: $tType,A6: set @ ( option @ A )] :
( ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ ( these @ A @ A6 ) )
= ( collect @ ( option @ A )
@ ^ [X4: option @ A] :
( ( member @ ( option @ A ) @ X4 @ A6 )
& ( X4
!= ( none @ A ) ) ) ) ) ).
% Some_image_these_eq
thf(fact_2059_Basic__BNF__LFPs_OPair__def__alt,axiom,
! [B: $tType,A: $tType] :
( ( product_Pair @ A @ B )
= ( ^ [A7: A,B5: B] : ( basic_BNF_xtor @ ( product_prod @ A @ B ) @ ( bNF_id_bnf @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A7 @ B5 ) ) ) ) ) ).
% Basic_BNF_LFPs.Pair_def_alt
thf(fact_2060_fold__map_Osimps_I1_J,axiom,
! [A: $tType,B: $tType,F: A > ( heap_Heap @ B )] :
( ( heap_fold_map @ A @ B @ F @ ( nil @ A ) )
= ( heap_return @ ( list @ B ) @ ( nil @ B ) ) ) ).
% fold_map.simps(1)
thf(fact_2061_success__def,axiom,
! [A: $tType] :
( ( heap_success @ A )
= ( ^ [F4: heap_Heap @ A,H6: heap_ext @ product_unit] :
( ( heap_execute @ A @ F4 @ H6 )
!= ( none @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) ) ) ) ).
% success_def
thf(fact_2062_successI,axiom,
! [A: $tType,F: heap_Heap @ A,H: heap_ext @ product_unit] :
( ( ( heap_execute @ A @ F @ H )
!= ( none @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) )
=> ( heap_success @ A @ F @ H ) ) ).
% successI
thf(fact_2063_execute__bind_I2_J,axiom,
! [A: $tType,B: $tType,F: heap_Heap @ A,H: heap_ext @ product_unit,G: A > ( heap_Heap @ B )] :
( ( ( heap_execute @ A @ F @ H )
= ( none @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) )
=> ( ( heap_execute @ B @ ( heap_bind @ A @ B @ F @ G ) @ H )
= ( none @ ( product_prod @ B @ ( heap_ext @ product_unit ) ) ) ) ) ).
% execute_bind(2)
thf(fact_2064_None__notin__image__Some,axiom,
! [A: $tType,A6: set @ A] :
~ ( member @ ( option @ A ) @ ( none @ A ) @ ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ A6 ) ) ).
% None_notin_image_Some
thf(fact_2065_execute__guard_I1_J,axiom,
! [A: $tType,P2: ( heap_ext @ product_unit ) > $o,H: heap_ext @ product_unit,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( heap_ext @ product_unit ) )] :
( ~ ( P2 @ H )
=> ( ( heap_execute @ A @ ( heap_guard @ A @ P2 @ F ) @ H )
= ( none @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) ) ) ).
% execute_guard(1)
thf(fact_2066_execute__assert_I2_J,axiom,
! [A: $tType,P2: A > $o,X2: A,H: heap_ext @ product_unit] :
( ~ ( P2 @ X2 )
=> ( ( heap_execute @ A @ ( heap_assert @ A @ P2 @ X2 ) @ H )
= ( none @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) ) ) ).
% execute_assert(2)
thf(fact_2067_execute__raise,axiom,
! [A: $tType,S2: literal] :
( ( heap_execute @ A @ ( heap_raise @ A @ S2 ) )
= ( ^ [Uu: heap_ext @ product_unit] : ( none @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) ) ) ).
% execute_raise
thf(fact_2068_fold__map_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,F: A > ( heap_Heap @ B ),X2: A,Xs: list @ A] :
( ( heap_fold_map @ A @ B @ F @ ( cons @ A @ X2 @ Xs ) )
= ( heap_bind @ B @ ( list @ B ) @ ( F @ X2 )
@ ^ [Y5: B] :
( heap_bind @ ( list @ B ) @ ( list @ B ) @ ( heap_fold_map @ A @ B @ F @ Xs )
@ ^ [Ys2: list @ B] : ( heap_return @ ( list @ B ) @ ( cons @ B @ Y5 @ Ys2 ) ) ) ) ) ).
% fold_map.simps(2)
thf(fact_2069_fold__map__append,axiom,
! [B: $tType,A: $tType,F: B > ( heap_Heap @ A ),Xs: list @ B,Ys: list @ B] :
( ( heap_fold_map @ B @ A @ F @ ( append @ B @ Xs @ Ys ) )
= ( heap_bind @ ( list @ A ) @ ( list @ A ) @ ( heap_fold_map @ B @ A @ F @ Xs )
@ ^ [Xs2: list @ A] :
( heap_bind @ ( list @ A ) @ ( list @ A ) @ ( heap_fold_map @ B @ A @ F @ Ys )
@ ^ [Ys2: list @ A] : ( heap_return @ ( list @ A ) @ ( append @ A @ Xs2 @ Ys2 ) ) ) ) ) ).
% fold_map_append
thf(fact_2070_weak__map__of__SomeI,axiom,
! [A: $tType,B: $tType,K: A,X2: B,L: list @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ X2 ) @ ( set3 @ ( product_prod @ A @ B ) @ L ) )
=> ? [X: B] :
( ( map_of @ A @ B @ L @ K )
= ( some @ B @ X ) ) ) ).
% weak_map_of_SomeI
thf(fact_2071_map__of__SomeD,axiom,
! [A: $tType,B: $tType,Xs: list @ ( product_prod @ B @ A ),K: B,Y3: A] :
( ( ( map_of @ B @ A @ Xs @ K )
= ( some @ A @ Y3 ) )
=> ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ K @ Y3 ) @ ( set3 @ ( product_prod @ B @ A ) @ Xs ) ) ) ).
% map_of_SomeD
thf(fact_2072_map__of__eqI,axiom,
! [B: $tType,A: $tType,Xs: list @ ( product_prod @ A @ B ),Ys: list @ ( product_prod @ A @ B )] :
( ( ( set3 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) )
= ( set3 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Ys ) ) )
=> ( ! [X: A] :
( ( member @ A @ X @ ( set3 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) ) )
=> ( ( map_of @ A @ B @ Xs @ X )
= ( map_of @ A @ B @ Ys @ X ) ) )
=> ( ( map_of @ A @ B @ Xs )
= ( map_of @ A @ B @ Ys ) ) ) ) ).
% map_of_eqI
thf(fact_2073_notin__range__Some,axiom,
! [A: $tType,X2: option @ A] :
( ( ~ ( member @ ( option @ A ) @ X2 @ ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ ( top_top @ ( set @ A ) ) ) ) )
= ( X2
= ( none @ A ) ) ) ).
% notin_range_Some
thf(fact_2074_UNIV__option__conv,axiom,
! [A: $tType] :
( ( top_top @ ( set @ ( option @ A ) ) )
= ( insert @ ( option @ A ) @ ( none @ A ) @ ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ).
% UNIV_option_conv
thf(fact_2075_map__of__eq__dom,axiom,
! [B: $tType,A: $tType,Xs: list @ ( product_prod @ A @ B ),Ys: list @ ( product_prod @ A @ B )] :
( ( ( map_of @ A @ B @ Xs )
= ( map_of @ A @ B @ Ys ) )
=> ( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( set3 @ ( product_prod @ A @ B ) @ Xs ) )
= ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( set3 @ ( product_prod @ A @ B ) @ Ys ) ) ) ) ).
% map_of_eq_dom
thf(fact_2076_Heap__cases,axiom,
! [A: $tType,F: heap_Heap @ A,H: heap_ext @ product_unit] :
( ! [X: A,H4: heap_ext @ product_unit] :
( ( heap_execute @ A @ F @ H )
!= ( some @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) @ ( product_Pair @ A @ ( heap_ext @ product_unit ) @ X @ H4 ) ) )
=> ( ( heap_execute @ A @ F @ H )
= ( none @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) ) ) ).
% Heap_cases
thf(fact_2077_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_2078_finite__range__map__of,axiom,
! [A: $tType,B: $tType,Xys: list @ ( product_prod @ B @ A )] : ( finite_finite2 @ ( option @ A ) @ ( image2 @ B @ ( option @ A ) @ ( map_of @ B @ A @ Xys ) @ ( top_top @ ( set @ B ) ) ) ) ).
% finite_range_map_of
thf(fact_2079_execute__nth_I2_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [H: heap_ext @ product_unit,A3: array @ A,I: nat] :
( ( ord_less_eq @ nat @ ( length @ A @ H @ A3 ) @ I )
=> ( ( heap_execute @ A @ ( nth @ A @ A3 @ I ) @ H )
= ( none @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) ) ) ) ).
% execute_nth(2)
thf(fact_2080_execute__swap_I2_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [H: heap_ext @ product_unit,A3: array @ A,I: nat,X2: A] :
( ( ord_less_eq @ nat @ ( length @ A @ H @ A3 ) @ I )
=> ( ( heap_execute @ A @ ( swap @ A @ I @ X2 @ A3 ) @ H )
= ( none @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) ) ) ) ).
% execute_swap(2)
thf(fact_2081_Basic__BNF__LFPs_Oxtor__rel__induct,axiom,
! [B: $tType,A: $tType,R3: A > B > $o,IR: A > B > $o] :
( ! [X: A,Y: B] :
( ( bNF_vimage2p @ A @ A @ B @ B @ $o @ ( bNF_id_bnf @ A ) @ ( bNF_id_bnf @ B ) @ R3 @ X @ Y )
=> ( IR @ ( basic_BNF_xtor @ A @ X ) @ ( basic_BNF_xtor @ B @ Y ) ) )
=> ( ord_less_eq @ ( A > B > $o ) @ R3 @ IR ) ) ).
% Basic_BNF_LFPs.xtor_rel_induct
thf(fact_2082_execute__upd_I2_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [H: heap_ext @ product_unit,A3: array @ A,I: nat,X2: A] :
( ( ord_less_eq @ nat @ ( length @ A @ H @ A3 ) @ I )
=> ( ( heap_execute @ ( array @ A ) @ ( upd @ A @ I @ X2 @ A3 ) @ H )
= ( none @ ( product_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) ) ) ) ) ) ).
% execute_upd(2)
thf(fact_2083_graph__map__upd,axiom,
! [A: $tType,B: $tType,M4: A > ( option @ B ),K: A,V: B] :
( ( graph @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M4 @ K @ ( some @ B @ V ) ) )
= ( insert @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V ) @ ( graph @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M4 @ K @ ( none @ B ) ) ) ) ) ).
% graph_map_upd
thf(fact_2084_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 )
@ ^ [K3: A] : ( product_Pair @ A @ B @ K3 @ ( F @ K3 ) )
@ Ks ) )
= ( restrict_map @ A @ B @ ( comp @ B @ ( option @ B ) @ A @ ( some @ B ) @ F ) @ ( set3 @ A @ Ks ) ) ) ).
% map_of_map_restrict
thf(fact_2085_map__of__is__SomeI,axiom,
! [A: $tType,B: $tType,Xys: list @ ( product_prod @ A @ B ),X2: A,Y3: B] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xys ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ ( set3 @ ( product_prod @ A @ B ) @ Xys ) )
=> ( ( map_of @ A @ B @ Xys @ X2 )
= ( some @ B @ Y3 ) ) ) ) ).
% map_of_is_SomeI
thf(fact_2086_Some__eq__map__of__iff,axiom,
! [B: $tType,A: $tType,Xys: list @ ( product_prod @ A @ B ),Y3: B,X2: A] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xys ) )
=> ( ( ( some @ B @ Y3 )
= ( map_of @ A @ B @ Xys @ X2 ) )
= ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ ( set3 @ ( product_prod @ A @ B ) @ Xys ) ) ) ) ).
% Some_eq_map_of_iff
thf(fact_2087_map__of__eq__Some__iff,axiom,
! [B: $tType,A: $tType,Xys: list @ ( product_prod @ A @ B ),X2: A,Y3: B] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xys ) )
=> ( ( ( map_of @ A @ B @ Xys @ X2 )
= ( some @ B @ Y3 ) )
= ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y3 ) @ ( set3 @ ( product_prod @ A @ B ) @ Xys ) ) ) ) ).
% map_of_eq_Some_iff
thf(fact_2088_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 ) )
=> ( ( set3 @ ( product_prod @ A @ B ) @ Xs )
= ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [K3: A,V3: B] :
( ( map_of @ A @ B @ Xs @ K3 )
= ( some @ B @ V3 ) ) ) ) ) ) ).
% set_map_of_compr
thf(fact_2089_map__of__map__keys,axiom,
! [B: $tType,A: $tType,Xs: list @ A,M4: A > ( option @ B )] :
( ( ( set3 @ A @ Xs )
= ( dom @ A @ B @ M4 ) )
=> ( ( map_of @ A @ B
@ ( map @ A @ ( product_prod @ A @ B )
@ ^ [K3: A] : ( product_Pair @ A @ B @ K3 @ ( the2 @ B @ ( M4 @ K3 ) ) )
@ Xs ) )
= M4 ) ) ).
% map_of_map_keys
thf(fact_2090_graph__restrictD_I1_J,axiom,
! [B: $tType,A: $tType,K: A,V: B,M4: A > ( option @ B ),A6: set @ A] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V ) @ ( graph @ A @ B @ ( restrict_map @ A @ B @ M4 @ A6 ) ) )
=> ( member @ A @ K @ A6 ) ) ).
% graph_restrictD(1)
thf(fact_2091_graph__domD,axiom,
! [B: $tType,A: $tType,X2: product_prod @ A @ B,M4: A > ( option @ B )] :
( ( member @ ( product_prod @ A @ B ) @ X2 @ ( graph @ A @ B @ M4 ) )
=> ( member @ A @ ( product_fst @ A @ B @ X2 ) @ ( dom @ A @ B @ M4 ) ) ) ).
% graph_domD
thf(fact_2092_fst__graph__eq__dom,axiom,
! [B: $tType,A: $tType,M4: A > ( option @ B )] :
( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( graph @ A @ B @ M4 ) )
= ( dom @ A @ B @ M4 ) ) ).
% fst_graph_eq_dom
thf(fact_2093_graph__restrictD_I2_J,axiom,
! [A: $tType,B: $tType,K: A,V: B,M4: A > ( option @ B ),A6: set @ A] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V ) @ ( graph @ A @ B @ ( restrict_map @ A @ B @ M4 @ A6 ) ) )
=> ( ( M4 @ K )
= ( some @ B @ V ) ) ) ).
% graph_restrictD(2)
thf(fact_2094_graph__eq__to__snd__dom,axiom,
! [B: $tType,A: $tType] :
( ( graph @ A @ B )
= ( ^ [M7: A > ( option @ B )] :
( image2 @ A @ ( product_prod @ A @ B )
@ ^ [X4: A] : ( product_Pair @ A @ B @ X4 @ ( the2 @ B @ ( M7 @ X4 ) ) )
@ ( dom @ A @ B @ M7 ) ) ) ) ).
% graph_eq_to_snd_dom
thf(fact_2095_graph__map__of__if__distinct__dom,axiom,
! [B: $tType,A: $tType,Al: list @ ( product_prod @ A @ B )] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Al ) )
=> ( ( graph @ A @ B @ ( map_of @ A @ B @ Al ) )
= ( set3 @ ( product_prod @ A @ B ) @ Al ) ) ) ).
% graph_map_of_if_distinct_dom
thf(fact_2096_in__graphI,axiom,
! [A: $tType,B: $tType,M4: B > ( option @ A ),K: B,V: A] :
( ( ( M4 @ K )
= ( some @ A @ V ) )
=> ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ K @ V ) @ ( graph @ B @ A @ M4 ) ) ) ).
% in_graphI
thf(fact_2097_in__graphD,axiom,
! [A: $tType,B: $tType,K: A,V: B,M4: A > ( option @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V ) @ ( graph @ A @ B @ M4 ) )
=> ( ( M4 @ K )
= ( some @ B @ V ) ) ) ).
% in_graphD
thf(fact_2098_inj__on__fst__graph,axiom,
! [B: $tType,A: $tType,M4: A > ( option @ B )] : ( inj_on @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( graph @ A @ B @ M4 ) ) ).
% inj_on_fst_graph
thf(fact_2099_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 ) @ ( set3 @ ( product_prod @ A @ B ) @ Xs ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V22 ) @ ( set3 @ ( product_prod @ A @ B ) @ Xs ) )
=> ( V1 = V22 ) ) ) ) ).
% eq_key_imp_eq_value
thf(fact_2100_map__of__inject__set,axiom,
! [B: $tType,A: $tType,Xs: list @ ( product_prod @ A @ B ),Ys: list @ ( product_prod @ A @ B )] :
( ( 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 ) @ Ys ) )
=> ( ( ( map_of @ A @ B @ Xs )
= ( map_of @ A @ B @ Ys ) )
= ( ( set3 @ ( product_prod @ A @ B ) @ Xs )
= ( set3 @ ( product_prod @ A @ B ) @ Ys ) ) ) ) ) ).
% map_of_inject_set
thf(fact_2101_graph__def,axiom,
! [B: $tType,A: $tType] :
( ( graph @ A @ B )
= ( ^ [M7: A > ( option @ B )] :
( collect @ ( product_prod @ A @ B )
@ ^ [Uu: product_prod @ A @ B] :
? [A7: A,B5: B] :
( ( Uu
= ( product_Pair @ A @ B @ A7 @ B5 ) )
& ( ( M7 @ A7 )
= ( some @ B @ B5 ) ) ) ) ) ) ).
% graph_def
thf(fact_2102_dom__map__of__conv__image__fst,axiom,
! [B: $tType,A: $tType,Xys: list @ ( product_prod @ A @ B )] :
( ( dom @ A @ B @ ( map_of @ A @ B @ Xys ) )
= ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( set3 @ ( product_prod @ A @ B ) @ Xys ) ) ) ).
% dom_map_of_conv_image_fst
thf(fact_2103_graph__fun__upd__None,axiom,
! [B: $tType,A: $tType,M4: A > ( option @ B ),K: A] :
( ( graph @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M4 @ K @ ( none @ B ) ) )
= ( collect @ ( product_prod @ A @ B )
@ ^ [E5: product_prod @ A @ B] :
( ( member @ ( product_prod @ A @ B ) @ E5 @ ( graph @ A @ B @ M4 ) )
& ( ( product_fst @ A @ B @ E5 )
!= K ) ) ) ) ).
% graph_fun_upd_None
thf(fact_2104_ran__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 ) @ ( set3 @ ( product_prod @ A @ B ) @ Al ) ) ) ) ).
% ran_distinct
thf(fact_2105_finite__range__map__of__map__add,axiom,
! [A: $tType,B: $tType,F: B > ( option @ A ),L: list @ ( product_prod @ B @ A )] :
( ( finite_finite2 @ ( option @ A ) @ ( image2 @ B @ ( option @ A ) @ F @ ( top_top @ ( set @ B ) ) ) )
=> ( finite_finite2 @ ( option @ A ) @ ( image2 @ B @ ( option @ A ) @ ( map_add @ B @ A @ F @ ( map_of @ B @ A @ L ) ) @ ( top_top @ ( set @ B ) ) ) ) ) ).
% finite_range_map_of_map_add
thf(fact_2106_heap__def,axiom,
! [A: $tType] :
( ( heap_heap @ A )
= ( ^ [F4: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( heap_ext @ product_unit ) )] : ( heap_Heap2 @ A @ ( comp @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) @ ( option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) @ ( heap_ext @ product_unit ) @ ( some @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) @ F4 ) ) ) ) ).
% heap_def
thf(fact_2107_execute__bind__case,axiom,
! [A: $tType,B: $tType,F: heap_Heap @ B,G: B > ( heap_Heap @ A ),H: heap_ext @ product_unit] :
( ( heap_execute @ A @ ( heap_bind @ B @ A @ F @ G ) @ H )
= ( case_option @ ( option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) @ ( product_prod @ B @ ( heap_ext @ product_unit ) ) @ ( none @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) )
@ ( product_case_prod @ B @ ( heap_ext @ product_unit ) @ ( option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) )
@ ^ [X4: B] : ( heap_execute @ A @ ( G @ X4 ) ) )
@ ( heap_execute @ B @ F @ H ) ) ) ).
% execute_bind_case
thf(fact_2108_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 )
@ ^ [K3: A,V3: C] : ( product_Pair @ A @ B @ K3 @ ( F @ V3 ) ) )
@ Xs ) )
= ( comp @ ( option @ C ) @ ( option @ B ) @ A @ ( map_option @ C @ B @ F ) @ ( map_of @ A @ C @ Xs ) ) ) ).
% map_of_map
thf(fact_2109_guard__def,axiom,
! [A: $tType] :
( ( heap_guard @ A )
= ( ^ [P4: ( heap_ext @ product_unit ) > $o,F4: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( heap_ext @ product_unit ) )] :
( heap_Heap2 @ A
@ ^ [H6: heap_ext @ product_unit] : ( if @ ( option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) @ ( P4 @ H6 ) @ ( some @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) @ ( F4 @ H6 ) ) @ ( none @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) ) ) ) ) ).
% guard_def
thf(fact_2110_tap__def,axiom,
! [A: $tType] :
( ( heap_tap @ A )
= ( ^ [F4: ( heap_ext @ product_unit ) > A] :
( heap_Heap2 @ A
@ ^ [H6: heap_ext @ product_unit] : ( some @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) @ ( product_Pair @ A @ ( heap_ext @ product_unit ) @ ( F4 @ H6 ) @ H6 ) ) ) ) ) ).
% tap_def
thf(fact_2111_Heap_Oinject,axiom,
! [A: $tType,X2: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ),Ya: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) )] :
( ( ( heap_Heap2 @ A @ X2 )
= ( heap_Heap2 @ A @ Ya ) )
= ( X2 = Ya ) ) ).
% Heap.inject
thf(fact_2112_Heap__execute,axiom,
! [A: $tType,F: heap_Heap @ A] :
( ( heap_Heap2 @ A @ ( heap_execute @ A @ F ) )
= F ) ).
% Heap_execute
thf(fact_2113_case__map__option,axiom,
! [B: $tType,A: $tType,C: $tType,G: A,H: B > A,F: C > B,X2: option @ C] :
( ( case_option @ A @ B @ G @ H @ ( map_option @ C @ B @ F @ X2 ) )
= ( case_option @ A @ C @ G @ ( comp @ B @ A @ C @ H @ F ) @ X2 ) ) ).
% case_map_option
thf(fact_2114_dom__map__option__comp,axiom,
! [B: $tType,C: $tType,A: $tType,G: C > B,M4: A > ( option @ C )] :
( ( dom @ A @ B @ ( comp @ ( option @ C ) @ ( option @ B ) @ A @ ( map_option @ C @ B @ G ) @ M4 ) )
= ( dom @ A @ C @ M4 ) ) ).
% dom_map_option_comp
thf(fact_2115_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 )
@ ^ [X4: A] : ( none @ C ) )
= ( ^ [X4: A] : ( none @ B ) ) ) ).
% map_option_o_empty
thf(fact_2116_map__option__o__map__upd,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > B,M4: A > ( option @ C ),A3: A,B2: C] :
( ( comp @ ( option @ C ) @ ( option @ B ) @ A @ ( map_option @ C @ B @ F ) @ ( fun_upd @ A @ ( option @ C ) @ M4 @ A3 @ ( some @ C @ B2 ) ) )
= ( fun_upd @ A @ ( option @ B ) @ ( comp @ ( option @ C ) @ ( option @ B ) @ A @ ( map_option @ C @ B @ F ) @ M4 ) @ A3 @ ( some @ B @ ( F @ B2 ) ) ) ) ).
% map_option_o_map_upd
thf(fact_2117_ran__map__option,axiom,
! [A: $tType,C: $tType,B: $tType,F: C > A,M4: B > ( option @ C )] :
( ( ran @ B @ A
@ ^ [X4: B] : ( map_option @ C @ A @ F @ ( M4 @ X4 ) ) )
= ( image2 @ C @ A @ F @ ( ran @ B @ C @ M4 ) ) ) ).
% ran_map_option
thf(fact_2118_Heap_Oexhaust,axiom,
! [A: $tType,Y3: heap_Heap @ A] :
~ ! [X: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) )] :
( Y3
!= ( heap_Heap2 @ A @ X ) ) ).
% Heap.exhaust
thf(fact_2119_disjE__realizer2,axiom,
! [B: $tType,A: $tType,P2: $o,Q2: A > $o,X2: option @ A,R3: B > $o,F: B,G: A > B] :
( ( case_option @ $o @ A @ P2 @ Q2 @ X2 )
=> ( ( P2
=> ( R3 @ F ) )
=> ( ! [Q7: A] :
( ( Q2 @ Q7 )
=> ( R3 @ ( G @ Q7 ) ) )
=> ( R3 @ ( case_option @ B @ A @ F @ G @ X2 ) ) ) ) ) ).
% disjE_realizer2
thf(fact_2120_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_2121_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_2122_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_2123_execute_Osimps,axiom,
! [A: $tType,F: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) )] :
( ( heap_execute @ A @ ( heap_Heap2 @ A @ F ) )
= F ) ).
% execute.simps
thf(fact_2124_Heap__Monad_Obind__def,axiom,
! [B: $tType,A: $tType] :
( ( heap_bind @ A @ B )
= ( ^ [F4: heap_Heap @ A,G4: A > ( heap_Heap @ B )] :
( heap_Heap2 @ B
@ ^ [H6: heap_ext @ product_unit] :
( case_option @ ( option @ ( product_prod @ B @ ( heap_ext @ product_unit ) ) ) @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) @ ( none @ ( product_prod @ B @ ( heap_ext @ product_unit ) ) )
@ ( product_case_prod @ A @ ( heap_ext @ product_unit ) @ ( option @ ( product_prod @ B @ ( heap_ext @ product_unit ) ) )
@ ^ [X4: A] : ( heap_execute @ B @ ( G4 @ X4 ) ) )
@ ( heap_execute @ A @ F4 @ H6 ) ) ) ) ) ).
% Heap_Monad.bind_def
thf(fact_2125_graph__ranD,axiom,
! [B: $tType,A: $tType,X2: product_prod @ A @ B,M4: A > ( option @ B )] :
( ( member @ ( product_prod @ A @ B ) @ X2 @ ( graph @ A @ B @ M4 ) )
=> ( member @ B @ ( product_snd @ A @ B @ X2 ) @ ( ran @ A @ B @ M4 ) ) ) ).
% graph_ranD
thf(fact_2126_raise__def,axiom,
! [A: $tType] :
( ( heap_raise @ A )
= ( ^ [S: literal] :
( heap_Heap2 @ A
@ ^ [Uu: heap_ext @ product_unit] : ( none @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) ) ) ) ).
% raise_def
thf(fact_2127_map__add__map__of__foldr,axiom,
! [B: $tType,A: $tType,M4: A > ( option @ B ),Ps: list @ ( product_prod @ A @ B )] :
( ( map_add @ A @ B @ M4 @ ( map_of @ A @ B @ Ps ) )
= ( foldr @ ( product_prod @ A @ B ) @ ( A > ( option @ B ) )
@ ( product_case_prod @ A @ B @ ( ( A > ( option @ B ) ) > A > ( option @ B ) )
@ ^ [K3: A,V3: B,M7: A > ( option @ B )] : ( fun_upd @ A @ ( option @ B ) @ M7 @ K3 @ ( some @ B @ V3 ) ) )
@ Ps
@ M4 ) ) ).
% map_add_map_of_foldr
thf(fact_2128_snd__graph__ran,axiom,
! [A: $tType,B: $tType,M4: B > ( option @ A )] :
( ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( graph @ B @ A @ M4 ) )
= ( ran @ B @ A @ M4 ) ) ).
% snd_graph_ran
thf(fact_2129_Heap_Osize__gen,axiom,
! [A: $tType,Xa2: A > nat,X2: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) )] :
( ( heap_size_Heap @ A @ Xa2 @ ( heap_Heap2 @ A @ X2 ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% Heap.size_gen
thf(fact_2130_Heap_Osize_I2_J,axiom,
! [A: $tType,X2: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) )] :
( ( size_size @ ( heap_Heap @ A ) @ ( heap_Heap2 @ A @ X2 ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% Heap.size(2)
thf(fact_2131_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_2132_Heap__lub__empty,axiom,
! [A: $tType] :
( ( heap_Heap_lub @ A @ ( bot_bot @ ( set @ ( heap_Heap @ A ) ) ) )
= ( heap_Heap2 @ A
@ ^ [X4: heap_ext @ product_unit] : ( none @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) ) ) ).
% Heap_lub_empty
thf(fact_2133_distinct__insort__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,X2: B,Xs: list @ B] :
( ( distinct @ A @ ( map @ B @ A @ F @ ( linorder_insort_key @ B @ A @ F @ X2 @ Xs ) ) )
= ( ~ ( member @ A @ ( F @ X2 ) @ ( image2 @ B @ A @ F @ ( set3 @ B @ Xs ) ) )
& ( distinct @ A @ ( map @ B @ A @ F @ Xs ) ) ) ) ) ).
% distinct_insort_key
thf(fact_2134_map__conv__bind__option,axiom,
! [A: $tType,B: $tType] :
( ( map_option @ B @ A )
= ( ^ [F4: B > A,X4: option @ B] : ( bind2 @ B @ A @ X4 @ ( comp @ A @ ( option @ A ) @ B @ ( some @ A ) @ F4 ) ) ) ) ).
% map_conv_bind_option
thf(fact_2135_assert__def,axiom,
! [A: $tType] :
( ( heap_assert @ A )
= ( ^ [P4: A > $o,X4: A] : ( if @ ( heap_Heap @ A ) @ ( P4 @ X4 ) @ ( heap_return @ A @ X4 ) @ ( heap_raise @ A @ ( literal2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( zero_zero @ literal ) ) ) ) ) ) ) ) ) ) ) ).
% assert_def
thf(fact_2136_sorted__list__of__set_Ofold__insort__key_Ocomp__fun__commute__on,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y3: A,X2: A] :
( ( comp @ ( list @ A ) @ ( list @ A ) @ ( list @ A )
@ ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ Y3 )
@ ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ X2 ) )
= ( comp @ ( list @ A ) @ ( list @ A ) @ ( list @ A )
@ ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ X2 )
@ ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ Y3 ) ) ) ) ).
% sorted_list_of_set.fold_insort_key.comp_fun_commute_on
thf(fact_2137_Heap_Osize__neq,axiom,
! [A: $tType,X2: heap_Heap @ A] :
( ( size_size @ ( heap_Heap @ A ) @ X2 )
!= ( zero_zero @ nat ) ) ).
% Heap.size_neq
thf(fact_2138_bind__map__option,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > B,X2: option @ C,G: B > ( option @ A )] :
( ( bind2 @ B @ A @ ( map_option @ C @ B @ F @ X2 ) @ G )
= ( bind2 @ C @ A @ X2 @ ( comp @ B @ ( option @ A ) @ C @ G @ F ) ) ) ).
% bind_map_option
thf(fact_2139_map__option__bind,axiom,
! [A: $tType,B: $tType,C: $tType,F: B > A,X2: option @ C,G: C > ( option @ B )] :
( ( map_option @ B @ A @ F @ ( bind2 @ C @ B @ X2 @ G ) )
= ( bind2 @ C @ A @ X2 @ ( comp @ ( option @ B ) @ ( option @ A ) @ C @ ( map_option @ B @ A @ F ) @ G ) ) ) ).
% map_option_bind
thf(fact_2140_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
@ ^ [X4: A] : ( Ga @ ( F @ X4 ) ) ) ) ).
% option.rec_o_map
thf(fact_2141_extract__Cons__code,axiom,
! [A: $tType,P2: A > $o,X2: A,Xs: list @ A] :
( ( ( P2 @ X2 )
=> ( ( extract @ A @ P2 @ ( cons @ A @ X2 @ 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 ) @ X2 @ Xs ) ) ) ) )
& ( ~ ( P2 @ X2 )
=> ( ( extract @ A @ P2 @ ( cons @ A @ X2 @ 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 ) ) ) )
@ ^ [Ys2: list @ A] :
( product_case_prod @ A @ ( list @ A ) @ ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) )
@ ^ [Y5: A,Zs2: list @ A] : ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( cons @ A @ X2 @ Ys2 ) @ ( product_Pair @ A @ ( list @ A ) @ Y5 @ Zs2 ) ) ) ) )
@ ( extract @ A @ P2 @ Xs ) ) ) ) ) ).
% extract_Cons_code
thf(fact_2142_insort__insert__insort__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,X2: B,Xs: list @ B] :
( ~ ( member @ A @ ( F @ X2 ) @ ( image2 @ B @ A @ F @ ( set3 @ B @ Xs ) ) )
=> ( ( linord329482645794927042rt_key @ B @ A @ F @ X2 @ Xs )
= ( linorder_insort_key @ B @ A @ F @ X2 @ Xs ) ) ) ) ).
% insort_insert_insort_key
thf(fact_2143_insort__insert__key__def,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ( ( linord329482645794927042rt_key @ B @ A )
= ( ^ [F4: B > A,X4: B,Xs2: list @ B] : ( if @ ( list @ B ) @ ( member @ A @ ( F4 @ X4 ) @ ( image2 @ B @ A @ F4 @ ( set3 @ B @ Xs2 ) ) ) @ Xs2 @ ( linorder_insort_key @ B @ A @ F4 @ X4 @ Xs2 ) ) ) ) ) ).
% insort_insert_key_def
thf(fact_2144_set__bind__option,axiom,
! [A: $tType,B: $tType,X2: option @ B,F: B > ( option @ A )] :
( ( set_option @ A @ ( bind2 @ B @ A @ X2 @ F ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ ( comp @ ( option @ A ) @ ( set @ A ) @ B @ ( set_option @ A ) @ F ) @ ( set_option @ B @ X2 ) ) ) ) ).
% set_bind_option
thf(fact_2145_image__uminus__greaterThanAtMost,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X2: A,Y3: A] :
( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_or3652927894154168847AtMost @ A @ X2 @ Y3 ) )
= ( set_or7035219750837199246ssThan @ A @ ( uminus_uminus @ A @ Y3 ) @ ( uminus_uminus @ A @ X2 ) ) ) ) ).
% image_uminus_greaterThanAtMost
thf(fact_2146_image__uminus__atLeastLessThan,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X2: A,Y3: A] :
( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_or7035219750837199246ssThan @ A @ X2 @ Y3 ) )
= ( set_or3652927894154168847AtMost @ A @ ( uminus_uminus @ A @ Y3 ) @ ( uminus_uminus @ A @ X2 ) ) ) ) ).
% image_uminus_atLeastLessThan
thf(fact_2147_image__add__greaterThanAtMost,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ C3 ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) )
= ( set_or3652927894154168847AtMost @ A @ ( plus_plus @ A @ C3 @ A3 ) @ ( plus_plus @ A @ C3 @ B2 ) ) ) ) ).
% image_add_greaterThanAtMost
thf(fact_2148_image__diff__atLeastLessThan,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( image2 @ A @ A @ ( minus_minus @ A @ C3 ) @ ( set_or7035219750837199246ssThan @ A @ A3 @ B2 ) )
= ( set_or3652927894154168847AtMost @ A @ ( minus_minus @ A @ C3 @ B2 ) @ ( minus_minus @ A @ C3 @ A3 ) ) ) ) ).
% image_diff_atLeastLessThan
thf(fact_2149_image__minus__const__greaterThanAtMost,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C3: A,A3: A,B2: A] :
( ( image2 @ A @ A @ ( minus_minus @ A @ C3 ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) )
= ( set_or7035219750837199246ssThan @ A @ ( minus_minus @ A @ C3 @ B2 ) @ ( minus_minus @ A @ C3 @ A3 ) ) ) ) ).
% image_minus_const_greaterThanAtMost
thf(fact_2150_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_2151_insort__insert__key__triv,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,X2: B,Xs: list @ B] :
( ( member @ A @ ( F @ X2 ) @ ( image2 @ B @ A @ F @ ( set3 @ B @ Xs ) ) )
=> ( ( linord329482645794927042rt_key @ B @ A @ F @ X2 @ Xs )
= Xs ) ) ) ).
% insort_insert_key_triv
thf(fact_2152_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 ) )
@ ^ [X4: option @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set_option @ ( product_prod @ A @ B ) @ X4 ) @ ( 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_2153_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 ) )
@ ^ [X4: option @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set_option @ ( product_prod @ A @ B ) @ X4 ) @ ( 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 ) )
@ ^ [X4: option @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set_option @ ( product_prod @ A @ B ) @ X4 ) @ ( 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_2154_inv__fn__o__fn__is__id,axiom,
! [A: $tType,F: A > A,N5: nat] :
( ( bij_betw @ A @ A @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) )
=> ( ( comp @ A @ A @ A @ ( compow @ ( A > A ) @ N5 @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F ) ) @ ( compow @ ( A > A ) @ N5 @ F ) )
= ( ^ [X4: A] : X4 ) ) ) ).
% inv_fn_o_fn_is_id
thf(fact_2155_fn__o__inv__fn__is__id,axiom,
! [A: $tType,F: A > A,N5: nat] :
( ( bij_betw @ A @ A @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) )
=> ( ( comp @ A @ A @ A @ ( compow @ ( A > A ) @ N5 @ F ) @ ( compow @ ( A > A ) @ N5 @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F ) ) )
= ( ^ [X4: A] : X4 ) ) ) ).
% fn_o_inv_fn_is_id
thf(fact_2156_lfp__induct2,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,F: ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ A @ B ) ),P2: 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,B4: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B4 ) @ ( 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 @ P2 ) ) ) ) )
=> ( P2 @ A5 @ B4 ) )
=> ( P2 @ A3 @ B2 ) ) ) ) ).
% lfp_induct2
thf(fact_2157_Heap_Osize__gen__o__map,axiom,
! [B: $tType,A: $tType,F: B > nat,G: A > B] :
( ( comp @ ( heap_Heap @ B ) @ nat @ ( heap_Heap @ A ) @ ( heap_size_Heap @ B @ F ) @ ( heap_map_Heap @ A @ B @ G ) )
= ( heap_size_Heap @ A @ ( comp @ B @ nat @ A @ F @ G ) ) ) ).
% Heap.size_gen_o_map
thf(fact_2158_lists__image,axiom,
! [A: $tType,B: $tType,F: B > A,A6: set @ B] :
( ( lists @ A @ ( image2 @ B @ A @ F @ A6 ) )
= ( image2 @ ( list @ B ) @ ( list @ A ) @ ( map @ B @ A @ F ) @ ( lists @ B @ A6 ) ) ) ).
% lists_image
thf(fact_2159_surj__fn,axiom,
! [A: $tType,F: A > A,N5: nat] :
( ( ( image2 @ A @ A @ F @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( image2 @ A @ A @ ( compow @ ( A > A ) @ N5 @ F ) @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surj_fn
thf(fact_2160_comp__funpow,axiom,
! [B: $tType,A: $tType,N5: nat,F: A > A] :
( ( compow @ ( ( B > A ) > B > A ) @ N5 @ ( comp @ A @ A @ B @ F ) )
= ( comp @ A @ A @ B @ ( compow @ ( A > A ) @ N5 @ F ) ) ) ).
% comp_funpow
thf(fact_2161_Heap_Omap__ident,axiom,
! [A: $tType,T3: heap_Heap @ A] :
( ( heap_map_Heap @ A @ A
@ ^ [X4: A] : X4
@ T3 )
= T3 ) ).
% Heap.map_ident
thf(fact_2162_relpowp__commute,axiom,
! [A: $tType,P2: A > A > $o,N5: nat] :
( ( relcompp @ A @ A @ A @ P2 @ ( compow @ ( A > A > $o ) @ N5 @ P2 ) )
= ( relcompp @ A @ A @ A @ ( compow @ ( A > A > $o ) @ N5 @ P2 ) @ P2 ) ) ).
% relpowp_commute
thf(fact_2163_option_Orel__compp,axiom,
! [A: $tType,C: $tType,B: $tType,R3: A > B > $o,S5: B > C > $o] :
( ( rel_option @ A @ C @ ( relcompp @ A @ B @ C @ R3 @ S5 ) )
= ( relcompp @ ( option @ A ) @ ( option @ B ) @ ( option @ C ) @ ( rel_option @ A @ B @ R3 ) @ ( rel_option @ B @ C @ S5 ) ) ) ).
% option.rel_compp
thf(fact_2164_Heap_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,G: B > C,F: A > B,V: heap_Heap @ A] :
( ( heap_map_Heap @ B @ C @ G @ ( heap_map_Heap @ A @ B @ F @ V ) )
= ( heap_map_Heap @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ V ) ) ).
% Heap.map_comp
thf(fact_2165_Heap_Omap__id,axiom,
! [A: $tType,T3: heap_Heap @ A] :
( ( heap_map_Heap @ A @ A @ ( id @ A ) @ T3 )
= T3 ) ).
% Heap.map_id
thf(fact_2166_Heap_Omap__id0,axiom,
! [A: $tType] :
( ( heap_map_Heap @ A @ A @ ( id @ A ) )
= ( id @ ( heap_Heap @ A ) ) ) ).
% Heap.map_id0
thf(fact_2167_Heap_Omap,axiom,
! [B: $tType,A: $tType,F: A > B,X2: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) )] :
( ( heap_map_Heap @ A @ B @ F @ ( heap_Heap2 @ A @ X2 ) )
= ( heap_Heap2 @ B @ ( comp @ ( option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) @ ( option @ ( product_prod @ B @ ( heap_ext @ product_unit ) ) ) @ ( heap_ext @ product_unit ) @ ( map_option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) @ ( product_prod @ B @ ( heap_ext @ product_unit ) ) @ ( product_map_prod @ A @ B @ ( heap_ext @ product_unit ) @ ( heap_ext @ product_unit ) @ F @ ( id @ ( heap_ext @ product_unit ) ) ) ) @ X2 ) ) ) ).
% Heap.map
thf(fact_2168_rel__option__iff,axiom,
! [B: $tType,A: $tType] :
( ( rel_option @ A @ B )
= ( ^ [R6: A > B > $o,X4: option @ A,Y5: 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
@ ^ [C6: B] : $false
@ B5 )
@ ^ [Z5: A] : ( case_option @ $o @ B @ $false @ ( R6 @ Z5 ) @ B5 )
@ A7 )
@ ( product_Pair @ ( option @ A ) @ ( option @ B ) @ X4 @ Y5 ) ) ) ) ).
% rel_option_iff
thf(fact_2169_relpowp_Osimps_I2_J,axiom,
! [A: $tType,N5: nat,R3: A > A > $o] :
( ( compow @ ( A > A > $o ) @ ( suc @ N5 ) @ R3 )
= ( relcompp @ A @ A @ A @ ( compow @ ( A > A > $o ) @ N5 @ R3 ) @ R3 ) ) ).
% relpowp.simps(2)
thf(fact_2170_relpowp__add,axiom,
! [A: $tType,M4: nat,N5: nat,P2: A > A > $o] :
( ( compow @ ( A > A > $o ) @ ( plus_plus @ nat @ M4 @ N5 ) @ P2 )
= ( relcompp @ A @ A @ A @ ( compow @ ( A > A > $o ) @ M4 @ P2 ) @ ( compow @ ( A > A > $o ) @ N5 @ P2 ) ) ) ).
% relpowp_add
thf(fact_2171_funpow__Suc__right,axiom,
! [A: $tType,N5: nat,F: A > A] :
( ( compow @ ( A > A ) @ ( suc @ N5 ) @ F )
= ( comp @ A @ A @ A @ ( compow @ ( A > A ) @ N5 @ F ) @ F ) ) ).
% funpow_Suc_right
thf(fact_2172_funpow_Osimps_I2_J,axiom,
! [A: $tType,N5: nat,F: A > A] :
( ( compow @ ( A > A ) @ ( suc @ N5 ) @ F )
= ( comp @ A @ A @ A @ F @ ( compow @ ( A > A ) @ N5 @ F ) ) ) ).
% funpow.simps(2)
thf(fact_2173_funpow__add,axiom,
! [A: $tType,M4: nat,N5: nat,F: A > A] :
( ( compow @ ( A > A ) @ ( plus_plus @ nat @ M4 @ N5 ) @ F )
= ( comp @ A @ A @ A @ ( compow @ ( A > A ) @ M4 @ F ) @ ( compow @ ( A > A ) @ N5 @ F ) ) ) ).
% funpow_add
thf(fact_2174_relpowp__relpow__eq,axiom,
! [A: $tType,N5: nat,R3: set @ ( product_prod @ A @ A )] :
( ( compow @ ( A > A > $o ) @ N5
@ ^ [X4: A,Y5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R3 ) )
= ( ^ [X4: A,Y5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N5 @ R3 ) ) ) ) ).
% relpowp_relpow_eq
thf(fact_2175_Heap_Oinj__map,axiom,
! [B: $tType,A: $tType,F: A > B] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( inj_on @ ( heap_Heap @ A ) @ ( heap_Heap @ B ) @ ( heap_map_Heap @ A @ B @ F ) @ ( top_top @ ( set @ ( heap_Heap @ A ) ) ) ) ) ).
% Heap.inj_map
thf(fact_2176_Heap_Orec__o__map,axiom,
! [B: $tType,C: $tType,A: $tType,G: ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ B @ ( heap_ext @ product_unit ) ) ) ) > C,F: A > B] :
( ( comp @ ( heap_Heap @ B ) @ C @ ( heap_Heap @ A ) @ ( heap_rec_Heap @ B @ C @ G ) @ ( heap_map_Heap @ A @ B @ F ) )
= ( heap_rec_Heap @ A @ C
@ ^ [X4: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) )] : ( G @ ( comp @ ( option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) @ ( option @ ( product_prod @ B @ ( heap_ext @ product_unit ) ) ) @ ( heap_ext @ product_unit ) @ ( map_option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) @ ( product_prod @ B @ ( heap_ext @ product_unit ) ) @ ( product_map_prod @ A @ B @ ( heap_ext @ product_unit ) @ ( heap_ext @ product_unit ) @ F @ ( id @ ( heap_ext @ product_unit ) ) ) ) @ X4 ) ) ) ) ).
% Heap.rec_o_map
thf(fact_2177_pcr__int__def,axiom,
( pcr_int
= ( relcompp @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ int
@ ( basic_rel_prod @ nat @ nat @ nat @ nat
@ ^ [Y2: nat,Z2: nat] : Y2 = Z2
@ ^ [Y2: nat,Z2: nat] : Y2 = Z2 )
@ cr_int ) ) ).
% pcr_int_def
thf(fact_2178_fold__graph__image,axiom,
! [C: $tType,B: $tType,A: $tType,G: A > B,A6: set @ A,F: B > C > C,Z3: C] :
( ( inj_on @ A @ B @ G @ A6 )
=> ( ( finite_fold_graph @ B @ C @ F @ Z3 @ ( image2 @ A @ B @ G @ A6 ) )
= ( finite_fold_graph @ A @ C @ ( comp @ B @ ( C > C ) @ A @ F @ G ) @ Z3 @ A6 ) ) ) ).
% fold_graph_image
thf(fact_2179_hom__Max__commute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [H: A > A,N: set @ A] :
( ! [X: A,Y: A] :
( ( H @ ( ord_max @ A @ X @ Y ) )
= ( ord_max @ A @ ( H @ X ) @ ( H @ Y ) ) )
=> ( ( finite_finite2 @ A @ N )
=> ( ( N
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H @ ( lattic643756798349783984er_Max @ A @ N ) )
= ( lattic643756798349783984er_Max @ A @ ( image2 @ A @ A @ H @ N ) ) ) ) ) ) ) ).
% hom_Max_commute
thf(fact_2180_fold__remove1__split,axiom,
! [B: $tType,A: $tType,Xs: list @ A,F: A > B > B,X2: A] :
( ! [X: A,Y: A] :
( ( member @ A @ X @ ( set3 @ A @ Xs ) )
=> ( ( member @ A @ Y @ ( set3 @ A @ Xs ) )
=> ( ( comp @ B @ B @ B @ ( F @ X ) @ ( F @ Y ) )
= ( comp @ B @ B @ B @ ( F @ Y ) @ ( F @ X ) ) ) ) )
=> ( ( member @ A @ X2 @ ( set3 @ A @ Xs ) )
=> ( ( fold @ A @ B @ F @ Xs )
= ( comp @ B @ B @ B @ ( fold @ A @ B @ F @ ( remove1 @ A @ X2 @ Xs ) ) @ ( F @ X2 ) ) ) ) ) ).
% fold_remove1_split
thf(fact_2181_Heap_Oset,axiom,
! [A: $tType,X2: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) )] :
( ( heap_set_Heap @ A @ ( heap_Heap2 @ A @ X2 ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ ( option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) @ ( set @ A )
@ ^ [Uu2: option @ ( product_prod @ A @ ( heap_ext @ product_unit ) )] : ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) @ ( set @ A ) @ ( basic_fsts @ A @ ( heap_ext @ product_unit ) ) @ ( set_option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) @ Uu2 ) ) )
@ ( image2 @ ( heap_ext @ product_unit ) @ ( option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) @ X2 @ ( top_top @ ( set @ ( heap_ext @ product_unit ) ) ) ) ) ) ) ).
% Heap.set
thf(fact_2182_trancl__def,axiom,
! [A: $tType] :
( ( transitive_trancl @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ( transitive_tranclp @ A
@ ^ [X4: A,Y5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R5 ) ) ) ) ) ) ).
% trancl_def
thf(fact_2183_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,P2: A > B > $o] :
( ( transitive_tranclp @ ( product_prod @ A @ B ) @ R @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) )
=> ( ! [A5: A,B4: B] :
( ( R @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A5 @ B4 ) )
=> ( P2 @ A5 @ B4 ) )
=> ( ! [A5: A,B4: B,Aa2: A,Ba: B] :
( ( transitive_tranclp @ ( product_prod @ A @ B ) @ R @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A5 @ B4 ) )
=> ( ( R @ ( product_Pair @ A @ B @ A5 @ B4 ) @ ( product_Pair @ A @ B @ Aa2 @ Ba ) )
=> ( ( P2 @ A5 @ B4 )
=> ( P2 @ Aa2 @ Ba ) ) ) )
=> ( P2 @ Bx @ By ) ) ) ) ).
% tranclp_induct2
thf(fact_2184_Heap_Oset__intros,axiom,
! [A: $tType,Y3: option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ),X2: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ),Ya: product_prod @ A @ ( heap_ext @ product_unit ),Yb2: A] :
( ( member @ ( option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) @ Y3 @ ( image2 @ ( heap_ext @ product_unit ) @ ( option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) @ X2 @ ( top_top @ ( set @ ( heap_ext @ product_unit ) ) ) ) )
=> ( ( member @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) @ Ya @ ( set_option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) @ Y3 ) )
=> ( ( member @ A @ Yb2 @ ( basic_fsts @ A @ ( heap_ext @ product_unit ) @ Ya ) )
=> ( member @ A @ Yb2 @ ( heap_set_Heap @ A @ ( heap_Heap2 @ A @ X2 ) ) ) ) ) ) ).
% Heap.set_intros
thf(fact_2185_Heap_Oset__cases,axiom,
! [A: $tType,E3: A,A3: heap_Heap @ A] :
( ( member @ A @ E3 @ ( heap_set_Heap @ A @ A3 ) )
=> ~ ! [Z4: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) )] :
( ( A3
= ( heap_Heap2 @ A @ Z4 ) )
=> ! [Xa: option @ ( product_prod @ A @ ( heap_ext @ product_unit ) )] :
( ( member @ ( option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) @ Xa @ ( image2 @ ( heap_ext @ product_unit ) @ ( option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) @ Z4 @ ( top_top @ ( set @ ( heap_ext @ product_unit ) ) ) ) )
=> ! [Xb2: product_prod @ A @ ( heap_ext @ product_unit )] :
( ( member @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) @ Xb2 @ ( set_option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) @ Xa ) )
=> ~ ( member @ A @ E3 @ ( basic_fsts @ A @ ( heap_ext @ product_unit ) @ Xb2 ) ) ) ) ) ) ).
% Heap.set_cases
thf(fact_2186_Heap_Oinj__map__strong,axiom,
! [B: $tType,A: $tType,X2: heap_Heap @ A,Xa2: heap_Heap @ A,F: A > B,Fa: A > B] :
( ! [Z4: A,Za: A] :
( ( member @ A @ Z4 @ ( heap_set_Heap @ A @ X2 ) )
=> ( ( member @ A @ Za @ ( heap_set_Heap @ A @ Xa2 ) )
=> ( ( ( F @ Z4 )
= ( Fa @ Za ) )
=> ( Z4 = Za ) ) ) )
=> ( ( ( heap_map_Heap @ A @ B @ F @ X2 )
= ( heap_map_Heap @ A @ B @ Fa @ Xa2 ) )
=> ( X2 = Xa2 ) ) ) ).
% Heap.inj_map_strong
thf(fact_2187_Heap_Omap__cong0,axiom,
! [B: $tType,A: $tType,X2: heap_Heap @ A,F: A > B,G: A > B] :
( ! [Z4: A] :
( ( member @ A @ Z4 @ ( heap_set_Heap @ A @ X2 ) )
=> ( ( F @ Z4 )
= ( G @ Z4 ) ) )
=> ( ( heap_map_Heap @ A @ B @ F @ X2 )
= ( heap_map_Heap @ A @ B @ G @ X2 ) ) ) ).
% Heap.map_cong0
thf(fact_2188_Heap_Omap__cong,axiom,
! [B: $tType,A: $tType,X2: heap_Heap @ A,Ya: heap_Heap @ A,F: A > B,G: A > B] :
( ( X2 = Ya )
=> ( ! [Z4: A] :
( ( member @ A @ Z4 @ ( heap_set_Heap @ A @ Ya ) )
=> ( ( F @ Z4 )
= ( G @ Z4 ) ) )
=> ( ( heap_map_Heap @ A @ B @ F @ X2 )
= ( heap_map_Heap @ A @ B @ G @ Ya ) ) ) ) ).
% Heap.map_cong
thf(fact_2189_tranclp__trancl__eq,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( transitive_tranclp @ A
@ ^ [X4: A,Y5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R ) )
= ( ^ [X4: A,Y5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ ( transitive_trancl @ A @ R ) ) ) ) ).
% tranclp_trancl_eq
thf(fact_2190_Heap_Orec,axiom,
! [C: $tType,A: $tType,F: ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) ) > C,X2: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) )] :
( ( heap_rec_Heap @ A @ C @ F @ ( heap_Heap2 @ A @ X2 ) )
= ( F @ X2 ) ) ).
% Heap.rec
thf(fact_2191_Heap_Oset__map,axiom,
! [B: $tType,A: $tType,F: A > B,V: heap_Heap @ A] :
( ( heap_set_Heap @ B @ ( heap_map_Heap @ A @ B @ F @ V ) )
= ( image2 @ A @ B @ F @ ( heap_set_Heap @ A @ V ) ) ) ).
% Heap.set_map
thf(fact_2192_Nitpick_Otranclp__unfold,axiom,
! [A: $tType] :
( ( transitive_tranclp @ A )
= ( ^ [R5: 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 @ R5 ) ) ) ) ) ) ).
% Nitpick.tranclp_unfold
thf(fact_2193_Heap_Orel__Grp,axiom,
! [B: $tType,A: $tType,A6: set @ A,F: A > B] :
( ( heap_rel_Heap @ A @ B @ ( bNF_Grp @ A @ B @ A6 @ F ) )
= ( bNF_Grp @ ( heap_Heap @ A ) @ ( heap_Heap @ B )
@ ( collect @ ( heap_Heap @ A )
@ ^ [X4: heap_Heap @ A] : ( ord_less_eq @ ( set @ A ) @ ( heap_set_Heap @ A @ X4 ) @ A6 ) )
@ ( heap_map_Heap @ A @ B @ F ) ) ) ).
% Heap.rel_Grp
thf(fact_2194_Heap_Oin__rel,axiom,
! [B: $tType,A: $tType] :
( ( heap_rel_Heap @ A @ B )
= ( ^ [R6: A > B > $o,A7: heap_Heap @ A,B5: heap_Heap @ B] :
? [Z5: heap_Heap @ ( product_prod @ A @ B )] :
( ( member @ ( heap_Heap @ ( product_prod @ A @ B ) ) @ Z5
@ ( collect @ ( heap_Heap @ ( product_prod @ A @ B ) )
@ ^ [X4: heap_Heap @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( heap_set_Heap @ ( product_prod @ A @ B ) @ X4 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) ) ) )
& ( ( heap_map_Heap @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Z5 )
= A7 )
& ( ( heap_map_Heap @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Z5 )
= B5 ) ) ) ) ).
% Heap.in_rel
thf(fact_2195_Heap_Orel__compp__Grp,axiom,
! [B: $tType,A: $tType] :
( ( heap_rel_Heap @ A @ B )
= ( ^ [R6: A > B > $o] :
( relcompp @ ( heap_Heap @ A ) @ ( heap_Heap @ ( product_prod @ A @ B ) ) @ ( heap_Heap @ B )
@ ( conversep @ ( heap_Heap @ ( product_prod @ A @ B ) ) @ ( heap_Heap @ A )
@ ( bNF_Grp @ ( heap_Heap @ ( product_prod @ A @ B ) ) @ ( heap_Heap @ A )
@ ( collect @ ( heap_Heap @ ( product_prod @ A @ B ) )
@ ^ [X4: heap_Heap @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( heap_set_Heap @ ( product_prod @ A @ B ) @ X4 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) ) )
@ ( heap_map_Heap @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) ) ) )
@ ( bNF_Grp @ ( heap_Heap @ ( product_prod @ A @ B ) ) @ ( heap_Heap @ B )
@ ( collect @ ( heap_Heap @ ( product_prod @ A @ B ) )
@ ^ [X4: heap_Heap @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( heap_set_Heap @ ( product_prod @ A @ B ) @ X4 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) ) )
@ ( heap_map_Heap @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) ) ) ) ) ) ).
% Heap.rel_compp_Grp
thf(fact_2196_reflp__refl__eq,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( reflp @ A
@ ^ [X4: A,Y5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R ) )
= ( refl_on @ A @ ( top_top @ ( set @ A ) ) @ R ) ) ).
% reflp_refl_eq
thf(fact_2197_update__change,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( update2 @ A )
= ( ^ [R5: ref @ A,E5: A] :
( heap_bind @ A @ product_unit
@ ( change @ A
@ ^ [Uu: A] : E5
@ R5 )
@ ^ [Uu: A] : ( heap_return @ product_unit @ product_Unity ) ) ) ) ) ).
% update_change
thf(fact_2198_old_Orec__bool__def,axiom,
! [T: $tType] :
( ( product_rec_bool @ T )
= ( ^ [F12: T,F22: T,X4: $o] : ( the @ T @ ( product_rec_set_bool @ T @ F12 @ F22 @ X4 ) ) ) ) ).
% old.rec_bool_def
thf(fact_2199_unit__abs__eta__conv,axiom,
! [A: $tType,F: product_unit > A] :
( ( ^ [U: product_unit] : ( F @ product_Unity ) )
= F ) ).
% unit_abs_eta_conv
thf(fact_2200_old_Obool_Osimps_I5_J,axiom,
! [T: $tType,F1: T,F23: T] :
( ( product_rec_bool @ T @ F1 @ F23 @ $true )
= F1 ) ).
% old.bool.simps(5)
thf(fact_2201_old_Obool_Osimps_I6_J,axiom,
! [T: $tType,F1: T,F23: T] :
( ( product_rec_bool @ T @ F1 @ F23 @ $false )
= F23 ) ).
% old.bool.simps(6)
thf(fact_2202_old_Ounit_Orec,axiom,
! [T: $tType,F1: T] :
( ( product_rec_unit @ T @ F1 @ product_Unity )
= F1 ) ).
% old.unit.rec
thf(fact_2203_UNIV__unit,axiom,
( ( top_top @ ( set @ product_unit ) )
= ( insert @ product_unit @ product_Unity @ ( bot_bot @ ( set @ product_unit ) ) ) ) ).
% UNIV_unit
thf(fact_2204_top__unit__def,axiom,
( ( top_top @ product_unit )
= product_Unity ) ).
% top_unit_def
thf(fact_2205_uminus__unit__def,axiom,
( ( uminus_uminus @ product_unit )
= ( ^ [Uu2: product_unit] : product_Unity ) ) ).
% uminus_unit_def
thf(fact_2206_inf__unit__def,axiom,
( ( inf_inf @ product_unit )
= ( ^ [Uu2: product_unit,Uv2: product_unit] : product_Unity ) ) ).
% inf_unit_def
thf(fact_2207_Heap_Orel__eq,axiom,
! [A: $tType] :
( ( heap_rel_Heap @ A @ A
@ ^ [Y2: A,Z2: A] : Y2 = Z2 )
= ( ^ [Y2: heap_Heap @ A,Z2: heap_Heap @ A] : Y2 = Z2 ) ) ).
% Heap.rel_eq
thf(fact_2208_Heap_Orel__refl,axiom,
! [B: $tType,Ra: B > B > $o,X2: heap_Heap @ B] :
( ! [X: B] : ( Ra @ X @ X )
=> ( heap_rel_Heap @ B @ B @ Ra @ X2 @ X2 ) ) ).
% Heap.rel_refl
thf(fact_2209_Heap_Orel__reflp,axiom,
! [A: $tType,R3: A > A > $o] :
( ( reflp @ A @ R3 )
=> ( reflp @ ( heap_Heap @ A ) @ ( heap_rel_Heap @ A @ A @ R3 ) ) ) ).
% Heap.rel_reflp
thf(fact_2210_old_Ounit_Oexhaust,axiom,
! [Y3: product_unit] : Y3 = product_Unity ).
% old.unit.exhaust
thf(fact_2211_Inf__unit__def,axiom,
( ( complete_Inf_Inf @ product_unit )
= ( ^ [Uu2: set @ product_unit] : product_Unity ) ) ).
% Inf_unit_def
thf(fact_2212_Sup__unit__def,axiom,
( ( complete_Sup_Sup @ product_unit )
= ( ^ [Uu2: set @ product_unit] : product_Unity ) ) ).
% Sup_unit_def
thf(fact_2213_sup__unit__def,axiom,
( ( sup_sup @ product_unit )
= ( ^ [Uu2: product_unit,Uv2: product_unit] : product_Unity ) ) ).
% sup_unit_def
thf(fact_2214_bot__unit__def,axiom,
( ( bot_bot @ product_unit )
= product_Unity ) ).
% bot_unit_def
thf(fact_2215_Heap_Orel__compp,axiom,
! [A: $tType,C: $tType,B: $tType,R3: A > B > $o,S5: B > C > $o] :
( ( heap_rel_Heap @ A @ C @ ( relcompp @ A @ B @ C @ R3 @ S5 ) )
= ( relcompp @ ( heap_Heap @ A ) @ ( heap_Heap @ B ) @ ( heap_Heap @ C ) @ ( heap_rel_Heap @ A @ B @ R3 ) @ ( heap_rel_Heap @ B @ C @ S5 ) ) ) ).
% Heap.rel_compp
thf(fact_2216_Heap_Orel__flip,axiom,
! [A: $tType,B: $tType,R3: A > B > $o,A3: heap_Heap @ B,B2: heap_Heap @ A] :
( ( heap_rel_Heap @ B @ A @ ( conversep @ A @ B @ R3 ) @ A3 @ B2 )
= ( heap_rel_Heap @ A @ B @ R3 @ B2 @ A3 ) ) ).
% Heap.rel_flip
thf(fact_2217_Heap_Orel__conversep,axiom,
! [B: $tType,A: $tType,R3: A > B > $o] :
( ( heap_rel_Heap @ B @ A @ ( conversep @ A @ B @ R3 ) )
= ( conversep @ ( heap_Heap @ A ) @ ( heap_Heap @ B ) @ ( heap_rel_Heap @ A @ B @ R3 ) ) ) ).
% Heap.rel_conversep
thf(fact_2218_Heap_Orel__mono,axiom,
! [B: $tType,A: $tType,R3: A > B > $o,Ra: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ R3 @ Ra )
=> ( ord_less_eq @ ( ( heap_Heap @ A ) > ( heap_Heap @ B ) > $o ) @ ( heap_rel_Heap @ A @ B @ R3 ) @ ( heap_rel_Heap @ A @ B @ Ra ) ) ) ).
% Heap.rel_mono
thf(fact_2219_Heap_Obi__total__rel,axiom,
! [B: $tType,A: $tType,R3: A > B > $o] :
( ( bi_total @ A @ B @ R3 )
=> ( bi_total @ ( heap_Heap @ A ) @ ( heap_Heap @ B ) @ ( heap_rel_Heap @ A @ B @ R3 ) ) ) ).
% Heap.bi_total_rel
thf(fact_2220_Heap_Oright__total__rel,axiom,
! [B: $tType,A: $tType,R3: A > B > $o] :
( ( right_total @ A @ B @ R3 )
=> ( right_total @ ( heap_Heap @ A ) @ ( heap_Heap @ B ) @ ( heap_rel_Heap @ A @ B @ R3 ) ) ) ).
% Heap.right_total_rel
thf(fact_2221_reflp__eq,axiom,
! [A: $tType] :
( ( reflp @ A )
= ( ord_less_eq @ ( A > A > $o )
@ ^ [Y2: A,Z2: A] : Y2 = Z2 ) ) ).
% reflp_eq
thf(fact_2222_Heap_Oright__unique__rel,axiom,
! [B: $tType,A: $tType,R3: A > B > $o] :
( ( right_unique @ A @ B @ R3 )
=> ( right_unique @ ( heap_Heap @ A ) @ ( heap_Heap @ B ) @ ( heap_rel_Heap @ A @ B @ R3 ) ) ) ).
% Heap.right_unique_rel
thf(fact_2223_Heap_Oleft__unique__rel,axiom,
! [B: $tType,A: $tType,R3: A > B > $o] :
( ( left_unique @ A @ B @ R3 )
=> ( left_unique @ ( heap_Heap @ A ) @ ( heap_Heap @ B ) @ ( heap_rel_Heap @ A @ B @ R3 ) ) ) ).
% Heap.left_unique_rel
thf(fact_2224_Heap_Oleft__total__rel,axiom,
! [B: $tType,A: $tType,R3: A > B > $o] :
( ( left_total @ A @ B @ R3 )
=> ( left_total @ ( heap_Heap @ A ) @ ( heap_Heap @ B ) @ ( heap_rel_Heap @ A @ B @ R3 ) ) ) ).
% Heap.left_total_rel
thf(fact_2225_Heap_Orel__cases,axiom,
! [A: $tType,B: $tType,R3: A > B > $o,A3: heap_Heap @ A,B2: heap_Heap @ B] :
( ( heap_rel_Heap @ A @ B @ R3 @ A3 @ B2 )
=> ~ ! [Xa: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) )] :
( ( A3
= ( heap_Heap2 @ A @ Xa ) )
=> ! [Ya2: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ B @ ( heap_ext @ product_unit ) ) )] :
( ( B2
= ( heap_Heap2 @ B @ Ya2 ) )
=> ~ ( bNF_rel_fun @ ( heap_ext @ product_unit ) @ ( heap_ext @ product_unit ) @ ( option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) @ ( option @ ( product_prod @ B @ ( heap_ext @ product_unit ) ) )
@ ^ [Y2: heap_ext @ product_unit,Z2: heap_ext @ product_unit] : Y2 = Z2
@ ( rel_option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) @ ( product_prod @ B @ ( heap_ext @ product_unit ) )
@ ( basic_rel_prod @ A @ B @ ( heap_ext @ product_unit ) @ ( heap_ext @ product_unit ) @ R3
@ ^ [Y2: heap_ext @ product_unit,Z2: heap_ext @ product_unit] : Y2 = Z2 ) )
@ Xa
@ Ya2 ) ) ) ) ).
% Heap.rel_cases
thf(fact_2226_Heap_Orel__induct,axiom,
! [A: $tType,B: $tType,R3: A > B > $o,X2: heap_Heap @ A,Y3: heap_Heap @ B,Q2: ( heap_Heap @ A ) > ( heap_Heap @ B ) > $o] :
( ( heap_rel_Heap @ A @ B @ R3 @ X2 @ Y3 )
=> ( ! [A5: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ),B4: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ B @ ( heap_ext @ product_unit ) ) )] :
( ( bNF_rel_fun @ ( heap_ext @ product_unit ) @ ( heap_ext @ product_unit ) @ ( option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) @ ( option @ ( product_prod @ B @ ( heap_ext @ product_unit ) ) )
@ ^ [Y2: heap_ext @ product_unit,Z2: heap_ext @ product_unit] : Y2 = Z2
@ ( rel_option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) @ ( product_prod @ B @ ( heap_ext @ product_unit ) )
@ ( basic_rel_prod @ A @ B @ ( heap_ext @ product_unit ) @ ( heap_ext @ product_unit ) @ R3
@ ^ [Y2: heap_ext @ product_unit,Z2: heap_ext @ product_unit] : Y2 = Z2 ) )
@ A5
@ B4 )
=> ( Q2 @ ( heap_Heap2 @ A @ A5 ) @ ( heap_Heap2 @ B @ B4 ) ) )
=> ( Q2 @ X2 @ Y3 ) ) ) ).
% Heap.rel_induct
thf(fact_2227_Heap_Orel__inject,axiom,
! [A: $tType,B: $tType,R3: A > B > $o,X2: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ),Y3: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ B @ ( heap_ext @ product_unit ) ) )] :
( ( heap_rel_Heap @ A @ B @ R3 @ ( heap_Heap2 @ A @ X2 ) @ ( heap_Heap2 @ B @ Y3 ) )
= ( bNF_rel_fun @ ( heap_ext @ product_unit ) @ ( heap_ext @ product_unit ) @ ( option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) @ ( option @ ( product_prod @ B @ ( heap_ext @ product_unit ) ) )
@ ^ [Y2: heap_ext @ product_unit,Z2: heap_ext @ product_unit] : Y2 = Z2
@ ( rel_option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) @ ( product_prod @ B @ ( heap_ext @ product_unit ) )
@ ( basic_rel_prod @ A @ B @ ( heap_ext @ product_unit ) @ ( heap_ext @ product_unit ) @ R3
@ ^ [Y2: heap_ext @ product_unit,Z2: heap_ext @ product_unit] : Y2 = Z2 ) )
@ X2
@ Y3 ) ) ).
% Heap.rel_inject
thf(fact_2228_Heap_Orel__intros,axiom,
! [A: $tType,B: $tType,R3: A > B > $o,X2: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ),Y3: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ B @ ( heap_ext @ product_unit ) ) )] :
( ( bNF_rel_fun @ ( heap_ext @ product_unit ) @ ( heap_ext @ product_unit ) @ ( option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) @ ( option @ ( product_prod @ B @ ( heap_ext @ product_unit ) ) )
@ ^ [Y2: heap_ext @ product_unit,Z2: heap_ext @ product_unit] : Y2 = Z2
@ ( rel_option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) @ ( product_prod @ B @ ( heap_ext @ product_unit ) )
@ ( basic_rel_prod @ A @ B @ ( heap_ext @ product_unit ) @ ( heap_ext @ product_unit ) @ R3
@ ^ [Y2: heap_ext @ product_unit,Z2: heap_ext @ product_unit] : Y2 = Z2 ) )
@ X2
@ Y3 )
=> ( heap_rel_Heap @ A @ B @ R3 @ ( heap_Heap2 @ A @ X2 ) @ ( heap_Heap2 @ B @ Y3 ) ) ) ).
% Heap.rel_intros
thf(fact_2229_Heap_Octr__transfer,axiom,
! [A: $tType,B: $tType,R3: A > B > $o] :
( bNF_rel_fun @ ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) ) @ ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ B @ ( heap_ext @ product_unit ) ) ) ) @ ( heap_Heap @ A ) @ ( heap_Heap @ B )
@ ( bNF_rel_fun @ ( heap_ext @ product_unit ) @ ( heap_ext @ product_unit ) @ ( option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) @ ( option @ ( product_prod @ B @ ( heap_ext @ product_unit ) ) )
@ ^ [Y2: heap_ext @ product_unit,Z2: heap_ext @ product_unit] : Y2 = Z2
@ ( rel_option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) @ ( product_prod @ B @ ( heap_ext @ product_unit ) )
@ ( basic_rel_prod @ A @ B @ ( heap_ext @ product_unit ) @ ( heap_ext @ product_unit ) @ R3
@ ^ [Y2: heap_ext @ product_unit,Z2: heap_ext @ product_unit] : Y2 = Z2 ) ) )
@ ( heap_rel_Heap @ A @ B @ R3 )
@ ( heap_Heap2 @ A )
@ ( heap_Heap2 @ B ) ) ).
% Heap.ctr_transfer
thf(fact_2230_Heap_Orel__cong,axiom,
! [A: $tType,B: $tType,X2: heap_Heap @ A,Ya: heap_Heap @ A,Y3: heap_Heap @ B,Xa2: heap_Heap @ B,R3: A > B > $o,Ra: A > B > $o] :
( ( X2 = Ya )
=> ( ( Y3 = Xa2 )
=> ( ! [Z4: A,Yb: B] :
( ( member @ A @ Z4 @ ( heap_set_Heap @ A @ Ya ) )
=> ( ( member @ B @ Yb @ ( heap_set_Heap @ B @ Xa2 ) )
=> ( ( R3 @ Z4 @ Yb )
= ( Ra @ Z4 @ Yb ) ) ) )
=> ( ( heap_rel_Heap @ A @ B @ R3 @ X2 @ Y3 )
= ( heap_rel_Heap @ A @ B @ Ra @ Ya @ Xa2 ) ) ) ) ) ).
% Heap.rel_cong
thf(fact_2231_Heap_Orel__mono__strong,axiom,
! [A: $tType,B: $tType,R3: A > B > $o,X2: heap_Heap @ A,Y3: heap_Heap @ B,Ra: A > B > $o] :
( ( heap_rel_Heap @ A @ B @ R3 @ X2 @ Y3 )
=> ( ! [Z4: A,Yb: B] :
( ( member @ A @ Z4 @ ( heap_set_Heap @ A @ X2 ) )
=> ( ( member @ B @ Yb @ ( heap_set_Heap @ B @ Y3 ) )
=> ( ( R3 @ Z4 @ Yb )
=> ( Ra @ Z4 @ Yb ) ) ) )
=> ( heap_rel_Heap @ A @ B @ Ra @ X2 @ Y3 ) ) ) ).
% Heap.rel_mono_strong
thf(fact_2232_Heap_Orel__refl__strong,axiom,
! [A: $tType,X2: heap_Heap @ A,Ra: A > A > $o] :
( ! [Z4: A] :
( ( member @ A @ Z4 @ ( heap_set_Heap @ A @ X2 ) )
=> ( Ra @ Z4 @ Z4 ) )
=> ( heap_rel_Heap @ A @ A @ Ra @ X2 @ X2 ) ) ).
% Heap.rel_refl_strong
thf(fact_2233_Heap_Orel__map_I2_J,axiom,
! [A: $tType,C: $tType,B: $tType,Sa: A > C > $o,X2: heap_Heap @ A,G: B > C,Y3: heap_Heap @ B] :
( ( heap_rel_Heap @ A @ C @ Sa @ X2 @ ( heap_map_Heap @ B @ C @ G @ Y3 ) )
= ( heap_rel_Heap @ A @ B
@ ^ [X4: A,Y5: B] : ( Sa @ X4 @ ( G @ Y5 ) )
@ X2
@ Y3 ) ) ).
% Heap.rel_map(2)
thf(fact_2234_Heap_Orel__map_I1_J,axiom,
! [A: $tType,C: $tType,B: $tType,Sb: C > B > $o,I: A > C,X2: heap_Heap @ A,Y3: heap_Heap @ B] :
( ( heap_rel_Heap @ C @ B @ Sb @ ( heap_map_Heap @ A @ C @ I @ X2 ) @ Y3 )
= ( heap_rel_Heap @ A @ B
@ ^ [X4: A] : ( Sb @ ( I @ X4 ) )
@ X2
@ Y3 ) ) ).
% Heap.rel_map(1)
thf(fact_2235_Heap_Orec__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,R3: A > B > $o,S5: C > D > $o] :
( bNF_rel_fun @ ( ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) ) > C ) @ ( ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ B @ ( heap_ext @ product_unit ) ) ) ) > D ) @ ( ( heap_Heap @ A ) > C ) @ ( ( heap_Heap @ B ) > D )
@ ( bNF_rel_fun @ ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) ) @ ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ B @ ( heap_ext @ product_unit ) ) ) ) @ C @ D
@ ( bNF_rel_fun @ ( heap_ext @ product_unit ) @ ( heap_ext @ product_unit ) @ ( option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) @ ( option @ ( product_prod @ B @ ( heap_ext @ product_unit ) ) )
@ ^ [Y2: heap_ext @ product_unit,Z2: heap_ext @ product_unit] : Y2 = Z2
@ ( rel_option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) @ ( product_prod @ B @ ( heap_ext @ product_unit ) )
@ ( basic_rel_prod @ A @ B @ ( heap_ext @ product_unit ) @ ( heap_ext @ product_unit ) @ R3
@ ^ [Y2: heap_ext @ product_unit,Z2: heap_ext @ product_unit] : Y2 = Z2 ) ) )
@ S5 )
@ ( bNF_rel_fun @ ( heap_Heap @ A ) @ ( heap_Heap @ B ) @ C @ D @ ( heap_rel_Heap @ A @ B @ R3 ) @ S5 )
@ ( heap_rec_Heap @ A @ C )
@ ( heap_rec_Heap @ B @ D ) ) ).
% Heap.rec_transfer
thf(fact_2236_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_Heap @ A ) > ( heap_Heap @ B ) > $o ) @ ( ( heap_Heap @ C ) > ( heap_Heap @ D ) > $o )
@ ( bNF_rel_fun @ A @ C @ ( B > $o ) @ ( D > $o ) @ Sa
@ ( bNF_rel_fun @ B @ D @ $o @ $o @ Sc
@ ^ [Y2: $o,Z2: $o] : Y2 = Z2 ) )
@ ( bNF_rel_fun @ ( heap_Heap @ A ) @ ( heap_Heap @ C ) @ ( ( heap_Heap @ B ) > $o ) @ ( ( heap_Heap @ D ) > $o ) @ ( heap_rel_Heap @ A @ C @ Sa )
@ ( bNF_rel_fun @ ( heap_Heap @ B ) @ ( heap_Heap @ D ) @ $o @ $o @ ( heap_rel_Heap @ B @ D @ Sc )
@ ^ [Y2: $o,Z2: $o] : Y2 = Z2 ) )
@ ( heap_rel_Heap @ A @ B )
@ ( heap_rel_Heap @ C @ D ) ) ).
% Heap.rel_transfer
thf(fact_2237_Heap_Omap__transfer,axiom,
! [A: $tType,B: $tType,F2: $tType,E: $tType,Rb: A > E > $o,Sd: B > F2 > $o] : ( bNF_rel_fun @ ( A > B ) @ ( E > F2 ) @ ( ( heap_Heap @ A ) > ( heap_Heap @ B ) ) @ ( ( heap_Heap @ E ) > ( heap_Heap @ F2 ) ) @ ( bNF_rel_fun @ A @ E @ B @ F2 @ Rb @ Sd ) @ ( bNF_rel_fun @ ( heap_Heap @ A ) @ ( heap_Heap @ E ) @ ( heap_Heap @ B ) @ ( heap_Heap @ F2 ) @ ( heap_rel_Heap @ A @ E @ Rb ) @ ( heap_rel_Heap @ B @ F2 @ Sd ) ) @ ( heap_map_Heap @ A @ B ) @ ( heap_map_Heap @ E @ F2 ) ) ).
% Heap.map_transfer
thf(fact_2238_Heap_OQuotient,axiom,
! [B: $tType,A: $tType,R3: A > A > $o,Abs: A > B,Rep: B > A,T4: A > B > $o] :
( ( quotient @ A @ B @ R3 @ Abs @ Rep @ T4 )
=> ( quotient @ ( heap_Heap @ A ) @ ( heap_Heap @ B ) @ ( heap_rel_Heap @ A @ A @ R3 ) @ ( heap_map_Heap @ A @ B @ Abs ) @ ( heap_map_Heap @ B @ A @ Rep ) @ ( heap_rel_Heap @ A @ B @ T4 ) ) ) ).
% Heap.Quotient
thf(fact_2239_execute__update,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R: ref @ A,V: A,H: heap_ext @ product_unit] :
( ( heap_execute @ product_unit @ ( update2 @ A @ R @ V ) @ H )
= ( some @ ( product_prod @ product_unit @ ( heap_ext @ product_unit ) ) @ ( product_Pair @ product_unit @ ( heap_ext @ product_unit ) @ product_Unity @ ( set4 @ A @ R @ V @ H ) ) ) ) ) ).
% execute_update
thf(fact_2240_Ref_Oupdate__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( update2 @ A )
= ( ^ [R5: ref @ A,V3: A] :
( heap_heap @ product_unit
@ ^ [H6: heap_ext @ product_unit] : ( product_Pair @ product_unit @ ( heap_ext @ product_unit ) @ product_Unity @ ( set4 @ A @ R5 @ V3 @ H6 ) ) ) ) ) ) ).
% Ref.update_def
thf(fact_2241_upd_H__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( upd2 @ A )
= ( ^ [A7: array @ A,I3: code_integer,X4: A] :
( heap_bind @ ( array @ A ) @ product_unit @ ( upd @ A @ ( code_nat_of_integer @ I3 ) @ X4 @ A7 )
@ ^ [Uu: array @ A] : ( heap_return @ product_unit @ product_Unity ) ) ) ) ) ).
% upd'_def
thf(fact_2242_default__unit__def,axiom,
( ( default_default @ product_unit )
= product_Unity ) ).
% default_unit_def
thf(fact_2243_rtranclp__is__Sup__relpowp,axiom,
! [A: $tType] :
( ( transitive_rtranclp @ A )
= ( ^ [P4: A > A > $o] :
( complete_Sup_Sup @ ( A > A > $o )
@ ( image2 @ nat @ ( A > A > $o )
@ ^ [N4: nat] : ( compow @ ( A > A > $o ) @ N4 @ P4 )
@ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% rtranclp_is_Sup_relpowp
thf(fact_2244_rtranclp__imp__Sup__relpowp,axiom,
! [A: $tType,P2: A > A > $o,X2: A,Y3: A] :
( ( transitive_rtranclp @ A @ P2 @ X2 @ Y3 )
=> ( complete_Sup_Sup @ ( A > A > $o )
@ ( image2 @ nat @ ( A > A > $o )
@ ^ [N4: nat] : ( compow @ ( A > A > $o ) @ N4 @ P2 )
@ ( top_top @ ( set @ nat ) ) )
@ X2
@ Y3 ) ) ).
% rtranclp_imp_Sup_relpowp
thf(fact_2245_map__option__o__case__sum,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F: D > C,G: A > ( option @ D ),H: 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 @ H ) )
= ( 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 ) @ H ) ) ) ).
% map_option_o_case_sum
thf(fact_2246_Heap_Opred__map,axiom,
! [B: $tType,A: $tType,Q2: B > $o,F: A > B,X2: heap_Heap @ A] :
( ( heap_pred_Heap @ B @ Q2 @ ( heap_map_Heap @ A @ B @ F @ X2 ) )
= ( heap_pred_Heap @ A @ ( comp @ B @ $o @ A @ Q2 @ F ) @ X2 ) ) ).
% Heap.pred_map
thf(fact_2247_Heap_Opred__True,axiom,
! [A: $tType] :
( ( heap_pred_Heap @ A
@ ^ [Uu: A] : $true )
= ( ^ [Uu: heap_Heap @ A] : $true ) ) ).
% Heap.pred_True
thf(fact_2248_disjE__realizer,axiom,
! [C: $tType,B: $tType,A: $tType,P2: A > $o,Q2: B > $o,X2: sum_sum @ A @ B,R3: C > $o,F: A > C,G: B > C] :
( ( sum_case_sum @ A @ $o @ B @ P2 @ Q2 @ X2 )
=> ( ! [P9: A] :
( ( P2 @ P9 )
=> ( R3 @ ( F @ P9 ) ) )
=> ( ! [Q7: B] :
( ( Q2 @ Q7 )
=> ( R3 @ ( G @ Q7 ) ) )
=> ( R3 @ ( sum_case_sum @ A @ C @ B @ F @ G @ X2 ) ) ) ) ) ).
% disjE_realizer
thf(fact_2249_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,P2: A > B > $o] :
( ( transitive_rtranclp @ ( product_prod @ A @ B ) @ R @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) )
=> ( ( P2 @ Bx @ By )
=> ( ! [A5: A,B4: B,Aa2: A,Ba: B] :
( ( R @ ( product_Pair @ A @ B @ A5 @ B4 ) @ ( product_Pair @ A @ B @ Aa2 @ Ba ) )
=> ( ( transitive_rtranclp @ ( product_prod @ A @ B ) @ R @ ( product_Pair @ A @ B @ Aa2 @ Ba ) @ ( product_Pair @ A @ B @ Bx @ By ) )
=> ( ( P2 @ Aa2 @ Ba )
=> ( P2 @ A5 @ B4 ) ) ) )
=> ( P2 @ Ax @ Ay ) ) ) ) ).
% converse_rtranclp_induct2
thf(fact_2250_converse__rtranclpE2,axiom,
! [A: $tType,B: $tType,R: ( product_prod @ A @ B ) > ( product_prod @ A @ B ) > $o,Xa2: A,Xb: B,Za2: A,Zb: B] :
( ( transitive_rtranclp @ ( product_prod @ A @ B ) @ R @ ( product_Pair @ A @ B @ Xa2 @ Xb ) @ ( product_Pair @ A @ B @ Za2 @ Zb ) )
=> ( ( ( product_Pair @ A @ B @ Xa2 @ Xb )
!= ( product_Pair @ A @ B @ Za2 @ Zb ) )
=> ~ ! [A5: A,B4: B] :
( ( R @ ( product_Pair @ A @ B @ Xa2 @ Xb ) @ ( product_Pair @ A @ B @ A5 @ B4 ) )
=> ~ ( transitive_rtranclp @ ( product_prod @ A @ B ) @ R @ ( product_Pair @ A @ B @ A5 @ B4 ) @ ( product_Pair @ A @ B @ Za2 @ Zb ) ) ) ) ) ).
% converse_rtranclpE2
thf(fact_2251_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,P2: A > B > $o] :
( ( transitive_rtranclp @ ( product_prod @ A @ B ) @ R @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) )
=> ( ( P2 @ Ax @ Ay )
=> ( ! [A5: A,B4: B,Aa2: A,Ba: B] :
( ( transitive_rtranclp @ ( product_prod @ A @ B ) @ R @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A5 @ B4 ) )
=> ( ( R @ ( product_Pair @ A @ B @ A5 @ B4 ) @ ( product_Pair @ A @ B @ Aa2 @ Ba ) )
=> ( ( P2 @ A5 @ B4 )
=> ( P2 @ Aa2 @ Ba ) ) ) )
=> ( P2 @ Bx @ By ) ) ) ) ).
% rtranclp_induct2
thf(fact_2252_o__case__sum,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,H: D > C,F: A > D,G: B > D] :
( ( comp @ D @ C @ ( sum_sum @ A @ B ) @ H @ ( sum_case_sum @ A @ D @ B @ F @ G ) )
= ( sum_case_sum @ A @ C @ B @ ( comp @ D @ C @ A @ H @ F ) @ ( comp @ D @ C @ B @ H @ G ) ) ) ).
% o_case_sum
thf(fact_2253_Heap_Opred__mono,axiom,
! [A: $tType,P2: A > $o,Pa: A > $o] :
( ( ord_less_eq @ ( A > $o ) @ P2 @ Pa )
=> ( ord_less_eq @ ( ( heap_Heap @ A ) > $o ) @ ( heap_pred_Heap @ A @ P2 ) @ ( heap_pred_Heap @ A @ Pa ) ) ) ).
% Heap.pred_mono
thf(fact_2254_Heap_Opred__mono__strong,axiom,
! [A: $tType,P2: A > $o,X2: heap_Heap @ A,Pa: A > $o] :
( ( heap_pred_Heap @ A @ P2 @ X2 )
=> ( ! [Z4: A] :
( ( member @ A @ Z4 @ ( heap_set_Heap @ A @ X2 ) )
=> ( ( P2 @ Z4 )
=> ( Pa @ Z4 ) ) )
=> ( heap_pred_Heap @ A @ Pa @ X2 ) ) ) ).
% Heap.pred_mono_strong
thf(fact_2255_Heap_Opred__cong,axiom,
! [A: $tType,X2: heap_Heap @ A,Ya: heap_Heap @ A,P2: A > $o,Pa: A > $o] :
( ( X2 = Ya )
=> ( ! [Z4: A] :
( ( member @ A @ Z4 @ ( heap_set_Heap @ A @ Ya ) )
=> ( ( P2 @ Z4 )
= ( Pa @ Z4 ) ) )
=> ( ( heap_pred_Heap @ A @ P2 @ X2 )
= ( heap_pred_Heap @ A @ Pa @ Ya ) ) ) ) ).
% Heap.pred_cong
thf(fact_2256_Transitive__Closure_Ortranclp__rtrancl__eq,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( transitive_rtranclp @ A
@ ^ [X4: A,Y5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R ) )
= ( ^ [X4: A,Y5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ).
% Transitive_Closure.rtranclp_rtrancl_eq
thf(fact_2257_Heap_Omap__cong__pred,axiom,
! [B: $tType,A: $tType,X2: heap_Heap @ A,Ya: heap_Heap @ A,F: A > B,G: A > B] :
( ( X2 = Ya )
=> ( ( heap_pred_Heap @ A
@ ^ [Z5: A] :
( ( F @ Z5 )
= ( G @ Z5 ) )
@ Ya )
=> ( ( heap_map_Heap @ A @ B @ F @ X2 )
= ( heap_map_Heap @ A @ B @ G @ Ya ) ) ) ) ).
% Heap.map_cong_pred
thf(fact_2258_rtrancl__def,axiom,
! [A: $tType] :
( ( transitive_rtrancl @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ( transitive_rtranclp @ A
@ ^ [X4: A,Y5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R5 ) ) ) ) ) ) ).
% rtrancl_def
thf(fact_2259_Enum_Ortranclp__rtrancl__eq,axiom,
! [A: $tType] :
( ( transitive_rtranclp @ A )
= ( ^ [R5: A > A > $o,X4: A,Y5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ ( transitive_rtrancl @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R5 ) ) ) ) ) ) ).
% Enum.rtranclp_rtrancl_eq
thf(fact_2260_Heap_Opred__transfer,axiom,
! [A: $tType,B: $tType,R3: A > B > $o] :
( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( heap_Heap @ A ) > $o ) @ ( ( heap_Heap @ B ) > $o )
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ R3
@ ^ [Y2: $o,Z2: $o] : Y2 = Z2 )
@ ( bNF_rel_fun @ ( heap_Heap @ A ) @ ( heap_Heap @ B ) @ $o @ $o @ ( heap_rel_Heap @ A @ B @ R3 )
@ ^ [Y2: $o,Z2: $o] : Y2 = Z2 )
@ ( heap_pred_Heap @ A )
@ ( heap_pred_Heap @ B ) ) ).
% Heap.pred_transfer
thf(fact_2261_Heap_Opred__set,axiom,
! [A: $tType] :
( ( heap_pred_Heap @ A )
= ( ^ [P4: A > $o,X4: heap_Heap @ A] :
! [Y5: A] :
( ( member @ A @ Y5 @ ( heap_set_Heap @ A @ X4 ) )
=> ( P4 @ Y5 ) ) ) ) ).
% Heap.pred_set
thf(fact_2262_Heap_Opred__rel,axiom,
! [A: $tType] :
( ( heap_pred_Heap @ A )
= ( ^ [P4: A > $o,X4: heap_Heap @ A] : ( heap_rel_Heap @ A @ A @ ( bNF_eq_onp @ A @ P4 ) @ X4 @ X4 ) ) ) ).
% Heap.pred_rel
thf(fact_2263_Heap_Orel__eq__onp,axiom,
! [A: $tType,P2: A > $o] :
( ( heap_rel_Heap @ A @ A @ ( bNF_eq_onp @ A @ P2 ) )
= ( bNF_eq_onp @ ( heap_Heap @ A ) @ ( heap_pred_Heap @ A @ P2 ) ) ) ).
% Heap.rel_eq_onp
thf(fact_2264_Heap_ODomainp__rel,axiom,
! [B: $tType,A: $tType,R3: A > B > $o] :
( ( domainp @ ( heap_Heap @ A ) @ ( heap_Heap @ B ) @ ( heap_rel_Heap @ A @ B @ R3 ) )
= ( heap_pred_Heap @ A @ ( domainp @ A @ B @ R3 ) ) ) ).
% Heap.Domainp_rel
thf(fact_2265_Heap_Ocase__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,R3: A > B > $o,S5: C > D > $o] :
( bNF_rel_fun @ ( ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) ) > C ) @ ( ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ B @ ( heap_ext @ product_unit ) ) ) ) > D ) @ ( ( heap_Heap @ A ) > C ) @ ( ( heap_Heap @ B ) > D )
@ ( bNF_rel_fun @ ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) ) @ ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ B @ ( heap_ext @ product_unit ) ) ) ) @ C @ D
@ ( bNF_rel_fun @ ( heap_ext @ product_unit ) @ ( heap_ext @ product_unit ) @ ( option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) @ ( option @ ( product_prod @ B @ ( heap_ext @ product_unit ) ) )
@ ^ [Y2: heap_ext @ product_unit,Z2: heap_ext @ product_unit] : Y2 = Z2
@ ( rel_option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) @ ( product_prod @ B @ ( heap_ext @ product_unit ) )
@ ( basic_rel_prod @ A @ B @ ( heap_ext @ product_unit ) @ ( heap_ext @ product_unit ) @ R3
@ ^ [Y2: heap_ext @ product_unit,Z2: heap_ext @ product_unit] : Y2 = Z2 ) ) )
@ S5 )
@ ( bNF_rel_fun @ ( heap_Heap @ A ) @ ( heap_Heap @ B ) @ C @ D @ ( heap_rel_Heap @ A @ B @ R3 ) @ S5 )
@ ( heap_case_Heap @ A @ C )
@ ( heap_case_Heap @ B @ D ) ) ).
% Heap.case_transfer
thf(fact_2266_iterate_Oelims,axiom,
! [A: $tType,B: $tType,X2: code_natural,Xa2: B > A > ( product_prod @ B @ A ),Xb: B,Y3: A > ( product_prod @ B @ A )] :
( ( ( iterate @ B @ A @ X2 @ Xa2 @ Xb )
= Y3 )
=> ( ( ( X2
= ( zero_zero @ code_natural ) )
=> ( Y3
= ( product_Pair @ B @ A @ Xb ) ) )
& ( ( X2
!= ( zero_zero @ code_natural ) )
=> ( Y3
= ( product_scomp @ A @ B @ A @ ( product_prod @ B @ A ) @ ( Xa2 @ Xb ) @ ( iterate @ B @ A @ ( minus_minus @ code_natural @ X2 @ ( one_one @ code_natural ) ) @ Xa2 ) ) ) ) ) ) ).
% iterate.elims
thf(fact_2267_iterate_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( iterate @ B @ A )
= ( ^ [K3: code_natural,F4: B > A > ( product_prod @ B @ A ),X4: B] :
( if @ ( A > ( product_prod @ B @ A ) )
@ ( K3
= ( zero_zero @ code_natural ) )
@ ( product_Pair @ B @ A @ X4 )
@ ( product_scomp @ A @ B @ A @ ( product_prod @ B @ A ) @ ( F4 @ X4 ) @ ( iterate @ B @ A @ ( minus_minus @ code_natural @ K3 @ ( one_one @ code_natural ) ) @ F4 ) ) ) ) ) ).
% iterate.simps
thf(fact_2268_r__OO__conversep__into__equivclp,axiom,
! [A: $tType,R: A > A > $o] : ( ord_less_eq @ ( A > A > $o ) @ ( relcompp @ A @ A @ A @ ( transitive_rtranclp @ A @ R ) @ ( transitive_rtranclp @ A @ ( conversep @ A @ A @ R ) ) ) @ ( equiv_equivclp @ A @ R ) ) ).
% r_OO_conversep_into_equivclp
thf(fact_2269_mult__inj__if__coprime__nat,axiom,
! [B: $tType,A: $tType,F: A > nat,A6: set @ A,G: B > nat,B6: set @ B] :
( ( inj_on @ A @ nat @ F @ A6 )
=> ( ( inj_on @ B @ nat @ G @ B6 )
=> ( ! [A5: A,B4: B] :
( ( member @ A @ A5 @ A6 )
=> ( ( member @ B @ B4 @ B6 )
=> ( algebr8660921524188924756oprime @ nat @ ( F @ A5 ) @ ( G @ B4 ) ) ) )
=> ( inj_on @ ( product_prod @ A @ B ) @ nat
@ ( product_case_prod @ A @ B @ nat
@ ^ [A7: A,B5: B] : ( times_times @ nat @ ( F @ A7 ) @ ( G @ B5 ) ) )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu: A] : B6 ) ) ) ) ) ).
% mult_inj_if_coprime_nat
thf(fact_2270_length__filter__map,axiom,
! [A: $tType,B: $tType,P2: A > $o,F: B > A,Xs: list @ B] :
( ( size_size @ ( list @ A ) @ ( filter2 @ A @ P2 @ ( map @ B @ A @ F @ Xs ) ) )
= ( size_size @ ( list @ B ) @ ( filter2 @ B @ ( comp @ A @ $o @ B @ P2 @ F ) @ Xs ) ) ) ).
% length_filter_map
thf(fact_2271_partition__filter__conv,axiom,
! [A: $tType] :
( ( partition @ A )
= ( ^ [F4: A > $o,Xs2: list @ A] : ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( filter2 @ A @ F4 @ Xs2 ) @ ( filter2 @ A @ ( comp @ $o @ $o @ A @ (~) @ F4 ) @ Xs2 ) ) ) ) ).
% partition_filter_conv
thf(fact_2272_partition__filter2,axiom,
! [A: $tType,P2: A > $o,Xs: list @ A] :
( ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( partition @ A @ P2 @ Xs ) )
= ( filter2 @ A @ ( comp @ $o @ $o @ A @ (~) @ P2 ) @ Xs ) ) ).
% partition_filter2
thf(fact_2273_Heap_Ocase__distrib,axiom,
! [C: $tType,B: $tType,A: $tType,H: B > C,F: ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) ) > B,Heap: heap_Heap @ A] :
( ( H @ ( heap_case_Heap @ A @ B @ F @ Heap ) )
= ( heap_case_Heap @ A @ C
@ ^ [X4: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) )] : ( H @ ( F @ X4 ) )
@ Heap ) ) ).
% Heap.case_distrib
thf(fact_2274_filter__map,axiom,
! [A: $tType,B: $tType,P2: A > $o,F: B > A,Xs: list @ B] :
( ( filter2 @ A @ P2 @ ( map @ B @ A @ F @ Xs ) )
= ( map @ B @ A @ F @ ( filter2 @ B @ ( comp @ A @ $o @ B @ P2 @ F ) @ Xs ) ) ) ).
% filter_map
thf(fact_2275_filter__shuffles,axiom,
! [A: $tType,P2: A > $o,Xs: list @ A,Ys: list @ A] :
( ( image2 @ ( list @ A ) @ ( list @ A ) @ ( filter2 @ A @ P2 ) @ ( shuffles @ A @ Xs @ Ys ) )
= ( shuffles @ A @ ( filter2 @ A @ P2 @ Xs ) @ ( filter2 @ A @ P2 @ Ys ) ) ) ).
% filter_shuffles
thf(fact_2276_equivclp__OO__equivclp__le__equivclp,axiom,
! [A: $tType,R: A > A > $o] : ( ord_less_eq @ ( A > A > $o ) @ ( relcompp @ A @ A @ A @ ( equiv_equivclp @ A @ R ) @ ( equiv_equivclp @ A @ R ) ) @ ( equiv_equivclp @ A @ R ) ) ).
% equivclp_OO_equivclp_le_equivclp
thf(fact_2277_map__of__filter__in,axiom,
! [B: $tType,A: $tType,Xs: list @ ( product_prod @ B @ A ),K: B,Z3: A,P2: B > A > $o] :
( ( ( map_of @ B @ A @ Xs @ K )
= ( some @ A @ Z3 ) )
=> ( ( P2 @ K @ Z3 )
=> ( ( map_of @ B @ A @ ( filter2 @ ( product_prod @ B @ A ) @ ( product_case_prod @ B @ A @ $o @ P2 ) @ Xs ) @ K )
= ( some @ A @ Z3 ) ) ) ) ).
% map_of_filter_in
thf(fact_2278_Heap_Ocase,axiom,
! [B: $tType,A: $tType,F: ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) ) > B,X2: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) )] :
( ( heap_case_Heap @ A @ B @ F @ ( heap_Heap2 @ A @ X2 ) )
= ( F @ X2 ) ) ).
% Heap.case
thf(fact_2279_map__filter__def,axiom,
! [B: $tType,A: $tType] :
( ( map_filter @ A @ B )
= ( ^ [F4: A > ( option @ B ),Xs2: list @ A] :
( map @ A @ B @ ( comp @ ( option @ B ) @ B @ A @ ( the2 @ B ) @ F4 )
@ ( filter2 @ A
@ ^ [X4: A] :
( ( F4 @ X4 )
!= ( none @ B ) )
@ Xs2 ) ) ) ) ).
% map_filter_def
thf(fact_2280_iterate_Opelims,axiom,
! [A: $tType,B: $tType,X2: code_natural,Xa2: B > A > ( product_prod @ B @ A ),Xb: B,Y3: A > ( product_prod @ B @ A )] :
( ( ( iterate @ B @ A @ X2 @ Xa2 @ Xb )
= Y3 )
=> ( ( 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 ) @ X2 @ ( product_Pair @ ( B > A > ( product_prod @ B @ A ) ) @ B @ Xa2 @ Xb ) ) )
=> ~ ( ( ( ( X2
= ( zero_zero @ code_natural ) )
=> ( Y3
= ( product_Pair @ B @ A @ Xb ) ) )
& ( ( X2
!= ( zero_zero @ code_natural ) )
=> ( Y3
= ( product_scomp @ A @ B @ A @ ( product_prod @ B @ A ) @ ( Xa2 @ Xb ) @ ( iterate @ B @ A @ ( minus_minus @ code_natural @ X2 @ ( one_one @ code_natural ) ) @ Xa2 ) ) ) ) )
=> ~ ( 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 ) @ X2 @ ( product_Pair @ ( B > A > ( product_prod @ B @ A ) ) @ B @ Xa2 @ Xb ) ) ) ) ) ) ).
% iterate.pelims
thf(fact_2281_Gcd__eq__Max,axiom,
! [M: set @ nat] :
( ( finite_finite2 @ nat @ M )
=> ( ( M
!= ( bot_bot @ ( set @ nat ) ) )
=> ( ~ ( member @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( gcd_Gcd @ nat @ M )
= ( lattic643756798349783984er_Max @ nat
@ ( complete_Inf_Inf @ ( set @ nat )
@ ( image2 @ nat @ ( set @ nat )
@ ^ [M7: nat] :
( collect @ nat
@ ^ [D5: nat] : ( dvd_dvd @ nat @ D5 @ M7 ) )
@ M ) ) ) ) ) ) ) ).
% Gcd_eq_Max
thf(fact_2282_select__weight__drop__zero,axiom,
! [A: $tType,Xs: list @ ( product_prod @ code_natural @ A )] :
( ( select_weight @ A
@ ( filter2 @ ( product_prod @ code_natural @ A )
@ ( product_case_prod @ code_natural @ A @ $o
@ ^ [K3: code_natural,Uu: A] : ( ord_less @ code_natural @ ( zero_zero @ code_natural ) @ K3 ) )
@ Xs ) )
= ( select_weight @ A @ Xs ) ) ).
% select_weight_drop_zero
thf(fact_2283_pick__drop__zero,axiom,
! [A: $tType,Xs: list @ ( product_prod @ code_natural @ A )] :
( ( pick @ A
@ ( filter2 @ ( product_prod @ code_natural @ A )
@ ( product_case_prod @ code_natural @ A @ $o
@ ^ [K3: code_natural,Uu: A] : ( ord_less @ code_natural @ ( zero_zero @ code_natural ) @ K3 ) )
@ Xs ) )
= ( pick @ A @ Xs ) ) ).
% pick_drop_zero
thf(fact_2284_case__sum__o__map__sum__id,axiom,
! [A: $tType,B: $tType,C: $tType,G: B > A,F: C > A,X2: 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 ) ) @ X2 )
= ( sum_case_sum @ C @ A @ B @ ( comp @ C @ A @ C @ F @ ( id @ C ) ) @ G @ X2 ) ) ).
% case_sum_o_map_sum_id
thf(fact_2285_Gcd__mono,axiom,
! [A: $tType,B: $tType] :
( ( semiring_Gcd @ A )
=> ! [A6: set @ B,F: B > A,G: B > A] :
( ! [X: B] :
( ( member @ B @ X @ A6 )
=> ( dvd_dvd @ A @ ( F @ X ) @ ( G @ X ) ) )
=> ( dvd_dvd @ A @ ( gcd_Gcd @ A @ ( image2 @ B @ A @ F @ A6 ) ) @ ( gcd_Gcd @ A @ ( image2 @ B @ A @ G @ A6 ) ) ) ) ) ).
% Gcd_mono
thf(fact_2286_sum_Omap__comp,axiom,
! [D: $tType,F2: $tType,E: $tType,C: $tType,B: $tType,A: $tType,G1: C > E,G22: D > F2,F1: A > C,F23: B > D,V: sum_sum @ A @ B] :
( ( sum_map_sum @ C @ E @ D @ F2 @ G1 @ G22 @ ( sum_map_sum @ A @ C @ B @ D @ F1 @ F23 @ V ) )
= ( sum_map_sum @ A @ E @ B @ F2 @ ( comp @ C @ E @ A @ G1 @ F1 ) @ ( comp @ D @ F2 @ B @ G22 @ F23 ) @ V ) ) ).
% sum.map_comp
thf(fact_2287_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_2288_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,X2: sum_sum @ D @ E] :
( ( sum_case_sum @ B @ A @ C @ L @ R @ ( sum_map_sum @ D @ B @ E @ C @ F @ G @ X2 ) )
= ( sum_case_sum @ D @ A @ E @ ( comp @ B @ A @ D @ L @ F ) @ ( comp @ C @ A @ E @ R @ G ) @ X2 ) ) ).
% case_sum_map_sum
thf(fact_2289_map__sum_Ocompositionality,axiom,
! [D: $tType,F2: $tType,E: $tType,C: $tType,B: $tType,A: $tType,F: C > E,G: D > F2,H: A > C,I: B > D,Sum: sum_sum @ A @ B] :
( ( sum_map_sum @ C @ E @ D @ F2 @ F @ G @ ( sum_map_sum @ A @ C @ B @ D @ H @ I @ Sum ) )
= ( sum_map_sum @ A @ E @ B @ F2 @ ( comp @ C @ E @ A @ F @ H ) @ ( comp @ D @ F2 @ B @ G @ I ) @ Sum ) ) ).
% map_sum.compositionality
thf(fact_2290_map__sum_Ocomp,axiom,
! [A: $tType,C: $tType,E: $tType,F2: $tType,D: $tType,B: $tType,F: C > E,G: D > F2,H: A > C,I: B > D] :
( ( comp @ ( sum_sum @ C @ D ) @ ( sum_sum @ E @ F2 ) @ ( sum_sum @ A @ B ) @ ( sum_map_sum @ C @ E @ D @ F2 @ F @ G ) @ ( sum_map_sum @ A @ C @ B @ D @ H @ I ) )
= ( sum_map_sum @ A @ E @ B @ F2 @ ( comp @ C @ E @ A @ F @ H ) @ ( comp @ D @ F2 @ B @ G @ I ) ) ) ).
% map_sum.comp
thf(fact_2291_sum_Osize__gen__o__map,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F: C > nat,Fa: D > nat,G: A > C,Ga: B > D] :
( ( comp @ ( sum_sum @ C @ D ) @ nat @ ( sum_sum @ A @ B ) @ ( basic_BNF_size_sum @ C @ D @ F @ Fa ) @ ( sum_map_sum @ A @ C @ B @ D @ G @ Ga ) )
= ( basic_BNF_size_sum @ A @ B @ ( comp @ C @ nat @ A @ F @ G ) @ ( comp @ D @ nat @ B @ Fa @ Ga ) ) ) ).
% sum.size_gen_o_map
thf(fact_2292_Gcd__int__eq,axiom,
! [N: set @ nat] :
( ( gcd_Gcd @ int @ ( image2 @ nat @ int @ ( semiring_1_of_nat @ int ) @ N ) )
= ( semiring_1_of_nat @ int @ ( gcd_Gcd @ nat @ N ) ) ) ).
% Gcd_int_eq
thf(fact_2293_extract__def,axiom,
! [A: $tType] :
( ( extract @ A )
= ( ^ [P4: A > $o,Xs2: 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 ) ) ) )
@ ^ [Y5: A,Ys2: 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 ) @ Xs2 ) @ ( product_Pair @ A @ ( list @ A ) @ Y5 @ Ys2 ) ) )
@ ( dropWhile @ A @ ( comp @ $o @ $o @ A @ (~) @ P4 ) @ Xs2 ) ) ) ) ).
% extract_def
thf(fact_2294_sym__INTER,axiom,
! [B: $tType,A: $tType,S5: set @ A,R: A > ( set @ ( product_prod @ B @ B ) )] :
( ! [X: A] :
( ( member @ A @ X @ S5 )
=> ( sym @ B @ ( R @ X ) ) )
=> ( sym @ B @ ( complete_Inf_Inf @ ( set @ ( product_prod @ B @ B ) ) @ ( image2 @ A @ ( set @ ( product_prod @ B @ B ) ) @ R @ S5 ) ) ) ) ).
% sym_INTER
thf(fact_2295_zip__takeWhile__fst,axiom,
! [A: $tType,B: $tType,P2: A > $o,Xs: list @ A,Ys: list @ B] :
( ( zip @ A @ B @ ( takeWhile @ A @ P2 @ Xs ) @ Ys )
= ( takeWhile @ ( product_prod @ A @ B ) @ ( comp @ A @ $o @ ( product_prod @ A @ B ) @ P2 @ ( product_fst @ A @ B ) ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) ).
% zip_takeWhile_fst
thf(fact_2296_zip__takeWhile__snd,axiom,
! [A: $tType,B: $tType,Xs: list @ A,P2: B > $o,Ys: list @ B] :
( ( zip @ A @ B @ Xs @ ( takeWhile @ B @ P2 @ Ys ) )
= ( takeWhile @ ( product_prod @ A @ B ) @ ( comp @ B @ $o @ ( product_prod @ A @ B ) @ P2 @ ( product_snd @ A @ B ) ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) ).
% zip_takeWhile_snd
thf(fact_2297_sym__def,axiom,
! [A: $tType] :
( ( sym @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
! [X4: A,Y5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R5 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X4 ) @ R5 ) ) ) ) ).
% sym_def
thf(fact_2298_symI,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ! [A5: A,B4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B4 ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ A5 ) @ R ) )
=> ( sym @ A @ R ) ) ).
% symI
thf(fact_2299_symE,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),B2: A,A3: A] :
( ( sym @ A @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R ) ) ) ).
% symE
thf(fact_2300_symD,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),B2: A,A3: A] :
( ( sym @ A @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R ) ) ) ).
% symD
thf(fact_2301_takeWhile__map,axiom,
! [A: $tType,B: $tType,P2: A > $o,F: B > A,Xs: list @ B] :
( ( takeWhile @ A @ P2 @ ( map @ B @ A @ F @ Xs ) )
= ( map @ B @ A @ F @ ( takeWhile @ B @ ( comp @ A @ $o @ B @ P2 @ F ) @ Xs ) ) ) ).
% takeWhile_map
thf(fact_2302_dropWhile__map,axiom,
! [A: $tType,B: $tType,P2: A > $o,F: B > A,Xs: list @ B] :
( ( dropWhile @ A @ P2 @ ( map @ B @ A @ F @ Xs ) )
= ( map @ B @ A @ F @ ( dropWhile @ B @ ( comp @ A @ $o @ B @ P2 @ F ) @ Xs ) ) ) ).
% dropWhile_map
thf(fact_2303_sym__UNION,axiom,
! [B: $tType,A: $tType,S5: set @ A,R: A > ( set @ ( product_prod @ B @ B ) )] :
( ! [X: A] :
( ( member @ A @ X @ S5 )
=> ( sym @ B @ ( R @ X ) ) )
=> ( sym @ B @ ( complete_Sup_Sup @ ( set @ ( product_prod @ B @ B ) ) @ ( image2 @ A @ ( set @ ( product_prod @ B @ B ) ) @ R @ S5 ) ) ) ) ).
% sym_UNION
thf(fact_2304_find__dropWhile,axiom,
! [A: $tType] :
( ( find @ A )
= ( ^ [P4: A > $o,Xs2: list @ A] :
( case_list @ ( option @ A ) @ A @ ( none @ A )
@ ^ [X4: A,Xa4: list @ A] : ( some @ A @ X4 )
@ ( dropWhile @ A @ ( comp @ $o @ $o @ A @ (~) @ P4 ) @ Xs2 ) ) ) ) ).
% find_dropWhile
thf(fact_2305_Gcd__int__def,axiom,
( ( gcd_Gcd @ int )
= ( ^ [K5: set @ int] : ( semiring_1_of_nat @ int @ ( gcd_Gcd @ nat @ ( image2 @ int @ nat @ ( comp @ int @ nat @ int @ nat2 @ ( abs_abs @ int ) ) @ K5 ) ) ) ) ) ).
% Gcd_int_def
thf(fact_2306_Gcd__nat__abs__eq,axiom,
! [K6: set @ int] :
( ( gcd_Gcd @ nat
@ ( image2 @ int @ nat
@ ^ [K3: int] : ( nat2 @ ( abs_abs @ int @ K3 ) )
@ K6 ) )
= ( nat2 @ ( gcd_Gcd @ int @ K6 ) ) ) ).
% Gcd_nat_abs_eq
thf(fact_2307_sum_Osize__gen_I1_J,axiom,
! [B: $tType,A: $tType,Xa2: A > nat,X2: B > nat,X1: A] :
( ( basic_BNF_size_sum @ A @ B @ Xa2 @ X2 @ ( sum_Inl @ A @ B @ X1 ) )
= ( plus_plus @ nat @ ( Xa2 @ X1 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% sum.size_gen(1)
thf(fact_2308_sum_Osize__gen_I2_J,axiom,
! [A: $tType,B: $tType,Xa2: A > nat,X2: B > nat,X22: B] :
( ( basic_BNF_size_sum @ A @ B @ Xa2 @ X2 @ ( sum_Inr @ B @ A @ X22 ) )
= ( plus_plus @ nat @ ( X2 @ X22 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% sum.size_gen(2)
thf(fact_2309_map__sum__o__inj_I1_J,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,F: A > B,G: D > C] :
( ( comp @ ( sum_sum @ A @ D ) @ ( sum_sum @ B @ C ) @ A @ ( sum_map_sum @ A @ B @ D @ C @ F @ G ) @ ( sum_Inl @ A @ D ) )
= ( comp @ B @ ( sum_sum @ B @ C ) @ A @ ( sum_Inl @ B @ C ) @ F ) ) ).
% map_sum_o_inj(1)
thf(fact_2310_range__abs__Nats,axiom,
( ( image2 @ int @ int @ ( abs_abs @ int ) @ ( top_top @ ( set @ int ) ) )
= ( semiring_1_Nats @ int ) ) ).
% range_abs_Nats
thf(fact_2311_Gcd__abs__eq,axiom,
! [K6: set @ int] :
( ( gcd_Gcd @ int @ ( image2 @ int @ int @ ( abs_abs @ int ) @ K6 ) )
= ( gcd_Gcd @ int @ K6 ) ) ).
% Gcd_abs_eq
thf(fact_2312_UNIV__sum,axiom,
! [A: $tType,B: $tType] :
( ( top_top @ ( set @ ( sum_sum @ A @ B ) ) )
= ( sup_sup @ ( set @ ( sum_sum @ A @ B ) ) @ ( image2 @ A @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B ) @ ( top_top @ ( set @ A ) ) ) @ ( image2 @ B @ ( sum_sum @ A @ B ) @ ( sum_Inr @ B @ A ) @ ( top_top @ ( set @ B ) ) ) ) ) ).
% UNIV_sum
thf(fact_2313_Basic__BNF__LFPs_OInr__def__alt,axiom,
! [B: $tType,A: $tType] :
( ( sum_Inr @ A @ B )
= ( ^ [A7: A] : ( basic_BNF_xtor @ ( sum_sum @ B @ A ) @ ( bNF_id_bnf @ ( sum_sum @ B @ A ) @ ( sum_Inr @ A @ B @ A7 ) ) ) ) ) ).
% Basic_BNF_LFPs.Inr_def_alt
thf(fact_2314_Basic__BNF__LFPs_OInl__def__alt,axiom,
! [B: $tType,A: $tType] :
( ( sum_Inl @ A @ B )
= ( ^ [A7: A] : ( basic_BNF_xtor @ ( sum_sum @ A @ B ) @ ( bNF_id_bnf @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B @ A7 ) ) ) ) ) ).
% Basic_BNF_LFPs.Inl_def_alt
thf(fact_2315_not__arg__cong__Inr,axiom,
! [B: $tType,A: $tType,X2: A,Y3: A] :
( ( X2 != Y3 )
=> ( ( sum_Inr @ A @ B @ X2 )
!= ( sum_Inr @ A @ B @ Y3 ) ) ) ).
% not_arg_cong_Inr
thf(fact_2316_obj__sumE,axiom,
! [A: $tType,B: $tType,S2: sum_sum @ A @ B] :
( ! [X: A] :
( S2
!= ( sum_Inl @ A @ B @ X ) )
=> ~ ! [X: B] :
( S2
!= ( sum_Inr @ B @ A @ X ) ) ) ).
% obj_sumE
thf(fact_2317_case__sum__expand__Inr,axiom,
! [B: $tType,C: $tType,A: $tType,F: ( sum_sum @ A @ C ) > B,G: A > B,X2: sum_sum @ A @ C] :
( ( ( comp @ ( sum_sum @ A @ C ) @ B @ A @ F @ ( sum_Inl @ A @ C ) )
= G )
=> ( ( F @ X2 )
= ( sum_case_sum @ A @ B @ C @ G @ ( comp @ ( sum_sum @ A @ C ) @ B @ C @ F @ ( sum_Inr @ C @ A ) ) @ X2 ) ) ) ).
% case_sum_expand_Inr
thf(fact_2318_case__sum__expand__Inr_H,axiom,
! [B: $tType,C: $tType,A: $tType,F: ( sum_sum @ A @ C ) > B,G: A > B,H: C > B] :
( ( ( comp @ ( sum_sum @ A @ C ) @ B @ A @ F @ ( sum_Inl @ A @ C ) )
= G )
=> ( ( H
= ( comp @ ( sum_sum @ A @ C ) @ B @ C @ F @ ( sum_Inr @ C @ A ) ) )
= ( ( sum_case_sum @ A @ B @ C @ G @ H )
= F ) ) ) ).
% case_sum_expand_Inr'
thf(fact_2319_case__sum__expand__Inr__pointfree,axiom,
! [B: $tType,C: $tType,A: $tType,F: ( sum_sum @ A @ C ) > B,G: A > B] :
( ( ( comp @ ( sum_sum @ A @ C ) @ B @ A @ F @ ( sum_Inl @ A @ C ) )
= G )
=> ( ( sum_case_sum @ A @ B @ C @ G @ ( comp @ ( sum_sum @ A @ C ) @ B @ C @ F @ ( sum_Inr @ C @ A ) ) )
= F ) ) ).
% case_sum_expand_Inr_pointfree
thf(fact_2320_finite__image__absD,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [S5: set @ A] :
( ( finite_finite2 @ A @ ( image2 @ A @ A @ ( abs_abs @ A ) @ S5 ) )
=> ( finite_finite2 @ A @ S5 ) ) ) ).
% finite_image_absD
thf(fact_2321_case__sum__o__inj_I2_J,axiom,
! [A: $tType,B: $tType,C: $tType,F: A > B,G: C > B] :
( ( comp @ ( sum_sum @ A @ C ) @ B @ C @ ( sum_case_sum @ A @ B @ C @ F @ G ) @ ( sum_Inr @ C @ A ) )
= G ) ).
% case_sum_o_inj(2)
thf(fact_2322_case__sum__o__inj_I1_J,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > B,G: C > B] :
( ( comp @ ( sum_sum @ A @ C ) @ B @ A @ ( sum_case_sum @ A @ B @ C @ F @ G ) @ ( sum_Inl @ A @ C ) )
= F ) ).
% case_sum_o_inj(1)
thf(fact_2323_int__in__range__abs,axiom,
! [N5: nat] : ( member @ int @ ( semiring_1_of_nat @ int @ N5 ) @ ( image2 @ int @ int @ ( abs_abs @ int ) @ ( top_top @ ( set @ int ) ) ) ) ).
% int_in_range_abs
thf(fact_2324_map__sum__o__inj_I2_J,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,F: A > B,G: D > C] :
( ( comp @ ( sum_sum @ A @ D ) @ ( sum_sum @ B @ C ) @ D @ ( sum_map_sum @ A @ B @ D @ C @ F @ G ) @ ( sum_Inr @ D @ A ) )
= ( comp @ C @ ( sum_sum @ B @ C ) @ D @ ( sum_Inr @ C @ B ) @ G ) ) ).
% map_sum_o_inj(2)
thf(fact_2325_measure__function__int,axiom,
fun_is_measure @ int @ ( comp @ int @ nat @ int @ nat2 @ ( abs_abs @ int ) ) ).
% measure_function_int
thf(fact_2326_sum_Osize_I3_J,axiom,
! [A: $tType,B: $tType,X1: A] :
( ( size_size @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B @ X1 ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% sum.size(3)
thf(fact_2327_sum_Osize_I4_J,axiom,
! [B: $tType,A: $tType,X22: B] :
( ( size_size @ ( sum_sum @ A @ B ) @ ( sum_Inr @ B @ A @ X22 ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% sum.size(4)
thf(fact_2328_prod_OPlus,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A6: set @ B,B6: set @ C,G: ( sum_sum @ B @ C ) > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ( finite_finite2 @ C @ B6 )
=> ( ( groups7121269368397514597t_prod @ ( sum_sum @ B @ C ) @ A @ G @ ( sum_Plus @ B @ C @ A6 @ B6 ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ ( comp @ ( sum_sum @ B @ C ) @ A @ B @ G @ ( sum_Inl @ B @ C ) ) @ A6 ) @ ( groups7121269368397514597t_prod @ C @ A @ ( comp @ ( sum_sum @ B @ C ) @ A @ C @ G @ ( sum_Inr @ C @ B ) ) @ B6 ) ) ) ) ) ) ).
% prod.Plus
thf(fact_2329_sum_OPlus,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [A6: set @ B,B6: set @ C,G: ( sum_sum @ B @ C ) > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ( finite_finite2 @ C @ B6 )
=> ( ( groups7311177749621191930dd_sum @ ( sum_sum @ B @ C ) @ A @ G @ ( sum_Plus @ B @ C @ A6 @ B6 ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ ( comp @ ( sum_sum @ B @ C ) @ A @ B @ G @ ( sum_Inl @ B @ C ) ) @ A6 ) @ ( groups7311177749621191930dd_sum @ C @ A @ ( comp @ ( sum_sum @ B @ C ) @ A @ C @ G @ ( sum_Inr @ C @ B ) ) @ B6 ) ) ) ) ) ) ).
% sum.Plus
thf(fact_2330_rat__abs__code,axiom,
! [P: rat] :
( ( quotient_of @ ( abs_abs @ rat @ P ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A7: int] : ( product_Pair @ int @ int @ ( abs_abs @ int @ A7 ) )
@ ( quotient_of @ P ) ) ) ).
% rat_abs_code
thf(fact_2331_is__measure_Osimps,axiom,
! [A: $tType] :
( ( fun_is_measure @ A )
= ( ^ [A7: A > nat] :
? [X10: A > nat] :
( ^ [Y2: A > nat,Z2: A > nat] : Y2 = Z2
@ A7
@ X10 ) ) ) ).
% is_measure.simps
thf(fact_2332_is__measure__trivial,axiom,
! [A: $tType,F: A > nat] : ( fun_is_measure @ A @ F ) ).
% is_measure_trivial
thf(fact_2333_measure__fst,axiom,
! [B: $tType,A: $tType,F: A > nat] :
( ( fun_is_measure @ A @ F )
=> ( fun_is_measure @ ( product_prod @ A @ B )
@ ^ [P3: product_prod @ A @ B] : ( F @ ( product_fst @ A @ B @ P3 ) ) ) ) ).
% measure_fst
thf(fact_2334_measure__snd,axiom,
! [A: $tType,B: $tType,F: A > nat] :
( ( fun_is_measure @ A @ F )
=> ( fun_is_measure @ ( product_prod @ B @ A )
@ ^ [P3: product_prod @ B @ A] : ( F @ ( product_snd @ B @ A @ P3 ) ) ) ) ).
% measure_snd
thf(fact_2335_Plus__def,axiom,
! [B: $tType,A: $tType] :
( ( sum_Plus @ A @ B )
= ( ^ [A8: set @ A,B7: set @ B] : ( sup_sup @ ( set @ ( sum_sum @ A @ B ) ) @ ( image2 @ A @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B ) @ A8 ) @ ( image2 @ B @ ( sum_sum @ A @ B ) @ ( sum_Inr @ B @ A ) @ B7 ) ) ) ) ).
% Plus_def
thf(fact_2336_measure__size,axiom,
! [A: $tType] :
( ( size @ A )
=> ( fun_is_measure @ A @ ( size_size @ A ) ) ) ).
% measure_size
thf(fact_2337_sum_Osize__neq,axiom,
! [A: $tType,B: $tType,X2: sum_sum @ A @ B] :
( ( size_size @ ( sum_sum @ A @ B ) @ X2 )
!= ( zero_zero @ nat ) ) ).
% sum.size_neq
thf(fact_2338_Field__csum,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ B @ B )] :
( ( field2 @ ( sum_sum @ A @ B ) @ ( bNF_Cardinal_csum @ A @ B @ R @ S2 ) )
= ( sup_sup @ ( set @ ( sum_sum @ A @ B ) ) @ ( image2 @ A @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B ) @ ( field2 @ A @ R ) ) @ ( image2 @ B @ ( sum_sum @ A @ B ) @ ( sum_Inr @ B @ A ) @ ( field2 @ B @ S2 ) ) ) ) ).
% Field_csum
thf(fact_2339_surj__int__encode,axiom,
( ( image2 @ int @ nat @ nat_int_encode @ ( top_top @ ( set @ int ) ) )
= ( top_top @ ( set @ nat ) ) ) ).
% surj_int_encode
thf(fact_2340_surj__int__decode,axiom,
( ( image2 @ nat @ int @ nat_int_decode @ ( top_top @ ( set @ nat ) ) )
= ( top_top @ ( set @ int ) ) ) ).
% surj_int_decode
thf(fact_2341_Ints__def,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_Ints @ A )
= ( image2 @ int @ A @ ( ring_1_of_int @ A ) @ ( top_top @ ( set @ int ) ) ) ) ) ).
% Ints_def
thf(fact_2342_Lcm__abs__eq,axiom,
! [K6: set @ int] :
( ( gcd_Lcm @ int @ ( image2 @ int @ int @ ( abs_abs @ int ) @ K6 ) )
= ( gcd_Lcm @ int @ K6 ) ) ).
% Lcm_abs_eq
thf(fact_2343_Lcm__int__def,axiom,
( ( gcd_Lcm @ int )
= ( ^ [K5: set @ int] : ( semiring_1_of_nat @ int @ ( gcd_Lcm @ nat @ ( image2 @ int @ nat @ ( comp @ int @ nat @ int @ nat2 @ ( abs_abs @ int ) ) @ K5 ) ) ) ) ) ).
% Lcm_int_def
thf(fact_2344_Lcm__int__eq,axiom,
! [N: set @ nat] :
( ( gcd_Lcm @ int @ ( image2 @ nat @ int @ ( semiring_1_of_nat @ int ) @ N ) )
= ( semiring_1_of_nat @ int @ ( gcd_Lcm @ nat @ N ) ) ) ).
% Lcm_int_eq
thf(fact_2345_Lcm__nat__abs__eq,axiom,
! [K6: set @ int] :
( ( gcd_Lcm @ nat
@ ( image2 @ int @ nat
@ ^ [K3: int] : ( nat2 @ ( abs_abs @ int @ K3 ) )
@ K6 ) )
= ( nat2 @ ( gcd_Lcm @ int @ K6 ) ) ) ).
% Lcm_nat_abs_eq
thf(fact_2346_Inr__Field__csum,axiom,
! [B: $tType,A: $tType,A3: A,S2: set @ ( product_prod @ A @ A ),R: set @ ( product_prod @ B @ B )] :
( ( member @ A @ A3 @ ( field2 @ A @ S2 ) )
=> ( member @ ( sum_sum @ B @ A ) @ ( sum_Inr @ A @ B @ A3 ) @ ( field2 @ ( sum_sum @ B @ A ) @ ( bNF_Cardinal_csum @ B @ A @ R @ S2 ) ) ) ) ).
% Inr_Field_csum
thf(fact_2347_Inl__Field__csum,axiom,
! [A: $tType,B: $tType,A3: A,R: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ B @ B )] :
( ( member @ A @ A3 @ ( field2 @ A @ R ) )
=> ( member @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B @ A3 ) @ ( field2 @ ( sum_sum @ A @ B ) @ ( bNF_Cardinal_csum @ A @ B @ R @ S2 ) ) ) ) ).
% Inl_Field_csum
thf(fact_2348_Lcm__mono,axiom,
! [A: $tType,B: $tType] :
( ( semiring_Gcd @ A )
=> ! [A6: set @ B,F: B > A,G: B > A] :
( ! [X: B] :
( ( member @ B @ X @ A6 )
=> ( dvd_dvd @ A @ ( F @ X ) @ ( G @ X ) ) )
=> ( dvd_dvd @ A @ ( gcd_Lcm @ A @ ( image2 @ B @ A @ F @ A6 ) ) @ ( gcd_Lcm @ A @ ( image2 @ B @ A @ G @ A6 ) ) ) ) ) ).
% Lcm_mono
thf(fact_2349_prod_Osize__neq,axiom,
! [A: $tType,B: $tType,X2: product_prod @ A @ B] :
( ( size_size @ ( product_prod @ A @ B ) @ X2 )
!= ( zero_zero @ nat ) ) ).
% prod.size_neq
thf(fact_2350_Lcm__mult,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A6: set @ A,C3: A] :
( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( gcd_Lcm @ A @ ( image2 @ A @ A @ ( times_times @ A @ C3 ) @ A6 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ C3 @ ( gcd_Lcm @ A @ A6 ) ) ) ) ) ) ).
% Lcm_mult
thf(fact_2351_option_Opred__map,axiom,
! [B: $tType,A: $tType,Q2: B > $o,F: A > B,X2: option @ A] :
( ( pred_option @ B @ Q2 @ ( map_option @ A @ B @ F @ X2 ) )
= ( pred_option @ A @ ( comp @ B @ $o @ A @ Q2 @ F ) @ X2 ) ) ).
% option.pred_map
thf(fact_2352_rotate__add,axiom,
! [A: $tType,M4: nat,N5: nat] :
( ( rotate @ A @ ( plus_plus @ nat @ M4 @ N5 ) )
= ( comp @ ( list @ A ) @ ( list @ A ) @ ( list @ A ) @ ( rotate @ A @ M4 ) @ ( rotate @ A @ N5 ) ) ) ).
% rotate_add
thf(fact_2353_old_Ounit_Ocase,axiom,
! [A: $tType,F: A] :
( ( product_case_unit @ A @ F @ product_Unity )
= F ) ).
% old.unit.case
thf(fact_2354_accp__eq__acc,axiom,
! [A: $tType] :
( ( accp @ A )
= ( ^ [R5: A > A > $o,X4: A] : ( member @ A @ X4 @ ( acc @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R5 ) ) ) ) ) ) ).
% accp_eq_acc
thf(fact_2355_rat__inverse__code,axiom,
! [P: rat] :
( ( quotient_of @ ( inverse_inverse @ rat @ P ) )
= ( 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 @ P ) ) ) ).
% rat_inverse_code
thf(fact_2356_Gcd__image__normalize,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A6: set @ A] :
( ( gcd_Gcd @ A @ ( image2 @ A @ A @ ( normal6383669964737779283malize @ A ) @ A6 ) )
= ( gcd_Gcd @ A @ A6 ) ) ) ).
% Gcd_image_normalize
thf(fact_2357_unit_Ocase__distrib,axiom,
! [A: $tType,B: $tType,H: A > B,F: A,Unit: product_unit] :
( ( H @ ( product_case_unit @ A @ F @ Unit ) )
= ( product_case_unit @ B @ ( H @ F ) @ Unit ) ) ).
% unit.case_distrib
thf(fact_2358_acc__induct__rule,axiom,
! [A: $tType,A3: A,R: set @ ( product_prod @ A @ A ),P2: A > $o] :
( ( member @ A @ A3 @ ( acc @ A @ R ) )
=> ( ! [X: A] :
( ( member @ A @ X @ ( acc @ A @ R ) )
=> ( ! [Y7: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y7 @ X ) @ R )
=> ( P2 @ Y7 ) )
=> ( P2 @ X ) ) )
=> ( P2 @ A3 ) ) ) ).
% acc_induct_rule
thf(fact_2359_not__acc__down,axiom,
! [A: $tType,X2: A,R3: set @ ( product_prod @ A @ A )] :
( ~ ( member @ A @ X2 @ ( acc @ A @ R3 ) )
=> ~ ! [Z4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z4 @ X2 ) @ R3 )
=> ( member @ A @ Z4 @ ( acc @ A @ R3 ) ) ) ) ).
% not_acc_down
thf(fact_2360_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_2361_acc__induct,axiom,
! [A: $tType,A3: A,R: set @ ( product_prod @ A @ A ),P2: A > $o] :
( ( member @ A @ A3 @ ( acc @ A @ R ) )
=> ( ! [X: A] :
( ( member @ A @ X @ ( acc @ A @ R ) )
=> ( ! [Y7: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y7 @ X ) @ R )
=> ( P2 @ Y7 ) )
=> ( P2 @ X ) ) )
=> ( P2 @ A3 ) ) ) ).
% acc_induct
thf(fact_2362_acc_Ointros,axiom,
! [A: $tType,X2: A,R: set @ ( product_prod @ A @ A )] :
( ! [Y: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X2 ) @ R )
=> ( member @ A @ Y @ ( acc @ A @ R ) ) )
=> ( member @ A @ X2 @ ( acc @ A @ R ) ) ) ).
% acc.intros
thf(fact_2363_acc_Osimps,axiom,
! [A: $tType,A3: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ A @ A3 @ ( acc @ A @ R ) )
= ( ? [X4: A] :
( ( A3 = X4 )
& ! [Y5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X4 ) @ R )
=> ( member @ A @ Y5 @ ( acc @ A @ R ) ) ) ) ) ) ).
% acc.simps
thf(fact_2364_acc_Ocases,axiom,
! [A: $tType,A3: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ A @ A3 @ ( acc @ A @ R ) )
=> ! [Y7: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y7 @ A3 ) @ R )
=> ( member @ A @ Y7 @ ( acc @ A @ R ) ) ) ) ).
% acc.cases
thf(fact_2365_prod__inversef,axiom,
! [A: $tType,B: $tType] :
( ( field @ A )
=> ! [F: B > A,A6: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A @ ( comp @ A @ A @ B @ ( inverse_inverse @ A ) @ F ) @ A6 )
= ( inverse_inverse @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A6 ) ) ) ) ).
% prod_inversef
thf(fact_2366_acc__subset__induct,axiom,
! [A: $tType,D3: set @ A,R3: set @ ( product_prod @ A @ A ),X2: A,P2: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ D3 @ ( acc @ A @ R3 ) )
=> ( ! [X: A,Z4: A] :
( ( member @ A @ X @ D3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z4 @ X ) @ R3 )
=> ( member @ A @ Z4 @ D3 ) ) )
=> ( ( member @ A @ X2 @ D3 )
=> ( ! [X: A] :
( ( member @ A @ X @ D3 )
=> ( ! [Z7: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z7 @ X ) @ R3 )
=> ( P2 @ Z7 ) )
=> ( P2 @ X ) ) )
=> ( P2 @ X2 ) ) ) ) ) ).
% acc_subset_induct
thf(fact_2367_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_2368_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_2369_Gcd__mult,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [C3: A,A6: set @ A] :
( ( gcd_Gcd @ A @ ( image2 @ A @ A @ ( times_times @ A @ C3 ) @ A6 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ C3 @ ( gcd_Gcd @ A @ A6 ) ) ) ) ) ).
% Gcd_mult
thf(fact_2370_accp__acc__eq,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( accp @ A
@ ^ [X4: A,Y5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R ) )
= ( ^ [X4: A] : ( member @ A @ X4 @ ( acc @ A @ R ) ) ) ) ).
% accp_acc_eq
thf(fact_2371_acc__def,axiom,
! [A: $tType] :
( ( acc @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
( collect @ A
@ ( accp @ A
@ ^ [X4: A,Y5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R5 ) ) ) ) ) ).
% acc_def
thf(fact_2372_Gcd__fin__mult,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A6: set @ A,B2: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( semiring_gcd_Gcd_fin @ A @ ( image2 @ A @ A @ ( times_times @ A @ B2 ) @ A6 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ B2 @ ( semiring_gcd_Gcd_fin @ A @ A6 ) ) ) ) ) ) ).
% Gcd_fin_mult
thf(fact_2373_Lcm__fin__mult,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A6: set @ A,B2: A] :
( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( semiring_gcd_Lcm_fin @ A @ ( image2 @ A @ A @ ( times_times @ A @ B2 ) @ A6 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ B2 @ ( semiring_gcd_Lcm_fin @ A @ A6 ) ) ) ) ) ) ).
% Lcm_fin_mult
thf(fact_2374_Heap_Opred__inject,axiom,
! [A: $tType,P2: A > $o,A3: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) )] :
( ( heap_pred_Heap @ A @ P2 @ ( heap_Heap2 @ A @ A3 ) )
= ( basic_pred_fun @ ( heap_ext @ product_unit ) @ ( option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) )
@ ^ [Uu: heap_ext @ product_unit] : $true
@ ( pred_option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) @ ( basic_pred_prod @ A @ ( heap_ext @ product_unit ) @ P2 @ ( top_top @ ( ( heap_ext @ product_unit ) > $o ) ) ) )
@ A3 ) ) ).
% Heap.pred_inject
thf(fact_2375_sum_Opred__map,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,Q1: C > $o,Q22: D > $o,F1: A > C,F23: B > D,X2: sum_sum @ A @ B] :
( ( basic_pred_sum @ C @ D @ Q1 @ Q22 @ ( sum_map_sum @ A @ C @ B @ D @ F1 @ F23 @ X2 ) )
= ( basic_pred_sum @ A @ B @ ( comp @ C @ $o @ A @ Q1 @ F1 ) @ ( comp @ D @ $o @ B @ Q22 @ F23 ) @ X2 ) ) ).
% sum.pred_map
thf(fact_2376_INF__parametric,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( complete_Inf @ C )
=> ! [A6: A > B > $o] :
( bNF_rel_fun @ ( set @ A ) @ ( set @ B ) @ ( ( A > C ) > C ) @ ( ( B > C ) > C ) @ ( bNF_rel_set @ A @ B @ A6 )
@ ( bNF_rel_fun @ ( A > C ) @ ( B > C ) @ C @ C
@ ( bNF_rel_fun @ A @ B @ C @ C @ A6
@ ^ [Y2: C,Z2: C] : Y2 = Z2 )
@ ^ [Y2: C,Z2: C] : Y2 = Z2 )
@ ^ [A8: set @ A,F4: A > C] : ( complete_Inf_Inf @ C @ ( image2 @ A @ C @ F4 @ A8 ) )
@ ^ [A8: set @ B,F4: B > C] : ( complete_Inf_Inf @ C @ ( image2 @ B @ C @ F4 @ A8 ) ) ) ) ).
% INF_parametric
thf(fact_2377_SUP__parametric,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( complete_Sup @ C )
=> ! [R3: A > B > $o] :
( bNF_rel_fun @ ( set @ A ) @ ( set @ B ) @ ( ( A > C ) > C ) @ ( ( B > C ) > C ) @ ( bNF_rel_set @ A @ B @ R3 )
@ ( bNF_rel_fun @ ( A > C ) @ ( B > C ) @ C @ C
@ ( bNF_rel_fun @ A @ B @ C @ C @ R3
@ ^ [Y2: C,Z2: C] : Y2 = Z2 )
@ ^ [Y2: C,Z2: C] : Y2 = Z2 )
@ ^ [A8: set @ A,F4: A > C] : ( complete_Sup_Sup @ C @ ( image2 @ A @ C @ F4 @ A8 ) )
@ ^ [A8: set @ B,F4: B > C] : ( complete_Sup_Sup @ C @ ( image2 @ B @ C @ F4 @ A8 ) ) ) ) ).
% SUP_parametric
thf(fact_2378_pred__prod__inject,axiom,
! [A: $tType,B: $tType,P12: A > $o,P23: B > $o,A3: A,B2: B] :
( ( basic_pred_prod @ A @ B @ P12 @ P23 @ ( product_Pair @ A @ B @ A3 @ B2 ) )
= ( ( P12 @ A3 )
& ( P23 @ B2 ) ) ) ).
% pred_prod_inject
thf(fact_2379_rel__setI,axiom,
! [A: $tType,B: $tType,A6: set @ A,B6: set @ B,R3: A > B > $o] :
( ! [X: A] :
( ( member @ A @ X @ A6 )
=> ? [Xa3: B] :
( ( member @ B @ Xa3 @ B6 )
& ( R3 @ X @ Xa3 ) ) )
=> ( ! [Y: B] :
( ( member @ B @ Y @ B6 )
=> ? [X5: A] :
( ( member @ A @ X5 @ A6 )
& ( R3 @ X5 @ Y ) ) )
=> ( bNF_rel_set @ A @ B @ R3 @ A6 @ B6 ) ) ) ).
% rel_setI
thf(fact_2380_rel__set__OO,axiom,
! [A: $tType,B: $tType,C: $tType,R3: A > C > $o,S5: C > B > $o] :
( ( relcompp @ ( set @ A ) @ ( set @ C ) @ ( set @ B ) @ ( bNF_rel_set @ A @ C @ R3 ) @ ( bNF_rel_set @ C @ B @ S5 ) )
= ( bNF_rel_set @ A @ B @ ( relcompp @ A @ C @ B @ R3 @ S5 ) ) ) ).
% rel_set_OO
thf(fact_2381_image__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,A6: A > C > $o,B6: B > D > $o] : ( bNF_rel_fun @ ( A > B ) @ ( C > D ) @ ( ( set @ A ) > ( set @ B ) ) @ ( ( set @ C ) > ( set @ D ) ) @ ( bNF_rel_fun @ A @ C @ B @ D @ A6 @ B6 ) @ ( bNF_rel_fun @ ( set @ A ) @ ( set @ C ) @ ( set @ B ) @ ( set @ D ) @ ( bNF_rel_set @ A @ C @ A6 ) @ ( bNF_rel_set @ B @ D @ B6 ) ) @ ( image2 @ A @ B ) @ ( image2 @ C @ D ) ) ).
% image_transfer
thf(fact_2382_rel__set__UNION,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,Q2: A > B > $o,A6: set @ A,B6: set @ B,R3: C > D > $o,F: A > ( set @ C ),G: B > ( set @ D )] :
( ( bNF_rel_set @ A @ B @ Q2 @ A6 @ B6 )
=> ( ( bNF_rel_fun @ A @ B @ ( set @ C ) @ ( set @ D ) @ Q2 @ ( bNF_rel_set @ C @ D @ R3 ) @ F @ G )
=> ( bNF_rel_set @ C @ D @ R3 @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ A @ ( set @ C ) @ F @ A6 ) ) @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ B @ ( set @ D ) @ G @ B6 ) ) ) ) ) ).
% rel_set_UNION
thf(fact_2383_Quotient__set,axiom,
! [B: $tType,A: $tType,R3: A > A > $o,Abs: A > B,Rep: B > A,T4: A > B > $o] :
( ( quotient @ A @ B @ R3 @ Abs @ Rep @ T4 )
=> ( quotient @ ( set @ A ) @ ( set @ B ) @ ( bNF_rel_set @ A @ A @ R3 ) @ ( image2 @ A @ B @ Abs ) @ ( image2 @ B @ A @ Rep ) @ ( bNF_rel_set @ A @ B @ T4 ) ) ) ).
% Quotient_set
thf(fact_2384_pred__prod_Ocases,axiom,
! [A: $tType,B: $tType,P12: A > $o,P23: B > $o,A3: product_prod @ A @ B] :
( ( basic_pred_prod @ A @ B @ P12 @ P23 @ A3 )
=> ~ ! [A5: A,B4: B] :
( ( A3
= ( product_Pair @ A @ B @ A5 @ B4 ) )
=> ( ( P12 @ A5 )
=> ~ ( P23 @ B4 ) ) ) ) ).
% pred_prod.cases
thf(fact_2385_pred__prod_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( basic_pred_prod @ A @ B )
= ( ^ [P13: A > $o,P25: B > $o,A7: product_prod @ A @ B] :
? [B5: A,C6: B] :
( ( A7
= ( product_Pair @ A @ B @ B5 @ C6 ) )
& ( P13 @ B5 )
& ( P25 @ C6 ) ) ) ) ).
% pred_prod.simps
thf(fact_2386_pred__prod_Ointros,axiom,
! [A: $tType,B: $tType,P12: A > $o,A3: A,P23: B > $o,B2: B] :
( ( P12 @ A3 )
=> ( ( P23 @ B2 )
=> ( basic_pred_prod @ A @ B @ P12 @ P23 @ ( product_Pair @ A @ B @ A3 @ B2 ) ) ) ) ).
% pred_prod.intros
thf(fact_2387_pred__prod__split,axiom,
! [B: $tType,A: $tType,P2: $o > $o,Q2: A > $o,R3: B > $o,Xy2: product_prod @ A @ B] :
( ( P2 @ ( basic_pred_prod @ A @ B @ Q2 @ R3 @ Xy2 ) )
= ( ! [X4: A,Y5: B] :
( ( Xy2
= ( product_Pair @ A @ B @ X4 @ Y5 ) )
=> ( P2
@ ( ( Q2 @ X4 )
& ( R3 @ Y5 ) ) ) ) ) ) ).
% pred_prod_split
thf(fact_2388_UNION__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A6: A > B > $o,B6: C > D > $o] :
( bNF_rel_fun @ ( set @ A ) @ ( set @ B ) @ ( ( A > ( set @ C ) ) > ( set @ C ) ) @ ( ( B > ( set @ D ) ) > ( set @ D ) ) @ ( bNF_rel_set @ A @ B @ A6 ) @ ( bNF_rel_fun @ ( A > ( set @ C ) ) @ ( B > ( set @ D ) ) @ ( set @ C ) @ ( set @ D ) @ ( bNF_rel_fun @ A @ B @ ( set @ C ) @ ( set @ D ) @ A6 @ ( bNF_rel_set @ C @ D @ B6 ) ) @ ( bNF_rel_set @ C @ D @ B6 ) )
@ ^ [A8: set @ A,F4: A > ( set @ C )] : ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ A @ ( set @ C ) @ F4 @ A8 ) )
@ ^ [A8: set @ B,F4: B > ( set @ D )] : ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ B @ ( set @ D ) @ F4 @ A8 ) ) ) ).
% UNION_transfer
thf(fact_2389_Heap_Oset__transfer,axiom,
! [A: $tType,B: $tType,R3: A > B > $o] : ( bNF_rel_fun @ ( heap_Heap @ A ) @ ( heap_Heap @ B ) @ ( set @ A ) @ ( set @ B ) @ ( heap_rel_Heap @ A @ B @ R3 ) @ ( bNF_rel_set @ A @ B @ R3 ) @ ( heap_set_Heap @ A ) @ ( heap_set_Heap @ B ) ) ).
% Heap.set_transfer
thf(fact_2390_rel__set__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_rel_set @ A @ B )
= ( ^ [R6: A > B > $o,A8: set @ A,B7: set @ B] :
( ! [X4: A] :
( ( member @ A @ X4 @ A8 )
=> ? [Y5: B] :
( ( member @ B @ Y5 @ B7 )
& ( R6 @ X4 @ Y5 ) ) )
& ! [X4: B] :
( ( member @ B @ X4 @ B7 )
=> ? [Y5: A] :
( ( member @ A @ Y5 @ A8 )
& ( R6 @ Y5 @ X4 ) ) ) ) ) ) ).
% rel_set_def
thf(fact_2391_pred__prod__beta,axiom,
! [B: $tType,A: $tType] :
( ( basic_pred_prod @ A @ B )
= ( ^ [P4: A > $o,Q3: B > $o,Xy: product_prod @ A @ B] :
( ( P4 @ ( product_fst @ A @ B @ Xy ) )
& ( Q3 @ ( product_snd @ A @ B @ Xy ) ) ) ) ) ).
% pred_prod_beta
thf(fact_2392_fun_Oset__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,R3: A > B > $o] :
( bNF_rel_fun @ ( D > A ) @ ( D > B ) @ ( set @ A ) @ ( set @ B )
@ ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y2: D,Z2: D] : Y2 = Z2
@ R3 )
@ ( bNF_rel_set @ A @ B @ R3 )
@ ^ [F4: D > A] : ( image2 @ D @ A @ F4 @ ( top_top @ ( set @ D ) ) )
@ ^ [F4: D > B] : ( image2 @ D @ B @ F4 @ ( top_top @ ( set @ D ) ) ) ) ).
% fun.set_transfer
thf(fact_2393_prod_Opred__map,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,Q1: C > $o,Q22: D > $o,F1: A > C,F23: B > D,X2: product_prod @ A @ B] :
( ( basic_pred_prod @ C @ D @ Q1 @ Q22 @ ( product_map_prod @ A @ C @ B @ D @ F1 @ F23 @ X2 ) )
= ( basic_pred_prod @ A @ B @ ( comp @ C @ $o @ A @ Q1 @ F1 ) @ ( comp @ D @ $o @ B @ Q22 @ F23 ) @ X2 ) ) ).
% prod.pred_map
thf(fact_2394_fold__rev,axiom,
! [B: $tType,A: $tType,Xs: list @ A,F: A > B > B] :
( ! [X: A,Y: A] :
( ( member @ A @ X @ ( set3 @ A @ Xs ) )
=> ( ( member @ A @ Y @ ( set3 @ A @ Xs ) )
=> ( ( comp @ B @ B @ B @ ( F @ Y ) @ ( F @ X ) )
= ( comp @ B @ B @ B @ ( F @ X ) @ ( F @ Y ) ) ) ) )
=> ( ( fold @ A @ B @ F @ ( rev @ A @ Xs ) )
= ( fold @ A @ B @ F @ Xs ) ) ) ).
% fold_rev
thf(fact_2395_dir__image__def,axiom,
! [A2: $tType,A: $tType] :
( ( bNF_We2720479622203943262_image @ A @ A2 )
= ( ^ [R5: set @ ( product_prod @ A @ A ),F4: A > A2] :
( collect @ ( product_prod @ A2 @ A2 )
@ ^ [Uu: product_prod @ A2 @ A2] :
? [A7: A,B5: A] :
( ( Uu
= ( product_Pair @ A2 @ A2 @ ( F4 @ A7 ) @ ( F4 @ B5 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B5 ) @ R5 ) ) ) ) ) ).
% dir_image_def
thf(fact_2396_card__order__csum__cone__cexp__def,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),A13: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( top_top @ ( set @ A ) ) @ R )
=> ( ( bNF_Cardinal_cexp @ ( sum_sum @ B @ product_unit ) @ A @ ( bNF_Cardinal_csum @ B @ product_unit @ ( bNF_Ca6860139660246222851ard_of @ B @ A13 ) @ bNF_Cardinal_cone ) @ R )
= ( bNF_Ca6860139660246222851ard_of @ ( A > ( sum_sum @ B @ product_unit ) ) @ ( bNF_Wellorder_Func @ A @ ( sum_sum @ B @ product_unit ) @ ( top_top @ ( set @ A ) ) @ ( sup_sup @ ( set @ ( sum_sum @ B @ product_unit ) ) @ ( image2 @ B @ ( sum_sum @ B @ product_unit ) @ ( sum_Inl @ B @ product_unit ) @ A13 ) @ ( insert @ ( sum_sum @ B @ product_unit ) @ ( sum_Inr @ product_unit @ B @ product_Unity ) @ ( bot_bot @ ( set @ ( sum_sum @ B @ product_unit ) ) ) ) ) ) ) ) ) ).
% card_order_csum_cone_cexp_def
thf(fact_2397_bi__unique__rel__set__lemma,axiom,
! [B: $tType,A: $tType,R3: A > B > $o,X6: set @ A,Y4: set @ B] :
( ( bi_unique @ A @ B @ R3 )
=> ( ( bNF_rel_set @ A @ B @ R3 @ X6 @ Y4 )
=> ~ ! [F3: A > B] :
( ( Y4
= ( image2 @ A @ B @ F3 @ X6 ) )
=> ( ( inj_on @ A @ B @ F3 @ X6 )
=> ~ ! [X5: A] :
( ( member @ A @ X5 @ X6 )
=> ( R3 @ X5 @ ( F3 @ X5 ) ) ) ) ) ) ) ).
% bi_unique_rel_set_lemma
thf(fact_2398_Heap__ordI,axiom,
! [A: $tType,X2: heap_Heap @ A,Y3: heap_Heap @ A] :
( ! [H8: heap_ext @ product_unit] :
( ( ( heap_execute @ A @ X2 @ H8 )
= ( none @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) )
| ( ( heap_execute @ A @ X2 @ H8 )
= ( heap_execute @ A @ Y3 @ H8 ) ) )
=> ( heap_Heap_ord @ A @ X2 @ Y3 ) ) ).
% Heap_ordI
thf(fact_2399_Heap__ordE,axiom,
! [A: $tType,X2: heap_Heap @ A,Y3: heap_Heap @ A,H: heap_ext @ product_unit] :
( ( heap_Heap_ord @ A @ X2 @ Y3 )
=> ( ( ( heap_execute @ A @ X2 @ H )
!= ( none @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) )
=> ( ( heap_execute @ A @ X2 @ H )
= ( heap_execute @ A @ Y3 @ H ) ) ) ) ).
% Heap_ordE
thf(fact_2400_heap_Oleq__refl,axiom,
! [A: $tType,X2: heap_Heap @ A] : ( heap_Heap_ord @ A @ X2 @ X2 ) ).
% heap.leq_refl
thf(fact_2401_heap_Oleq__trans,axiom,
! [A: $tType,X2: heap_Heap @ A,Y3: heap_Heap @ A,Z3: heap_Heap @ A] :
( ( heap_Heap_ord @ A @ X2 @ Y3 )
=> ( ( heap_Heap_ord @ A @ Y3 @ Z3 )
=> ( heap_Heap_ord @ A @ X2 @ Z3 ) ) ) ).
% heap.leq_trans
thf(fact_2402_heap_Oleq__antisym,axiom,
! [A: $tType,X2: heap_Heap @ A,Y3: heap_Heap @ A] :
( ( heap_Heap_ord @ A @ X2 @ Y3 )
=> ( ( heap_Heap_ord @ A @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ).
% heap.leq_antisym
thf(fact_2403_Heap_Obi__unique__rel,axiom,
! [B: $tType,A: $tType,R3: A > B > $o] :
( ( bi_unique @ A @ B @ R3 )
=> ( bi_unique @ ( heap_Heap @ A ) @ ( heap_Heap @ B ) @ ( heap_rel_Heap @ A @ B @ R3 ) ) ) ).
% Heap.bi_unique_rel
thf(fact_2404_bi__unique__OO,axiom,
! [A: $tType,C: $tType,B: $tType,A6: A > B > $o,B6: B > C > $o] :
( ( bi_unique @ A @ B @ A6 )
=> ( ( bi_unique @ B @ C @ B6 )
=> ( bi_unique @ A @ C @ ( relcompp @ A @ B @ C @ A6 @ B6 ) ) ) ) ).
% bi_unique_OO
thf(fact_2405_dir__image__Field,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ B @ B ),F: B > A] :
( ( field2 @ A @ ( bNF_We2720479622203943262_image @ B @ A @ R @ F ) )
= ( image2 @ B @ A @ F @ ( field2 @ B @ R ) ) ) ).
% dir_image_Field
thf(fact_2406_heap__interpretation,axiom,
! [A: $tType] : ( partia7178651479351089652itions @ ( heap_Heap @ A ) @ ( heap_Heap_ord @ A ) @ ( heap_Heap_lub @ A ) ) ).
% heap_interpretation
thf(fact_2407_scomp__fcomp,axiom,
! [A: $tType,C: $tType,B: $tType,E: $tType,D: $tType,F: A > ( product_prod @ D @ E ),G: D > E > C,H: C > B] :
( ( fcomp @ A @ C @ B @ ( product_scomp @ A @ D @ E @ C @ F @ G ) @ H )
= ( product_scomp @ A @ D @ E @ B @ F
@ ^ [X4: D] : ( fcomp @ E @ C @ B @ ( G @ X4 ) @ H ) ) ) ).
% scomp_fcomp
thf(fact_2408_image__uminus__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X2: A,Y3: A] :
( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_or5935395276787703475ssThan @ A @ X2 @ Y3 ) )
= ( set_or5935395276787703475ssThan @ A @ ( uminus_uminus @ A @ Y3 ) @ ( uminus_uminus @ A @ X2 ) ) ) ) ).
% image_uminus_greaterThanLessThan
thf(fact_2409_euclidean__size__int__def,axiom,
( ( euclid6346220572633701492n_size @ int )
= ( comp @ int @ nat @ int @ nat2 @ ( abs_abs @ int ) ) ) ).
% euclidean_size_int_def
thf(fact_2410_symp__INF,axiom,
! [B: $tType,A: $tType,S5: set @ A,R: A > B > B > $o] :
( ! [X: A] :
( ( member @ A @ X @ S5 )
=> ( symp @ B @ ( R @ X ) ) )
=> ( symp @ B @ ( complete_Inf_Inf @ ( B > B > $o ) @ ( image2 @ A @ ( B > B > $o ) @ R @ S5 ) ) ) ) ).
% symp_INF
thf(fact_2411_transp__INF,axiom,
! [B: $tType,A: $tType,S5: set @ A,R: A > B > B > $o] :
( ! [X: A] :
( ( member @ A @ X @ S5 )
=> ( transp @ B @ ( R @ X ) ) )
=> ( transp @ B @ ( complete_Inf_Inf @ ( B > B > $o ) @ ( image2 @ A @ ( B > B > $o ) @ R @ S5 ) ) ) ) ).
% transp_INF
thf(fact_2412_Heap_Orel__symp,axiom,
! [A: $tType,R3: A > A > $o] :
( ( symp @ A @ R3 )
=> ( symp @ ( heap_Heap @ A ) @ ( heap_rel_Heap @ A @ A @ R3 ) ) ) ).
% Heap.rel_symp
thf(fact_2413_Heap_Orel__transp,axiom,
! [A: $tType,R3: A > A > $o] :
( ( transp @ A @ R3 )
=> ( transp @ ( heap_Heap @ A ) @ ( heap_rel_Heap @ A @ A @ R3 ) ) ) ).
% Heap.rel_transp
thf(fact_2414_fcomp__scomp,axiom,
! [A: $tType,E: $tType,B: $tType,D: $tType,C: $tType,F: A > E,G: E > ( product_prod @ C @ D ),H: C > D > B] :
( ( product_scomp @ A @ C @ D @ B @ ( fcomp @ A @ E @ ( product_prod @ C @ D ) @ F @ G ) @ H )
= ( fcomp @ A @ E @ B @ F @ ( product_scomp @ E @ C @ D @ B @ G @ H ) ) ) ).
% fcomp_scomp
thf(fact_2415_fcomp__comp,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( fcomp @ A @ C @ B )
= ( ^ [F4: A > C,G4: C > B] : ( comp @ C @ B @ A @ G4 @ F4 ) ) ) ).
% fcomp_comp
thf(fact_2416_transp__relcompp__less__eq,axiom,
! [A: $tType,R: A > A > $o] :
( ( transp @ A @ R )
=> ( ord_less_eq @ ( A > A > $o ) @ ( relcompp @ A @ A @ A @ R @ R ) @ R ) ) ).
% transp_relcompp_less_eq
thf(fact_2417_transp__relcompp,axiom,
! [A: $tType] :
( ( transp @ A )
= ( ^ [R5: A > A > $o] : ( ord_less_eq @ ( A > A > $o ) @ ( relcompp @ A @ A @ A @ R5 @ R5 ) @ R5 ) ) ) ).
% transp_relcompp
thf(fact_2418_symp__conversep,axiom,
! [A: $tType] :
( ( symp @ A )
= ( ^ [R6: A > A > $o] : ( ord_less_eq @ ( A > A > $o ) @ ( conversep @ A @ A @ R6 ) @ R6 ) ) ) ).
% symp_conversep
thf(fact_2419_transp__trans__eq,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( transp @ A
@ ^ [X4: A,Y5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R ) )
= ( trans @ A @ R ) ) ).
% transp_trans_eq
thf(fact_2420_symp__sym__eq,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( symp @ A
@ ^ [X4: A,Y5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R ) )
= ( sym @ A @ R ) ) ).
% symp_sym_eq
thf(fact_2421_transp__trans,axiom,
! [A: $tType] :
( ( transp @ A )
= ( ^ [R5: A > A > $o] : ( trans @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R5 ) ) ) ) ) ).
% transp_trans
thf(fact_2422_symp__SUP,axiom,
! [B: $tType,A: $tType,S5: set @ A,R: A > B > B > $o] :
( ! [X: A] :
( ( member @ A @ X @ S5 )
=> ( symp @ B @ ( R @ X ) ) )
=> ( symp @ B @ ( complete_Sup_Sup @ ( B > B > $o ) @ ( image2 @ A @ ( B > B > $o ) @ R @ S5 ) ) ) ) ).
% symp_SUP
thf(fact_2423_sum_Oset__map_I1_J,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,F1: A > C,F23: B > D,V: sum_sum @ A @ B] :
( ( basic_setl @ C @ D @ ( sum_map_sum @ A @ C @ B @ D @ F1 @ F23 @ V ) )
= ( image2 @ A @ C @ F1 @ ( basic_setl @ A @ B @ V ) ) ) ).
% sum.set_map(1)
thf(fact_2424_sum_Oset__map_I2_J,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,F1: A > C,F23: B > D,V: sum_sum @ A @ B] :
( ( basic_setr @ C @ D @ ( sum_map_sum @ A @ C @ B @ D @ F1 @ F23 @ V ) )
= ( image2 @ B @ D @ F23 @ ( basic_setr @ A @ B @ V ) ) ) ).
% sum.set_map(2)
thf(fact_2425_heap_Olub__least,axiom,
! [A: $tType,A6: set @ ( heap_Heap @ A ),Z3: heap_Heap @ A] :
( ( comple1602240252501008431_chain @ ( heap_Heap @ A ) @ ( heap_Heap_ord @ A ) @ A6 )
=> ( ! [X: heap_Heap @ A] :
( ( member @ ( heap_Heap @ A ) @ X @ A6 )
=> ( heap_Heap_ord @ A @ X @ Z3 ) )
=> ( heap_Heap_ord @ A @ ( heap_Heap_lub @ A @ A6 ) @ Z3 ) ) ) ).
% heap.lub_least
thf(fact_2426_heap_Olub__upper,axiom,
! [A: $tType,A6: set @ ( heap_Heap @ A ),X2: heap_Heap @ A] :
( ( comple1602240252501008431_chain @ ( heap_Heap @ A ) @ ( heap_Heap_ord @ A ) @ A6 )
=> ( ( member @ ( heap_Heap @ A ) @ X2 @ A6 )
=> ( heap_Heap_ord @ A @ X2 @ ( heap_Heap_lub @ A @ A6 ) ) ) ) ).
% heap.lub_upper
thf(fact_2427_folding__idem__axioms_Ointro,axiom,
! [B: $tType,A: $tType,F: A > B > B] :
( ! [X: A] :
( ( comp @ B @ B @ B @ ( F @ X ) @ ( F @ X ) )
= ( F @ X ) )
=> ( finite7837460588564673216axioms @ A @ B @ F ) ) ).
% folding_idem_axioms.intro
thf(fact_2428_comp__fun__idem__axioms_Ointro,axiom,
! [B: $tType,A: $tType,F: A > B > B] :
( ! [X: A] :
( ( comp @ B @ B @ B @ ( F @ X ) @ ( F @ X ) )
= ( F @ X ) )
=> ( finite1981172056906289455axioms @ A @ B @ F ) ) ).
% comp_fun_idem_axioms.intro
thf(fact_2429_comp__fun__idem__axioms__def,axiom,
! [B: $tType,A: $tType] :
( ( finite1981172056906289455axioms @ A @ B )
= ( ^ [F4: A > B > B] :
! [X4: A] :
( ( comp @ B @ B @ B @ ( F4 @ X4 ) @ ( F4 @ X4 ) )
= ( F4 @ X4 ) ) ) ) ).
% comp_fun_idem_axioms_def
thf(fact_2430_folding__idem__axioms__def,axiom,
! [B: $tType,A: $tType] :
( ( finite7837460588564673216axioms @ A @ B )
= ( ^ [F4: A > B > B] :
! [X4: A] :
( ( comp @ B @ B @ B @ ( F4 @ X4 ) @ ( F4 @ X4 ) )
= ( F4 @ X4 ) ) ) ) ).
% folding_idem_axioms_def
thf(fact_2431_chain__imageI,axiom,
! [B: $tType,A: $tType,Le_a: A > A > $o,Y4: set @ A,Le_b: B > B > $o,F: A > B] :
( ( comple1602240252501008431_chain @ A @ Le_a @ Y4 )
=> ( ! [X: A,Y: A] :
( ( member @ A @ X @ Y4 )
=> ( ( member @ A @ Y @ Y4 )
=> ( ( Le_a @ X @ Y )
=> ( Le_b @ ( F @ X ) @ ( F @ Y ) ) ) ) )
=> ( comple1602240252501008431_chain @ B @ Le_b @ ( image2 @ A @ B @ F @ Y4 ) ) ) ) ).
% chain_imageI
thf(fact_2432_sum_Orel__compp__Grp,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType] :
( ( bNF_rel_sum @ A @ C @ B @ D )
= ( ^ [R14: A > C > $o,R25: B > D > $o] :
( relcompp @ ( sum_sum @ A @ B ) @ ( sum_sum @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) @ ( sum_sum @ C @ D )
@ ( conversep @ ( sum_sum @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) @ ( sum_sum @ A @ B )
@ ( bNF_Grp @ ( sum_sum @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) @ ( sum_sum @ A @ B )
@ ( collect @ ( sum_sum @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) )
@ ^ [X4: sum_sum @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ C ) ) @ ( basic_setl @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ X4 ) @ ( collect @ ( product_prod @ A @ C ) @ ( product_case_prod @ A @ C @ $o @ R14 ) ) )
& ( ord_less_eq @ ( set @ ( product_prod @ B @ D ) ) @ ( basic_setr @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ X4 ) @ ( collect @ ( product_prod @ B @ D ) @ ( product_case_prod @ B @ D @ $o @ R25 ) ) ) ) )
@ ( sum_map_sum @ ( product_prod @ A @ C ) @ A @ ( product_prod @ B @ D ) @ B @ ( product_fst @ A @ C ) @ ( product_fst @ B @ D ) ) ) )
@ ( bNF_Grp @ ( sum_sum @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) @ ( sum_sum @ C @ D )
@ ( collect @ ( sum_sum @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) )
@ ^ [X4: sum_sum @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ C ) ) @ ( basic_setl @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ X4 ) @ ( collect @ ( product_prod @ A @ C ) @ ( product_case_prod @ A @ C @ $o @ R14 ) ) )
& ( ord_less_eq @ ( set @ ( product_prod @ B @ D ) ) @ ( basic_setr @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ X4 ) @ ( collect @ ( product_prod @ B @ D ) @ ( product_case_prod @ B @ D @ $o @ R25 ) ) ) ) )
@ ( sum_map_sum @ ( product_prod @ A @ C ) @ C @ ( product_prod @ B @ D ) @ D @ ( product_snd @ A @ C ) @ ( product_snd @ B @ D ) ) ) ) ) ) ).
% sum.rel_compp_Grp
thf(fact_2433_sum_Oin__rel,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType] :
( ( bNF_rel_sum @ A @ C @ B @ D )
= ( ^ [R14: A > C > $o,R25: B > D > $o,A7: sum_sum @ A @ B,B5: sum_sum @ C @ D] :
? [Z5: sum_sum @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D )] :
( ( member @ ( sum_sum @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) @ Z5
@ ( collect @ ( sum_sum @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) )
@ ^ [X4: sum_sum @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ C ) ) @ ( basic_setl @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ X4 ) @ ( collect @ ( product_prod @ A @ C ) @ ( product_case_prod @ A @ C @ $o @ R14 ) ) )
& ( ord_less_eq @ ( set @ ( product_prod @ B @ D ) ) @ ( basic_setr @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ X4 ) @ ( collect @ ( product_prod @ B @ D ) @ ( product_case_prod @ B @ D @ $o @ R25 ) ) ) ) ) )
& ( ( sum_map_sum @ ( product_prod @ A @ C ) @ A @ ( product_prod @ B @ D ) @ B @ ( product_fst @ A @ C ) @ ( product_fst @ B @ D ) @ Z5 )
= A7 )
& ( ( sum_map_sum @ ( product_prod @ A @ C ) @ C @ ( product_prod @ B @ D ) @ D @ ( product_snd @ A @ C ) @ ( product_snd @ B @ D ) @ Z5 )
= B5 ) ) ) ) ).
% sum.in_rel
thf(fact_2434_comp__fun__idem__on__axioms__def,axiom,
! [B: $tType,A: $tType] :
( ( finite4980608107308702382axioms @ A @ B )
= ( ^ [S4: set @ A,F4: A > B > B] :
! [X4: A] :
( ( member @ A @ X4 @ S4 )
=> ( ( comp @ B @ B @ B @ ( F4 @ X4 ) @ ( F4 @ X4 ) )
= ( F4 @ X4 ) ) ) ) ) ).
% comp_fun_idem_on_axioms_def
thf(fact_2435_folding__idem__on__axioms__def,axiom,
! [B: $tType,A: $tType] :
( ( finite6916993218817215295axioms @ A @ B )
= ( ^ [S4: set @ A,F4: A > B > B] :
! [X4: A,Y5: A] :
( ( member @ A @ X4 @ S4 )
=> ( ( member @ A @ Y5 @ S4 )
=> ( ( comp @ B @ B @ B @ ( F4 @ X4 ) @ ( F4 @ X4 ) )
= ( F4 @ X4 ) ) ) ) ) ) ).
% folding_idem_on_axioms_def
thf(fact_2436_comp__fun__idem__on__axioms_Ointro,axiom,
! [B: $tType,A: $tType,S5: set @ A,F: A > B > B] :
( ! [X: A] :
( ( member @ A @ X @ S5 )
=> ( ( comp @ B @ B @ B @ ( F @ X ) @ ( F @ X ) )
= ( F @ X ) ) )
=> ( finite4980608107308702382axioms @ A @ B @ S5 @ F ) ) ).
% comp_fun_idem_on_axioms.intro
thf(fact_2437_sum_Orel__compp,axiom,
! [B: $tType,A: $tType,C: $tType,E: $tType,F2: $tType,D: $tType,R1: A > C > $o,S1: C > E > $o,R22: B > D > $o,S22: D > F2 > $o] :
( ( bNF_rel_sum @ A @ E @ B @ F2 @ ( relcompp @ A @ C @ E @ R1 @ S1 ) @ ( relcompp @ B @ D @ F2 @ R22 @ S22 ) )
= ( relcompp @ ( sum_sum @ A @ B ) @ ( sum_sum @ C @ D ) @ ( sum_sum @ E @ F2 ) @ ( bNF_rel_sum @ A @ C @ B @ D @ R1 @ R22 ) @ ( bNF_rel_sum @ C @ E @ D @ F2 @ S1 @ S22 ) ) ) ).
% sum.rel_compp
thf(fact_2438_rel__sum_Ointros_I2_J,axiom,
! [B: $tType,A: $tType,C: $tType,D: $tType,R22: B > D > $o,B2: B,D4: D,R1: A > C > $o] :
( ( R22 @ B2 @ D4 )
=> ( bNF_rel_sum @ A @ C @ B @ D @ R1 @ R22 @ ( sum_Inr @ B @ A @ B2 ) @ ( sum_Inr @ D @ C @ D4 ) ) ) ).
% rel_sum.intros(2)
thf(fact_2439_rel__sum_Ointros_I1_J,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,R1: A > C > $o,A3: A,C3: C,R22: B > D > $o] :
( ( R1 @ A3 @ C3 )
=> ( bNF_rel_sum @ A @ C @ B @ D @ R1 @ R22 @ ( sum_Inl @ A @ B @ A3 ) @ ( sum_Inl @ C @ D @ C3 ) ) ) ).
% rel_sum.intros(1)
thf(fact_2440_rel__sum_Osimps,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType] :
( ( bNF_rel_sum @ A @ C @ B @ D )
= ( ^ [R14: A > C > $o,R25: B > D > $o,A12: sum_sum @ A @ B,A23: sum_sum @ C @ D] :
( ? [A7: A,C6: C] :
( ( A12
= ( sum_Inl @ A @ B @ A7 ) )
& ( A23
= ( sum_Inl @ C @ D @ C6 ) )
& ( R14 @ A7 @ C6 ) )
| ? [B5: B,D5: D] :
( ( A12
= ( sum_Inr @ B @ A @ B5 ) )
& ( A23
= ( sum_Inr @ D @ C @ D5 ) )
& ( R25 @ B5 @ D5 ) ) ) ) ) ).
% rel_sum.simps
thf(fact_2441_rel__sum_Ocases,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,R1: A > C > $o,R22: B > D > $o,A1: sum_sum @ A @ B,A22: sum_sum @ C @ D] :
( ( bNF_rel_sum @ A @ C @ B @ D @ R1 @ R22 @ A1 @ A22 )
=> ( ! [A5: A] :
( ( A1
= ( sum_Inl @ A @ B @ A5 ) )
=> ! [C2: C] :
( ( A22
= ( sum_Inl @ C @ D @ C2 ) )
=> ~ ( R1 @ A5 @ C2 ) ) )
=> ~ ! [B4: B] :
( ( A1
= ( sum_Inr @ B @ A @ B4 ) )
=> ! [D2: D] :
( ( A22
= ( sum_Inr @ D @ C @ D2 ) )
=> ~ ( R22 @ B4 @ D2 ) ) ) ) ) ).
% rel_sum.cases
thf(fact_2442_folding__idem__on__axioms_Ointro,axiom,
! [B: $tType,A: $tType,S5: set @ A,F: A > B > B] :
( ! [X: A,Y: A] :
( ( member @ A @ X @ S5 )
=> ( ( member @ A @ Y @ S5 )
=> ( ( comp @ B @ B @ B @ ( F @ X ) @ ( F @ X ) )
= ( F @ X ) ) ) )
=> ( finite6916993218817215295axioms @ A @ B @ S5 @ F ) ) ).
% folding_idem_on_axioms.intro
thf(fact_2443_sndsp_Ointros,axiom,
! [B: $tType,A: $tType,P: product_prod @ A @ B] : ( basic_sndsp @ A @ B @ P @ ( product_snd @ A @ B @ P ) ) ).
% sndsp.intros
thf(fact_2444_sndsp_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( basic_sndsp @ A @ B )
= ( ^ [P3: product_prod @ A @ B,A7: B] :
( A7
= ( product_snd @ A @ B @ P3 ) ) ) ) ).
% sndsp.simps
thf(fact_2445_sndsp_Ocases,axiom,
! [B: $tType,A: $tType,P: product_prod @ A @ B,A3: B] :
( ( basic_sndsp @ A @ B @ P @ A3 )
=> ( A3
= ( product_snd @ A @ B @ P ) ) ) ).
% sndsp.cases
thf(fact_2446_fstsp_Ointros,axiom,
! [B: $tType,A: $tType,P: product_prod @ A @ B] : ( basic_fstsp @ A @ B @ P @ ( product_fst @ A @ B @ P ) ) ).
% fstsp.intros
thf(fact_2447_fstsp_Ocases,axiom,
! [B: $tType,A: $tType,P: product_prod @ A @ B,A3: A] :
( ( basic_fstsp @ A @ B @ P @ A3 )
=> ( A3
= ( product_fst @ A @ B @ P ) ) ) ).
% fstsp.cases
thf(fact_2448_fstsp_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( basic_fstsp @ A @ B )
= ( ^ [P3: product_prod @ A @ B,A7: A] :
( A7
= ( product_fst @ A @ B @ P3 ) ) ) ) ).
% fstsp.simps
thf(fact_2449_comm__monoid__set_OatLeastAtMost__shift__0,axiom,
! [A: $tType,F: A > A > A,Z3: A,M4: nat,N5: nat,G: nat > A] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( ord_less_eq @ nat @ M4 @ N5 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ M4 @ N5 ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M4 ) ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N5 @ M4 ) ) ) ) ) ) ).
% comm_monoid_set.atLeastAtMost_shift_0
thf(fact_2450_list_Opred__map,axiom,
! [B: $tType,A: $tType,Q2: B > $o,F: A > B,X2: list @ A] :
( ( list_all @ B @ Q2 @ ( map @ A @ B @ F @ X2 ) )
= ( list_all @ A @ ( comp @ B @ $o @ A @ Q2 @ F ) @ X2 ) ) ).
% list.pred_map
thf(fact_2451_Unity__def,axiom,
( product_Unity
= ( product_Abs_unit @ $true ) ) ).
% Unity_def
thf(fact_2452_prod__encode__def,axiom,
( nat_prod_encode
= ( product_case_prod @ nat @ nat @ nat
@ ^ [M7: nat,N4: nat] : ( plus_plus @ nat @ ( nat_triangle @ ( plus_plus @ nat @ M7 @ N4 ) ) @ M7 ) ) ) ).
% prod_encode_def
thf(fact_2453_select__weight__def,axiom,
! [A: $tType] :
( ( select_weight @ A )
= ( ^ [Xs2: 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 ) @ Xs2 ) ) )
@ ^ [K3: code_natural] : ( product_Pair @ A @ ( product_prod @ code_natural @ code_natural ) @ ( pick @ A @ Xs2 @ K3 ) ) ) ) ) ).
% select_weight_def
thf(fact_2454_adm__wf__def,axiom,
! [B: $tType,A: $tType] :
( ( adm_wf @ A @ B )
= ( ^ [R6: set @ ( product_prod @ A @ A ),F8: ( A > B ) > A > B] :
! [F4: A > B,G4: A > B,X4: A] :
( ! [Z5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z5 @ X4 ) @ R6 )
=> ( ( F4 @ Z5 )
= ( G4 @ Z5 ) ) )
=> ( ( F8 @ F4 @ X4 )
= ( F8 @ G4 @ X4 ) ) ) ) ) ).
% adm_wf_def
thf(fact_2455_comm__monoid__set_OUnion__comp,axiom,
! [A: $tType,B: $tType,F: A > A > A,Z3: A,B6: set @ ( set @ B ),G: B > A] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ! [X: set @ B] :
( ( member @ ( set @ B ) @ X @ B6 )
=> ( finite_finite2 @ B @ X ) )
=> ( ! [A18: set @ B] :
( ( member @ ( set @ B ) @ A18 @ B6 )
=> ! [A26: set @ B] :
( ( member @ ( set @ B ) @ A26 @ B6 )
=> ( ( A18 != A26 )
=> ! [X: B] :
( ( member @ B @ X @ A18 )
=> ( ( member @ B @ X @ A26 )
=> ( ( G @ X )
= Z3 ) ) ) ) ) )
=> ( ( groups_comm_monoid_F @ A @ B @ F @ Z3 @ G @ ( complete_Sup_Sup @ ( set @ B ) @ B6 ) )
= ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups_comm_monoid_F @ A @ ( set @ B ) @ F @ Z3 ) @ ( groups_comm_monoid_F @ A @ B @ F @ Z3 ) @ G @ B6 ) ) ) ) ) ).
% comm_monoid_set.Union_comp
thf(fact_2456_comm__monoid__set_Oreindex__cong,axiom,
! [C: $tType,A: $tType,B: $tType,F: A > A > A,Z3: A,L: B > C,B6: set @ B,A6: set @ C,G: C > A,H: B > A] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( inj_on @ B @ C @ L @ B6 )
=> ( ( A6
= ( image2 @ B @ C @ L @ B6 ) )
=> ( ! [X: B] :
( ( member @ B @ X @ B6 )
=> ( ( G @ ( L @ X ) )
= ( H @ X ) ) )
=> ( ( groups_comm_monoid_F @ A @ C @ F @ Z3 @ G @ A6 )
= ( groups_comm_monoid_F @ A @ B @ F @ Z3 @ H @ B6 ) ) ) ) ) ) ).
% comm_monoid_set.reindex_cong
thf(fact_2457_Abs__unit__cases,axiom,
! [X2: product_unit] :
~ ! [Y: $o] :
( ( X2
= ( product_Abs_unit @ Y ) )
=> ~ ( member @ $o @ Y @ ( insert @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) ) ) ).
% Abs_unit_cases
thf(fact_2458_Abs__unit__induct,axiom,
! [P2: product_unit > $o,X2: product_unit] :
( ! [Y: $o] :
( ( member @ $o @ Y @ ( insert @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) )
=> ( P2 @ ( product_Abs_unit @ Y ) ) )
=> ( P2 @ X2 ) ) ).
% Abs_unit_induct
thf(fact_2459_Abs__unit__inject,axiom,
! [X2: $o,Y3: $o] :
( ( member @ $o @ X2 @ ( insert @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) )
=> ( ( member @ $o @ Y3 @ ( insert @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) )
=> ( ( ( product_Abs_unit @ X2 )
= ( product_Abs_unit @ Y3 ) )
= ( X2 = Y3 ) ) ) ) ).
% Abs_unit_inject
thf(fact_2460_comm__monoid__set_Ocartesian__product,axiom,
! [A: $tType,C: $tType,B: $tType,F: A > A > A,Z3: A,G: B > C > A,B6: set @ C,A6: set @ B] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ B @ F @ Z3
@ ^ [X4: B] : ( groups_comm_monoid_F @ A @ C @ F @ Z3 @ ( G @ X4 ) @ B6 )
@ A6 )
= ( groups_comm_monoid_F @ A @ ( product_prod @ B @ C ) @ F @ Z3 @ ( product_case_prod @ B @ C @ A @ G )
@ ( product_Sigma @ B @ C @ A6
@ ^ [Uu: B] : B6 ) ) ) ) ).
% comm_monoid_set.cartesian_product
thf(fact_2461_comm__monoid__set_OSigma,axiom,
! [A: $tType,C: $tType,B: $tType,F: A > A > A,Z3: A,A6: set @ B,B6: B > ( set @ C ),G: B > C > A] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( finite_finite2 @ B @ A6 )
=> ( ! [X: B] :
( ( member @ B @ X @ A6 )
=> ( finite_finite2 @ C @ ( B6 @ X ) ) )
=> ( ( groups_comm_monoid_F @ A @ B @ F @ Z3
@ ^ [X4: B] : ( groups_comm_monoid_F @ A @ C @ F @ Z3 @ ( G @ X4 ) @ ( B6 @ X4 ) )
@ A6 )
= ( groups_comm_monoid_F @ A @ ( product_prod @ B @ C ) @ F @ Z3 @ ( product_case_prod @ B @ C @ A @ G ) @ ( product_Sigma @ B @ C @ A6 @ B6 ) ) ) ) ) ) ).
% comm_monoid_set.Sigma
thf(fact_2462_comm__monoid__set_Oeq__fold,axiom,
! [A: $tType,B: $tType,F: A > A > A,Z3: A,G: B > A,A6: set @ B] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ B @ F @ Z3 @ G @ A6 )
= ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ F @ G ) @ Z3 @ A6 ) ) ) ).
% comm_monoid_set.eq_fold
thf(fact_2463_comm__monoid__set_Oimage__gen,axiom,
! [A: $tType,C: $tType,B: $tType,F: A > A > A,Z3: A,S5: set @ B,H: B > A,G: B > C] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( finite_finite2 @ B @ S5 )
=> ( ( groups_comm_monoid_F @ A @ B @ F @ Z3 @ H @ S5 )
= ( groups_comm_monoid_F @ A @ C @ F @ Z3
@ ^ [Y5: C] :
( groups_comm_monoid_F @ A @ B @ F @ Z3 @ H
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ S5 )
& ( ( G @ X4 )
= Y5 ) ) ) )
@ ( image2 @ B @ C @ G @ S5 ) ) ) ) ) ).
% comm_monoid_set.image_gen
thf(fact_2464_comm__monoid__set_Oimage__eq,axiom,
! [A: $tType,B: $tType,F: A > A > A,Z3: A,G: B > A,A6: set @ B] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( inj_on @ B @ A @ G @ A6 )
=> ( ( groups_comm_monoid_F @ A @ A @ F @ Z3
@ ^ [X4: A] : X4
@ ( image2 @ B @ A @ G @ A6 ) )
= ( groups_comm_monoid_F @ A @ B @ F @ Z3 @ G @ A6 ) ) ) ) ).
% comm_monoid_set.image_eq
thf(fact_2465_comm__monoid__set_OatLeastAtMost__reindex,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > A > A,Z3: A,H: nat > B,M4: nat,N5: nat,G: B > A] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( bij_betw @ nat @ B @ H @ ( set_or1337092689740270186AtMost @ nat @ M4 @ N5 ) @ ( set_or1337092689740270186AtMost @ B @ ( H @ M4 ) @ ( H @ N5 ) ) )
=> ( ( groups_comm_monoid_F @ A @ B @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ B @ ( H @ M4 ) @ ( H @ N5 ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ ( comp @ B @ A @ nat @ G @ H ) @ ( set_or1337092689740270186AtMost @ nat @ M4 @ N5 ) ) ) ) ) ) ).
% comm_monoid_set.atLeastAtMost_reindex
thf(fact_2466_comm__monoid__set_OatLeastLessThan__reindex,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > A > A,Z3: A,H: nat > B,M4: nat,N5: nat,G: B > A] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( bij_betw @ nat @ B @ H @ ( set_or7035219750837199246ssThan @ nat @ M4 @ N5 ) @ ( set_or7035219750837199246ssThan @ B @ ( H @ M4 ) @ ( H @ N5 ) ) )
=> ( ( groups_comm_monoid_F @ A @ B @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ B @ ( H @ M4 ) @ ( H @ N5 ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ ( comp @ B @ A @ nat @ G @ H ) @ ( set_or7035219750837199246ssThan @ nat @ M4 @ N5 ) ) ) ) ) ) ).
% comm_monoid_set.atLeastLessThan_reindex
thf(fact_2467_comm__monoid__set_Oreindex__nontrivial,axiom,
! [C: $tType,A: $tType,B: $tType,F: A > A > A,Z3: A,A6: set @ B,H: B > C,G: C > A] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( finite_finite2 @ B @ A6 )
=> ( ! [X: B,Y: B] :
( ( member @ B @ X @ A6 )
=> ( ( member @ B @ Y @ A6 )
=> ( ( X != Y )
=> ( ( ( H @ X )
= ( H @ Y ) )
=> ( ( G @ ( H @ X ) )
= Z3 ) ) ) ) )
=> ( ( groups_comm_monoid_F @ A @ C @ F @ Z3 @ G @ ( image2 @ B @ C @ H @ A6 ) )
= ( groups_comm_monoid_F @ A @ B @ F @ Z3 @ ( comp @ C @ A @ B @ G @ H ) @ A6 ) ) ) ) ) ).
% comm_monoid_set.reindex_nontrivial
thf(fact_2468_comm__monoid__set_Oreindex,axiom,
! [C: $tType,A: $tType,B: $tType,F: A > A > A,Z3: A,H: B > C,A6: set @ B,G: C > A] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( inj_on @ B @ C @ H @ A6 )
=> ( ( groups_comm_monoid_F @ A @ C @ F @ Z3 @ G @ ( image2 @ B @ C @ H @ A6 ) )
= ( groups_comm_monoid_F @ A @ B @ F @ Z3 @ ( comp @ C @ A @ B @ G @ H ) @ A6 ) ) ) ) ).
% comm_monoid_set.reindex
thf(fact_2469_comm__monoid__set_OatLeast__Suc__atMost__Suc__shift,axiom,
! [A: $tType,F: A > A > A,Z3: A,G: nat > A,M4: nat,N5: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M4 ) @ ( suc @ N5 ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ M4 @ N5 ) ) ) ) ).
% comm_monoid_set.atLeast_Suc_atMost_Suc_shift
thf(fact_2470_comm__monoid__set_OatLeast__Suc__lessThan__Suc__shift,axiom,
! [A: $tType,F: A > A > A,Z3: A,G: nat > A,M4: nat,N5: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M4 ) @ ( suc @ N5 ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ M4 @ N5 ) ) ) ) ).
% comm_monoid_set.atLeast_Suc_lessThan_Suc_shift
thf(fact_2471_comm__monoid__set_OatLeastAtMost__shift__bounds,axiom,
! [A: $tType,F: A > A > A,Z3: A,G: nat > A,M4: nat,K: nat,N5: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M4 @ K ) @ ( plus_plus @ nat @ N5 @ K ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ K ) ) @ ( set_or1337092689740270186AtMost @ nat @ M4 @ N5 ) ) ) ) ).
% comm_monoid_set.atLeastAtMost_shift_bounds
thf(fact_2472_comm__monoid__set_OatLeastLessThan__shift__bounds,axiom,
! [A: $tType,F: A > A > A,Z3: A,G: nat > A,M4: nat,K: nat,N5: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M4 @ K ) @ ( plus_plus @ nat @ N5 @ K ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ K ) ) @ ( set_or7035219750837199246ssThan @ nat @ M4 @ N5 ) ) ) ) ).
% comm_monoid_set.atLeastLessThan_shift_bounds
thf(fact_2473_comm__monoid__set_Ogroup,axiom,
! [C: $tType,A: $tType,B: $tType,F: A > A > A,Z3: A,S5: set @ B,T4: set @ C,G: B > C,H: B > A] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( finite_finite2 @ B @ S5 )
=> ( ( finite_finite2 @ C @ T4 )
=> ( ( ord_less_eq @ ( set @ C ) @ ( image2 @ B @ C @ G @ S5 ) @ T4 )
=> ( ( groups_comm_monoid_F @ A @ C @ F @ Z3
@ ^ [Y5: C] :
( groups_comm_monoid_F @ A @ B @ F @ Z3 @ H
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ S5 )
& ( ( G @ X4 )
= Y5 ) ) ) )
@ T4 )
= ( groups_comm_monoid_F @ A @ B @ F @ Z3 @ H @ S5 ) ) ) ) ) ) ).
% comm_monoid_set.group
thf(fact_2474_comm__monoid__set_OPlus,axiom,
! [B: $tType,A: $tType,C: $tType,F: A > A > A,Z3: A,A6: set @ B,B6: set @ C,G: ( sum_sum @ B @ C ) > A] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( finite_finite2 @ B @ A6 )
=> ( ( finite_finite2 @ C @ B6 )
=> ( ( groups_comm_monoid_F @ A @ ( sum_sum @ B @ C ) @ F @ Z3 @ G @ ( sum_Plus @ B @ C @ A6 @ B6 ) )
= ( F @ ( groups_comm_monoid_F @ A @ B @ F @ Z3 @ ( comp @ ( sum_sum @ B @ C ) @ A @ B @ G @ ( sum_Inl @ B @ C ) ) @ A6 ) @ ( groups_comm_monoid_F @ A @ C @ F @ Z3 @ ( comp @ ( sum_sum @ B @ C ) @ A @ C @ G @ ( sum_Inr @ C @ B ) ) @ B6 ) ) ) ) ) ) ).
% comm_monoid_set.Plus
thf(fact_2475_comm__monoid__set_OUnion__disjoint,axiom,
! [A: $tType,B: $tType,F: A > A > A,Z3: A,C5: set @ ( set @ B ),G: B > A] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ! [X: set @ B] :
( ( member @ ( set @ B ) @ X @ C5 )
=> ( finite_finite2 @ B @ X ) )
=> ( ! [X: set @ B] :
( ( member @ ( set @ B ) @ X @ C5 )
=> ! [Xa: set @ B] :
( ( member @ ( set @ B ) @ Xa @ C5 )
=> ( ( X != Xa )
=> ( ( inf_inf @ ( set @ B ) @ X @ Xa )
= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ( groups_comm_monoid_F @ A @ B @ F @ Z3 @ G @ ( complete_Sup_Sup @ ( set @ B ) @ C5 ) )
= ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups_comm_monoid_F @ A @ ( set @ B ) @ F @ Z3 ) @ ( groups_comm_monoid_F @ A @ B @ F @ Z3 ) @ G @ C5 ) ) ) ) ) ).
% comm_monoid_set.Union_disjoint
thf(fact_2476_comm__monoid__set_OatLeast0__atMost__Suc__shift,axiom,
! [A: $tType,F: A > A > A,Z3: A,G: nat > A,N5: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N5 ) ) )
= ( F @ ( G @ ( zero_zero @ nat ) ) @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N5 ) ) ) ) ) ).
% comm_monoid_set.atLeast0_atMost_Suc_shift
thf(fact_2477_comm__monoid__set_OatLeast0__lessThan__Suc__shift,axiom,
! [A: $tType,F: A > A > A,Z3: A,G: nat > A,N5: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N5 ) ) )
= ( F @ ( G @ ( zero_zero @ nat ) ) @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N5 ) ) ) ) ) ).
% comm_monoid_set.atLeast0_lessThan_Suc_shift
thf(fact_2478_surj__prod__encode,axiom,
( ( image2 @ ( product_prod @ nat @ nat ) @ nat @ nat_prod_encode @ ( top_top @ ( set @ ( product_prod @ nat @ nat ) ) ) )
= ( top_top @ ( set @ nat ) ) ) ).
% surj_prod_encode
thf(fact_2479_comm__monoid__set_OatLeastLessThan__shift__0,axiom,
! [A: $tType,F: A > A > A,Z3: A,G: nat > A,M4: nat,N5: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ M4 @ N5 ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M4 ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N5 @ M4 ) ) ) ) ) ).
% comm_monoid_set.atLeastLessThan_shift_0
thf(fact_2480_comm__monoid__set_OUNION__disjoint,axiom,
! [A: $tType,C: $tType,B: $tType,F: A > A > A,Z3: A,I4: set @ B,A6: B > ( set @ C ),G: C > A] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( finite_finite2 @ B @ I4 )
=> ( ! [X: B] :
( ( member @ B @ X @ I4 )
=> ( finite_finite2 @ C @ ( A6 @ X ) ) )
=> ( ! [X: B] :
( ( member @ B @ X @ I4 )
=> ! [Xa: B] :
( ( member @ B @ Xa @ I4 )
=> ( ( X != Xa )
=> ( ( inf_inf @ ( set @ C ) @ ( A6 @ X ) @ ( A6 @ Xa ) )
= ( bot_bot @ ( set @ C ) ) ) ) ) )
=> ( ( groups_comm_monoid_F @ A @ C @ F @ Z3 @ G @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ A6 @ I4 ) ) )
= ( groups_comm_monoid_F @ A @ B @ F @ Z3
@ ^ [X4: B] : ( groups_comm_monoid_F @ A @ C @ F @ Z3 @ G @ ( A6 @ X4 ) )
@ I4 ) ) ) ) ) ) ).
% comm_monoid_set.UNION_disjoint
thf(fact_2481_comm__monoid__set_OatLeast__atMost__pred__shift,axiom,
! [A: $tType,F: A > A > A,Z3: A,G: nat > A,M4: nat,N5: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3
@ ( comp @ nat @ A @ nat @ G
@ ^ [N4: nat] : ( minus_minus @ nat @ N4 @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M4 ) @ ( suc @ N5 ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ M4 @ N5 ) ) ) ) ).
% comm_monoid_set.atLeast_atMost_pred_shift
thf(fact_2482_comm__monoid__set_OatLeast__lessThan__pred__shift,axiom,
! [A: $tType,F: A > A > A,Z3: A,G: nat > A,M4: nat,N5: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3
@ ( comp @ nat @ A @ nat @ G
@ ^ [N4: nat] : ( minus_minus @ nat @ N4 @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M4 ) @ ( suc @ N5 ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ M4 @ N5 ) ) ) ) ).
% comm_monoid_set.atLeast_lessThan_pred_shift
thf(fact_2483_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_2484_comm__monoid__set_Otriangle__reindex,axiom,
! [A: $tType,F: A > A > A,Z3: A,G: nat > nat > A,N5: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ ( product_prod @ nat @ nat ) @ F @ Z3 @ ( product_case_prod @ nat @ nat @ A @ G )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I3: nat,J: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I3 @ J ) @ N5 ) ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z3
@ ^ [K3: nat] :
( groups_comm_monoid_F @ A @ nat @ F @ Z3
@ ^ [I3: nat] : ( G @ I3 @ ( minus_minus @ nat @ K3 @ I3 ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_lessThan @ nat @ N5 ) ) ) ) ).
% comm_monoid_set.triangle_reindex
thf(fact_2485_comm__monoid__set_Otriangle__reindex__eq,axiom,
! [A: $tType,F: A > A > A,Z3: A,G: nat > nat > A,N5: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ ( product_prod @ nat @ nat ) @ F @ Z3 @ ( product_case_prod @ nat @ nat @ A @ G )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I3: nat,J: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I3 @ J ) @ N5 ) ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z3
@ ^ [K3: nat] :
( groups_comm_monoid_F @ A @ nat @ F @ Z3
@ ^ [I3: nat] : ( G @ I3 @ ( minus_minus @ nat @ K3 @ I3 ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_atMost @ nat @ N5 ) ) ) ) ).
% comm_monoid_set.triangle_reindex_eq
thf(fact_2486_comm__monoid__set_OatLeast__int__lessThan__int__shift,axiom,
! [A: $tType,F: A > A > A,Z3: A,G: int > A,M4: nat,N5: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ int @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ int @ ( semiring_1_of_nat @ int @ M4 ) @ ( semiring_1_of_nat @ int @ N5 ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ ( comp @ int @ A @ nat @ G @ ( semiring_1_of_nat @ int ) ) @ ( set_or7035219750837199246ssThan @ nat @ M4 @ N5 ) ) ) ) ).
% comm_monoid_set.atLeast_int_lessThan_int_shift
thf(fact_2487_comm__monoid__set_OatLeast__int__atMost__int__shift,axiom,
! [A: $tType,F: A > A > A,Z3: A,G: int > A,M4: nat,N5: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ int @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ M4 ) @ ( semiring_1_of_nat @ int @ N5 ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ ( comp @ int @ A @ nat @ G @ ( semiring_1_of_nat @ int ) ) @ ( set_or1337092689740270186AtMost @ nat @ M4 @ N5 ) ) ) ) ).
% comm_monoid_set.atLeast_int_atMost_int_shift
thf(fact_2488_type__definition__unit,axiom,
type_definition @ product_unit @ $o @ product_Rep_unit @ product_Abs_unit @ ( insert @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) ).
% type_definition_unit
thf(fact_2489_Rep__unit__inject,axiom,
! [X2: product_unit,Y3: product_unit] :
( ( ( product_Rep_unit @ X2 )
= ( product_Rep_unit @ Y3 ) )
= ( X2 = Y3 ) ) ).
% Rep_unit_inject
thf(fact_2490_Rep__unit__induct,axiom,
! [Y3: $o,P2: $o > $o] :
( ( member @ $o @ Y3 @ ( insert @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) )
=> ( ! [X: product_unit] : ( P2 @ ( product_Rep_unit @ X ) )
=> ( P2 @ Y3 ) ) ) ).
% Rep_unit_induct
thf(fact_2491_Rep__unit__cases,axiom,
! [Y3: $o] :
( ( member @ $o @ Y3 @ ( insert @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) )
=> ~ ! [X: product_unit] :
( Y3
= ( ~ ( product_Rep_unit @ X ) ) ) ) ).
% Rep_unit_cases
thf(fact_2492_Rep__unit,axiom,
! [X2: product_unit] : ( member @ $o @ ( product_Rep_unit @ X2 ) @ ( insert @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) ) ).
% Rep_unit
thf(fact_2493_Rep__unit__inverse,axiom,
! [X2: product_unit] :
( ( product_Abs_unit @ ( product_Rep_unit @ X2 ) )
= X2 ) ).
% Rep_unit_inverse
thf(fact_2494_Abs__unit__inverse,axiom,
! [Y3: $o] :
( ( member @ $o @ Y3 @ ( insert @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) )
=> ( ( product_Rep_unit @ ( product_Abs_unit @ Y3 ) )
= Y3 ) ) ).
% Abs_unit_inverse
thf(fact_2495_surj__prod__decode,axiom,
( ( image2 @ nat @ ( product_prod @ nat @ nat ) @ nat_prod_decode @ ( top_top @ ( set @ nat ) ) )
= ( top_top @ ( set @ ( product_prod @ nat @ nat ) ) ) ) ).
% surj_prod_decode
thf(fact_2496_select__def,axiom,
! [A: $tType] :
( ( select @ A )
= ( ^ [Xs2: 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 ) @ Xs2 ) ) )
@ ^ [K3: code_natural] : ( product_Pair @ A @ ( product_prod @ code_natural @ code_natural ) @ ( nth3 @ A @ Xs2 @ ( code_nat_of_natural @ K3 ) ) ) ) ) ) ).
% select_def
thf(fact_2497_folding__idem_Ocomp__fun__idem,axiom,
! [B: $tType,A: $tType,F: A > B > B,X2: A] :
( ( finite_folding_idem @ A @ B @ F )
=> ( ( comp @ B @ B @ B @ ( F @ X2 ) @ ( F @ X2 ) )
= ( F @ X2 ) ) ) ).
% folding_idem.comp_fun_idem
thf(fact_2498_pair__less__def,axiom,
( fun_pair_less
= ( lex_prod @ nat @ nat @ less_than @ less_than ) ) ).
% pair_less_def
thf(fact_2499_map__upds__fold__map__upd,axiom,
! [B: $tType,A: $tType] :
( ( map_upds @ A @ B )
= ( ^ [M7: A > ( option @ B ),Ks2: list @ A,Vs: list @ B] :
( foldl @ ( A > ( option @ B ) ) @ ( product_prod @ A @ B )
@ ^ [N4: A > ( option @ B )] :
( product_case_prod @ A @ B @ ( A > ( option @ B ) )
@ ^ [K3: A,V3: B] : ( fun_upd @ A @ ( option @ B ) @ N4 @ K3 @ ( some @ B @ V3 ) ) )
@ M7
@ ( zip @ A @ B @ Ks2 @ Vs ) ) ) ) ).
% map_upds_fold_map_upd
thf(fact_2500_rec__nat__0__imp,axiom,
! [A: $tType,F: nat > A,F1: A,F23: nat > A > A] :
( ( F
= ( rec_nat @ A @ F1 @ F23 ) )
=> ( ( F @ ( zero_zero @ nat ) )
= F1 ) ) ).
% rec_nat_0_imp
thf(fact_2501_rec__nat__Suc__imp,axiom,
! [A: $tType,F: nat > A,F1: A,F23: nat > A > A,N5: nat] :
( ( F
= ( rec_nat @ A @ F1 @ F23 ) )
=> ( ( F @ ( suc @ N5 ) )
= ( F23 @ N5 @ ( F @ N5 ) ) ) ) ).
% rec_nat_Suc_imp
thf(fact_2502_list__decode_Oelims,axiom,
! [X2: nat,Y3: list @ nat] :
( ( ( nat_list_decode @ X2 )
= Y3 )
=> ( ( ( X2
= ( zero_zero @ nat ) )
=> ( Y3
!= ( nil @ nat ) ) )
=> ~ ! [N2: nat] :
( ( X2
= ( suc @ N2 ) )
=> ( Y3
!= ( product_case_prod @ nat @ nat @ ( list @ nat )
@ ^ [X4: nat,Y5: nat] : ( cons @ nat @ X4 @ ( nat_list_decode @ Y5 ) )
@ ( nat_prod_decode @ N2 ) ) ) ) ) ) ).
% list_decode.elims
thf(fact_2503_nat__to__rat__surj__def,axiom,
( nat_to_rat_surj
= ( ^ [N4: nat] :
( product_case_prod @ nat @ nat @ rat
@ ^ [A7: nat,B5: nat] : ( fract @ ( nat_int_decode @ A7 ) @ ( nat_int_decode @ B5 ) )
@ ( nat_prod_decode @ N4 ) ) ) ) ).
% nat_to_rat_surj_def
thf(fact_2504_list__decode_Osimps_I2_J,axiom,
! [N5: nat] :
( ( nat_list_decode @ ( suc @ N5 ) )
= ( product_case_prod @ nat @ nat @ ( list @ nat )
@ ^ [X4: nat,Y5: nat] : ( cons @ nat @ X4 @ ( nat_list_decode @ Y5 ) )
@ ( nat_prod_decode @ N5 ) ) ) ).
% list_decode.simps(2)
thf(fact_2505_list__decode_Opelims,axiom,
! [X2: nat,Y3: list @ nat] :
( ( ( nat_list_decode @ X2 )
= Y3 )
=> ( ( accp @ nat @ nat_list_decode_rel @ X2 )
=> ( ( ( X2
= ( zero_zero @ nat ) )
=> ( ( Y3
= ( nil @ nat ) )
=> ~ ( accp @ nat @ nat_list_decode_rel @ ( zero_zero @ nat ) ) ) )
=> ~ ! [N2: nat] :
( ( X2
= ( suc @ N2 ) )
=> ( ( Y3
= ( product_case_prod @ nat @ nat @ ( list @ nat )
@ ^ [X4: nat,Y5: nat] : ( cons @ nat @ X4 @ ( nat_list_decode @ Y5 ) )
@ ( nat_prod_decode @ N2 ) ) )
=> ~ ( accp @ nat @ nat_list_decode_rel @ ( suc @ N2 ) ) ) ) ) ) ) ).
% list_decode.pelims
thf(fact_2506_folding_Ointro,axiom,
! [B: $tType,A: $tType,F: A > B > B] :
( ! [Y: A,X: A] :
( ( comp @ B @ B @ B @ ( F @ Y ) @ ( F @ X ) )
= ( comp @ B @ B @ B @ ( F @ X ) @ ( F @ Y ) ) )
=> ( finite_folding @ A @ B @ F ) ) ).
% folding.intro
thf(fact_2507_folding_Ocomp__fun__commute,axiom,
! [B: $tType,A: $tType,F: A > B > B,Y3: A,X2: A] :
( ( finite_folding @ A @ B @ F )
=> ( ( comp @ B @ B @ B @ ( F @ Y3 ) @ ( F @ X2 ) )
= ( comp @ B @ B @ B @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ) ).
% folding.comp_fun_commute
thf(fact_2508_surj__list__decode,axiom,
( ( image2 @ nat @ ( list @ nat ) @ nat_list_decode @ ( top_top @ ( set @ nat ) ) )
= ( top_top @ ( set @ ( list @ nat ) ) ) ) ).
% surj_list_decode
thf(fact_2509_list__decode_Opsimps_I2_J,axiom,
! [N5: nat] :
( ( accp @ nat @ nat_list_decode_rel @ ( suc @ N5 ) )
=> ( ( nat_list_decode @ ( suc @ N5 ) )
= ( product_case_prod @ nat @ nat @ ( list @ nat )
@ ^ [X4: nat,Y5: nat] : ( cons @ nat @ X4 @ ( nat_list_decode @ Y5 ) )
@ ( nat_prod_decode @ N5 ) ) ) ) ).
% list_decode.psimps(2)
thf(fact_2510_folding__def,axiom,
! [B: $tType,A: $tType] :
( ( finite_folding @ A @ B )
= ( ^ [F4: A > B > B] :
! [Y5: A,X4: A] :
( ( comp @ B @ B @ B @ ( F4 @ Y5 ) @ ( F4 @ X4 ) )
= ( comp @ B @ B @ B @ ( F4 @ X4 ) @ ( F4 @ Y5 ) ) ) ) ) ).
% folding_def
thf(fact_2511_folding__idem__on_Ocomp__fun__idem__on,axiom,
! [B: $tType,A: $tType,S5: set @ A,F: A > B > B,X2: A,Y3: A] :
( ( finite1890593828518410140dem_on @ A @ B @ S5 @ F )
=> ( ( member @ A @ X2 @ S5 )
=> ( ( member @ A @ Y3 @ S5 )
=> ( ( comp @ B @ B @ B @ ( F @ X2 ) @ ( F @ X2 ) )
= ( F @ X2 ) ) ) ) ) ).
% folding_idem_on.comp_fun_idem_on
thf(fact_2512_Heap__lub__def,axiom,
! [A: $tType] :
( ( heap_Heap_lub @ A )
= ( partial_img_lub @ ( heap_Heap @ A ) @ ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) ) @ ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) ) @ ( heap_Heap @ A ) @ ( heap_execute @ A ) @ ( heap_Heap2 @ A ) @ ( partial_fun_lub @ ( option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) @ ( option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) @ ( heap_ext @ product_unit ) @ ( partial_flat_lub @ ( option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) @ ( none @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) ) ) ) ) ).
% Heap_lub_def
thf(fact_2513_heap_Oconst__mono,axiom,
! [A: $tType,B: $tType,Ord: B > B > $o,C3: heap_Heap @ A] :
( comple7038119648293358887notone @ B @ ( heap_Heap @ A ) @ Ord @ ( heap_Heap_ord @ A )
@ ^ [F4: B] : C3 ) ).
% heap.const_mono
thf(fact_2514_Ref_Oget__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( get2 @ A )
= ( ^ [H6: heap_ext @ product_unit] : ( comp @ nat @ A @ ( ref @ A ) @ ( comp @ nat @ A @ nat @ ( from_nat @ A ) @ ( refs @ product_unit @ H6 @ ( typerep_typerep @ A @ ( type2 @ A ) ) ) ) @ ( addr_of_ref @ A ) ) ) ) ) ).
% Ref.get_def
thf(fact_2515_Array_Oget__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( get @ A )
= ( ^ [H6: heap_ext @ product_unit,A7: array @ A] : ( map @ nat @ A @ ( from_nat @ A ) @ ( arrays @ product_unit @ H6 @ ( typerep_typerep @ A @ ( type2 @ A ) ) @ ( addr_of_array @ A @ A7 ) ) ) ) ) ) ).
% Array.get_def
thf(fact_2516_surj__sum__encode,axiom,
( ( image2 @ ( sum_sum @ nat @ nat ) @ nat @ nat_sum_encode @ ( top_top @ ( set @ ( sum_sum @ nat @ nat ) ) ) )
= ( top_top @ ( set @ nat ) ) ) ).
% surj_sum_encode
thf(fact_2517_Array_Onoteq__def,axiom,
! [B: $tType,A: $tType] :
( ( ( heap @ A )
& ( heap @ B ) )
=> ( ( noteq @ A @ B )
= ( ^ [R5: array @ A,S: array @ B] :
( ( ( typerep_typerep @ A @ ( type2 @ A ) )
!= ( typerep_typerep @ B @ ( type2 @ B ) ) )
| ( ( addr_of_array @ A @ R5 )
!= ( addr_of_array @ B @ S ) ) ) ) ) ) ).
% Array.noteq_def
thf(fact_2518_Ref_Onoteq__def,axiom,
! [B: $tType,A: $tType] :
( ( ( heap @ A )
& ( heap @ B ) )
=> ( ( noteq2 @ A @ B )
= ( ^ [R5: ref @ A,S: ref @ B] :
( ( ( typerep_typerep @ A @ ( type2 @ A ) )
!= ( typerep_typerep @ B @ ( type2 @ B ) ) )
| ( ( addr_of_ref @ A @ R5 )
!= ( addr_of_ref @ B @ S ) ) ) ) ) ) ).
% Ref.noteq_def
thf(fact_2519_equality,axiom,
! [Z: $tType,R: heap_ext @ Z,R2: heap_ext @ Z] :
( ( ( arrays @ Z @ R )
= ( arrays @ Z @ R2 ) )
=> ( ( ( refs @ Z @ R )
= ( refs @ Z @ R2 ) )
=> ( ( ( lim @ Z @ R )
= ( lim @ Z @ R2 ) )
=> ( ( ( more @ Z @ R )
= ( more @ Z @ R2 ) )
=> ( R = R2 ) ) ) ) ) ).
% equality
thf(fact_2520_img__lub__def,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType] :
( ( partial_img_lub @ A @ C @ D @ B )
= ( ^ [F4: A > C,G4: D > B,Lub: ( set @ C ) > D,A8: set @ A] : ( G4 @ ( Lub @ ( image2 @ A @ C @ F4 @ A8 ) ) ) ) ) ).
% img_lub_def
thf(fact_2521_typerep__Heap__def,axiom,
! [A: $tType] :
( ( typerep2 @ A )
=> ( ( typerep_typerep @ ( heap_Heap @ A ) )
= ( ^ [T5: itself @ ( heap_Heap @ A )] : ( typerep3 @ ( literal2 @ $false @ $false @ $false @ $true @ $false @ $false @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $true @ $false @ $true @ ( literal2 @ $true @ $false @ $true @ $true @ $false @ $false @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $false @ ( literal2 @ $false @ $false @ $false @ $true @ $false @ $false @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( zero_zero @ literal ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( cons @ typerep @ ( typerep_typerep @ A @ ( type2 @ A ) ) @ ( nil @ typerep ) ) ) ) ) ) ).
% typerep_Heap_def
thf(fact_2522_Array_Oset__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( set2 @ A )
= ( ^ [A7: array @ A,X4: list @ A] :
( arrays_update @ product_unit
@ ^ [H6: typerep > nat > ( list @ nat )] : ( fun_upd @ typerep @ ( nat > ( list @ nat ) ) @ H6 @ ( typerep_typerep @ A @ ( type2 @ A ) ) @ ( fun_upd @ nat @ ( list @ nat ) @ ( H6 @ ( typerep_typerep @ A @ ( type2 @ A ) ) ) @ ( addr_of_array @ A @ A7 ) @ ( map @ A @ nat @ ( to_nat @ A ) @ X4 ) ) ) ) ) ) ) ).
% Array.set_def
thf(fact_2523_Ref_Oset__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( set4 @ A )
= ( ^ [R5: ref @ A,X4: A] :
( refs_update @ product_unit
@ ^ [H6: typerep > nat > nat] : ( fun_upd @ typerep @ ( nat > nat ) @ H6 @ ( typerep_typerep @ A @ ( type2 @ A ) ) @ ( fun_upd @ nat @ nat @ ( H6 @ ( typerep_typerep @ A @ ( type2 @ A ) ) ) @ ( addr_of_ref @ A @ R5 ) @ ( to_nat @ A @ X4 ) ) ) ) ) ) ) ).
% Ref.set_def
thf(fact_2524_surj__list__encode,axiom,
( ( image2 @ ( list @ nat ) @ nat @ nat_list_encode @ ( top_top @ ( set @ ( list @ nat ) ) ) )
= ( top_top @ ( set @ nat ) ) ) ).
% surj_list_encode
thf(fact_2525_surjective,axiom,
! [Z: $tType,R: heap_ext @ Z] :
( R
= ( heap_ext2 @ Z @ ( arrays @ Z @ R ) @ ( refs @ Z @ R ) @ ( lim @ Z @ R ) @ ( more @ Z @ R ) ) ) ).
% surjective
thf(fact_2526_admissible__image,axiom,
! [A: $tType,B: $tType,Le: A > A > $o,Lub2: ( set @ A ) > A,P2: B > $o,G: A > B,F: B > A] :
( ( partia7178651479351089652itions @ A @ Le @ Lub2 )
=> ( ( comple1908693960933563346ssible @ A @ Lub2 @ Le @ ( comp @ B @ $o @ A @ P2 @ G ) )
=> ( ! [X: B,Y: B] :
( ( ( F @ X )
= ( F @ Y ) )
=> ( X = Y ) )
=> ( ! [X: A] :
( ( F @ ( G @ X ) )
= X )
=> ( comple1908693960933563346ssible @ B @ ( partial_img_lub @ B @ A @ A @ B @ F @ G @ Lub2 ) @ ( partial_img_ord @ B @ A @ $o @ F @ Le ) @ P2 ) ) ) ) ) ).
% admissible_image
thf(fact_2527_surj__sum__decode,axiom,
( ( image2 @ nat @ ( sum_sum @ nat @ nat ) @ nat_sum_decode @ ( top_top @ ( set @ nat ) ) )
= ( top_top @ ( set @ ( sum_sum @ nat @ nat ) ) ) ) ).
% surj_sum_decode
thf(fact_2528_Heap__Monad_Obind__mono,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,B6: ( A > ( heap_Heap @ B ) ) > ( heap_Heap @ C ),C5: C > ( A > ( heap_Heap @ B ) ) > ( heap_Heap @ D )] :
( ( comple7038119648293358887notone @ ( A > ( heap_Heap @ B ) ) @ ( heap_Heap @ C ) @ ( partial_fun_ord @ ( heap_Heap @ B ) @ ( heap_Heap @ B ) @ A @ ( heap_Heap_ord @ B ) ) @ ( heap_Heap_ord @ C ) @ B6 )
=> ( ! [Y: C] : ( comple7038119648293358887notone @ ( A > ( heap_Heap @ B ) ) @ ( heap_Heap @ D ) @ ( partial_fun_ord @ ( heap_Heap @ B ) @ ( heap_Heap @ B ) @ A @ ( heap_Heap_ord @ B ) ) @ ( heap_Heap_ord @ D ) @ ( C5 @ Y ) )
=> ( comple7038119648293358887notone @ ( A > ( heap_Heap @ B ) ) @ ( heap_Heap @ D ) @ ( partial_fun_ord @ ( heap_Heap @ B ) @ ( heap_Heap @ B ) @ A @ ( heap_Heap_ord @ B ) ) @ ( heap_Heap_ord @ D )
@ ^ [F4: A > ( heap_Heap @ B )] :
( heap_bind @ C @ D @ ( B6 @ F4 )
@ ^ [Y5: C] : ( C5 @ Y5 @ F4 ) ) ) ) ) ).
% Heap_Monad.bind_mono
thf(fact_2529_heap__step__admissible,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,P2: D > A > C > B > $o,X2: D] :
( comple1908693960933563346ssible @ ( A > ( option @ ( product_prod @ B @ C ) ) ) @ ( partial_fun_lub @ ( option @ ( product_prod @ B @ C ) ) @ ( option @ ( product_prod @ B @ C ) ) @ A @ ( partial_flat_lub @ ( option @ ( product_prod @ B @ C ) ) @ ( none @ ( product_prod @ B @ C ) ) ) ) @ ( partial_fun_ord @ ( option @ ( product_prod @ B @ C ) ) @ ( option @ ( product_prod @ B @ C ) ) @ A @ ( partial_flat_ord @ ( option @ ( product_prod @ B @ C ) ) @ ( none @ ( product_prod @ B @ C ) ) ) )
@ ^ [F4: A > ( option @ ( product_prod @ B @ C ) )] :
! [H6: A,H9: C,R5: B] :
( ( ( F4 @ H6 )
= ( some @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ R5 @ H9 ) ) )
=> ( P2 @ X2 @ H6 @ H9 @ R5 ) ) ) ).
% heap_step_admissible
thf(fact_2530_semilattice__map2,axiom,
! [A: $tType,F: A > A > A] :
( ( semilattice @ A @ F )
=> ( semilattice @ ( list @ A )
@ ^ [Xs2: list @ A,Ys2: list @ A] : ( map @ ( product_prod @ A @ A ) @ A @ ( product_case_prod @ A @ A @ A @ F ) @ ( zip @ A @ A @ Xs2 @ Ys2 ) ) ) ) ).
% semilattice_map2
thf(fact_2531_Heap__ord__def,axiom,
! [A: $tType] :
( ( heap_Heap_ord @ A )
= ( partial_img_ord @ ( heap_Heap @ A ) @ ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) ) @ $o @ ( heap_execute @ A ) @ ( partial_fun_ord @ ( option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) @ ( option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) @ ( heap_ext @ product_unit ) @ ( partial_flat_ord @ ( option @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) @ ( none @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) ) ) ) ) ).
% Heap_ord_def
thf(fact_2532_select__convs_I3_J,axiom,
! [Z: $tType,Arrays: typerep > nat > ( list @ nat ),Refs: typerep > nat > nat,Lim: nat,More: Z] :
( ( lim @ Z @ ( heap_ext2 @ Z @ Arrays @ Refs @ Lim @ More ) )
= Lim ) ).
% select_convs(3)
thf(fact_2533_admissible__heap,axiom,
! [B: $tType,A: $tType,P2: A > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) > B > $o] :
( comple1908693960933563346ssible @ ( A > ( heap_Heap @ B ) ) @ ( partial_fun_lub @ ( heap_Heap @ B ) @ ( heap_Heap @ B ) @ A @ ( heap_Heap_lub @ B ) ) @ ( partial_fun_ord @ ( heap_Heap @ B ) @ ( heap_Heap @ B ) @ A @ ( heap_Heap_ord @ B ) )
@ ^ [F4: A > ( heap_Heap @ B )] :
! [X4: A,H6: heap_ext @ product_unit,H9: heap_ext @ product_unit,R5: B] :
( ( heap_effect @ B @ ( F4 @ X4 ) @ H6 @ H9 @ R5 )
=> ( P2 @ X4 @ H6 @ H9 @ R5 ) ) ) ).
% admissible_heap
thf(fact_2534_heap_Ofixp__induct__uc,axiom,
! [B: $tType,A: $tType,C: $tType,U5: C > B > ( heap_Heap @ A ),F7: C > C,C5: ( B > ( heap_Heap @ A ) ) > C,F: C,P2: ( B > ( heap_Heap @ A ) ) > $o] :
( ! [X: B] :
( comple7038119648293358887notone @ ( B > ( heap_Heap @ A ) ) @ ( heap_Heap @ A ) @ ( partial_fun_ord @ ( heap_Heap @ A ) @ ( heap_Heap @ A ) @ B @ ( heap_Heap_ord @ A ) ) @ ( heap_Heap_ord @ A )
@ ^ [F4: B > ( heap_Heap @ A )] : ( U5 @ ( F7 @ ( C5 @ F4 ) ) @ X ) )
=> ( ( F
= ( C5
@ ( comple187402453842119260l_fixp @ ( B > ( heap_Heap @ A ) ) @ ( partial_fun_lub @ ( heap_Heap @ A ) @ ( heap_Heap @ A ) @ B @ ( heap_Heap_lub @ A ) ) @ ( partial_fun_ord @ ( heap_Heap @ A ) @ ( heap_Heap @ A ) @ B @ ( heap_Heap_ord @ A ) )
@ ^ [F4: B > ( heap_Heap @ A )] : ( U5 @ ( F7 @ ( C5 @ F4 ) ) ) ) ) )
=> ( ! [F3: B > ( heap_Heap @ A )] :
( ( U5 @ ( C5 @ F3 ) )
= F3 )
=> ( ( comple1908693960933563346ssible @ ( B > ( heap_Heap @ A ) ) @ ( partial_fun_lub @ ( heap_Heap @ A ) @ ( heap_Heap @ A ) @ B @ ( heap_Heap_lub @ A ) ) @ ( partial_fun_ord @ ( heap_Heap @ A ) @ ( heap_Heap @ A ) @ B @ ( heap_Heap_ord @ A ) ) @ P2 )
=> ( ( P2
@ ^ [Uu: B] :
( heap_Heap2 @ A
@ ^ [X4: heap_ext @ product_unit] : ( none @ ( product_prod @ A @ ( heap_ext @ product_unit ) ) ) ) )
=> ( ! [F3: C] :
( ( P2 @ ( U5 @ F3 ) )
=> ( P2 @ ( U5 @ ( F7 @ F3 ) ) ) )
=> ( P2 @ ( U5 @ F ) ) ) ) ) ) ) ) ).
% heap.fixp_induct_uc
thf(fact_2535_fixp__induct__heap,axiom,
! [C: $tType,B: $tType,A: $tType,U5: C > B > ( heap_Heap @ A ),F7: C > C,C5: ( B > ( heap_Heap @ A ) ) > C,F: C,P2: B > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) > A > $o,X2: B,H: heap_ext @ product_unit,H2: heap_ext @ product_unit,R: A] :
( ! [X: B] :
( comple7038119648293358887notone @ ( B > ( heap_Heap @ A ) ) @ ( heap_Heap @ A ) @ ( partial_fun_ord @ ( heap_Heap @ A ) @ ( heap_Heap @ A ) @ B @ ( heap_Heap_ord @ A ) ) @ ( heap_Heap_ord @ A )
@ ^ [F4: B > ( heap_Heap @ A )] : ( U5 @ ( F7 @ ( C5 @ F4 ) ) @ X ) )
=> ( ( F
= ( C5
@ ( comple187402453842119260l_fixp @ ( B > ( heap_Heap @ A ) ) @ ( partial_fun_lub @ ( heap_Heap @ A ) @ ( heap_Heap @ A ) @ B @ ( heap_Heap_lub @ A ) ) @ ( partial_fun_ord @ ( heap_Heap @ A ) @ ( heap_Heap @ A ) @ B @ ( heap_Heap_ord @ A ) )
@ ^ [F4: B > ( heap_Heap @ A )] : ( U5 @ ( F7 @ ( C5 @ F4 ) ) ) ) ) )
=> ( ! [F3: B > ( heap_Heap @ A )] :
( ( U5 @ ( C5 @ F3 ) )
= F3 )
=> ( ! [F3: C,X: B,H8: heap_ext @ product_unit,H4: heap_ext @ product_unit,R4: A] :
( ! [Xa3: B,Ha: heap_ext @ product_unit,H_a: heap_ext @ product_unit,Ra3: A] :
( ( heap_effect @ A @ ( U5 @ F3 @ Xa3 ) @ Ha @ H_a @ Ra3 )
=> ( P2 @ Xa3 @ Ha @ H_a @ Ra3 ) )
=> ( ( heap_effect @ A @ ( U5 @ ( F7 @ F3 ) @ X ) @ H8 @ H4 @ R4 )
=> ( P2 @ X @ H8 @ H4 @ R4 ) ) )
=> ( ( heap_effect @ A @ ( U5 @ F @ X2 ) @ H @ H2 @ R )
=> ( P2 @ X2 @ H @ H2 @ R ) ) ) ) ) ) ).
% fixp_induct_heap
thf(fact_2536_heap_Omono__body__fixp,axiom,
! [A: $tType,B: $tType,F7: ( B > ( heap_Heap @ A ) ) > B > ( heap_Heap @ A )] :
( ! [X: B] :
( comple7038119648293358887notone @ ( B > ( heap_Heap @ A ) ) @ ( heap_Heap @ A ) @ ( partial_fun_ord @ ( heap_Heap @ A ) @ ( heap_Heap @ A ) @ B @ ( heap_Heap_ord @ A ) ) @ ( heap_Heap_ord @ A )
@ ^ [F4: B > ( heap_Heap @ A )] : ( F7 @ F4 @ X ) )
=> ( ( comple187402453842119260l_fixp @ ( B > ( heap_Heap @ A ) ) @ ( partial_fun_lub @ ( heap_Heap @ A ) @ ( heap_Heap @ A ) @ B @ ( heap_Heap_lub @ A ) ) @ ( partial_fun_ord @ ( heap_Heap @ A ) @ ( heap_Heap @ A ) @ B @ ( heap_Heap_ord @ A ) ) @ F7 )
= ( F7 @ ( comple187402453842119260l_fixp @ ( B > ( heap_Heap @ A ) ) @ ( partial_fun_lub @ ( heap_Heap @ A ) @ ( heap_Heap @ A ) @ B @ ( heap_Heap_lub @ A ) ) @ ( partial_fun_ord @ ( heap_Heap @ A ) @ ( heap_Heap @ A ) @ B @ ( heap_Heap_ord @ A ) ) @ F7 ) ) ) ) ).
% heap.mono_body_fixp
thf(fact_2537_heap_Ofixp__rule__uc,axiom,
! [B: $tType,A: $tType,C: $tType,U5: C > B > ( heap_Heap @ A ),F7: C > C,C5: ( B > ( heap_Heap @ A ) ) > C,F: C] :
( ! [X: B] :
( comple7038119648293358887notone @ ( B > ( heap_Heap @ A ) ) @ ( heap_Heap @ A ) @ ( partial_fun_ord @ ( heap_Heap @ A ) @ ( heap_Heap @ A ) @ B @ ( heap_Heap_ord @ A ) ) @ ( heap_Heap_ord @ A )
@ ^ [F4: B > ( heap_Heap @ A )] : ( U5 @ ( F7 @ ( C5 @ F4 ) ) @ X ) )
=> ( ( F
= ( C5
@ ( comple187402453842119260l_fixp @ ( B > ( heap_Heap @ A ) ) @ ( partial_fun_lub @ ( heap_Heap @ A ) @ ( heap_Heap @ A ) @ B @ ( heap_Heap_lub @ A ) ) @ ( partial_fun_ord @ ( heap_Heap @ A ) @ ( heap_Heap @ A ) @ B @ ( heap_Heap_ord @ A ) )
@ ^ [F4: B > ( heap_Heap @ A )] : ( U5 @ ( F7 @ ( C5 @ F4 ) ) ) ) ) )
=> ( ! [F3: C] :
( ( C5 @ ( U5 @ F3 ) )
= F3 )
=> ( F
= ( F7 @ F ) ) ) ) ) ).
% heap.fixp_rule_uc
thf(fact_2538_divmod__step__int__def,axiom,
( ( unique1321980374590559556d_step @ int )
= ( ^ [L3: num] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [Q6: int,R5: int] : ( if @ ( product_prod @ int @ int ) @ ( ord_less_eq @ int @ ( numeral_numeral @ int @ L3 ) @ R5 ) @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one ) ) @ Q6 ) @ ( one_one @ int ) ) @ ( minus_minus @ int @ R5 @ ( numeral_numeral @ int @ L3 ) ) ) @ ( product_Pair @ int @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one ) ) @ Q6 ) @ R5 ) ) ) ) ) ).
% divmod_step_int_def
thf(fact_2539_divmod__step__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [L: num,R: A,Q: A] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ L ) @ R )
=> ( ( unique1321980374590559556d_step @ A @ L @ ( product_Pair @ A @ A @ Q @ R ) )
= ( product_Pair @ A @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one ) ) @ Q ) @ ( 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 @ Q @ R ) )
= ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one ) ) @ Q ) @ R ) ) ) ) ) ).
% divmod_step_eq
thf(fact_2540_divmod__step__def,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique1321980374590559556d_step @ A )
= ( ^ [L3: num] :
( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q6: A,R5: A] : ( if @ ( product_prod @ A @ A ) @ ( ord_less_eq @ A @ ( numeral_numeral @ A @ L3 ) @ R5 ) @ ( product_Pair @ A @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one ) ) @ Q6 ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ R5 @ ( numeral_numeral @ A @ L3 ) ) ) @ ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one ) ) @ Q6 ) @ R5 ) ) ) ) ) ) ).
% divmod_step_def
thf(fact_2541_divmod__algorithm__code_I5_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M4: num,N5: num] :
( ( unique8689654367752047608divmod @ A @ ( bit0 @ M4 ) @ ( bit0 @ N5 ) )
= ( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q6: A,R5: A] : ( product_Pair @ A @ A @ Q6 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one ) ) @ R5 ) )
@ ( unique8689654367752047608divmod @ A @ M4 @ N5 ) ) ) ) ).
% divmod_algorithm_code(5)
thf(fact_2542_one__mod__numeral,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N5: num] :
( ( modulo_modulo @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N5 ) )
= ( product_snd @ A @ A @ ( unique8689654367752047608divmod @ A @ one @ N5 ) ) ) ) ).
% one_mod_numeral
thf(fact_2543_one__div__numeral,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N5: num] :
( ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N5 ) )
= ( product_fst @ A @ A @ ( unique8689654367752047608divmod @ A @ one @ N5 ) ) ) ) ).
% one_div_numeral
thf(fact_2544_divmod__algorithm__code_I3_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N5: num] :
( ( unique8689654367752047608divmod @ A @ one @ ( bit0 @ N5 ) )
= ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ one ) ) ) ) ).
% divmod_algorithm_code(3)
thf(fact_2545_divmod__divmod__step,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique8689654367752047608divmod @ A )
= ( ^ [M7: num,N4: num] : ( if @ ( product_prod @ A @ A ) @ ( ord_less @ num @ M7 @ N4 ) @ ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ M7 ) ) @ ( unique1321980374590559556d_step @ A @ N4 @ ( unique8689654367752047608divmod @ A @ M7 @ ( bit0 @ N4 ) ) ) ) ) ) ) ).
% divmod_divmod_step
thf(fact_2546_divmod__algorithm__code_I2_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M4: num] :
( ( unique8689654367752047608divmod @ A @ M4 @ one )
= ( product_Pair @ A @ A @ ( numeral_numeral @ A @ M4 ) @ ( zero_zero @ A ) ) ) ) ).
% divmod_algorithm_code(2)
thf(fact_2547_numeral__div__numeral,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [K: num,L: num] :
( ( divide_divide @ A @ ( numeral_numeral @ A @ K ) @ ( numeral_numeral @ A @ L ) )
= ( product_fst @ A @ A @ ( unique8689654367752047608divmod @ A @ K @ L ) ) ) ) ).
% numeral_div_numeral
thf(fact_2548_numeral__mod__numeral,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [K: num,L: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ K ) @ ( numeral_numeral @ A @ L ) )
= ( product_snd @ A @ A @ ( unique8689654367752047608divmod @ A @ K @ L ) ) ) ) ).
% numeral_mod_numeral
thf(fact_2549_fst__divmod,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M4: num,N5: num] :
( ( product_fst @ A @ A @ ( unique8689654367752047608divmod @ A @ M4 @ N5 ) )
= ( divide_divide @ A @ ( numeral_numeral @ A @ M4 ) @ ( numeral_numeral @ A @ N5 ) ) ) ) ).
% fst_divmod
thf(fact_2550_snd__divmod,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M4: num,N5: num] :
( ( product_snd @ A @ A @ ( unique8689654367752047608divmod @ A @ M4 @ N5 ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ M4 ) @ ( numeral_numeral @ A @ N5 ) ) ) ) ).
% snd_divmod
thf(fact_2551_divmod__def,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique8689654367752047608divmod @ A )
= ( ^ [M7: num,N4: num] : ( product_Pair @ A @ A @ ( divide_divide @ A @ ( numeral_numeral @ A @ M7 ) @ ( numeral_numeral @ A @ N4 ) ) @ ( modulo_modulo @ A @ ( numeral_numeral @ A @ M7 ) @ ( numeral_numeral @ A @ N4 ) ) ) ) ) ) ).
% divmod_def
thf(fact_2552_divmod__step__nat__def,axiom,
( ( unique1321980374590559556d_step @ nat )
= ( ^ [L3: num] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [Q6: nat,R5: nat] : ( if @ ( product_prod @ nat @ nat ) @ ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ L3 ) @ R5 ) @ ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one ) ) @ Q6 ) @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ R5 @ ( numeral_numeral @ nat @ L3 ) ) ) @ ( product_Pair @ nat @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one ) ) @ Q6 ) @ R5 ) ) ) ) ) ).
% divmod_step_nat_def
thf(fact_2553_divmod__algorithm__code_I6_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M4: num,N5: num] :
( ( unique8689654367752047608divmod @ A @ ( bit1 @ M4 ) @ ( bit0 @ N5 ) )
= ( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q6: A,R5: A] : ( product_Pair @ A @ A @ Q6 @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one ) ) @ R5 ) @ ( one_one @ A ) ) )
@ ( unique8689654367752047608divmod @ A @ M4 @ N5 ) ) ) ) ).
% divmod_algorithm_code(6)
thf(fact_2554_divmod__algorithm__code_I8_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M4: num,N5: num] :
( ( ( ord_less @ num @ M4 @ N5 )
=> ( ( unique8689654367752047608divmod @ A @ ( bit1 @ M4 ) @ ( bit1 @ N5 ) )
= ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ ( bit1 @ M4 ) ) ) ) )
& ( ~ ( ord_less @ num @ M4 @ N5 )
=> ( ( unique8689654367752047608divmod @ A @ ( bit1 @ M4 ) @ ( bit1 @ N5 ) )
= ( unique1321980374590559556d_step @ A @ ( bit1 @ N5 ) @ ( unique8689654367752047608divmod @ A @ ( bit1 @ M4 ) @ ( bit0 @ ( bit1 @ N5 ) ) ) ) ) ) ) ) ).
% divmod_algorithm_code(8)
thf(fact_2555_divmod__step__integer__def,axiom,
( ( unique1321980374590559556d_step @ code_integer )
= ( ^ [L3: num] :
( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [Q6: code_integer,R5: code_integer] : ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less_eq @ code_integer @ ( numeral_numeral @ code_integer @ L3 ) @ R5 ) @ ( product_Pair @ code_integer @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one ) ) @ Q6 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ R5 @ ( numeral_numeral @ code_integer @ L3 ) ) ) @ ( product_Pair @ code_integer @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one ) ) @ Q6 ) @ R5 ) ) ) ) ) ).
% divmod_step_integer_def
thf(fact_2556_nat__of__integer__code,axiom,
( code_nat_of_integer
= ( ^ [K3: code_integer] :
( if @ nat @ ( ord_less_eq @ code_integer @ K3 @ ( zero_zero @ code_integer ) ) @ ( zero_zero @ nat )
@ ( product_case_prod @ code_integer @ code_integer @ nat
@ ^ [L3: code_integer,J: code_integer] :
( if @ nat
@ ( J
= ( 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 @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one ) ) ) ) ) ) ) ).
% nat_of_integer_code
thf(fact_2557_divmod__algorithm__code_I7_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M4: num,N5: num] :
( ( ( ord_less_eq @ num @ M4 @ N5 )
=> ( ( unique8689654367752047608divmod @ A @ ( bit0 @ M4 ) @ ( bit1 @ N5 ) )
= ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ ( bit0 @ M4 ) ) ) ) )
& ( ~ ( ord_less_eq @ num @ M4 @ N5 )
=> ( ( unique8689654367752047608divmod @ A @ ( bit0 @ M4 ) @ ( bit1 @ N5 ) )
= ( unique1321980374590559556d_step @ A @ ( bit1 @ N5 ) @ ( unique8689654367752047608divmod @ A @ ( bit0 @ M4 ) @ ( bit0 @ ( bit1 @ N5 ) ) ) ) ) ) ) ) ).
% divmod_algorithm_code(7)
thf(fact_2558_divmod__algorithm__code_I4_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N5: num] :
( ( unique8689654367752047608divmod @ A @ one @ ( bit1 @ N5 ) )
= ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ one ) ) ) ) ).
% divmod_algorithm_code(4)
thf(fact_2559_num__of__integer__code,axiom,
( code_num_of_integer
= ( ^ [K3: code_integer] :
( if @ num @ ( ord_less_eq @ code_integer @ K3 @ ( one_one @ code_integer ) ) @ one
@ ( product_case_prod @ code_integer @ code_integer @ num
@ ^ [L3: code_integer,J: code_integer] :
( if @ num
@ ( J
= ( 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 ) ) @ one ) )
@ ( code_divmod_integer @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one ) ) ) ) ) ) ) ).
% num_of_integer_code
thf(fact_2560_divmod__integer__code,axiom,
( code_divmod_integer
= ( ^ [K3: code_integer,L3: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( K3
= ( 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 ) @ K3 ) @ ( code_divmod_abs @ K3 @ L3 )
@ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [R5: code_integer,S: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( S
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ L3 @ S ) ) )
@ ( code_divmod_abs @ K3 @ L3 ) ) )
@ ( if @ ( product_prod @ code_integer @ code_integer )
@ ( L3
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ K3 )
@ ( product_apsnd @ code_integer @ code_integer @ code_integer @ ( uminus_uminus @ code_integer )
@ ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less @ code_integer @ K3 @ ( zero_zero @ code_integer ) ) @ ( code_divmod_abs @ K3 @ L3 )
@ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [R5: code_integer,S: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( S
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ L3 ) @ S ) ) )
@ ( code_divmod_abs @ K3 @ L3 ) ) ) ) ) ) ) ) ) ).
% divmod_integer_code
thf(fact_2561_int__of__integer__code,axiom,
( code_int_of_integer
= ( ^ [K3: code_integer] :
( if @ int @ ( ord_less @ code_integer @ K3 @ ( zero_zero @ code_integer ) ) @ ( uminus_uminus @ int @ ( code_int_of_integer @ ( uminus_uminus @ code_integer @ K3 ) ) )
@ ( if @ int
@ ( K3
= ( zero_zero @ code_integer ) )
@ ( zero_zero @ int )
@ ( product_case_prod @ code_integer @ code_integer @ int
@ ^ [L3: code_integer,J: code_integer] :
( if @ int
@ ( J
= ( zero_zero @ code_integer ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L3 ) )
@ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L3 ) ) @ ( one_one @ int ) ) )
@ ( code_divmod_integer @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% int_of_integer_code
thf(fact_2562_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_2563_bit__cut__integer__code,axiom,
( code_bit_cut_integer
= ( ^ [K3: code_integer] :
( if @ ( product_prod @ code_integer @ $o )
@ ( K3
= ( 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 )
@ ^ [R5: code_integer,S: code_integer] :
( product_Pair @ code_integer @ $o @ ( if @ code_integer @ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ K3 ) @ R5 @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ S ) )
@ ( S
= ( one_one @ code_integer ) ) )
@ ( code_divmod_abs @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one ) ) ) ) ) ) ) ).
% bit_cut_integer_code
thf(fact_2564_divmod__integer__eq__cases,axiom,
( code_divmod_integer
= ( ^ [K3: code_integer,L3: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( K3
= ( 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 ) @ K3 )
@ ( 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 @ K3 )
= ( sgn_sgn @ code_integer @ L3 ) )
@ ( code_divmod_abs @ K3 @ L3 )
@ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [R5: code_integer,S: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( S
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ ( abs_abs @ code_integer @ L3 ) @ S ) ) )
@ ( code_divmod_abs @ K3 @ L3 ) ) ) ) ) ) ) ) ).
% divmod_integer_eq_cases
thf(fact_2565_UNIV__char__of__nat,axiom,
( ( top_top @ ( set @ char ) )
= ( image2 @ nat @ char @ ( unique5772411509450598832har_of @ nat ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% UNIV_char_of_nat
thf(fact_2566_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_2567_range__nat__of__char,axiom,
( ( image2 @ char @ nat @ ( comm_s6883823935334413003f_char @ nat ) @ ( top_top @ ( set @ char ) ) )
= ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% range_nat_of_char
thf(fact_2568_weak__coinduct__image,axiom,
! [A: $tType,B: $tType,A3: A,X6: set @ A,G: A > B,F: ( set @ B ) > ( set @ B )] :
( ( member @ A @ A3 @ X6 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ G @ X6 ) @ ( F @ ( image2 @ A @ B @ G @ X6 ) ) )
=> ( member @ B @ ( G @ A3 ) @ ( complete_lattice_gfp @ ( set @ B ) @ F ) ) ) ) ).
% weak_coinduct_image
thf(fact_2569_last__zip,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] :
( ( Xs
!= ( nil @ A ) )
=> ( ( Ys
!= ( nil @ B ) )
=> ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( last @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) )
= ( product_Pair @ A @ B @ ( last @ A @ Xs ) @ ( last @ B @ Ys ) ) ) ) ) ) ).
% last_zip
thf(fact_2570_cuts__eq,axiom,
! [B: $tType,A: $tType,F: A > B,R3: set @ ( product_prod @ A @ A ),X2: A,G: A > B] :
( ( ( cut @ A @ B @ F @ R3 @ X2 )
= ( cut @ A @ B @ G @ R3 @ X2 ) )
= ( ! [Y5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X2 ) @ R3 )
=> ( ( F @ Y5 )
= ( G @ Y5 ) ) ) ) ) ).
% cuts_eq
thf(fact_2571_char__of__comp__of__char,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ( ( comp @ A @ char @ char @ ( unique5772411509450598832har_of @ A ) @ ( comm_s6883823935334413003f_char @ A ) )
= ( id @ char ) ) ) ).
% char_of_comp_of_char
thf(fact_2572_num__of__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ num @ num
@ ^ [Y2: int,Z2: int] : Y2 = Z2
@ ^ [Y2: num,Z2: num] : Y2 = Z2
@ ( comp @ nat @ num @ int @ num_of_nat @ nat2 )
@ ( comp @ nat @ num @ int @ num_of_nat @ nat2 ) ) ).
% num_of_integer.rsp
thf(fact_2573_cut__apply,axiom,
! [B: $tType,A: $tType,X2: A,A3: A,R3: set @ ( product_prod @ A @ A ),F: A > B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ A3 ) @ R3 )
=> ( ( cut @ A @ B @ F @ R3 @ A3 @ X2 )
= ( F @ X2 ) ) ) ).
% cut_apply
thf(fact_2574_num__of__integer_Otransfer,axiom,
( bNF_rel_fun @ int @ code_integer @ num @ num @ code_pcr_integer
@ ^ [Y2: num,Z2: num] : Y2 = Z2
@ ( comp @ nat @ num @ int @ num_of_nat @ nat2 )
@ code_num_of_integer ) ).
% num_of_integer.transfer
thf(fact_2575_cut__def,axiom,
! [B: $tType,A: $tType] :
( ( cut @ A @ B )
= ( ^ [F4: A > B,R6: set @ ( product_prod @ A @ A ),X4: A,Y5: A] : ( if @ B @ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X4 ) @ R6 ) @ ( F4 @ Y5 ) @ ( undefined @ B ) ) ) ) ).
% cut_def
thf(fact_2576_NEG__pcr__rule,axiom,
! [C: $tType,A: $tType,D: $tType,B: $tType,A6: A > C > $o,B6: C > B > $o,C5: A > B > $o,X6: B > D > $o] :
( ( neg @ A @ B @ ( relcompp @ A @ C @ B @ A6 @ B6 ) @ C5 )
=> ( neg @ A @ D @ ( relcompp @ A @ C @ D @ A6 @ ( relcompp @ C @ B @ D @ B6 @ X6 ) ) @ ( relcompp @ A @ B @ D @ C5 @ X6 ) ) ) ).
% NEG_pcr_rule
thf(fact_2577_neg__eq__OO,axiom,
! [B: $tType,A: $tType,A6: A > B > $o] :
( neg @ A @ B
@ ( relcompp @ A @ A @ B
@ ^ [Y2: A,Z2: A] : Y2 = Z2
@ A6 )
@ A6 ) ).
% neg_eq_OO
thf(fact_2578_neg__OO__eq,axiom,
! [B: $tType,A: $tType,A6: A > B > $o] :
( neg @ A @ B
@ ( relcompp @ A @ B @ B @ A6
@ ^ [Y2: B,Z2: B] : Y2 = Z2 )
@ A6 ) ).
% neg_OO_eq
thf(fact_2579_Product__Type_Oprod__def,axiom,
! [B: $tType,A: $tType] :
( ( product_prod2 @ A @ B )
= ( collect @ ( A > B > $o )
@ ^ [F4: A > B > $o] :
? [A7: A,B5: B] :
( F4
= ( product_Pair_Rep @ A @ B @ A7 @ B5 ) ) ) ) ).
% Product_Type.prod_def
thf(fact_2580_Pair__Rep__def,axiom,
! [B: $tType,A: $tType] :
( ( product_Pair_Rep @ A @ B )
= ( ^ [A7: A,B5: B,X4: A,Y5: B] :
( ( X4 = A7 )
& ( Y5 = B5 ) ) ) ) ).
% Pair_Rep_def
thf(fact_2581_Pair__def,axiom,
! [B: $tType,A: $tType] :
( ( product_Pair @ A @ B )
= ( ^ [A7: A,B5: B] : ( product_Abs_prod @ A @ B @ ( product_Pair_Rep @ A @ B @ A7 @ B5 ) ) ) ) ).
% Pair_def
thf(fact_2582_pcr__integer__def,axiom,
( code_pcr_integer
= ( relcompp @ int @ int @ code_integer
@ ^ [Y2: int,Z2: int] : Y2 = Z2
@ code_cr_integer ) ) ).
% pcr_integer_def
thf(fact_2583_curr__def,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( bNF_Wellorder_curr @ A @ B @ C )
= ( ^ [A8: set @ A,F4: ( product_prod @ A @ B ) > C,A7: A] :
( if @ ( B > C ) @ ( member @ A @ A7 @ A8 )
@ ^ [B5: B] : ( F4 @ ( product_Pair @ A @ B @ A7 @ B5 ) )
@ ( undefined @ ( B > C ) ) ) ) ) ).
% curr_def
thf(fact_2584_Rep__prod__induct,axiom,
! [B: $tType,A: $tType,Y3: A > B > $o,P2: ( A > B > $o ) > $o] :
( ( member @ ( A > B > $o ) @ Y3 @ ( product_prod2 @ A @ B ) )
=> ( ! [X: product_prod @ A @ B] : ( P2 @ ( product_Rep_prod @ A @ B @ X ) )
=> ( P2 @ Y3 ) ) ) ).
% Rep_prod_induct
thf(fact_2585_Rep__prod__cases,axiom,
! [B: $tType,A: $tType,Y3: A > B > $o] :
( ( member @ ( A > B > $o ) @ Y3 @ ( product_prod2 @ A @ B ) )
=> ~ ! [X: product_prod @ A @ B] :
( Y3
!= ( product_Rep_prod @ A @ B @ X ) ) ) ).
% Rep_prod_cases
thf(fact_2586_Rep__prod,axiom,
! [B: $tType,A: $tType,X2: product_prod @ A @ B] : ( member @ ( A > B > $o ) @ ( product_Rep_prod @ A @ B @ X2 ) @ ( product_prod2 @ A @ B ) ) ).
% Rep_prod
thf(fact_2587_Rep__prod__inverse,axiom,
! [B: $tType,A: $tType,X2: product_prod @ A @ B] :
( ( product_Abs_prod @ A @ B @ ( product_Rep_prod @ A @ B @ X2 ) )
= X2 ) ).
% Rep_prod_inverse
thf(fact_2588_Rep__prod__inject,axiom,
! [B: $tType,A: $tType,X2: product_prod @ A @ B,Y3: product_prod @ A @ B] :
( ( ( product_Rep_prod @ A @ B @ X2 )
= ( product_Rep_prod @ A @ B @ Y3 ) )
= ( X2 = Y3 ) ) ).
% Rep_prod_inject
thf(fact_2589_Abs__prod__inverse,axiom,
! [B: $tType,A: $tType,Y3: A > B > $o] :
( ( member @ ( A > B > $o ) @ Y3 @ ( product_prod2 @ A @ B ) )
=> ( ( product_Rep_prod @ A @ B @ ( product_Abs_prod @ A @ B @ Y3 ) )
= Y3 ) ) ).
% Abs_prod_inverse
thf(fact_2590_type__definition__prod,axiom,
! [B: $tType,A: $tType] : ( type_definition @ ( product_prod @ A @ B ) @ ( A > B > $o ) @ ( product_Rep_prod @ A @ B ) @ ( product_Abs_prod @ A @ B ) @ ( product_prod2 @ A @ B ) ) ).
% type_definition_prod
thf(fact_2591_Abs__prod__cases,axiom,
! [B: $tType,A: $tType,X2: product_prod @ A @ B] :
~ ! [Y: A > B > $o] :
( ( X2
= ( product_Abs_prod @ A @ B @ Y ) )
=> ~ ( member @ ( A > B > $o ) @ Y @ ( product_prod2 @ A @ B ) ) ) ).
% Abs_prod_cases
thf(fact_2592_Abs__prod__induct,axiom,
! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,X2: product_prod @ A @ B] :
( ! [Y: A > B > $o] :
( ( member @ ( A > B > $o ) @ Y @ ( product_prod2 @ A @ B ) )
=> ( P2 @ ( product_Abs_prod @ A @ B @ Y ) ) )
=> ( P2 @ X2 ) ) ).
% Abs_prod_induct
thf(fact_2593_Abs__prod__inject,axiom,
! [B: $tType,A: $tType,X2: A > B > $o,Y3: A > B > $o] :
( ( member @ ( A > B > $o ) @ X2 @ ( product_prod2 @ A @ B ) )
=> ( ( member @ ( A > B > $o ) @ Y3 @ ( product_prod2 @ A @ B ) )
=> ( ( ( product_Abs_prod @ A @ B @ X2 )
= ( product_Abs_prod @ A @ B @ Y3 ) )
= ( X2 = Y3 ) ) ) ) ).
% Abs_prod_inject
thf(fact_2594_ntrunc__UN1,axiom,
! [A: $tType,B: $tType,C: $tType,K: nat,F: C > ( set @ ( old_node @ A @ B ) )] :
( ( old_ntrunc @ A @ B @ K @ ( complete_Sup_Sup @ ( set @ ( old_node @ A @ B ) ) @ ( image2 @ C @ ( set @ ( old_node @ A @ B ) ) @ F @ ( top_top @ ( set @ C ) ) ) ) )
= ( complete_Sup_Sup @ ( set @ ( old_node @ A @ B ) )
@ ( image2 @ C @ ( set @ ( old_node @ A @ B ) )
@ ^ [X4: C] : ( old_ntrunc @ A @ B @ K @ ( F @ X4 ) )
@ ( top_top @ ( set @ C ) ) ) ) ) ).
% ntrunc_UN1
thf(fact_2595_POS__pcr__rule,axiom,
! [C: $tType,A: $tType,D: $tType,B: $tType,A6: A > C > $o,B6: C > B > $o,C5: A > B > $o,X6: B > D > $o] :
( ( pos @ A @ B @ ( relcompp @ A @ C @ B @ A6 @ B6 ) @ C5 )
=> ( pos @ A @ D @ ( relcompp @ A @ C @ D @ A6 @ ( relcompp @ C @ B @ D @ B6 @ X6 ) ) @ ( relcompp @ A @ B @ D @ C5 @ X6 ) ) ) ).
% POS_pcr_rule
thf(fact_2596_pos__eq__OO,axiom,
! [B: $tType,A: $tType,A6: A > B > $o] :
( pos @ A @ B
@ ( relcompp @ A @ A @ B
@ ^ [Y2: A,Z2: A] : Y2 = Z2
@ A6 )
@ A6 ) ).
% pos_eq_OO
thf(fact_2597_pos__OO__eq,axiom,
! [B: $tType,A: $tType,A6: A > B > $o] :
( pos @ A @ B
@ ( relcompp @ A @ B @ B @ A6
@ ^ [Y2: B,Z2: B] : Y2 = Z2 )
@ A6 ) ).
% pos_OO_eq
thf(fact_2598_comp__fun__idem_Ocomp__comp__fun__idem,axiom,
! [B: $tType,A: $tType,C: $tType,F: A > B > B,G: C > A] :
( ( finite_comp_fun_idem @ A @ B @ F )
=> ( finite_comp_fun_idem @ C @ B @ ( comp @ A @ ( B > B ) @ C @ F @ G ) ) ) ).
% comp_fun_idem.comp_comp_fun_idem
thf(fact_2599_In1__UN1,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > ( set @ ( old_node @ A @ B ) )] :
( ( old_In1 @ A @ B @ ( complete_Sup_Sup @ ( set @ ( old_node @ A @ B ) ) @ ( image2 @ C @ ( set @ ( old_node @ A @ B ) ) @ F @ ( top_top @ ( set @ C ) ) ) ) )
= ( complete_Sup_Sup @ ( set @ ( old_node @ A @ B ) )
@ ( image2 @ C @ ( set @ ( old_node @ A @ B ) )
@ ^ [X4: C] : ( old_In1 @ A @ B @ ( F @ X4 ) )
@ ( top_top @ ( set @ C ) ) ) ) ) ).
% In1_UN1
thf(fact_2600_ntrunc__o__equality,axiom,
! [C: $tType,B: $tType,A: $tType,H1: A > ( set @ ( old_node @ B @ C ) ),H22: A > ( set @ ( old_node @ B @ C ) )] :
( ! [K4: nat] :
( ( comp @ ( set @ ( old_node @ B @ C ) ) @ ( set @ ( old_node @ B @ C ) ) @ A @ ( old_ntrunc @ B @ C @ K4 ) @ H1 )
= ( comp @ ( set @ ( old_node @ B @ C ) ) @ ( set @ ( old_node @ B @ C ) ) @ A @ ( old_ntrunc @ B @ C @ K4 ) @ H22 ) )
=> ( H1 = H22 ) ) ).
% ntrunc_o_equality
thf(fact_2601_comp__fun__idem_Ocomp__fun__idem,axiom,
! [B: $tType,A: $tType,F: A > B > B,X2: A] :
( ( finite_comp_fun_idem @ A @ B @ F )
=> ( ( comp @ B @ B @ B @ ( F @ X2 ) @ ( F @ X2 ) )
= ( F @ X2 ) ) ) ).
% comp_fun_idem.comp_fun_idem
thf(fact_2602_In0__UN1,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > ( set @ ( old_node @ A @ B ) )] :
( ( old_In0 @ A @ B @ ( complete_Sup_Sup @ ( set @ ( old_node @ A @ B ) ) @ ( image2 @ C @ ( set @ ( old_node @ A @ B ) ) @ F @ ( top_top @ ( set @ C ) ) ) ) )
= ( complete_Sup_Sup @ ( set @ ( old_node @ A @ B ) )
@ ( image2 @ C @ ( set @ ( old_node @ A @ B ) )
@ ^ [X4: C] : ( old_In0 @ A @ B @ ( F @ X4 ) )
@ ( top_top @ ( set @ C ) ) ) ) ) ).
% In0_UN1
thf(fact_2603_Scons__UN1__x,axiom,
! [B: $tType,A: $tType,C: $tType,F: C > ( set @ ( old_node @ A @ B ) ),M: set @ ( old_node @ A @ B )] :
( ( old_Scons @ A @ B @ ( complete_Sup_Sup @ ( set @ ( old_node @ A @ B ) ) @ ( image2 @ C @ ( set @ ( old_node @ A @ B ) ) @ F @ ( top_top @ ( set @ C ) ) ) ) @ M )
= ( complete_Sup_Sup @ ( set @ ( old_node @ A @ B ) )
@ ( image2 @ C @ ( set @ ( old_node @ A @ B ) )
@ ^ [X4: C] : ( old_Scons @ A @ B @ ( F @ X4 ) @ M )
@ ( top_top @ ( set @ C ) ) ) ) ) ).
% Scons_UN1_x
thf(fact_2604_Scons__UN1__y,axiom,
! [A: $tType,B: $tType,C: $tType,M: set @ ( old_node @ A @ B ),F: C > ( set @ ( old_node @ A @ B ) )] :
( ( old_Scons @ A @ B @ M @ ( complete_Sup_Sup @ ( set @ ( old_node @ A @ B ) ) @ ( image2 @ C @ ( set @ ( old_node @ A @ B ) ) @ F @ ( top_top @ ( set @ C ) ) ) ) )
= ( complete_Sup_Sup @ ( set @ ( old_node @ A @ B ) )
@ ( image2 @ C @ ( set @ ( old_node @ A @ B ) )
@ ^ [X4: C] : ( old_Scons @ A @ B @ M @ ( F @ X4 ) )
@ ( top_top @ ( set @ C ) ) ) ) ) ).
% Scons_UN1_y
thf(fact_2605_Lim__def,axiom,
! [A: $tType,B: $tType] :
( ( old_Lim @ B @ A )
= ( ^ [F4: B > ( set @ ( old_node @ A @ B ) )] :
( complete_Sup_Sup @ ( set @ ( old_node @ A @ B ) )
@ ( collect @ ( set @ ( old_node @ A @ B ) )
@ ^ [Z5: set @ ( old_node @ A @ B )] :
? [X4: B] :
( Z5
= ( image2 @ ( old_node @ A @ B ) @ ( old_node @ A @ B ) @ ( old_Push_Node @ B @ A @ ( sum_Inl @ B @ nat @ X4 ) ) @ ( F4 @ X4 ) ) ) ) ) ) ) ).
% Lim_def
thf(fact_2606_iso__tuple__snd__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( iso_tuple_snd @ A @ B @ C )
= ( ^ [Isom: tuple_isomorphism @ A @ B @ C] : ( comp @ ( product_prod @ B @ C ) @ C @ A @ ( product_snd @ B @ C ) @ ( repr @ A @ B @ C @ Isom ) ) ) ) ).
% iso_tuple_snd_def
thf(fact_2607_iso__tuple__fst__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( iso_tuple_fst @ A @ B @ C )
= ( ^ [Isom: tuple_isomorphism @ A @ B @ C] : ( comp @ ( product_prod @ B @ C ) @ B @ A @ ( product_fst @ B @ C ) @ ( repr @ A @ B @ C @ Isom ) ) ) ) ).
% iso_tuple_fst_def
thf(fact_2608_Scons__def,axiom,
! [B: $tType,A: $tType] :
( ( old_Scons @ A @ B )
= ( ^ [M9: set @ ( old_node @ A @ B ),N10: 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 ) ) ) @ M9 ) @ ( image2 @ ( old_node @ A @ B ) @ ( old_node @ A @ B ) @ ( old_Push_Node @ B @ A @ ( sum_Inr @ nat @ B @ ( suc @ ( one_one @ nat ) ) ) ) @ N10 ) ) ) ) ).
% Scons_def
thf(fact_2609_dsum__def,axiom,
! [B: $tType,A: $tType] :
( ( old_dsum @ A @ B )
= ( ^ [R5: 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 ) ) )] :
( 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 ) ) ) )
@ ^ [X4: set @ ( old_node @ A @ B ),X7: set @ ( old_node @ A @ B )] : ( insert @ ( 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 @ X4 ) @ ( old_In0 @ A @ B @ X7 ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) ) ) ) )
@ R5 ) )
@ ( 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 ) ) ) )
@ ^ [Y5: set @ ( old_node @ A @ B ),Y6: set @ ( old_node @ A @ B )] : ( insert @ ( 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 @ Y5 ) @ ( old_In1 @ A @ B @ Y6 ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) ) ) ) )
@ S ) ) ) ) ) ).
% dsum_def
thf(fact_2610_usum__def,axiom,
! [B: $tType,A: $tType] :
( ( old_usum @ A @ B )
= ( ^ [A8: set @ ( set @ ( old_node @ A @ B ) ),B7: set @ ( set @ ( old_node @ A @ B ) )] : ( sup_sup @ ( set @ ( set @ ( old_node @ A @ B ) ) ) @ ( image2 @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( old_In0 @ A @ B ) @ A8 ) @ ( image2 @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( old_In1 @ A @ B ) @ B7 ) ) ) ) ).
% usum_def
thf(fact_2611_nth__item_Oelims,axiom,
! [A: $tType] :
( ( countable @ A )
=> ! [X2: nat,Y3: set @ ( old_node @ A @ product_unit )] :
( ( ( nth_item @ A @ X2 )
= Y3 )
=> ( ( ( X2
= ( zero_zero @ nat ) )
=> ( Y3
!= ( undefined @ ( set @ ( old_node @ A @ product_unit ) ) ) ) )
=> ~ ! [N2: nat] :
( ( X2
= ( suc @ N2 ) )
=> ( Y3
!= ( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
@ ^ [I3: nat] :
( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
@ ^ [J: nat] : ( old_In0 @ A @ product_unit @ ( nth_item @ A @ J ) )
@ ^ [J: nat] : ( old_In1 @ A @ product_unit @ ( nth_item @ A @ J ) )
@ ( nat_sum_decode @ I3 ) )
@ ^ [I3: nat] :
( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
@ ^ [J: nat] : ( old_Leaf @ A @ product_unit @ ( from_nat @ A @ J ) )
@ ^ [J: 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 @ J ) )
@ ( nat_sum_decode @ I3 ) )
@ ( nat_sum_decode @ N2 ) ) ) ) ) ) ) ).
% nth_item.elims
thf(fact_2612_nth__item_Opelims,axiom,
! [A: $tType] :
( ( countable @ A )
=> ! [X2: nat,Y3: set @ ( old_node @ A @ product_unit )] :
( ( ( nth_item @ A @ X2 )
= Y3 )
=> ( ( accp @ nat @ nth_item_rel @ X2 )
=> ( ( ( X2
= ( zero_zero @ nat ) )
=> ( ( Y3
= ( undefined @ ( set @ ( old_node @ A @ product_unit ) ) ) )
=> ~ ( accp @ nat @ nth_item_rel @ ( zero_zero @ nat ) ) ) )
=> ~ ! [N2: nat] :
( ( X2
= ( suc @ N2 ) )
=> ( ( Y3
= ( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
@ ^ [I3: nat] :
( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
@ ^ [J: nat] : ( old_In0 @ A @ product_unit @ ( nth_item @ A @ J ) )
@ ^ [J: nat] : ( old_In1 @ A @ product_unit @ ( nth_item @ A @ J ) )
@ ( nat_sum_decode @ I3 ) )
@ ^ [I3: nat] :
( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
@ ^ [J: nat] : ( old_Leaf @ A @ product_unit @ ( from_nat @ A @ J ) )
@ ^ [J: 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 @ J ) )
@ ( nat_sum_decode @ I3 ) )
@ ( nat_sum_decode @ N2 ) ) )
=> ~ ( accp @ nat @ nth_item_rel @ ( suc @ N2 ) ) ) ) ) ) ) ) ).
% nth_item.pelims
thf(fact_2613_iso__tuple__snd__update__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( iso_tuple_snd_update @ A @ B @ C )
= ( ^ [Isom: tuple_isomorphism @ A @ B @ C,F4: C > C] : ( comp @ ( product_prod @ B @ C ) @ A @ A @ ( comp @ ( product_prod @ B @ C ) @ A @ ( product_prod @ B @ C ) @ ( abst @ A @ B @ C @ Isom ) @ ( product_apsnd @ C @ C @ B @ F4 ) ) @ ( repr @ A @ B @ C @ Isom ) ) ) ) ).
% iso_tuple_snd_update_def
thf(fact_2614_iso__tuple__fst__update__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( iso_tuple_fst_update @ A @ B @ C )
= ( ^ [Isom: tuple_isomorphism @ A @ B @ C,F4: B > B] : ( comp @ ( product_prod @ B @ C ) @ A @ A @ ( comp @ ( product_prod @ B @ C ) @ A @ ( product_prod @ B @ C ) @ ( abst @ A @ B @ C @ Isom ) @ ( product_apfst @ B @ B @ C @ F4 ) ) @ ( repr @ A @ B @ C @ Isom ) ) ) ) ).
% iso_tuple_fst_update_def
thf(fact_2615_nth__item_Opsimps_I2_J,axiom,
! [A: $tType] :
( ( countable @ A )
=> ! [N5: nat] :
( ( accp @ nat @ nth_item_rel @ ( suc @ N5 ) )
=> ( ( nth_item @ A @ ( suc @ N5 ) )
= ( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
@ ^ [I3: nat] :
( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
@ ^ [J: nat] : ( old_In0 @ A @ product_unit @ ( nth_item @ A @ J ) )
@ ^ [J: nat] : ( old_In1 @ A @ product_unit @ ( nth_item @ A @ J ) )
@ ( nat_sum_decode @ I3 ) )
@ ^ [I3: nat] :
( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
@ ^ [J: nat] : ( old_Leaf @ A @ product_unit @ ( from_nat @ A @ J ) )
@ ^ [J: 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 @ J ) )
@ ( nat_sum_decode @ I3 ) )
@ ( nat_sum_decode @ N5 ) ) ) ) ) ).
% nth_item.psimps(2)
thf(fact_2616_nth__item_Osimps_I2_J,axiom,
! [A: $tType] :
( ( countable @ A )
=> ! [N5: nat] :
( ( nth_item @ A @ ( suc @ N5 ) )
= ( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
@ ^ [I3: nat] :
( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
@ ^ [J: nat] : ( old_In0 @ A @ product_unit @ ( nth_item @ A @ J ) )
@ ^ [J: nat] : ( old_In1 @ A @ product_unit @ ( nth_item @ A @ J ) )
@ ( nat_sum_decode @ I3 ) )
@ ^ [I3: nat] :
( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
@ ^ [J: nat] : ( old_Leaf @ A @ product_unit @ ( from_nat @ A @ J ) )
@ ^ [J: 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 @ J ) )
@ ( nat_sum_decode @ I3 ) )
@ ( nat_sum_decode @ N5 ) ) ) ) ).
% nth_item.simps(2)
thf(fact_2617_dprod__def,axiom,
! [B: $tType,A: $tType] :
( ( old_dprod @ A @ B )
= ( ^ [R5: 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 ) ) )] :
( 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 ) ) ) )
@ ^ [X4: set @ ( old_node @ A @ B ),X7: 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 ) ) ) )
@ ^ [Y5: set @ ( old_node @ A @ B ),Y6: set @ ( old_node @ A @ B )] : ( insert @ ( 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 @ X4 @ Y5 ) @ ( old_Scons @ A @ B @ X7 @ Y6 ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) ) ) ) )
@ S ) ) )
@ R5 ) ) ) ) ).
% dprod_def
thf(fact_2618_isomorphic__tuple_Oiso__tuple__access__update__snd__fst,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,Isom2: tuple_isomorphism @ A @ B @ C,F: C > D,H: E > B > B,G: E] :
( ( isomorphic_tuple @ A @ B @ C @ Isom2 )
=> ( ( comp @ A @ D @ A @ ( comp @ C @ D @ A @ F @ ( iso_tuple_snd @ A @ B @ C @ Isom2 ) ) @ ( comp @ ( B > B ) @ ( A > A ) @ E @ ( iso_tuple_fst_update @ A @ B @ C @ Isom2 ) @ H @ G ) )
= ( comp @ D @ D @ A @ ( id @ D ) @ ( comp @ C @ D @ A @ F @ ( iso_tuple_snd @ A @ B @ C @ Isom2 ) ) ) ) ) ).
% isomorphic_tuple.iso_tuple_access_update_snd_fst
thf(fact_2619_isomorphic__tuple_Oiso__tuple__access__update__fst__snd,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,Isom2: tuple_isomorphism @ A @ B @ C,F: B > D,H: E > C > C,G: E] :
( ( isomorphic_tuple @ A @ B @ C @ Isom2 )
=> ( ( comp @ A @ D @ A @ ( comp @ B @ D @ A @ F @ ( iso_tuple_fst @ A @ B @ C @ Isom2 ) ) @ ( comp @ ( C > C ) @ ( A > A ) @ E @ ( iso_tuple_snd_update @ A @ B @ C @ Isom2 ) @ H @ G ) )
= ( comp @ D @ D @ A @ ( id @ D ) @ ( comp @ B @ D @ A @ F @ ( iso_tuple_fst @ A @ B @ C @ Isom2 ) ) ) ) ) ).
% isomorphic_tuple.iso_tuple_access_update_fst_snd
thf(fact_2620_isomorphic__tuple_Oiso__tuple__access__update__snd__snd,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,Isom2: tuple_isomorphism @ A @ B @ C,F: C > D,H: E > C > C,G: E,J5: D > D] :
( ( isomorphic_tuple @ A @ B @ C @ Isom2 )
=> ( ( ( comp @ C @ D @ C @ F @ ( H @ G ) )
= ( comp @ D @ D @ C @ J5 @ F ) )
=> ( ( comp @ A @ D @ A @ ( comp @ C @ D @ A @ F @ ( iso_tuple_snd @ A @ B @ C @ Isom2 ) ) @ ( comp @ ( C > C ) @ ( A > A ) @ E @ ( iso_tuple_snd_update @ A @ B @ C @ Isom2 ) @ H @ G ) )
= ( comp @ D @ D @ A @ J5 @ ( comp @ C @ D @ A @ F @ ( iso_tuple_snd @ A @ B @ C @ Isom2 ) ) ) ) ) ) ).
% isomorphic_tuple.iso_tuple_access_update_snd_snd
thf(fact_2621_isomorphic__tuple_Oiso__tuple__access__update__fst__fst,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,Isom2: tuple_isomorphism @ A @ B @ C,F: B > D,H: E > B > B,G: E,J5: D > D] :
( ( isomorphic_tuple @ A @ B @ C @ Isom2 )
=> ( ( ( comp @ B @ D @ B @ F @ ( H @ G ) )
= ( comp @ D @ D @ B @ J5 @ F ) )
=> ( ( comp @ A @ D @ A @ ( comp @ B @ D @ A @ F @ ( iso_tuple_fst @ A @ B @ C @ Isom2 ) ) @ ( comp @ ( B > B ) @ ( A > A ) @ E @ ( iso_tuple_fst_update @ A @ B @ C @ Isom2 ) @ H @ G ) )
= ( comp @ D @ D @ A @ J5 @ ( comp @ B @ D @ A @ F @ ( iso_tuple_fst @ A @ B @ C @ Isom2 ) ) ) ) ) ) ).
% isomorphic_tuple.iso_tuple_access_update_fst_fst
thf(fact_2622_isomorphic__tuple_Oiso__tuple__update__swap__snd__fst,axiom,
! [E: $tType,B: $tType,C: $tType,A: $tType,D: $tType,Isom2: tuple_isomorphism @ A @ B @ C,H: D > B > B,F: D,J5: E > C > C,G: E] :
( ( isomorphic_tuple @ A @ B @ C @ Isom2 )
=> ( ( comp @ A @ A @ A @ ( comp @ ( B > B ) @ ( A > A ) @ D @ ( iso_tuple_fst_update @ A @ B @ C @ Isom2 ) @ H @ F ) @ ( comp @ ( C > C ) @ ( A > A ) @ E @ ( iso_tuple_snd_update @ A @ B @ C @ Isom2 ) @ J5 @ G ) )
= ( comp @ A @ A @ A @ ( comp @ ( C > C ) @ ( A > A ) @ E @ ( iso_tuple_snd_update @ A @ B @ C @ Isom2 ) @ J5 @ G ) @ ( comp @ ( B > B ) @ ( A > A ) @ D @ ( iso_tuple_fst_update @ A @ B @ C @ Isom2 ) @ H @ F ) ) ) ) ).
% isomorphic_tuple.iso_tuple_update_swap_snd_fst
thf(fact_2623_isomorphic__tuple_Oiso__tuple__update__swap__fst__fst,axiom,
! [E: $tType,B: $tType,C: $tType,A: $tType,D: $tType,Isom2: tuple_isomorphism @ A @ B @ C,H: D > B > B,F: D,J5: E > B > B,G: E] :
( ( isomorphic_tuple @ A @ B @ C @ Isom2 )
=> ( ( ( comp @ B @ B @ B @ ( H @ F ) @ ( J5 @ G ) )
= ( comp @ B @ B @ B @ ( J5 @ G ) @ ( H @ F ) ) )
=> ( ( comp @ A @ A @ A @ ( comp @ ( B > B ) @ ( A > A ) @ D @ ( iso_tuple_fst_update @ A @ B @ C @ Isom2 ) @ H @ F ) @ ( comp @ ( B > B ) @ ( A > A ) @ E @ ( iso_tuple_fst_update @ A @ B @ C @ Isom2 ) @ J5 @ G ) )
= ( comp @ A @ A @ A @ ( comp @ ( B > B ) @ ( A > A ) @ E @ ( iso_tuple_fst_update @ A @ B @ C @ Isom2 ) @ J5 @ G ) @ ( comp @ ( B > B ) @ ( A > A ) @ D @ ( iso_tuple_fst_update @ A @ B @ C @ Isom2 ) @ H @ F ) ) ) ) ) ).
% isomorphic_tuple.iso_tuple_update_swap_fst_fst
thf(fact_2624_isomorphic__tuple_Oiso__tuple__update__swap__snd__snd,axiom,
! [E: $tType,C: $tType,B: $tType,A: $tType,D: $tType,Isom2: tuple_isomorphism @ A @ B @ C,H: D > C > C,F: D,J5: E > C > C,G: E] :
( ( isomorphic_tuple @ A @ B @ C @ Isom2 )
=> ( ( ( comp @ C @ C @ C @ ( H @ F ) @ ( J5 @ G ) )
= ( comp @ C @ C @ C @ ( J5 @ G ) @ ( H @ F ) ) )
=> ( ( comp @ A @ A @ A @ ( comp @ ( C > C ) @ ( A > A ) @ D @ ( iso_tuple_snd_update @ A @ B @ C @ Isom2 ) @ H @ F ) @ ( comp @ ( C > C ) @ ( A > A ) @ E @ ( iso_tuple_snd_update @ A @ B @ C @ Isom2 ) @ J5 @ G ) )
= ( comp @ A @ A @ A @ ( comp @ ( C > C ) @ ( A > A ) @ E @ ( iso_tuple_snd_update @ A @ B @ C @ Isom2 ) @ J5 @ G ) @ ( comp @ ( C > C ) @ ( A > A ) @ D @ ( iso_tuple_snd_update @ A @ B @ C @ Isom2 ) @ H @ F ) ) ) ) ) ).
% isomorphic_tuple.iso_tuple_update_swap_snd_snd
thf(fact_2625_isomorphic__tuple_Oiso__tuple__update__compose__fst__fst,axiom,
! [B: $tType,C: $tType,A: $tType,E: $tType,D: $tType,F2: $tType,Isom2: tuple_isomorphism @ A @ B @ C,H: ( D > E ) > B > B,F: D > E,J5: ( F2 > D ) > B > B,G: F2 > D,K: ( F2 > E ) > B > B] :
( ( isomorphic_tuple @ A @ B @ C @ Isom2 )
=> ( ( ( comp @ B @ B @ B @ ( H @ F ) @ ( J5 @ G ) )
= ( K @ ( comp @ D @ E @ F2 @ F @ G ) ) )
=> ( ( comp @ A @ A @ A @ ( comp @ ( B > B ) @ ( A > A ) @ ( D > E ) @ ( iso_tuple_fst_update @ A @ B @ C @ Isom2 ) @ H @ F ) @ ( comp @ ( B > B ) @ ( A > A ) @ ( F2 > D ) @ ( iso_tuple_fst_update @ A @ B @ C @ Isom2 ) @ J5 @ G ) )
= ( comp @ ( B > B ) @ ( A > A ) @ ( F2 > E ) @ ( iso_tuple_fst_update @ A @ B @ C @ Isom2 ) @ K @ ( comp @ D @ E @ F2 @ F @ G ) ) ) ) ) ).
% isomorphic_tuple.iso_tuple_update_compose_fst_fst
thf(fact_2626_isomorphic__tuple_Oiso__tuple__update__compose__snd__snd,axiom,
! [C: $tType,B: $tType,A: $tType,E: $tType,D: $tType,F2: $tType,Isom2: tuple_isomorphism @ A @ B @ C,H: ( D > E ) > C > C,F: D > E,J5: ( F2 > D ) > C > C,G: F2 > D,K: ( F2 > E ) > C > C] :
( ( isomorphic_tuple @ A @ B @ C @ Isom2 )
=> ( ( ( comp @ C @ C @ C @ ( H @ F ) @ ( J5 @ G ) )
= ( K @ ( comp @ D @ E @ F2 @ F @ G ) ) )
=> ( ( comp @ A @ A @ A @ ( comp @ ( C > C ) @ ( A > A ) @ ( D > E ) @ ( iso_tuple_snd_update @ A @ B @ C @ Isom2 ) @ H @ F ) @ ( comp @ ( C > C ) @ ( A > A ) @ ( F2 > D ) @ ( iso_tuple_snd_update @ A @ B @ C @ Isom2 ) @ J5 @ G ) )
= ( comp @ ( C > C ) @ ( A > A ) @ ( F2 > E ) @ ( iso_tuple_snd_update @ A @ B @ C @ Isom2 ) @ K @ ( comp @ D @ E @ F2 @ F @ G ) ) ) ) ) ).
% isomorphic_tuple.iso_tuple_update_compose_snd_snd
thf(fact_2627_isomorphic__tuple_Oiso__tuple__update__swap__fst__snd,axiom,
! [E: $tType,C: $tType,B: $tType,A: $tType,D: $tType,Isom2: tuple_isomorphism @ A @ B @ C,H: D > C > C,F: D,J5: E > B > B,G: E] :
( ( isomorphic_tuple @ A @ B @ C @ Isom2 )
=> ( ( comp @ A @ A @ A @ ( comp @ ( C > C ) @ ( A > A ) @ D @ ( iso_tuple_snd_update @ A @ B @ C @ Isom2 ) @ H @ F ) @ ( comp @ ( B > B ) @ ( A > A ) @ E @ ( iso_tuple_fst_update @ A @ B @ C @ Isom2 ) @ J5 @ G ) )
= ( comp @ A @ A @ A @ ( comp @ ( B > B ) @ ( A > A ) @ E @ ( iso_tuple_fst_update @ A @ B @ C @ Isom2 ) @ J5 @ G ) @ ( comp @ ( C > C ) @ ( A > A ) @ D @ ( iso_tuple_snd_update @ A @ B @ C @ Isom2 ) @ H @ F ) ) ) ) ).
% isomorphic_tuple.iso_tuple_update_swap_fst_snd
thf(fact_2628_isomorphic__tuple_Oiso__tuple__snd__update__accessor__cong__assist,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,Isom2: tuple_isomorphism @ A @ B @ C,F: ( D > D ) > C > C,G: C > D] :
( ( isomorphic_tuple @ A @ B @ C @ Isom2 )
=> ( ( iso_tu6617205062836021118assist @ D @ C @ F @ G )
=> ( iso_tu6617205062836021118assist @ D @ A @ ( comp @ ( C > C ) @ ( A > A ) @ ( D > D ) @ ( iso_tuple_snd_update @ A @ B @ C @ Isom2 ) @ F ) @ ( comp @ C @ D @ A @ G @ ( iso_tuple_snd @ A @ B @ C @ Isom2 ) ) ) ) ) ).
% isomorphic_tuple.iso_tuple_snd_update_accessor_cong_assist
thf(fact_2629_isomorphic__tuple_Oiso__tuple__fst__update__accessor__cong__assist,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,Isom2: tuple_isomorphism @ A @ B @ C,F: ( D > D ) > B > B,G: B > D] :
( ( isomorphic_tuple @ A @ B @ C @ Isom2 )
=> ( ( iso_tu6617205062836021118assist @ D @ B @ F @ G )
=> ( iso_tu6617205062836021118assist @ D @ A @ ( comp @ ( B > B ) @ ( A > A ) @ ( D > D ) @ ( iso_tuple_fst_update @ A @ B @ C @ Isom2 ) @ F ) @ ( comp @ B @ D @ A @ G @ ( iso_tuple_fst @ A @ B @ C @ Isom2 ) ) ) ) ) ).
% isomorphic_tuple.iso_tuple_fst_update_accessor_cong_assist
thf(fact_2630_iso__tuple__cons__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( iso_tuple_cons @ A @ B @ C )
= ( ^ [Isom: tuple_isomorphism @ A @ B @ C] : ( product_curry @ B @ C @ A @ ( abst @ A @ B @ C @ Isom ) ) ) ) ).
% iso_tuple_cons_def
thf(fact_2631_isomorphic__tuple_Oiso__tuple__snd__update__accessor__eq__assist,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,Isom2: tuple_isomorphism @ A @ B @ C,F: ( D > D ) > C > C,G: C > D,B2: C,U2: D > D,B3: C,V: D,A3: B] :
( ( isomorphic_tuple @ A @ B @ C @ Isom2 )
=> ( ( iso_tu8710522896132340869assist @ D @ C @ F @ G @ B2 @ U2 @ B3 @ V )
=> ( iso_tu8710522896132340869assist @ D @ A @ ( comp @ ( C > C ) @ ( A > A ) @ ( D > D ) @ ( iso_tuple_snd_update @ A @ B @ C @ Isom2 ) @ F ) @ ( comp @ C @ D @ A @ G @ ( iso_tuple_snd @ A @ B @ C @ Isom2 ) ) @ ( iso_tuple_cons @ A @ B @ C @ Isom2 @ A3 @ B2 ) @ U2 @ ( iso_tuple_cons @ A @ B @ C @ Isom2 @ A3 @ B3 ) @ V ) ) ) ).
% isomorphic_tuple.iso_tuple_snd_update_accessor_eq_assist
thf(fact_2632_isomorphic__tuple_Oiso__tuple__fst__update__accessor__eq__assist,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,Isom2: tuple_isomorphism @ A @ B @ C,F: ( D > D ) > B > B,G: B > D,A3: B,U2: D > D,A4: B,V: D,B2: C] :
( ( isomorphic_tuple @ A @ B @ C @ Isom2 )
=> ( ( iso_tu8710522896132340869assist @ D @ B @ F @ G @ A3 @ U2 @ A4 @ V )
=> ( iso_tu8710522896132340869assist @ D @ A @ ( comp @ ( B > B ) @ ( A > A ) @ ( D > D ) @ ( iso_tuple_fst_update @ A @ B @ C @ Isom2 ) @ F ) @ ( comp @ B @ D @ A @ G @ ( iso_tuple_fst @ A @ B @ C @ Isom2 ) ) @ ( iso_tuple_cons @ A @ B @ C @ Isom2 @ A3 @ B2 ) @ U2 @ ( iso_tuple_cons @ A @ B @ C @ Isom2 @ A4 @ B2 ) @ V ) ) ) ).
% isomorphic_tuple.iso_tuple_fst_update_accessor_eq_assist
thf(fact_2633_Leaf__def,axiom,
! [B: $tType,A: $tType] :
( ( old_Leaf @ A @ B )
= ( comp @ ( sum_sum @ A @ nat ) @ ( set @ ( old_node @ A @ B ) ) @ A @ ( old_Atom @ A @ B ) @ ( sum_Inl @ A @ nat ) ) ) ).
% Leaf_def
thf(fact_2634_update__accessor__noop__compE,axiom,
! [A: $tType,B: $tType,Upd: ( A > A ) > B > B,Ac2: B > A,F: A > A,X2: B,G: A > A] :
( ( iso_tu6617205062836021118assist @ A @ B @ Upd @ Ac2 )
=> ( ( ( F @ ( Ac2 @ X2 ) )
= ( Ac2 @ X2 ) )
=> ( ( Upd @ ( comp @ A @ A @ A @ G @ F ) @ X2 )
= ( Upd @ G @ X2 ) ) ) ) ).
% update_accessor_noop_compE
thf(fact_2635_isomorphic__tuple_Oiso__tuple__surjective__proof__assist__step,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType,Isom2: tuple_isomorphism @ A @ B @ C,V: D,A3: B,F: D > A,B2: C] :
( ( isomorphic_tuple @ A @ B @ C @ Isom2 )
=> ( ( iso_tu562120763810524177assist @ D @ B @ V @ A3 @ ( comp @ A @ B @ D @ ( iso_tuple_fst @ A @ B @ C @ Isom2 ) @ F ) )
=> ( ( iso_tu562120763810524177assist @ D @ C @ V @ B2 @ ( comp @ A @ C @ D @ ( iso_tuple_snd @ A @ B @ C @ Isom2 ) @ F ) )
=> ( iso_tu562120763810524177assist @ D @ A @ V @ ( iso_tuple_cons @ A @ B @ C @ Isom2 @ A3 @ B2 ) @ F ) ) ) ) ).
% isomorphic_tuple.iso_tuple_surjective_proof_assist_step
thf(fact_2636_Numb__def,axiom,
! [B: $tType,A: $tType] :
( ( old_Numb @ A @ B )
= ( comp @ ( sum_sum @ A @ nat ) @ ( set @ ( old_node @ A @ B ) ) @ nat @ ( old_Atom @ A @ B ) @ ( sum_Inr @ nat @ A ) ) ) ).
% Numb_def
thf(fact_2637_ndepth__def,axiom,
! [B: $tType,A: $tType] :
( ( old_ndepth @ A @ B )
= ( ^ [N4: old_node @ A @ B] :
( product_case_prod @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat ) @ nat
@ ^ [F4: nat > ( sum_sum @ B @ nat ),X4: sum_sum @ A @ nat] :
( ord_Least @ nat
@ ^ [K3: nat] :
( ( F4 @ K3 )
= ( sum_Inr @ nat @ B @ ( zero_zero @ nat ) ) ) )
@ ( old_Rep_Node @ A @ B @ N4 ) ) ) ) ).
% ndepth_def
thf(fact_2638_uprod__def,axiom,
! [B: $tType,A: $tType] :
( ( old_uprod @ A @ B )
= ( ^ [A8: set @ ( set @ ( old_node @ A @ B ) ),B7: set @ ( set @ ( old_node @ A @ B ) )] :
( complete_Sup_Sup @ ( set @ ( set @ ( old_node @ A @ B ) ) )
@ ( image2 @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( set @ ( old_node @ A @ B ) ) )
@ ^ [X4: set @ ( old_node @ A @ B )] :
( complete_Sup_Sup @ ( set @ ( set @ ( old_node @ A @ B ) ) )
@ ( image2 @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( set @ ( old_node @ A @ B ) ) )
@ ^ [Y5: set @ ( old_node @ A @ B )] : ( insert @ ( set @ ( old_node @ A @ B ) ) @ ( old_Scons @ A @ B @ X4 @ Y5 ) @ ( bot_bot @ ( set @ ( set @ ( old_node @ A @ B ) ) ) ) )
@ B7 ) )
@ A8 ) ) ) ) ).
% uprod_def
thf(fact_2639_ext__surjective,axiom,
! [Z: $tType,X2: heap_ext @ Z] :
( X2
= ( heap_ext2 @ Z @ ( comp @ ( product_prod @ ( typerep > nat > ( list @ nat ) ) @ ( product_prod @ ( typerep > nat > nat ) @ nat ) ) @ ( typerep > nat > ( list @ nat ) ) @ ( heap_ext @ Z ) @ ( iso_tuple_fst @ ( product_prod @ ( typerep > nat > ( list @ nat ) ) @ ( product_prod @ ( typerep > nat > nat ) @ nat ) ) @ ( typerep > nat > ( list @ nat ) ) @ ( product_prod @ ( typerep > nat > nat ) @ nat ) @ ( tuple_iso_tuple @ ( typerep > nat > ( list @ nat ) ) @ ( product_prod @ ( typerep > nat > nat ) @ nat ) ) ) @ ( comp @ ( heap_ext @ Z ) @ ( product_prod @ ( typerep > nat > ( list @ nat ) ) @ ( product_prod @ ( typerep > nat > nat ) @ nat ) ) @ ( heap_ext @ Z ) @ ( iso_tuple_fst @ ( heap_ext @ Z ) @ ( product_prod @ ( typerep > nat > ( list @ nat ) ) @ ( product_prod @ ( typerep > nat > nat ) @ nat ) ) @ Z @ ( heap_h6279449762549114721le_Iso @ Z ) ) @ ( id @ ( heap_ext @ Z ) ) ) @ X2 ) @ ( comp @ ( product_prod @ ( typerep > nat > nat ) @ nat ) @ ( typerep > nat > nat ) @ ( heap_ext @ Z ) @ ( iso_tuple_fst @ ( product_prod @ ( typerep > nat > nat ) @ nat ) @ ( typerep > nat > nat ) @ nat @ ( tuple_iso_tuple @ ( typerep > nat > nat ) @ nat ) ) @ ( comp @ ( product_prod @ ( typerep > nat > ( list @ nat ) ) @ ( product_prod @ ( typerep > nat > nat ) @ nat ) ) @ ( product_prod @ ( typerep > nat > nat ) @ nat ) @ ( heap_ext @ Z ) @ ( iso_tuple_snd @ ( product_prod @ ( typerep > nat > ( list @ nat ) ) @ ( product_prod @ ( typerep > nat > nat ) @ nat ) ) @ ( typerep > nat > ( list @ nat ) ) @ ( product_prod @ ( typerep > nat > nat ) @ nat ) @ ( tuple_iso_tuple @ ( typerep > nat > ( list @ nat ) ) @ ( product_prod @ ( typerep > nat > nat ) @ nat ) ) ) @ ( comp @ ( heap_ext @ Z ) @ ( product_prod @ ( typerep > nat > ( list @ nat ) ) @ ( product_prod @ ( typerep > nat > nat ) @ nat ) ) @ ( heap_ext @ Z ) @ ( iso_tuple_fst @ ( heap_ext @ Z ) @ ( product_prod @ ( typerep > nat > ( list @ nat ) ) @ ( product_prod @ ( typerep > nat > nat ) @ nat ) ) @ Z @ ( heap_h6279449762549114721le_Iso @ Z ) ) @ ( id @ ( heap_ext @ Z ) ) ) ) @ X2 ) @ ( comp @ ( product_prod @ ( typerep > nat > nat ) @ nat ) @ nat @ ( heap_ext @ Z ) @ ( iso_tuple_snd @ ( product_prod @ ( typerep > nat > nat ) @ nat ) @ ( typerep > nat > nat ) @ nat @ ( tuple_iso_tuple @ ( typerep > nat > nat ) @ nat ) ) @ ( comp @ ( product_prod @ ( typerep > nat > ( list @ nat ) ) @ ( product_prod @ ( typerep > nat > nat ) @ nat ) ) @ ( product_prod @ ( typerep > nat > nat ) @ nat ) @ ( heap_ext @ Z ) @ ( iso_tuple_snd @ ( product_prod @ ( typerep > nat > ( list @ nat ) ) @ ( product_prod @ ( typerep > nat > nat ) @ nat ) ) @ ( typerep > nat > ( list @ nat ) ) @ ( product_prod @ ( typerep > nat > nat ) @ nat ) @ ( tuple_iso_tuple @ ( typerep > nat > ( list @ nat ) ) @ ( product_prod @ ( typerep > nat > nat ) @ nat ) ) ) @ ( comp @ ( heap_ext @ Z ) @ ( product_prod @ ( typerep > nat > ( list @ nat ) ) @ ( product_prod @ ( typerep > nat > nat ) @ nat ) ) @ ( heap_ext @ Z ) @ ( iso_tuple_fst @ ( heap_ext @ Z ) @ ( product_prod @ ( typerep > nat > ( list @ nat ) ) @ ( product_prod @ ( typerep > nat > nat ) @ nat ) ) @ Z @ ( heap_h6279449762549114721le_Iso @ Z ) ) @ ( id @ ( heap_ext @ Z ) ) ) ) @ X2 ) @ ( comp @ ( heap_ext @ Z ) @ Z @ ( heap_ext @ Z ) @ ( iso_tuple_snd @ ( heap_ext @ Z ) @ ( product_prod @ ( typerep > nat > ( list @ nat ) ) @ ( product_prod @ ( typerep > nat > nat ) @ nat ) ) @ Z @ ( heap_h6279449762549114721le_Iso @ Z ) ) @ ( id @ ( heap_ext @ Z ) ) @ X2 ) ) ) ).
% ext_surjective
thf(fact_2640_eval__SUP,axiom,
! [A: $tType,B: $tType,F: B > ( pred @ A ),A6: set @ B] :
( ( eval @ A @ ( complete_Sup_Sup @ ( pred @ A ) @ ( image2 @ B @ ( pred @ A ) @ F @ A6 ) ) )
= ( complete_Sup_Sup @ ( A > $o ) @ ( image2 @ B @ ( A > $o ) @ ( comp @ ( pred @ A ) @ ( A > $o ) @ B @ ( eval @ A ) @ F ) @ A6 ) ) ) ).
% eval_SUP
thf(fact_2641_eval__Inf,axiom,
! [A: $tType,A6: set @ ( pred @ A )] :
( ( eval @ A @ ( complete_Inf_Inf @ ( pred @ A ) @ A6 ) )
= ( complete_Inf_Inf @ ( A > $o ) @ ( image2 @ ( pred @ A ) @ ( A > $o ) @ ( eval @ A ) @ A6 ) ) ) ).
% eval_Inf
thf(fact_2642_eval__Sup,axiom,
! [A: $tType,A6: set @ ( pred @ A )] :
( ( eval @ A @ ( complete_Sup_Sup @ ( pred @ A ) @ A6 ) )
= ( complete_Sup_Sup @ ( A > $o ) @ ( image2 @ ( pred @ A ) @ ( A > $o ) @ ( eval @ A ) @ A6 ) ) ) ).
% eval_Sup
thf(fact_2643_eval__INF,axiom,
! [A: $tType,B: $tType,F: B > ( pred @ A ),A6: set @ B] :
( ( eval @ A @ ( complete_Inf_Inf @ ( pred @ A ) @ ( image2 @ B @ ( pred @ A ) @ F @ A6 ) ) )
= ( complete_Inf_Inf @ ( A > $o ) @ ( image2 @ B @ ( A > $o ) @ ( comp @ ( pred @ A ) @ ( A > $o ) @ B @ ( eval @ A ) @ F ) @ A6 ) ) ) ).
% eval_INF
thf(fact_2644_eval__map,axiom,
! [A: $tType,B: $tType,F: B > A,P2: pred @ B] :
( ( eval @ A @ ( map2 @ B @ A @ F @ P2 ) )
= ( complete_Sup_Sup @ ( A > $o )
@ ( image2 @ B @ ( A > $o )
@ ^ [X4: B] :
( ^ [Y2: A,Z2: A] : Y2 = Z2
@ ( F @ X4 ) )
@ ( collect @ B @ ( eval @ B @ P2 ) ) ) ) ) ).
% eval_map
thf(fact_2645_eval__bind,axiom,
! [A: $tType,B: $tType,P2: pred @ B,F: B > ( pred @ A )] :
( ( eval @ A @ ( bind3 @ B @ A @ P2 @ F ) )
= ( eval @ A @ ( complete_Sup_Sup @ ( pred @ A ) @ ( image2 @ B @ ( pred @ A ) @ F @ ( collect @ B @ ( eval @ B @ P2 ) ) ) ) ) ) ).
% eval_bind
thf(fact_2646_Predicate_Obind__def,axiom,
! [B: $tType,A: $tType] :
( ( bind3 @ A @ B )
= ( ^ [P4: pred @ A,F4: A > ( pred @ B )] : ( complete_Sup_Sup @ ( pred @ B ) @ ( image2 @ A @ ( pred @ B ) @ F4 @ ( collect @ A @ ( eval @ A @ P4 ) ) ) ) ) ) ).
% Predicate.bind_def
thf(fact_2647_Sup__pred__def,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( pred @ A ) )
= ( ^ [A8: set @ ( pred @ A )] : ( pred2 @ A @ ( complete_Sup_Sup @ ( A > $o ) @ ( image2 @ ( pred @ A ) @ ( A > $o ) @ ( eval @ A ) @ A8 ) ) ) ) ) ).
% Sup_pred_def
thf(fact_2648_Inf__pred__def,axiom,
! [A: $tType] :
( ( complete_Inf_Inf @ ( pred @ A ) )
= ( ^ [A8: set @ ( pred @ A )] : ( pred2 @ A @ ( complete_Inf_Inf @ ( A > $o ) @ ( image2 @ ( pred @ A ) @ ( A > $o ) @ ( eval @ A ) @ A8 ) ) ) ) ) ).
% Inf_pred_def
thf(fact_2649_Sup__bind,axiom,
! [A: $tType,B: $tType,A6: set @ ( pred @ B ),F: B > ( pred @ A )] :
( ( bind3 @ B @ A @ ( complete_Sup_Sup @ ( pred @ B ) @ A6 ) @ F )
= ( complete_Sup_Sup @ ( pred @ A )
@ ( image2 @ ( pred @ B ) @ ( pred @ A )
@ ^ [X4: pred @ B] : ( bind3 @ B @ A @ X4 @ F )
@ A6 ) ) ) ).
% Sup_bind
thf(fact_2650_Predicate_Omap_Ocomp,axiom,
! [C: $tType,B: $tType,A: $tType,F: B > C,G: A > B] :
( ( comp @ ( pred @ B ) @ ( pred @ C ) @ ( pred @ A ) @ ( map2 @ B @ C @ F ) @ ( map2 @ A @ B @ G ) )
= ( map2 @ A @ C @ ( comp @ B @ C @ A @ F @ G ) ) ) ).
% Predicate.map.comp
thf(fact_2651_Predicate_Omap_Ocompositionality,axiom,
! [B: $tType,C: $tType,A: $tType,F: B > C,G: A > B,Pred: pred @ A] :
( ( map2 @ B @ C @ F @ ( map2 @ A @ B @ G @ Pred ) )
= ( map2 @ A @ C @ ( comp @ B @ C @ A @ F @ G ) @ Pred ) ) ).
% Predicate.map.compositionality
thf(fact_2652_Predicate_Omap__def,axiom,
! [B: $tType,A: $tType] :
( ( map2 @ A @ B )
= ( ^ [F4: A > B,P4: pred @ A] : ( bind3 @ A @ B @ P4 @ ( comp @ B @ ( pred @ B ) @ A @ ( single @ B ) @ F4 ) ) ) ) ).
% Predicate.map_def
thf(fact_2653_pred__of__set__def,axiom,
! [A: $tType] :
( ( pred_of_set @ A )
= ( comp @ ( A > $o ) @ ( pred @ A ) @ ( set @ A ) @ ( pred2 @ A )
@ ^ [A8: set @ A,X4: A] : ( member @ A @ X4 @ A8 ) ) ) ).
% pred_of_set_def
thf(fact_2654_set__of__pred__def,axiom,
! [A: $tType] :
( ( set_of_pred @ A )
= ( comp @ ( A > $o ) @ ( set @ A ) @ ( pred @ A ) @ ( collect @ A ) @ ( eval @ A ) ) ) ).
% set_of_pred_def
thf(fact_2655_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 ) ),F4: A > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ B ) @ ( product_prod @ code_natural @ code_natural ) ),S: 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 )
@ ( bind3 @ A @ B @ P4
@ ^ [A7: A] : ( product_fst @ ( pred @ B ) @ ( product_prod @ code_natural @ code_natural ) @ ( F4 @ A7 @ S12 ) ) ) )
@ ( split_seed @ S8 ) )
@ ( R6 @ S ) ) ) ) ).
% Random_Pred.bind_def
thf(fact_2656_pred__of__set__fold__sup,axiom,
! [A: $tType,A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( pred_of_set @ A @ A6 )
= ( finite_fold @ ( pred @ A ) @ ( pred @ A ) @ ( sup_sup @ ( pred @ A ) ) @ ( bot_bot @ ( pred @ A ) ) @ ( image2 @ A @ ( pred @ A ) @ ( single @ A ) @ A6 ) ) ) ) ).
% pred_of_set_fold_sup
thf(fact_2657_wfrec__rel_Ocases,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),F7: ( A > B ) > A > B,A1: A,A22: B] :
( ( wfrec_rel @ A @ B @ R3 @ F7 @ A1 @ A22 )
=> ~ ! [G3: A > B] :
( ( A22
= ( F7 @ G3 @ A1 ) )
=> ~ ! [Z7: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z7 @ A1 ) @ R3 )
=> ( wfrec_rel @ A @ B @ R3 @ F7 @ Z7 @ ( G3 @ Z7 ) ) ) ) ) ).
% wfrec_rel.cases
thf(fact_2658_wfrecI,axiom,
! [B: $tType,A: $tType,X2: A,R3: set @ ( product_prod @ A @ A ),F7: ( A > B ) > A > B,G: A > B] :
( ! [Z4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z4 @ X2 ) @ R3 )
=> ( wfrec_rel @ A @ B @ R3 @ F7 @ Z4 @ ( G @ Z4 ) ) )
=> ( wfrec_rel @ A @ B @ R3 @ F7 @ X2 @ ( F7 @ G @ X2 ) ) ) ).
% wfrecI
thf(fact_2659_wfrec__rel_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( wfrec_rel @ A @ B )
= ( ^ [R6: set @ ( product_prod @ A @ A ),F8: ( A > B ) > A > B,A12: A,A23: B] :
? [X4: A,G4: A > B] :
( ( A12 = X4 )
& ( A23
= ( F8 @ G4 @ X4 ) )
& ! [Y5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X4 ) @ R6 )
=> ( wfrec_rel @ A @ B @ R6 @ F8 @ Y5 @ ( G4 @ Y5 ) ) ) ) ) ) ).
% wfrec_rel.simps
thf(fact_2660_Random__Pred_Ounion__def,axiom,
! [A: $tType] :
( ( random_union @ A )
= ( ^ [R14: ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ),R25: ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ),S: 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 ) )
@ ^ [P13: 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 ) )
@ ^ [P25: pred @ A] : ( product_Pair @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) @ ( sup_sup @ ( pred @ A ) @ P13 @ P25 ) )
@ ( R25 @ S8 ) )
@ ( R14 @ S ) ) ) ) ).
% Random_Pred.union_def
thf(fact_2661_Random__Pred_ORandom__def,axiom,
! [A: $tType] :
( ( random_Random @ A )
= ( ^ [G4: ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( product_prod @ A @ ( product_unit > code_term ) ) @ ( product_prod @ code_natural @ code_natural ) )] : ( product_scomp @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ A @ ( product_unit > code_term ) ) @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ) @ G4 @ ( comp @ ( pred @ A ) @ ( ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ) ) @ ( product_prod @ A @ ( product_unit > code_term ) ) @ ( product_Pair @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ) @ ( comp @ A @ ( pred @ A ) @ ( product_prod @ A @ ( product_unit > code_term ) ) @ ( single @ A ) @ ( product_fst @ A @ ( product_unit > code_term ) ) ) ) ) ) ) ).
% Random_Pred.Random_def
thf(fact_2662_Random__Pred_Onot__randompred__def,axiom,
( random6974930770145893639ompred
= ( ^ [P4: ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ product_unit ) @ ( product_prod @ code_natural @ code_natural ) ),S: product_prod @ code_natural @ code_natural] :
( product_case_prod @ ( pred @ product_unit ) @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ ( pred @ product_unit ) @ ( product_prod @ code_natural @ code_natural ) )
@ ^ [P10: pred @ product_unit,S8: product_prod @ code_natural @ code_natural] : ( if @ ( product_prod @ ( pred @ product_unit ) @ ( product_prod @ code_natural @ code_natural ) ) @ ( eval @ product_unit @ P10 @ product_Unity ) @ ( product_Pair @ ( pred @ product_unit ) @ ( product_prod @ code_natural @ code_natural ) @ ( bot_bot @ ( pred @ product_unit ) ) @ S8 ) @ ( product_Pair @ ( pred @ product_unit ) @ ( product_prod @ code_natural @ code_natural ) @ ( single @ product_unit @ product_Unity ) @ S8 ) )
@ ( P4 @ S ) ) ) ) ).
% Random_Pred.not_randompred_def
thf(fact_2663_Random__Pred_Omap__def,axiom,
! [B: $tType,A: $tType] :
( ( random_map @ A @ B )
= ( ^ [F4: A > B,P4: ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) )] : ( random_bind @ A @ B @ P4 @ ( comp @ B @ ( ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ B ) @ ( product_prod @ code_natural @ code_natural ) ) ) @ A @ ( random_single @ B ) @ F4 ) ) ) ) ).
% Random_Pred.map_def
thf(fact_2664_folding__on__def,axiom,
! [B: $tType,A: $tType] :
( ( finite_folding_on @ A @ B )
= ( ^ [S4: set @ A,F4: A > B > B] :
! [X4: A,Y5: A] :
( ( member @ A @ X4 @ S4 )
=> ( ( member @ A @ Y5 @ S4 )
=> ( ( comp @ B @ B @ B @ ( F4 @ Y5 ) @ ( F4 @ X4 ) )
= ( comp @ B @ B @ B @ ( F4 @ X4 ) @ ( F4 @ Y5 ) ) ) ) ) ) ) ).
% folding_on_def
thf(fact_2665_folding__on_Ocomp__fun__commute__on,axiom,
! [B: $tType,A: $tType,S5: set @ A,F: A > B > B,X2: A,Y3: A] :
( ( finite_folding_on @ A @ B @ S5 @ F )
=> ( ( member @ A @ X2 @ S5 )
=> ( ( member @ A @ Y3 @ S5 )
=> ( ( comp @ B @ B @ B @ ( F @ Y3 ) @ ( F @ X2 ) )
= ( comp @ B @ B @ B @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ) ) ) ).
% folding_on.comp_fun_commute_on
thf(fact_2666_folding__on_Ointro,axiom,
! [B: $tType,A: $tType,S5: set @ A,F: A > B > B] :
( ! [X: A,Y: A] :
( ( member @ A @ X @ S5 )
=> ( ( member @ A @ Y @ S5 )
=> ( ( comp @ B @ B @ B @ ( F @ Y ) @ ( F @ X ) )
= ( comp @ B @ B @ B @ ( F @ X ) @ ( F @ Y ) ) ) ) )
=> ( finite_folding_on @ A @ B @ S5 @ F ) ) ).
% folding_on.intro
thf(fact_2667_asym_Ocases,axiom,
! [A: $tType,A3: set @ ( product_prod @ A @ A )] :
( ( asym @ A @ A3 )
=> ! [A20: A,B14: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A20 @ B14 ) @ A3 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B14 @ A20 ) @ A3 ) ) ) ).
% asym.cases
thf(fact_2668_asym_Osimps,axiom,
! [A: $tType] :
( ( asym @ A )
= ( ^ [A7: set @ ( product_prod @ A @ A )] :
? [R6: set @ ( product_prod @ A @ A )] :
( ( A7 = R6 )
& ! [X4: A,Y5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R6 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X4 ) @ R6 ) ) ) ) ) ).
% asym.simps
thf(fact_2669_asym_Ointros,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ! [A5: A,B4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B4 ) @ R3 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ A5 ) @ R3 ) )
=> ( asym @ A @ R3 ) ) ).
% asym.intros
thf(fact_2670_asymD,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),X2: A,Y3: A] :
( ( asym @ A @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y3 ) @ R3 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X2 ) @ R3 ) ) ) ).
% asymD
thf(fact_2671_asym__iff,axiom,
! [A: $tType] :
( ( asym @ A )
= ( ^ [R6: set @ ( product_prod @ A @ A )] :
! [X4: A,Y5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y5 ) @ R6 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X4 ) @ R6 ) ) ) ) ).
% asym_iff
thf(fact_2672_asymp__asym__eq,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( asymp @ A
@ ^ [A7: A,B5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B5 ) @ R3 ) )
= ( asym @ A @ R3 ) ) ).
% asymp_asym_eq
thf(fact_2673_semilattice__set_Ohom__commute,axiom,
! [A: $tType,F: A > A > A,H: A > A,N: set @ A] :
( ( lattic149705377957585745ce_set @ A @ F )
=> ( ! [X: A,Y: A] :
( ( H @ ( F @ X @ Y ) )
= ( F @ ( H @ X ) @ ( H @ Y ) ) )
=> ( ( finite_finite2 @ A @ N )
=> ( ( N
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H @ ( lattic1715443433743089157tice_F @ A @ F @ N ) )
= ( lattic1715443433743089157tice_F @ A @ F @ ( image2 @ A @ A @ H @ N ) ) ) ) ) ) ) ).
% semilattice_set.hom_commute
thf(fact_2674_ref_Orel__compp__Grp,axiom,
! [B: $tType,A: $tType] :
( ( rel_ref @ A @ B )
= ( ^ [R6: A > B > $o] :
( relcompp @ ( ref @ A ) @ ( ref @ ( product_prod @ A @ B ) ) @ ( ref @ B )
@ ( conversep @ ( ref @ ( product_prod @ A @ B ) ) @ ( ref @ A )
@ ( bNF_Grp @ ( ref @ ( product_prod @ A @ B ) ) @ ( ref @ A )
@ ( collect @ ( ref @ ( product_prod @ A @ B ) )
@ ^ [X4: ref @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set_ref @ ( product_prod @ A @ B ) @ X4 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) ) )
@ ( map_ref @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) ) ) )
@ ( bNF_Grp @ ( ref @ ( product_prod @ A @ B ) ) @ ( ref @ B )
@ ( collect @ ( ref @ ( product_prod @ A @ B ) )
@ ^ [X4: ref @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set_ref @ ( product_prod @ A @ B ) @ X4 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) ) )
@ ( map_ref @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) ) ) ) ) ) ).
% ref.rel_compp_Grp
thf(fact_2675_array_Orel__compp__Grp,axiom,
! [B: $tType,A: $tType] :
( ( rel_array @ A @ B )
= ( ^ [R6: A > B > $o] :
( relcompp @ ( array @ A ) @ ( array @ ( product_prod @ A @ B ) ) @ ( array @ B )
@ ( conversep @ ( array @ ( product_prod @ A @ B ) ) @ ( array @ A )
@ ( bNF_Grp @ ( array @ ( product_prod @ A @ B ) ) @ ( array @ A )
@ ( collect @ ( array @ ( product_prod @ A @ B ) )
@ ^ [X4: array @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set_array @ ( product_prod @ A @ B ) @ X4 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) ) )
@ ( map_array @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) ) ) )
@ ( bNF_Grp @ ( array @ ( product_prod @ A @ B ) ) @ ( array @ B )
@ ( collect @ ( array @ ( product_prod @ A @ B ) )
@ ^ [X4: array @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set_array @ ( product_prod @ A @ B ) @ X4 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) ) )
@ ( map_array @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) ) ) ) ) ) ).
% array.rel_compp_Grp
thf(fact_2676_array_Oin__rel,axiom,
! [B: $tType,A: $tType] :
( ( rel_array @ A @ B )
= ( ^ [R6: A > B > $o,A7: array @ A,B5: array @ B] :
? [Z5: array @ ( product_prod @ A @ B )] :
( ( member @ ( array @ ( product_prod @ A @ B ) ) @ Z5
@ ( collect @ ( array @ ( product_prod @ A @ B ) )
@ ^ [X4: array @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set_array @ ( product_prod @ A @ B ) @ X4 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) ) ) )
& ( ( map_array @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Z5 )
= A7 )
& ( ( map_array @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Z5 )
= B5 ) ) ) ) ).
% array.in_rel
thf(fact_2677_ref_Oin__rel,axiom,
! [B: $tType,A: $tType] :
( ( rel_ref @ A @ B )
= ( ^ [R6: A > B > $o,A7: ref @ A,B5: ref @ B] :
? [Z5: ref @ ( product_prod @ A @ B )] :
( ( member @ ( ref @ ( product_prod @ A @ B ) ) @ Z5
@ ( collect @ ( ref @ ( product_prod @ A @ B ) )
@ ^ [X4: ref @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set_ref @ ( product_prod @ A @ B ) @ X4 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) ) ) )
& ( ( map_ref @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Z5 )
= A7 )
& ( ( map_ref @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Z5 )
= B5 ) ) ) ) ).
% ref.in_rel
thf(fact_2678_ref_Oset__map,axiom,
! [B: $tType,A: $tType,F: A > B,V: ref @ A] :
( ( set_ref @ B @ ( map_ref @ A @ B @ F @ V ) )
= ( image2 @ A @ B @ F @ ( set_ref @ A @ V ) ) ) ).
% ref.set_map
thf(fact_2679_array_Oset__map,axiom,
! [B: $tType,A: $tType,F: A > B,V: array @ A] :
( ( set_array @ B @ ( map_array @ A @ B @ F @ V ) )
= ( image2 @ A @ B @ F @ ( set_array @ A @ V ) ) ) ).
% array.set_map
thf(fact_2680_array_Orel__compp,axiom,
! [A: $tType,C: $tType,B: $tType,R3: A > B > $o,S5: B > C > $o] :
( ( rel_array @ A @ C @ ( relcompp @ A @ B @ C @ R3 @ S5 ) )
= ( relcompp @ ( array @ A ) @ ( array @ B ) @ ( array @ C ) @ ( rel_array @ A @ B @ R3 ) @ ( rel_array @ B @ C @ S5 ) ) ) ).
% array.rel_compp
thf(fact_2681_ref_Orel__compp,axiom,
! [A: $tType,C: $tType,B: $tType,R3: A > B > $o,S5: B > C > $o] :
( ( rel_ref @ A @ C @ ( relcompp @ A @ B @ C @ R3 @ S5 ) )
= ( relcompp @ ( ref @ A ) @ ( ref @ B ) @ ( ref @ C ) @ ( rel_ref @ A @ B @ R3 ) @ ( rel_ref @ B @ C @ S5 ) ) ) ).
% ref.rel_compp
thf(fact_2682_ref_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,G: B > C,F: A > B,V: ref @ A] :
( ( map_ref @ B @ C @ G @ ( map_ref @ A @ B @ F @ V ) )
= ( map_ref @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ V ) ) ).
% ref.map_comp
thf(fact_2683_array_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,G: B > C,F: A > B,V: array @ A] :
( ( map_array @ B @ C @ G @ ( map_array @ A @ B @ F @ V ) )
= ( map_array @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ V ) ) ).
% array.map_comp
thf(fact_2684_ref_Osize__gen__o__map,axiom,
! [B: $tType,A: $tType,F: B > nat,G: A > B] :
( ( comp @ ( ref @ B ) @ nat @ ( ref @ A ) @ ( size_ref @ B @ F ) @ ( map_ref @ A @ B @ G ) )
= ( size_ref @ A @ ( comp @ B @ nat @ A @ F @ G ) ) ) ).
% ref.size_gen_o_map
thf(fact_2685_array_Osize__gen__o__map,axiom,
! [B: $tType,A: $tType,F: B > nat,G: A > B] :
( ( comp @ ( array @ B ) @ nat @ ( array @ A ) @ ( size_array @ B @ F ) @ ( map_array @ A @ B @ G ) )
= ( size_array @ A @ ( comp @ B @ nat @ A @ F @ G ) ) ) ).
% array.size_gen_o_map
thf(fact_2686_ref_Orec__o__map,axiom,
! [B: $tType,A: $tType,C: $tType,G: nat > C,F: A > B] :
( ( comp @ ( ref @ B ) @ C @ ( ref @ A ) @ ( rec_ref @ C @ B @ G ) @ ( map_ref @ A @ B @ F ) )
= ( rec_ref @ C @ A @ G ) ) ).
% ref.rec_o_map
thf(fact_2687_array_Orec__o__map,axiom,
! [B: $tType,A: $tType,C: $tType,G: nat > C,F: A > B] :
( ( comp @ ( array @ B ) @ C @ ( array @ A ) @ ( rec_array @ C @ B @ G ) @ ( map_array @ A @ B @ F ) )
= ( rec_array @ C @ A @ G ) ) ).
% array.rec_o_map
thf(fact_2688_ref_Opred__map,axiom,
! [B: $tType,A: $tType,Q2: B > $o,F: A > B,X2: ref @ A] :
( ( pred_ref @ B @ Q2 @ ( map_ref @ A @ B @ F @ X2 ) )
= ( pred_ref @ A @ ( comp @ B @ $o @ A @ Q2 @ F ) @ X2 ) ) ).
% ref.pred_map
thf(fact_2689_array_Opred__map,axiom,
! [B: $tType,A: $tType,Q2: B > $o,F: A > B,X2: array @ A] :
( ( pred_array @ B @ Q2 @ ( map_array @ A @ B @ F @ X2 ) )
= ( pred_array @ A @ ( comp @ B @ $o @ A @ Q2 @ F ) @ X2 ) ) ).
% array.pred_map
thf(fact_2690_isl__map__sum,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,F: C > A,G: D > B,S2: sum_sum @ C @ D] :
( ( sum_isl @ A @ B @ ( sum_map_sum @ C @ A @ D @ B @ F @ G @ S2 ) )
= ( sum_isl @ C @ D @ S2 ) ) ).
% isl_map_sum
thf(fact_2691_isl__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,A6: A > C > $o,B6: B > D > $o] :
( bNF_rel_fun @ ( sum_sum @ A @ B ) @ ( sum_sum @ C @ D ) @ $o @ $o @ ( bNF_rel_sum @ A @ C @ B @ D @ A6 @ B6 )
@ ^ [Y2: $o,Z2: $o] : Y2 = Z2
@ ( sum_isl @ A @ B )
@ ( sum_isl @ C @ D ) ) ).
% isl_transfer
thf(fact_2692_rat__floor__code,axiom,
( ( archim6421214686448440834_floor @ rat )
= ( ^ [P3: rat] : ( product_case_prod @ int @ int @ int @ ( divide_divide @ int ) @ ( quotient_of @ P3 ) ) ) ) ).
% rat_floor_code
thf(fact_2693_Divides_Oadjust__div__def,axiom,
( adjust_div
= ( product_case_prod @ int @ int @ int
@ ^ [Q6: int,R5: int] :
( plus_plus @ int @ Q6
@ ( zero_neq_one_of_bool @ int
@ ( R5
!= ( zero_zero @ int ) ) ) ) ) ) ).
% Divides.adjust_div_def
thf(fact_2694_hd__zip,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] :
( ( Xs
!= ( nil @ A ) )
=> ( ( Ys
!= ( nil @ B ) )
=> ( ( hd @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) )
= ( product_Pair @ A @ B @ ( hd @ A @ Xs ) @ ( hd @ B @ Ys ) ) ) ) ) ).
% hd_zip
thf(fact_2695_set__of__seq__def,axiom,
! [A: $tType] :
( ( set_of_seq @ A )
= ( comp @ ( pred @ A ) @ ( set @ A ) @ ( seq @ A ) @ ( set_of_pred @ A ) @ ( pred_of_seq @ A ) ) ) ).
% set_of_seq_def
thf(fact_2696_set__sum__sel_I1_J,axiom,
! [B: $tType,A: $tType,S2: sum_sum @ A @ B] :
( ( sum_isl @ A @ B @ S2 )
=> ( member @ A @ ( sum_projl @ A @ B @ S2 ) @ ( basic_setl @ A @ B @ S2 ) ) ) ).
% set_sum_sel(1)
thf(fact_2697_set__sum__sel_I2_J,axiom,
! [B: $tType,A: $tType,S2: sum_sum @ A @ B] :
( ~ ( sum_isl @ A @ B @ S2 )
=> ( member @ B @ ( sum_projr @ A @ B @ S2 ) @ ( basic_setr @ A @ B @ S2 ) ) ) ).
% set_sum_sel(2)
thf(fact_2698_map__sum__sel_I1_J,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,S2: sum_sum @ A @ B,F: A > C,G: B > D] :
( ( sum_isl @ A @ B @ S2 )
=> ( ( sum_projl @ C @ D @ ( sum_map_sum @ A @ C @ B @ D @ F @ G @ S2 ) )
= ( F @ ( sum_projl @ A @ B @ S2 ) ) ) ) ).
% map_sum_sel(1)
thf(fact_2699_map__sum__sel_I2_J,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,S2: sum_sum @ A @ B,F: A > C,G: B > D] :
( ~ ( sum_isl @ A @ B @ S2 )
=> ( ( sum_projr @ C @ D @ ( sum_map_sum @ A @ C @ B @ D @ F @ G @ S2 ) )
= ( G @ ( sum_projr @ A @ B @ S2 ) ) ) ) ).
% map_sum_sel(2)
thf(fact_2700_rel__sum__sel,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType] :
( ( bNF_rel_sum @ A @ B @ C @ D )
= ( ^ [R14: A > B > $o,R25: C > D > $o,A7: sum_sum @ A @ C,B5: sum_sum @ B @ D] :
( ( ( sum_isl @ A @ C @ A7 )
= ( sum_isl @ B @ D @ B5 ) )
& ( ( sum_isl @ A @ C @ A7 )
=> ( ( sum_isl @ B @ D @ B5 )
=> ( R14 @ ( sum_projl @ A @ C @ A7 ) @ ( sum_projl @ B @ D @ B5 ) ) ) )
& ( ~ ( sum_isl @ A @ C @ A7 )
=> ( ~ ( sum_isl @ B @ D @ B5 )
=> ( R25 @ ( sum_projr @ A @ C @ A7 ) @ ( sum_projr @ B @ D @ B5 ) ) ) ) ) ) ) ).
% rel_sum_sel
thf(fact_2701_take__bit__num__code,axiom,
( bit_take_bit_num
= ( ^ [N4: nat,M7: num] :
( product_case_prod @ nat @ num @ ( option @ num )
@ ^ [A7: nat,X4: num] :
( case_nat @ ( option @ num ) @ ( none @ num )
@ ^ [O2: nat] :
( case_num @ ( option @ num ) @ ( some @ num @ one )
@ ^ [P3: num] :
( case_option @ ( option @ num ) @ num @ ( none @ num )
@ ^ [Q6: num] : ( some @ num @ ( bit0 @ Q6 ) )
@ ( bit_take_bit_num @ O2 @ P3 ) )
@ ^ [P3: num] : ( some @ num @ ( case_option @ num @ num @ one @ bit1 @ ( bit_take_bit_num @ O2 @ P3 ) ) )
@ X4 )
@ A7 )
@ ( product_Pair @ nat @ num @ N4 @ M7 ) ) ) ) ).
% take_bit_num_code
thf(fact_2702_nths__drop,axiom,
! [A: $tType,N5: nat,Xs: list @ A,I4: set @ nat] :
( ( nths @ A @ ( drop @ A @ N5 @ Xs ) @ I4 )
= ( nths @ A @ Xs @ ( image2 @ nat @ nat @ ( plus_plus @ nat @ N5 ) @ I4 ) ) ) ).
% nths_drop
thf(fact_2703_folding__insort__key_Oinsort__key__commute,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S5: set @ B,F: B > A,X2: B,Y3: B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S5 @ F )
=> ( ( member @ B @ X2 @ S5 )
=> ( ( member @ B @ Y3 @ S5 )
=> ( ( comp @ ( list @ B ) @ ( list @ B ) @ ( list @ B ) @ ( insort_key @ A @ B @ Less_eq @ F @ Y3 ) @ ( insort_key @ A @ B @ Less_eq @ F @ X2 ) )
= ( comp @ ( list @ B ) @ ( list @ B ) @ ( list @ B ) @ ( insort_key @ A @ B @ Less_eq @ F @ X2 ) @ ( insort_key @ A @ B @ Less_eq @ F @ Y3 ) ) ) ) ) ) ).
% folding_insort_key.insort_key_commute
thf(fact_2704_list__ex__iff__not__all__inverval__nat,axiom,
! [P2: nat > $o,I: nat,J5: nat] :
( ( list_ex @ nat @ P2 @ ( upt @ I @ J5 ) )
= ( ~ ( all_interval_nat @ ( comp @ $o @ $o @ nat @ (~) @ P2 ) @ I @ J5 ) ) ) ).
% list_ex_iff_not_all_inverval_nat
thf(fact_2705_Inl__def,axiom,
! [B: $tType,A: $tType] :
( ( sum_Inl @ A @ B )
= ( comp @ ( A > B > $o > $o ) @ ( sum_sum @ A @ B ) @ A @ ( sum_Abs_sum @ A @ B ) @ ( sum_Inl_Rep @ A @ B ) ) ) ).
% Inl_def
thf(fact_2706_fundef__ex1__iff,axiom,
! [A: $tType,B: $tType,F: A > B,D4: A > B,G6: A > B > $o,X2: A,Y3: B] :
( ( F
= ( ^ [X4: A] : ( fun_THE_default @ B @ ( D4 @ X4 ) @ ( G6 @ X4 ) ) ) )
=> ( ? [X5: B] :
( ( G6 @ X2 @ X5 )
& ! [Y: B] :
( ( G6 @ X2 @ Y )
=> ( Y = X5 ) ) )
=> ( ( G6 @ X2 @ Y3 )
= ( ( F @ X2 )
= Y3 ) ) ) ) ).
% fundef_ex1_iff
thf(fact_2707_fundef__default__value,axiom,
! [B: $tType,A: $tType,F: A > B,D4: A > B,G6: A > B > $o,D3: A > $o,X2: A] :
( ( F
= ( ^ [X4: A] : ( fun_THE_default @ B @ ( D4 @ X4 ) @ ( G6 @ X4 ) ) ) )
=> ( ! [X: A,Y: B] :
( ( G6 @ X @ Y )
=> ( D3 @ X ) )
=> ( ~ ( D3 @ X2 )
=> ( ( F @ X2 )
= ( D4 @ X2 ) ) ) ) ) ).
% fundef_default_value
thf(fact_2708_THE__default1__equality,axiom,
! [A: $tType,P2: A > $o,A3: A,D4: A] :
( ? [X5: A] :
( ( P2 @ X5 )
& ! [Y: A] :
( ( P2 @ Y )
=> ( Y = X5 ) ) )
=> ( ( P2 @ A3 )
=> ( ( fun_THE_default @ A @ D4 @ P2 )
= A3 ) ) ) ).
% THE_default1_equality
thf(fact_2709_THE__default__none,axiom,
! [A: $tType,P2: A > $o,D4: A] :
( ~ ? [X: A] :
( ( P2 @ X )
& ! [Y7: A] :
( ( P2 @ Y7 )
=> ( Y7 = X ) ) )
=> ( ( fun_THE_default @ A @ D4 @ P2 )
= D4 ) ) ).
% THE_default_none
thf(fact_2710_THE__defaultI_H,axiom,
! [A: $tType,P2: A > $o,D4: A] :
( ? [X5: A] :
( ( P2 @ X5 )
& ! [Y: A] :
( ( P2 @ Y )
=> ( Y = X5 ) ) )
=> ( P2 @ ( fun_THE_default @ A @ D4 @ P2 ) ) ) ).
% THE_defaultI'
thf(fact_2711_fundef__ex1__uniqueness,axiom,
! [B: $tType,A: $tType,F: A > B,D4: A > B,G6: A > B > $o,X2: A,H: A > B] :
( ( F
= ( ^ [X4: A] : ( fun_THE_default @ B @ ( D4 @ X4 ) @ ( G6 @ X4 ) ) ) )
=> ( ? [X5: B] :
( ( G6 @ X2 @ X5 )
& ! [Y: B] :
( ( G6 @ X2 @ Y )
=> ( Y = X5 ) ) )
=> ( ( G6 @ X2 @ ( H @ X2 ) )
=> ( ( H @ X2 )
= ( F @ X2 ) ) ) ) ) ).
% fundef_ex1_uniqueness
thf(fact_2712_fundef__ex1__existence,axiom,
! [B: $tType,A: $tType,F: A > B,D4: A > B,G6: A > B > $o,X2: A] :
( ( F
= ( ^ [X4: A] : ( fun_THE_default @ B @ ( D4 @ X4 ) @ ( G6 @ X4 ) ) ) )
=> ( ? [X5: B] :
( ( G6 @ X2 @ X5 )
& ! [Y: B] :
( ( G6 @ X2 @ Y )
=> ( Y = X5 ) ) )
=> ( G6 @ X2 @ ( F @ X2 ) ) ) ) ).
% fundef_ex1_existence
thf(fact_2713_Inr__def,axiom,
! [A: $tType,B: $tType] :
( ( sum_Inr @ B @ A )
= ( comp @ ( A > B > $o > $o ) @ ( sum_sum @ A @ B ) @ B @ ( sum_Abs_sum @ A @ B ) @ ( sum_Inr_Rep @ B @ A ) ) ) ).
% Inr_def
thf(fact_2714_THE__default__def,axiom,
! [A: $tType] :
( ( fun_THE_default @ A )
= ( ^ [D5: A,P4: A > $o] :
( if @ A
@ ? [X4: A] :
( ( P4 @ X4 )
& ! [Y5: A] :
( ( P4 @ Y5 )
=> ( Y5 = X4 ) ) )
@ ( the @ A @ P4 )
@ D5 ) ) ) ).
% THE_default_def
% Subclasses (3)
thf(subcl_Heap_Oheap___HOL_Otype,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( type @ A ) ) ).
thf(subcl_Heap_Oheap___Typerep_Otyperep,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( typerep2 @ A ) ) ).
thf(subcl_Heap_Oheap___Countable_Ocountable,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( countable @ A ) ) ).
% Type constructors (282)
thf(tcon_Product__Type_Ounit___Finite__Set_Ofinite,axiom,
finite_finite @ product_unit ).
thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_1,axiom,
! [A21: $tType,A27: $tType] :
( ( ( finite_finite @ A21 )
& ( finite_finite @ A27 ) )
=> ( finite_finite @ ( product_prod @ A21 @ A27 ) ) ) ).
thf(tcon_Option_Ooption___Finite__Set_Ofinite_2,axiom,
! [A21: $tType] :
( ( finite_finite @ A21 )
=> ( finite_finite @ ( option @ A21 ) ) ) ).
thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_3,axiom,
! [A21: $tType,A27: $tType] :
( ( ( finite_finite @ A21 )
& ( finite_finite @ A27 ) )
=> ( finite_finite @ ( sum_sum @ A21 @ A27 ) ) ) ).
thf(tcon_String_Ochar___Finite__Set_Ofinite_4,axiom,
finite_finite @ char ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_5,axiom,
finite_finite @ $o ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_6,axiom,
! [A21: $tType] :
( ( finite_finite @ A21 )
=> ( finite_finite @ ( set @ A21 ) ) ) ).
thf(tcon_fun___Finite__Set_Ofinite_7,axiom,
! [A21: $tType,A27: $tType] :
( ( ( finite_finite @ A21 )
& ( finite_finite @ A27 ) )
=> ( finite_finite @ ( A21 > A27 ) ) ) ).
thf(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
! [A21: $tType,A27: $tType] :
( ( comple6319245703460814977attice @ A27 )
=> ( condit1219197933456340205attice @ ( A21 > A27 ) ) ) ).
thf(tcon_fun___Complete__Lattices_Ocomplete__distrib__lattice,axiom,
! [A21: $tType,A27: $tType] :
( ( comple592849572758109894attice @ A27 )
=> ( comple592849572758109894attice @ ( A21 > A27 ) ) ) ).
thf(tcon_fun___Complete__Lattices_Ocomplete__boolean__algebra,axiom,
! [A21: $tType,A27: $tType] :
( ( comple489889107523837845lgebra @ A27 )
=> ( comple489889107523837845lgebra @ ( A21 > A27 ) ) ) ).
thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
! [A21: $tType,A27: $tType] :
( ( comple6319245703460814977attice @ A27 )
=> ( comple6319245703460814977attice @ ( A21 > A27 ) ) ) ).
thf(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
! [A21: $tType,A27: $tType] :
( ( boolea8198339166811842893lgebra @ A27 )
=> ( boolea8198339166811842893lgebra @ ( A21 > A27 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A21: $tType,A27: $tType] :
( ( semilattice_sup @ A27 )
=> ( semilattice_sup @ ( A21 > A27 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__inf,axiom,
! [A21: $tType,A27: $tType] :
( ( semilattice_inf @ A27 )
=> ( semilattice_inf @ ( A21 > A27 ) ) ) ).
thf(tcon_fun___Complete__Lattices_OSup,axiom,
! [A21: $tType,A27: $tType] :
( ( complete_Sup @ A27 )
=> ( complete_Sup @ ( A21 > A27 ) ) ) ).
thf(tcon_fun___Complete__Lattices_OInf,axiom,
! [A21: $tType,A27: $tType] :
( ( complete_Inf @ A27 )
=> ( complete_Inf @ ( A21 > A27 ) ) ) ).
thf(tcon_fun___Countable_Ocountable,axiom,
! [A21: $tType,A27: $tType] :
( ( ( finite_finite @ A21 )
& ( countable @ A27 ) )
=> ( countable @ ( A21 > A27 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A21: $tType,A27: $tType] :
( ( preorder @ A27 )
=> ( preorder @ ( A21 > A27 ) ) ) ).
thf(tcon_fun___Lattices_Olattice,axiom,
! [A21: $tType,A27: $tType] :
( ( lattice @ A27 )
=> ( lattice @ ( A21 > A27 ) ) ) ).
thf(tcon_fun___Typerep_Otyperep,axiom,
! [A21: $tType,A27: $tType] :
( ( ( typerep2 @ A21 )
& ( typerep2 @ A27 ) )
=> ( typerep2 @ ( A21 > A27 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A21: $tType,A27: $tType] :
( ( order @ A27 )
=> ( order @ ( A21 > A27 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A21: $tType,A27: $tType] :
( ( ord @ A27 )
=> ( ord @ ( A21 > A27 ) ) ) ).
thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
condit6923001295902523014norder @ int ).
thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_8,axiom,
condit1219197933456340205attice @ int ).
thf(tcon_Int_Oint___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations,axiom,
bit_un5681908812861735899ations @ int ).
thf(tcon_Int_Oint___Divides_Ounique__euclidean__semiring__numeral,axiom,
unique1627219031080169319umeral @ int ).
thf(tcon_Int_Oint___Groups_Olinordered__ab__semigroup__add,axiom,
linord4140545234300271783up_add @ 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_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___Lattices_Osemilattice__sup_9,axiom,
semilattice_sup @ int ).
thf(tcon_Int_Oint___Lattices_Osemilattice__inf_10,axiom,
semilattice_inf @ int ).
thf(tcon_Int_Oint___Groups_Ocomm__monoid__mult,axiom,
comm_monoid_mult @ int ).
thf(tcon_Int_Oint___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add @ 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___Complete__Lattices_OSup_11,axiom,
complete_Sup @ int ).
thf(tcon_Int_Oint___Complete__Lattices_OInf_12,axiom,
complete_Inf @ 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_13,axiom,
countable @ int ).
thf(tcon_Int_Oint___Orderings_Opreorder_14,axiom,
preorder @ int ).
thf(tcon_Int_Oint___Orderings_Olinorder,axiom,
linorder @ 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___Lattices_Olattice_15,axiom,
lattice @ 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___Typerep_Otyperep_16,axiom,
typerep2 @ int ).
thf(tcon_Int_Oint___Orderings_Oorder_17,axiom,
order @ int ).
thf(tcon_Int_Oint___Orderings_Oord_18,axiom,
ord @ int ).
thf(tcon_Int_Oint___Rings_Oring__1,axiom,
ring_1 @ int ).
thf(tcon_Int_Oint___Heap_Oheap,axiom,
heap @ int ).
thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_19,axiom,
condit6923001295902523014norder @ nat ).
thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_20,axiom,
condit1219197933456340205attice @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_21,axiom,
bit_un5681908812861735899ations @ nat ).
thf(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_22,axiom,
unique1627219031080169319umeral @ nat ).
thf(tcon_Nat_Onat___Groups_Olinordered__ab__semigroup__add_23,axiom,
linord4140545234300271783up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_24,axiom,
ordere6911136660526730532id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_25,axiom,
cancel_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semidom_26,axiom,
linordered_semidom @ nat ).
thf(tcon_Nat_Onat___Lattices_Osemilattice__sup_27,axiom,
semilattice_sup @ nat ).
thf(tcon_Nat_Onat___Lattices_Osemilattice__inf_28,axiom,
semilattice_inf @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_29,axiom,
comm_monoid_mult @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add_30,axiom,
comm_monoid_add @ nat ).
thf(tcon_Nat_Onat___Complete__Lattices_OSup_31,axiom,
complete_Sup @ nat ).
thf(tcon_Nat_Onat___Complete__Lattices_OInf_32,axiom,
complete_Inf @ nat ).
thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
wellorder @ nat ).
thf(tcon_Nat_Onat___Nat_Osemiring__char__0_33,axiom,
semiring_char_0 @ nat ).
thf(tcon_Nat_Onat___Countable_Ocountable_34,axiom,
countable @ nat ).
thf(tcon_Nat_Onat___Orderings_Opreorder_35,axiom,
preorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Olinorder_36,axiom,
linorder @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1_37,axiom,
semiring_1 @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__0_38,axiom,
semiring_0 @ nat ).
thf(tcon_Nat_Onat___Lattices_Olattice_39,axiom,
lattice @ nat ).
thf(tcon_Nat_Onat___GCD_Osemiring__gcd_40,axiom,
semiring_gcd @ nat ).
thf(tcon_Nat_Onat___GCD_Osemiring__Gcd_41,axiom,
semiring_Gcd @ nat ).
thf(tcon_Nat_Onat___Typerep_Otyperep_42,axiom,
typerep2 @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder_43,axiom,
order @ nat ).
thf(tcon_Nat_Onat___Orderings_Oord_44,axiom,
ord @ nat ).
thf(tcon_Nat_Onat___Heap_Oheap_45,axiom,
heap @ nat ).
thf(tcon_Nat_Onat___Nat_Osize,axiom,
size @ nat ).
thf(tcon_Num_Onum___Orderings_Opreorder_46,axiom,
preorder @ num ).
thf(tcon_Num_Onum___Orderings_Olinorder_47,axiom,
linorder @ num ).
thf(tcon_Num_Onum___Typerep_Otyperep_48,axiom,
typerep2 @ num ).
thf(tcon_Num_Onum___Orderings_Oorder_49,axiom,
order @ num ).
thf(tcon_Num_Onum___Orderings_Oord_50,axiom,
ord @ num ).
thf(tcon_Num_Onum___Nat_Osize_51,axiom,
size @ num ).
thf(tcon_Rat_Orat___Groups_Olinordered__ab__semigroup__add_52,axiom,
linord4140545234300271783up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_53,axiom,
ordere6911136660526730532id_add @ rat ).
thf(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_54,axiom,
linord5086331880401160121up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add_55,axiom,
ordered_ab_group_add @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_56,axiom,
cancel_semigroup_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semidom_57,axiom,
linordered_semidom @ rat ).
thf(tcon_Rat_Orat___Lattices_Osemilattice__sup_58,axiom,
semilattice_sup @ rat ).
thf(tcon_Rat_Orat___Lattices_Osemilattice__inf_59,axiom,
semilattice_inf @ rat ).
thf(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_60,axiom,
comm_monoid_mult @ rat ).
thf(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
linordered_field @ rat ).
thf(tcon_Rat_Orat___Groups_Ocomm__monoid__add_61,axiom,
comm_monoid_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__ring_62,axiom,
linordered_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__idom_63,axiom,
linordered_idom @ rat ).
thf(tcon_Rat_Orat___Nat_Osemiring__char__0_64,axiom,
semiring_char_0 @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__group__add_65,axiom,
ab_group_add @ rat ).
thf(tcon_Rat_Orat___Fields_Ofield__char__0,axiom,
field_char_0 @ rat ).
thf(tcon_Rat_Orat___Countable_Ocountable_66,axiom,
countable @ rat ).
thf(tcon_Rat_Orat___Orderings_Opreorder_67,axiom,
preorder @ rat ).
thf(tcon_Rat_Orat___Orderings_Olinorder_68,axiom,
linorder @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1_69,axiom,
semiring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__0_70,axiom,
semiring_0 @ rat ).
thf(tcon_Rat_Orat___Lattices_Olattice_71,axiom,
lattice @ rat ).
thf(tcon_Rat_Orat___Groups_Ogroup__add_72,axiom,
group_add @ rat ).
thf(tcon_Rat_Orat___Typerep_Otyperep_73,axiom,
typerep2 @ rat ).
thf(tcon_Rat_Orat___Orderings_Oorder_74,axiom,
order @ rat ).
thf(tcon_Rat_Orat___Orderings_Oord_75,axiom,
ord @ rat ).
thf(tcon_Rat_Orat___Rings_Oring__1_76,axiom,
ring_1 @ rat ).
thf(tcon_Rat_Orat___Fields_Ofield,axiom,
field @ rat ).
thf(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_77,axiom,
! [A21: $tType] : ( condit1219197933456340205attice @ ( set @ A21 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__distrib__lattice_78,axiom,
! [A21: $tType] : ( comple592849572758109894attice @ ( set @ A21 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__boolean__algebra_79,axiom,
! [A21: $tType] : ( comple489889107523837845lgebra @ ( set @ A21 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_80,axiom,
! [A21: $tType] : ( comple6319245703460814977attice @ ( set @ A21 ) ) ).
thf(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_81,axiom,
! [A21: $tType] : ( boolea8198339166811842893lgebra @ ( set @ A21 ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__sup_82,axiom,
! [A21: $tType] : ( semilattice_sup @ ( set @ A21 ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__inf_83,axiom,
! [A21: $tType] : ( semilattice_inf @ ( set @ A21 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_OSup_84,axiom,
! [A21: $tType] : ( complete_Sup @ ( set @ A21 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_OInf_85,axiom,
! [A21: $tType] : ( complete_Inf @ ( set @ A21 ) ) ).
thf(tcon_Set_Oset___Countable_Ocountable_86,axiom,
! [A21: $tType] :
( ( finite_finite @ A21 )
=> ( countable @ ( set @ A21 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_87,axiom,
! [A21: $tType] : ( preorder @ ( set @ A21 ) ) ).
thf(tcon_Set_Oset___Lattices_Olattice_88,axiom,
! [A21: $tType] : ( lattice @ ( set @ A21 ) ) ).
thf(tcon_Set_Oset___Typerep_Otyperep_89,axiom,
! [A21: $tType] :
( ( typerep2 @ A21 )
=> ( typerep2 @ ( set @ A21 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_90,axiom,
! [A21: $tType] : ( order @ ( set @ A21 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_91,axiom,
! [A21: $tType] : ( ord @ ( set @ A21 ) ) ).
thf(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_92,axiom,
condit1219197933456340205attice @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__distrib__lattice_93,axiom,
comple592849572758109894attice @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__boolean__algebra_94,axiom,
comple489889107523837845lgebra @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_95,axiom,
comple6319245703460814977attice @ $o ).
thf(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_96,axiom,
boolea8198339166811842893lgebra @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_97,axiom,
semilattice_sup @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_98,axiom,
semilattice_inf @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_OSup_99,axiom,
complete_Sup @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_OInf_100,axiom,
complete_Inf @ $o ).
thf(tcon_HOL_Obool___Countable_Ocountable_101,axiom,
countable @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_102,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_103,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Lattices_Olattice_104,axiom,
lattice @ $o ).
thf(tcon_HOL_Obool___Typerep_Otyperep_105,axiom,
typerep2 @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_106,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_107,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Heap_Oheap_108,axiom,
heap @ $o ).
thf(tcon_Heap_Oref___Countable_Ocountable_109,axiom,
! [A21: $tType] : ( countable @ ( ref @ A21 ) ) ).
thf(tcon_Heap_Oref___Typerep_Otyperep_110,axiom,
! [A21: $tType] : ( typerep2 @ ( ref @ A21 ) ) ).
thf(tcon_Heap_Oref___Heap_Oheap_111,axiom,
! [A21: $tType] : ( heap @ ( ref @ A21 ) ) ).
thf(tcon_Heap_Oref___Nat_Osize_112,axiom,
! [A21: $tType] : ( size @ ( ref @ A21 ) ) ).
thf(tcon_List_Olist___Countable_Ocountable_113,axiom,
! [A21: $tType] :
( ( countable @ A21 )
=> ( countable @ ( list @ A21 ) ) ) ).
thf(tcon_List_Olist___Typerep_Otyperep_114,axiom,
! [A21: $tType] :
( ( typerep2 @ A21 )
=> ( typerep2 @ ( list @ A21 ) ) ) ).
thf(tcon_List_Olist___Heap_Oheap_115,axiom,
! [A21: $tType] :
( ( heap @ A21 )
=> ( heap @ ( list @ A21 ) ) ) ).
thf(tcon_List_Olist___Nat_Osize_116,axiom,
! [A21: $tType] : ( size @ ( list @ A21 ) ) ).
thf(tcon_Heap_Oarray___Countable_Ocountable_117,axiom,
! [A21: $tType] : ( countable @ ( array @ A21 ) ) ).
thf(tcon_Heap_Oarray___Typerep_Otyperep_118,axiom,
! [A21: $tType] : ( typerep2 @ ( array @ A21 ) ) ).
thf(tcon_Heap_Oarray___Heap_Oheap_119,axiom,
! [A21: $tType] : ( heap @ ( array @ A21 ) ) ).
thf(tcon_Heap_Oarray___Nat_Osize_120,axiom,
! [A21: $tType] : ( size @ ( array @ A21 ) ) ).
thf(tcon_String_Ochar___Countable_Ocountable_121,axiom,
countable @ char ).
thf(tcon_String_Ochar___Typerep_Otyperep_122,axiom,
typerep2 @ char ).
thf(tcon_String_Ochar___Heap_Oheap_123,axiom,
heap @ char ).
thf(tcon_String_Ochar___Nat_Osize_124,axiom,
size @ char ).
thf(tcon_Sum__Type_Osum___Countable_Ocountable_125,axiom,
! [A21: $tType,A27: $tType] :
( ( ( countable @ A21 )
& ( countable @ A27 ) )
=> ( countable @ ( sum_sum @ A21 @ A27 ) ) ) ).
thf(tcon_Sum__Type_Osum___Typerep_Otyperep_126,axiom,
! [A21: $tType,A27: $tType] :
( ( ( typerep2 @ A21 )
& ( typerep2 @ A27 ) )
=> ( typerep2 @ ( sum_sum @ A21 @ A27 ) ) ) ).
thf(tcon_Sum__Type_Osum___Heap_Oheap_127,axiom,
! [A21: $tType,A27: $tType] :
( ( ( heap @ A21 )
& ( heap @ A27 ) )
=> ( heap @ ( sum_sum @ A21 @ A27 ) ) ) ).
thf(tcon_Sum__Type_Osum___Nat_Osize_128,axiom,
! [A21: $tType,A27: $tType] : ( size @ ( sum_sum @ A21 @ A27 ) ) ).
thf(tcon_Filter_Ofilter___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_129,axiom,
! [A21: $tType] : ( condit1219197933456340205attice @ ( filter @ A21 ) ) ).
thf(tcon_Filter_Ofilter___Complete__Lattices_Ocomplete__lattice_130,axiom,
! [A21: $tType] : ( comple6319245703460814977attice @ ( filter @ A21 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Osemilattice__sup_131,axiom,
! [A21: $tType] : ( semilattice_sup @ ( filter @ A21 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Osemilattice__inf_132,axiom,
! [A21: $tType] : ( semilattice_inf @ ( filter @ A21 ) ) ).
thf(tcon_Filter_Ofilter___Complete__Lattices_OSup_133,axiom,
! [A21: $tType] : ( complete_Sup @ ( filter @ A21 ) ) ).
thf(tcon_Filter_Ofilter___Complete__Lattices_OInf_134,axiom,
! [A21: $tType] : ( complete_Inf @ ( filter @ A21 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Opreorder_135,axiom,
! [A21: $tType] : ( preorder @ ( filter @ A21 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Olattice_136,axiom,
! [A21: $tType] : ( lattice @ ( filter @ A21 ) ) ).
thf(tcon_Filter_Ofilter___Typerep_Otyperep_137,axiom,
! [A21: $tType] :
( ( typerep2 @ A21 )
=> ( typerep2 @ ( filter @ A21 ) ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder_138,axiom,
! [A21: $tType] : ( order @ ( filter @ A21 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oord_139,axiom,
! [A21: $tType] : ( ord @ ( filter @ A21 ) ) ).
thf(tcon_Option_Ooption___Countable_Ocountable_140,axiom,
! [A21: $tType] :
( ( countable @ A21 )
=> ( countable @ ( option @ A21 ) ) ) ).
thf(tcon_Option_Ooption___Typerep_Otyperep_141,axiom,
! [A21: $tType] :
( ( typerep2 @ A21 )
=> ( typerep2 @ ( option @ A21 ) ) ) ).
thf(tcon_Option_Ooption___Heap_Oheap_142,axiom,
! [A21: $tType] :
( ( heap @ A21 )
=> ( heap @ ( option @ A21 ) ) ) ).
thf(tcon_Option_Ooption___Nat_Osize_143,axiom,
! [A21: $tType] : ( size @ ( option @ A21 ) ) ).
thf(tcon_Predicate_Oseq___Typerep_Otyperep_144,axiom,
! [A21: $tType] :
( ( typerep2 @ A21 )
=> ( typerep2 @ ( seq @ A21 ) ) ) ).
thf(tcon_Predicate_Opred___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_145,axiom,
! [A21: $tType] : ( condit1219197933456340205attice @ ( pred @ A21 ) ) ).
thf(tcon_Predicate_Opred___Complete__Lattices_Ocomplete__distrib__lattice_146,axiom,
! [A21: $tType] : ( comple592849572758109894attice @ ( pred @ A21 ) ) ).
thf(tcon_Predicate_Opred___Complete__Lattices_Ocomplete__boolean__algebra_147,axiom,
! [A21: $tType] : ( comple489889107523837845lgebra @ ( pred @ A21 ) ) ).
thf(tcon_Predicate_Opred___Complete__Lattices_Ocomplete__lattice_148,axiom,
! [A21: $tType] : ( comple6319245703460814977attice @ ( pred @ A21 ) ) ).
thf(tcon_Predicate_Opred___Boolean__Algebras_Oboolean__algebra_149,axiom,
! [A21: $tType] : ( boolea8198339166811842893lgebra @ ( pred @ A21 ) ) ).
thf(tcon_Predicate_Opred___Lattices_Osemilattice__sup_150,axiom,
! [A21: $tType] : ( semilattice_sup @ ( pred @ A21 ) ) ).
thf(tcon_Predicate_Opred___Lattices_Osemilattice__inf_151,axiom,
! [A21: $tType] : ( semilattice_inf @ ( pred @ A21 ) ) ).
thf(tcon_Predicate_Opred___Complete__Lattices_OSup_152,axiom,
! [A21: $tType] : ( complete_Sup @ ( pred @ A21 ) ) ).
thf(tcon_Predicate_Opred___Complete__Lattices_OInf_153,axiom,
! [A21: $tType] : ( complete_Inf @ ( pred @ A21 ) ) ).
thf(tcon_Predicate_Opred___Orderings_Opreorder_154,axiom,
! [A21: $tType] : ( preorder @ ( pred @ A21 ) ) ).
thf(tcon_Predicate_Opred___Lattices_Olattice_155,axiom,
! [A21: $tType] : ( lattice @ ( pred @ A21 ) ) ).
thf(tcon_Predicate_Opred___Typerep_Otyperep_156,axiom,
! [A21: $tType] :
( ( typerep2 @ A21 )
=> ( typerep2 @ ( pred @ A21 ) ) ) ).
thf(tcon_Predicate_Opred___Orderings_Oorder_157,axiom,
! [A21: $tType] : ( order @ ( pred @ A21 ) ) ).
thf(tcon_Predicate_Opred___Orderings_Oord_158,axiom,
! [A21: $tType] : ( ord @ ( pred @ A21 ) ) ).
thf(tcon_String_Oliteral___Countable_Ocountable_159,axiom,
countable @ literal ).
thf(tcon_String_Oliteral___Orderings_Opreorder_160,axiom,
preorder @ literal ).
thf(tcon_String_Oliteral___Orderings_Olinorder_161,axiom,
linorder @ literal ).
thf(tcon_String_Oliteral___Typerep_Otyperep_162,axiom,
typerep2 @ literal ).
thf(tcon_String_Oliteral___Orderings_Oorder_163,axiom,
order @ literal ).
thf(tcon_String_Oliteral___Orderings_Oord_164,axiom,
ord @ literal ).
thf(tcon_String_Oliteral___Heap_Oheap_165,axiom,
heap @ literal ).
thf(tcon_String_Oliteral___Nat_Osize_166,axiom,
size @ literal ).
thf(tcon_Heap__Monad_OHeap___Typerep_Otyperep_167,axiom,
! [A21: $tType] :
( ( typerep2 @ A21 )
=> ( typerep2 @ ( heap_Heap @ A21 ) ) ) ).
thf(tcon_Heap__Monad_OHeap___Nat_Osize_168,axiom,
! [A21: $tType] : ( size @ ( heap_Heap @ A21 ) ) ).
thf(tcon_Typerep_Otyperep___Countable_Ocountable_169,axiom,
countable @ typerep ).
thf(tcon_Typerep_Otyperep___Typerep_Otyperep_170,axiom,
typerep2 @ typerep ).
thf(tcon_Typerep_Otyperep___Heap_Oheap_171,axiom,
heap @ typerep ).
thf(tcon_Typerep_Otyperep___Nat_Osize_172,axiom,
size @ typerep ).
thf(tcon_Old__Datatype_Onode___Typerep_Otyperep_173,axiom,
! [A21: $tType,A27: $tType] :
( ( ( typerep2 @ A21 )
& ( typerep2 @ A27 ) )
=> ( typerep2 @ ( old_node @ A21 @ A27 ) ) ) ).
thf(tcon_Product__Type_Oprod___Countable_Ocountable_174,axiom,
! [A21: $tType,A27: $tType] :
( ( ( countable @ A21 )
& ( countable @ A27 ) )
=> ( countable @ ( product_prod @ A21 @ A27 ) ) ) ).
thf(tcon_Product__Type_Oprod___Typerep_Otyperep_175,axiom,
! [A21: $tType,A27: $tType] :
( ( ( typerep2 @ A21 )
& ( typerep2 @ A27 ) )
=> ( typerep2 @ ( product_prod @ A21 @ A27 ) ) ) ).
thf(tcon_Product__Type_Oprod___Heap_Oheap_176,axiom,
! [A21: $tType,A27: $tType] :
( ( ( heap @ A21 )
& ( heap @ A27 ) )
=> ( heap @ ( product_prod @ A21 @ A27 ) ) ) ).
thf(tcon_Product__Type_Oprod___Nat_Osize_177,axiom,
! [A21: $tType,A27: $tType] : ( size @ ( product_prod @ A21 @ A27 ) ) ).
thf(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_178,axiom,
condit6923001295902523014norder @ product_unit ).
thf(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_179,axiom,
condit1219197933456340205attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__distrib__lattice_180,axiom,
comple592849572758109894attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__boolean__algebra_181,axiom,
comple489889107523837845lgebra @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__linorder,axiom,
comple5582772986160207858norder @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_182,axiom,
comple6319245703460814977attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_183,axiom,
boolea8198339166811842893lgebra @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_184,axiom,
semilattice_sup @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__inf_185,axiom,
semilattice_inf @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_OSup_186,axiom,
complete_Sup @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_OInf_187,axiom,
complete_Inf @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Owellorder_188,axiom,
wellorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Countable_Ocountable_189,axiom,
countable @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Opreorder_190,axiom,
preorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Olinorder_191,axiom,
linorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Olattice_192,axiom,
lattice @ product_unit ).
thf(tcon_Product__Type_Ounit___Typerep_Otyperep_193,axiom,
typerep2 @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder_194,axiom,
order @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oord_195,axiom,
ord @ product_unit ).
thf(tcon_Product__Type_Ounit___Heap_Oheap_196,axiom,
heap @ product_unit ).
thf(tcon_Heap_Oheap_Oheap__ext___Typerep_Otyperep_197,axiom,
! [A21: $tType] :
( ( typerep2 @ A21 )
=> ( typerep2 @ ( heap_ext @ A21 ) ) ) ).
thf(tcon_Code__Evaluation_Oterm___Typerep_Otyperep_198,axiom,
typerep2 @ code_term ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_199,axiom,
bit_un5681908812861735899ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_200,axiom,
unique1627219031080169319umeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__semigroup__add_201,axiom,
linord4140545234300271783up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_202,axiom,
ordere6911136660526730532id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_203,axiom,
linord5086331880401160121up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_204,axiom,
ordered_ab_group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_205,axiom,
cancel_semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_206,axiom,
linordered_semidom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_207,axiom,
comm_monoid_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_208,axiom,
comm_monoid_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_209,axiom,
linordered_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_210,axiom,
linordered_idom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_211,axiom,
semiring_char_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_212,axiom,
ab_group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_213,axiom,
preorder @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_214,axiom,
linorder @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_215,axiom,
semiring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_216,axiom,
semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_217,axiom,
group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Typerep_Otyperep_218,axiom,
typerep2 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Oorder_219,axiom,
order @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Oord_220,axiom,
ord @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1_221,axiom,
ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Onatural___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_222,axiom,
bit_un5681908812861735899ations @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Olinordered__ab__semigroup__add_223,axiom,
linord4140545234300271783up_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__comm__monoid__add_224,axiom,
ordere6911136660526730532id_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__semigroup__add_225,axiom,
cancel_semigroup_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__semidom_226,axiom,
linordered_semidom @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__mult_227,axiom,
comm_monoid_mult @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__add_228,axiom,
comm_monoid_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Nat_Osemiring__char__0_229,axiom,
semiring_char_0 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Orderings_Opreorder_230,axiom,
preorder @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Orderings_Olinorder_231,axiom,
linorder @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__1_232,axiom,
semiring_1 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__0_233,axiom,
semiring_0 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Typerep_Otyperep_234,axiom,
typerep2 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Orderings_Oorder_235,axiom,
order @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Orderings_Oord_236,axiom,
ord @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Nat_Osize_237,axiom,
size @ code_natural ).
% Helper facts (4)
thf(help_If_3_1_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X2: A,Y3: A] :
( ( if @ A @ $false @ X2 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X2: A,Y3: A] :
( ( if @ A @ $true @ X2 @ Y3 )
= X2 ) ).
thf(help_fChoice_1_1_T,axiom,
! [A: $tType,P2: A > $o] :
( ( P2 @ ( fChoice @ A @ P2 ) )
= ( ? [X10: A] : ( P2 @ X10 ) ) ) ).
% Free types (1)
thf(tfree_0,hypothesis,
heap @ a ).
% Conjectures (2)
thf(conj_0,hypothesis,
$true ).
thf(conj_1,conjecture,
thesis ).
%------------------------------------------------------------------------------