TPTP Problem File: ITP206_2.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : ITP206_2 : 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 : 0.50 v8.1.0
% Syntax : Number of formulae : 4919 (1457 unt;1300 typ; 0 def)
% Number of atoms : 8450 (3025 equ)
% Maximal formula atoms : 15 ( 2 avg)
% Number of connectives : 5410 ( 579 ~; 64 |; 548 &)
% ( 592 <=>;3627 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 7 avg)
% Maximal term depth : 18 ( 2 avg)
% Number of types : 14 ( 13 usr)
% Number of type conns : 1273 ( 953 >; 320 *; 0 +; 0 <<)
% Number of predicates : 163 ( 160 usr; 3 prp; 0-9 aty)
% Number of functors : 1127 (1127 usr; 48 con; 0-8 aty)
% Number of variables : 18145 (15794 !; 258 ?;18145 :)
% (2093 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% from the van Emde Boas Trees session in the Archive of Formal
% proofs -
% www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
% 2022-02-17 13:38:36.224
%------------------------------------------------------------------------------
% Could-be-implicit typings (29)
tff(ty_t_Record_Otuple__isomorphism,type,
tuple_isomorphism: ( $tType * $tType * $tType ) > $tType ).
tff(ty_t_Code__Numeral_Onatural,type,
code_natural: $tType ).
tff(ty_t_Code__Numeral_Ointeger,type,
code_integer: $tType ).
tff(ty_t_Code__Evaluation_Oterm,type,
code_term: $tType ).
tff(ty_t_Heap_Oheap_Oheap__ext,type,
heap_ext: $tType > $tType ).
tff(ty_t_Product__Type_Ounit,type,
product_unit: $tType ).
tff(ty_t_Product__Type_Oprod,type,
product_prod: ( $tType * $tType ) > $tType ).
tff(ty_t_Old__Datatype_Onode,type,
old_node: ( $tType * $tType ) > $tType ).
tff(ty_t_Heap__Monad_OHeap,type,
heap_Heap: $tType > $tType ).
tff(ty_t_Typerep_Otyperep,type,
typerep: $tType ).
tff(ty_t_String_Oliteral,type,
literal: $tType ).
tff(ty_t_Predicate_Opred,type,
pred: $tType > $tType ).
tff(ty_t_Sum__Type_Osum,type,
sum_sum: ( $tType * $tType ) > $tType ).
tff(ty_t_Predicate_Oseq,type,
seq: $tType > $tType ).
tff(ty_t_Option_Ooption,type,
option: $tType > $tType ).
tff(ty_t_Filter_Ofilter,type,
filter: $tType > $tType ).
tff(ty_t_String_Ochar,type,
char: $tType ).
tff(ty_t_Heap_Oarray,type,
array: $tType > $tType ).
tff(ty_t_List_Olist,type,
list: $tType > $tType ).
tff(ty_t_Heap_Oref,type,
ref: $tType > $tType ).
tff(ty_t_HOL_Obool,type,
bool: $tType ).
tff(ty_t_Set_Oset,type,
set: $tType > $tType ).
tff(ty_t_Rat_Orat,type,
rat: $tType ).
tff(ty_t_Num_Onum,type,
num: $tType ).
tff(ty_t_Nat_Onat,type,
nat: $tType ).
tff(ty_t_Int_Oint,type,
int: $tType ).
tff(ty_t_itself,type,
itself: $tType > $tType ).
tff(ty_t_fun,type,
fun: ( $tType * $tType ) > $tType ).
tff(ty_tf_a,type,
a: $tType ).
% Explicit typings (1271)
tff(sy_cl_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : $o ).
tff(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
tff(sy_cl_Nat_Osize,type,
size:
!>[A: $tType] : $o ).
tff(sy_cl_Heap_Oheap,type,
heap:
!>[A: $tType] : $o ).
tff(sy_cl_Fields_Ofield,type,
field:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Oring__1,type,
ring_1:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
tff(sy_cl_Typerep_Otyperep,type,
typerep2:
!>[A: $tType] : $o ).
tff(sy_cl_GCD_Osemiring__Gcd,type,
semiring_Gcd:
!>[A: $tType] : $o ).
tff(sy_cl_GCD_Osemiring__gcd,type,
semiring_gcd:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : $o ).
tff(sy_cl_Lattices_Olattice,type,
lattice:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Osemiring__0,type,
semiring_0:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Osemiring__1,type,
semiring_1:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
tff(sy_cl_Countable_Ocountable,type,
countable:
!>[A: $tType] : $o ).
tff(sy_cl_Fields_Ofield__char__0,type,
field_char_0:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oab__group__add,type,
ab_group_add:
!>[A: $tType] : $o ).
tff(sy_cl_Nat_Osemiring__char__0,type,
semiring_char_0:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Owellorder,type,
wellorder:
!>[A: $tType] : $o ).
tff(sy_cl_Complete__Lattices_OInf,type,
complete_Inf:
!>[A: $tType] : $o ).
tff(sy_cl_Complete__Lattices_OSup,type,
complete_Sup:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__ring,type,
linordered_ring:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocomm__monoid__add,type,
comm_monoid_add:
!>[A: $tType] : $o ).
tff(sy_cl_Fields_Olinordered__field,type,
linordered_field:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocomm__monoid__mult,type,
comm_monoid_mult:
!>[A: $tType] : $o ).
tff(sy_cl_Lattices_Osemilattice__inf,type,
semilattice_inf:
!>[A: $tType] : $o ).
tff(sy_cl_Lattices_Osemilattice__sup,type,
semilattice_sup:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[A: $tType] : $o ).
tff(sy_cl_Enum_Ofinite__distrib__lattice,type,
finite8700451911770168679attice:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Olinordered__ab__group__add,type,
linord5086331880401160121up_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oordered__comm__monoid__add,type,
ordere6911136660526730532id_add:
!>[A: $tType] : $o ).
tff(sy_cl_Boolean__Algebras_Oboolean__algebra,type,
boolea8198339166811842893lgebra:
!>[A: $tType] : $o ).
tff(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
comple6319245703460814977attice:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Olinordered__ab__semigroup__add,type,
linord4140545234300271783up_add:
!>[A: $tType] : $o ).
tff(sy_cl_Complete__Lattices_Ocomplete__linorder,type,
comple5582772986160207858norder:
!>[A: $tType] : $o ).
tff(sy_cl_Divides_Ounique__euclidean__semiring__numeral,type,
unique1627219031080169319umeral:
!>[A: $tType] : $o ).
tff(sy_cl_Complete__Lattices_Ocomplete__boolean__algebra,type,
comple489889107523837845lgebra:
!>[A: $tType] : $o ).
tff(sy_cl_Complete__Lattices_Ocomplete__distrib__lattice,type,
comple592849572758109894attice:
!>[A: $tType] : $o ).
tff(sy_cl_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations,type,
bit_un5681908812861735899ations:
!>[A: $tType] : $o ).
tff(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__lattice,type,
condit1219197933456340205attice:
!>[A: $tType] : $o ).
tff(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
condit6923001295902523014norder:
!>[A: $tType] : $o ).
tff(sy_c_ATP_058Lamp__a____,type,
aTP_Lamp_a:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__aa____,type,
aTP_Lamp_aa:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ab____,type,
aTP_Lamp_ab:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ac____,type,
aTP_Lamp_ac:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ad____,type,
aTP_Lamp_ad:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ae____,type,
aTP_Lamp_ae:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__af____,type,
aTP_Lamp_af:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ag____,type,
aTP_Lamp_ag:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ah____,type,
aTP_Lamp_ah:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ai____,type,
aTP_Lamp_ai:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__aj____,type,
aTP_Lamp_aj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ak____,type,
aTP_Lamp_ak:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__al____,type,
aTP_Lamp_al:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__am____,type,
aTP_Lamp_am:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__an____,type,
aTP_Lamp_an:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ao____,type,
aTP_Lamp_ao:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ap____,type,
aTP_Lamp_ap:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__aq____,type,
aTP_Lamp_aq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ar____,type,
aTP_Lamp_ar:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__as____,type,
aTP_Lamp_as:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__at____,type,
aTP_Lamp_at:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__au____,type,
aTP_Lamp_au:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__av____,type,
aTP_Lamp_av:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__aw____,type,
aTP_Lamp_aw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ax____,type,
aTP_Lamp_ax:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ay____,type,
aTP_Lamp_ay:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__az____,type,
aTP_Lamp_az:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ba____,type,
aTP_Lamp_ba:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__bb____,type,
aTP_Lamp_bb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bc____,type,
aTP_Lamp_bc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bd____,type,
aTP_Lamp_bd:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__be____,type,
aTP_Lamp_be:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bf____,type,
aTP_Lamp_bf:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bg____,type,
aTP_Lamp_bg:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__bh____,type,
aTP_Lamp_bh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bi____,type,
aTP_Lamp_bi:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__bj____,type,
aTP_Lamp_bj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bk____,type,
aTP_Lamp_bk:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__bl____,type,
aTP_Lamp_bl:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bm____,type,
aTP_Lamp_bm:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__bn____,type,
aTP_Lamp_bn:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bo____,type,
aTP_Lamp_bo:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bp____,type,
aTP_Lamp_bp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bq____,type,
aTP_Lamp_bq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__br____,type,
aTP_Lamp_br:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bs____,type,
aTP_Lamp_bs:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bt____,type,
aTP_Lamp_bt:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bu____,type,
aTP_Lamp_bu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bv____,type,
aTP_Lamp_bv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bw____,type,
aTP_Lamp_bw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bx____,type,
aTP_Lamp_bx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__by____,type,
aTP_Lamp_by:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__bz____,type,
aTP_Lamp_bz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ca____,type,
aTP_Lamp_ca:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__cb____,type,
aTP_Lamp_cb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cc____,type,
aTP_Lamp_cc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cd____,type,
aTP_Lamp_cd:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ce____,type,
aTP_Lamp_ce:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cf____,type,
aTP_Lamp_cf:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cg____,type,
aTP_Lamp_cg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ch____,type,
aTP_Lamp_ch:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ci____,type,
aTP_Lamp_ci:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__cj____,type,
aTP_Lamp_cj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ck____,type,
aTP_Lamp_ck:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cl____,type,
aTP_Lamp_cl:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cm____,type,
aTP_Lamp_cm:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__cn____,type,
aTP_Lamp_cn:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__co____,type,
aTP_Lamp_co:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cp____,type,
aTP_Lamp_cp:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__cq____,type,
aTP_Lamp_cq:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__cr____,type,
aTP_Lamp_cr:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cs____,type,
aTP_Lamp_cs:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ct____,type,
aTP_Lamp_ct:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__cu____,type,
aTP_Lamp_cu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cv____,type,
aTP_Lamp_cv:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__cw____,type,
aTP_Lamp_cw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cx____,type,
aTP_Lamp_cx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cy____,type,
aTP_Lamp_cy:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__cz____,type,
aTP_Lamp_cz:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__da____,type,
aTP_Lamp_da:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__db____,type,
aTP_Lamp_db:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dc____,type,
aTP_Lamp_dc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dd____,type,
aTP_Lamp_dd:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__de____,type,
aTP_Lamp_de:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__df____,type,
aTP_Lamp_df:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dg____,type,
aTP_Lamp_dg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dh____,type,
aTP_Lamp_dh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__di____,type,
aTP_Lamp_di:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dj____,type,
aTP_Lamp_dj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dk____,type,
aTP_Lamp_dk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dl____,type,
aTP_Lamp_dl:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dm____,type,
aTP_Lamp_dm:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__dn____,type,
aTP_Lamp_dn:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__do____,type,
aTP_Lamp_do:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dp____,type,
aTP_Lamp_dp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dq____,type,
aTP_Lamp_dq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dr____,type,
aTP_Lamp_dr:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ds____,type,
aTP_Lamp_ds:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dt____,type,
aTP_Lamp_dt:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__du____,type,
aTP_Lamp_du:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dv____,type,
aTP_Lamp_dv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dw____,type,
aTP_Lamp_dw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dx____,type,
aTP_Lamp_dx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dy____,type,
aTP_Lamp_dy:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__dz____,type,
aTP_Lamp_dz:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ea____,type,
aTP_Lamp_ea:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__eb____,type,
aTP_Lamp_eb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ec____,type,
aTP_Lamp_ec:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ed____,type,
aTP_Lamp_ed:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ee____,type,
aTP_Lamp_ee:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ef____,type,
aTP_Lamp_ef:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__eg____,type,
aTP_Lamp_eg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__eh____,type,
aTP_Lamp_eh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ei____,type,
aTP_Lamp_ei:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ej____,type,
aTP_Lamp_ej:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ek____,type,
aTP_Lamp_ek:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__el____,type,
aTP_Lamp_el:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__em____,type,
aTP_Lamp_em:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__en____,type,
aTP_Lamp_en:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__eo____,type,
aTP_Lamp_eo:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ep____,type,
aTP_Lamp_ep:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__eq____,type,
aTP_Lamp_eq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__er____,type,
aTP_Lamp_er:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__es____,type,
aTP_Lamp_es:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__et____,type,
aTP_Lamp_et:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__eu____,type,
aTP_Lamp_eu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ev____,type,
aTP_Lamp_ev:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ew____,type,
aTP_Lamp_ew:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ex____,type,
aTP_Lamp_ex:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ey____,type,
aTP_Lamp_ey:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ez____,type,
aTP_Lamp_ez:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fa____,type,
aTP_Lamp_fa:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fb____,type,
aTP_Lamp_fb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fc____,type,
aTP_Lamp_fc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fd____,type,
aTP_Lamp_fd:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fe____,type,
aTP_Lamp_fe:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ff____,type,
aTP_Lamp_ff:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fg____,type,
aTP_Lamp_fg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fh____,type,
aTP_Lamp_fh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fi____,type,
aTP_Lamp_fi:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fj____,type,
aTP_Lamp_fj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fk____,type,
aTP_Lamp_fk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fl____,type,
aTP_Lamp_fl:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fm____,type,
aTP_Lamp_fm:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fn____,type,
aTP_Lamp_fn:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fo____,type,
aTP_Lamp_fo:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fp____,type,
aTP_Lamp_fp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fq____,type,
aTP_Lamp_fq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fr____,type,
aTP_Lamp_fr:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fs____,type,
aTP_Lamp_fs:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ft____,type,
aTP_Lamp_ft:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fu____,type,
aTP_Lamp_fu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fv____,type,
aTP_Lamp_fv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fw____,type,
aTP_Lamp_fw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fx____,type,
aTP_Lamp_fx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fy____,type,
aTP_Lamp_fy:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__fz____,type,
aTP_Lamp_fz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ga____,type,
aTP_Lamp_ga:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gb____,type,
aTP_Lamp_gb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gc____,type,
aTP_Lamp_gc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gd____,type,
aTP_Lamp_gd:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ge____,type,
aTP_Lamp_ge:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gf____,type,
aTP_Lamp_gf:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gg____,type,
aTP_Lamp_gg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gh____,type,
aTP_Lamp_gh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gi____,type,
aTP_Lamp_gi:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gj____,type,
aTP_Lamp_gj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gk____,type,
aTP_Lamp_gk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gl____,type,
aTP_Lamp_gl:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gm____,type,
aTP_Lamp_gm:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gn____,type,
aTP_Lamp_gn:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__go____,type,
aTP_Lamp_go:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gp____,type,
aTP_Lamp_gp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gq____,type,
aTP_Lamp_gq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gr____,type,
aTP_Lamp_gr:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gs____,type,
aTP_Lamp_gs:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gt____,type,
aTP_Lamp_gt:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gu____,type,
aTP_Lamp_gu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gv____,type,
aTP_Lamp_gv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gw____,type,
aTP_Lamp_gw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gx____,type,
aTP_Lamp_gx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gy____,type,
aTP_Lamp_gy:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__gz____,type,
aTP_Lamp_gz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ha____,type,
aTP_Lamp_ha:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hb____,type,
aTP_Lamp_hb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hc____,type,
aTP_Lamp_hc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hd____,type,
aTP_Lamp_hd:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__he____,type,
aTP_Lamp_he:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__hf____,type,
aTP_Lamp_hf:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hg____,type,
aTP_Lamp_hg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hh____,type,
aTP_Lamp_hh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hi____,type,
aTP_Lamp_hi:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hj____,type,
aTP_Lamp_hj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hk____,type,
aTP_Lamp_hk:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__hl____,type,
aTP_Lamp_hl:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hm____,type,
aTP_Lamp_hm:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__hn____,type,
aTP_Lamp_hn:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ho____,type,
aTP_Lamp_ho:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hp____,type,
aTP_Lamp_hp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hq____,type,
aTP_Lamp_hq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hr____,type,
aTP_Lamp_hr:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hs____,type,
aTP_Lamp_hs:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ht____,type,
aTP_Lamp_ht:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hu____,type,
aTP_Lamp_hu:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__hv____,type,
aTP_Lamp_hv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hw____,type,
aTP_Lamp_hw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hx____,type,
aTP_Lamp_hx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hy____,type,
aTP_Lamp_hy:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__hz____,type,
aTP_Lamp_hz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ia____,type,
aTP_Lamp_ia:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ib____,type,
aTP_Lamp_ib:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ic____,type,
aTP_Lamp_ic:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__id____,type,
aTP_Lamp_id:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ie____,type,
aTP_Lamp_ie:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__if____,type,
aTP_Lamp_if:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ig____,type,
aTP_Lamp_ig:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ih____,type,
aTP_Lamp_ih:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ii____,type,
aTP_Lamp_ii:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ij____,type,
aTP_Lamp_ij:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ik____,type,
aTP_Lamp_ik:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__il____,type,
aTP_Lamp_il:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__im____,type,
aTP_Lamp_im:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__in____,type,
aTP_Lamp_in:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__io____,type,
aTP_Lamp_io:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ip____,type,
aTP_Lamp_ip:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__iq____,type,
aTP_Lamp_iq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ir____,type,
aTP_Lamp_ir:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__is____,type,
aTP_Lamp_is:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__it____,type,
aTP_Lamp_it:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__iu____,type,
aTP_Lamp_iu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__iv____,type,
aTP_Lamp_iv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__iw____,type,
aTP_Lamp_iw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ix____,type,
aTP_Lamp_ix:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__iy____,type,
aTP_Lamp_iy:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__iz____,type,
aTP_Lamp_iz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ja____,type,
aTP_Lamp_ja:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jb____,type,
aTP_Lamp_jb:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__jc____,type,
aTP_Lamp_jc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jd____,type,
aTP_Lamp_jd:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__je____,type,
aTP_Lamp_je:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jf____,type,
aTP_Lamp_jf:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jg____,type,
aTP_Lamp_jg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jh____,type,
aTP_Lamp_jh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ji____,type,
aTP_Lamp_ji:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jj____,type,
aTP_Lamp_jj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jk____,type,
aTP_Lamp_jk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jl____,type,
aTP_Lamp_jl:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jm____,type,
aTP_Lamp_jm:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jn____,type,
aTP_Lamp_jn:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jo____,type,
aTP_Lamp_jo:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jp____,type,
aTP_Lamp_jp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jq____,type,
aTP_Lamp_jq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jr____,type,
aTP_Lamp_jr:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__js____,type,
aTP_Lamp_js:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jt____,type,
aTP_Lamp_jt:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ju____,type,
aTP_Lamp_ju:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jv____,type,
aTP_Lamp_jv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jw____,type,
aTP_Lamp_jw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jx____,type,
aTP_Lamp_jx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jy____,type,
aTP_Lamp_jy:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__jz____,type,
aTP_Lamp_jz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ka____,type,
aTP_Lamp_ka:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kb____,type,
aTP_Lamp_kb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kc____,type,
aTP_Lamp_kc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kd____,type,
aTP_Lamp_kd:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ke____,type,
aTP_Lamp_ke:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kf____,type,
aTP_Lamp_kf:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kg____,type,
aTP_Lamp_kg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kh____,type,
aTP_Lamp_kh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ki____,type,
aTP_Lamp_ki:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kj____,type,
aTP_Lamp_kj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kk____,type,
aTP_Lamp_kk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kl____,type,
aTP_Lamp_kl:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__km____,type,
aTP_Lamp_km:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kn____,type,
aTP_Lamp_kn:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ko____,type,
aTP_Lamp_ko:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kp____,type,
aTP_Lamp_kp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kq____,type,
aTP_Lamp_kq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kr____,type,
aTP_Lamp_kr:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ks____,type,
aTP_Lamp_ks:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kt____,type,
aTP_Lamp_kt:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ku____,type,
aTP_Lamp_ku:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kv____,type,
aTP_Lamp_kv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kw____,type,
aTP_Lamp_kw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kx____,type,
aTP_Lamp_kx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ky____,type,
aTP_Lamp_ky:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__kz____,type,
aTP_Lamp_kz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__la____,type,
aTP_Lamp_la:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lb____,type,
aTP_Lamp_lb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lc____,type,
aTP_Lamp_lc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ld____,type,
aTP_Lamp_ld:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__le____,type,
aTP_Lamp_le:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lf____,type,
aTP_Lamp_lf:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lg____,type,
aTP_Lamp_lg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lh____,type,
aTP_Lamp_lh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__li____,type,
aTP_Lamp_li:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lj____,type,
aTP_Lamp_lj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lk____,type,
aTP_Lamp_lk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ll____,type,
aTP_Lamp_ll:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lm____,type,
aTP_Lamp_lm:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ln____,type,
aTP_Lamp_ln:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lo____,type,
aTP_Lamp_lo:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lp____,type,
aTP_Lamp_lp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lq____,type,
aTP_Lamp_lq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lr____,type,
aTP_Lamp_lr:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ls____,type,
aTP_Lamp_ls:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lt____,type,
aTP_Lamp_lt:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__lu____,type,
aTP_Lamp_lu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lv____,type,
aTP_Lamp_lv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lw____,type,
aTP_Lamp_lw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lx____,type,
aTP_Lamp_lx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ly____,type,
aTP_Lamp_ly:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__lz____,type,
aTP_Lamp_lz:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ma____,type,
aTP_Lamp_ma:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mb____,type,
aTP_Lamp_mb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mc____,type,
aTP_Lamp_mc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__md____,type,
aTP_Lamp_md:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__me____,type,
aTP_Lamp_me:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mf____,type,
aTP_Lamp_mf:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__mg____,type,
aTP_Lamp_mg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mh____,type,
aTP_Lamp_mh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mi____,type,
aTP_Lamp_mi:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mj____,type,
aTP_Lamp_mj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mk____,type,
aTP_Lamp_mk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ml____,type,
aTP_Lamp_ml:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mm____,type,
aTP_Lamp_mm:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mn____,type,
aTP_Lamp_mn:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mo____,type,
aTP_Lamp_mo:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mp____,type,
aTP_Lamp_mp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mq____,type,
aTP_Lamp_mq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mr____,type,
aTP_Lamp_mr:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ms____,type,
aTP_Lamp_ms:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mt____,type,
aTP_Lamp_mt:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mu____,type,
aTP_Lamp_mu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mv____,type,
aTP_Lamp_mv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mw____,type,
aTP_Lamp_mw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mx____,type,
aTP_Lamp_mx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__my____,type,
aTP_Lamp_my:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__mz____,type,
aTP_Lamp_mz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__na____,type,
aTP_Lamp_na:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__nb____,type,
aTP_Lamp_nb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nc____,type,
aTP_Lamp_nc:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__nd____,type,
aTP_Lamp_nd:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ne____,type,
aTP_Lamp_ne:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nf____,type,
aTP_Lamp_nf:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ng____,type,
aTP_Lamp_ng:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nh____,type,
aTP_Lamp_nh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ni____,type,
aTP_Lamp_ni:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nj____,type,
aTP_Lamp_nj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nk____,type,
aTP_Lamp_nk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nl____,type,
aTP_Lamp_nl:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nm____,type,
aTP_Lamp_nm:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nn____,type,
aTP_Lamp_nn:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__no____,type,
aTP_Lamp_no:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__np____,type,
aTP_Lamp_np:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nq____,type,
aTP_Lamp_nq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nr____,type,
aTP_Lamp_nr:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ns____,type,
aTP_Lamp_ns:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nt____,type,
aTP_Lamp_nt:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nu____,type,
aTP_Lamp_nu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nv____,type,
aTP_Lamp_nv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nw____,type,
aTP_Lamp_nw:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__nx____,type,
aTP_Lamp_nx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ny____,type,
aTP_Lamp_ny:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__nz____,type,
aTP_Lamp_nz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__oa____,type,
aTP_Lamp_oa:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ob____,type,
aTP_Lamp_ob:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__oc____,type,
aTP_Lamp_oc:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__od____,type,
aTP_Lamp_od:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__oe____,type,
aTP_Lamp_oe:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__of____,type,
aTP_Lamp_of:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__og____,type,
aTP_Lamp_og:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__oh____,type,
aTP_Lamp_oh:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__oi____,type,
aTP_Lamp_oi:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__oj____,type,
aTP_Lamp_oj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ok____,type,
aTP_Lamp_ok:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ol____,type,
aTP_Lamp_ol:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__om____,type,
aTP_Lamp_om:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__on____,type,
aTP_Lamp_on:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__oo____,type,
aTP_Lamp_oo:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__op____,type,
aTP_Lamp_op:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__oq____,type,
aTP_Lamp_oq:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__or____,type,
aTP_Lamp_or:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__os____,type,
aTP_Lamp_os:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ot____,type,
aTP_Lamp_ot:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ou____,type,
aTP_Lamp_ou:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ov____,type,
aTP_Lamp_ov:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ow____,type,
aTP_Lamp_ow:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ox____,type,
aTP_Lamp_ox:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__oy____,type,
aTP_Lamp_oy:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__oz____,type,
aTP_Lamp_oz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pa____,type,
aTP_Lamp_pa:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pb____,type,
aTP_Lamp_pb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pc____,type,
aTP_Lamp_pc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pd____,type,
aTP_Lamp_pd:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pe____,type,
aTP_Lamp_pe:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pf____,type,
aTP_Lamp_pf:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__pg____,type,
aTP_Lamp_pg:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ph____,type,
aTP_Lamp_ph:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pi____,type,
aTP_Lamp_pi:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pj____,type,
aTP_Lamp_pj:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__pk____,type,
aTP_Lamp_pk:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__pl____,type,
aTP_Lamp_pl:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pm____,type,
aTP_Lamp_pm:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pn____,type,
aTP_Lamp_pn:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__po____,type,
aTP_Lamp_po:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pp____,type,
aTP_Lamp_pp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pq____,type,
aTP_Lamp_pq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pr____,type,
aTP_Lamp_pr:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ps____,type,
aTP_Lamp_ps:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__pt____,type,
aTP_Lamp_pt:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pu____,type,
aTP_Lamp_pu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pv____,type,
aTP_Lamp_pv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pw____,type,
aTP_Lamp_pw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__px____,type,
aTP_Lamp_px:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__py____,type,
aTP_Lamp_py:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__pz____,type,
aTP_Lamp_pz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qa____,type,
aTP_Lamp_qa:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qb____,type,
aTP_Lamp_qb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qc____,type,
aTP_Lamp_qc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qd____,type,
aTP_Lamp_qd:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qe____,type,
aTP_Lamp_qe:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qf____,type,
aTP_Lamp_qf:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__qg____,type,
aTP_Lamp_qg:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__qh____,type,
aTP_Lamp_qh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qi____,type,
aTP_Lamp_qi:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__qj____,type,
aTP_Lamp_qj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qk____,type,
aTP_Lamp_qk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ql____,type,
aTP_Lamp_ql:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qm____,type,
aTP_Lamp_qm:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qn____,type,
aTP_Lamp_qn:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qo____,type,
aTP_Lamp_qo:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qp____,type,
aTP_Lamp_qp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qq____,type,
aTP_Lamp_qq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qr____,type,
aTP_Lamp_qr:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__qs____,type,
aTP_Lamp_qs:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qt____,type,
aTP_Lamp_qt:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__qu____,type,
aTP_Lamp_qu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qv____,type,
aTP_Lamp_qv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qw____,type,
aTP_Lamp_qw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qx____,type,
aTP_Lamp_qx:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__qy____,type,
aTP_Lamp_qy:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__qz____,type,
aTP_Lamp_qz:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ra____,type,
aTP_Lamp_ra:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__rb____,type,
aTP_Lamp_rb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rc____,type,
aTP_Lamp_rc:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__rd____,type,
aTP_Lamp_rd:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__re____,type,
aTP_Lamp_re:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rf____,type,
aTP_Lamp_rf:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__rg____,type,
aTP_Lamp_rg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rh____,type,
aTP_Lamp_rh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ri____,type,
aTP_Lamp_ri:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rj____,type,
aTP_Lamp_rj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rk____,type,
aTP_Lamp_rk:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__rl____,type,
aTP_Lamp_rl:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rm____,type,
aTP_Lamp_rm:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rn____,type,
aTP_Lamp_rn:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ro____,type,
aTP_Lamp_ro:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rp____,type,
aTP_Lamp_rp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rq____,type,
aTP_Lamp_rq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rr____,type,
aTP_Lamp_rr:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rs____,type,
aTP_Lamp_rs:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rt____,type,
aTP_Lamp_rt:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ru____,type,
aTP_Lamp_ru:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rv____,type,
aTP_Lamp_rv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rw____,type,
aTP_Lamp_rw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rx____,type,
aTP_Lamp_rx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ry____,type,
aTP_Lamp_ry:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__rz____,type,
aTP_Lamp_rz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__sa____,type,
aTP_Lamp_sa:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__sb____,type,
aTP_Lamp_sb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__sc____,type,
aTP_Lamp_sc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__sd____,type,
aTP_Lamp_sd:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__se____,type,
aTP_Lamp_se:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__sf____,type,
aTP_Lamp_sf:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__sg____,type,
aTP_Lamp_sg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__sh____,type,
aTP_Lamp_sh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__si____,type,
aTP_Lamp_si:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__sj____,type,
aTP_Lamp_sj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__sk____,type,
aTP_Lamp_sk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__sl____,type,
aTP_Lamp_sl:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__sm____,type,
aTP_Lamp_sm:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__sn____,type,
aTP_Lamp_sn:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__so____,type,
aTP_Lamp_so:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__sp____,type,
aTP_Lamp_sp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__sq____,type,
aTP_Lamp_sq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__sr____,type,
aTP_Lamp_sr:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ss____,type,
aTP_Lamp_ss:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__st____,type,
aTP_Lamp_st:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__su____,type,
aTP_Lamp_su:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__sv____,type,
aTP_Lamp_sv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__sw____,type,
aTP_Lamp_sw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__sx____,type,
aTP_Lamp_sx:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__sy____,type,
aTP_Lamp_sy:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__sz____,type,
aTP_Lamp_sz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ta____,type,
aTP_Lamp_ta:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__tb____,type,
aTP_Lamp_tb:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__tc____,type,
aTP_Lamp_tc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__td____,type,
aTP_Lamp_td:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__te____,type,
aTP_Lamp_te:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__tf____,type,
aTP_Lamp_tf:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__tg____,type,
aTP_Lamp_tg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__th____,type,
aTP_Lamp_th:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ti____,type,
aTP_Lamp_ti:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__tj____,type,
aTP_Lamp_tj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__tk____,type,
aTP_Lamp_tk:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__tl____,type,
aTP_Lamp_tl:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__tm____,type,
aTP_Lamp_tm:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__tn____,type,
aTP_Lamp_tn:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__to____,type,
aTP_Lamp_to:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__tp____,type,
aTP_Lamp_tp:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__tq____,type,
aTP_Lamp_tq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__tr____,type,
aTP_Lamp_tr:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ts____,type,
aTP_Lamp_ts:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__tt____,type,
aTP_Lamp_tt:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__tu____,type,
aTP_Lamp_tu:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__tv____,type,
aTP_Lamp_tv:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__tw____,type,
aTP_Lamp_tw:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__tx____,type,
aTP_Lamp_tx:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ty____,type,
aTP_Lamp_ty:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__tz____,type,
aTP_Lamp_tz:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ua____,type,
aTP_Lamp_ua:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ub____,type,
aTP_Lamp_ub:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__uc____,type,
aTP_Lamp_uc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ud____,type,
aTP_Lamp_ud:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ue____,type,
aTP_Lamp_ue:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__uf____,type,
aTP_Lamp_uf:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ug____,type,
aTP_Lamp_ug:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__uh____,type,
aTP_Lamp_uh:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ui____,type,
aTP_Lamp_ui:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__uj____,type,
aTP_Lamp_uj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__uk____,type,
aTP_Lamp_uk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ul____,type,
aTP_Lamp_ul:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__um____,type,
aTP_Lamp_um:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__un____,type,
aTP_Lamp_un:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__uo____,type,
aTP_Lamp_uo:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__up____,type,
aTP_Lamp_up:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__uq____,type,
aTP_Lamp_uq:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ur____,type,
aTP_Lamp_ur:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__us____,type,
aTP_Lamp_us:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ut____,type,
aTP_Lamp_ut:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__uu____,type,
aTP_Lamp_uu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__uv____,type,
aTP_Lamp_uv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__uw____,type,
aTP_Lamp_uw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ux____,type,
aTP_Lamp_ux:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__uy____,type,
aTP_Lamp_uy:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__uz____,type,
aTP_Lamp_uz:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__va____,type,
aTP_Lamp_va:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__vb____,type,
aTP_Lamp_vb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__vc____,type,
aTP_Lamp_vc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__vd____,type,
aTP_Lamp_vd:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ve____,type,
aTP_Lamp_ve:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__vf____,type,
aTP_Lamp_vf:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__vg____,type,
aTP_Lamp_vg:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__vh____,type,
aTP_Lamp_vh:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__vi____,type,
aTP_Lamp_vi:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__vj____,type,
aTP_Lamp_vj:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__vk____,type,
aTP_Lamp_vk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__vl____,type,
aTP_Lamp_vl:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__vm____,type,
aTP_Lamp_vm:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__vn____,type,
aTP_Lamp_vn:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__vo____,type,
aTP_Lamp_vo:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__vp____,type,
aTP_Lamp_vp:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__vq____,type,
aTP_Lamp_vq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__vr____,type,
aTP_Lamp_vr:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__vs____,type,
aTP_Lamp_vs:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__vt____,type,
aTP_Lamp_vt:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__vu____,type,
aTP_Lamp_vu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__vv____,type,
aTP_Lamp_vv:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__vw____,type,
aTP_Lamp_vw:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__vx____,type,
aTP_Lamp_vx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__vy____,type,
aTP_Lamp_vy:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__vz____,type,
aTP_Lamp_vz:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__wa____,type,
aTP_Lamp_wa:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__wb____,type,
aTP_Lamp_wb:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__wc____,type,
aTP_Lamp_wc:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__wd____,type,
aTP_Lamp_wd:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__we____,type,
aTP_Lamp_we:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__wf____,type,
aTP_Lamp_wf:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__wg____,type,
aTP_Lamp_wg:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__wh____,type,
aTP_Lamp_wh:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__wi____,type,
aTP_Lamp_wi:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__wj____,type,
aTP_Lamp_wj:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__wk____,type,
aTP_Lamp_wk:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__wl____,type,
aTP_Lamp_wl:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__wm____,type,
aTP_Lamp_wm:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__wn____,type,
aTP_Lamp_wn:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__wo____,type,
aTP_Lamp_wo:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__wp____,type,
aTP_Lamp_wp:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__wq____,type,
aTP_Lamp_wq:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__wr____,type,
aTP_Lamp_wr:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__ws____,type,
aTP_Lamp_ws:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__wt____,type,
aTP_Lamp_wt:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__wu____,type,
aTP_Lamp_wu:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__wv____,type,
aTP_Lamp_wv:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__ww____,type,
aTP_Lamp_ww:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__wx____,type,
aTP_Lamp_wx:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__wy____,type,
aTP_Lamp_wy:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__wz____,type,
aTP_Lamp_wz:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_ATP_058Lamp__xa____,type,
aTP_Lamp_xa:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__xb____,type,
aTP_Lamp_xb:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__xc____,type,
aTP_Lamp_xc:
!>[A: $tType,B: $tType] : ( A > B ) ).
tff(sy_c_ATP_058Lamp__xd____,type,
aTP_Lamp_xd:
!>[A: $tType,B: $tType] : fun(A,B) ).
tff(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor,type,
archim6421214686448440834_floor:
!>[A: $tType] : ( A > int ) ).
tff(sy_c_Array_Oalloc,type,
alloc:
!>[A: $tType] : ( list(A) > fun(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit))) ) ).
tff(sy_c_Array_Ofreeze,type,
freeze:
!>[A: $tType] : ( array(A) > heap_Heap(list(A)) ) ).
tff(sy_c_Array_Oget,type,
get:
!>[A: $tType] : ( heap_ext(product_unit) > fun(array(A),list(A)) ) ).
tff(sy_c_Array_Olen,type,
len:
!>[A: $tType] : ( array(A) > heap_Heap(nat) ) ).
tff(sy_c_Array_Olen_H,type,
len2:
!>[B: $tType,A: $tType] : ( array(B) > heap_Heap(A) ) ).
tff(sy_c_Array_Olength,type,
length:
!>[A: $tType] : ( heap_ext(product_unit) > fun(array(A),nat) ) ).
tff(sy_c_Array_Omake,type,
make:
!>[A: $tType] : ( ( nat * fun(nat,A) ) > heap_Heap(array(A)) ) ).
tff(sy_c_Array_Omake_H,type,
make2:
!>[B: $tType,A: $tType] : ( ( code_integer * fun(B,A) ) > heap_Heap(array(A)) ) ).
tff(sy_c_Array_Omap__entry,type,
map_entry:
!>[A: $tType] : ( ( nat * fun(A,A) * array(A) ) > heap_Heap(array(A)) ) ).
tff(sy_c_Array_Onew,type,
new:
!>[A: $tType] : fun(nat,fun(A,heap_Heap(array(A)))) ).
tff(sy_c_Array_Onew_H,type,
new2:
!>[A: $tType] : fun(code_integer,fun(A,heap_Heap(array(A)))) ).
tff(sy_c_Array_Onoteq,type,
noteq:
!>[A: $tType,B: $tType] : ( ( array(A) * array(B) ) > $o ) ).
tff(sy_c_Array_Onth,type,
nth:
!>[A: $tType] : ( array(A) > fun(nat,heap_Heap(A)) ) ).
tff(sy_c_Array_Onth_H,type,
nth2:
!>[A: $tType] : ( array(A) > fun(code_integer,heap_Heap(A)) ) ).
tff(sy_c_Array_Oof__list,type,
of_list:
!>[A: $tType] : ( list(A) > heap_Heap(array(A)) ) ).
tff(sy_c_Array_Opresent,type,
present:
!>[A: $tType] : ( heap_ext(product_unit) > fun(array(A),bool) ) ).
tff(sy_c_Array_Oset,type,
set2:
!>[A: $tType] : ( ( array(A) * list(A) ) > fun(heap_ext(product_unit),heap_ext(product_unit)) ) ).
tff(sy_c_Array_Oswap,type,
swap:
!>[A: $tType] : ( ( nat * A * array(A) ) > heap_Heap(A) ) ).
tff(sy_c_Array_Oupd,type,
upd:
!>[A: $tType] : ( ( nat * A * array(A) ) > heap_Heap(array(A)) ) ).
tff(sy_c_Array_Oupd_H,type,
upd2:
!>[A: $tType] : ( ( array(A) * code_integer * A ) > heap_Heap(product_unit) ) ).
tff(sy_c_Array_Oupdate,type,
update:
!>[A: $tType] : ( ( array(A) * nat * A * heap_ext(product_unit) ) > heap_ext(product_unit) ) ).
tff(sy_c_BNF__Cardinal__Arithmetic_Ocexp,type,
bNF_Cardinal_cexp:
!>[B: $tType,A: $tType] : ( ( set(product_prod(B,B)) * set(product_prod(A,A)) ) > set(product_prod(fun(A,B),fun(A,B))) ) ).
tff(sy_c_BNF__Cardinal__Arithmetic_Ocinfinite,type,
bNF_Ca4139267488887388095finite:
!>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).
tff(sy_c_BNF__Cardinal__Arithmetic_Ocone,type,
bNF_Cardinal_cone: set(product_prod(product_unit,product_unit)) ).
tff(sy_c_BNF__Cardinal__Arithmetic_Ocprod,type,
bNF_Cardinal_cprod:
!>[A: $tType,B: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(B,B)) ) > set(product_prod(product_prod(A,B),product_prod(A,B))) ) ).
tff(sy_c_BNF__Cardinal__Arithmetic_Ocsum,type,
bNF_Cardinal_csum:
!>[A: $tType,B: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(B,B)) ) > set(product_prod(sum_sum(A,B),sum_sum(A,B))) ) ).
tff(sy_c_BNF__Cardinal__Order__Relation_OcardSuc,type,
bNF_Ca8387033319878233205ardSuc:
!>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(set(A),set(A))) ) ).
tff(sy_c_BNF__Cardinal__Order__Relation_Ocard__of,type,
bNF_Ca6860139660246222851ard_of:
!>[A: $tType] : ( set(A) > set(product_prod(A,A)) ) ).
tff(sy_c_BNF__Cardinal__Order__Relation_Ocard__order__on,type,
bNF_Ca8970107618336181345der_on:
!>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).
tff(sy_c_BNF__Cardinal__Order__Relation_Ocofinal,type,
bNF_Ca7293521722713021262ofinal:
!>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).
tff(sy_c_BNF__Cardinal__Order__Relation_OnatLeq,type,
bNF_Ca8665028551170535155natLeq: set(product_prod(nat,nat)) ).
tff(sy_c_BNF__Cardinal__Order__Relation_OnatLess,type,
bNF_Ca8459412986667044542atLess: set(product_prod(nat,nat)) ).
tff(sy_c_BNF__Cardinal__Order__Relation_OregularCard,type,
bNF_Ca7133664381575040944arCard:
!>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).
tff(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
bNF_Ca3754400796208372196lChain:
!>[A: $tType,B: $tType] : ( ( set(product_prod(A,A)) * fun(A,B) ) > $o ) ).
tff(sy_c_BNF__Composition_Oid__bnf,type,
bNF_id_bnf:
!>[A: $tType] : fun(A,A) ).
tff(sy_c_BNF__Def_OGr,type,
bNF_Gr:
!>[A: $tType,B: $tType] : ( ( set(A) * fun(A,B) ) > set(product_prod(A,B)) ) ).
tff(sy_c_BNF__Def_OGrp,type,
bNF_Grp:
!>[A: $tType,B: $tType] : ( ( set(A) * fun(A,B) ) > fun(A,fun(B,bool)) ) ).
tff(sy_c_BNF__Def_Ocollect,type,
bNF_collect:
!>[B: $tType,A: $tType] : ( set(fun(B,set(A))) > fun(B,set(A)) ) ).
tff(sy_c_BNF__Def_Oconvol,type,
bNF_convol:
!>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,B) * fun(A,C) ) > fun(A,product_prod(B,C)) ) ).
tff(sy_c_BNF__Def_Ocsquare,type,
bNF_csquare:
!>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( set(A) * fun(B,C) * fun(D,C) * fun(A,B) * fun(A,D) ) > $o ) ).
tff(sy_c_BNF__Def_Oeq__onp,type,
bNF_eq_onp:
!>[A: $tType] : ( fun(A,bool) > fun(A,fun(A,bool)) ) ).
tff(sy_c_BNF__Def_OfstOp,type,
bNF_fstOp:
!>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,fun(B,bool)) * fun(B,fun(C,bool)) ) > fun(product_prod(A,C),product_prod(A,B)) ) ).
tff(sy_c_BNF__Def_Opick__middlep,type,
bNF_pick_middlep:
!>[B: $tType,A: $tType,C: $tType] : ( ( fun(B,fun(A,bool)) * fun(A,fun(C,bool)) * B * C ) > A ) ).
tff(sy_c_BNF__Def_Orel__fun,type,
bNF_rel_fun:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( fun(A,fun(C,bool)) * fun(B,fun(D,bool)) ) > fun(fun(A,B),fun(fun(C,D),bool)) ) ).
tff(sy_c_BNF__Def_Orel__set,type,
bNF_rel_set:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > fun(set(A),fun(set(B),bool)) ) ).
tff(sy_c_BNF__Def_Orel__sum,type,
bNF_rel_sum:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( fun(A,fun(C,bool)) * fun(B,fun(D,bool)) ) > fun(sum_sum(A,B),fun(sum_sum(C,D),bool)) ) ).
tff(sy_c_BNF__Def_OsndOp,type,
bNF_sndOp:
!>[C: $tType,A: $tType,B: $tType] : ( ( fun(C,fun(A,bool)) * fun(A,fun(B,bool)) ) > fun(product_prod(C,B),product_prod(A,B)) ) ).
tff(sy_c_BNF__Def_Ovimage2p,type,
bNF_vimage2p:
!>[A: $tType,D: $tType,B: $tType,E: $tType,C: $tType] : ( ( fun(A,D) * fun(B,E) ) > fun(fun(D,fun(E,C)),fun(A,fun(B,C))) ) ).
tff(sy_c_BNF__Greatest__Fixpoint_OShift,type,
bNF_Greatest_Shift:
!>[A: $tType] : ( ( set(list(A)) * A ) > set(list(A)) ) ).
tff(sy_c_BNF__Greatest__Fixpoint_OSucc,type,
bNF_Greatest_Succ:
!>[A: $tType] : ( ( set(list(A)) * list(A) ) > set(A) ) ).
tff(sy_c_BNF__Greatest__Fixpoint_OfromCard,type,
bNF_Gr5436034075474128252omCard:
!>[A: $tType,B: $tType] : ( ( set(A) * set(product_prod(B,B)) * B ) > A ) ).
tff(sy_c_BNF__Greatest__Fixpoint_Oimage2,type,
bNF_Greatest_image2:
!>[C: $tType,A: $tType,B: $tType] : ( ( set(C) * fun(C,A) * fun(C,B) ) > set(product_prod(A,B)) ) ).
tff(sy_c_BNF__Greatest__Fixpoint_Oimage2p,type,
bNF_Greatest_image2p:
!>[C: $tType,A: $tType,D: $tType,B: $tType] : ( ( fun(C,A) * fun(D,B) * fun(C,fun(D,bool)) ) > fun(A,fun(B,bool)) ) ).
tff(sy_c_BNF__Greatest__Fixpoint_OrelImage,type,
bNF_Gr4221423524335903396lImage:
!>[B: $tType,A: $tType] : ( ( set(product_prod(B,B)) * fun(B,A) ) > set(product_prod(A,A)) ) ).
tff(sy_c_BNF__Greatest__Fixpoint_OrelInvImage,type,
bNF_Gr7122648621184425601vImage:
!>[A: $tType,B: $tType] : ( ( set(A) * set(product_prod(B,B)) * fun(A,B) ) > set(product_prod(A,A)) ) ).
tff(sy_c_BNF__Greatest__Fixpoint_Oshift,type,
bNF_Greatest_shift:
!>[A: $tType,B: $tType] : ( ( fun(list(A),B) * A * list(A) ) > B ) ).
tff(sy_c_BNF__Greatest__Fixpoint_OtoCard,type,
bNF_Greatest_toCard:
!>[A: $tType,B: $tType] : ( ( set(A) * set(product_prod(B,B)) ) > fun(A,B) ) ).
tff(sy_c_BNF__Greatest__Fixpoint_OtoCard__pred,type,
bNF_Gr1419584066657907630d_pred:
!>[A: $tType,B: $tType] : ( ( set(A) * set(product_prod(B,B)) ) > fun(fun(A,B),bool) ) ).
tff(sy_c_BNF__Greatest__Fixpoint_Ouniv,type,
bNF_Greatest_univ:
!>[B: $tType,A: $tType] : ( ( fun(B,A) * set(B) ) > A ) ).
tff(sy_c_BNF__Wellorder__Constructions_OFunc,type,
bNF_Wellorder_Func:
!>[A: $tType,B: $tType] : ( ( set(A) * set(B) ) > set(fun(A,B)) ) ).
tff(sy_c_BNF__Wellorder__Constructions_OFunc__map,type,
bNF_We4925052301507509544nc_map:
!>[B: $tType,C: $tType,A: $tType,D: $tType] : ( ( set(B) * fun(C,A) * fun(B,D) ) > fun(fun(D,C),fun(B,A)) ) ).
tff(sy_c_BNF__Wellorder__Constructions_Obsqr,type,
bNF_Wellorder_bsqr:
!>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(product_prod(A,A),product_prod(A,A))) ) ).
tff(sy_c_BNF__Wellorder__Constructions_Ocurr,type,
bNF_Wellorder_curr:
!>[A: $tType,B: $tType,C: $tType] : ( ( set(A) * fun(product_prod(A,B),C) * A ) > fun(B,C) ) ).
tff(sy_c_BNF__Wellorder__Constructions_Odir__image,type,
bNF_We2720479622203943262_image:
!>[A: $tType,A2: $tType] : ( ( set(product_prod(A,A)) * fun(A,A2) ) > set(product_prod(A2,A2)) ) ).
tff(sy_c_BNF__Wellorder__Constructions_OofilterIncl,type,
bNF_We413866401316099525erIncl:
!>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(set(A),set(A))) ) ).
tff(sy_c_BNF__Wellorder__Constructions_OordIso,type,
bNF_Wellorder_ordIso:
!>[A: $tType,A2: $tType] : set(product_prod(set(product_prod(A,A)),set(product_prod(A2,A2)))) ).
tff(sy_c_BNF__Wellorder__Constructions_OordLeq,type,
bNF_Wellorder_ordLeq:
!>[A: $tType,A2: $tType] : set(product_prod(set(product_prod(A,A)),set(product_prod(A2,A2)))) ).
tff(sy_c_BNF__Wellorder__Constructions_OordLess,type,
bNF_We4044943003108391690rdLess:
!>[A: $tType,A2: $tType] : set(product_prod(set(product_prod(A,A)),set(product_prod(A2,A2)))) ).
tff(sy_c_BNF__Wellorder__Constructions_Oord__to__filter,type,
bNF_We8469521843155493636filter:
!>[A: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(A,A)) ) > set(A) ) ).
tff(sy_c_BNF__Wellorder__Embedding_Ocompat,type,
bNF_Wellorder_compat:
!>[A: $tType,A2: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(A2,A2)) * fun(A,A2) ) > $o ) ).
tff(sy_c_BNF__Wellorder__Embedding_Oembed,type,
bNF_Wellorder_embed:
!>[A: $tType,A2: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(A2,A2)) ) > fun(fun(A,A2),bool) ) ).
tff(sy_c_BNF__Wellorder__Embedding_OembedS,type,
bNF_Wellorder_embedS:
!>[A: $tType,A2: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(A2,A2)) * fun(A,A2) ) > $o ) ).
tff(sy_c_BNF__Wellorder__Embedding_Oiso,type,
bNF_Wellorder_iso:
!>[A: $tType,A2: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(A2,A2)) * fun(A,A2) ) > $o ) ).
tff(sy_c_BNF__Wellorder__Relation_Owo__rel,type,
bNF_Wellorder_wo_rel:
!>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).
tff(sy_c_BNF__Wellorder__Relation_Owo__rel_OisMinim,type,
bNF_We4791949203932849705sMinim:
!>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) * A ) > $o ) ).
tff(sy_c_BNF__Wellorder__Relation_Owo__rel_Omax2,type,
bNF_We1388413361240627857o_max2:
!>[A: $tType] : ( ( set(product_prod(A,A)) * A * A ) > A ) ).
tff(sy_c_BNF__Wellorder__Relation_Owo__rel_Ominim,type,
bNF_We6954850376910717587_minim:
!>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > A ) ).
tff(sy_c_BNF__Wellorder__Relation_Owo__rel_Osuc,type,
bNF_Wellorder_wo_suc:
!>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > A ) ).
tff(sy_c_Basic__BNF__LFPs_Octor__rec,type,
basic_BNF_ctor_rec:
!>[A: $tType] : fun(A,A) ).
tff(sy_c_Basic__BNF__LFPs_Oprod_Osize__prod,type,
basic_BNF_size_prod:
!>[A: $tType,B: $tType] : ( ( fun(A,nat) * fun(B,nat) ) > fun(product_prod(A,B),nat) ) ).
tff(sy_c_Basic__BNF__LFPs_Osum_Osize__sum,type,
basic_BNF_size_sum:
!>[A: $tType,B: $tType] : ( ( fun(A,nat) * fun(B,nat) ) > fun(sum_sum(A,B),nat) ) ).
tff(sy_c_Basic__BNF__LFPs_Oxtor,type,
basic_BNF_xtor:
!>[A: $tType] : fun(A,A) ).
tff(sy_c_Basic__BNFs_Ofsts,type,
basic_fsts:
!>[A: $tType,B: $tType] : fun(product_prod(A,B),set(A)) ).
tff(sy_c_Basic__BNFs_Ofstsp,type,
basic_fstsp:
!>[A: $tType,B: $tType] : ( ( product_prod(A,B) * A ) > $o ) ).
tff(sy_c_Basic__BNFs_Opred__fun,type,
basic_pred_fun:
!>[A: $tType,B: $tType] : ( ( fun(A,bool) * fun(B,bool) * fun(A,B) ) > $o ) ).
tff(sy_c_Basic__BNFs_Opred__prod,type,
basic_pred_prod:
!>[A: $tType,B: $tType] : ( ( fun(A,bool) * fun(B,bool) ) > fun(product_prod(A,B),bool) ) ).
tff(sy_c_Basic__BNFs_Opred__sum,type,
basic_pred_sum:
!>[A: $tType,B: $tType] : ( ( fun(A,bool) * fun(B,bool) * sum_sum(A,B) ) > $o ) ).
tff(sy_c_Basic__BNFs_Orel__prod,type,
basic_rel_prod:
!>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( fun(A,fun(B,bool)) * fun(C,fun(D,bool)) ) > fun(product_prod(A,C),fun(product_prod(B,D),bool)) ) ).
tff(sy_c_Basic__BNFs_Osetl,type,
basic_setl:
!>[A: $tType,B: $tType] : ( sum_sum(A,B) > set(A) ) ).
tff(sy_c_Basic__BNFs_Osetr,type,
basic_setr:
!>[A: $tType,B: $tType] : ( sum_sum(A,B) > set(B) ) ).
tff(sy_c_Basic__BNFs_Osnds,type,
basic_snds:
!>[A: $tType,B: $tType] : ( product_prod(A,B) > set(B) ) ).
tff(sy_c_Basic__BNFs_Osndsp,type,
basic_sndsp:
!>[A: $tType,B: $tType] : ( ( product_prod(A,B) * B ) > $o ) ).
tff(sy_c_Bit__Operations_Otake__bit__num,type,
bit_take_bit_num: ( nat * num ) > option(num) ).
tff(sy_c_Code__Numeral_Obit__cut__integer,type,
code_bit_cut_integer: code_integer > product_prod(code_integer,bool) ).
tff(sy_c_Code__Numeral_Ocr__integer,type,
code_cr_integer: fun(int,fun(code_integer,bool)) ).
tff(sy_c_Code__Numeral_Odivmod__abs,type,
code_divmod_abs: ( code_integer * code_integer ) > product_prod(code_integer,code_integer) ).
tff(sy_c_Code__Numeral_Odivmod__integer,type,
code_divmod_integer: ( code_integer * code_integer ) > product_prod(code_integer,code_integer) ).
tff(sy_c_Code__Numeral_Ointeger_Oint__of__integer,type,
code_int_of_integer: fun(code_integer,int) ).
tff(sy_c_Code__Numeral_Onat__of__integer,type,
code_nat_of_integer: fun(code_integer,nat) ).
tff(sy_c_Code__Numeral_Onatural_Onat__of__natural,type,
code_nat_of_natural: code_natural > nat ).
tff(sy_c_Code__Numeral_Onatural_Onatural__of__nat,type,
code_natural_of_nat: nat > code_natural ).
tff(sy_c_Code__Numeral_Onegative,type,
code_negative: fun(num,code_integer) ).
tff(sy_c_Code__Numeral_Onum__of__integer,type,
code_num_of_integer: fun(code_integer,num) ).
tff(sy_c_Code__Numeral_Opcr__integer,type,
code_pcr_integer: fun(int,fun(code_integer,bool)) ).
tff(sy_c_Complete__Lattices_OInf__class_OInf,type,
complete_Inf_Inf:
!>[A: $tType] : fun(set(A),A) ).
tff(sy_c_Complete__Lattices_OSup__class_OSup,type,
complete_Sup_Sup:
!>[A: $tType] : fun(set(A),A) ).
tff(sy_c_Complete__Partial__Order_Occpo_Oadmissible,type,
comple1908693960933563346ssible:
!>[A: $tType] : ( ( fun(set(A),A) * fun(A,fun(A,bool)) * fun(A,bool) ) > $o ) ).
tff(sy_c_Complete__Partial__Order_Occpo_Ofixp,type,
comple187402453842119260l_fixp:
!>[A: $tType] : ( ( fun(set(A),A) * fun(A,fun(A,bool)) * fun(A,A) ) > A ) ).
tff(sy_c_Complete__Partial__Order_Ochain,type,
comple1602240252501008431_chain:
!>[A: $tType] : ( ( fun(A,fun(A,bool)) * set(A) ) > $o ) ).
tff(sy_c_Complete__Partial__Order_Omonotone,type,
comple7038119648293358887notone:
!>[A: $tType,B: $tType] : ( ( fun(A,fun(A,bool)) * fun(B,fun(B,bool)) * fun(A,B) ) > $o ) ).
tff(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above,type,
condit941137186595557371_above:
!>[A: $tType] : ( set(A) > $o ) ).
tff(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__below,type,
condit1013018076250108175_below:
!>[A: $tType] : ( set(A) > $o ) ).
tff(sy_c_Countable_Ofrom__nat,type,
from_nat:
!>[A: $tType] : fun(nat,A) ).
tff(sy_c_Countable_Onat__to__rat__surj,type,
nat_to_rat_surj: fun(nat,rat) ).
tff(sy_c_Countable_Onth__item,type,
nth_item:
!>[A: $tType] : ( nat > set(old_node(A,product_unit)) ) ).
tff(sy_c_Countable_Onth__item__rel,type,
nth_item_rel: fun(nat,fun(nat,bool)) ).
tff(sy_c_Countable_Oto__nat,type,
to_nat:
!>[A: $tType] : fun(A,nat) ).
tff(sy_c_Divides_Oadjust__div,type,
adjust_div: product_prod(int,int) > int ).
tff(sy_c_Divides_Odivmod__nat,type,
divmod_nat: ( nat * nat ) > product_prod(nat,nat) ).
tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux,type,
unique5940410009612947441es_aux:
!>[A: $tType] : ( product_prod(A,A) > $o ) ).
tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod,type,
unique8689654367752047608divmod:
!>[A: $tType] : ( ( num * num ) > product_prod(A,A) ) ).
tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step,type,
unique1321980374590559556d_step:
!>[A: $tType] : ( ( num * product_prod(A,A) ) > product_prod(A,A) ) ).
tff(sy_c_Equiv__Relations_Ocongruent,type,
equiv_congruent:
!>[A: $tType,B: $tType] : ( ( set(product_prod(A,A)) * fun(A,B) ) > $o ) ).
tff(sy_c_Equiv__Relations_Ocongruent2,type,
equiv_congruent2:
!>[A: $tType,B: $tType,C: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(B,B)) * fun(A,fun(B,C)) ) > $o ) ).
tff(sy_c_Equiv__Relations_Oequiv,type,
equiv_equiv:
!>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).
tff(sy_c_Equiv__Relations_Oequivclp,type,
equiv_equivclp:
!>[A: $tType] : ( fun(A,fun(A,bool)) > fun(A,fun(A,bool)) ) ).
tff(sy_c_Equiv__Relations_Oequivp,type,
equiv_equivp:
!>[A: $tType] : ( fun(A,fun(A,bool)) > $o ) ).
tff(sy_c_Equiv__Relations_Oproj,type,
equiv_proj:
!>[B: $tType,A: $tType] : ( set(product_prod(B,A)) > fun(B,set(A)) ) ).
tff(sy_c_Equiv__Relations_Oquotient,type,
equiv_quotient:
!>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > set(set(A)) ) ).
tff(sy_c_Euclidean__Division_Oeuclidean__semiring__class_Oeuclidean__size,type,
euclid6346220572633701492n_size:
!>[A: $tType] : fun(A,nat) ).
tff(sy_c_Fields_Oinverse__class_Oinverse,type,
inverse_inverse:
!>[A: $tType] : fun(A,A) ).
tff(sy_c_Filter_Oat__bot,type,
at_bot:
!>[A: $tType] : filter(A) ).
tff(sy_c_Filter_Oat__top,type,
at_top:
!>[A: $tType] : filter(A) ).
tff(sy_c_Filter_Oeventually,type,
eventually:
!>[A: $tType] : ( ( fun(A,bool) * filter(A) ) > $o ) ).
tff(sy_c_Filter_Ofiltercomap,type,
filtercomap:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * filter(B) ) > filter(A) ) ).
tff(sy_c_Filter_Ofilterlim,type,
filterlim:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * filter(B) * filter(A) ) > $o ) ).
tff(sy_c_Filter_Ofiltermap,type,
filtermap:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * filter(A) ) > filter(B) ) ).
tff(sy_c_Filter_Ofinite__subsets__at__top,type,
finite5375528669736107172at_top:
!>[A: $tType] : ( set(A) > filter(set(A)) ) ).
tff(sy_c_Filter_Omap__filter__on,type,
map_filter_on:
!>[A: $tType,B: $tType] : ( ( set(A) * fun(A,B) * filter(A) ) > filter(B) ) ).
tff(sy_c_Filter_Oprincipal,type,
principal:
!>[A: $tType] : ( set(A) > filter(A) ) ).
tff(sy_c_Filter_Oprod__filter,type,
prod_filter:
!>[A: $tType,B: $tType] : ( ( filter(A) * filter(B) ) > filter(product_prod(A,B)) ) ).
tff(sy_c_Filter_Orel__filter,type,
rel_filter:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > fun(filter(A),fun(filter(B),bool)) ) ).
tff(sy_c_Finite__Set_OFpow,type,
finite_Fpow:
!>[A: $tType] : ( set(A) > set(set(A)) ) ).
tff(sy_c_Finite__Set_Ocard,type,
finite_card:
!>[B: $tType] : fun(set(B),nat) ).
tff(sy_c_Finite__Set_Ocomp__fun__commute,type,
finite6289374366891150609ommute:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,B)) > $o ) ).
tff(sy_c_Finite__Set_Ocomp__fun__commute__on,type,
finite4664212375090638736ute_on:
!>[A: $tType,B: $tType] : ( ( set(A) * fun(A,fun(B,B)) ) > $o ) ).
tff(sy_c_Finite__Set_Ocomp__fun__idem,type,
finite_comp_fun_idem:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,B)) > $o ) ).
tff(sy_c_Finite__Set_Ocomp__fun__idem__axioms,type,
finite1981172056906289455axioms:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,B)) > $o ) ).
tff(sy_c_Finite__Set_Ocomp__fun__idem__on,type,
finite673082921795544331dem_on:
!>[A: $tType,B: $tType] : ( ( set(A) * fun(A,fun(B,B)) ) > $o ) ).
tff(sy_c_Finite__Set_Ocomp__fun__idem__on__axioms,type,
finite4980608107308702382axioms:
!>[A: $tType,B: $tType] : ( ( set(A) * fun(A,fun(B,B)) ) > $o ) ).
tff(sy_c_Finite__Set_Ofinite,type,
finite_finite2:
!>[A: $tType] : ( set(A) > $o ) ).
tff(sy_c_Finite__Set_Ofold,type,
finite_fold:
!>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * B * set(A) ) > B ) ).
tff(sy_c_Finite__Set_Ofold__graph,type,
finite_fold_graph:
!>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * B * set(A) ) > fun(B,bool) ) ).
tff(sy_c_Finite__Set_Ofolding,type,
finite_folding:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,B)) > $o ) ).
tff(sy_c_Finite__Set_Ofolding__idem,type,
finite_folding_idem:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,B)) > $o ) ).
tff(sy_c_Finite__Set_Ofolding__idem__axioms,type,
finite7837460588564673216axioms:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,B)) > $o ) ).
tff(sy_c_Finite__Set_Ofolding__idem__on,type,
finite1890593828518410140dem_on:
!>[A: $tType,B: $tType] : ( ( set(A) * fun(A,fun(B,B)) ) > $o ) ).
tff(sy_c_Finite__Set_Ofolding__idem__on__axioms,type,
finite6916993218817215295axioms:
!>[A: $tType,B: $tType] : ( ( set(A) * fun(A,fun(B,B)) ) > $o ) ).
tff(sy_c_Finite__Set_Ofolding__on,type,
finite_folding_on:
!>[A: $tType,B: $tType] : ( ( set(A) * fun(A,fun(B,B)) ) > $o ) ).
tff(sy_c_Fun_Obij__betw,type,
bij_betw:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * set(A) * set(B) ) > $o ) ).
tff(sy_c_Fun_Ocomp,type,
comp:
!>[B: $tType,C: $tType,A: $tType] : fun(fun(B,C),fun(fun(A,B),fun(A,C))) ).
tff(sy_c_Fun_Ofcomp,type,
fcomp:
!>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,B) * fun(B,C) ) > fun(A,C) ) ).
tff(sy_c_Fun_Ofun__upd,type,
fun_upd:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * A * B ) > fun(A,B) ) ).
tff(sy_c_Fun_Oid,type,
id:
!>[A: $tType] : fun(A,A) ).
tff(sy_c_Fun_Oinj__on,type,
inj_on:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * set(A) ) > $o ) ).
tff(sy_c_Fun_Omap__fun,type,
map_fun:
!>[C: $tType,A: $tType,B: $tType,D: $tType] : ( ( fun(C,A) * fun(B,D) ) > fun(fun(A,B),fun(C,D)) ) ).
tff(sy_c_Fun_Ostrict__mono__on,type,
strict_mono_on:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * set(A) ) > $o ) ).
tff(sy_c_Fun_Othe__inv__into,type,
the_inv_into:
!>[A: $tType,B: $tType] : ( ( set(A) * fun(A,B) ) > fun(B,A) ) ).
tff(sy_c_Fun__Def_OTHE__default,type,
fun_THE_default:
!>[A: $tType] : ( ( A * fun(A,bool) ) > A ) ).
tff(sy_c_Fun__Def_Oin__rel,type,
fun_in_rel:
!>[A: $tType,B: $tType] : ( set(product_prod(A,B)) > fun(A,fun(B,bool)) ) ).
tff(sy_c_Fun__Def_Ois__measure,type,
fun_is_measure:
!>[A: $tType] : ( fun(A,nat) > $o ) ).
tff(sy_c_Fun__Def_Omax__strict,type,
fun_max_strict: set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))) ).
tff(sy_c_Fun__Def_Omax__weak,type,
fun_max_weak: set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))) ).
tff(sy_c_Fun__Def_Omin__strict,type,
fun_min_strict: set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))) ).
tff(sy_c_Fun__Def_Omin__weak,type,
fun_min_weak: set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))) ).
tff(sy_c_Fun__Def_Opair__leq,type,
fun_pair_leq: set(product_prod(product_prod(nat,nat),product_prod(nat,nat))) ).
tff(sy_c_Fun__Def_Opair__less,type,
fun_pair_less: set(product_prod(product_prod(nat,nat),product_prod(nat,nat))) ).
tff(sy_c_Fun__Def_Oreduction__pair,type,
fun_reduction_pair:
!>[A: $tType] : ( product_prod(set(product_prod(A,A)),set(product_prod(A,A))) > $o ) ).
tff(sy_c_Fun__Def_Orp__inv__image,type,
fun_rp_inv_image:
!>[A: $tType,B: $tType] : fun(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),fun(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B))))) ).
tff(sy_c_GCD_OGcd__class_OGcd,type,
gcd_Gcd:
!>[A: $tType] : ( set(A) > A ) ).
tff(sy_c_GCD_OGcd__class_OLcm,type,
gcd_Lcm:
!>[A: $tType] : ( set(A) > A ) ).
tff(sy_c_GCD_Osemiring__gcd__class_OGcd__fin,type,
semiring_gcd_Gcd_fin:
!>[A: $tType] : ( set(A) > A ) ).
tff(sy_c_GCD_Osemiring__gcd__class_OLcm__fin,type,
semiring_gcd_Lcm_fin:
!>[A: $tType] : ( set(A) > A ) ).
tff(sy_c_Groups_Oabs__class_Oabs,type,
abs_abs:
!>[A: $tType] : fun(A,A) ).
tff(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : fun(A,fun(A,A)) ).
tff(sy_c_Groups_Oone__class_Oone,type,
one_one:
!>[A: $tType] : A ).
tff(sy_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : fun(A,fun(A,A)) ).
tff(sy_c_Groups_Osgn__class_Osgn,type,
sgn_sgn:
!>[A: $tType] : fun(A,A) ).
tff(sy_c_Groups_Otimes__class_Otimes,type,
times_times:
!>[A: $tType] : fun(A,fun(A,A)) ).
tff(sy_c_Groups_Ouminus__class_Ouminus,type,
uminus_uminus:
!>[A: $tType] : fun(A,A) ).
tff(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
tff(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum,type,
groups7311177749621191930dd_sum:
!>[B: $tType,A: $tType] : fun(fun(B,A),fun(set(B),A)) ).
tff(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod,type,
groups7121269368397514597t_prod:
!>[B: $tType,A: $tType] : fun(fun(B,A),fun(set(B),A)) ).
tff(sy_c_Groups__Big_Ocomm__monoid__set,type,
groups778175481326437816id_set:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * A ) > $o ) ).
tff(sy_c_Groups__Big_Ocomm__monoid__set_OF,type,
groups_comm_monoid_F:
!>[A: $tType,B: $tType] : ( ( fun(A,fun(A,A)) * A ) > fun(fun(B,A),fun(set(B),A)) ) ).
tff(sy_c_Groups__List_Omonoid__add__class_Osum__list,type,
groups8242544230860333062m_list:
!>[A: $tType] : ( list(A) > A ) ).
tff(sy_c_HOL_ONO__MATCH,type,
nO_MATCH:
!>[A: $tType,B: $tType] : ( ( A * B ) > $o ) ).
tff(sy_c_HOL_OThe,type,
the:
!>[A: $tType] : ( fun(A,bool) > A ) ).
tff(sy_c_HOL_Odefault__class_Odefault,type,
default_default:
!>[A: $tType] : A ).
tff(sy_c_HOL_Oundefined,type,
undefined:
!>[A: $tType] : A ).
tff(sy_c_Heap_Oaddr__of__array,type,
addr_of_array:
!>[A: $tType] : ( array(A) > nat ) ).
tff(sy_c_Heap_Oaddr__of__ref,type,
addr_of_ref:
!>[A: $tType] : fun(ref(A),nat) ).
tff(sy_c_Heap_Oarray_OArray,type,
array2:
!>[A: $tType] : ( nat > array(A) ) ).
tff(sy_c_Heap_Oarray_Omap__array,type,
map_array:
!>[A: $tType,Aa: $tType] : ( fun(A,Aa) > fun(array(A),array(Aa)) ) ).
tff(sy_c_Heap_Oarray_Opred__array,type,
pred_array:
!>[A: $tType] : ( ( fun(A,bool) * array(A) ) > $o ) ).
tff(sy_c_Heap_Oarray_Orec__array,type,
rec_array:
!>[C: $tType,A: $tType] : ( fun(nat,C) > fun(array(A),C) ) ).
tff(sy_c_Heap_Oarray_Orel__array,type,
rel_array:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > fun(array(A),fun(array(B),bool)) ) ).
tff(sy_c_Heap_Oarray_Oset__array,type,
set_array:
!>[A: $tType] : ( array(A) > set(A) ) ).
tff(sy_c_Heap_Oarray_Osize__array,type,
size_array:
!>[A: $tType] : ( fun(A,nat) > fun(array(A),nat) ) ).
tff(sy_c_Heap_Oheap_Oarrays,type,
arrays:
!>[Z: $tType] : ( heap_ext(Z) > fun(typerep,fun(nat,list(nat))) ) ).
tff(sy_c_Heap_Oheap_Oarrays__update,type,
arrays_update:
!>[Z: $tType] : ( fun(fun(typerep,fun(nat,list(nat))),fun(typerep,fun(nat,list(nat)))) > fun(heap_ext(Z),heap_ext(Z)) ) ).
tff(sy_c_Heap_Oheap_Oheap__ext,type,
heap_ext2:
!>[Z: $tType] : ( ( fun(typerep,fun(nat,list(nat))) * fun(typerep,fun(nat,nat)) * nat * Z ) > heap_ext(Z) ) ).
tff(sy_c_Heap_Oheap_Oheap__ext_Oheap__ext__Tuple__Iso,type,
heap_h6279449762549114721le_Iso:
!>[Z: $tType] : tuple_isomorphism(heap_ext(Z),product_prod(fun(typerep,fun(nat,list(nat))),product_prod(fun(typerep,fun(nat,nat)),nat)),Z) ).
tff(sy_c_Heap_Oheap_Olim,type,
lim:
!>[Z: $tType] : ( heap_ext(Z) > nat ) ).
tff(sy_c_Heap_Oheap_Olim__update,type,
lim_update:
!>[Z: $tType] : ( ( fun(nat,nat) * heap_ext(Z) ) > heap_ext(Z) ) ).
tff(sy_c_Heap_Oheap_Omore,type,
more:
!>[Z: $tType] : ( heap_ext(Z) > Z ) ).
tff(sy_c_Heap_Oheap_Orefs,type,
refs:
!>[Z: $tType] : ( heap_ext(Z) > fun(typerep,fun(nat,nat)) ) ).
tff(sy_c_Heap_Oheap_Orefs__update,type,
refs_update:
!>[Z: $tType] : ( fun(fun(typerep,fun(nat,nat)),fun(typerep,fun(nat,nat))) > fun(heap_ext(Z),heap_ext(Z)) ) ).
tff(sy_c_Heap_Oref_ORef,type,
ref2:
!>[A: $tType] : ( nat > ref(A) ) ).
tff(sy_c_Heap_Oref_Omap__ref,type,
map_ref:
!>[A: $tType,Aa: $tType] : ( fun(A,Aa) > fun(ref(A),ref(Aa)) ) ).
tff(sy_c_Heap_Oref_Opred__ref,type,
pred_ref:
!>[A: $tType] : ( ( fun(A,bool) * ref(A) ) > $o ) ).
tff(sy_c_Heap_Oref_Orec__ref,type,
rec_ref:
!>[C: $tType,A: $tType] : ( fun(nat,C) > fun(ref(A),C) ) ).
tff(sy_c_Heap_Oref_Orel__ref,type,
rel_ref:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > fun(ref(A),fun(ref(B),bool)) ) ).
tff(sy_c_Heap_Oref_Oset__ref,type,
set_ref:
!>[A: $tType] : ( ref(A) > set(A) ) ).
tff(sy_c_Heap_Oref_Osize__ref,type,
size_ref:
!>[A: $tType] : ( fun(A,nat) > fun(ref(A),nat) ) ).
tff(sy_c_Heap__Monad_OHeap_OHeap,type,
heap_Heap2:
!>[A: $tType] : fun(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_Heap(A)) ).
tff(sy_c_Heap__Monad_OHeap_Ocase__Heap,type,
heap_case_Heap:
!>[A: $tType,B: $tType] : fun(fun(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),B),fun(heap_Heap(A),B)) ).
tff(sy_c_Heap__Monad_OHeap_Omap__Heap,type,
heap_map_Heap:
!>[A: $tType,Aa: $tType] : fun(fun(A,Aa),fun(heap_Heap(A),heap_Heap(Aa))) ).
tff(sy_c_Heap__Monad_OHeap_Opred__Heap,type,
heap_pred_Heap:
!>[A: $tType] : fun(fun(A,bool),fun(heap_Heap(A),bool)) ).
tff(sy_c_Heap__Monad_OHeap_Orec__Heap,type,
heap_rec_Heap:
!>[A: $tType,C: $tType] : fun(fun(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),C),fun(heap_Heap(A),C)) ).
tff(sy_c_Heap__Monad_OHeap_Orel__Heap,type,
heap_rel_Heap:
!>[A: $tType,B: $tType] : fun(fun(A,fun(B,bool)),fun(heap_Heap(A),fun(heap_Heap(B),bool))) ).
tff(sy_c_Heap__Monad_OHeap_Oset__Heap,type,
heap_set_Heap:
!>[A: $tType] : fun(heap_Heap(A),set(A)) ).
tff(sy_c_Heap__Monad_OHeap_Osize__Heap,type,
heap_size_Heap:
!>[A: $tType] : ( fun(A,nat) > fun(heap_Heap(A),nat) ) ).
tff(sy_c_Heap__Monad_OHeap__lub,type,
heap_Heap_lub:
!>[A: $tType] : fun(set(heap_Heap(A)),heap_Heap(A)) ).
tff(sy_c_Heap__Monad_OHeap__ord,type,
heap_Heap_ord:
!>[A: $tType] : fun(heap_Heap(A),fun(heap_Heap(A),bool)) ).
tff(sy_c_Heap__Monad_Oassert,type,
heap_assert:
!>[A: $tType] : ( ( fun(A,bool) * A ) > heap_Heap(A) ) ).
tff(sy_c_Heap__Monad_Obind,type,
heap_bind:
!>[A: $tType,B: $tType] : ( ( heap_Heap(A) * fun(A,heap_Heap(B)) ) > heap_Heap(B) ) ).
tff(sy_c_Heap__Monad_Oeffect,type,
heap_effect:
!>[A: $tType] : ( ( heap_Heap(A) * heap_ext(product_unit) * heap_ext(product_unit) * A ) > $o ) ).
tff(sy_c_Heap__Monad_Oexecute,type,
heap_execute:
!>[A: $tType] : fun(heap_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))))) ).
tff(sy_c_Heap__Monad_Ofold__map,type,
heap_fold_map:
!>[A: $tType,B: $tType] : ( ( fun(A,heap_Heap(B)) * list(A) ) > heap_Heap(list(B)) ) ).
tff(sy_c_Heap__Monad_Oguard,type,
heap_guard:
!>[A: $tType] : ( ( fun(heap_ext(product_unit),bool) * fun(heap_ext(product_unit),product_prod(A,heap_ext(product_unit))) ) > heap_Heap(A) ) ).
tff(sy_c_Heap__Monad_Oheap,type,
heap_heap:
!>[A: $tType] : ( fun(heap_ext(product_unit),product_prod(A,heap_ext(product_unit))) > heap_Heap(A) ) ).
tff(sy_c_Heap__Monad_Olift,type,
heap_lift:
!>[A: $tType,B: $tType] : ( fun(A,B) > fun(A,heap_Heap(B)) ) ).
tff(sy_c_Heap__Monad_Oraise,type,
heap_raise:
!>[A: $tType] : ( literal > heap_Heap(A) ) ).
tff(sy_c_Heap__Monad_Oreturn,type,
heap_return:
!>[A: $tType] : fun(A,heap_Heap(A)) ).
tff(sy_c_Heap__Monad_Osuccess,type,
heap_success:
!>[A: $tType] : ( ( heap_Heap(A) * heap_ext(product_unit) ) > $o ) ).
tff(sy_c_Heap__Monad_Otap,type,
heap_tap:
!>[A: $tType] : ( fun(heap_ext(product_unit),A) > heap_Heap(A) ) ).
tff(sy_c_Hilbert__Choice_Obijection,type,
hilbert_bijection:
!>[A: $tType] : ( fun(A,A) > $o ) ).
tff(sy_c_Hilbert__Choice_Oinv__into,type,
hilbert_inv_into:
!>[A: $tType,B: $tType] : ( ( set(A) * fun(A,B) ) > fun(B,A) ) ).
tff(sy_c_If,type,
if:
!>[A: $tType] : ( ( bool * A * A ) > A ) ).
tff(sy_c_Inductive_Ocomplete__lattice__class_Ogfp,type,
complete_lattice_gfp:
!>[A: $tType] : ( fun(A,A) > A ) ).
tff(sy_c_Inductive_Ocomplete__lattice__class_Olfp,type,
complete_lattice_lfp:
!>[A: $tType] : ( fun(A,A) > A ) ).
tff(sy_c_Int_OAbs__Integ,type,
abs_Integ: fun(product_prod(nat,nat),int) ).
tff(sy_c_Int_ORep__Integ,type,
rep_Integ: fun(int,product_prod(nat,nat)) ).
tff(sy_c_Int_Ocr__int,type,
cr_int: fun(product_prod(nat,nat),fun(int,bool)) ).
tff(sy_c_Int_Oint__ge__less__than,type,
int_ge_less_than: int > set(product_prod(int,int)) ).
tff(sy_c_Int_Oint__ge__less__than2,type,
int_ge_less_than2: int > set(product_prod(int,int)) ).
tff(sy_c_Int_Ointrel,type,
intrel: fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)) ).
tff(sy_c_Int_Onat,type,
nat2: fun(int,nat) ).
tff(sy_c_Int_Opcr__int,type,
pcr_int: fun(product_prod(nat,nat),fun(int,bool)) ).
tff(sy_c_Int_Oring__1__class_OInts,type,
ring_1_Ints:
!>[A: $tType] : set(A) ).
tff(sy_c_Int_Oring__1__class_Oof__int,type,
ring_1_of_int:
!>[A: $tType] : fun(int,A) ).
tff(sy_c_Lattices_Oinf__class_Oinf,type,
inf_inf:
!>[A: $tType] : fun(A,fun(A,A)) ).
tff(sy_c_Lattices_Osemilattice,type,
semilattice:
!>[A: $tType] : ( fun(A,fun(A,A)) > $o ) ).
tff(sy_c_Lattices_Osup__class_Osup,type,
sup_sup:
!>[A: $tType] : fun(A,fun(A,A)) ).
tff(sy_c_Lattices__Big_Olinorder__class_OMax,type,
lattic643756798349783984er_Max:
!>[A: $tType] : ( set(A) > A ) ).
tff(sy_c_Lattices__Big_Olinorder__class_OMin,type,
lattic643756798350308766er_Min:
!>[A: $tType] : ( set(A) > A ) ).
tff(sy_c_Lattices__Big_Oord__class_Oarg__min__on,type,
lattic7623131987881927897min_on:
!>[B: $tType,A: $tType] : ( ( fun(B,A) * set(B) ) > B ) ).
tff(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin,type,
lattic7752659483105999362nf_fin:
!>[A: $tType] : ( set(A) > A ) ).
tff(sy_c_Lattices__Big_Osemilattice__set,type,
lattic149705377957585745ce_set:
!>[A: $tType] : ( fun(A,fun(A,A)) > $o ) ).
tff(sy_c_Lattices__Big_Osemilattice__set_OF,type,
lattic1715443433743089157tice_F:
!>[A: $tType] : ( ( fun(A,fun(A,A)) * set(A) ) > A ) ).
tff(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin,type,
lattic5882676163264333800up_fin:
!>[A: $tType] : ( set(A) > A ) ).
tff(sy_c_Lifting_ONEG,type,
neg:
!>[A: $tType,B: $tType] : ( ( fun(A,fun(B,bool)) * fun(A,fun(B,bool)) ) > $o ) ).
tff(sy_c_Lifting_OPOS,type,
pos:
!>[A: $tType,B: $tType] : ( ( fun(A,fun(B,bool)) * fun(A,fun(B,bool)) ) > $o ) ).
tff(sy_c_Lifting_OQuotient,type,
quotient:
!>[A: $tType,B: $tType] : ( ( fun(A,fun(A,bool)) * fun(A,B) * fun(B,A) * fun(A,fun(B,bool)) ) > $o ) ).
tff(sy_c_List_Oall__interval__nat,type,
all_interval_nat: ( fun(nat,bool) * nat * nat ) > $o ).
tff(sy_c_List_Oappend,type,
append:
!>[A: $tType] : ( ( list(A) * list(A) ) > list(A) ) ).
tff(sy_c_List_Oarg__min__list,type,
arg_min_list:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * list(A) ) > A ) ).
tff(sy_c_List_Obind,type,
bind:
!>[A: $tType,B: $tType] : ( ( list(A) * fun(A,list(B)) ) > list(B) ) ).
tff(sy_c_List_Oconcat,type,
concat:
!>[A: $tType] : ( list(list(A)) > list(A) ) ).
tff(sy_c_List_Odistinct,type,
distinct:
!>[A: $tType] : ( list(A) > $o ) ).
tff(sy_c_List_Odrop,type,
drop:
!>[A: $tType] : ( ( nat * list(A) ) > list(A) ) ).
tff(sy_c_List_OdropWhile,type,
dropWhile:
!>[A: $tType] : ( ( fun(A,bool) * list(A) ) > list(A) ) ).
tff(sy_c_List_Oextract,type,
extract:
!>[A: $tType] : ( ( fun(A,bool) * list(A) ) > option(product_prod(list(A),product_prod(A,list(A)))) ) ).
tff(sy_c_List_Ofilter,type,
filter2:
!>[A: $tType] : ( fun(A,bool) > fun(list(A),list(A)) ) ).
tff(sy_c_List_Ofind,type,
find:
!>[A: $tType] : ( ( fun(A,bool) * list(A) ) > option(A) ) ).
tff(sy_c_List_Ofold,type,
fold:
!>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * list(A) ) > fun(B,B) ) ).
tff(sy_c_List_Ofolding__insort__key,type,
folding_insort_key:
!>[A: $tType,B: $tType] : ( ( fun(A,fun(A,bool)) * fun(A,fun(A,bool)) * set(B) * fun(B,A) ) > $o ) ).
tff(sy_c_List_Ofoldl,type,
foldl:
!>[B: $tType,A: $tType] : ( ( fun(B,fun(A,B)) * B * list(A) ) > B ) ).
tff(sy_c_List_Ofoldr,type,
foldr:
!>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * list(A) ) > fun(B,B) ) ).
tff(sy_c_List_Olast,type,
last:
!>[A: $tType] : ( list(A) > A ) ).
tff(sy_c_List_Olenlex,type,
lenlex:
!>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(list(A),list(A))) ) ).
tff(sy_c_List_Olex,type,
lex:
!>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(list(A),list(A))) ) ).
tff(sy_c_List_Olexn,type,
lexn:
!>[A: $tType] : ( set(product_prod(A,A)) > fun(nat,set(product_prod(list(A),list(A)))) ) ).
tff(sy_c_List_Olexord,type,
lexord:
!>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(list(A),list(A))) ) ).
tff(sy_c_List_Olexordp,type,
lexordp:
!>[A: $tType] : ( ( fun(A,fun(A,bool)) * list(A) * list(A) ) > $o ) ).
tff(sy_c_List_Olinorder_Oinsort__key,type,
insort_key:
!>[A: $tType,B: $tType] : ( ( fun(A,fun(A,bool)) * fun(B,A) * B ) > fun(list(B),list(B)) ) ).
tff(sy_c_List_Olinorder__class_Oinsort__insert__key,type,
linord329482645794927042rt_key:
!>[B: $tType,A: $tType] : ( ( fun(B,A) * B * list(B) ) > list(B) ) ).
tff(sy_c_List_Olinorder__class_Oinsort__key,type,
linorder_insort_key:
!>[B: $tType,A: $tType] : ( ( fun(B,A) * B ) > fun(list(B),list(B)) ) ).
tff(sy_c_List_Olist_OCons,type,
cons:
!>[A: $tType] : fun(A,fun(list(A),list(A))) ).
tff(sy_c_List_Olist_ONil,type,
nil:
!>[A: $tType] : list(A) ).
tff(sy_c_List_Olist_Ocase__list,type,
case_list:
!>[B: $tType,A: $tType] : ( ( B * fun(A,fun(list(A),B)) * list(A) ) > B ) ).
tff(sy_c_List_Olist_Ohd,type,
hd:
!>[A: $tType] : ( list(A) > A ) ).
tff(sy_c_List_Olist_Olist__all,type,
list_all:
!>[A: $tType] : ( ( fun(A,bool) * list(A) ) > $o ) ).
tff(sy_c_List_Olist_Olist__all2,type,
list_all2:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > fun(list(A),fun(list(B),bool)) ) ).
tff(sy_c_List_Olist_Omap,type,
map:
!>[A: $tType,Aa: $tType] : ( fun(A,Aa) > fun(list(A),list(Aa)) ) ).
tff(sy_c_List_Olist_Orec__list,type,
rec_list:
!>[C: $tType,A: $tType] : ( ( C * fun(A,fun(list(A),fun(C,C))) ) > fun(list(A),C) ) ).
tff(sy_c_List_Olist_Oset,type,
set3:
!>[A: $tType] : fun(list(A),set(A)) ).
tff(sy_c_List_Olist_Osize__list,type,
size_list:
!>[A: $tType] : ( fun(A,nat) > fun(list(A),nat) ) ).
tff(sy_c_List_Olist__ex,type,
list_ex:
!>[A: $tType] : ( ( fun(A,bool) * list(A) ) > $o ) ).
tff(sy_c_List_Olist__update,type,
list_update:
!>[A: $tType] : ( ( list(A) * nat * A ) > list(A) ) ).
tff(sy_c_List_Olistrel,type,
listrel:
!>[A: $tType,B: $tType] : ( set(product_prod(A,B)) > set(product_prod(list(A),list(B))) ) ).
tff(sy_c_List_Olistrel1,type,
listrel1:
!>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(list(A),list(A))) ) ).
tff(sy_c_List_Olistrel1p,type,
listrel1p:
!>[A: $tType] : ( ( fun(A,fun(A,bool)) * list(A) * list(A) ) > $o ) ).
tff(sy_c_List_Olistrelp,type,
listrelp:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > fun(list(A),fun(list(B),bool)) ) ).
tff(sy_c_List_Olists,type,
lists:
!>[A: $tType] : ( set(A) > set(list(A)) ) ).
tff(sy_c_List_Omap__filter,type,
map_filter:
!>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * list(A) ) > list(B) ) ).
tff(sy_c_List_Omeasures,type,
measures:
!>[A: $tType] : ( list(fun(A,nat)) > set(product_prod(A,A)) ) ).
tff(sy_c_List_Onth,type,
nth3:
!>[A: $tType] : ( list(A) > fun(nat,A) ) ).
tff(sy_c_List_Onths,type,
nths:
!>[A: $tType] : ( ( list(A) * set(nat) ) > list(A) ) ).
tff(sy_c_List_Oord__class_Olexordp,type,
ord_lexordp:
!>[A: $tType] : ( ( list(A) * list(A) ) > $o ) ).
tff(sy_c_List_Opartition,type,
partition:
!>[A: $tType] : ( ( fun(A,bool) * list(A) ) > product_prod(list(A),list(A)) ) ).
tff(sy_c_List_Oproduct,type,
product:
!>[A: $tType,B: $tType] : ( ( list(A) * list(B) ) > list(product_prod(A,B)) ) ).
tff(sy_c_List_Oremdups,type,
remdups:
!>[A: $tType] : ( list(A) > list(A) ) ).
tff(sy_c_List_Oremdups__adj,type,
remdups_adj:
!>[A: $tType] : ( list(A) > list(A) ) ).
tff(sy_c_List_Oremove1,type,
remove1:
!>[A: $tType] : ( ( A * list(A) ) > list(A) ) ).
tff(sy_c_List_Oreplicate,type,
replicate:
!>[A: $tType] : ( ( nat * A ) > list(A) ) ).
tff(sy_c_List_Orev,type,
rev:
!>[A: $tType] : ( list(A) > list(A) ) ).
tff(sy_c_List_Orotate,type,
rotate:
!>[A: $tType] : ( nat > fun(list(A),list(A)) ) ).
tff(sy_c_List_Oset__Cons,type,
set_Cons:
!>[A: $tType] : ( ( set(A) * set(list(A)) ) > set(list(A)) ) ).
tff(sy_c_List_Oshuffles,type,
shuffles:
!>[A: $tType] : ( ( list(A) * list(A) ) > set(list(A)) ) ).
tff(sy_c_List_Oshuffles__rel,type,
shuffles_rel:
!>[A: $tType] : fun(product_prod(list(A),list(A)),fun(product_prod(list(A),list(A)),bool)) ).
tff(sy_c_List_Osubseqs,type,
subseqs:
!>[A: $tType] : ( list(A) > list(list(A)) ) ).
tff(sy_c_List_Otake,type,
take:
!>[A: $tType] : ( ( nat * list(A) ) > list(A) ) ).
tff(sy_c_List_OtakeWhile,type,
takeWhile:
!>[A: $tType] : ( ( fun(A,bool) * list(A) ) > list(A) ) ).
tff(sy_c_List_Oupt,type,
upt: ( nat * nat ) > list(nat) ).
tff(sy_c_List_Ozip,type,
zip:
!>[A: $tType,B: $tType] : ( ( list(A) * list(B) ) > list(product_prod(A,B)) ) ).
tff(sy_c_Map_Odom,type,
dom:
!>[A: $tType,B: $tType] : ( fun(A,option(B)) > set(A) ) ).
tff(sy_c_Map_Ograph,type,
graph:
!>[A: $tType,B: $tType] : ( fun(A,option(B)) > set(product_prod(A,B)) ) ).
tff(sy_c_Map_Omap__add,type,
map_add:
!>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * fun(A,option(B)) ) > fun(A,option(B)) ) ).
tff(sy_c_Map_Omap__of,type,
map_of:
!>[A: $tType,B: $tType] : ( list(product_prod(A,B)) > fun(A,option(B)) ) ).
tff(sy_c_Map_Omap__upds,type,
map_upds:
!>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * list(A) * list(B) ) > fun(A,option(B)) ) ).
tff(sy_c_Map_Oran,type,
ran:
!>[A: $tType,B: $tType] : ( fun(A,option(B)) > set(B) ) ).
tff(sy_c_Map_Orestrict__map,type,
restrict_map:
!>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * set(A) ) > fun(A,option(B)) ) ).
tff(sy_c_Nat_OSuc,type,
suc: fun(nat,nat) ).
tff(sy_c_Nat_Ocompow,type,
compow:
!>[A: $tType] : ( ( nat * A ) > A ) ).
tff(sy_c_Nat_Onat_Ocase__nat,type,
case_nat:
!>[A: $tType] : ( ( A * fun(nat,A) * nat ) > A ) ).
tff(sy_c_Nat_Oold_Onat_Orec__nat,type,
rec_nat:
!>[T: $tType] : ( ( T * fun(nat,fun(T,T)) ) > fun(nat,T) ) ).
tff(sy_c_Nat_Osemiring__1__class_ONats,type,
semiring_1_Nats:
!>[A: $tType] : set(A) ).
tff(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
semiring_1_of_nat:
!>[A: $tType] : fun(nat,A) ).
tff(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : fun(A,nat) ).
tff(sy_c_Nat__Bijection_Oint__decode,type,
nat_int_decode: fun(nat,int) ).
tff(sy_c_Nat__Bijection_Oint__encode,type,
nat_int_encode: fun(int,nat) ).
tff(sy_c_Nat__Bijection_Olist__decode,type,
nat_list_decode: fun(nat,list(nat)) ).
tff(sy_c_Nat__Bijection_Olist__decode__rel,type,
nat_list_decode_rel: fun(nat,fun(nat,bool)) ).
tff(sy_c_Nat__Bijection_Olist__encode,type,
nat_list_encode: fun(list(nat),nat) ).
tff(sy_c_Nat__Bijection_Oprod__decode,type,
nat_prod_decode: fun(nat,product_prod(nat,nat)) ).
tff(sy_c_Nat__Bijection_Oprod__encode,type,
nat_prod_encode: fun(product_prod(nat,nat),nat) ).
tff(sy_c_Nat__Bijection_Osum__decode,type,
nat_sum_decode: fun(nat,sum_sum(nat,nat)) ).
tff(sy_c_Nat__Bijection_Osum__encode,type,
nat_sum_encode: fun(sum_sum(nat,nat),nat) ).
tff(sy_c_Nat__Bijection_Otriangle,type,
nat_triangle: nat > nat ).
tff(sy_c_Num_Onum_OBit0,type,
bit0: num > num ).
tff(sy_c_Num_Onum_OBit1,type,
bit1: fun(num,num) ).
tff(sy_c_Num_Onum_OOne,type,
one: num ).
tff(sy_c_Num_Onum_Ocase__num,type,
case_num:
!>[A: $tType] : ( ( A * fun(num,A) * fun(num,A) * num ) > A ) ).
tff(sy_c_Num_Onum__of__nat,type,
num_of_nat: fun(nat,num) ).
tff(sy_c_Num_Onumeral__class_Onumeral,type,
numeral_numeral:
!>[A: $tType] : fun(num,A) ).
tff(sy_c_Old__Datatype_OAtom,type,
old_Atom:
!>[A: $tType,B: $tType] : fun(sum_sum(A,nat),set(old_node(A,B))) ).
tff(sy_c_Old__Datatype_OIn0,type,
old_In0:
!>[A: $tType,B: $tType] : fun(set(old_node(A,B)),set(old_node(A,B))) ).
tff(sy_c_Old__Datatype_OIn1,type,
old_In1:
!>[A: $tType,B: $tType] : fun(set(old_node(A,B)),set(old_node(A,B))) ).
tff(sy_c_Old__Datatype_OLeaf,type,
old_Leaf:
!>[A: $tType,B: $tType] : fun(A,set(old_node(A,B))) ).
tff(sy_c_Old__Datatype_OLim,type,
old_Lim:
!>[B: $tType,A: $tType] : ( fun(B,set(old_node(A,B))) > set(old_node(A,B)) ) ).
tff(sy_c_Old__Datatype_ONumb,type,
old_Numb:
!>[A: $tType,B: $tType] : fun(nat,set(old_node(A,B))) ).
tff(sy_c_Old__Datatype_OPush__Node,type,
old_Push_Node:
!>[B: $tType,A: $tType] : ( sum_sum(B,nat) > fun(old_node(A,B),old_node(A,B)) ) ).
tff(sy_c_Old__Datatype_OScons,type,
old_Scons:
!>[A: $tType,B: $tType] : ( ( set(old_node(A,B)) * set(old_node(A,B)) ) > set(old_node(A,B)) ) ).
tff(sy_c_Old__Datatype_Odprod,type,
old_dprod:
!>[A: $tType,B: $tType] : ( ( set(product_prod(set(old_node(A,B)),set(old_node(A,B)))) * set(product_prod(set(old_node(A,B)),set(old_node(A,B)))) ) > set(product_prod(set(old_node(A,B)),set(old_node(A,B)))) ) ).
tff(sy_c_Old__Datatype_Odsum,type,
old_dsum:
!>[A: $tType,B: $tType] : ( ( set(product_prod(set(old_node(A,B)),set(old_node(A,B)))) * set(product_prod(set(old_node(A,B)),set(old_node(A,B)))) ) > set(product_prod(set(old_node(A,B)),set(old_node(A,B)))) ) ).
tff(sy_c_Old__Datatype_Ondepth,type,
old_ndepth:
!>[A: $tType,B: $tType] : ( old_node(A,B) > nat ) ).
tff(sy_c_Old__Datatype_Onode_ORep__Node,type,
old_Rep_Node:
!>[A: $tType,B: $tType] : ( old_node(A,B) > product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat)) ) ).
tff(sy_c_Old__Datatype_Ontrunc,type,
old_ntrunc:
!>[A: $tType,B: $tType] : ( nat > fun(set(old_node(A,B)),set(old_node(A,B))) ) ).
tff(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))) ) ).
tff(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))) ) ).
tff(sy_c_Option_Obind,type,
bind2:
!>[A: $tType,B: $tType] : ( ( option(A) * fun(A,option(B)) ) > option(B) ) ).
tff(sy_c_Option_Ooption_ONone,type,
none:
!>[A: $tType] : option(A) ).
tff(sy_c_Option_Ooption_OSome,type,
some:
!>[A: $tType] : fun(A,option(A)) ).
tff(sy_c_Option_Ooption_Ocase__option,type,
case_option:
!>[B: $tType,A: $tType] : ( ( B * fun(A,B) * option(A) ) > B ) ).
tff(sy_c_Option_Ooption_Omap__option,type,
map_option:
!>[A: $tType,Aa: $tType] : ( fun(A,Aa) > fun(option(A),option(Aa)) ) ).
tff(sy_c_Option_Ooption_Opred__option,type,
pred_option:
!>[A: $tType] : ( fun(A,bool) > fun(option(A),bool) ) ).
tff(sy_c_Option_Ooption_Orec__option,type,
rec_option:
!>[C: $tType,A: $tType] : ( ( C * fun(A,C) ) > fun(option(A),C) ) ).
tff(sy_c_Option_Ooption_Orel__option,type,
rel_option:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > fun(option(A),fun(option(B),bool)) ) ).
tff(sy_c_Option_Ooption_Oset__option,type,
set_option:
!>[A: $tType] : fun(option(A),set(A)) ).
tff(sy_c_Option_Ooption_Osize__option,type,
size_option:
!>[A: $tType] : ( fun(A,nat) > fun(option(A),nat) ) ).
tff(sy_c_Option_Ooption_Othe,type,
the2:
!>[A: $tType] : fun(option(A),A) ).
tff(sy_c_Option_Othese,type,
these:
!>[A: $tType] : ( set(option(A)) > set(A) ) ).
tff(sy_c_Order__Relation_OAbove,type,
order_Above:
!>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > set(A) ) ).
tff(sy_c_Order__Relation_OAboveS,type,
order_AboveS:
!>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > set(A) ) ).
tff(sy_c_Order__Relation_OUnder,type,
order_Under:
!>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > set(A) ) ).
tff(sy_c_Order__Relation_OUnderS,type,
order_UnderS:
!>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > set(A) ) ).
tff(sy_c_Order__Relation_Oabove,type,
order_above:
!>[A: $tType] : ( ( set(product_prod(A,A)) * A ) > set(A) ) ).
tff(sy_c_Order__Relation_OaboveS,type,
order_aboveS:
!>[A: $tType] : ( ( set(product_prod(A,A)) * A ) > set(A) ) ).
tff(sy_c_Order__Relation_Olinear__order__on,type,
order_679001287576687338der_on:
!>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).
tff(sy_c_Order__Relation_Oofilter,type,
order_ofilter:
!>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > $o ) ).
tff(sy_c_Order__Relation_Opartial__order__on,type,
order_7125193373082350890der_on:
!>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).
tff(sy_c_Order__Relation_Opreorder__on,type,
order_preorder_on:
!>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).
tff(sy_c_Order__Relation_Orelation__of,type,
order_relation_of:
!>[A: $tType] : ( ( fun(A,fun(A,bool)) * set(A) ) > set(product_prod(A,A)) ) ).
tff(sy_c_Order__Relation_Ounder,type,
order_under:
!>[A: $tType] : ( set(product_prod(A,A)) > fun(A,set(A)) ) ).
tff(sy_c_Order__Relation_OunderS,type,
order_underS:
!>[A: $tType] : ( ( set(product_prod(A,A)) * A ) > set(A) ) ).
tff(sy_c_Order__Relation_Owell__order__on,type,
order_well_order_on:
!>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).
tff(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
tff(sy_c_Orderings_Oord__class_OLeast,type,
ord_Least:
!>[A: $tType] : ( fun(A,bool) > A ) ).
tff(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : fun(A,fun(A,bool)) ).
tff(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : fun(A,fun(A,bool)) ).
tff(sy_c_Orderings_Oord__class_Omax,type,
ord_max:
!>[A: $tType] : ( ( A * A ) > A ) ).
tff(sy_c_Orderings_Oord__class_Omin,type,
ord_min:
!>[A: $tType] : ( ( A * A ) > A ) ).
tff(sy_c_Orderings_Oorder__class_Oantimono,type,
order_antimono:
!>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).
tff(sy_c_Orderings_Oorder__class_Omono,type,
order_mono:
!>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).
tff(sy_c_Orderings_Oorder__class_Ostrict__mono,type,
order_strict_mono:
!>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).
tff(sy_c_Orderings_Otop__class_Otop,type,
top_top:
!>[A: $tType] : A ).
tff(sy_c_Partial__Function_Oflat__lub,type,
partial_flat_lub:
!>[A: $tType] : ( A > fun(set(A),A) ) ).
tff(sy_c_Partial__Function_Oflat__ord,type,
partial_flat_ord:
!>[A: $tType] : ( A > fun(A,fun(A,bool)) ) ).
tff(sy_c_Partial__Function_Ofun__lub,type,
partial_fun_lub:
!>[C: $tType,B: $tType,A: $tType] : ( fun(set(C),B) > fun(set(fun(A,C)),fun(A,B)) ) ).
tff(sy_c_Partial__Function_Ofun__ord,type,
partial_fun_ord:
!>[A: $tType,B: $tType,C: $tType] : ( fun(A,fun(B,bool)) > fun(fun(C,A),fun(fun(C,B),bool)) ) ).
tff(sy_c_Partial__Function_Oimg__lub,type,
partial_img_lub:
!>[A: $tType,C: $tType,D: $tType,B: $tType] : ( ( fun(A,C) * fun(D,B) * fun(set(C),D) ) > fun(set(A),B) ) ).
tff(sy_c_Partial__Function_Oimg__ord,type,
partial_img_ord:
!>[A: $tType,C: $tType,B: $tType] : ( ( fun(A,C) * fun(C,fun(C,B)) ) > fun(A,fun(A,B)) ) ).
tff(sy_c_Partial__Function_Opartial__function__definitions,type,
partia7178651479351089652itions:
!>[A: $tType] : ( ( fun(A,fun(A,bool)) * fun(set(A),A) ) > $o ) ).
tff(sy_c_Predicate_Obind,type,
bind3:
!>[A: $tType,B: $tType] : ( ( pred(A) * fun(A,pred(B)) ) > pred(B) ) ).
tff(sy_c_Predicate_Omap,type,
map2:
!>[A: $tType,B: $tType] : ( fun(A,B) > fun(pred(A),pred(B)) ) ).
tff(sy_c_Predicate_Opred_OPred,type,
pred2:
!>[A: $tType] : fun(fun(A,bool),pred(A)) ).
tff(sy_c_Predicate_Opred_Oeval,type,
eval:
!>[A: $tType] : fun(pred(A),fun(A,bool)) ).
tff(sy_c_Predicate_Opred__of__seq,type,
pred_of_seq:
!>[A: $tType] : fun(seq(A),pred(A)) ).
tff(sy_c_Predicate_Opred__of__set,type,
pred_of_set:
!>[A: $tType] : fun(set(A),pred(A)) ).
tff(sy_c_Predicate_Oset__of__pred,type,
set_of_pred:
!>[A: $tType] : fun(pred(A),set(A)) ).
tff(sy_c_Predicate_Oset__of__seq,type,
set_of_seq:
!>[A: $tType] : fun(seq(A),set(A)) ).
tff(sy_c_Predicate_Osingle,type,
single:
!>[A: $tType] : fun(A,pred(A)) ).
tff(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : fun(A,fun(B,product_prod(A,B))) ).
tff(sy_c_Product__Type_OPair__Rep,type,
product_Pair_Rep:
!>[A: $tType,B: $tType] : ( ( A * B ) > fun(A,fun(B,bool)) ) ).
tff(sy_c_Product__Type_OSigma,type,
product_Sigma:
!>[A: $tType,B: $tType] : ( ( set(A) * fun(A,set(B)) ) > set(product_prod(A,B)) ) ).
tff(sy_c_Product__Type_OUnity,type,
product_Unity: product_unit ).
tff(sy_c_Product__Type_Oapfst,type,
product_apfst:
!>[A: $tType,C: $tType,B: $tType] : ( fun(A,C) > fun(product_prod(A,B),product_prod(C,B)) ) ).
tff(sy_c_Product__Type_Oapsnd,type,
product_apsnd:
!>[B: $tType,C: $tType,A: $tType] : fun(fun(B,C),fun(product_prod(A,B),product_prod(A,C))) ).
tff(sy_c_Product__Type_Ocurry,type,
product_curry:
!>[A: $tType,B: $tType,C: $tType] : fun(fun(product_prod(A,B),C),fun(A,fun(B,C))) ).
tff(sy_c_Product__Type_Ointernal__case__prod,type,
produc5280177257484947105e_prod:
!>[A: $tType,B: $tType,C: $tType] : fun(fun(A,fun(B,C)),fun(product_prod(A,B),C)) ).
tff(sy_c_Product__Type_Omap__prod,type,
product_map_prod:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( fun(A,C) * fun(B,D) ) > fun(product_prod(A,B),product_prod(C,D)) ) ).
tff(sy_c_Product__Type_Oold_Obool_Orec__bool,type,
product_rec_bool:
!>[T: $tType] : ( ( T * T * bool ) > T ) ).
tff(sy_c_Product__Type_Oold_Obool_Orec__set__bool,type,
product_rec_set_bool:
!>[T: $tType] : ( ( T * T * bool ) > fun(T,bool) ) ).
tff(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
product_rec_prod:
!>[A: $tType,B: $tType,T: $tType] : ( ( fun(A,fun(B,T)) * product_prod(A,B) ) > T ) ).
tff(sy_c_Product__Type_Oold_Oprod_Orec__set__prod,type,
product_rec_set_prod:
!>[A: $tType,B: $tType,T: $tType] : ( ( fun(A,fun(B,T)) * product_prod(A,B) ) > fun(T,bool) ) ).
tff(sy_c_Product__Type_Oold_Ounit_Orec__set__unit,type,
product_rec_set_unit:
!>[T: $tType] : ( ( T * product_unit ) > fun(T,bool) ) ).
tff(sy_c_Product__Type_Oold_Ounit_Orec__unit,type,
product_rec_unit:
!>[T: $tType] : ( ( T * product_unit ) > T ) ).
tff(sy_c_Product__Type_Oprod,type,
product_prod2:
!>[A: $tType,B: $tType] : set(fun(A,fun(B,bool))) ).
tff(sy_c_Product__Type_Oprod_OAbs__prod,type,
product_Abs_prod:
!>[A: $tType,B: $tType] : fun(fun(A,fun(B,bool)),product_prod(A,B)) ).
tff(sy_c_Product__Type_Oprod_ORep__prod,type,
product_Rep_prod:
!>[A: $tType,B: $tType] : fun(product_prod(A,B),fun(A,fun(B,bool))) ).
tff(sy_c_Product__Type_Oprod_Ocase__prod,type,
product_case_prod:
!>[A: $tType,B: $tType,C: $tType] : fun(fun(A,fun(B,C)),fun(product_prod(A,B),C)) ).
tff(sy_c_Product__Type_Oprod_Ofst,type,
product_fst:
!>[A: $tType,B: $tType] : fun(product_prod(A,B),A) ).
tff(sy_c_Product__Type_Oprod_Osnd,type,
product_snd:
!>[A: $tType,B: $tType] : fun(product_prod(A,B),B) ).
tff(sy_c_Product__Type_Oprod_Oswap,type,
product_swap:
!>[A: $tType,B: $tType] : fun(product_prod(A,B),product_prod(B,A)) ).
tff(sy_c_Product__Type_Oproduct,type,
product_product:
!>[A: $tType,B: $tType] : ( ( set(A) * set(B) ) > set(product_prod(A,B)) ) ).
tff(sy_c_Product__Type_Oscomp,type,
product_scomp:
!>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( fun(A,product_prod(B,C)) * fun(B,fun(C,D)) ) > fun(A,D) ) ).
tff(sy_c_Product__Type_Ounit_OAbs__unit,type,
product_Abs_unit: fun(bool,product_unit) ).
tff(sy_c_Product__Type_Ounit_ORep__unit,type,
product_Rep_unit: fun(product_unit,bool) ).
tff(sy_c_Product__Type_Ounit_Ocase__unit,type,
product_case_unit:
!>[A: $tType] : ( ( A * product_unit ) > A ) ).
tff(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : itself(A) ).
tff(sy_c_Quotient_OQuotient3,type,
quotient3:
!>[A: $tType,B: $tType] : ( ( fun(A,fun(A,bool)) * fun(A,B) * fun(B,A) ) > $o ) ).
tff(sy_c_Random_Oiterate,type,
iterate:
!>[B: $tType,A: $tType] : ( ( code_natural * fun(B,fun(A,product_prod(B,A))) ) > fun(B,fun(A,product_prod(B,A))) ) ).
tff(sy_c_Random_Oiterate__rel,type,
iterate_rel:
!>[B: $tType,A: $tType] : fun(product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),fun(product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),bool)) ).
tff(sy_c_Random_Opick,type,
pick:
!>[A: $tType] : ( list(product_prod(code_natural,A)) > fun(code_natural,A) ) ).
tff(sy_c_Random_Orange,type,
range: code_natural > fun(product_prod(code_natural,code_natural),product_prod(code_natural,product_prod(code_natural,code_natural))) ).
tff(sy_c_Random_Oselect,type,
select:
!>[A: $tType] : ( list(A) > fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural))) ) ).
tff(sy_c_Random_Oselect__weight,type,
select_weight:
!>[A: $tType] : ( list(product_prod(code_natural,A)) > fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural))) ) ).
tff(sy_c_Random_Osplit__seed,type,
split_seed: product_prod(code_natural,code_natural) > product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)) ).
tff(sy_c_Random__Pred_ORandom,type,
random_Random:
!>[A: $tType] : ( fun(product_prod(code_natural,code_natural),product_prod(product_prod(A,fun(product_unit,code_term)),product_prod(code_natural,code_natural))) > fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))) ) ).
tff(sy_c_Random__Pred_Obind,type,
random_bind:
!>[A: $tType,B: $tType] : ( ( fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))) * fun(A,fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)))) ) > fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural))) ) ).
tff(sy_c_Random__Pred_Omap,type,
random_map:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))) ) > fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural))) ) ).
tff(sy_c_Random__Pred_Onot__randompred,type,
random6974930770145893639ompred: ( fun(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)) ).
tff(sy_c_Random__Pred_Osingle,type,
random_single:
!>[A: $tType] : fun(A,fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))) ).
tff(sy_c_Random__Pred_Ounion,type,
random_union:
!>[A: $tType] : ( ( fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))) * fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))) * product_prod(code_natural,code_natural) ) > product_prod(pred(A),product_prod(code_natural,code_natural)) ) ).
tff(sy_c_Rat_OFract,type,
fract: ( int * int ) > rat ).
tff(sy_c_Rat_Ofield__char__0__class_ORats,type,
field_char_0_Rats:
!>[A: $tType] : set(A) ).
tff(sy_c_Rat_Ofield__char__0__class_Oof__rat,type,
field_char_0_of_rat:
!>[A: $tType] : fun(rat,A) ).
tff(sy_c_Rat_Onormalize,type,
normalize: product_prod(int,int) > product_prod(int,int) ).
tff(sy_c_Rat_Oquotient__of,type,
quotient_of: rat > product_prod(int,int) ).
tff(sy_c_Record_Oabst,type,
abst:
!>[A: $tType,B: $tType,C: $tType] : ( tuple_isomorphism(A,B,C) > fun(product_prod(B,C),A) ) ).
tff(sy_c_Record_Oiso__tuple__cons,type,
iso_tuple_cons:
!>[A: $tType,B: $tType,C: $tType] : ( tuple_isomorphism(A,B,C) > fun(B,fun(C,A)) ) ).
tff(sy_c_Record_Oiso__tuple__fst,type,
iso_tuple_fst:
!>[A: $tType,B: $tType,C: $tType] : ( tuple_isomorphism(A,B,C) > fun(A,B) ) ).
tff(sy_c_Record_Oiso__tuple__fst__update,type,
iso_tuple_fst_update:
!>[A: $tType,B: $tType,C: $tType] : ( tuple_isomorphism(A,B,C) > fun(fun(B,B),fun(A,A)) ) ).
tff(sy_c_Record_Oiso__tuple__snd,type,
iso_tuple_snd:
!>[A: $tType,B: $tType,C: $tType] : ( tuple_isomorphism(A,B,C) > fun(A,C) ) ).
tff(sy_c_Record_Oiso__tuple__snd__update,type,
iso_tuple_snd_update:
!>[A: $tType,B: $tType,C: $tType] : ( tuple_isomorphism(A,B,C) > fun(fun(C,C),fun(A,A)) ) ).
tff(sy_c_Record_Oiso__tuple__surjective__proof__assist,type,
iso_tu562120763810524177assist:
!>[A: $tType,B: $tType] : ( ( A * B * fun(A,B) ) > $o ) ).
tff(sy_c_Record_Oiso__tuple__update__accessor__cong__assist,type,
iso_tu6617205062836021118assist:
!>[B: $tType,A: $tType] : ( ( fun(fun(B,B),fun(A,A)) * fun(A,B) ) > $o ) ).
tff(sy_c_Record_Oiso__tuple__update__accessor__eq__assist,type,
iso_tu8710522896132340869assist:
!>[B: $tType,A: $tType] : ( ( fun(fun(B,B),fun(A,A)) * fun(A,B) * A * fun(B,B) * A * B ) > $o ) ).
tff(sy_c_Record_Oisomorphic__tuple,type,
isomorphic_tuple:
!>[A: $tType,B: $tType,C: $tType] : ( tuple_isomorphism(A,B,C) > $o ) ).
tff(sy_c_Record_Orepr,type,
repr:
!>[A: $tType,B: $tType,C: $tType] : ( tuple_isomorphism(A,B,C) > fun(A,product_prod(B,C)) ) ).
tff(sy_c_Record_Otuple__iso__tuple,type,
tuple_iso_tuple:
!>[A: $tType,B: $tType] : tuple_isomorphism(product_prod(A,B),A,B) ).
tff(sy_c_Ref_Oalloc,type,
alloc2:
!>[A: $tType] : ( A > fun(heap_ext(product_unit),product_prod(ref(A),heap_ext(product_unit))) ) ).
tff(sy_c_Ref_Ochange,type,
change:
!>[A: $tType] : ( ( fun(A,A) * ref(A) ) > heap_Heap(A) ) ).
tff(sy_c_Ref_Oget,type,
get2:
!>[A: $tType] : ( heap_ext(product_unit) > fun(ref(A),A) ) ).
tff(sy_c_Ref_Olookup,type,
lookup:
!>[A: $tType] : ( ref(A) > heap_Heap(A) ) ).
tff(sy_c_Ref_Onoteq,type,
noteq2:
!>[A: $tType,B: $tType] : ( ( ref(A) * ref(B) ) > $o ) ).
tff(sy_c_Ref_Opresent,type,
present2:
!>[A: $tType] : ( heap_ext(product_unit) > fun(ref(A),bool) ) ).
tff(sy_c_Ref_Oref,type,
ref3:
!>[A: $tType] : fun(A,heap_Heap(ref(A))) ).
tff(sy_c_Ref_Oref_H,type,
ref4:
!>[A: $tType] : fun(A,heap_Heap(ref(A))) ).
tff(sy_c_Ref_Oset,type,
set4:
!>[A: $tType] : ( ( ref(A) * A ) > fun(heap_ext(product_unit),heap_ext(product_unit)) ) ).
tff(sy_c_Ref_Oupdate,type,
update2:
!>[A: $tType] : ( ( ref(A) * A ) > heap_Heap(product_unit) ) ).
tff(sy_c_Relation_ODomain,type,
domain:
!>[A: $tType,B: $tType] : fun(set(product_prod(A,B)),set(A)) ).
tff(sy_c_Relation_ODomainp,type,
domainp:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > fun(A,bool) ) ).
tff(sy_c_Relation_OField,type,
field2:
!>[A: $tType] : fun(set(product_prod(A,A)),set(A)) ).
tff(sy_c_Relation_OId,type,
id2:
!>[A: $tType] : set(product_prod(A,A)) ).
tff(sy_c_Relation_OId__on,type,
id_on:
!>[A: $tType] : ( set(A) > set(product_prod(A,A)) ) ).
tff(sy_c_Relation_OImage,type,
image:
!>[A: $tType,B: $tType] : ( set(product_prod(A,B)) > fun(set(A),set(B)) ) ).
tff(sy_c_Relation_ORange,type,
range2:
!>[A: $tType,B: $tType] : fun(set(product_prod(A,B)),set(B)) ).
tff(sy_c_Relation_ORangep,type,
rangep:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > fun(B,bool) ) ).
tff(sy_c_Relation_Oantisym,type,
antisym:
!>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).
tff(sy_c_Relation_Oantisymp,type,
antisymp:
!>[A: $tType] : ( fun(A,fun(A,bool)) > $o ) ).
tff(sy_c_Relation_Oasym,type,
asym:
!>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).
tff(sy_c_Relation_Oasymp,type,
asymp:
!>[A: $tType] : ( fun(A,fun(A,bool)) > $o ) ).
tff(sy_c_Relation_Oconverse,type,
converse:
!>[A: $tType,B: $tType] : ( set(product_prod(A,B)) > set(product_prod(B,A)) ) ).
tff(sy_c_Relation_Oconversep,type,
conversep:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > fun(B,fun(A,bool)) ) ).
tff(sy_c_Relation_Oinv__image,type,
inv_image:
!>[B: $tType,A: $tType] : ( ( set(product_prod(B,B)) * fun(A,B) ) > set(product_prod(A,A)) ) ).
tff(sy_c_Relation_Oirrefl,type,
irrefl:
!>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).
tff(sy_c_Relation_Oirreflp,type,
irreflp:
!>[A: $tType] : ( fun(A,fun(A,bool)) > $o ) ).
tff(sy_c_Relation_Orefl__on,type,
refl_on:
!>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).
tff(sy_c_Relation_Oreflp,type,
reflp:
!>[A: $tType] : ( fun(A,fun(A,bool)) > $o ) ).
tff(sy_c_Relation_Orelcomp,type,
relcomp:
!>[A: $tType,B: $tType,C: $tType] : ( ( set(product_prod(A,B)) * set(product_prod(B,C)) ) > set(product_prod(A,C)) ) ).
tff(sy_c_Relation_Orelcompp,type,
relcompp:
!>[A: $tType,B: $tType,C: $tType] : fun(fun(A,fun(B,bool)),fun(fun(B,fun(C,bool)),fun(A,fun(C,bool)))) ).
tff(sy_c_Relation_Osingle__valued,type,
single_valued:
!>[A: $tType,B: $tType] : ( set(product_prod(A,B)) > $o ) ).
tff(sy_c_Relation_Osingle__valuedp,type,
single_valuedp:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > $o ) ).
tff(sy_c_Relation_Osym,type,
sym:
!>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).
tff(sy_c_Relation_Osymp,type,
symp:
!>[A: $tType] : ( fun(A,fun(A,bool)) > $o ) ).
tff(sy_c_Relation_Ototal__on,type,
total_on:
!>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).
tff(sy_c_Relation_Otrans,type,
trans:
!>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).
tff(sy_c_Relation_Otransp,type,
transp:
!>[A: $tType] : ( fun(A,fun(A,bool)) > $o ) ).
tff(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime,type,
algebr8660921524188924756oprime:
!>[A: $tType] : ( ( A * A ) > $o ) ).
tff(sy_c_Rings_Odivide__class_Odivide,type,
divide_divide:
!>[A: $tType] : fun(A,fun(A,A)) ).
tff(sy_c_Rings_Odvd__class_Odvd,type,
dvd_dvd:
!>[A: $tType] : ( ( A * A ) > $o ) ).
tff(sy_c_Rings_Omodulo__class_Omodulo,type,
modulo_modulo:
!>[A: $tType] : ( ( A * A ) > A ) ).
tff(sy_c_Rings_Onormalization__semidom__class_Onormalize,type,
normal6383669964737779283malize:
!>[A: $tType] : fun(A,A) ).
tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool,type,
zero_neq_one_of_bool:
!>[A: $tType] : ( bool > A ) ).
tff(sy_c_Set_OBall,type,
ball:
!>[A: $tType] : fun(set(A),fun(fun(A,bool),bool)) ).
tff(sy_c_Set_OBex,type,
bex:
!>[A: $tType] : fun(set(A),fun(fun(A,bool),bool)) ).
tff(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : fun(fun(A,bool),set(A)) ).
tff(sy_c_Set_OPow,type,
pow:
!>[A: $tType] : ( set(A) > set(set(A)) ) ).
tff(sy_c_Set_Obind,type,
bind4:
!>[A: $tType,B: $tType] : ( ( set(A) * fun(A,set(B)) ) > set(B) ) ).
tff(sy_c_Set_Odisjnt,type,
disjnt:
!>[A: $tType] : fun(set(A),fun(set(A),bool)) ).
tff(sy_c_Set_Oimage,type,
image2:
!>[A: $tType,B: $tType] : fun(fun(A,B),fun(set(A),set(B))) ).
tff(sy_c_Set_Oinsert,type,
insert:
!>[A: $tType] : ( A > fun(set(A),set(A)) ) ).
tff(sy_c_Set_Opairwise,type,
pairwise:
!>[A: $tType] : ( ( fun(A,fun(A,bool)) * set(A) ) > $o ) ).
tff(sy_c_Set_Othe__elem,type,
the_elem:
!>[A: $tType] : ( set(A) > A ) ).
tff(sy_c_Set_Ovimage,type,
vimage:
!>[A: $tType,B: $tType] : ( fun(A,B) > fun(set(B),set(A)) ) ).
tff(sy_c_Set__Interval_Oord__class_OatLeast,type,
set_ord_atLeast:
!>[A: $tType] : fun(A,set(A)) ).
tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost,type,
set_or1337092689740270186AtMost:
!>[A: $tType] : ( ( A * A ) > set(A) ) ).
tff(sy_c_Set__Interval_Oord__class_OatLeastLessThan,type,
set_or7035219750837199246ssThan:
!>[A: $tType] : ( ( A * A ) > set(A) ) ).
tff(sy_c_Set__Interval_Oord__class_OatMost,type,
set_ord_atMost:
!>[A: $tType] : fun(A,set(A)) ).
tff(sy_c_Set__Interval_Oord__class_OgreaterThan,type,
set_ord_greaterThan:
!>[A: $tType] : fun(A,set(A)) ).
tff(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost,type,
set_or3652927894154168847AtMost:
!>[A: $tType] : ( ( A * A ) > set(A) ) ).
tff(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan,type,
set_or5935395276787703475ssThan:
!>[A: $tType] : ( ( A * A ) > set(A) ) ).
tff(sy_c_Set__Interval_Oord__class_OlessThan,type,
set_ord_lessThan:
!>[A: $tType] : fun(A,set(A)) ).
tff(sy_c_String_OLiteral,type,
literal2: ( bool * bool * bool * bool * bool * bool * bool * literal ) > literal ).
tff(sy_c_String_Ocomm__semiring__1__class_Oof__char,type,
comm_s6883823935334413003f_char:
!>[A: $tType] : fun(char,A) ).
tff(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of,type,
unique5772411509450598832har_of:
!>[A: $tType] : fun(A,char) ).
tff(sy_c_Sum__Type_OInl,type,
sum_Inl:
!>[A: $tType,B: $tType] : fun(A,sum_sum(A,B)) ).
tff(sy_c_Sum__Type_OInl__Rep,type,
sum_Inl_Rep:
!>[A: $tType,B: $tType] : fun(A,fun(A,fun(B,fun(bool,bool)))) ).
tff(sy_c_Sum__Type_OInr,type,
sum_Inr:
!>[B: $tType,A: $tType] : fun(B,sum_sum(A,B)) ).
tff(sy_c_Sum__Type_OInr__Rep,type,
sum_Inr_Rep:
!>[B: $tType,A: $tType] : fun(B,fun(A,fun(B,fun(bool,bool)))) ).
tff(sy_c_Sum__Type_OPlus,type,
sum_Plus:
!>[A: $tType,B: $tType] : ( ( set(A) * set(B) ) > set(sum_sum(A,B)) ) ).
tff(sy_c_Sum__Type_Omap__sum,type,
sum_map_sum:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( fun(A,C) * fun(B,D) ) > fun(sum_sum(A,B),sum_sum(C,D)) ) ).
tff(sy_c_Sum__Type_Osum_OAbs__sum,type,
sum_Abs_sum:
!>[A: $tType,B: $tType] : fun(fun(A,fun(B,fun(bool,bool))),sum_sum(A,B)) ).
tff(sy_c_Sum__Type_Osum_Ocase__sum,type,
sum_case_sum:
!>[A: $tType,C: $tType,B: $tType] : ( ( fun(A,C) * fun(B,C) ) > fun(sum_sum(A,B),C) ) ).
tff(sy_c_Sum__Type_Osum_Oisl,type,
sum_isl:
!>[A: $tType,B: $tType] : fun(sum_sum(A,B),bool) ).
tff(sy_c_Sum__Type_Osum_Oprojl,type,
sum_projl:
!>[A: $tType,B: $tType] : ( sum_sum(A,B) > A ) ).
tff(sy_c_Sum__Type_Osum_Oprojr,type,
sum_projr:
!>[A: $tType,B: $tType] : ( sum_sum(A,B) > B ) ).
tff(sy_c_Transfer_Obi__total,type,
bi_total:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > $o ) ).
tff(sy_c_Transfer_Obi__unique,type,
bi_unique:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > $o ) ).
tff(sy_c_Transfer_Oleft__total,type,
left_total:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > $o ) ).
tff(sy_c_Transfer_Oleft__unique,type,
left_unique:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > $o ) ).
tff(sy_c_Transfer_Oright__total,type,
right_total:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > $o ) ).
tff(sy_c_Transfer_Oright__unique,type,
right_unique:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > $o ) ).
tff(sy_c_Transitive__Closure_Oacyclic,type,
transitive_acyclic:
!>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).
tff(sy_c_Transitive__Closure_Ontrancl,type,
transitive_ntrancl:
!>[A: $tType] : ( ( nat * set(product_prod(A,A)) ) > set(product_prod(A,A)) ) ).
tff(sy_c_Transitive__Closure_Ortrancl,type,
transitive_rtrancl:
!>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(A,A)) ) ).
tff(sy_c_Transitive__Closure_Ortranclp,type,
transitive_rtranclp:
!>[A: $tType] : ( fun(A,fun(A,bool)) > fun(A,fun(A,bool)) ) ).
tff(sy_c_Transitive__Closure_Otrancl,type,
transitive_trancl:
!>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(A,A)) ) ).
tff(sy_c_Transitive__Closure_Otranclp,type,
transitive_tranclp:
!>[A: $tType] : ( fun(A,fun(A,bool)) > fun(A,fun(A,bool)) ) ).
tff(sy_c_Typedef_Otype__definition,type,
type_definition:
!>[B: $tType,A: $tType] : ( ( fun(B,A) * fun(A,B) * set(A) ) > $o ) ).
tff(sy_c_Typerep_Otyperep_OTyperep,type,
typerep3: ( literal * list(typerep) ) > typerep ).
tff(sy_c_Typerep_Otyperep__class_Otyperep,type,
typerep_typerep:
!>[A: $tType] : ( itself(A) > typerep ) ).
tff(sy_c_Wellfounded_Oacc,type,
acc:
!>[A: $tType] : ( set(product_prod(A,A)) > set(A) ) ).
tff(sy_c_Wellfounded_Oaccp,type,
accp:
!>[A: $tType] : ( fun(A,fun(A,bool)) > fun(A,bool) ) ).
tff(sy_c_Wellfounded_Ofinite__psubset,type,
finite_psubset:
!>[A: $tType] : set(product_prod(set(A),set(A))) ).
tff(sy_c_Wellfounded_Oless__than,type,
less_than: set(product_prod(nat,nat)) ).
tff(sy_c_Wellfounded_Olex__prod,type,
lex_prod:
!>[A: $tType,B: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(B,B)) ) > set(product_prod(product_prod(A,B),product_prod(A,B))) ) ).
tff(sy_c_Wellfounded_Omax__ext,type,
max_ext:
!>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(set(A),set(A))) ) ).
tff(sy_c_Wellfounded_Omax__extp,type,
max_extp:
!>[A: $tType] : ( fun(A,fun(A,bool)) > fun(set(A),fun(set(A),bool)) ) ).
tff(sy_c_Wellfounded_Omeasure,type,
measure:
!>[A: $tType] : ( fun(A,nat) > set(product_prod(A,A)) ) ).
tff(sy_c_Wellfounded_Omin__ext,type,
min_ext:
!>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(set(A),set(A))) ) ).
tff(sy_c_Wellfounded_Omlex__prod,type,
mlex_prod:
!>[A: $tType] : ( ( fun(A,nat) * set(product_prod(A,A)) ) > set(product_prod(A,A)) ) ).
tff(sy_c_Wellfounded_Opred__nat,type,
pred_nat: set(product_prod(nat,nat)) ).
tff(sy_c_Wellfounded_Owf,type,
wf:
!>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).
tff(sy_c_Wellfounded_OwfP,type,
wfP:
!>[A: $tType] : ( fun(A,fun(A,bool)) > $o ) ).
tff(sy_c_Wfrec_Oadm__wf,type,
adm_wf:
!>[A: $tType,B: $tType] : ( ( set(product_prod(A,A)) * fun(fun(A,B),fun(A,B)) ) > $o ) ).
tff(sy_c_Wfrec_Ocut,type,
cut:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * set(product_prod(A,A)) * A ) > fun(A,B) ) ).
tff(sy_c_Wfrec_Osame__fst,type,
same_fst:
!>[A: $tType,B: $tType] : ( ( fun(A,bool) * fun(A,set(product_prod(B,B))) ) > set(product_prod(product_prod(A,B),product_prod(A,B))) ) ).
tff(sy_c_Wfrec_Owfrec__rel,type,
wfrec_rel:
!>[A: $tType,B: $tType] : ( ( set(product_prod(A,A)) * fun(fun(A,B),fun(A,B)) * A * B ) > $o ) ).
tff(sy_c_Zorn_OChains,type,
chains:
!>[A: $tType] : ( set(product_prod(A,A)) > set(set(A)) ) ).
tff(sy_c_Zorn_Oinit__seg__of,type,
init_seg_of:
!>[A: $tType] : set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))) ).
tff(sy_c_Zorn_Opred__on_Ochain,type,
pred_chain:
!>[A: $tType] : ( ( set(A) * fun(A,fun(A,bool)) ) > fun(set(A),bool) ) ).
tff(sy_c_aa,type,
aa:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * A ) > B ) ).
tff(sy_c_fChoice,type,
fChoice:
!>[A: $tType] : ( fun(A,bool) > A ) ).
tff(sy_c_fEx,type,
fEx:
!>[A: $tType] : fun(fun(A,bool),bool) ).
tff(sy_c_fFalse,type,
fFalse: bool ).
tff(sy_c_fNot,type,
fNot: fun(bool,bool) ).
tff(sy_c_fTrue,type,
fTrue: bool ).
tff(sy_c_fconj,type,
fconj: ( bool * bool ) > bool ).
tff(sy_c_fequal,type,
fequal:
!>[A: $tType] : fun(A,fun(A,bool)) ).
tff(sy_c_member,type,
member:
!>[A: $tType] : ( A > fun(set(A),bool) ) ).
tff(sy_c_pp,type,
pp: bool > $o ).
tff(sy_v_h,type,
h: heap_ext(product_unit) ).
tff(sy_v_h_H,type,
h2: heap_ext(product_unit) ).
tff(sy_v_r,type,
r: ref(a) ).
tff(sy_v_thesis,type,
thesis: $o ).
tff(sy_v_x,type,
x: a ).
% Relevant facts (3318)
tff(fact_0_that,axiom,
( ( r = aa(product_prod(ref(a),heap_ext(product_unit)),ref(a),product_fst(ref(a),heap_ext(product_unit)),aa(heap_ext(product_unit),product_prod(ref(a),heap_ext(product_unit)),alloc2(a,x),h)) )
=> ( ( h2 = aa(product_prod(ref(a),heap_ext(product_unit)),heap_ext(product_unit),product_snd(ref(a),heap_ext(product_unit)),aa(heap_ext(product_unit),product_prod(ref(a),heap_ext(product_unit)),alloc2(a,x),h)) )
=> thesis ) ) ).
% that
tff(fact_1_assms,axiom,
heap_effect(ref(a),aa(a,heap_Heap(ref(a)),ref3(a),x),h,h2,r) ).
% assms
tff(fact_2_prod_Oexpand,axiom,
! [B: $tType,A: $tType,Prod: product_prod(A,B),Prod2: product_prod(A,B)] :
( ( ( aa(product_prod(A,B),A,product_fst(A,B),Prod) = aa(product_prod(A,B),A,product_fst(A,B),Prod2) )
& ( aa(product_prod(A,B),B,product_snd(A,B),Prod) = aa(product_prod(A,B),B,product_snd(A,B),Prod2) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
tff(fact_3_prod__eqI,axiom,
! [B: $tType,A: $tType,P: product_prod(A,B),Q: product_prod(A,B)] :
( ( aa(product_prod(A,B),A,product_fst(A,B),P) = aa(product_prod(A,B),A,product_fst(A,B),Q) )
=> ( ( aa(product_prod(A,B),B,product_snd(A,B),P) = aa(product_prod(A,B),B,product_snd(A,B),Q) )
=> ( P = Q ) ) ) ).
% prod_eqI
tff(fact_4_exE__realizer_H,axiom,
! [A: $tType,B: $tType,P2: fun(A,fun(B,bool)),P: product_prod(B,A)] :
( pp(aa(B,bool,aa(A,fun(B,bool),P2,aa(product_prod(B,A),A,product_snd(B,A),P)),aa(product_prod(B,A),B,product_fst(B,A),P)))
=> ~ ! [X: B,Y: A] : ~ pp(aa(B,bool,aa(A,fun(B,bool),P2,Y),X)) ) ).
% exE_realizer'
tff(fact_5_prod__eq__iff,axiom,
! [B: $tType,A: $tType,S: product_prod(A,B),T2: product_prod(A,B)] :
( ( S = T2 )
<=> ( ( aa(product_prod(A,B),A,product_fst(A,B),S) = aa(product_prod(A,B),A,product_fst(A,B),T2) )
& ( aa(product_prod(A,B),B,product_snd(A,B),S) = aa(product_prod(A,B),B,product_snd(A,B),T2) ) ) ) ).
% prod_eq_iff
tff(fact_6_Ref_Oget__alloc,axiom,
! [A: $tType] :
( heap(A)
=> ! [X2: A,H: heap_ext(product_unit),X3: A] : aa(ref(A),A,get2(A,aa(product_prod(ref(A),heap_ext(product_unit)),heap_ext(product_unit),product_snd(ref(A),heap_ext(product_unit)),aa(heap_ext(product_unit),product_prod(ref(A),heap_ext(product_unit)),alloc2(A,X2),H))),aa(product_prod(ref(A),heap_ext(product_unit)),ref(A),product_fst(ref(A),heap_ext(product_unit)),aa(heap_ext(product_unit),product_prod(ref(A),heap_ext(product_unit)),alloc2(A,X3),H))) = X2 ) ).
% Ref.get_alloc
tff(fact_7_set__alloc,axiom,
! [A: $tType] :
( heap(A)
=> ! [V: A,H: heap_ext(product_unit),V2: A] : aa(heap_ext(product_unit),heap_ext(product_unit),set4(A,aa(product_prod(ref(A),heap_ext(product_unit)),ref(A),product_fst(ref(A),heap_ext(product_unit)),aa(heap_ext(product_unit),product_prod(ref(A),heap_ext(product_unit)),alloc2(A,V),H)),V2),aa(product_prod(ref(A),heap_ext(product_unit)),heap_ext(product_unit),product_snd(ref(A),heap_ext(product_unit)),aa(heap_ext(product_unit),product_prod(ref(A),heap_ext(product_unit)),alloc2(A,V),H))) = aa(product_prod(ref(A),heap_ext(product_unit)),heap_ext(product_unit),product_snd(ref(A),heap_ext(product_unit)),aa(heap_ext(product_unit),product_prod(ref(A),heap_ext(product_unit)),alloc2(A,V2),H)) ) ).
% set_alloc
tff(fact_8_Ref_Opresent__alloc,axiom,
! [B: $tType,A: $tType] :
( ( heap(A)
& heap(B) )
=> ! [H: heap_ext(product_unit),R: ref(A),V: B] :
( pp(aa(ref(A),bool,present2(A,H),R))
=> pp(aa(ref(A),bool,present2(A,aa(product_prod(ref(B),heap_ext(product_unit)),heap_ext(product_unit),product_snd(ref(B),heap_ext(product_unit)),aa(heap_ext(product_unit),product_prod(ref(B),heap_ext(product_unit)),alloc2(B,V),H))),R)) ) ) ).
% Ref.present_alloc
tff(fact_9_Ref_Onot__present__alloc,axiom,
! [A: $tType] :
( heap(A)
=> ! [H: heap_ext(product_unit),V: A] : ~ pp(aa(ref(A),bool,present2(A,H),aa(product_prod(ref(A),heap_ext(product_unit)),ref(A),product_fst(ref(A),heap_ext(product_unit)),aa(heap_ext(product_unit),product_prod(ref(A),heap_ext(product_unit)),alloc2(A,V),H)))) ) ).
% Ref.not_present_alloc
tff(fact_10_array__present__alloc,axiom,
! [B: $tType,A: $tType] :
( ( heap(A)
& heap(B) )
=> ! [H: heap_ext(product_unit),A3: array(A),V: B] :
( pp(aa(array(A),bool,present(A,H),A3))
=> pp(aa(array(A),bool,present(A,aa(product_prod(ref(B),heap_ext(product_unit)),heap_ext(product_unit),product_snd(ref(B),heap_ext(product_unit)),aa(heap_ext(product_unit),product_prod(ref(B),heap_ext(product_unit)),alloc2(B,V),H))),A3)) ) ) ).
% array_present_alloc
tff(fact_11_Ref_Olength__alloc,axiom,
! [B: $tType,A: $tType] :
( ( heap(A)
& heap(B) )
=> ! [V: B,H: heap_ext(product_unit),A3: array(A)] : aa(array(A),nat,length(A,aa(product_prod(ref(B),heap_ext(product_unit)),heap_ext(product_unit),product_snd(ref(B),heap_ext(product_unit)),aa(heap_ext(product_unit),product_prod(ref(B),heap_ext(product_unit)),alloc2(B,V),H))),A3) = aa(array(A),nat,length(A,H),A3) ) ).
% Ref.length_alloc
tff(fact_12_array__get__alloc,axiom,
! [B: $tType,A: $tType] :
( ( heap(A)
& heap(B) )
=> ! [V: B,H: heap_ext(product_unit)] : get(A,aa(product_prod(ref(B),heap_ext(product_unit)),heap_ext(product_unit),product_snd(ref(B),heap_ext(product_unit)),aa(heap_ext(product_unit),product_prod(ref(B),heap_ext(product_unit)),alloc2(B,V),H))) = get(A,H) ) ).
% array_get_alloc
tff(fact_13_Ref_Oset__same,axiom,
! [A: $tType] :
( heap(A)
=> ! [R: ref(A),X2: A,Y2: A,H: heap_ext(product_unit)] : aa(heap_ext(product_unit),heap_ext(product_unit),set4(A,R,X2),aa(heap_ext(product_unit),heap_ext(product_unit),set4(A,R,Y2),H)) = aa(heap_ext(product_unit),heap_ext(product_unit),set4(A,R,X2),H) ) ).
% Ref.set_same
tff(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,aa(heap_ext(product_unit),heap_ext(product_unit),set4(B,R,V),H)) = get(A,H) ) ).
% array_get_set
tff(fact_15_Ref_Oget__set__eq,axiom,
! [A: $tType] :
( heap(A)
=> ! [R: ref(A),X2: A,H: heap_ext(product_unit)] : aa(ref(A),A,get2(A,aa(heap_ext(product_unit),heap_ext(product_unit),set4(A,R,X2),H)),R) = X2 ) ).
% Ref.get_set_eq
tff(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,aa(heap_ext(product_unit),heap_ext(product_unit),set4(B,R,V),H)) = present(A,H) ) ).
% array_present_set
tff(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,aa(heap_ext(product_unit),heap_ext(product_unit),set4(B,R,V),H)) = present2(A,H) ) ).
% present_set
tff(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),aa(A,heap_Heap(ref(A)),ref3(A),V),H,H2,R)
=> ~ ( ( aa(ref(A),A,get2(A,H2),R) = V )
=> ( pp(aa(ref(A),bool,present2(A,H2),R))
=> pp(aa(ref(A),bool,present2(A,H),R)) ) ) ) ) ).
% effect_refE
tff(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)] : aa(product_prod(ref(A),heap_ext(product_unit)),ref(A),product_fst(ref(A),heap_ext(product_unit)),aa(heap_ext(product_unit),product_prod(ref(A),heap_ext(product_unit)),alloc2(A,X2),aa(heap_ext(product_unit),heap_ext(product_unit),set4(B,R,X3),H))) = aa(product_prod(ref(A),heap_ext(product_unit)),ref(A),product_fst(ref(A),heap_ext(product_unit)),aa(heap_ext(product_unit),product_prod(ref(A),heap_ext(product_unit)),alloc2(A,X2),H)) ) ).
% alloc_set
tff(fact_20_ref_H__def,axiom,
! [A: $tType] :
( heap(A)
=> ( ref4(A) = ref3(A) ) ) ).
% ref'_def
tff(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,aa(product_prod(ref(B),heap_ext(product_unit)),ref(B),product_fst(ref(B),heap_ext(product_unit)),aa(heap_ext(product_unit),product_prod(ref(B),heap_ext(product_unit)),alloc2(B,V),H)))
=> ( aa(ref(A),A,get2(A,aa(product_prod(ref(B),heap_ext(product_unit)),heap_ext(product_unit),product_snd(ref(B),heap_ext(product_unit)),aa(heap_ext(product_unit),product_prod(ref(B),heap_ext(product_unit)),alloc2(B,V),H))),R) = aa(ref(A),A,get2(A,H),R) ) ) ) ).
% get_alloc_neq
tff(fact_22_effect__changeE,axiom,
! [A: $tType] :
( heap(A)
=> ! [F: fun(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 = aa(heap_ext(product_unit),heap_ext(product_unit),set4(A,R2,aa(A,A,F,aa(ref(A),A,get2(A,H),R2))),H) )
=> ( R != aa(A,A,F,aa(ref(A),A,get2(A,H),R2)) ) ) ) ) ).
% effect_changeE
tff(fact_23_effect__changeI,axiom,
! [A: $tType] :
( heap(A)
=> ! [H2: heap_ext(product_unit),R: ref(A),F: fun(A,A),H: heap_ext(product_unit),X2: A] :
( ( H2 = aa(heap_ext(product_unit),heap_ext(product_unit),set4(A,R,aa(A,A,F,aa(ref(A),A,get2(A,H),R))),H) )
=> ( ( X2 = aa(A,A,F,aa(ref(A),A,get2(A,H),R)) )
=> heap_effect(A,change(A,F,R),H,H2,X2) ) ) ) ).
% effect_changeI
tff(fact_24_present__alloc__neq,axiom,
! [A: $tType,B: $tType] :
( ( heap(B)
& heap(A) )
=> ! [H: heap_ext(product_unit),R: ref(A),V: B] :
( pp(aa(ref(A),bool,present2(A,H),R))
=> noteq2(A,B,R,aa(product_prod(ref(B),heap_ext(product_unit)),ref(B),product_fst(ref(B),heap_ext(product_unit)),aa(heap_ext(product_unit),product_prod(ref(B),heap_ext(product_unit)),alloc2(B,V),H))) ) ) ).
% present_alloc_neq
tff(fact_25_Ref_Oget__set__neq,axiom,
! [B: $tType,A: $tType] :
( ( heap(A)
& heap(B) )
=> ! [R: ref(A),S: ref(B),X2: B,H: heap_ext(product_unit)] :
( noteq2(A,B,R,S)
=> ( aa(ref(A),A,get2(A,aa(heap_ext(product_unit),heap_ext(product_unit),set4(B,S,X2),H)),R) = aa(ref(A),A,get2(A,H),R) ) ) ) ).
% Ref.get_set_neq
tff(fact_26_ref__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [V: A] : aa(A,heap_Heap(ref(A)),ref3(A),V) = heap_heap(ref(A),alloc2(A,V)) ) ).
% ref_def
tff(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 != aa(ref(A),A,get2(A,H),R) ) ) ) ) ).
% effect_lookupE
tff(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 = aa(ref(A),A,get2(A,H),R) )
=> heap_effect(A,lookup(A,R),H,H2,X2) ) ) ) ).
% effect_lookupI
tff(fact_29_effect__refI,axiom,
! [A: $tType] :
( heap(A)
=> ! [R: ref(A),H2: heap_ext(product_unit),V: A,H: heap_ext(product_unit)] :
( ( aa(heap_ext(product_unit),product_prod(ref(A),heap_ext(product_unit)),aa(ref(A),fun(heap_ext(product_unit),product_prod(ref(A),heap_ext(product_unit))),product_Pair(ref(A),heap_ext(product_unit)),R),H2) = aa(heap_ext(product_unit),product_prod(ref(A),heap_ext(product_unit)),alloc2(A,V),H) )
=> heap_effect(ref(A),aa(A,heap_Heap(ref(A)),ref3(A),V),H,H2,R) ) ) ).
% effect_refI
tff(fact_30_snd__swap,axiom,
! [B: $tType,A: $tType,X2: product_prod(A,B)] : aa(product_prod(B,A),A,product_snd(B,A),aa(product_prod(A,B),product_prod(B,A),product_swap(A,B),X2)) = aa(product_prod(A,B),A,product_fst(A,B),X2) ).
% snd_swap
tff(fact_31_fst__swap,axiom,
! [A: $tType,B: $tType,X2: product_prod(B,A)] : aa(product_prod(A,B),A,product_fst(A,B),aa(product_prod(B,A),product_prod(A,B),product_swap(B,A),X2)) = aa(product_prod(B,A),A,product_snd(B,A),X2) ).
% fst_swap
tff(fact_32_prod_Oinject,axiom,
! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y22: B] :
( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X1),X22) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y1),Y22) )
<=> ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
tff(fact_33_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,A4: A,B3: B] :
( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3) )
<=> ( ( A3 = A4 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
tff(fact_34_swap__swap,axiom,
! [B: $tType,A: $tType,P: product_prod(A,B)] : aa(product_prod(B,A),product_prod(A,B),product_swap(B,A),aa(product_prod(A,B),product_prod(B,A),product_swap(A,B),P)) = P ).
% swap_swap
tff(fact_35_Ref_Ounequal,axiom,
! [A: $tType] :
( heap(A)
=> ! [R: ref(A),R2: ref(A)] :
( ( R != R2 )
<=> noteq2(A,A,R,R2) ) ) ).
% Ref.unequal
tff(fact_36_swap__simp,axiom,
! [A: $tType,B: $tType,X2: B,Y2: A] : aa(product_prod(B,A),product_prod(A,B),product_swap(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X2),Y2)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y2),X2) ).
% swap_simp
tff(fact_37_prod_Ocollapse,axiom,
! [B: $tType,A: $tType,Prod: product_prod(A,B)] : aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(A,B),A,product_fst(A,B),Prod)),aa(product_prod(A,B),B,product_snd(A,B),Prod)) = Prod ).
% prod.collapse
tff(fact_38_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y2: product_prod(A,B)] :
~ ! [A5: A,B4: B] : Y2 != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4) ).
% old.prod.exhaust
tff(fact_39_surj__pair,axiom,
! [A: $tType,B: $tType,P: product_prod(A,B)] :
? [X: A,Y: B] : P = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y) ).
% surj_pair
tff(fact_40_prod__cases,axiom,
! [B: $tType,A: $tType,P2: fun(product_prod(A,B),bool),P: product_prod(A,B)] :
( ! [A5: A,B4: B] : pp(aa(product_prod(A,B),bool,P2,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4)))
=> pp(aa(product_prod(A,B),bool,P2,P)) ) ).
% prod_cases
tff(fact_41_Pair__inject,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,A4: A,B3: B] :
( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3) )
=> ~ ( ( A3 = A4 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
tff(fact_42_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y2: product_prod(A,product_prod(B,C))] :
~ ! [A5: A,B4: B,C2: C] : Y2 != aa(product_prod(B,C),product_prod(A,product_prod(B,C)),aa(A,fun(product_prod(B,C),product_prod(A,product_prod(B,C))),product_Pair(A,product_prod(B,C)),A5),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B4),C2)) ).
% prod_cases3
tff(fact_43_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y2: product_prod(A,product_prod(B,product_prod(C,D)))] :
~ ! [A5: A,B4: B,C2: C,D2: D] : Y2 != aa(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D))),aa(A,fun(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D)))),product_Pair(A,product_prod(B,product_prod(C,D))),A5),aa(product_prod(C,D),product_prod(B,product_prod(C,D)),aa(B,fun(product_prod(C,D),product_prod(B,product_prod(C,D))),product_Pair(B,product_prod(C,D)),B4),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),C2),D2))) ).
% prod_cases4
tff(fact_44_mem__Collect__eq,axiom,
! [A: $tType,A3: A,P2: fun(A,bool)] :
( pp(aa(set(A),bool,member(A,A3),aa(fun(A,bool),set(A),collect(A),P2)))
<=> pp(aa(A,bool,P2,A3)) ) ).
% mem_Collect_eq
tff(fact_45_Collect__mem__eq,axiom,
! [A: $tType,A6: set(A)] : aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool)),A6)) = A6 ).
% Collect_mem_eq
tff(fact_46_Collect__cong,axiom,
! [A: $tType,P2: fun(A,bool),Q2: fun(A,bool)] :
( ! [X: A] :
( pp(aa(A,bool,P2,X))
<=> pp(aa(A,bool,Q2,X)) )
=> ( aa(fun(A,bool),set(A),collect(A),P2) = aa(fun(A,bool),set(A),collect(A),Q2) ) ) ).
% Collect_cong
tff(fact_47_ext,axiom,
! [B: $tType,A: $tType,F: fun(A,B),G: fun(A,B)] :
( ! [X: A] : aa(A,B,F,X) = aa(A,B,G,X)
=> ( F = G ) ) ).
% ext
tff(fact_48_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y2: product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))] :
~ ! [A5: A,B4: B,C2: C,D2: D,E2: E] : Y2 != aa(product_prod(B,product_prod(C,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,E)))),A5),aa(product_prod(C,product_prod(D,E)),product_prod(B,product_prod(C,product_prod(D,E))),aa(B,fun(product_prod(C,product_prod(D,E)),product_prod(B,product_prod(C,product_prod(D,E)))),product_Pair(B,product_prod(C,product_prod(D,E))),B4),aa(product_prod(D,E),product_prod(C,product_prod(D,E)),aa(C,fun(product_prod(D,E),product_prod(C,product_prod(D,E))),product_Pair(C,product_prod(D,E)),C2),aa(E,product_prod(D,E),aa(D,fun(E,product_prod(D,E)),product_Pair(D,E),D2),E2)))) ).
% prod_cases5
tff(fact_49_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F2: $tType,Y2: 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] : Y2 != aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2)))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2))))),A5),aa(product_prod(C,product_prod(D,product_prod(E,F2))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2)))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,F2))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,F2)))),B4),aa(product_prod(D,product_prod(E,F2)),product_prod(C,product_prod(D,product_prod(E,F2))),aa(C,fun(product_prod(D,product_prod(E,F2)),product_prod(C,product_prod(D,product_prod(E,F2)))),product_Pair(C,product_prod(D,product_prod(E,F2))),C2),aa(product_prod(E,F2),product_prod(D,product_prod(E,F2)),aa(D,fun(product_prod(E,F2),product_prod(D,product_prod(E,F2))),product_Pair(D,product_prod(E,F2)),D2),aa(F2,product_prod(E,F2),aa(E,fun(F2,product_prod(E,F2)),product_Pair(E,F2),E2),F3))))) ).
% prod_cases6
tff(fact_50_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F2: $tType,G2: $tType,Y2: 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] : Y2 != aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2)))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2))))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2)))))),A5),aa(product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2))))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2)))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2))))),B4),aa(product_prod(D,product_prod(E,product_prod(F2,G2))),product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2)))),aa(C,fun(product_prod(D,product_prod(E,product_prod(F2,G2))),product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2))))),product_Pair(C,product_prod(D,product_prod(E,product_prod(F2,G2)))),C2),aa(product_prod(E,product_prod(F2,G2)),product_prod(D,product_prod(E,product_prod(F2,G2))),aa(D,fun(product_prod(E,product_prod(F2,G2)),product_prod(D,product_prod(E,product_prod(F2,G2)))),product_Pair(D,product_prod(E,product_prod(F2,G2))),D2),aa(product_prod(F2,G2),product_prod(E,product_prod(F2,G2)),aa(E,fun(product_prod(F2,G2),product_prod(E,product_prod(F2,G2))),product_Pair(E,product_prod(F2,G2)),E2),aa(G2,product_prod(F2,G2),aa(F2,fun(G2,product_prod(F2,G2)),product_Pair(F2,G2),F3),G3)))))) ).
% prod_cases7
tff(fact_51_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P2: fun(product_prod(A,product_prod(B,C)),bool),X2: product_prod(A,product_prod(B,C))] :
( ! [A5: A,B4: B,C2: C] : pp(aa(product_prod(A,product_prod(B,C)),bool,P2,aa(product_prod(B,C),product_prod(A,product_prod(B,C)),aa(A,fun(product_prod(B,C),product_prod(A,product_prod(B,C))),product_Pair(A,product_prod(B,C)),A5),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B4),C2))))
=> pp(aa(product_prod(A,product_prod(B,C)),bool,P2,X2)) ) ).
% prod_induct3
tff(fact_52_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,P2: fun(product_prod(A,product_prod(B,product_prod(C,D))),bool),X2: product_prod(A,product_prod(B,product_prod(C,D)))] :
( ! [A5: A,B4: B,C2: C,D2: D] : pp(aa(product_prod(A,product_prod(B,product_prod(C,D))),bool,P2,aa(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D))),aa(A,fun(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D)))),product_Pair(A,product_prod(B,product_prod(C,D))),A5),aa(product_prod(C,D),product_prod(B,product_prod(C,D)),aa(B,fun(product_prod(C,D),product_prod(B,product_prod(C,D))),product_Pair(B,product_prod(C,D)),B4),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),C2),D2)))))
=> pp(aa(product_prod(A,product_prod(B,product_prod(C,D))),bool,P2,X2)) ) ).
% prod_induct4
tff(fact_53_prod__induct5,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),bool),X2: product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))] :
( ! [A5: A,B4: B,C2: C,D2: D,E2: E] : pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),bool,P2,aa(product_prod(B,product_prod(C,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,E)))),A5),aa(product_prod(C,product_prod(D,E)),product_prod(B,product_prod(C,product_prod(D,E))),aa(B,fun(product_prod(C,product_prod(D,E)),product_prod(B,product_prod(C,product_prod(D,E)))),product_Pair(B,product_prod(C,product_prod(D,E))),B4),aa(product_prod(D,E),product_prod(C,product_prod(D,E)),aa(C,fun(product_prod(D,E),product_prod(C,product_prod(D,E))),product_Pair(C,product_prod(D,E)),C2),aa(E,product_prod(D,E),aa(D,fun(E,product_prod(D,E)),product_Pair(D,E),D2),E2))))))
=> pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),bool,P2,X2)) ) ).
% prod_induct5
tff(fact_54_prod__induct6,axiom,
! [F2: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2))))),bool),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] : pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2))))),bool,P2,aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2)))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2))))),A5),aa(product_prod(C,product_prod(D,product_prod(E,F2))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2)))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,F2))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,F2)))),B4),aa(product_prod(D,product_prod(E,F2)),product_prod(C,product_prod(D,product_prod(E,F2))),aa(C,fun(product_prod(D,product_prod(E,F2)),product_prod(C,product_prod(D,product_prod(E,F2)))),product_Pair(C,product_prod(D,product_prod(E,F2))),C2),aa(product_prod(E,F2),product_prod(D,product_prod(E,F2)),aa(D,fun(product_prod(E,F2),product_prod(D,product_prod(E,F2))),product_Pair(D,product_prod(E,F2)),D2),aa(F2,product_prod(E,F2),aa(E,fun(F2,product_prod(E,F2)),product_Pair(E,F2),E2),F3)))))))
=> pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2))))),bool,P2,X2)) ) ).
% prod_induct6
tff(fact_55_prod__induct7,axiom,
! [G2: $tType,F2: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2)))))),bool),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] : pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2)))))),bool,P2,aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2)))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2))))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2)))))),A5),aa(product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2))))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2)))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2))))),B4),aa(product_prod(D,product_prod(E,product_prod(F2,G2))),product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2)))),aa(C,fun(product_prod(D,product_prod(E,product_prod(F2,G2))),product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2))))),product_Pair(C,product_prod(D,product_prod(E,product_prod(F2,G2)))),C2),aa(product_prod(E,product_prod(F2,G2)),product_prod(D,product_prod(E,product_prod(F2,G2))),aa(D,fun(product_prod(E,product_prod(F2,G2)),product_prod(D,product_prod(E,product_prod(F2,G2)))),product_Pair(D,product_prod(E,product_prod(F2,G2))),D2),aa(product_prod(F2,G2),product_prod(E,product_prod(F2,G2)),aa(E,fun(product_prod(F2,G2),product_prod(E,product_prod(F2,G2))),product_Pair(E,product_prod(F2,G2)),E2),aa(G2,product_prod(F2,G2),aa(F2,fun(G2,product_prod(F2,G2)),product_Pair(F2,G2),F3),G3))))))))
=> pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F2,G2)))))),bool,P2,X2)) ) ).
% prod_induct7
tff(fact_56_Ref_Onoteq__irrefl,axiom,
! [A: $tType] :
( heap(A)
=> ! [R: ref(A)] : ~ noteq2(A,A,R,R) ) ).
% Ref.noteq_irrefl
tff(fact_57_Ref_Onoteq__sym,axiom,
! [B: $tType,A: $tType] :
( ( heap(A)
& heap(B) )
=> ! [R: ref(A),S: ref(B)] :
( noteq2(A,B,R,S)
=> noteq2(B,A,S,R) ) ) ).
% Ref.noteq_sym
tff(fact_58_fst__eqD,axiom,
! [B: $tType,A: $tType,X2: A,Y2: B,A3: A] :
( ( aa(product_prod(A,B),A,product_fst(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y2)) = A3 )
=> ( X2 = A3 ) ) ).
% fst_eqD
tff(fact_59_fst__conv,axiom,
! [B: $tType,A: $tType,X1: A,X22: B] : aa(product_prod(A,B),A,product_fst(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X1),X22)) = X1 ).
% fst_conv
tff(fact_60_snd__eqD,axiom,
! [B: $tType,A: $tType,X2: B,Y2: A,A3: A] :
( ( aa(product_prod(B,A),A,product_snd(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X2),Y2)) = A3 )
=> ( Y2 = A3 ) ) ).
% snd_eqD
tff(fact_61_snd__conv,axiom,
! [Aa: $tType,A: $tType,X1: Aa,X22: A] : aa(product_prod(Aa,A),A,product_snd(Aa,A),aa(A,product_prod(Aa,A),aa(Aa,fun(A,product_prod(Aa,A)),product_Pair(Aa,A),X1),X22)) = X22 ).
% snd_conv
tff(fact_62_prod_Oswap__def,axiom,
! [B: $tType,A: $tType,P: product_prod(A,B)] : aa(product_prod(A,B),product_prod(B,A),product_swap(A,B),P) = aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),aa(product_prod(A,B),B,product_snd(A,B),P)),aa(product_prod(A,B),A,product_fst(A,B),P)) ).
% prod.swap_def
tff(fact_63_prod_Oexhaust__sel,axiom,
! [B: $tType,A: $tType,Prod: product_prod(A,B)] : Prod = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(A,B),A,product_fst(A,B),Prod)),aa(product_prod(A,B),B,product_snd(A,B),Prod)) ).
% prod.exhaust_sel
tff(fact_64_exI__realizer,axiom,
! [B: $tType,A: $tType,P2: fun(A,fun(B,bool)),Y2: A,X2: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),P2,Y2),X2))
=> pp(aa(B,bool,aa(A,fun(B,bool),P2,aa(product_prod(B,A),A,product_snd(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X2),Y2))),aa(product_prod(B,A),B,product_fst(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X2),Y2)))) ) ).
% exI_realizer
tff(fact_65_conjI__realizer,axiom,
! [A: $tType,B: $tType,P2: fun(A,bool),P: A,Q2: fun(B,bool),Q: B] :
( pp(aa(A,bool,P2,P))
=> ( pp(aa(B,bool,Q2,Q))
=> ( pp(aa(A,bool,P2,aa(product_prod(A,B),A,product_fst(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),P),Q))))
& pp(aa(B,bool,Q2,aa(product_prod(A,B),B,product_snd(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),P),Q)))) ) ) ) ).
% conjI_realizer
tff(fact_66_surjective__pairing,axiom,
! [B: $tType,A: $tType,T2: product_prod(A,B)] : T2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(A,B),A,product_fst(A,B),T2)),aa(product_prod(A,B),B,product_snd(A,B),T2)) ).
% surjective_pairing
tff(fact_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)
=> ( aa(heap_ext(product_unit),heap_ext(product_unit),set4(A,R,X2),aa(heap_ext(product_unit),heap_ext(product_unit),set4(B,R2,X3),H)) = aa(heap_ext(product_unit),heap_ext(product_unit),set4(B,R2,X3),aa(heap_ext(product_unit),heap_ext(product_unit),set4(A,R,X2),H)) ) ) ) ).
% Ref.set_set_swap
tff(fact_68_next__fresh,axiom,
! [A: $tType] :
( heap(A)
=> ! [R: ref(A),H2: heap_ext(product_unit),X2: A,H: heap_ext(product_unit)] :
( ( aa(heap_ext(product_unit),product_prod(ref(A),heap_ext(product_unit)),aa(ref(A),fun(heap_ext(product_unit),product_prod(ref(A),heap_ext(product_unit))),product_Pair(ref(A),heap_ext(product_unit)),R),H2) = aa(heap_ext(product_unit),product_prod(ref(A),heap_ext(product_unit)),alloc2(A,X2),H) )
=> ~ pp(aa(ref(A),bool,present2(A,H),R)) ) ) ).
% next_fresh
tff(fact_69_next__present,axiom,
! [A: $tType] :
( heap(A)
=> ! [R: ref(A),H2: heap_ext(product_unit),X2: A,H: heap_ext(product_unit)] :
( ( aa(heap_ext(product_unit),product_prod(ref(A),heap_ext(product_unit)),aa(ref(A),fun(heap_ext(product_unit),product_prod(ref(A),heap_ext(product_unit))),product_Pair(ref(A),heap_ext(product_unit)),R),H2) = aa(heap_ext(product_unit),product_prod(ref(A),heap_ext(product_unit)),alloc2(A,X2),H) )
=> pp(aa(ref(A),bool,present2(A,H2),R)) ) ) ).
% next_present
tff(fact_70_noteq__I,axiom,
! [A: $tType,B: $tType] :
( ( heap(B)
& heap(A) )
=> ! [H: heap_ext(product_unit),R: ref(A),R2: ref(B)] :
( pp(aa(ref(A),bool,present2(A,H),R))
=> ( ~ pp(aa(ref(B),bool,present2(B,H),R2))
=> noteq2(A,B,R,R2) ) ) ) ).
% noteq_I
tff(fact_71_effect__heapE,axiom,
! [A: $tType,F: fun(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 = aa(product_prod(A,heap_ext(product_unit)),heap_ext(product_unit),product_snd(A,heap_ext(product_unit)),aa(heap_ext(product_unit),product_prod(A,heap_ext(product_unit)),F,H)) )
=> ( R != aa(product_prod(A,heap_ext(product_unit)),A,product_fst(A,heap_ext(product_unit)),aa(heap_ext(product_unit),product_prod(A,heap_ext(product_unit)),F,H)) ) ) ) ).
% effect_heapE
tff(fact_72_effect__heapI,axiom,
! [A: $tType,H2: heap_ext(product_unit),F: fun(heap_ext(product_unit),product_prod(A,heap_ext(product_unit))),H: heap_ext(product_unit),R: A] :
( ( H2 = aa(product_prod(A,heap_ext(product_unit)),heap_ext(product_unit),product_snd(A,heap_ext(product_unit)),aa(heap_ext(product_unit),product_prod(A,heap_ext(product_unit)),F,H)) )
=> ( ( R = aa(product_prod(A,heap_ext(product_unit)),A,product_fst(A,heap_ext(product_unit)),aa(heap_ext(product_unit),product_prod(A,heap_ext(product_unit)),F,H)) )
=> heap_effect(A,heap_heap(A,F),H,H2,R) ) ) ).
% effect_heapI
tff(fact_73_old_Oprod_Orec,axiom,
! [A: $tType,T: $tType,B: $tType,F1: fun(A,fun(B,T)),A3: A,B2: B] : product_rec_prod(A,B,T,F1,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)) = aa(B,T,aa(A,fun(B,T),F1,A3),B2) ).
% old.prod.rec
tff(fact_74_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [B: $tType,A: $tType,P2: fun(A,fun(B,bool)),X2: A,Y2: B,A3: product_prod(A,B)] :
( pp(aa(B,bool,aa(A,fun(B,bool),P2,X2),Y2))
=> ( ( A3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y2) )
=> pp(aa(B,bool,aa(A,fun(B,bool),P2,aa(product_prod(A,B),A,product_fst(A,B),A3)),aa(product_prod(A,B),B,product_snd(A,B),A3))) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
tff(fact_75_sndI,axiom,
! [A: $tType,B: $tType,X2: product_prod(A,B),Y2: A,Z2: B] :
( ( X2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y2),Z2) )
=> ( aa(product_prod(A,B),B,product_snd(A,B),X2) = Z2 ) ) ).
% sndI
tff(fact_76_eq__snd__iff,axiom,
! [B: $tType,A: $tType,B2: A,P: product_prod(B,A)] :
( ( B2 = aa(product_prod(B,A),A,product_snd(B,A),P) )
<=> ? [A7: B] : P = aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A7),B2) ) ).
% eq_snd_iff
tff(fact_77_fstI,axiom,
! [B: $tType,A: $tType,X2: product_prod(A,B),Y2: A,Z2: B] :
( ( X2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y2),Z2) )
=> ( aa(product_prod(A,B),A,product_fst(A,B),X2) = Y2 ) ) ).
% fstI
tff(fact_78_eq__fst__iff,axiom,
! [A: $tType,B: $tType,A3: A,P: product_prod(A,B)] :
( ( A3 = aa(product_prod(A,B),A,product_fst(A,B),P) )
<=> ? [B5: B] : P = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B5) ) ).
% eq_fst_iff
tff(fact_79_internal__case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,C3: fun(B,fun(C,A)),A3: B,B2: C] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),produc5280177257484947105e_prod(B,C,A),C3),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A3),B2)) = aa(C,A,aa(B,fun(C,A),C3,A3),B2) ).
% internal_case_prod_conv
tff(fact_80_effect__guardE,axiom,
! [A: $tType,P2: fun(heap_ext(product_unit),bool),F: fun(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 = aa(product_prod(A,heap_ext(product_unit)),heap_ext(product_unit),product_snd(A,heap_ext(product_unit)),aa(heap_ext(product_unit),product_prod(A,heap_ext(product_unit)),F,H)) )
=> ( ( R = aa(product_prod(A,heap_ext(product_unit)),A,product_fst(A,heap_ext(product_unit)),aa(heap_ext(product_unit),product_prod(A,heap_ext(product_unit)),F,H)) )
=> ~ pp(aa(heap_ext(product_unit),bool,P2,H)) ) ) ) ).
% effect_guardE
tff(fact_81_effect__guardI,axiom,
! [A: $tType,P2: fun(heap_ext(product_unit),bool),H: heap_ext(product_unit),H2: heap_ext(product_unit),F: fun(heap_ext(product_unit),product_prod(A,heap_ext(product_unit))),R: A] :
( pp(aa(heap_ext(product_unit),bool,P2,H))
=> ( ( H2 = aa(product_prod(A,heap_ext(product_unit)),heap_ext(product_unit),product_snd(A,heap_ext(product_unit)),aa(heap_ext(product_unit),product_prod(A,heap_ext(product_unit)),F,H)) )
=> ( ( R = aa(product_prod(A,heap_ext(product_unit)),A,product_fst(A,heap_ext(product_unit)),aa(heap_ext(product_unit),product_prod(A,heap_ext(product_unit)),F,H)) )
=> heap_effect(A,heap_guard(A,P2,F),H,H2,R) ) ) ) ).
% effect_guardI
tff(fact_82_effect__ifI,axiom,
! [A: $tType,C3: bool,T2: heap_Heap(A),H: heap_ext(product_unit),H2: heap_ext(product_unit),R: A,E3: heap_Heap(A)] :
( ( pp(C3)
=> heap_effect(A,T2,H,H2,R) )
=> ( ( ~ pp(C3)
=> heap_effect(A,E3,H,H2,R) )
=> heap_effect(A,if(heap_Heap(A),C3,T2,E3),H,H2,R) ) ) ).
% effect_ifI
tff(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)
tff(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)
tff(fact_85_effect__ifE,axiom,
! [A: $tType,C3: bool,T2: 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,T2,E3),H,H2,R)
=> ( ( pp(C3)
=> ~ heap_effect(A,T2,H,H2,R) )
=> ~ ( ~ pp(C3)
=> ~ heap_effect(A,E3,H,H2,R) ) ) ) ).
% effect_ifE
tff(fact_86_return__def,axiom,
! [A: $tType,X2: A] : aa(A,heap_Heap(A),heap_return(A),X2) = heap_heap(A,aa(A,fun(heap_ext(product_unit),product_prod(A,heap_ext(product_unit))),product_Pair(A,heap_ext(product_unit)),X2)) ).
% return_def
tff(fact_87_sndOp__def,axiom,
! [A: $tType,B: $tType,C: $tType,P2: fun(C,fun(A,bool)),Q2: fun(A,fun(B,bool)),Ac: product_prod(C,B)] : aa(product_prod(C,B),product_prod(A,B),bNF_sndOp(C,A,B,P2,Q2),Ac) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),bNF_pick_middlep(C,A,B,P2,Q2,aa(product_prod(C,B),C,product_fst(C,B),Ac),aa(product_prod(C,B),B,product_snd(C,B),Ac))),aa(product_prod(C,B),B,product_snd(C,B),Ac)) ).
% sndOp_def
tff(fact_88_fstOp__def,axiom,
! [B: $tType,C: $tType,A: $tType,P2: fun(A,fun(B,bool)),Q2: fun(B,fun(C,bool)),Ac: product_prod(A,C)] : aa(product_prod(A,C),product_prod(A,B),bNF_fstOp(A,B,C,P2,Q2),Ac) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(A,C),A,product_fst(A,C),Ac)),bNF_pick_middlep(A,B,C,P2,Q2,aa(product_prod(A,C),A,product_fst(A,C),Ac),aa(product_prod(A,C),C,product_snd(A,C),Ac))) ).
% fstOp_def
tff(fact_89_image2__eqI,axiom,
! [A: $tType,C: $tType,B: $tType,B2: A,F: fun(B,A),X2: B,C3: C,G: fun(B,C),A6: set(B)] :
( ( B2 = aa(B,A,F,X2) )
=> ( ( C3 = aa(B,C,G,X2) )
=> ( pp(aa(set(B),bool,member(B,X2),A6))
=> pp(aa(set(product_prod(A,C)),bool,member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),B2),C3)),bNF_Greatest_image2(B,A,C,A6,F,G))) ) ) ) ).
% image2_eqI
tff(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)] : aa(product_prod(ref(A),heap_ext(product_unit)),ref(A),product_fst(ref(A),heap_ext(product_unit)),aa(heap_ext(product_unit),product_prod(ref(A),heap_ext(product_unit)),alloc2(A,V),update(B,A3,I,V2,H))) = aa(product_prod(ref(A),heap_ext(product_unit)),ref(A),product_fst(ref(A),heap_ext(product_unit)),aa(heap_ext(product_unit),product_prod(ref(A),heap_ext(product_unit)),alloc2(A,V),H)) ) ).
% alloc_update
tff(fact_91_lim__set,axiom,
! [A: $tType] :
( heap(A)
=> ! [R: ref(A),V: A,H: heap_ext(product_unit)] : lim(product_unit,aa(heap_ext(product_unit),heap_ext(product_unit),set4(A,R,V),H)) = lim(product_unit,H) ) ).
% lim_set
tff(fact_92_prod_Osplit__sel__asm,axiom,
! [C: $tType,B: $tType,A: $tType,P2: fun(C,bool),F: fun(A,fun(B,C)),Prod: product_prod(A,B)] :
( pp(aa(C,bool,P2,aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F),Prod)))
<=> ~ ( ( Prod = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(A,B),A,product_fst(A,B),Prod)),aa(product_prod(A,B),B,product_snd(A,B),Prod)) )
& ~ pp(aa(C,bool,P2,aa(B,C,aa(A,fun(B,C),F,aa(product_prod(A,B),A,product_fst(A,B),Prod)),aa(product_prod(A,B),B,product_snd(A,B),Prod)))) ) ) ).
% prod.split_sel_asm
tff(fact_93_prod_Osplit__sel,axiom,
! [C: $tType,B: $tType,A: $tType,P2: fun(C,bool),F: fun(A,fun(B,C)),Prod: product_prod(A,B)] :
( pp(aa(C,bool,P2,aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F),Prod)))
<=> ( ( Prod = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(A,B),A,product_fst(A,B),Prod)),aa(product_prod(A,B),B,product_snd(A,B),Prod)) )
=> pp(aa(C,bool,P2,aa(B,C,aa(A,fun(B,C),F,aa(product_prod(A,B),A,product_fst(A,B),Prod)),aa(product_prod(A,B),B,product_snd(A,B),Prod)))) ) ) ).
% prod.split_sel
tff(fact_94_ssubst__Pair__rhs,axiom,
! [B: $tType,A: $tType,R: A,S: B,R3: set(product_prod(A,B)),S2: B] :
( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),R),S)),R3))
=> ( ( S2 = S )
=> pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),R),S2)),R3)) ) ) ).
% ssubst_Pair_rhs
tff(fact_95_curry__conv,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(product_prod(B,C),A),A3: B,B2: C] : aa(C,A,aa(B,fun(C,A),aa(fun(product_prod(B,C),A),fun(B,fun(C,A)),product_curry(B,C,A),F),A3),B2) = aa(product_prod(B,C),A,F,aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A3),B2)) ).
% curry_conv
tff(fact_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
tff(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 = aa(heap_ext(product_unit),heap_ext(product_unit),set4(A,R2,V),H) ) ) ) ).
% effect_updateE
tff(fact_98_curryI,axiom,
! [A: $tType,B: $tType,F: fun(product_prod(A,B),bool),A3: A,B2: B] :
( pp(aa(product_prod(A,B),bool,F,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)))
=> pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(product_prod(A,B),bool),fun(A,fun(B,bool)),product_curry(A,B,bool),F),A3),B2)) ) ).
% curryI
tff(fact_99_case__prod__curry,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(product_prod(A,B),C)] : aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),aa(fun(product_prod(A,B),C),fun(A,fun(B,C)),product_curry(A,B,C),F)) = F ).
% case_prod_curry
tff(fact_100_curry__case__prod,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(A,fun(B,C))] : aa(fun(product_prod(A,B),C),fun(A,fun(B,C)),product_curry(A,B,C),aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F)) = F ).
% curry_case_prod
tff(fact_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)] : aa(ref(A),A,get2(A,update(B,A3,I,V,H)),R) = aa(ref(A),A,get2(A,H),R) ) ).
% get_update
tff(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
tff(fact_103_mem__case__prodE,axiom,
! [B: $tType,A: $tType,C: $tType,Z2: A,C3: fun(B,fun(C,set(A))),P: product_prod(B,C)] :
( pp(aa(set(A),bool,member(A,Z2),aa(product_prod(B,C),set(A),aa(fun(B,fun(C,set(A))),fun(product_prod(B,C),set(A)),product_case_prod(B,C,set(A)),C3),P)))
=> ~ ! [X: B,Y: C] :
( ( P = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X),Y) )
=> ~ pp(aa(set(A),bool,member(A,Z2),aa(C,set(A),aa(B,fun(C,set(A)),C3,X),Y))) ) ) ).
% mem_case_prodE
tff(fact_104_old_Oprod_Ocase,axiom,
! [A: $tType,C: $tType,B: $tType,F: fun(A,fun(B,C)),X1: A,X22: B] : aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X1),X22)) = aa(B,C,aa(A,fun(B,C),F,X1),X22) ).
% old.prod.case
tff(fact_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,aa(heap_ext(product_unit),heap_ext(product_unit),set4(B,R,V2),H)) = aa(heap_ext(product_unit),heap_ext(product_unit),set4(B,R,V2),update(A,A3,I,V,H)) ) ).
% update_set_swap
tff(fact_106_curryD,axiom,
! [A: $tType,B: $tType,F: fun(product_prod(A,B),bool),A3: A,B2: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(product_prod(A,B),bool),fun(A,fun(B,bool)),product_curry(A,B,bool),F),A3),B2))
=> pp(aa(product_prod(A,B),bool,F,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2))) ) ).
% curryD
tff(fact_107_curryE,axiom,
! [A: $tType,B: $tType,F: fun(product_prod(A,B),bool),A3: A,B2: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(product_prod(A,B),bool),fun(A,fun(B,bool)),product_curry(A,B,bool),F),A3),B2))
=> pp(aa(product_prod(A,B),bool,F,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2))) ) ).
% curryE
tff(fact_108_Product__Type_OCollect__case__prodD,axiom,
! [B: $tType,A: $tType,X2: product_prod(A,B),A6: fun(A,fun(B,bool))] :
( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),X2),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),A6))))
=> pp(aa(B,bool,aa(A,fun(B,bool),A6,aa(product_prod(A,B),A,product_fst(A,B),X2)),aa(product_prod(A,B),B,product_snd(A,B),X2))) ) ).
% Product_Type.Collect_case_prodD
tff(fact_109_effect__returnI,axiom,
! [A: $tType,H: heap_ext(product_unit),H2: heap_ext(product_unit),X2: A] :
( ( H = H2 )
=> heap_effect(A,aa(A,heap_Heap(A),heap_return(A),X2),H,H2,X2) ) ).
% effect_returnI
tff(fact_110_effect__returnE,axiom,
! [A: $tType,X2: A,H: heap_ext(product_unit),H2: heap_ext(product_unit),R: A] :
( heap_effect(A,aa(A,heap_Heap(A),heap_return(A),X2),H,H2,R)
=> ~ ( ( R = X2 )
=> ( H2 != H ) ) ) ).
% effect_returnE
tff(fact_111_split__beta,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(A,fun(B,C)),Prod: product_prod(A,B)] : aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F),Prod) = aa(B,C,aa(A,fun(B,C),F,aa(product_prod(A,B),A,product_fst(A,B),Prod)),aa(product_prod(A,B),B,product_snd(A,B),Prod)) ).
% split_beta
tff(fact_112_case__prod__beta,axiom,
! [A: $tType,C: $tType,B: $tType,F: fun(B,fun(C,A)),P: product_prod(B,C)] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),F),P) = aa(C,A,aa(B,fun(C,A),F,aa(product_prod(B,C),B,product_fst(B,C),P)),aa(product_prod(B,C),C,product_snd(B,C),P)) ).
% case_prod_beta
tff(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
tff(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 = aa(heap_ext(product_unit),heap_ext(product_unit),set4(A,R,V),H) )
=> heap_effect(product_unit,update2(A,R,V),H,H2,X2) ) ) ).
% effect_updateI
tff(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
tff(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
tff(fact_117_lift__collapse,axiom,
! [A: $tType,B: $tType,F: fun(B,A),X2: B] : aa(B,heap_Heap(A),heap_lift(B,A,F),X2) = aa(A,heap_Heap(A),heap_return(A),aa(B,A,F,X2)) ).
% lift_collapse
tff(fact_118_split__cong,axiom,
! [C: $tType,B: $tType,A: $tType,Q: product_prod(A,B),F: fun(A,fun(B,C)),G: fun(A,fun(B,C)),P: product_prod(A,B)] :
( ! [X: A,Y: B] :
( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y) = Q )
=> ( aa(B,C,aa(A,fun(B,C),F,X),Y) = aa(B,C,aa(A,fun(B,C),G,X),Y) ) )
=> ( ( P = Q )
=> ( aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F),P) = aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),G),Q) ) ) ) ).
% split_cong
tff(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
tff(fact_120_csquare__fstOp__sndOp,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(fun(A,fun(B,bool)),fun(product_prod(A,B),bool)),P2: fun(A,fun(C,bool)),Q2: fun(C,fun(B,bool))] : bNF_csquare(product_prod(A,B),product_prod(A,C),C,product_prod(C,B),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),F,aa(fun(C,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(C,bool)),fun(fun(C,fun(B,bool)),fun(A,fun(B,bool))),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
tff(fact_121_effect__tapI,axiom,
! [A: $tType,H2: heap_ext(product_unit),H: heap_ext(product_unit),R: A,F: fun(heap_ext(product_unit),A)] :
( ( H2 = H )
=> ( ( R = aa(heap_ext(product_unit),A,F,H) )
=> heap_effect(A,heap_tap(A,F),H,H2,R) ) ) ).
% effect_tapI
tff(fact_122_effect__tapE,axiom,
! [A: $tType,F: fun(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 != aa(heap_ext(product_unit),A,F,H) ) ) ) ).
% effect_tapE
tff(fact_123_scomp__apply,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F: fun(B,product_prod(C,D)),G: fun(C,fun(D,A)),X2: B] : aa(B,A,product_scomp(B,C,D,A,F,G),X2) = aa(product_prod(C,D),A,aa(fun(C,fun(D,A)),fun(product_prod(C,D),A),product_case_prod(C,D,A),G),aa(B,product_prod(C,D),F,X2)) ).
% scomp_apply
tff(fact_124_pair__in__swap__image,axiom,
! [A: $tType,B: $tType,Y2: A,X2: B,A6: set(product_prod(B,A))] :
( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y2),X2)),aa(set(product_prod(B,A)),set(product_prod(A,B)),aa(fun(product_prod(B,A),product_prod(A,B)),fun(set(product_prod(B,A)),set(product_prod(A,B))),image2(product_prod(B,A),product_prod(A,B)),product_swap(B,A)),A6)))
<=> pp(aa(set(product_prod(B,A)),bool,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X2),Y2)),A6)) ) ).
% pair_in_swap_image
tff(fact_125_raise__bind,axiom,
! [B: $tType,A: $tType,E3: literal,F: fun(B,heap_Heap(A))] : heap_bind(B,A,heap_raise(B,E3),F) = heap_raise(A,E3) ).
% raise_bind
tff(fact_126_snd__sndOp,axiom,
! [C: $tType,A: $tType,B: $tType,Bc: product_prod(B,A),P2: fun(B,fun(C,bool)),Q2: fun(C,fun(A,bool))] : aa(product_prod(B,A),A,product_snd(B,A),Bc) = aa(product_prod(B,A),A,aa(fun(product_prod(B,A),product_prod(C,A)),fun(product_prod(B,A),A),aa(fun(product_prod(C,A),A),fun(fun(product_prod(B,A),product_prod(C,A)),fun(product_prod(B,A),A)),comp(product_prod(C,A),A,product_prod(B,A)),product_snd(C,A)),bNF_sndOp(B,C,A,P2,Q2)),Bc) ).
% snd_sndOp
tff(fact_127_return__bind,axiom,
! [A: $tType,B: $tType,X2: B,F: fun(B,heap_Heap(A))] : heap_bind(B,A,aa(B,heap_Heap(B),heap_return(B),X2),F) = aa(B,heap_Heap(A),F,X2) ).
% return_bind
tff(fact_128_bind__return,axiom,
! [A: $tType,F: heap_Heap(A)] : heap_bind(A,A,F,heap_return(A)) = F ).
% bind_return
tff(fact_129_nchotomy__relcomppE,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType,F: fun(B,A),R: fun(C,fun(A,bool)),S: fun(A,fun(D,bool)),A3: C,C3: D] :
( ! [Y: A] :
? [X4: B] : Y = aa(B,A,F,X4)
=> ( pp(aa(D,bool,aa(C,fun(D,bool),aa(fun(A,fun(D,bool)),fun(C,fun(D,bool)),aa(fun(C,fun(A,bool)),fun(fun(A,fun(D,bool)),fun(C,fun(D,bool))),relcompp(C,A,D),R),S),A3),C3))
=> ~ ! [B4: B] :
( pp(aa(A,bool,aa(C,fun(A,bool),R,A3),aa(B,A,F,B4)))
=> ~ pp(aa(D,bool,aa(A,fun(D,bool),S,aa(B,A,F,B4)),C3)) ) ) ) ).
% nchotomy_relcomppE
tff(fact_130_comp__cong,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,E: $tType,F: fun(B,A),G: fun(C,B),X2: C,F4: fun(D,A),G4: fun(E,D),X3: E] :
( ( aa(B,A,F,aa(C,B,G,X2)) = aa(D,A,F4,aa(E,D,G4,X3)) )
=> ( aa(C,A,aa(fun(C,B),fun(C,A),aa(fun(B,A),fun(fun(C,B),fun(C,A)),comp(B,A,C),F),G),X2) = aa(E,A,aa(fun(E,D),fun(E,A),aa(fun(D,A),fun(fun(E,D),fun(E,A)),comp(D,A,E),F4),G4),X3) ) ) ).
% comp_cong
tff(fact_131_comp__apply__eq,axiom,
! [B: $tType,D: $tType,A: $tType,C: $tType,F: fun(B,A),G: fun(C,B),X2: C,H: fun(D,A),K: fun(C,D)] :
( ( aa(B,A,F,aa(C,B,G,X2)) = aa(D,A,H,aa(C,D,K,X2)) )
=> ( aa(C,A,aa(fun(C,B),fun(C,A),aa(fun(B,A),fun(fun(C,B),fun(C,A)),comp(B,A,C),F),G),X2) = aa(C,A,aa(fun(C,D),fun(C,A),aa(fun(D,A),fun(fun(C,D),fun(C,A)),comp(D,A,C),H),K),X2) ) ) ).
% comp_apply_eq
tff(fact_132_pick__middlep,axiom,
! [B: $tType,A: $tType,C: $tType,P2: fun(A,fun(B,bool)),Q2: fun(B,fun(C,bool)),A3: A,C3: C] :
( pp(aa(C,bool,aa(A,fun(C,bool),aa(fun(B,fun(C,bool)),fun(A,fun(C,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(C,bool)),fun(A,fun(C,bool))),relcompp(A,B,C),P2),Q2),A3),C3))
=> ( pp(aa(B,bool,aa(A,fun(B,bool),P2,A3),bNF_pick_middlep(A,B,C,P2,Q2,A3,C3)))
& pp(aa(C,bool,aa(B,fun(C,bool),Q2,bNF_pick_middlep(A,B,C,P2,Q2,A3,C3)),C3)) ) ) ).
% pick_middlep
tff(fact_133_sndOp__in,axiom,
! [A: $tType,B: $tType,C: $tType,Ac: product_prod(A,B),P2: fun(A,fun(C,bool)),Q2: fun(C,fun(B,bool))] :
( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),Ac),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aa(fun(C,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(C,bool)),fun(fun(C,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,C,B),P2),Q2)))))
=> pp(aa(set(product_prod(C,B)),bool,member(product_prod(C,B),aa(product_prod(A,B),product_prod(C,B),bNF_sndOp(A,C,B,P2,Q2),Ac)),aa(fun(product_prod(C,B),bool),set(product_prod(C,B)),collect(product_prod(C,B)),aa(fun(C,fun(B,bool)),fun(product_prod(C,B),bool),product_case_prod(C,B,bool),Q2)))) ) ).
% sndOp_in
tff(fact_134_fstOp__in,axiom,
! [B: $tType,C: $tType,A: $tType,Ac: product_prod(A,B),P2: fun(A,fun(C,bool)),Q2: fun(C,fun(B,bool))] :
( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),Ac),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aa(fun(C,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(C,bool)),fun(fun(C,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,C,B),P2),Q2)))))
=> pp(aa(set(product_prod(A,C)),bool,member(product_prod(A,C),aa(product_prod(A,B),product_prod(A,C),bNF_fstOp(A,C,B,P2,Q2),Ac)),aa(fun(product_prod(A,C),bool),set(product_prod(A,C)),collect(product_prod(A,C)),aa(fun(A,fun(C,bool)),fun(product_prod(A,C),bool),product_case_prod(A,C,bool),P2)))) ) ).
% fstOp_in
tff(fact_135_effect__bindE,axiom,
! [B: $tType,A: $tType,F: heap_Heap(B),G: fun(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,aa(B,heap_Heap(A),G,R4),H4,H3,R2) ) ) ).
% effect_bindE
tff(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: fun(A,heap_Heap(B)),H3: heap_ext(product_unit),R2: B] :
( heap_effect(A,F,H,H2,R)
=> ( heap_effect(B,aa(A,heap_Heap(B),G,R),H2,H3,R2)
=> heap_effect(B,heap_bind(A,B,F,G),H,H3,R2) ) ) ).
% effect_bindI
tff(fact_137_scomp__Pair,axiom,
! [C: $tType,B: $tType,A: $tType,X2: fun(A,product_prod(B,C))] : product_scomp(A,B,C,product_prod(B,C),X2,product_Pair(B,C)) = X2 ).
% scomp_Pair
tff(fact_138_Pair__scomp,axiom,
! [A: $tType,B: $tType,C: $tType,X2: C,F: fun(C,fun(A,B))] : product_scomp(A,C,A,B,aa(C,fun(A,product_prod(C,A)),product_Pair(C,A),X2),F) = aa(C,fun(A,B),F,X2) ).
% Pair_scomp
tff(fact_139_csquare__def,axiom,
! [B: $tType,C: $tType,D: $tType,A: $tType,A6: set(A),F1: fun(B,C),F22: fun(D,C),P1: fun(A,B),P22: fun(A,D)] :
( bNF_csquare(A,B,C,D,A6,F1,F22,P1,P22)
<=> ! [X5: A] :
( pp(aa(set(A),bool,member(A,X5),A6))
=> ( aa(B,C,F1,aa(A,B,P1,X5)) = aa(D,C,F22,aa(A,D,P22,X5)) ) ) ) ).
% csquare_def
tff(fact_140_scomp__def,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,F: fun(A,product_prod(B,C)),G: fun(B,fun(C,D)),X4: A] : aa(A,D,product_scomp(A,B,C,D,F,G),X4) = aa(product_prod(B,C),D,aa(fun(B,fun(C,D)),fun(product_prod(B,C),D),product_case_prod(B,C,D),G),aa(A,product_prod(B,C),F,X4)) ).
% scomp_def
tff(fact_141_fst__fstOp,axiom,
! [C: $tType,B: $tType,A: $tType,Bc: product_prod(A,B),P2: fun(A,fun(C,bool)),Q2: fun(C,fun(B,bool))] : aa(product_prod(A,B),A,product_fst(A,B),Bc) = aa(product_prod(A,B),A,aa(fun(product_prod(A,B),product_prod(A,C)),fun(product_prod(A,B),A),aa(fun(product_prod(A,C),A),fun(fun(product_prod(A,B),product_prod(A,C)),fun(product_prod(A,B),A)),comp(product_prod(A,C),A,product_prod(A,B)),product_fst(A,C)),bNF_fstOp(A,C,B,P2,Q2)),Bc) ).
% fst_fstOp
tff(fact_142_comp__apply,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(B,A),G: fun(C,B),X2: C] : aa(C,A,aa(fun(C,B),fun(C,A),aa(fun(B,A),fun(fun(C,B),fun(C,A)),comp(B,A,C),F),G),X2) = aa(B,A,F,aa(C,B,G,X2)) ).
% comp_apply
tff(fact_143_image__eqI,axiom,
! [A: $tType,B: $tType,B2: A,F: fun(B,A),X2: B,A6: set(B)] :
( ( B2 = aa(B,A,F,X2) )
=> ( pp(aa(set(B),bool,member(B,X2),A6))
=> pp(aa(set(A),bool,member(A,B2),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))) ) ) ).
% image_eqI
tff(fact_144_Inf_OINF__image,axiom,
! [B: $tType,A: $tType,C: $tType,Inf: fun(set(A),A),G: fun(B,A),F: fun(C,B),A6: set(C)] : aa(set(A),A,Inf,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),G),aa(set(C),set(B),aa(fun(C,B),fun(set(C),set(B)),image2(C,B),F),A6))) = aa(set(A),A,Inf,aa(set(C),set(A),aa(fun(C,A),fun(set(C),set(A)),image2(C,A),aa(fun(C,B),fun(C,A),aa(fun(B,A),fun(fun(C,B),fun(C,A)),comp(B,A,C),G),F)),A6)) ).
% Inf.INF_image
tff(fact_145_Sup_OSUP__image,axiom,
! [B: $tType,A: $tType,C: $tType,Sup: fun(set(A),A),G: fun(B,A),F: fun(C,B),A6: set(C)] : aa(set(A),A,Sup,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),G),aa(set(C),set(B),aa(fun(C,B),fun(set(C),set(B)),image2(C,B),F),A6))) = aa(set(A),A,Sup,aa(set(C),set(A),aa(fun(C,A),fun(set(C),set(A)),image2(C,A),aa(fun(C,B),fun(C,A),aa(fun(B,A),fun(fun(C,B),fun(C,A)),comp(B,A,C),G),F)),A6)) ).
% Sup.SUP_image
tff(fact_146_image__comp,axiom,
! [B: $tType,A: $tType,C: $tType,F: fun(B,A),G: fun(C,B),R: set(C)] : aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),aa(set(C),set(B),aa(fun(C,B),fun(set(C),set(B)),image2(C,B),G),R)) = aa(set(C),set(A),aa(fun(C,A),fun(set(C),set(A)),image2(C,A),aa(fun(C,B),fun(C,A),aa(fun(B,A),fun(fun(C,B),fun(C,A)),comp(B,A,C),F),G)),R) ).
% image_comp
tff(fact_147_image__eq__imp__comp,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F: fun(B,A),A6: set(B),G: fun(C,A),B6: set(C),H: fun(A,D)] :
( ( aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6) = aa(set(C),set(A),aa(fun(C,A),fun(set(C),set(A)),image2(C,A),G),B6) )
=> ( aa(set(B),set(D),aa(fun(B,D),fun(set(B),set(D)),image2(B,D),aa(fun(B,A),fun(B,D),aa(fun(A,D),fun(fun(B,A),fun(B,D)),comp(A,D,B),H),F)),A6) = aa(set(C),set(D),aa(fun(C,D),fun(set(C),set(D)),image2(C,D),aa(fun(C,A),fun(C,D),aa(fun(A,D),fun(fun(C,A),fun(C,D)),comp(A,D,C),H),G)),B6) ) ) ).
% image_eq_imp_comp
tff(fact_148_snd__comp__apsnd,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(B,C)] : aa(fun(product_prod(A,B),product_prod(A,C)),fun(product_prod(A,B),C),aa(fun(product_prod(A,C),C),fun(fun(product_prod(A,B),product_prod(A,C)),fun(product_prod(A,B),C)),comp(product_prod(A,C),C,product_prod(A,B)),product_snd(A,C)),aa(fun(B,C),fun(product_prod(A,B),product_prod(A,C)),product_apsnd(B,C,A),F)) = aa(fun(product_prod(A,B),B),fun(product_prod(A,B),C),aa(fun(B,C),fun(fun(product_prod(A,B),B),fun(product_prod(A,B),C)),comp(B,C,product_prod(A,B)),F),product_snd(A,B)) ).
% snd_comp_apsnd
tff(fact_149_fst__comp__apfst,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(A,C)] : aa(fun(product_prod(A,B),product_prod(C,B)),fun(product_prod(A,B),C),aa(fun(product_prod(C,B),C),fun(fun(product_prod(A,B),product_prod(C,B)),fun(product_prod(A,B),C)),comp(product_prod(C,B),C,product_prod(A,B)),product_fst(C,B)),product_apfst(A,C,B,F)) = aa(fun(product_prod(A,B),A),fun(product_prod(A,B),C),aa(fun(A,C),fun(fun(product_prod(A,B),A),fun(product_prod(A,B),C)),comp(A,C,product_prod(A,B)),F),product_fst(A,B)) ).
% fst_comp_apfst
tff(fact_150_Collect__case__prod__Grp__eqD,axiom,
! [B: $tType,A: $tType,Z2: product_prod(A,B),A6: set(A),F: fun(A,B)] :
( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),Z2),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),bNF_Grp(A,B,A6,F)))))
=> ( aa(product_prod(A,B),B,aa(fun(product_prod(A,B),A),fun(product_prod(A,B),B),aa(fun(A,B),fun(fun(product_prod(A,B),A),fun(product_prod(A,B),B)),comp(A,B,product_prod(A,B)),F),product_fst(A,B)),Z2) = aa(product_prod(A,B),B,product_snd(A,B),Z2) ) ) ).
% Collect_case_prod_Grp_eqD
tff(fact_151_Collect__split__mono__strong,axiom,
! [B: $tType,A: $tType,X6: set(A),A6: set(product_prod(A,B)),Y3: set(B),P2: fun(A,fun(B,bool)),Q2: fun(A,fun(B,bool))] :
( ( X6 = aa(set(product_prod(A,B)),set(A),aa(fun(product_prod(A,B),A),fun(set(product_prod(A,B)),set(A)),image2(product_prod(A,B),A),product_fst(A,B)),A6) )
=> ( ( Y3 = aa(set(product_prod(A,B)),set(B),aa(fun(product_prod(A,B),B),fun(set(product_prod(A,B)),set(B)),image2(product_prod(A,B),B),product_snd(A,B)),A6) )
=> ( ! [X: A] :
( pp(aa(set(A),bool,member(A,X),X6))
=> ! [Xa: B] :
( pp(aa(set(B),bool,member(B,Xa),Y3))
=> ( pp(aa(B,bool,aa(A,fun(B,bool),P2,X),Xa))
=> pp(aa(B,bool,aa(A,fun(B,bool),Q2,X),Xa)) ) ) )
=> ( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),A6),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),P2))))
=> pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),A6),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),Q2)))) ) ) ) ) ).
% Collect_split_mono_strong
tff(fact_152_snd__comp__map__prod,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,F: fun(A,D),G: fun(B,C)] : aa(fun(product_prod(A,B),product_prod(D,C)),fun(product_prod(A,B),C),aa(fun(product_prod(D,C),C),fun(fun(product_prod(A,B),product_prod(D,C)),fun(product_prod(A,B),C)),comp(product_prod(D,C),C,product_prod(A,B)),product_snd(D,C)),product_map_prod(A,D,B,C,F,G)) = aa(fun(product_prod(A,B),B),fun(product_prod(A,B),C),aa(fun(B,C),fun(fun(product_prod(A,B),B),fun(product_prod(A,B),C)),comp(B,C,product_prod(A,B)),G),product_snd(A,B)) ).
% snd_comp_map_prod
tff(fact_153_fst__comp__map__prod,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,F: fun(A,C),G: fun(B,D)] : aa(fun(product_prod(A,B),product_prod(C,D)),fun(product_prod(A,B),C),aa(fun(product_prod(C,D),C),fun(fun(product_prod(A,B),product_prod(C,D)),fun(product_prod(A,B),C)),comp(product_prod(C,D),C,product_prod(A,B)),product_fst(C,D)),product_map_prod(A,C,B,D,F,G)) = aa(fun(product_prod(A,B),A),fun(product_prod(A,B),C),aa(fun(A,C),fun(fun(product_prod(A,B),A),fun(product_prod(A,B),C)),comp(A,C,product_prod(A,B)),F),product_fst(A,B)) ).
% fst_comp_map_prod
tff(fact_154_map__prod__simp,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F: fun(C,A),G: fun(D,B),A3: C,B2: D] : aa(product_prod(C,D),product_prod(A,B),product_map_prod(C,A,D,B,F,G),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),A3),B2)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F,A3)),aa(D,B,G,B2)) ).
% map_prod_simp
tff(fact_155_fst__map__prod,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F: fun(C,A),G: fun(D,B),X2: product_prod(C,D)] : aa(product_prod(A,B),A,product_fst(A,B),aa(product_prod(C,D),product_prod(A,B),product_map_prod(C,A,D,B,F,G),X2)) = aa(C,A,F,aa(product_prod(C,D),C,product_fst(C,D),X2)) ).
% fst_map_prod
tff(fact_156_snd__map__prod,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F: fun(C,B),G: fun(D,A),X2: product_prod(C,D)] : aa(product_prod(B,A),A,product_snd(B,A),aa(product_prod(C,D),product_prod(B,A),product_map_prod(C,B,D,A,F,G),X2)) = aa(D,A,G,aa(product_prod(C,D),D,product_snd(C,D),X2)) ).
% snd_map_prod
tff(fact_157_apfst__conv,axiom,
! [C: $tType,A: $tType,B: $tType,F: fun(C,A),X2: C,Y2: B] : aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,F),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),X2),Y2)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F,X2)),Y2) ).
% apfst_conv
tff(fact_158_apsnd__conv,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(C,B),X2: A,Y2: C] : aa(product_prod(A,C),product_prod(A,B),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),F),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),X2),Y2)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),aa(C,B,F,Y2)) ).
% apsnd_conv
tff(fact_159_fst__apfst,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(C,A),X2: product_prod(C,B)] : aa(product_prod(A,B),A,product_fst(A,B),aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,F),X2)) = aa(C,A,F,aa(product_prod(C,B),C,product_fst(C,B),X2)) ).
% fst_apfst
tff(fact_160_apfst__eq__conv,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(C,A),X2: product_prod(C,B),G: fun(C,A)] :
( ( aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,F),X2) = aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,G),X2) )
<=> ( aa(C,A,F,aa(product_prod(C,B),C,product_fst(C,B),X2)) = aa(C,A,G,aa(product_prod(C,B),C,product_fst(C,B),X2)) ) ) ).
% apfst_eq_conv
tff(fact_161_snd__apfst,axiom,
! [B: $tType,A: $tType,C: $tType,F: fun(C,B),X2: product_prod(C,A)] : aa(product_prod(B,A),A,product_snd(B,A),aa(product_prod(C,A),product_prod(B,A),product_apfst(C,B,A,F),X2)) = aa(product_prod(C,A),A,product_snd(C,A),X2) ).
% snd_apfst
tff(fact_162_fst__apsnd,axiom,
! [B: $tType,C: $tType,A: $tType,F: fun(C,B),X2: product_prod(A,C)] : aa(product_prod(A,B),A,product_fst(A,B),aa(product_prod(A,C),product_prod(A,B),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),F),X2)) = aa(product_prod(A,C),A,product_fst(A,C),X2) ).
% fst_apsnd
tff(fact_163_snd__apsnd,axiom,
! [A: $tType,C: $tType,B: $tType,F: fun(C,A),X2: product_prod(B,C)] : aa(product_prod(B,A),A,product_snd(B,A),aa(product_prod(B,C),product_prod(B,A),aa(fun(C,A),fun(product_prod(B,C),product_prod(B,A)),product_apsnd(C,A,B),F),X2)) = aa(C,A,F,aa(product_prod(B,C),C,product_snd(B,C),X2)) ).
% snd_apsnd
tff(fact_164_apsnd__eq__conv,axiom,
! [B: $tType,C: $tType,A: $tType,F: fun(C,B),X2: product_prod(A,C),G: fun(C,B)] :
( ( aa(product_prod(A,C),product_prod(A,B),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),F),X2) = aa(product_prod(A,C),product_prod(A,B),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),G),X2) )
<=> ( aa(C,B,F,aa(product_prod(A,C),C,product_snd(A,C),X2)) = aa(C,B,G,aa(product_prod(A,C),C,product_snd(A,C),X2)) ) ) ).
% apsnd_eq_conv
tff(fact_165_map__prod__imageI,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,A3: A,B2: B,R3: set(product_prod(A,B)),F: fun(A,C),G: fun(B,D)] :
( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),R3))
=> pp(aa(set(product_prod(C,D)),bool,member(product_prod(C,D),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),aa(A,C,F,A3)),aa(B,D,G,B2))),aa(set(product_prod(A,B)),set(product_prod(C,D)),aa(fun(product_prod(A,B),product_prod(C,D)),fun(set(product_prod(A,B)),set(product_prod(C,D))),image2(product_prod(A,B),product_prod(C,D)),product_map_prod(A,C,B,D,F,G)),R3))) ) ).
% map_prod_imageI
tff(fact_166_snd__comp__apfst,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(A,C)] : aa(fun(product_prod(A,B),product_prod(C,B)),fun(product_prod(A,B),B),aa(fun(product_prod(C,B),B),fun(fun(product_prod(A,B),product_prod(C,B)),fun(product_prod(A,B),B)),comp(product_prod(C,B),B,product_prod(A,B)),product_snd(C,B)),product_apfst(A,C,B,F)) = product_snd(A,B) ).
% snd_comp_apfst
tff(fact_167_fst__comp__apsnd,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(B,C)] : aa(fun(product_prod(A,B),product_prod(A,C)),fun(product_prod(A,B),A),aa(fun(product_prod(A,C),A),fun(fun(product_prod(A,B),product_prod(A,C)),fun(product_prod(A,B),A)),comp(product_prod(A,C),A,product_prod(A,B)),product_fst(A,C)),aa(fun(B,C),fun(product_prod(A,B),product_prod(A,C)),product_apsnd(B,C,A),F)) = product_fst(A,B) ).
% fst_comp_apsnd
tff(fact_168_apfst__apsnd,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,F: fun(C,A),G: fun(D,B),X2: product_prod(C,D)] : aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,F),aa(product_prod(C,D),product_prod(C,B),aa(fun(D,B),fun(product_prod(C,D),product_prod(C,B)),product_apsnd(D,B,C),G),X2)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F,aa(product_prod(C,D),C,product_fst(C,D),X2))),aa(D,B,G,aa(product_prod(C,D),D,product_snd(C,D),X2))) ).
% apfst_apsnd
tff(fact_169_apsnd__apfst,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,F: fun(C,B),G: fun(D,A),X2: product_prod(D,C)] : aa(product_prod(A,C),product_prod(A,B),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),F),aa(product_prod(D,C),product_prod(A,C),product_apfst(D,A,C,G),X2)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(D,A,G,aa(product_prod(D,C),D,product_fst(D,C),X2))),aa(C,B,F,aa(product_prod(D,C),C,product_snd(D,C),X2))) ).
% apsnd_apfst
tff(fact_170_GrpE,axiom,
! [B: $tType,A: $tType,A6: set(A),F: fun(A,B),X2: A,Y2: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),bNF_Grp(A,B,A6,F),X2),Y2))
=> ~ ( ( aa(A,B,F,X2) = Y2 )
=> ~ pp(aa(set(A),bool,member(A,X2),A6)) ) ) ).
% GrpE
tff(fact_171_GrpI,axiom,
! [B: $tType,A: $tType,F: fun(B,A),X2: B,Y2: A,A6: set(B)] :
( ( aa(B,A,F,X2) = Y2 )
=> ( pp(aa(set(B),bool,member(B,X2),A6))
=> pp(aa(A,bool,aa(B,fun(A,bool),bNF_Grp(B,A,A6,F),X2),Y2)) ) ) ).
% GrpI
tff(fact_172_apsnd__apfst__commute,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,F: fun(C,B),G: fun(D,A),P: product_prod(D,C)] : aa(product_prod(A,C),product_prod(A,B),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),F),aa(product_prod(D,C),product_prod(A,C),product_apfst(D,A,C,G),P)) = aa(product_prod(D,B),product_prod(A,B),product_apfst(D,A,B,G),aa(product_prod(D,C),product_prod(D,B),aa(fun(C,B),fun(product_prod(D,C),product_prod(D,B)),product_apsnd(C,B,D),F),P)) ).
% apsnd_apfst_commute
tff(fact_173_subset__image__iff,axiom,
! [A: $tType,B: $tType,B6: set(A),F: fun(B,A),A6: set(B)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B6),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6)))
<=> ? [AA: set(B)] :
( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),AA),A6))
& ( B6 = aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),AA) ) ) ) ).
% subset_image_iff
tff(fact_174_image__subset__iff,axiom,
! [B: $tType,A: $tType,F: fun(B,A),A6: set(B),B6: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6)),B6))
<=> ! [X5: B] :
( pp(aa(set(B),bool,member(B,X5),A6))
=> pp(aa(set(A),bool,member(A,aa(B,A,F,X5)),B6)) ) ) ).
% image_subset_iff
tff(fact_175_subset__imageE,axiom,
! [A: $tType,B: $tType,B6: set(A),F: fun(B,A),A6: set(B)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B6),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6)))
=> ~ ! [C4: set(B)] :
( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),C4),A6))
=> ( B6 != aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),C4) ) ) ) ).
% subset_imageE
tff(fact_176_image__subsetI,axiom,
! [A: $tType,B: $tType,A6: set(A),F: fun(A,B),B6: set(B)] :
( ! [X: A] :
( pp(aa(set(A),bool,member(A,X),A6))
=> pp(aa(set(B),bool,member(B,aa(A,B,F,X)),B6)) )
=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6)),B6)) ) ).
% image_subsetI
tff(fact_177_image__mono,axiom,
! [B: $tType,A: $tType,A6: set(A),B6: set(A),F: fun(A,B)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B6))
=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6)),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),B6))) ) ).
% image_mono
tff(fact_178_map__prod_Ocomp,axiom,
! [A: $tType,C: $tType,E: $tType,F2: $tType,D: $tType,B: $tType,F: fun(C,E),G: fun(D,F2),H: fun(A,C),I: fun(B,D)] : aa(fun(product_prod(A,B),product_prod(C,D)),fun(product_prod(A,B),product_prod(E,F2)),aa(fun(product_prod(C,D),product_prod(E,F2)),fun(fun(product_prod(A,B),product_prod(C,D)),fun(product_prod(A,B),product_prod(E,F2))),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,aa(fun(A,C),fun(A,E),aa(fun(C,E),fun(fun(A,C),fun(A,E)),comp(C,E,A),F),H),aa(fun(B,D),fun(B,F2),aa(fun(D,F2),fun(fun(B,D),fun(B,F2)),comp(D,F2,B),G),I)) ).
% map_prod.comp
tff(fact_179_map__prod_Ocompositionality,axiom,
! [D: $tType,F2: $tType,E: $tType,C: $tType,B: $tType,A: $tType,F: fun(C,E),G: fun(D,F2),H: fun(A,C),I: fun(B,D),Prod: product_prod(A,B)] : aa(product_prod(C,D),product_prod(E,F2),product_map_prod(C,E,D,F2,F,G),aa(product_prod(A,B),product_prod(C,D),product_map_prod(A,C,B,D,H,I),Prod)) = aa(product_prod(A,B),product_prod(E,F2),product_map_prod(A,E,B,F2,aa(fun(A,C),fun(A,E),aa(fun(C,E),fun(fun(A,C),fun(A,E)),comp(C,E,A),F),H),aa(fun(B,D),fun(B,F2),aa(fun(D,F2),fun(fun(B,D),fun(B,F2)),comp(D,F2,B),G),I)),Prod) ).
% map_prod.compositionality
tff(fact_180_map__prod__compose,axiom,
! [D: $tType,C: $tType,A: $tType,E: $tType,F2: $tType,B: $tType,F1: fun(E,C),F22: fun(A,E),G1: fun(F2,D),G22: fun(B,F2)] : product_map_prod(A,C,B,D,aa(fun(A,E),fun(A,C),aa(fun(E,C),fun(fun(A,E),fun(A,C)),comp(E,C,A),F1),F22),aa(fun(B,F2),fun(B,D),aa(fun(F2,D),fun(fun(B,F2),fun(B,D)),comp(F2,D,B),G1),G22)) = aa(fun(product_prod(A,B),product_prod(E,F2)),fun(product_prod(A,B),product_prod(C,D)),aa(fun(product_prod(E,F2),product_prod(C,D)),fun(fun(product_prod(A,B),product_prod(E,F2)),fun(product_prod(A,B),product_prod(C,D))),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,F22,G22)) ).
% map_prod_compose
tff(fact_181_apfst__compose,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F: fun(C,A),G: fun(D,C),X2: product_prod(D,B)] : aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,F),aa(product_prod(D,B),product_prod(C,B),product_apfst(D,C,B,G),X2)) = aa(product_prod(D,B),product_prod(A,B),product_apfst(D,A,B,aa(fun(D,C),fun(D,A),aa(fun(C,A),fun(fun(D,C),fun(D,A)),comp(C,A,D),F),G)),X2) ).
% apfst_compose
tff(fact_182_apsnd__compose,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,F: fun(C,B),G: fun(D,C),X2: product_prod(A,D)] : aa(product_prod(A,C),product_prod(A,B),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),F),aa(product_prod(A,D),product_prod(A,C),aa(fun(D,C),fun(product_prod(A,D),product_prod(A,C)),product_apsnd(D,C,A),G),X2)) = aa(product_prod(A,D),product_prod(A,B),aa(fun(D,B),fun(product_prod(A,D),product_prod(A,B)),product_apsnd(D,B,A),aa(fun(D,C),fun(D,B),aa(fun(C,B),fun(fun(D,C),fun(D,B)),comp(C,B,D),F),G)),X2) ).
% apsnd_compose
tff(fact_183_Collect__case__prod__Grp__in,axiom,
! [B: $tType,A: $tType,Z2: product_prod(A,B),A6: set(A),F: fun(A,B)] :
( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),Z2),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),bNF_Grp(A,B,A6,F)))))
=> pp(aa(set(A),bool,member(A,aa(product_prod(A,B),A,product_fst(A,B),Z2)),A6)) ) ).
% Collect_case_prod_Grp_in
tff(fact_184_prod__fun__imageE,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,C3: product_prod(A,B),F: fun(C,A),G: fun(D,B),R3: set(product_prod(C,D))] :
( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),C3),aa(set(product_prod(C,D)),set(product_prod(A,B)),aa(fun(product_prod(C,D),product_prod(A,B)),fun(set(product_prod(C,D)),set(product_prod(A,B))),image2(product_prod(C,D),product_prod(A,B)),product_map_prod(C,A,D,B,F,G)),R3)))
=> ~ ! [X: C,Y: D] :
( ( C3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F,X)),aa(D,B,G,Y)) )
=> ~ pp(aa(set(product_prod(C,D)),bool,member(product_prod(C,D),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),X),Y)),R3)) ) ) ).
% prod_fun_imageE
tff(fact_185_lift__def,axiom,
! [B: $tType,A: $tType,F: fun(A,B)] : heap_lift(A,B,F) = aa(fun(A,B),fun(A,heap_Heap(B)),aa(fun(B,heap_Heap(B)),fun(fun(A,B),fun(A,heap_Heap(B))),comp(B,heap_Heap(B),A),heap_return(B)),F) ).
% lift_def
tff(fact_186_rev__image__eqI,axiom,
! [B: $tType,A: $tType,X2: A,A6: set(A),B2: B,F: fun(A,B)] :
( pp(aa(set(A),bool,member(A,X2),A6))
=> ( ( B2 = aa(A,B,F,X2) )
=> pp(aa(set(B),bool,member(B,B2),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6))) ) ) ).
% rev_image_eqI
tff(fact_187_ball__imageD,axiom,
! [A: $tType,B: $tType,F: fun(B,A),A6: set(B),P2: fun(A,bool)] :
( ! [X: A] :
( pp(aa(set(A),bool,member(A,X),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6)))
=> pp(aa(A,bool,P2,X)) )
=> ! [X4: B] :
( pp(aa(set(B),bool,member(B,X4),A6))
=> pp(aa(A,bool,P2,aa(B,A,F,X4))) ) ) ).
% ball_imageD
tff(fact_188_image__cong,axiom,
! [B: $tType,A: $tType,M: set(A),N: set(A),F: fun(A,B),G: fun(A,B)] :
( ( M = N )
=> ( ! [X: A] :
( pp(aa(set(A),bool,member(A,X),N))
=> ( aa(A,B,F,X) = aa(A,B,G,X) ) )
=> ( aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),M) = aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),G),N) ) ) ) ).
% image_cong
tff(fact_189_bex__imageD,axiom,
! [A: $tType,B: $tType,F: fun(B,A),A6: set(B),P2: fun(A,bool)] :
( ? [X4: A] :
( pp(aa(set(A),bool,member(A,X4),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6)))
& pp(aa(A,bool,P2,X4)) )
=> ? [X: B] :
( pp(aa(set(B),bool,member(B,X),A6))
& pp(aa(A,bool,P2,aa(B,A,F,X))) ) ) ).
% bex_imageD
tff(fact_190_image__iff,axiom,
! [A: $tType,B: $tType,Z2: A,F: fun(B,A),A6: set(B)] :
( pp(aa(set(A),bool,member(A,Z2),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6)))
<=> ? [X5: B] :
( pp(aa(set(B),bool,member(B,X5),A6))
& ( Z2 = aa(B,A,F,X5) ) ) ) ).
% image_iff
tff(fact_191_imageI,axiom,
! [B: $tType,A: $tType,X2: A,A6: set(A),F: fun(A,B)] :
( pp(aa(set(A),bool,member(A,X2),A6))
=> pp(aa(set(B),bool,member(B,aa(A,B,F,X2)),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6))) ) ).
% imageI
tff(fact_192_Sup_OSUP__cong,axiom,
! [A: $tType,B: $tType,A6: set(B),B6: set(B),C5: fun(B,A),D3: fun(B,A),Sup: fun(set(A),A)] :
( ( A6 = B6 )
=> ( ! [X: B] :
( pp(aa(set(B),bool,member(B,X),B6))
=> ( aa(B,A,C5,X) = aa(B,A,D3,X) ) )
=> ( aa(set(A),A,Sup,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),C5),A6)) = aa(set(A),A,Sup,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),D3),B6)) ) ) ) ).
% Sup.SUP_cong
tff(fact_193_Inf_OINF__cong,axiom,
! [A: $tType,B: $tType,A6: set(B),B6: set(B),C5: fun(B,A),D3: fun(B,A),Inf: fun(set(A),A)] :
( ( A6 = B6 )
=> ( ! [X: B] :
( pp(aa(set(B),bool,member(B,X),B6))
=> ( aa(B,A,C5,X) = aa(B,A,D3,X) ) )
=> ( aa(set(A),A,Inf,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),C5),A6)) = aa(set(A),A,Inf,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),D3),B6)) ) ) ) ).
% Inf.INF_cong
tff(fact_194_comp__eq__dest__lhs,axiom,
! [C: $tType,B: $tType,A: $tType,A3: fun(C,B),B2: fun(A,C),C3: fun(A,B),V: A] :
( ( aa(fun(A,C),fun(A,B),aa(fun(C,B),fun(fun(A,C),fun(A,B)),comp(C,B,A),A3),B2) = C3 )
=> ( aa(C,B,A3,aa(A,C,B2,V)) = aa(A,B,C3,V) ) ) ).
% comp_eq_dest_lhs
tff(fact_195_comp__eq__elim,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,A3: fun(C,B),B2: fun(A,C),C3: fun(D,B),D4: fun(A,D)] :
( ( aa(fun(A,C),fun(A,B),aa(fun(C,B),fun(fun(A,C),fun(A,B)),comp(C,B,A),A3),B2) = aa(fun(A,D),fun(A,B),aa(fun(D,B),fun(fun(A,D),fun(A,B)),comp(D,B,A),C3),D4) )
=> ! [V3: A] : aa(C,B,A3,aa(A,C,B2,V3)) = aa(D,B,C3,aa(A,D,D4,V3)) ) ).
% comp_eq_elim
tff(fact_196_comp__eq__dest,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,A3: fun(C,B),B2: fun(A,C),C3: fun(D,B),D4: fun(A,D),V: A] :
( ( aa(fun(A,C),fun(A,B),aa(fun(C,B),fun(fun(A,C),fun(A,B)),comp(C,B,A),A3),B2) = aa(fun(A,D),fun(A,B),aa(fun(D,B),fun(fun(A,D),fun(A,B)),comp(D,B,A),C3),D4) )
=> ( aa(C,B,A3,aa(A,C,B2,V)) = aa(D,B,C3,aa(A,D,D4,V)) ) ) ).
% comp_eq_dest
tff(fact_197_comp__assoc,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType,F: fun(D,B),G: fun(C,D),H: fun(A,C)] : aa(fun(A,C),fun(A,B),aa(fun(C,B),fun(fun(A,C),fun(A,B)),comp(C,B,A),aa(fun(C,D),fun(C,B),aa(fun(D,B),fun(fun(C,D),fun(C,B)),comp(D,B,C),F),G)),H) = aa(fun(A,D),fun(A,B),aa(fun(D,B),fun(fun(A,D),fun(A,B)),comp(D,B,A),F),aa(fun(A,C),fun(A,D),aa(fun(C,D),fun(fun(A,C),fun(A,D)),comp(C,D,A),G),H)) ).
% comp_assoc
tff(fact_198_comp__def,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(B,C),G: fun(A,B),X4: A] : aa(A,C,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),F),G),X4) = aa(B,C,F,aa(A,B,G,X4)) ).
% comp_def
tff(fact_199_apfst__convE,axiom,
! [C: $tType,A: $tType,B: $tType,Q: product_prod(A,B),F: fun(C,A),P: product_prod(C,B)] :
( ( Q = aa(product_prod(C,B),product_prod(A,B),product_apfst(C,A,B,F),P) )
=> ~ ! [X: C,Y: B] :
( ( P = aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),X),Y) )
=> ( Q != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F,X)),Y) ) ) ) ).
% apfst_convE
tff(fact_200_prod_Omap__comp,axiom,
! [D: $tType,F2: $tType,E: $tType,C: $tType,B: $tType,A: $tType,G1: fun(C,E),G22: fun(D,F2),F1: fun(A,C),F22: fun(B,D),V: product_prod(A,B)] : aa(product_prod(C,D),product_prod(E,F2),product_map_prod(C,E,D,F2,G1,G22),aa(product_prod(A,B),product_prod(C,D),product_map_prod(A,C,B,D,F1,F22),V)) = aa(product_prod(A,B),product_prod(E,F2),product_map_prod(A,E,B,F2,aa(fun(A,C),fun(A,E),aa(fun(C,E),fun(fun(A,C),fun(A,E)),comp(C,E,A),G1),F1),aa(fun(B,D),fun(B,F2),aa(fun(D,F2),fun(fun(B,D),fun(B,F2)),comp(D,F2,B),G22),F22)),V) ).
% prod.map_comp
tff(fact_201_subrelI,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,B)),S: set(product_prod(A,B))] :
( ! [X: A,Y: B] :
( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)),R))
=> pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)),S)) )
=> pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),R),S)) ) ).
% subrelI
tff(fact_202_all__subset__image,axiom,
! [A: $tType,B: $tType,F: fun(B,A),A6: set(B),P2: fun(set(A),bool)] :
( ! [B7: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B7),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6)))
=> pp(aa(set(A),bool,P2,B7)) )
<=> ! [B7: set(B)] :
( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B7),A6))
=> pp(aa(set(A),bool,P2,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),B7))) ) ) ).
% all_subset_image
tff(fact_203_rel__fun__Collect__case__prodD,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,A6: fun(A,fun(B,bool)),B6: fun(C,fun(D,bool)),F: fun(A,C),G: fun(B,D),X6: set(product_prod(A,B)),X2: product_prod(A,B)] :
( pp(aa(fun(B,D),bool,aa(fun(A,C),fun(fun(B,D),bool),bNF_rel_fun(A,B,C,D,A6,B6),F),G))
=> ( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),X6),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),A6))))
=> ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),X2),X6))
=> pp(aa(D,bool,aa(C,fun(D,bool),B6,aa(product_prod(A,B),C,aa(fun(product_prod(A,B),A),fun(product_prod(A,B),C),aa(fun(A,C),fun(fun(product_prod(A,B),A),fun(product_prod(A,B),C)),comp(A,C,product_prod(A,B)),F),product_fst(A,B)),X2)),aa(product_prod(A,B),D,aa(fun(product_prod(A,B),B),fun(product_prod(A,B),D),aa(fun(B,D),fun(fun(product_prod(A,B),B),fun(product_prod(A,B),D)),comp(B,D,product_prod(A,B)),G),product_snd(A,B)),X2))) ) ) ) ).
% rel_fun_Collect_case_prodD
tff(fact_204_convol__image__vimage2p,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,F: fun(C,A),G: fun(D,B),R3: fun(A,fun(B,bool))] : pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),aa(set(product_prod(C,D)),set(product_prod(A,B)),aa(fun(product_prod(C,D),product_prod(A,B)),fun(set(product_prod(C,D)),set(product_prod(A,B))),image2(product_prod(C,D),product_prod(A,B)),bNF_convol(product_prod(C,D),A,B,aa(fun(product_prod(C,D),C),fun(product_prod(C,D),A),aa(fun(C,A),fun(fun(product_prod(C,D),C),fun(product_prod(C,D),A)),comp(C,A,product_prod(C,D)),F),product_fst(C,D)),aa(fun(product_prod(C,D),D),fun(product_prod(C,D),B),aa(fun(D,B),fun(fun(product_prod(C,D),D),fun(product_prod(C,D),B)),comp(D,B,product_prod(C,D)),G),product_snd(C,D)))),aa(fun(product_prod(C,D),bool),set(product_prod(C,D)),collect(product_prod(C,D)),aa(fun(C,fun(D,bool)),fun(product_prod(C,D),bool),product_case_prod(C,D,bool),aa(fun(A,fun(B,bool)),fun(C,fun(D,bool)),bNF_vimage2p(C,A,D,B,bool,F,G),R3))))),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),R3)))) ).
% convol_image_vimage2p
tff(fact_205_prod_Osize__gen__o__map,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F: fun(C,nat),Fa: fun(D,nat),G: fun(A,C),Ga: fun(B,D)] : aa(fun(product_prod(A,B),product_prod(C,D)),fun(product_prod(A,B),nat),aa(fun(product_prod(C,D),nat),fun(fun(product_prod(A,B),product_prod(C,D)),fun(product_prod(A,B),nat)),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,aa(fun(A,C),fun(A,nat),aa(fun(C,nat),fun(fun(A,C),fun(A,nat)),comp(C,nat,A),F),G),aa(fun(B,D),fun(B,nat),aa(fun(D,nat),fun(fun(B,D),fun(B,nat)),comp(D,nat,B),Fa),Ga)) ).
% prod.size_gen_o_map
tff(fact_206_relChain__def,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [R: set(product_prod(A,A)),As: fun(A,B)] :
( bNF_Ca3754400796208372196lChain(A,B,R,As)
<=> ! [I3: A,J: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),I3),J)),R))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,As,I3)),aa(A,B,As,J))) ) ) ) ).
% relChain_def
tff(fact_207_Collect__case__prod__in__rel__leI,axiom,
! [B: $tType,A: $tType,X6: set(product_prod(A,B)),Y3: set(product_prod(A,B))] :
( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),X6),Y3))
=> pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),X6),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),fun_in_rel(A,B,Y3))))) ) ).
% Collect_case_prod_in_rel_leI
tff(fact_208_Collect__case__prod__in__rel__leE,axiom,
! [B: $tType,A: $tType,X6: set(product_prod(A,B)),Y3: set(product_prod(A,B))] :
( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),X6),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),fun_in_rel(A,B,Y3)))))
=> pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),X6),Y3)) ) ).
% Collect_case_prod_in_rel_leE
tff(fact_209_relcompp__assoc,axiom,
! [A: $tType,D: $tType,B: $tType,C: $tType,R: fun(A,fun(D,bool)),S: fun(D,fun(C,bool)),T2: fun(C,fun(B,bool))] : aa(fun(C,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(C,bool)),fun(fun(C,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,C,B),aa(fun(D,fun(C,bool)),fun(A,fun(C,bool)),aa(fun(A,fun(D,bool)),fun(fun(D,fun(C,bool)),fun(A,fun(C,bool))),relcompp(A,D,C),R),S)),T2) = aa(fun(D,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(D,bool)),fun(fun(D,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,D,B),R),aa(fun(C,fun(B,bool)),fun(D,fun(B,bool)),aa(fun(D,fun(C,bool)),fun(fun(C,fun(B,bool)),fun(D,fun(B,bool))),relcompp(D,C,B),S),T2)) ).
% relcompp_assoc
tff(fact_210_relcompp__apply,axiom,
! [A: $tType,B: $tType,C: $tType,R3: fun(A,fun(B,bool)),S3: fun(B,fun(C,bool)),A3: A,C3: C] :
( pp(aa(C,bool,aa(A,fun(C,bool),aa(fun(B,fun(C,bool)),fun(A,fun(C,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(C,bool)),fun(A,fun(C,bool))),relcompp(A,B,C),R3),S3),A3),C3))
<=> ? [B5: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),R3,A3),B5))
& pp(aa(C,bool,aa(B,fun(C,bool),S3,B5),C3)) ) ) ).
% relcompp_apply
tff(fact_211_rel__funI,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A6: fun(A,fun(B,bool)),B6: fun(C,fun(D,bool)),F: fun(A,C),G: fun(B,D)] :
( ! [X: A,Y: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),A6,X),Y))
=> pp(aa(D,bool,aa(C,fun(D,bool),B6,aa(A,C,F,X)),aa(B,D,G,Y))) )
=> pp(aa(fun(B,D),bool,aa(fun(A,C),fun(fun(B,D),bool),bNF_rel_fun(A,B,C,D,A6,B6),F),G)) ) ).
% rel_funI
tff(fact_212_fun_Omap__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,G2: $tType,F2: $tType,Rb: fun(A,fun(F2,bool)),Sd: fun(B,fun(G2,bool))] : pp(aa(fun(fun(F2,G2),fun(fun(D,F2),fun(D,G2))),bool,aa(fun(fun(A,B),fun(fun(D,A),fun(D,B))),fun(fun(fun(F2,G2),fun(fun(D,F2),fun(D,G2))),bool),bNF_rel_fun(fun(A,B),fun(F2,G2),fun(fun(D,A),fun(D,B)),fun(fun(D,F2),fun(D,G2)),bNF_rel_fun(A,F2,B,G2,Rb,Sd),bNF_rel_fun(fun(D,A),fun(D,F2),fun(D,B),fun(D,G2),bNF_rel_fun(D,D,A,F2,fequal(D),Rb),bNF_rel_fun(D,D,B,G2,fequal(D),Sd))),comp(A,B,D)),comp(F2,G2,D))) ).
% fun.map_transfer
tff(fact_213_rel__fun__iff__leq__vimage2p,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,R3: fun(A,fun(B,bool)),S3: fun(C,fun(D,bool)),F: fun(A,C),G: fun(B,D)] :
( pp(aa(fun(B,D),bool,aa(fun(A,C),fun(fun(B,D),bool),bNF_rel_fun(A,B,C,D,R3,S3),F),G))
<=> pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),R3),aa(fun(C,fun(D,bool)),fun(A,fun(B,bool)),bNF_vimage2p(A,C,B,D,bool,F,G),S3))) ) ).
% rel_fun_iff_leq_vimage2p
tff(fact_214_predicate2D__vimage2p,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,R3: fun(A,fun(B,bool)),F: fun(A,C),G: fun(B,D),S3: fun(C,fun(D,bool)),X2: A,Y2: B] :
( pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),R3),aa(fun(C,fun(D,bool)),fun(A,fun(B,bool)),bNF_vimage2p(A,C,B,D,bool,F,G),S3)))
=> ( pp(aa(B,bool,aa(A,fun(B,bool),R3,X2),Y2))
=> pp(aa(D,bool,aa(C,fun(D,bool),S3,aa(A,C,F,X2)),aa(B,D,G,Y2))) ) ) ).
% predicate2D_vimage2p
tff(fact_215_rel__fun__mono_H,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,Y3: fun(A,fun(B,bool)),X6: fun(A,fun(B,bool)),A6: fun(C,fun(D,bool)),B6: fun(C,fun(D,bool)),F: fun(A,C),G: fun(B,D)] :
( ! [X: A,Y: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),Y3,X),Y))
=> pp(aa(B,bool,aa(A,fun(B,bool),X6,X),Y)) )
=> ( ! [X: C,Y: D] :
( pp(aa(D,bool,aa(C,fun(D,bool),A6,X),Y))
=> pp(aa(D,bool,aa(C,fun(D,bool),B6,X),Y)) )
=> ( pp(aa(fun(B,D),bool,aa(fun(A,C),fun(fun(B,D),bool),bNF_rel_fun(A,B,C,D,X6,A6),F),G))
=> pp(aa(fun(B,D),bool,aa(fun(A,C),fun(fun(B,D),bool),bNF_rel_fun(A,B,C,D,Y3,B6),F),G)) ) ) ) ).
% rel_fun_mono'
tff(fact_216_rel__fun__mono,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,X6: fun(A,fun(B,bool)),A6: fun(C,fun(D,bool)),F: fun(A,C),G: fun(B,D),Y3: fun(A,fun(B,bool)),B6: fun(C,fun(D,bool))] :
( pp(aa(fun(B,D),bool,aa(fun(A,C),fun(fun(B,D),bool),bNF_rel_fun(A,B,C,D,X6,A6),F),G))
=> ( ! [X: A,Y: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),Y3,X),Y))
=> pp(aa(B,bool,aa(A,fun(B,bool),X6,X),Y)) )
=> ( ! [X: C,Y: D] :
( pp(aa(D,bool,aa(C,fun(D,bool),A6,X),Y))
=> pp(aa(D,bool,aa(C,fun(D,bool),B6,X),Y)) )
=> pp(aa(fun(B,D),bool,aa(fun(A,C),fun(fun(B,D),bool),bNF_rel_fun(A,B,C,D,Y3,B6),F),G)) ) ) ) ).
% rel_fun_mono
tff(fact_217_vimage2pI,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType,R3: fun(A,fun(B,bool)),F: fun(C,A),X2: C,G: fun(D,B),Y2: D] :
( pp(aa(B,bool,aa(A,fun(B,bool),R3,aa(C,A,F,X2)),aa(D,B,G,Y2)))
=> pp(aa(D,bool,aa(C,fun(D,bool),aa(fun(A,fun(B,bool)),fun(C,fun(D,bool)),bNF_vimage2p(C,A,D,B,bool,F,G),R3),X2),Y2)) ) ).
% vimage2pI
tff(fact_218_rel__funD,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A6: fun(A,fun(B,bool)),B6: fun(C,fun(D,bool)),F: fun(A,C),G: fun(B,D),X2: A,Y2: B] :
( pp(aa(fun(B,D),bool,aa(fun(A,C),fun(fun(B,D),bool),bNF_rel_fun(A,B,C,D,A6,B6),F),G))
=> ( pp(aa(B,bool,aa(A,fun(B,bool),A6,X2),Y2))
=> pp(aa(D,bool,aa(C,fun(D,bool),B6,aa(A,C,F,X2)),aa(B,D,G,Y2))) ) ) ).
% rel_funD
tff(fact_219_fun_Orel__compp,axiom,
! [A: $tType,D: $tType,C: $tType,B: $tType,R3: fun(A,fun(B,bool)),S3: fun(B,fun(C,bool))] : bNF_rel_fun(D,D,A,C,fequal(D),aa(fun(B,fun(C,bool)),fun(A,fun(C,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(C,bool)),fun(A,fun(C,bool))),relcompp(A,B,C),R3),S3)) = aa(fun(fun(D,B),fun(fun(D,C),bool)),fun(fun(D,A),fun(fun(D,C),bool)),aa(fun(fun(D,A),fun(fun(D,B),bool)),fun(fun(fun(D,B),fun(fun(D,C),bool)),fun(fun(D,A),fun(fun(D,C),bool))),relcompp(fun(D,A),fun(D,B),fun(D,C)),bNF_rel_fun(D,D,A,B,fequal(D),R3)),bNF_rel_fun(D,D,B,C,fequal(D),S3)) ).
% fun.rel_compp
tff(fact_220_relcompp__mono,axiom,
! [A: $tType,C: $tType,B: $tType,R2: fun(A,fun(B,bool)),R: fun(A,fun(B,bool)),S2: fun(B,fun(C,bool)),S: fun(B,fun(C,bool))] :
( pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),R2),R))
=> ( pp(aa(fun(B,fun(C,bool)),bool,aa(fun(B,fun(C,bool)),fun(fun(B,fun(C,bool)),bool),ord_less_eq(fun(B,fun(C,bool))),S2),S))
=> pp(aa(fun(A,fun(C,bool)),bool,aa(fun(A,fun(C,bool)),fun(fun(A,fun(C,bool)),bool),ord_less_eq(fun(A,fun(C,bool))),aa(fun(B,fun(C,bool)),fun(A,fun(C,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(C,bool)),fun(A,fun(C,bool))),relcompp(A,B,C),R2),S2)),aa(fun(B,fun(C,bool)),fun(A,fun(C,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(C,bool)),fun(A,fun(C,bool))),relcompp(A,B,C),R),S))) ) ) ).
% relcompp_mono
tff(fact_221_leq__OOI,axiom,
! [A: $tType,R3: fun(A,fun(A,bool))] :
( ( R3 = fequal(A) )
=> pp(aa(fun(A,fun(A,bool)),bool,aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),bool),ord_less_eq(fun(A,fun(A,bool))),R3),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),relcompp(A,A,A),R3),R3))) ) ).
% leq_OOI
tff(fact_222_convol__def,axiom,
! [B: $tType,C: $tType,A: $tType,F: fun(A,B),G: fun(A,C),X4: A] : aa(A,product_prod(B,C),bNF_convol(A,B,C,F,G),X4) = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(A,B,F,X4)),aa(A,C,G,X4)) ).
% convol_def
tff(fact_223_in__rel__def,axiom,
! [B: $tType,A: $tType,R3: set(product_prod(A,B)),X2: A,Y2: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),fun_in_rel(A,B,R3),X2),Y2))
<=> pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y2)),R3)) ) ).
% in_rel_def
tff(fact_224_in__rel__Collect__case__prod__eq,axiom,
! [B: $tType,A: $tType,X6: fun(A,fun(B,bool))] : fun_in_rel(A,B,aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),X6))) = X6 ).
% in_rel_Collect_case_prod_eq
tff(fact_225_Grp__mono,axiom,
! [B: $tType,A: $tType,A6: set(A),B6: set(A),F: fun(A,B)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B6))
=> pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),bNF_Grp(A,B,A6,F)),bNF_Grp(A,B,B6,F))) ) ).
% Grp_mono
tff(fact_226_Collect__case__prod__mono,axiom,
! [B: $tType,A: $tType,A6: fun(A,fun(B,bool)),B6: fun(A,fun(B,bool))] :
( pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),A6),B6))
=> pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),A6))),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),B6)))) ) ).
% Collect_case_prod_mono
tff(fact_227_fst__convol,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(A,B),G: fun(A,C)] : aa(fun(A,product_prod(B,C)),fun(A,B),aa(fun(product_prod(B,C),B),fun(fun(A,product_prod(B,C)),fun(A,B)),comp(product_prod(B,C),B,A),product_fst(B,C)),bNF_convol(A,B,C,F,G)) = F ).
% fst_convol
tff(fact_228_snd__convol,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(A,C),G: fun(A,B)] : aa(fun(A,product_prod(C,B)),fun(A,B),aa(fun(product_prod(C,B),B),fun(fun(A,product_prod(C,B)),fun(A,B)),comp(product_prod(C,B),B,A),product_snd(C,B)),bNF_convol(A,C,B,F,G)) = G ).
% snd_convol
tff(fact_229_fun_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,D: $tType,G: fun(B,C),F: fun(A,B),V: fun(D,A)] : aa(fun(D,B),fun(D,C),aa(fun(B,C),fun(fun(D,B),fun(D,C)),comp(B,C,D),G),aa(fun(D,A),fun(D,B),aa(fun(A,B),fun(fun(D,A),fun(D,B)),comp(A,B,D),F),V)) = aa(fun(D,A),fun(D,C),aa(fun(A,C),fun(fun(D,A),fun(D,C)),comp(A,C,D),aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),G),F)),V) ).
% fun.map_comp
tff(fact_230_relcompp_Ocases,axiom,
! [A: $tType,B: $tType,C: $tType,R: fun(A,fun(B,bool)),S: fun(B,fun(C,bool)),A1: A,A22: C] :
( pp(aa(C,bool,aa(A,fun(C,bool),aa(fun(B,fun(C,bool)),fun(A,fun(C,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(C,bool)),fun(A,fun(C,bool))),relcompp(A,B,C),R),S),A1),A22))
=> ~ ! [B4: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),R,A1),B4))
=> ~ pp(aa(C,bool,aa(B,fun(C,bool),S,B4),A22)) ) ) ).
% relcompp.cases
tff(fact_231_relcompp_Osimps,axiom,
! [A: $tType,B: $tType,C: $tType,R: fun(A,fun(B,bool)),S: fun(B,fun(C,bool)),A1: A,A22: C] :
( pp(aa(C,bool,aa(A,fun(C,bool),aa(fun(B,fun(C,bool)),fun(A,fun(C,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(C,bool)),fun(A,fun(C,bool))),relcompp(A,B,C),R),S),A1),A22))
<=> ? [A7: A,B5: B,C6: C] :
( ( A1 = A7 )
& ( A22 = C6 )
& pp(aa(B,bool,aa(A,fun(B,bool),R,A7),B5))
& pp(aa(C,bool,aa(B,fun(C,bool),S,B5),C6)) ) ) ).
% relcompp.simps
tff(fact_232_OO__eq,axiom,
! [B: $tType,A: $tType,R3: fun(A,fun(B,bool))] : aa(fun(B,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,B,B),R3),fequal(B)) = R3 ).
% OO_eq
tff(fact_233_eq__OO,axiom,
! [B: $tType,A: $tType,R3: fun(A,fun(B,bool))] : aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(A,bool)),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,A,B),fequal(A)),R3) = R3 ).
% eq_OO
tff(fact_234_relcomppE,axiom,
! [A: $tType,B: $tType,C: $tType,R: fun(A,fun(B,bool)),S: fun(B,fun(C,bool)),A3: A,C3: C] :
( pp(aa(C,bool,aa(A,fun(C,bool),aa(fun(B,fun(C,bool)),fun(A,fun(C,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(C,bool)),fun(A,fun(C,bool))),relcompp(A,B,C),R),S),A3),C3))
=> ~ ! [B4: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),R,A3),B4))
=> ~ pp(aa(C,bool,aa(B,fun(C,bool),S,B4),C3)) ) ) ).
% relcomppE
tff(fact_235_relcomppI,axiom,
! [A: $tType,B: $tType,C: $tType,R: fun(A,fun(B,bool)),A3: A,B2: B,S: fun(B,fun(C,bool)),C3: C] :
( pp(aa(B,bool,aa(A,fun(B,bool),R,A3),B2))
=> ( pp(aa(C,bool,aa(B,fun(C,bool),S,B2),C3))
=> pp(aa(C,bool,aa(A,fun(C,bool),aa(fun(B,fun(C,bool)),fun(A,fun(C,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(C,bool)),fun(A,fun(C,bool))),relcompp(A,B,C),R),S),A3),C3)) ) ) ).
% relcomppI
tff(fact_236_map__prod__o__convol,axiom,
! [D: $tType,B: $tType,C: $tType,E: $tType,A: $tType,H1: fun(D,B),H22: fun(E,C),F: fun(A,D),G: fun(A,E)] : aa(fun(A,product_prod(D,E)),fun(A,product_prod(B,C)),aa(fun(product_prod(D,E),product_prod(B,C)),fun(fun(A,product_prod(D,E)),fun(A,product_prod(B,C))),comp(product_prod(D,E),product_prod(B,C),A),product_map_prod(D,B,E,C,H1,H22)),bNF_convol(A,D,E,F,G)) = bNF_convol(A,B,C,aa(fun(A,D),fun(A,B),aa(fun(D,B),fun(fun(A,D),fun(A,B)),comp(D,B,A),H1),F),aa(fun(A,E),fun(A,C),aa(fun(E,C),fun(fun(A,E),fun(A,C)),comp(E,C,A),H22),G)) ).
% map_prod_o_convol
tff(fact_237_convol__expand__snd_H,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(A,product_prod(B,C)),G: fun(A,B),H: fun(A,C)] :
( ( aa(fun(A,product_prod(B,C)),fun(A,B),aa(fun(product_prod(B,C),B),fun(fun(A,product_prod(B,C)),fun(A,B)),comp(product_prod(B,C),B,A),product_fst(B,C)),F) = G )
=> ( ( H = aa(fun(A,product_prod(B,C)),fun(A,C),aa(fun(product_prod(B,C),C),fun(fun(A,product_prod(B,C)),fun(A,C)),comp(product_prod(B,C),C,A),product_snd(B,C)),F) )
<=> ( bNF_convol(A,B,C,G,H) = F ) ) ) ).
% convol_expand_snd'
tff(fact_238_convol__expand__snd,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(A,product_prod(B,C)),G: fun(A,B)] :
( ( aa(fun(A,product_prod(B,C)),fun(A,B),aa(fun(product_prod(B,C),B),fun(fun(A,product_prod(B,C)),fun(A,B)),comp(product_prod(B,C),B,A),product_fst(B,C)),F) = G )
=> ( bNF_convol(A,B,C,G,aa(fun(A,product_prod(B,C)),fun(A,C),aa(fun(product_prod(B,C),C),fun(fun(A,product_prod(B,C)),fun(A,C)),comp(product_prod(B,C),C,A),product_snd(B,C)),F)) = F ) ) ).
% convol_expand_snd
tff(fact_239_convol__o,axiom,
! [B: $tType,C: $tType,D: $tType,A: $tType,F: fun(D,B),G: fun(D,C),H: fun(A,D)] : aa(fun(A,D),fun(A,product_prod(B,C)),aa(fun(D,product_prod(B,C)),fun(fun(A,D),fun(A,product_prod(B,C))),comp(D,product_prod(B,C),A),bNF_convol(D,B,C,F,G)),H) = bNF_convol(A,B,C,aa(fun(A,D),fun(A,B),aa(fun(D,B),fun(fun(A,D),fun(A,B)),comp(D,B,A),F),H),aa(fun(A,D),fun(A,C),aa(fun(D,C),fun(fun(A,D),fun(A,C)),comp(D,C,A),G),H)) ).
% convol_o
tff(fact_240_vimage2p__relcompp__mono,axiom,
! [C: $tType,F2: $tType,E: $tType,D: $tType,B: $tType,A: $tType,R3: fun(A,fun(C,bool)),S3: fun(C,fun(B,bool)),T3: fun(A,fun(B,bool)),F: fun(D,A),G: fun(F2,C),H: fun(E,B)] :
( pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),aa(fun(C,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(C,bool)),fun(fun(C,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,C,B),R3),S3)),T3))
=> pp(aa(fun(D,fun(E,bool)),bool,aa(fun(D,fun(E,bool)),fun(fun(D,fun(E,bool)),bool),ord_less_eq(fun(D,fun(E,bool))),aa(fun(F2,fun(E,bool)),fun(D,fun(E,bool)),aa(fun(D,fun(F2,bool)),fun(fun(F2,fun(E,bool)),fun(D,fun(E,bool))),relcompp(D,F2,E),aa(fun(A,fun(C,bool)),fun(D,fun(F2,bool)),bNF_vimage2p(D,A,F2,C,bool,F,G),R3)),aa(fun(C,fun(B,bool)),fun(F2,fun(E,bool)),bNF_vimage2p(F2,C,E,B,bool,G,H),S3))),aa(fun(A,fun(B,bool)),fun(D,fun(E,bool)),bNF_vimage2p(D,A,E,B,bool,F,H),T3))) ) ).
% vimage2p_relcompp_mono
tff(fact_241_snd__convol_H,axiom,
! [B: $tType,A: $tType,C: $tType,F: fun(C,B),G: fun(C,A),X2: C] : aa(product_prod(B,A),A,product_snd(B,A),aa(C,product_prod(B,A),bNF_convol(C,B,A,F,G),X2)) = aa(C,A,G,X2) ).
% snd_convol'
tff(fact_242_fst__convol_H,axiom,
! [B: $tType,A: $tType,C: $tType,F: fun(C,A),G: fun(C,B),X2: C] : aa(product_prod(A,B),A,product_fst(A,B),aa(C,product_prod(A,B),bNF_convol(C,A,B,F,G),X2)) = aa(C,A,F,X2) ).
% fst_convol'
tff(fact_243_vimage2p__comp,axiom,
! [E: $tType,D: $tType,F2: $tType,A: $tType,C: $tType,B: $tType,G2: $tType,F1: fun(F2,A),F22: fun(D,F2),G1: fun(G2,B),G22: fun(E,G2)] : bNF_vimage2p(D,A,E,B,C,aa(fun(D,F2),fun(D,A),aa(fun(F2,A),fun(fun(D,F2),fun(D,A)),comp(F2,A,D),F1),F22),aa(fun(E,G2),fun(E,B),aa(fun(G2,B),fun(fun(E,G2),fun(E,B)),comp(G2,B,E),G1),G22)) = aa(fun(fun(A,fun(B,C)),fun(F2,fun(G2,C))),fun(fun(A,fun(B,C)),fun(D,fun(E,C))),aa(fun(fun(F2,fun(G2,C)),fun(D,fun(E,C))),fun(fun(fun(A,fun(B,C)),fun(F2,fun(G2,C))),fun(fun(A,fun(B,C)),fun(D,fun(E,C)))),comp(fun(F2,fun(G2,C)),fun(D,fun(E,C)),fun(A,fun(B,C))),bNF_vimage2p(D,F2,E,G2,C,F22,G22)),bNF_vimage2p(F2,A,G2,B,C,F1,G1)) ).
% vimage2p_comp
tff(fact_244_pos__fun__distr,axiom,
! [E: $tType,C: $tType,A: $tType,B: $tType,D: $tType,F2: $tType,R3: fun(A,fun(E,bool)),S3: fun(B,fun(F2,bool)),R5: fun(E,fun(C,bool)),S4: fun(F2,fun(D,bool))] : pp(aa(fun(fun(A,B),fun(fun(C,D),bool)),bool,aa(fun(fun(A,B),fun(fun(C,D),bool)),fun(fun(fun(A,B),fun(fun(C,D),bool)),bool),ord_less_eq(fun(fun(A,B),fun(fun(C,D),bool))),aa(fun(fun(E,F2),fun(fun(C,D),bool)),fun(fun(A,B),fun(fun(C,D),bool)),aa(fun(fun(A,B),fun(fun(E,F2),bool)),fun(fun(fun(E,F2),fun(fun(C,D),bool)),fun(fun(A,B),fun(fun(C,D),bool))),relcompp(fun(A,B),fun(E,F2),fun(C,D)),bNF_rel_fun(A,E,B,F2,R3,S3)),bNF_rel_fun(E,C,F2,D,R5,S4))),bNF_rel_fun(A,C,B,D,aa(fun(E,fun(C,bool)),fun(A,fun(C,bool)),aa(fun(A,fun(E,bool)),fun(fun(E,fun(C,bool)),fun(A,fun(C,bool))),relcompp(A,E,C),R3),R5),aa(fun(F2,fun(D,bool)),fun(B,fun(D,bool)),aa(fun(B,fun(F2,bool)),fun(fun(F2,fun(D,bool)),fun(B,fun(D,bool))),relcompp(B,F2,D),S3),S4)))) ).
% pos_fun_distr
tff(fact_245_o__rsp_I2_J,axiom,
! [E: $tType,F2: $tType,H5: $tType,G2: $tType,R1: fun(E,fun(F2,bool))] : pp(aa(fun(fun(G2,H5),fun(fun(F2,G2),fun(F2,H5))),bool,aa(fun(fun(G2,H5),fun(fun(E,G2),fun(E,H5))),fun(fun(fun(G2,H5),fun(fun(F2,G2),fun(F2,H5))),bool),bNF_rel_fun(fun(G2,H5),fun(G2,H5),fun(fun(E,G2),fun(E,H5)),fun(fun(F2,G2),fun(F2,H5)),fequal(fun(G2,H5)),bNF_rel_fun(fun(E,G2),fun(F2,G2),fun(E,H5),fun(F2,H5),bNF_rel_fun(E,F2,G2,G2,R1,fequal(G2)),bNF_rel_fun(E,F2,H5,H5,R1,fequal(H5)))),comp(G2,H5,E)),comp(G2,H5,F2))) ).
% o_rsp(2)
tff(fact_246_o__rsp_I1_J,axiom,
! [A: $tType,B: $tType,E: $tType,F2: $tType,D: $tType,C: $tType,R22: fun(A,fun(C,bool)),R32: fun(B,fun(D,bool)),R1: fun(E,fun(F2,bool))] : pp(aa(fun(fun(C,D),fun(fun(F2,C),fun(F2,D))),bool,aa(fun(fun(A,B),fun(fun(E,A),fun(E,B))),fun(fun(fun(C,D),fun(fun(F2,C),fun(F2,D))),bool),bNF_rel_fun(fun(A,B),fun(C,D),fun(fun(E,A),fun(E,B)),fun(fun(F2,C),fun(F2,D)),bNF_rel_fun(A,C,B,D,R22,R32),bNF_rel_fun(fun(E,A),fun(F2,C),fun(E,B),fun(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)
tff(fact_247_ge__eq__refl,axiom,
! [A: $tType,R3: fun(A,fun(A,bool)),X2: A] :
( pp(aa(fun(A,fun(A,bool)),bool,aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),bool),ord_less_eq(fun(A,fun(A,bool))),fequal(A)),R3))
=> pp(aa(A,bool,aa(A,fun(A,bool),R3,X2),X2)) ) ).
% ge_eq_refl
tff(fact_248_refl__ge__eq,axiom,
! [A: $tType,R3: fun(A,fun(A,bool))] :
( ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),R3,X),X))
=> pp(aa(fun(A,fun(A,bool)),bool,aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),bool),ord_less_eq(fun(A,fun(A,bool))),fequal(A)),R3)) ) ).
% refl_ge_eq
tff(fact_249_rewriteR__comp__comp2,axiom,
! [C: $tType,B: $tType,E: $tType,D: $tType,A: $tType,G: fun(C,B),H: fun(A,C),R12: fun(D,B),R23: fun(A,D),F: fun(B,E),L: fun(D,E)] :
( ( aa(fun(A,C),fun(A,B),aa(fun(C,B),fun(fun(A,C),fun(A,B)),comp(C,B,A),G),H) = aa(fun(A,D),fun(A,B),aa(fun(D,B),fun(fun(A,D),fun(A,B)),comp(D,B,A),R12),R23) )
=> ( ( aa(fun(D,B),fun(D,E),aa(fun(B,E),fun(fun(D,B),fun(D,E)),comp(B,E,D),F),R12) = L )
=> ( aa(fun(A,C),fun(A,E),aa(fun(C,E),fun(fun(A,C),fun(A,E)),comp(C,E,A),aa(fun(C,B),fun(C,E),aa(fun(B,E),fun(fun(C,B),fun(C,E)),comp(B,E,C),F),G)),H) = aa(fun(A,D),fun(A,E),aa(fun(D,E),fun(fun(A,D),fun(A,E)),comp(D,E,A),L),R23) ) ) ) ).
% rewriteR_comp_comp2
tff(fact_250_rewriteL__comp__comp2,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,E: $tType,F: fun(C,B),G: fun(A,C),L1: fun(D,B),L2: fun(A,D),H: fun(E,A),R: fun(E,D)] :
( ( aa(fun(A,C),fun(A,B),aa(fun(C,B),fun(fun(A,C),fun(A,B)),comp(C,B,A),F),G) = aa(fun(A,D),fun(A,B),aa(fun(D,B),fun(fun(A,D),fun(A,B)),comp(D,B,A),L1),L2) )
=> ( ( aa(fun(E,A),fun(E,D),aa(fun(A,D),fun(fun(E,A),fun(E,D)),comp(A,D,E),L2),H) = R )
=> ( aa(fun(E,C),fun(E,B),aa(fun(C,B),fun(fun(E,C),fun(E,B)),comp(C,B,E),F),aa(fun(E,A),fun(E,C),aa(fun(A,C),fun(fun(E,A),fun(E,C)),comp(A,C,E),G),H)) = aa(fun(E,D),fun(E,B),aa(fun(D,B),fun(fun(E,D),fun(E,B)),comp(D,B,E),L1),R) ) ) ) ).
% rewriteL_comp_comp2
tff(fact_251_rewriteR__comp__comp,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,G: fun(C,B),H: fun(A,C),R: fun(A,B),F: fun(B,D)] :
( ( aa(fun(A,C),fun(A,B),aa(fun(C,B),fun(fun(A,C),fun(A,B)),comp(C,B,A),G),H) = R )
=> ( aa(fun(A,C),fun(A,D),aa(fun(C,D),fun(fun(A,C),fun(A,D)),comp(C,D,A),aa(fun(C,B),fun(C,D),aa(fun(B,D),fun(fun(C,B),fun(C,D)),comp(B,D,C),F),G)),H) = aa(fun(A,B),fun(A,D),aa(fun(B,D),fun(fun(A,B),fun(A,D)),comp(B,D,A),F),R) ) ) ).
% rewriteR_comp_comp
tff(fact_252_rewriteL__comp__comp,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType,F: fun(C,B),G: fun(A,C),L: fun(A,B),H: fun(D,A)] :
( ( aa(fun(A,C),fun(A,B),aa(fun(C,B),fun(fun(A,C),fun(A,B)),comp(C,B,A),F),G) = L )
=> ( aa(fun(D,C),fun(D,B),aa(fun(C,B),fun(fun(D,C),fun(D,B)),comp(C,B,D),F),aa(fun(D,A),fun(D,C),aa(fun(A,C),fun(fun(D,A),fun(D,C)),comp(A,C,D),G),H)) = aa(fun(D,A),fun(D,B),aa(fun(A,B),fun(fun(D,A),fun(D,B)),comp(A,B,D),L),H) ) ) ).
% rewriteL_comp_comp
tff(fact_253_type__copy__map__cong0,axiom,
! [B: $tType,D: $tType,E: $tType,A: $tType,C: $tType,M: fun(B,A),G: fun(C,B),X2: C,N: fun(D,A),H: fun(C,D),F: fun(A,E)] :
( ( aa(B,A,M,aa(C,B,G,X2)) = aa(D,A,N,aa(C,D,H,X2)) )
=> ( aa(C,E,aa(fun(C,B),fun(C,E),aa(fun(B,E),fun(fun(C,B),fun(C,E)),comp(B,E,C),aa(fun(B,A),fun(B,E),aa(fun(A,E),fun(fun(B,A),fun(B,E)),comp(A,E,B),F),M)),G),X2) = aa(C,E,aa(fun(C,D),fun(C,E),aa(fun(D,E),fun(fun(C,D),fun(C,E)),comp(D,E,C),aa(fun(D,A),fun(D,E),aa(fun(A,E),fun(fun(D,A),fun(D,E)),comp(A,E,D),F),N)),H),X2) ) ) ).
% type_copy_map_cong0
tff(fact_254_eq__comp__r,axiom,
! [A: $tType,R3: fun(A,fun(A,bool))] : aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),relcompp(A,A,A),fequal(A)),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),relcompp(A,A,A),R3),fequal(A))) = R3 ).
% eq_comp_r
tff(fact_255_comp__transfer,axiom,
! [A: $tType,B: $tType,E: $tType,F2: $tType,D: $tType,C: $tType,B6: fun(A,fun(C,bool)),C5: fun(B,fun(D,bool)),A6: fun(E,fun(F2,bool))] : pp(aa(fun(fun(C,D),fun(fun(F2,C),fun(F2,D))),bool,aa(fun(fun(A,B),fun(fun(E,A),fun(E,B))),fun(fun(fun(C,D),fun(fun(F2,C),fun(F2,D))),bool),bNF_rel_fun(fun(A,B),fun(C,D),fun(fun(E,A),fun(E,B)),fun(fun(F2,C),fun(F2,D)),bNF_rel_fun(A,C,B,D,B6,C5),bNF_rel_fun(fun(E,A),fun(F2,C),fun(E,B),fun(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
tff(fact_256_rel__fun__iff__geq__image2p,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,R3: fun(A,fun(B,bool)),S3: fun(C,fun(D,bool)),F: fun(A,C),G: fun(B,D)] :
( pp(aa(fun(B,D),bool,aa(fun(A,C),fun(fun(B,D),bool),bNF_rel_fun(A,B,C,D,R3,S3),F),G))
<=> pp(aa(fun(C,fun(D,bool)),bool,aa(fun(C,fun(D,bool)),fun(fun(C,fun(D,bool)),bool),ord_less_eq(fun(C,fun(D,bool))),bNF_Greatest_image2p(A,C,B,D,F,G,R3)),S3)) ) ).
% rel_fun_iff_geq_image2p
tff(fact_257_map__prod__o__convol__id,axiom,
! [B: $tType,A: $tType,C: $tType,F: fun(C,A),G: fun(C,B),X2: C] : aa(C,product_prod(A,B),aa(fun(C,product_prod(C,B)),fun(C,product_prod(A,B)),aa(fun(product_prod(C,B),product_prod(A,B)),fun(fun(C,product_prod(C,B)),fun(C,product_prod(A,B))),comp(product_prod(C,B),product_prod(A,B),C),product_map_prod(C,A,B,B,F,id(B))),bNF_convol(C,C,B,id(C),G)),X2) = aa(C,product_prod(A,B),bNF_convol(C,A,B,aa(fun(C,A),fun(C,A),aa(fun(A,A),fun(fun(C,A),fun(C,A)),comp(A,A,C),id(A)),F),G),X2) ).
% map_prod_o_convol_id
tff(fact_258_in__rel__Gr,axiom,
! [B: $tType,A: $tType,A6: set(A),F: fun(A,B)] : fun_in_rel(A,B,bNF_Gr(A,B,A6,F)) = bNF_Grp(A,B,A6,F) ).
% in_rel_Gr
tff(fact_259_image__Fpow__mono,axiom,
! [B: $tType,A: $tType,F: fun(B,A),A6: set(B),B6: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6)),B6))
=> pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),aa(set(set(B)),set(set(A)),aa(fun(set(B),set(A)),fun(set(set(B)),set(set(A))),image2(set(B),set(A)),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F)),finite_Fpow(B,A6))),finite_Fpow(A,B6))) ) ).
% image_Fpow_mono
tff(fact_260_swap__comp__swap,axiom,
! [B: $tType,A: $tType] : aa(fun(product_prod(A,B),product_prod(B,A)),fun(product_prod(A,B),product_prod(A,B)),aa(fun(product_prod(B,A),product_prod(A,B)),fun(fun(product_prod(A,B),product_prod(B,A)),fun(product_prod(A,B),product_prod(A,B))),comp(product_prod(B,A),product_prod(A,B),product_prod(A,B)),product_swap(B,A)),product_swap(A,B)) = id(product_prod(A,B)) ).
% swap_comp_swap
tff(fact_261_surj__swap,axiom,
! [B: $tType,A: $tType] : aa(set(product_prod(B,A)),set(product_prod(A,B)),aa(fun(product_prod(B,A),product_prod(A,B)),fun(set(product_prod(B,A)),set(product_prod(A,B))),image2(product_prod(B,A),product_prod(A,B)),product_swap(B,A)),top_top(set(product_prod(B,A)))) = top_top(set(product_prod(A,B))) ).
% surj_swap
tff(fact_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 = aa(array(A),list(A),get(A,H),A3) )
=> heap_effect(list(A),freeze(A,A3),H,H2,R) ) ) ) ).
% effect_freezeI
tff(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 != aa(array(A),list(A),get(A,H),A3) ) ) ) ) ).
% effect_freezeE
tff(fact_264_convol__mem__GrpI,axiom,
! [B: $tType,A: $tType,X2: A,A6: set(A),G: fun(A,B)] :
( pp(aa(set(A),bool,member(A,X2),A6))
=> pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(A,product_prod(A,B),bNF_convol(A,A,B,id(A),G),X2)),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),bNF_Grp(A,B,A6,G))))) ) ).
% convol_mem_GrpI
tff(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: fun(A,heap_Heap(B))] :
( heap_effect(A,F,H,H2,X2)
=> ( heap_success(B,aa(A,heap_Heap(B),G,X2),H2)
=> heap_success(B,heap_bind(A,B,F,G),H) ) ) ).
% success_bind_effectI
tff(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 = aa(array(A),nat,length(A,H),A3) )
=> heap_effect(nat,len(A,A3),H,H2,R) ) ) ) ).
% effect_lengthI
tff(fact_267_image__id,axiom,
! [A: $tType] : aa(fun(A,A),fun(set(A),set(A)),image2(A,A),id(A)) = id(set(A)) ).
% image_id
tff(fact_268_fun_Omap__id,axiom,
! [A: $tType,D: $tType,T2: fun(D,A)] : aa(fun(D,A),fun(D,A),aa(fun(A,A),fun(fun(D,A),fun(D,A)),comp(A,A,D),id(A)),T2) = T2 ).
% fun.map_id
tff(fact_269_comp__id,axiom,
! [B: $tType,A: $tType,F: fun(A,B)] : aa(fun(A,A),fun(A,B),aa(fun(A,B),fun(fun(A,A),fun(A,B)),comp(A,B,A),F),id(A)) = F ).
% comp_id
tff(fact_270_id__comp,axiom,
! [B: $tType,A: $tType,G: fun(A,B)] : aa(fun(A,B),fun(A,B),aa(fun(B,B),fun(fun(A,B),fun(A,B)),comp(B,B,A),id(B)),G) = G ).
% id_comp
tff(fact_271_case__prod__Pair,axiom,
! [B: $tType,A: $tType] : aa(fun(A,fun(B,product_prod(A,B))),fun(product_prod(A,B),product_prod(A,B)),product_case_prod(A,B,product_prod(A,B)),product_Pair(A,B)) = id(product_prod(A,B)) ).
% case_prod_Pair
tff(fact_272_apfst__id,axiom,
! [B: $tType,A: $tType] : product_apfst(A,A,B,id(A)) = id(product_prod(A,B)) ).
% apfst_id
tff(fact_273_apsnd__id,axiom,
! [B: $tType,A: $tType] : aa(fun(B,B),fun(product_prod(A,B),product_prod(A,B)),product_apsnd(B,B,A),id(B)) = id(product_prod(A,B)) ).
% apsnd_id
tff(fact_274_range__fst,axiom,
! [B: $tType,A: $tType] : aa(set(product_prod(A,B)),set(A),aa(fun(product_prod(A,B),A),fun(set(product_prod(A,B)),set(A)),image2(product_prod(A,B),A),product_fst(A,B)),top_top(set(product_prod(A,B)))) = top_top(set(A)) ).
% range_fst
tff(fact_275_range__snd,axiom,
! [B: $tType,A: $tType] : aa(set(product_prod(B,A)),set(A),aa(fun(product_prod(B,A),A),fun(set(product_prod(B,A)),set(A)),image2(product_prod(B,A),A),product_snd(B,A)),top_top(set(product_prod(B,A)))) = top_top(set(A)) ).
% range_snd
tff(fact_276_success__ifI,axiom,
! [A: $tType,C3: bool,T2: heap_Heap(A),H: heap_ext(product_unit),E3: heap_Heap(A)] :
( ( pp(C3)
=> heap_success(A,T2,H) )
=> ( ( ~ pp(C3)
=> heap_success(A,E3,H) )
=> heap_success(A,if(heap_Heap(A),C3,T2,E3),H) ) ) ).
% success_ifI
tff(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
tff(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
tff(fact_279_fun_Omap__id0,axiom,
! [A: $tType,D: $tType] : aa(fun(A,A),fun(fun(D,A),fun(D,A)),comp(A,A,D),id(A)) = id(fun(D,A)) ).
% fun.map_id0
tff(fact_280_surj__id,axiom,
! [A: $tType] : aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),id(A)),top_top(set(A))) = top_top(set(A)) ).
% surj_id
tff(fact_281_Grp__UNIV__idI,axiom,
! [A: $tType,X2: A,Y2: A] :
( ( X2 = Y2 )
=> pp(aa(A,bool,aa(A,fun(A,bool),bNF_Grp(A,A,top_top(set(A)),id(A)),X2),Y2)) ) ).
% Grp_UNIV_idI
tff(fact_282_eq__alt,axiom,
! [A: $tType] : fequal(A) = bNF_Grp(A,A,top_top(set(A)),id(A)) ).
% eq_alt
tff(fact_283_image2pE,axiom,
! [D: $tType,B: $tType,A: $tType,C: $tType,F: fun(A,B),G: fun(C,D),R3: fun(A,fun(C,bool)),Fx: B,Gy: D] :
( pp(aa(D,bool,aa(B,fun(D,bool),bNF_Greatest_image2p(A,B,C,D,F,G,R3),Fx),Gy))
=> ~ ! [X: A] :
( ( Fx = aa(A,B,F,X) )
=> ! [Y: C] :
( ( Gy = aa(C,D,G,Y) )
=> ~ pp(aa(C,bool,aa(A,fun(C,bool),R3,X),Y)) ) ) ) ).
% image2pE
tff(fact_284_image2pI,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,R3: fun(A,fun(B,bool)),X2: A,Y2: B,F: fun(A,C),G: fun(B,D)] :
( pp(aa(B,bool,aa(A,fun(B,bool),R3,X2),Y2))
=> pp(aa(D,bool,aa(C,fun(D,bool),bNF_Greatest_image2p(A,C,B,D,F,G,R3),aa(A,C,F,X2)),aa(B,D,G,Y2))) ) ).
% image2pI
tff(fact_285_Inf_OINF__id__eq,axiom,
! [A: $tType,Inf: fun(set(A),A),A6: set(A)] : aa(set(A),A,Inf,aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),id(A)),A6)) = aa(set(A),A,Inf,A6) ).
% Inf.INF_id_eq
tff(fact_286_Sup_OSUP__id__eq,axiom,
! [A: $tType,Sup: fun(set(A),A),A6: set(A)] : aa(set(A),A,Sup,aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),id(A)),A6)) = aa(set(A),A,Sup,A6) ).
% Sup.SUP_id_eq
tff(fact_287_comp__eq__id__dest,axiom,
! [C: $tType,B: $tType,A: $tType,A3: fun(C,B),B2: fun(A,C),C3: fun(A,B),V: A] :
( ( aa(fun(A,C),fun(A,B),aa(fun(C,B),fun(fun(A,C),fun(A,B)),comp(C,B,A),A3),B2) = aa(fun(A,B),fun(A,B),aa(fun(B,B),fun(fun(A,B),fun(A,B)),comp(B,B,A),id(B)),C3) )
=> ( aa(C,B,A3,aa(A,C,B2,V)) = aa(A,B,C3,V) ) ) ).
% comp_eq_id_dest
tff(fact_288_pointfree__idE,axiom,
! [B: $tType,A: $tType,F: fun(B,A),G: fun(A,B),X2: A] :
( ( aa(fun(A,B),fun(A,A),aa(fun(B,A),fun(fun(A,B),fun(A,A)),comp(B,A,A),F),G) = id(A) )
=> ( aa(B,A,F,aa(A,B,G,X2)) = X2 ) ) ).
% pointfree_idE
tff(fact_289_surjD,axiom,
! [A: $tType,B: $tType,F: fun(B,A),Y2: A] :
( ( aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),top_top(set(B))) = top_top(set(A)) )
=> ? [X: B] : Y2 = aa(B,A,F,X) ) ).
% surjD
tff(fact_290_surjE,axiom,
! [A: $tType,B: $tType,F: fun(B,A),Y2: A] :
( ( aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),top_top(set(B))) = top_top(set(A)) )
=> ~ ! [X: B] : Y2 != aa(B,A,F,X) ) ).
% surjE
tff(fact_291_surjI,axiom,
! [B: $tType,A: $tType,G: fun(B,A),F: fun(A,B)] :
( ! [X: A] : aa(B,A,G,aa(A,B,F,X)) = X
=> ( aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),G),top_top(set(B))) = top_top(set(A)) ) ) ).
% surjI
tff(fact_292_rangeI,axiom,
! [A: $tType,B: $tType,F: fun(B,A),X2: B] : pp(aa(set(A),bool,member(A,aa(B,A,F,X2)),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),top_top(set(B))))) ).
% rangeI
tff(fact_293_surj__def,axiom,
! [A: $tType,B: $tType,F: fun(B,A)] :
( ( aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),top_top(set(B))) = top_top(set(A)) )
<=> ! [Y4: A] :
? [X5: B] : Y4 = aa(B,A,F,X5) ) ).
% surj_def
tff(fact_294_range__eqI,axiom,
! [A: $tType,B: $tType,B2: A,F: fun(B,A),X2: B] :
( ( B2 = aa(B,A,F,X2) )
=> pp(aa(set(A),bool,member(A,B2),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),top_top(set(B))))) ) ).
% range_eqI
tff(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
tff(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
tff(fact_297_map__prod__surj,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F: fun(A,B),G: fun(C,D)] :
( ( aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),top_top(set(A))) = top_top(set(B)) )
=> ( ( aa(set(C),set(D),aa(fun(C,D),fun(set(C),set(D)),image2(C,D),G),top_top(set(C))) = top_top(set(D)) )
=> ( aa(set(product_prod(A,C)),set(product_prod(B,D)),aa(fun(product_prod(A,C),product_prod(B,D)),fun(set(product_prod(A,C)),set(product_prod(B,D))),image2(product_prod(A,C),product_prod(B,D)),product_map_prod(A,B,C,D,F,G)),top_top(set(product_prod(A,C)))) = top_top(set(product_prod(B,D))) ) ) ) ).
% map_prod_surj
tff(fact_298_GrD1,axiom,
! [B: $tType,A: $tType,X2: A,Fx: B,A6: set(A),F: fun(A,B)] :
( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Fx)),bNF_Gr(A,B,A6,F)))
=> pp(aa(set(A),bool,member(A,X2),A6)) ) ).
% GrD1
tff(fact_299_GrD2,axiom,
! [A: $tType,B: $tType,X2: A,Fx: B,A6: set(A),F: fun(A,B)] :
( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Fx)),bNF_Gr(A,B,A6,F)))
=> ( aa(A,B,F,X2) = Fx ) ) ).
% GrD2
tff(fact_300_success__returnI,axiom,
! [A: $tType,X2: A,H: heap_ext(product_unit)] : heap_success(A,aa(A,heap_Heap(A),heap_return(A),X2),H) ).
% success_returnI
tff(fact_301_success__heapI,axiom,
! [A: $tType,F: fun(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
tff(fact_302_success__guardI,axiom,
! [A: $tType,P2: fun(heap_ext(product_unit),bool),H: heap_ext(product_unit),F: fun(heap_ext(product_unit),product_prod(A,heap_ext(product_unit)))] :
( pp(aa(heap_ext(product_unit),bool,P2,H))
=> heap_success(A,heap_guard(A,P2,F),H) ) ).
% success_guardI
tff(fact_303_success__tapI,axiom,
! [A: $tType,F: fun(heap_ext(product_unit),A),H: heap_ext(product_unit)] : heap_success(A,heap_tap(A,F),H) ).
% success_tapI
tff(fact_304_success__refI,axiom,
! [A: $tType] :
( heap(A)
=> ! [V: A,H: heap_ext(product_unit)] : heap_success(ref(A),aa(A,heap_Heap(ref(A)),ref3(A),V),H) ) ).
% success_refI
tff(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
tff(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
tff(fact_307_fun_Orel__cong,axiom,
! [A: $tType,B: $tType,D: $tType,X2: fun(D,A),Ya: fun(D,A),Y2: fun(D,B),Xa2: fun(D,B),R3: fun(A,fun(B,bool)),Ra: fun(A,fun(B,bool))] :
( ( X2 = Ya )
=> ( ( Y2 = Xa2 )
=> ( ! [Z3: A,Yb: B] :
( pp(aa(set(A),bool,member(A,Z3),aa(set(D),set(A),aa(fun(D,A),fun(set(D),set(A)),image2(D,A),Ya),top_top(set(D)))))
=> ( pp(aa(set(B),bool,member(B,Yb),aa(set(D),set(B),aa(fun(D,B),fun(set(D),set(B)),image2(D,B),Xa2),top_top(set(D)))))
=> ( pp(aa(B,bool,aa(A,fun(B,bool),R3,Z3),Yb))
<=> pp(aa(B,bool,aa(A,fun(B,bool),Ra,Z3),Yb)) ) ) )
=> ( pp(aa(fun(D,B),bool,aa(fun(D,A),fun(fun(D,B),bool),bNF_rel_fun(D,D,A,B,fequal(D),R3),X2),Y2))
<=> pp(aa(fun(D,B),bool,aa(fun(D,A),fun(fun(D,B),bool),bNF_rel_fun(D,D,A,B,fequal(D),Ra),Ya),Xa2)) ) ) ) ) ).
% fun.rel_cong
tff(fact_308_fun_Orel__mono__strong,axiom,
! [A: $tType,B: $tType,D: $tType,R3: fun(A,fun(B,bool)),X2: fun(D,A),Y2: fun(D,B),Ra: fun(A,fun(B,bool))] :
( pp(aa(fun(D,B),bool,aa(fun(D,A),fun(fun(D,B),bool),bNF_rel_fun(D,D,A,B,fequal(D),R3),X2),Y2))
=> ( ! [Z3: A,Yb: B] :
( pp(aa(set(A),bool,member(A,Z3),aa(set(D),set(A),aa(fun(D,A),fun(set(D),set(A)),image2(D,A),X2),top_top(set(D)))))
=> ( pp(aa(set(B),bool,member(B,Yb),aa(set(D),set(B),aa(fun(D,B),fun(set(D),set(B)),image2(D,B),Y2),top_top(set(D)))))
=> ( pp(aa(B,bool,aa(A,fun(B,bool),R3,Z3),Yb))
=> pp(aa(B,bool,aa(A,fun(B,bool),Ra,Z3),Yb)) ) ) )
=> pp(aa(fun(D,B),bool,aa(fun(D,A),fun(fun(D,B),bool),bNF_rel_fun(D,D,A,B,fequal(D),Ra),X2),Y2)) ) ) ).
% fun.rel_mono_strong
tff(fact_309_fun_Orel__refl__strong,axiom,
! [A: $tType,B: $tType,X2: fun(B,A),Ra: fun(A,fun(A,bool))] :
( ! [Z3: A] :
( pp(aa(set(A),bool,member(A,Z3),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),X2),top_top(set(B)))))
=> pp(aa(A,bool,aa(A,fun(A,bool),Ra,Z3),Z3)) )
=> pp(aa(fun(B,A),bool,aa(fun(B,A),fun(fun(B,A),bool),bNF_rel_fun(B,B,A,A,fequal(B),Ra),X2),X2)) ) ).
% fun.rel_refl_strong
tff(fact_310_range__subsetD,axiom,
! [B: $tType,A: $tType,F: fun(B,A),B6: set(A),I: B] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),top_top(set(B)))),B6))
=> pp(aa(set(A),bool,member(A,aa(B,A,F,I)),B6)) ) ).
% range_subsetD
tff(fact_311_fun_Oset__map,axiom,
! [B: $tType,A: $tType,D: $tType,F: fun(A,B),V: fun(D,A)] : aa(set(D),set(B),aa(fun(D,B),fun(set(D),set(B)),image2(D,B),aa(fun(D,A),fun(D,B),aa(fun(A,B),fun(fun(D,A),fun(D,B)),comp(A,B,D),F),V)),top_top(set(D))) = aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),aa(set(D),set(A),aa(fun(D,A),fun(set(D),set(A)),image2(D,A),V),top_top(set(D)))) ).
% fun.set_map
tff(fact_312_fun_Omap__cong,axiom,
! [B: $tType,A: $tType,D: $tType,X2: fun(D,A),Ya: fun(D,A),F: fun(A,B),G: fun(A,B)] :
( ( X2 = Ya )
=> ( ! [Z3: A] :
( pp(aa(set(A),bool,member(A,Z3),aa(set(D),set(A),aa(fun(D,A),fun(set(D),set(A)),image2(D,A),Ya),top_top(set(D)))))
=> ( aa(A,B,F,Z3) = aa(A,B,G,Z3) ) )
=> ( aa(fun(D,A),fun(D,B),aa(fun(A,B),fun(fun(D,A),fun(D,B)),comp(A,B,D),F),X2) = aa(fun(D,A),fun(D,B),aa(fun(A,B),fun(fun(D,A),fun(D,B)),comp(A,B,D),G),Ya) ) ) ) ).
% fun.map_cong
tff(fact_313_fun_Omap__cong0,axiom,
! [B: $tType,A: $tType,D: $tType,X2: fun(D,A),F: fun(A,B),G: fun(A,B)] :
( ! [Z3: A] :
( pp(aa(set(A),bool,member(A,Z3),aa(set(D),set(A),aa(fun(D,A),fun(set(D),set(A)),image2(D,A),X2),top_top(set(D)))))
=> ( aa(A,B,F,Z3) = aa(A,B,G,Z3) ) )
=> ( aa(fun(D,A),fun(D,B),aa(fun(A,B),fun(fun(D,A),fun(D,B)),comp(A,B,D),F),X2) = aa(fun(D,A),fun(D,B),aa(fun(A,B),fun(fun(D,A),fun(D,B)),comp(A,B,D),G),X2) ) ) ).
% fun.map_cong0
tff(fact_314_fun_Oinj__map__strong,axiom,
! [B: $tType,A: $tType,D: $tType,X2: fun(D,A),Xa2: fun(D,A),F: fun(A,B),Fa: fun(A,B)] :
( ! [Z3: A,Za: A] :
( pp(aa(set(A),bool,member(A,Z3),aa(set(D),set(A),aa(fun(D,A),fun(set(D),set(A)),image2(D,A),X2),top_top(set(D)))))
=> ( pp(aa(set(A),bool,member(A,Za),aa(set(D),set(A),aa(fun(D,A),fun(set(D),set(A)),image2(D,A),Xa2),top_top(set(D)))))
=> ( ( aa(A,B,F,Z3) = aa(A,B,Fa,Za) )
=> ( Z3 = Za ) ) ) )
=> ( ( aa(fun(D,A),fun(D,B),aa(fun(A,B),fun(fun(D,A),fun(D,B)),comp(A,B,D),F),X2) = aa(fun(D,A),fun(D,B),aa(fun(A,B),fun(fun(D,A),fun(D,B)),comp(A,B,D),Fa),Xa2) )
=> ( X2 = Xa2 ) ) ) ).
% fun.inj_map_strong
tff(fact_315_surj__fun__eq,axiom,
! [B: $tType,C: $tType,A: $tType,F: fun(B,A),X6: set(B),G1: fun(A,C),G22: fun(A,C)] :
( ( aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),X6) = top_top(set(A)) )
=> ( ! [X: B] :
( pp(aa(set(B),bool,member(B,X),X6))
=> ( aa(B,C,aa(fun(B,A),fun(B,C),aa(fun(A,C),fun(fun(B,A),fun(B,C)),comp(A,C,B),G1),F),X) = aa(B,C,aa(fun(B,A),fun(B,C),aa(fun(A,C),fun(fun(B,A),fun(B,C)),comp(A,C,B),G22),F),X) ) )
=> ( G1 = G22 ) ) ) ).
% surj_fun_eq
tff(fact_316_comp__surj,axiom,
! [B: $tType,A: $tType,C: $tType,F: fun(B,A),G: fun(A,C)] :
( ( aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),top_top(set(B))) = top_top(set(A)) )
=> ( ( aa(set(A),set(C),aa(fun(A,C),fun(set(A),set(C)),image2(A,C),G),top_top(set(A))) = top_top(set(C)) )
=> ( aa(set(B),set(C),aa(fun(B,C),fun(set(B),set(C)),image2(B,C),aa(fun(B,A),fun(B,C),aa(fun(A,C),fun(fun(B,A),fun(B,C)),comp(A,C,B),G),F)),top_top(set(B))) = top_top(set(C)) ) ) ) ).
% comp_surj
tff(fact_317_success__changeI,axiom,
! [A: $tType] :
( heap(A)
=> ! [F: fun(A,A),R: ref(A),H: heap_ext(product_unit)] : heap_success(A,change(A,F,R),H) ) ).
% success_changeI
tff(fact_318_apfst__def,axiom,
! [A: $tType,C: $tType,B: $tType,F: fun(A,C)] : product_apfst(A,C,B,F) = product_map_prod(A,C,B,B,F,id(B)) ).
% apfst_def
tff(fact_319_apsnd__def,axiom,
! [A: $tType,C: $tType,B: $tType,F: fun(B,C)] : aa(fun(B,C),fun(product_prod(A,B),product_prod(A,C)),product_apsnd(B,C,A),F) = product_map_prod(A,A,B,C,id(A),F) ).
% apsnd_def
tff(fact_320_rel__fun__image2p,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,R3: fun(A,fun(B,bool)),F: fun(A,C),G: fun(B,D)] : pp(aa(fun(B,D),bool,aa(fun(A,C),fun(fun(B,D),bool),bNF_rel_fun(A,B,C,D,R3,bNF_Greatest_image2p(A,C,B,D,F,G,R3)),F),G)) ).
% rel_fun_image2p
tff(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 = aa(array(A),nat,length(A,H2),A3) )
=> ( H2 != H ) ) ) ) ).
% effect_lengthE
tff(fact_322_comp__fun__idem__on_Ocomp__comp__fun__idem__on,axiom,
! [B: $tType,A: $tType,C: $tType,S3: set(A),F: fun(A,fun(B,B)),G: fun(C,A),R3: set(C)] :
( finite673082921795544331dem_on(A,B,S3,F)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(C),set(A),aa(fun(C,A),fun(set(C),set(A)),image2(C,A),G),top_top(set(C)))),S3))
=> finite673082921795544331dem_on(C,B,R3,aa(fun(C,A),fun(C,fun(B,B)),aa(fun(A,fun(B,B)),fun(fun(C,A),fun(C,fun(B,B))),comp(A,fun(B,B),C),F),G)) ) ) ).
% comp_fun_idem_on.comp_comp_fun_idem_on
tff(fact_323_comp__fun__commute__on_Ocomp__comp__fun__commute__on,axiom,
! [B: $tType,A: $tType,C: $tType,S3: set(A),F: fun(A,fun(B,B)),G: fun(C,A),R3: set(C)] :
( finite4664212375090638736ute_on(A,B,S3,F)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(C),set(A),aa(fun(C,A),fun(set(C),set(A)),image2(C,A),G),top_top(set(C)))),S3))
=> finite4664212375090638736ute_on(C,B,R3,aa(fun(C,A),fun(C,fun(B,B)),aa(fun(A,fun(B,B)),fun(fun(C,A),fun(C,fun(B,B))),comp(A,fun(B,B),C),F),G)) ) ) ).
% comp_fun_commute_on.comp_comp_fun_commute_on
tff(fact_324_image__Pow__mono,axiom,
! [B: $tType,A: $tType,F: fun(B,A),A6: set(B),B6: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6)),B6))
=> pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),aa(set(set(B)),set(set(A)),aa(fun(set(B),set(A)),fun(set(set(B)),set(set(A))),image2(set(B),set(A)),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F)),pow(B,A6))),pow(A,B6))) ) ).
% image_Pow_mono
tff(fact_325_vimage2p__Grp,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F: fun(A,C),G: fun(B,D),P2: fun(C,fun(D,bool))] : aa(fun(C,fun(D,bool)),fun(A,fun(B,bool)),bNF_vimage2p(A,C,B,D,bool,F,G),P2) = aa(fun(C,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(C,bool)),fun(fun(C,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,C,B),bNF_Grp(A,C,top_top(set(A)),F)),aa(fun(D,fun(B,bool)),fun(C,fun(B,bool)),aa(fun(C,fun(D,bool)),fun(fun(D,fun(B,bool)),fun(C,fun(B,bool))),relcompp(C,D,B),P2),conversep(B,D,bNF_Grp(B,D,top_top(set(B)),G)))) ).
% vimage2p_Grp
tff(fact_326_fun_Oin__rel,axiom,
! [A: $tType,B: $tType,D: $tType,R3: fun(A,fun(B,bool)),A3: fun(D,A),B2: fun(D,B)] :
( pp(aa(fun(D,B),bool,aa(fun(D,A),fun(fun(D,B),bool),bNF_rel_fun(D,D,A,B,fequal(D),R3),A3),B2))
<=> ? [Z4: fun(D,product_prod(A,B))] :
( pp(aa(set(fun(D,product_prod(A,B))),bool,member(fun(D,product_prod(A,B)),Z4),aa(fun(fun(D,product_prod(A,B)),bool),set(fun(D,product_prod(A,B))),collect(fun(D,product_prod(A,B))),aTP_Lamp_aa(fun(A,fun(B,bool)),fun(fun(D,product_prod(A,B)),bool),R3))))
& ( aa(fun(D,product_prod(A,B)),fun(D,A),aa(fun(product_prod(A,B),A),fun(fun(D,product_prod(A,B)),fun(D,A)),comp(product_prod(A,B),A,D),product_fst(A,B)),Z4) = A3 )
& ( aa(fun(D,product_prod(A,B)),fun(D,B),aa(fun(product_prod(A,B),B),fun(fun(D,product_prod(A,B)),fun(D,B)),comp(product_prod(A,B),B,D),product_snd(A,B)),Z4) = B2 ) ) ) ).
% fun.in_rel
tff(fact_327_Grp__UNIV__id,axiom,
! [A: $tType,F: fun(A,A)] :
( ( F = id(A) )
=> ( aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),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
tff(fact_328_strict__mono__inv,axiom,
! [A: $tType,B: $tType] :
( ( linorder(B)
& linorder(A) )
=> ! [F: fun(A,B),G: fun(B,A)] :
( order_strict_mono(A,B,F)
=> ( ( aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),top_top(set(A))) = top_top(set(B)) )
=> ( ! [X: A] : aa(B,A,G,aa(A,B,F,X)) = X
=> order_strict_mono(B,A,G) ) ) ) ) ).
% strict_mono_inv
tff(fact_329_Id__on__Gr,axiom,
! [A: $tType,A6: set(A)] : id_on(A,A6) = bNF_Gr(A,A,A6,id(A)) ).
% Id_on_Gr
tff(fact_330_type__copy__map__id0,axiom,
! [B: $tType,A: $tType,Rep: fun(A,B),Abs: fun(B,A),M: fun(B,B)] :
( type_definition(A,B,Rep,Abs,top_top(set(B)))
=> ( ( M = id(B) )
=> ( aa(fun(A,B),fun(A,A),aa(fun(B,A),fun(fun(A,B),fun(A,A)),comp(B,A,A),aa(fun(B,B),fun(B,A),aa(fun(B,A),fun(fun(B,B),fun(B,A)),comp(B,A,B),Abs),M)),Rep) = id(A) ) ) ) ).
% type_copy_map_id0
tff(fact_331_type__copy__Abs__o__Rep,axiom,
! [B: $tType,A: $tType,Rep: fun(A,B),Abs: fun(B,A)] :
( type_definition(A,B,Rep,Abs,top_top(set(B)))
=> ( aa(fun(A,B),fun(A,A),aa(fun(B,A),fun(fun(A,B),fun(A,A)),comp(B,A,A),Abs),Rep) = id(A) ) ) ).
% type_copy_Abs_o_Rep
tff(fact_332_type__copy__Rep__o__Abs,axiom,
! [A: $tType,B: $tType,Rep: fun(A,B),Abs: fun(B,A)] :
( type_definition(A,B,Rep,Abs,top_top(set(B)))
=> ( aa(fun(B,A),fun(B,B),aa(fun(A,B),fun(fun(B,A),fun(B,B)),comp(A,B,B),Rep),Abs) = id(B) ) ) ).
% type_copy_Rep_o_Abs
tff(fact_333_case__prodI2_H,axiom,
! [A: $tType,B: $tType,C: $tType,P: product_prod(A,B),C3: fun(A,fun(B,fun(C,bool))),X2: C] :
( ! [A5: A,B4: B] :
( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4) = P )
=> pp(aa(C,bool,aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),C3,A5),B4),X2)) )
=> pp(aa(C,bool,aa(product_prod(A,B),fun(C,bool),aa(fun(A,fun(B,fun(C,bool))),fun(product_prod(A,B),fun(C,bool)),product_case_prod(A,B,fun(C,bool)),C3),P),X2)) ) ).
% case_prodI2'
tff(fact_334_image__ident,axiom,
! [A: $tType,Y3: set(A)] : aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aTP_Lamp_ab(A,A)),Y3) = Y3 ).
% image_ident
tff(fact_335_split__part,axiom,
! [B: $tType,A: $tType,P2: bool,Q2: fun(A,fun(B,bool)),X4: product_prod(A,B)] :
( pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aTP_Lamp_ac(bool,fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),P2),Q2)),X4))
<=> ( pp(P2)
& pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),Q2),X4)) ) ) ).
% split_part
tff(fact_336_map__prod__ident,axiom,
! [B: $tType,A: $tType,X4: product_prod(A,B)] : aa(product_prod(A,B),product_prod(A,B),product_map_prod(A,A,B,B,aTP_Lamp_ab(A,A),aTP_Lamp_ad(B,B)),X4) = X4 ).
% map_prod_ident
tff(fact_337_Heap__Monad_Obind__bind,axiom,
! [A: $tType,B: $tType,C: $tType,F: heap_Heap(A),G: fun(A,heap_Heap(C)),K: fun(C,heap_Heap(B))] : heap_bind(C,B,heap_bind(A,C,F,G),K) = heap_bind(A,B,F,aa(fun(C,heap_Heap(B)),fun(A,heap_Heap(B)),aTP_Lamp_ae(fun(A,heap_Heap(C)),fun(fun(C,heap_Heap(B)),fun(A,heap_Heap(B))),G),K)) ).
% Heap_Monad.bind_bind
tff(fact_338_Id__onI,axiom,
! [A: $tType,A3: A,A6: set(A)] :
( pp(aa(set(A),bool,member(A,A3),A6))
=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A3)),id_on(A,A6))) ) ).
% Id_onI
tff(fact_339_case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,F: fun(B,fun(C,A)),A3: B,B2: C] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),F),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A3),B2)) = aa(C,A,aa(B,fun(C,A),F,A3),B2) ).
% case_prod_conv
tff(fact_340_case__prodI,axiom,
! [A: $tType,B: $tType,F: fun(A,fun(B,bool)),A3: A,B2: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),F,A3),B2))
=> pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),F),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2))) ) ).
% case_prodI
tff(fact_341_case__prodI2,axiom,
! [B: $tType,A: $tType,P: product_prod(A,B),C3: fun(A,fun(B,bool))] :
( ! [A5: A,B4: B] :
( ( P = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4) )
=> pp(aa(B,bool,aa(A,fun(B,bool),C3,A5),B4)) )
=> pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),C3),P)) ) ).
% case_prodI2
tff(fact_342_mem__case__prodI,axiom,
! [A: $tType,B: $tType,C: $tType,Z2: A,C3: fun(B,fun(C,set(A))),A3: B,B2: C] :
( pp(aa(set(A),bool,member(A,Z2),aa(C,set(A),aa(B,fun(C,set(A)),C3,A3),B2)))
=> pp(aa(set(A),bool,member(A,Z2),aa(product_prod(B,C),set(A),aa(fun(B,fun(C,set(A))),fun(product_prod(B,C),set(A)),product_case_prod(B,C,set(A)),C3),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A3),B2)))) ) ).
% mem_case_prodI
tff(fact_343_mem__case__prodI2,axiom,
! [C: $tType,B: $tType,A: $tType,P: product_prod(A,B),Z2: C,C3: fun(A,fun(B,set(C)))] :
( ! [A5: A,B4: B] :
( ( P = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4) )
=> pp(aa(set(C),bool,member(C,Z2),aa(B,set(C),aa(A,fun(B,set(C)),C3,A5),B4))) )
=> pp(aa(set(C),bool,member(C,Z2),aa(product_prod(A,B),set(C),aa(fun(A,fun(B,set(C))),fun(product_prod(A,B),set(C)),product_case_prod(A,B,set(C)),C3),P))) ) ).
% mem_case_prodI2
tff(fact_344_case__swap,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(C,fun(B,A)),P: product_prod(C,B)] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),aTP_Lamp_af(fun(C,fun(B,A)),fun(B,fun(C,A)),F)),aa(product_prod(C,B),product_prod(B,C),product_swap(C,B),P)) = aa(product_prod(C,B),A,aa(fun(C,fun(B,A)),fun(product_prod(C,B),A),product_case_prod(C,B,A),F),P) ).
% case_swap
tff(fact_345_pair__imageI,axiom,
! [C: $tType,B: $tType,A: $tType,A3: A,B2: B,A6: set(product_prod(A,B)),F: fun(A,fun(B,C))] :
( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),A6))
=> pp(aa(set(C),bool,member(C,aa(B,C,aa(A,fun(B,C),F,A3),B2)),aa(set(product_prod(A,B)),set(C),aa(fun(product_prod(A,B),C),fun(set(product_prod(A,B)),set(C)),image2(product_prod(A,B),C),aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F)),A6))) ) ).
% pair_imageI
tff(fact_346_Inf_OINF__identity__eq,axiom,
! [A: $tType,Inf: fun(set(A),A),A6: set(A)] : aa(set(A),A,Inf,aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aTP_Lamp_ab(A,A)),A6)) = aa(set(A),A,Inf,A6) ).
% Inf.INF_identity_eq
tff(fact_347_Sup_OSUP__identity__eq,axiom,
! [A: $tType,Sup: fun(set(A),A),A6: set(A)] : aa(set(A),A,Sup,aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aTP_Lamp_ab(A,A)),A6)) = aa(set(A),A,Sup,A6) ).
% Sup.SUP_identity_eq
tff(fact_348_imageE,axiom,
! [A: $tType,B: $tType,B2: A,F: fun(B,A),A6: set(B)] :
( pp(aa(set(A),bool,member(A,B2),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6)))
=> ~ ! [X: B] :
( ( B2 = aa(B,A,F,X) )
=> ~ pp(aa(set(B),bool,member(B,X),A6)) ) ) ).
% imageE
tff(fact_349_image__image,axiom,
! [B: $tType,A: $tType,C: $tType,F: fun(B,A),G: fun(C,B),A6: set(C)] : aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),aa(set(C),set(B),aa(fun(C,B),fun(set(C),set(B)),image2(C,B),G),A6)) = aa(set(C),set(A),aa(fun(C,A),fun(set(C),set(A)),image2(C,A),aa(fun(C,B),fun(C,A),aTP_Lamp_ag(fun(B,A),fun(fun(C,B),fun(C,A)),F),G)),A6) ).
% image_image
tff(fact_350_Compr__image__eq,axiom,
! [B: $tType,A: $tType,F: fun(B,A),A6: set(B),P2: fun(A,bool)] : aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aa(set(B),fun(fun(A,bool),fun(A,bool)),aTP_Lamp_ah(fun(B,A),fun(set(B),fun(fun(A,bool),fun(A,bool))),F),A6),P2)) = aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),aa(fun(B,bool),set(B),collect(B),aa(fun(A,bool),fun(B,bool),aa(set(B),fun(fun(A,bool),fun(B,bool)),aTP_Lamp_ai(fun(B,A),fun(set(B),fun(fun(A,bool),fun(B,bool))),F),A6),P2))) ).
% Compr_image_eq
tff(fact_351_Id__on__def_H,axiom,
! [A: $tType,A6: fun(A,bool)] : id_on(A,aa(fun(A,bool),set(A),collect(A),A6)) = aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aTP_Lamp_aj(fun(A,bool),fun(A,fun(A,bool)),A6))) ).
% Id_on_def'
tff(fact_352_pred__equals__eq2,axiom,
! [B: $tType,A: $tType,R3: set(product_prod(A,B)),S3: set(product_prod(A,B))] :
( ! [X5: A,Xa3: B] :
( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X5),Xa3)),R3))
<=> pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X5),Xa3)),S3)) )
<=> ( R3 = S3 ) ) ).
% pred_equals_eq2
tff(fact_353_case__prodE_H,axiom,
! [B: $tType,A: $tType,C: $tType,C3: fun(A,fun(B,fun(C,bool))),P: product_prod(A,B),Z2: C] :
( pp(aa(C,bool,aa(product_prod(A,B),fun(C,bool),aa(fun(A,fun(B,fun(C,bool))),fun(product_prod(A,B),fun(C,bool)),product_case_prod(A,B,fun(C,bool)),C3),P),Z2))
=> ~ ! [X: A,Y: B] :
( ( P = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y) )
=> ~ pp(aa(C,bool,aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),C3,X),Y),Z2)) ) ) ).
% case_prodE'
tff(fact_354_case__prodD_H,axiom,
! [B: $tType,A: $tType,C: $tType,R3: fun(A,fun(B,fun(C,bool))),A3: A,B2: B,C3: C] :
( pp(aa(C,bool,aa(product_prod(A,B),fun(C,bool),aa(fun(A,fun(B,fun(C,bool))),fun(product_prod(A,B),fun(C,bool)),product_case_prod(A,B,fun(C,bool)),R3),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),C3))
=> pp(aa(C,bool,aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),R3,A3),B2),C3)) ) ).
% case_prodD'
tff(fact_355_case__prod__Pair__iden,axiom,
! [B: $tType,A: $tType,P: product_prod(A,B)] : aa(product_prod(A,B),product_prod(A,B),aa(fun(A,fun(B,product_prod(A,B))),fun(product_prod(A,B),product_prod(A,B)),product_case_prod(A,B,product_prod(A,B)),product_Pair(A,B)),P) = P ).
% case_prod_Pair_iden
tff(fact_356_subset__CollectI,axiom,
! [A: $tType,B6: set(A),A6: set(A),Q2: fun(A,bool),P2: fun(A,bool)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B6),A6))
=> ( ! [X: A] :
( pp(aa(set(A),bool,member(A,X),B6))
=> ( pp(aa(A,bool,Q2,X))
=> pp(aa(A,bool,P2,X)) ) )
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_ak(set(A),fun(fun(A,bool),fun(A,bool)),B6),Q2))),aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_ak(set(A),fun(fun(A,bool),fun(A,bool)),A6),P2)))) ) ) ).
% subset_CollectI
tff(fact_357_subset__Collect__iff,axiom,
! [A: $tType,B6: set(A),A6: set(A),P2: fun(A,bool)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B6),A6))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B6),aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_ak(set(A),fun(fun(A,bool),fun(A,bool)),A6),P2))))
<=> ! [X5: A] :
( pp(aa(set(A),bool,member(A,X5),B6))
=> pp(aa(A,bool,P2,X5)) ) ) ) ).
% subset_Collect_iff
tff(fact_358_prop__restrict,axiom,
! [A: $tType,X2: A,Z5: set(A),X6: set(A),P2: fun(A,bool)] :
( pp(aa(set(A),bool,member(A,X2),Z5))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Z5),aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_ak(set(A),fun(fun(A,bool),fun(A,bool)),X6),P2))))
=> pp(aa(A,bool,P2,X2)) ) ) ).
% prop_restrict
tff(fact_359_Collect__restrict,axiom,
! [A: $tType,X6: set(A),P2: fun(A,bool)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(fun(A,bool),set(A),collect(A),aa(fun(A,bool),fun(A,bool),aTP_Lamp_ak(set(A),fun(fun(A,bool),fun(A,bool)),X6),P2))),X6)) ).
% Collect_restrict
tff(fact_360_fun_Omap__ident,axiom,
! [A: $tType,D: $tType,T2: fun(D,A)] : aa(fun(D,A),fun(D,A),aa(fun(A,A),fun(fun(D,A),fun(D,A)),comp(A,A,D),aTP_Lamp_ab(A,A)),T2) = T2 ).
% fun.map_ident
tff(fact_361_case__prod__app,axiom,
! [D: $tType,A: $tType,C: $tType,B: $tType,F: fun(B,fun(C,fun(D,A))),X2: product_prod(B,C),Y2: D] : aa(D,A,aa(product_prod(B,C),fun(D,A),aa(fun(B,fun(C,fun(D,A))),fun(product_prod(B,C),fun(D,A)),product_case_prod(B,C,fun(D,A)),F),X2),Y2) = aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),aa(D,fun(B,fun(C,A)),aTP_Lamp_al(fun(B,fun(C,fun(D,A))),fun(D,fun(B,fun(C,A))),F),Y2)),X2) ).
% case_prod_app
tff(fact_362_prod_Ocase__distrib,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,H: fun(C,D),F: fun(A,fun(B,C)),Prod: product_prod(A,B)] : aa(C,D,H,aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F),Prod)) = aa(product_prod(A,B),D,aa(fun(A,fun(B,D)),fun(product_prod(A,B),D),product_case_prod(A,B,D),aa(fun(A,fun(B,C)),fun(A,fun(B,D)),aTP_Lamp_am(fun(C,D),fun(fun(A,fun(B,C)),fun(A,fun(B,D))),H),F)),Prod) ).
% prod.case_distrib
tff(fact_363_prod_Odisc__eq__case,axiom,
! [B: $tType,A: $tType,Prod: product_prod(A,B)] : pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aTP_Lamp_an(A,fun(B,bool))),Prod)) ).
% prod.disc_eq_case
tff(fact_364_Grp__def,axiom,
! [B: $tType,A: $tType,A6: set(A),F: fun(A,B),X4: A,Xa4: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),bNF_Grp(A,B,A6,F),X4),Xa4))
<=> ( ( Xa4 = aa(A,B,F,X4) )
& pp(aa(set(A),bool,member(A,X4),A6)) ) ) ).
% Grp_def
tff(fact_365_effect__LetI,axiom,
! [A: $tType,B: $tType,X2: A,T2: A,F: fun(A,heap_Heap(B)),H: heap_ext(product_unit),H2: heap_ext(product_unit),R: B] :
( ( X2 = T2 )
=> ( heap_effect(B,aa(A,heap_Heap(B),F,X2),H,H2,R)
=> heap_effect(B,aa(A,heap_Heap(B),F,T2),H,H2,R) ) ) ).
% effect_LetI
tff(fact_366_success__LetI,axiom,
! [A: $tType,B: $tType,X2: A,T2: A,F: fun(A,heap_Heap(B)),H: heap_ext(product_unit)] :
( ( X2 = T2 )
=> ( heap_success(B,aa(A,heap_Heap(B),F,X2),H)
=> heap_success(B,aa(A,heap_Heap(B),F,T2),H) ) ) ).
% success_LetI
tff(fact_367_vimage2p__def,axiom,
! [A: $tType,D: $tType,C: $tType,E: $tType,B: $tType,F: fun(A,D),G: fun(B,E),R3: fun(D,fun(E,C)),X4: A,Xa4: B] : aa(B,C,aa(A,fun(B,C),aa(fun(D,fun(E,C)),fun(A,fun(B,C)),bNF_vimage2p(A,D,B,E,C,F,G),R3),X4),Xa4) = aa(E,C,aa(D,fun(E,C),R3,aa(A,D,F,X4)),aa(B,E,G,Xa4)) ).
% vimage2p_def
tff(fact_368_curry__K,axiom,
! [B: $tType,A: $tType,C: $tType,C3: C,X4: A,Xa4: B] : aa(B,C,aa(A,fun(B,C),aa(fun(product_prod(A,B),C),fun(A,fun(B,C)),product_curry(A,B,C),aTP_Lamp_ao(C,fun(product_prod(A,B),C),C3)),X4),Xa4) = C3 ).
% curry_K
tff(fact_369_scomp__unfold,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,X4: fun(A,product_prod(B,C)),Xa4: fun(B,fun(C,D)),Xb: A] : aa(A,D,product_scomp(A,B,C,D,X4,Xa4),Xb) = aa(C,D,aa(B,fun(C,D),Xa4,aa(product_prod(B,C),B,product_fst(B,C),aa(A,product_prod(B,C),X4,Xb))),aa(product_prod(B,C),C,product_snd(B,C),aa(A,product_prod(B,C),X4,Xb))) ).
% scomp_unfold
tff(fact_370_image__Collect__subsetI,axiom,
! [A: $tType,B: $tType,P2: fun(A,bool),F: fun(A,B),B6: set(B)] :
( ! [X: A] :
( pp(aa(A,bool,P2,X))
=> pp(aa(set(B),bool,member(B,aa(A,B,F,X)),B6)) )
=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),aa(fun(A,bool),set(A),collect(A),P2))),B6)) ) ).
% image_Collect_subsetI
tff(fact_371_range__composition,axiom,
! [A: $tType,C: $tType,B: $tType,F: fun(C,A),G: fun(B,C)] : aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),aa(fun(B,C),fun(B,A),aTP_Lamp_ap(fun(C,A),fun(fun(B,C),fun(B,A)),F),G)),top_top(set(B))) = aa(set(C),set(A),aa(fun(C,A),fun(set(C),set(A)),image2(C,A),F),aa(set(B),set(C),aa(fun(B,C),fun(set(B),set(C)),image2(B,C),G),top_top(set(B)))) ).
% range_composition
tff(fact_372_rangeE,axiom,
! [A: $tType,B: $tType,B2: A,F: fun(B,A)] :
( pp(aa(set(A),bool,member(A,B2),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),top_top(set(B)))))
=> ~ ! [X: B] : B2 != aa(B,A,F,X) ) ).
% rangeE
tff(fact_373_fun_Orel__map_I1_J,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,Sb: fun(C,fun(B,bool)),I: fun(A,C),X2: fun(D,A),Y2: fun(D,B)] :
( pp(aa(fun(D,B),bool,aa(fun(D,C),fun(fun(D,B),bool),bNF_rel_fun(D,D,C,B,fequal(D),Sb),aa(fun(D,A),fun(D,C),aa(fun(A,C),fun(fun(D,A),fun(D,C)),comp(A,C,D),I),X2)),Y2))
<=> pp(aa(fun(D,B),bool,aa(fun(D,A),fun(fun(D,B),bool),bNF_rel_fun(D,D,A,B,fequal(D),aa(fun(A,C),fun(A,fun(B,bool)),aTP_Lamp_aq(fun(C,fun(B,bool)),fun(fun(A,C),fun(A,fun(B,bool))),Sb),I)),X2),Y2)) ) ).
% fun.rel_map(1)
tff(fact_374_fun_Orel__map_I2_J,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,Sa: fun(A,fun(C,bool)),X2: fun(D,A),G: fun(B,C),Y2: fun(D,B)] :
( pp(aa(fun(D,C),bool,aa(fun(D,A),fun(fun(D,C),bool),bNF_rel_fun(D,D,A,C,fequal(D),Sa),X2),aa(fun(D,B),fun(D,C),aa(fun(B,C),fun(fun(D,B),fun(D,C)),comp(B,C,D),G),Y2)))
<=> pp(aa(fun(D,B),bool,aa(fun(D,A),fun(fun(D,B),bool),bNF_rel_fun(D,D,A,B,fequal(D),aa(fun(B,C),fun(A,fun(B,bool)),aTP_Lamp_ar(fun(A,fun(C,bool)),fun(fun(B,C),fun(A,fun(B,bool))),Sa),G)),X2),Y2)) ) ).
% fun.rel_map(2)
tff(fact_375_case__prodE2,axiom,
! [B: $tType,A: $tType,C: $tType,Q2: fun(A,bool),P2: fun(B,fun(C,A)),Z2: product_prod(B,C)] :
( pp(aa(A,bool,Q2,aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),P2),Z2)))
=> ~ ! [X: B,Y: C] :
( ( Z2 = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X),Y) )
=> ~ pp(aa(A,bool,Q2,aa(C,A,aa(B,fun(C,A),P2,X),Y))) ) ) ).
% case_prodE2
tff(fact_376_case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(product_prod(A,B),C)] : aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),aTP_Lamp_as(fun(product_prod(A,B),C),fun(A,fun(B,C)),F)) = F ).
% case_prod_eta
tff(fact_377_cond__case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(A,fun(B,C)),G: fun(product_prod(A,B),C)] :
( ! [X: A,Y: B] : aa(B,C,aa(A,fun(B,C),F,X),Y) = aa(product_prod(A,B),C,G,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y))
=> ( aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F) = G ) ) ).
% cond_case_prod_eta
tff(fact_378_case__prodD,axiom,
! [A: $tType,B: $tType,F: fun(A,fun(B,bool)),A3: A,B2: B] :
( pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),F),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)))
=> pp(aa(B,bool,aa(A,fun(B,bool),F,A3),B2)) ) ).
% case_prodD
tff(fact_379_case__prodE,axiom,
! [A: $tType,B: $tType,C3: fun(A,fun(B,bool)),P: product_prod(A,B)] :
( pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),C3),P))
=> ~ ! [X: A,Y: B] :
( ( P = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y) )
=> ~ pp(aa(B,bool,aa(A,fun(B,bool),C3,X),Y)) ) ) ).
% case_prodE
tff(fact_380_fst__def,axiom,
! [B: $tType,A: $tType,Prod: product_prod(A,B)] : aa(product_prod(A,B),A,product_fst(A,B),Prod) = aa(product_prod(A,B),A,aa(fun(A,fun(B,A)),fun(product_prod(A,B),A),product_case_prod(A,B,A),aTP_Lamp_at(A,fun(B,A))),Prod) ).
% fst_def
tff(fact_381_snd__def,axiom,
! [B: $tType,A: $tType,Prod: product_prod(A,B)] : aa(product_prod(A,B),B,product_snd(A,B),Prod) = aa(product_prod(A,B),B,aa(fun(A,fun(B,B)),fun(product_prod(A,B),B),product_case_prod(A,B,B),aTP_Lamp_au(A,fun(B,B))),Prod) ).
% snd_def
tff(fact_382_map__prod__def,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F: fun(A,C),G: fun(B,D)] : product_map_prod(A,C,B,D,F,G) = aa(fun(A,fun(B,product_prod(C,D))),fun(product_prod(A,B),product_prod(C,D)),product_case_prod(A,B,product_prod(C,D)),aa(fun(B,D),fun(A,fun(B,product_prod(C,D))),aTP_Lamp_av(fun(A,C),fun(fun(B,D),fun(A,fun(B,product_prod(C,D)))),F),G)) ).
% map_prod_def
tff(fact_383_case__prod__map__prod,axiom,
! [C: $tType,A: $tType,B: $tType,E: $tType,D: $tType,H: fun(B,fun(C,A)),F: fun(D,B),G: fun(E,C),X2: product_prod(D,E)] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),H),aa(product_prod(D,E),product_prod(B,C),product_map_prod(D,B,E,C,F,G),X2)) = aa(product_prod(D,E),A,aa(fun(D,fun(E,A)),fun(product_prod(D,E),A),product_case_prod(D,E,A),aa(fun(E,C),fun(D,fun(E,A)),aa(fun(D,B),fun(fun(E,C),fun(D,fun(E,A))),aTP_Lamp_aw(fun(B,fun(C,A)),fun(fun(D,B),fun(fun(E,C),fun(D,fun(E,A)))),H),F),G)),X2) ).
% case_prod_map_prod
tff(fact_384_curry__def,axiom,
! [C: $tType,A: $tType,B: $tType,X4: fun(product_prod(A,B),C),Xa4: A,Xb: B] : aa(B,C,aa(A,fun(B,C),aa(fun(product_prod(A,B),C),fun(A,fun(B,C)),product_curry(A,B,C),X4),Xa4),Xb) = aa(product_prod(A,B),C,X4,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa4),Xb)) ).
% curry_def
tff(fact_385_converse__relcompp,axiom,
! [A: $tType,C: $tType,B: $tType,R: fun(B,fun(C,bool)),S: fun(C,fun(A,bool))] : conversep(B,A,aa(fun(C,fun(A,bool)),fun(B,fun(A,bool)),aa(fun(B,fun(C,bool)),fun(fun(C,fun(A,bool)),fun(B,fun(A,bool))),relcompp(B,C,A),R),S)) = aa(fun(C,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(C,bool)),fun(fun(C,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,C,B),conversep(C,A,S)),conversep(B,C,R)) ).
% converse_relcompp
tff(fact_386_leq__conversepI,axiom,
! [A: $tType,R3: fun(A,fun(A,bool))] :
( ( R3 = fequal(A) )
=> pp(aa(fun(A,fun(A,bool)),bool,aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),bool),ord_less_eq(fun(A,fun(A,bool))),R3),conversep(A,A,R3))) ) ).
% leq_conversepI
tff(fact_387_map__prod_Oidentity,axiom,
! [B: $tType,A: $tType] : product_map_prod(A,A,B,B,aTP_Lamp_ab(A,A),aTP_Lamp_ad(B,B)) = id(product_prod(A,B)) ).
% map_prod.identity
tff(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),aTP_Lamp_ax(heap_Heap(A),fun(B,heap_Heap(A)),F)) = F ) ).
% lookup_chain
tff(fact_389_scomp__scomp,axiom,
! [A: $tType,E: $tType,F2: $tType,B: $tType,D: $tType,C: $tType,F: fun(A,product_prod(E,F2)),G: fun(E,fun(F2,product_prod(C,D))),H: fun(C,fun(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,aa(fun(C,fun(D,B)),fun(E,fun(F2,B)),aTP_Lamp_ay(fun(E,fun(F2,product_prod(C,D))),fun(fun(C,fun(D,B)),fun(E,fun(F2,B))),G),H)) ).
% scomp_scomp
tff(fact_390_len__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [A3: array(A)] : len(A,A3) = heap_tap(nat,aTP_Lamp_az(array(A),fun(heap_ext(product_unit),nat),A3)) ) ).
% len_def
tff(fact_391_flip__pred,axiom,
! [A: $tType,B: $tType,A6: set(product_prod(A,B)),R3: fun(B,fun(A,bool))] :
( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),A6),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),conversep(B,A,R3)))))
=> pp(aa(set(product_prod(B,A)),bool,aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),bool),ord_less_eq(set(product_prod(B,A))),aa(set(product_prod(A,B)),set(product_prod(B,A)),aa(fun(product_prod(A,B),product_prod(B,A)),fun(set(product_prod(A,B)),set(product_prod(B,A))),image2(product_prod(A,B),product_prod(B,A)),aa(fun(A,fun(B,product_prod(B,A))),fun(product_prod(A,B),product_prod(B,A)),product_case_prod(A,B,product_prod(B,A)),aTP_Lamp_ba(A,fun(B,product_prod(B,A))))),A6)),aa(fun(product_prod(B,A),bool),set(product_prod(B,A)),collect(product_prod(B,A)),aa(fun(B,fun(A,bool)),fun(product_prod(B,A),bool),product_case_prod(B,A,bool),R3)))) ) ).
% flip_pred
tff(fact_392_freeze__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [A3: array(A)] : freeze(A,A3) = heap_tap(list(A),aTP_Lamp_bb(array(A),fun(heap_ext(product_unit),list(A)),A3)) ) ).
% freeze_def
tff(fact_393_type__copy__wit,axiom,
! [A: $tType,C: $tType,B: $tType,Rep: fun(A,B),Abs: fun(B,A),X2: C,S3: fun(B,set(C)),Y2: B] :
( type_definition(A,B,Rep,Abs,top_top(set(B)))
=> ( pp(aa(set(C),bool,member(C,X2),aa(A,set(C),aa(fun(A,B),fun(A,set(C)),aa(fun(B,set(C)),fun(fun(A,B),fun(A,set(C))),comp(B,set(C),A),S3),Rep),aa(B,A,Abs,Y2))))
=> pp(aa(set(C),bool,member(C,X2),aa(B,set(C),S3,Y2))) ) ) ).
% type_copy_wit
tff(fact_394_Cantors__paradox,axiom,
! [A: $tType,A6: set(A)] :
~ ? [F5: fun(A,set(A))] : aa(set(A),set(set(A)),aa(fun(A,set(A)),fun(set(A),set(set(A))),image2(A,set(A)),F5),A6) = pow(A,A6) ).
% Cantors_paradox
tff(fact_395_type__copy__vimage2p__Grp__Rep,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,Rep: fun(A,B),Abs: fun(B,A),F: fun(C,D),P2: fun(D,bool),H: fun(D,B)] :
( type_definition(A,B,Rep,Abs,top_top(set(B)))
=> ( aa(fun(D,fun(B,bool)),fun(C,fun(A,bool)),bNF_vimage2p(C,D,A,B,bool,F,Rep),bNF_Grp(D,B,aa(fun(D,bool),set(D),collect(D),P2),H)) = bNF_Grp(C,A,aa(fun(C,bool),set(C),collect(C),aa(fun(D,bool),fun(C,bool),aTP_Lamp_bc(fun(C,D),fun(fun(D,bool),fun(C,bool)),F),P2)),aa(fun(C,D),fun(C,A),aa(fun(D,A),fun(fun(C,D),fun(C,A)),comp(D,A,C),aa(fun(D,B),fun(D,A),aa(fun(B,A),fun(fun(D,B),fun(D,A)),comp(B,A,D),Abs),H)),F)) ) ) ).
% type_copy_vimage2p_Grp_Rep
tff(fact_396_type__copy__vimage2p__Grp__Abs,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Rep: fun(A,B),Abs: fun(B,A),G: fun(D,C),P2: fun(C,bool),H: fun(C,A)] :
( type_definition(A,B,Rep,Abs,top_top(set(B)))
=> ( aa(fun(C,fun(A,bool)),fun(D,fun(B,bool)),bNF_vimage2p(D,C,B,A,bool,G,Abs),bNF_Grp(C,A,aa(fun(C,bool),set(C),collect(C),P2),H)) = bNF_Grp(D,B,aa(fun(D,bool),set(D),collect(D),aa(fun(C,bool),fun(D,bool),aTP_Lamp_bd(fun(D,C),fun(fun(C,bool),fun(D,bool)),G),P2)),aa(fun(D,C),fun(D,B),aa(fun(C,B),fun(fun(D,C),fun(D,B)),comp(C,B,D),aa(fun(C,A),fun(C,B),aa(fun(A,B),fun(fun(C,A),fun(C,B)),comp(A,B,C),Rep),H)),G)) ) ) ).
% type_copy_vimage2p_Grp_Abs
tff(fact_397_exE__realizer,axiom,
! [C: $tType,A: $tType,B: $tType,P2: fun(A,fun(B,bool)),P: product_prod(B,A),Q2: fun(C,bool),F: fun(B,fun(A,C))] :
( pp(aa(B,bool,aa(A,fun(B,bool),P2,aa(product_prod(B,A),A,product_snd(B,A),P)),aa(product_prod(B,A),B,product_fst(B,A),P)))
=> ( ! [X: B,Y: A] :
( pp(aa(B,bool,aa(A,fun(B,bool),P2,Y),X))
=> pp(aa(C,bool,Q2,aa(A,C,aa(B,fun(A,C),F,X),Y))) )
=> pp(aa(C,bool,Q2,aa(product_prod(B,A),C,aa(fun(B,fun(A,C)),fun(product_prod(B,A),C),product_case_prod(B,A,C),F),P))) ) ) ).
% exE_realizer
tff(fact_398_split__comp__eq,axiom,
! [B: $tType,C: $tType,A: $tType,D: $tType,F: fun(A,fun(B,C)),G: fun(D,A)] : aa(fun(D,A),fun(product_prod(D,B),C),aTP_Lamp_be(fun(A,fun(B,C)),fun(fun(D,A),fun(product_prod(D,B),C)),F),G) = aa(fun(D,fun(B,C)),fun(product_prod(D,B),C),product_case_prod(D,B,C),aa(fun(D,A),fun(D,fun(B,C)),aTP_Lamp_bf(fun(A,fun(B,C)),fun(fun(D,A),fun(D,fun(B,C))),F),G)) ).
% split_comp_eq
tff(fact_399_case__prod__beta_H,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(A,fun(B,C)),X4: product_prod(A,B)] : aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),F),X4) = aa(B,C,aa(A,fun(B,C),F,aa(product_prod(A,B),A,product_fst(A,B),X4)),aa(product_prod(A,B),B,product_snd(A,B),X4)) ).
% case_prod_beta'
tff(fact_400_case__prod__unfold,axiom,
! [C: $tType,B: $tType,A: $tType,X4: fun(A,fun(B,C)),Xa4: product_prod(A,B)] : aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),X4),Xa4) = aa(B,C,aa(A,fun(B,C),X4,aa(product_prod(A,B),A,product_fst(A,B),Xa4)),aa(product_prod(A,B),B,product_snd(A,B),Xa4)) ).
% case_prod_unfold
tff(fact_401_pred__subset__eq2,axiom,
! [B: $tType,A: $tType,R3: set(product_prod(A,B)),S3: set(product_prod(A,B))] :
( pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_bg(set(product_prod(A,B)),fun(A,fun(B,bool))),R3)),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_bg(set(product_prod(A,B)),fun(A,fun(B,bool))),S3)))
<=> pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),R3),S3)) ) ).
% pred_subset_eq2
tff(fact_402_comp__fun__commute__on__def,axiom,
! [B: $tType,A: $tType,S3: set(A),F: fun(A,fun(B,B))] :
( finite4664212375090638736ute_on(A,B,S3,F)
<=> ! [X5: A,Y4: A] :
( pp(aa(set(A),bool,member(A,X5),S3))
=> ( pp(aa(set(A),bool,member(A,Y4),S3))
=> ( aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,Y4)),aa(A,fun(B,B),F,X5)) = aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X5)),aa(A,fun(B,B),F,Y4)) ) ) ) ) ).
% comp_fun_commute_on_def
tff(fact_403_comp__fun__commute__on_Ocomp__fun__commute__on,axiom,
! [B: $tType,A: $tType,S3: set(A),F: fun(A,fun(B,B)),X2: A,Y2: A] :
( finite4664212375090638736ute_on(A,B,S3,F)
=> ( pp(aa(set(A),bool,member(A,X2),S3))
=> ( pp(aa(set(A),bool,member(A,Y2),S3))
=> ( aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,Y2)),aa(A,fun(B,B),F,X2)) = aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X2)),aa(A,fun(B,B),F,Y2)) ) ) ) ) ).
% comp_fun_commute_on.comp_fun_commute_on
tff(fact_404_comp__fun__commute__on_Ocommute__left__comp,axiom,
! [A: $tType,B: $tType,C: $tType,S3: set(A),F: fun(A,fun(B,B)),X2: A,Y2: A,G: fun(C,B)] :
( finite4664212375090638736ute_on(A,B,S3,F)
=> ( pp(aa(set(A),bool,member(A,X2),S3))
=> ( pp(aa(set(A),bool,member(A,Y2),S3))
=> ( aa(fun(C,B),fun(C,B),aa(fun(B,B),fun(fun(C,B),fun(C,B)),comp(B,B,C),aa(A,fun(B,B),F,Y2)),aa(fun(C,B),fun(C,B),aa(fun(B,B),fun(fun(C,B),fun(C,B)),comp(B,B,C),aa(A,fun(B,B),F,X2)),G)) = aa(fun(C,B),fun(C,B),aa(fun(B,B),fun(fun(C,B),fun(C,B)),comp(B,B,C),aa(A,fun(B,B),F,X2)),aa(fun(C,B),fun(C,B),aa(fun(B,B),fun(fun(C,B),fun(C,B)),comp(B,B,C),aa(A,fun(B,B),F,Y2)),G)) ) ) ) ) ).
% comp_fun_commute_on.commute_left_comp
tff(fact_405_comp__fun__commute__on_Ointro,axiom,
! [B: $tType,A: $tType,S3: set(A),F: fun(A,fun(B,B))] :
( ! [X: A,Y: A] :
( pp(aa(set(A),bool,member(A,X),S3))
=> ( pp(aa(set(A),bool,member(A,Y),S3))
=> ( aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,Y)),aa(A,fun(B,B),F,X)) = aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X)),aa(A,fun(B,B),F,Y)) ) ) )
=> finite4664212375090638736ute_on(A,B,S3,F) ) ).
% comp_fun_commute_on.intro
tff(fact_406_case__prod__o__map__prod,axiom,
! [A: $tType,D: $tType,C: $tType,E: $tType,B: $tType,F: fun(D,fun(E,C)),G1: fun(A,D),G22: fun(B,E)] : aa(fun(product_prod(A,B),product_prod(D,E)),fun(product_prod(A,B),C),aa(fun(product_prod(D,E),C),fun(fun(product_prod(A,B),product_prod(D,E)),fun(product_prod(A,B),C)),comp(product_prod(D,E),C,product_prod(A,B)),aa(fun(D,fun(E,C)),fun(product_prod(D,E),C),product_case_prod(D,E,C),F)),product_map_prod(A,D,B,E,G1,G22)) = aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),aa(fun(B,E),fun(A,fun(B,C)),aa(fun(A,D),fun(fun(B,E),fun(A,fun(B,C))),aTP_Lamp_bh(fun(D,fun(E,C)),fun(fun(A,D),fun(fun(B,E),fun(A,fun(B,C)))),F),G1),G22)) ).
% case_prod_o_map_prod
tff(fact_407_vimage2p__relcompp__converse,axiom,
! [E: $tType,C: $tType,D: $tType,A: $tType,F2: $tType,B: $tType,Rep: fun(A,B),Abs: fun(B,A),F: fun(C,E),G: fun(D,F2),R3: fun(B,fun(E,bool)),S3: fun(B,fun(F2,bool))] :
( type_definition(A,B,Rep,Abs,top_top(set(B)))
=> ( aa(fun(E,fun(F2,bool)),fun(C,fun(D,bool)),bNF_vimage2p(C,E,D,F2,bool,F,G),aa(fun(B,fun(F2,bool)),fun(E,fun(F2,bool)),aa(fun(E,fun(B,bool)),fun(fun(B,fun(F2,bool)),fun(E,fun(F2,bool))),relcompp(E,B,F2),conversep(B,E,R3)),S3)) = aa(fun(A,fun(D,bool)),fun(C,fun(D,bool)),aa(fun(C,fun(A,bool)),fun(fun(A,fun(D,bool)),fun(C,fun(D,bool))),relcompp(C,A,D),conversep(A,C,aa(fun(B,fun(E,bool)),fun(A,fun(C,bool)),bNF_vimage2p(A,B,C,E,bool,Rep,F),R3))),aa(fun(B,fun(F2,bool)),fun(A,fun(D,bool)),bNF_vimage2p(A,B,D,F2,bool,Rep,G),S3)) ) ) ).
% vimage2p_relcompp_converse
tff(fact_408_Id__on__iff,axiom,
! [A: $tType,X2: A,Y2: A,A6: set(A)] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),id_on(A,A6)))
<=> ( ( X2 = Y2 )
& pp(aa(set(A),bool,member(A,X2),A6)) ) ) ).
% Id_on_iff
tff(fact_409_Id__on__eqI,axiom,
! [A: $tType,A3: A,B2: A,A6: set(A)] :
( ( A3 = B2 )
=> ( pp(aa(set(A),bool,member(A,A3),A6))
=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),id_on(A,A6))) ) ) ).
% Id_on_eqI
tff(fact_410_Id__onE,axiom,
! [A: $tType,C3: product_prod(A,A),A6: set(A)] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),C3),id_on(A,A6)))
=> ~ ! [X: A] :
( pp(aa(set(A),bool,member(A,X),A6))
=> ( C3 != aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X) ) ) ) ).
% Id_onE
tff(fact_411_predicate2__transferD,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,R1: fun(A,fun(B,bool)),R22: fun(C,fun(D,bool)),P2: fun(A,fun(C,bool)),Q2: fun(B,fun(D,bool)),A3: product_prod(A,B),A6: set(product_prod(A,B)),B2: product_prod(C,D),B6: set(product_prod(C,D))] :
( pp(aa(fun(B,fun(D,bool)),bool,aa(fun(A,fun(C,bool)),fun(fun(B,fun(D,bool)),bool),bNF_rel_fun(A,B,fun(C,bool),fun(D,bool),R1,bNF_rel_fun(C,D,bool,bool,R22,fequal(bool))),P2),Q2))
=> ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),A3),A6))
=> ( pp(aa(set(product_prod(C,D)),bool,member(product_prod(C,D),B2),B6))
=> ( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),A6),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),R1))))
=> ( pp(aa(set(product_prod(C,D)),bool,aa(set(product_prod(C,D)),fun(set(product_prod(C,D)),bool),ord_less_eq(set(product_prod(C,D))),B6),aa(fun(product_prod(C,D),bool),set(product_prod(C,D)),collect(product_prod(C,D)),aa(fun(C,fun(D,bool)),fun(product_prod(C,D),bool),product_case_prod(C,D,bool),R22))))
=> ( pp(aa(C,bool,aa(A,fun(C,bool),P2,aa(product_prod(A,B),A,product_fst(A,B),A3)),aa(product_prod(C,D),C,product_fst(C,D),B2)))
<=> pp(aa(D,bool,aa(B,fun(D,bool),Q2,aa(product_prod(A,B),B,product_snd(A,B),A3)),aa(product_prod(C,D),D,product_snd(C,D),B2))) ) ) ) ) ) ) ).
% predicate2_transferD
tff(fact_412_fst__diag__id,axiom,
! [A: $tType,Z2: A] : aa(A,A,aa(fun(A,product_prod(A,A)),fun(A,A),aa(fun(product_prod(A,A),A),fun(fun(A,product_prod(A,A)),fun(A,A)),comp(product_prod(A,A),A,A),product_fst(A,A)),aTP_Lamp_bi(A,product_prod(A,A))),Z2) = aa(A,A,id(A),Z2) ).
% fst_diag_id
tff(fact_413_snd__diag__id,axiom,
! [A: $tType,Z2: A] : aa(A,A,aa(fun(A,product_prod(A,A)),fun(A,A),aa(fun(product_prod(A,A),A),fun(fun(A,product_prod(A,A)),fun(A,A)),comp(product_prod(A,A),A,A),product_snd(A,A)),aTP_Lamp_bi(A,product_prod(A,A))),Z2) = aa(A,A,id(A),Z2) ).
% snd_diag_id
tff(fact_414_case__prod__comp,axiom,
! [D: $tType,A: $tType,C: $tType,B: $tType,F: fun(D,fun(C,A)),G: fun(B,D),X2: product_prod(B,C)] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),aa(fun(B,D),fun(B,fun(C,A)),aa(fun(D,fun(C,A)),fun(fun(B,D),fun(B,fun(C,A))),comp(D,fun(C,A),B),F),G)),X2) = aa(C,A,aa(D,fun(C,A),F,aa(B,D,G,aa(product_prod(B,C),B,product_fst(B,C),X2))),aa(product_prod(B,C),C,product_snd(B,C),X2)) ).
% case_prod_comp
tff(fact_415_lookup__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [R: ref(A)] : lookup(A,R) = heap_tap(A,aTP_Lamp_bj(ref(A),fun(heap_ext(product_unit),A),R)) ) ).
% lookup_def
tff(fact_416_fst__snd__flip,axiom,
! [B: $tType,A: $tType,Xy: product_prod(A,B)] : aa(product_prod(A,B),A,product_fst(A,B),Xy) = aa(product_prod(A,B),A,aa(fun(product_prod(A,B),product_prod(B,A)),fun(product_prod(A,B),A),aa(fun(product_prod(B,A),A),fun(fun(product_prod(A,B),product_prod(B,A)),fun(product_prod(A,B),A)),comp(product_prod(B,A),A,product_prod(A,B)),product_snd(B,A)),aa(fun(A,fun(B,product_prod(B,A))),fun(product_prod(A,B),product_prod(B,A)),product_case_prod(A,B,product_prod(B,A)),aTP_Lamp_ba(A,fun(B,product_prod(B,A))))),Xy) ).
% fst_snd_flip
tff(fact_417_snd__fst__flip,axiom,
! [A: $tType,B: $tType,Xy: product_prod(B,A)] : aa(product_prod(B,A),A,product_snd(B,A),Xy) = aa(product_prod(B,A),A,aa(fun(product_prod(B,A),product_prod(A,B)),fun(product_prod(B,A),A),aa(fun(product_prod(A,B),A),fun(fun(product_prod(B,A),product_prod(A,B)),fun(product_prod(B,A),A)),comp(product_prod(A,B),A,product_prod(B,A)),product_fst(A,B)),aa(fun(B,fun(A,product_prod(A,B))),fun(product_prod(B,A),product_prod(A,B)),product_case_prod(B,A,product_prod(A,B)),aTP_Lamp_bk(B,fun(A,product_prod(A,B))))),Xy) ).
% snd_fst_flip
tff(fact_418_comp__fun__idem__on_Ocomp__fun__idem__on,axiom,
! [B: $tType,A: $tType,S3: set(A),F: fun(A,fun(B,B)),X2: A] :
( finite673082921795544331dem_on(A,B,S3,F)
=> ( pp(aa(set(A),bool,member(A,X2),S3))
=> ( aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X2)),aa(A,fun(B,B),F,X2)) = aa(A,fun(B,B),F,X2) ) ) ) ).
% comp_fun_idem_on.comp_fun_idem_on
tff(fact_419_bind__lift,axiom,
! [A: $tType,B: $tType,F: heap_Heap(B),G: fun(B,A)] : heap_bind(B,A,F,heap_lift(B,A,G)) = heap_bind(B,A,F,aTP_Lamp_bl(fun(B,A),fun(B,heap_Heap(A)),G)) ).
% bind_lift
tff(fact_420_OO__Grp__cong,axiom,
! [B: $tType,C: $tType,A: $tType,A6: set(A),B6: set(A),F: fun(A,B),G: fun(A,C)] :
( ( A6 = B6 )
=> ( aa(fun(A,fun(C,bool)),fun(B,fun(C,bool)),aa(fun(B,fun(A,bool)),fun(fun(A,fun(C,bool)),fun(B,fun(C,bool))),relcompp(B,A,C),conversep(A,B,bNF_Grp(A,B,A6,F))),bNF_Grp(A,C,A6,G)) = aa(fun(A,fun(C,bool)),fun(B,fun(C,bool)),aa(fun(B,fun(A,bool)),fun(fun(A,fun(C,bool)),fun(B,fun(C,bool))),relcompp(B,A,C),conversep(A,B,bNF_Grp(A,B,B6,F))),bNF_Grp(A,C,B6,G)) ) ) ).
% OO_Grp_cong
tff(fact_421_fst__diag__fst,axiom,
! [B: $tType,A: $tType] : aa(fun(product_prod(A,B),product_prod(A,A)),fun(product_prod(A,B),A),aa(fun(product_prod(A,A),A),fun(fun(product_prod(A,B),product_prod(A,A)),fun(product_prod(A,B),A)),comp(product_prod(A,A),A,product_prod(A,B)),product_fst(A,A)),aa(fun(product_prod(A,B),A),fun(product_prod(A,B),product_prod(A,A)),aa(fun(A,product_prod(A,A)),fun(fun(product_prod(A,B),A),fun(product_prod(A,B),product_prod(A,A))),comp(A,product_prod(A,A),product_prod(A,B)),aTP_Lamp_bi(A,product_prod(A,A))),product_fst(A,B))) = product_fst(A,B) ).
% fst_diag_fst
tff(fact_422_snd__diag__snd,axiom,
! [B: $tType,A: $tType] : aa(fun(product_prod(A,B),product_prod(B,B)),fun(product_prod(A,B),B),aa(fun(product_prod(B,B),B),fun(fun(product_prod(A,B),product_prod(B,B)),fun(product_prod(A,B),B)),comp(product_prod(B,B),B,product_prod(A,B)),product_snd(B,B)),aa(fun(product_prod(A,B),B),fun(product_prod(A,B),product_prod(B,B)),aa(fun(B,product_prod(B,B)),fun(fun(product_prod(A,B),B),fun(product_prod(A,B),product_prod(B,B))),comp(B,product_prod(B,B),product_prod(A,B)),aTP_Lamp_bm(B,product_prod(B,B))),product_snd(A,B))) = product_snd(A,B) ).
% snd_diag_snd
tff(fact_423_type__copy__map__comp0,axiom,
! [F2: $tType,D: $tType,B: $tType,A: $tType,C: $tType,E: $tType,Rep: fun(A,B),Abs: fun(B,A),M: fun(C,D),M1: fun(B,D),M2: fun(C,B),F: fun(D,F2),G: fun(E,C)] :
( type_definition(A,B,Rep,Abs,top_top(set(B)))
=> ( ( M = aa(fun(C,B),fun(C,D),aa(fun(B,D),fun(fun(C,B),fun(C,D)),comp(B,D,C),M1),M2) )
=> ( aa(fun(E,C),fun(E,F2),aa(fun(C,F2),fun(fun(E,C),fun(E,F2)),comp(C,F2,E),aa(fun(C,D),fun(C,F2),aa(fun(D,F2),fun(fun(C,D),fun(C,F2)),comp(D,F2,C),F),M)),G) = aa(fun(E,A),fun(E,F2),aa(fun(A,F2),fun(fun(E,A),fun(E,F2)),comp(A,F2,E),aa(fun(A,B),fun(A,F2),aa(fun(B,F2),fun(fun(A,B),fun(A,F2)),comp(B,F2,A),aa(fun(B,D),fun(B,F2),aa(fun(D,F2),fun(fun(B,D),fun(B,F2)),comp(D,F2,B),F),M1)),Rep)),aa(fun(E,C),fun(E,A),aa(fun(C,A),fun(fun(E,C),fun(E,A)),comp(C,A,E),aa(fun(C,B),fun(C,A),aa(fun(B,A),fun(fun(C,B),fun(C,A)),comp(B,A,C),Abs),M2)),G)) ) ) ) ).
% type_copy_map_comp0
tff(fact_424_type__copy__map__comp0__undo,axiom,
! [E: $tType,A: $tType,C: $tType,B: $tType,D: $tType,F2: $tType,Rep: fun(A,B),Abs: fun(B,A),Rep2: fun(C,D),Abs2: fun(D,C),Rep3: fun(E,F2),Abs3: fun(F2,E),M: fun(F2,D),M1: fun(B,D),M2: fun(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)))
=> ( ( aa(fun(E,F2),fun(E,C),aa(fun(F2,C),fun(fun(E,F2),fun(E,C)),comp(F2,C,E),aa(fun(F2,D),fun(F2,C),aa(fun(D,C),fun(fun(F2,D),fun(F2,C)),comp(D,C,F2),Abs2),M)),Rep3) = aa(fun(E,A),fun(E,C),aa(fun(A,C),fun(fun(E,A),fun(E,C)),comp(A,C,E),aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),aa(fun(B,D),fun(B,C),aa(fun(D,C),fun(fun(B,D),fun(B,C)),comp(D,C,B),Abs2),M1)),Rep)),aa(fun(E,F2),fun(E,A),aa(fun(F2,A),fun(fun(E,F2),fun(E,A)),comp(F2,A,E),aa(fun(F2,B),fun(F2,A),aa(fun(B,A),fun(fun(F2,B),fun(F2,A)),comp(B,A,F2),Abs),M2)),Rep3)) )
=> ( aa(fun(F2,B),fun(F2,D),aa(fun(B,D),fun(fun(F2,B),fun(F2,D)),comp(B,D,F2),M1),M2) = M ) ) ) ) ) ).
% type_copy_map_comp0_undo
tff(fact_425_fun_Orel__compp__Grp,axiom,
! [D: $tType,B: $tType,A: $tType,R3: fun(A,fun(B,bool))] : bNF_rel_fun(D,D,A,B,fequal(D),R3) = aa(fun(fun(D,product_prod(A,B)),fun(fun(D,B),bool)),fun(fun(D,A),fun(fun(D,B),bool)),aa(fun(fun(D,A),fun(fun(D,product_prod(A,B)),bool)),fun(fun(fun(D,product_prod(A,B)),fun(fun(D,B),bool)),fun(fun(D,A),fun(fun(D,B),bool))),relcompp(fun(D,A),fun(D,product_prod(A,B)),fun(D,B)),conversep(fun(D,product_prod(A,B)),fun(D,A),bNF_Grp(fun(D,product_prod(A,B)),fun(D,A),aa(fun(fun(D,product_prod(A,B)),bool),set(fun(D,product_prod(A,B))),collect(fun(D,product_prod(A,B))),aTP_Lamp_aa(fun(A,fun(B,bool)),fun(fun(D,product_prod(A,B)),bool),R3)),aa(fun(product_prod(A,B),A),fun(fun(D,product_prod(A,B)),fun(D,A)),comp(product_prod(A,B),A,D),product_fst(A,B))))),bNF_Grp(fun(D,product_prod(A,B)),fun(D,B),aa(fun(fun(D,product_prod(A,B)),bool),set(fun(D,product_prod(A,B))),collect(fun(D,product_prod(A,B))),aTP_Lamp_aa(fun(A,fun(B,bool)),fun(fun(D,product_prod(A,B)),bool),R3)),aa(fun(product_prod(A,B),B),fun(fun(D,product_prod(A,B)),fun(D,B)),comp(product_prod(A,B),B,D),product_snd(A,B)))) ).
% fun.rel_compp_Grp
tff(fact_426_image__Pow__surj,axiom,
! [B: $tType,A: $tType,F: fun(B,A),A6: set(B),B6: set(A)] :
( ( aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6) = B6 )
=> ( aa(set(set(B)),set(set(A)),aa(fun(set(B),set(A)),fun(set(set(B)),set(set(A))),image2(set(B),set(A)),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F)),pow(B,A6)) = pow(A,B6) ) ) ).
% image_Pow_surj
tff(fact_427_fst__diag__snd,axiom,
! [B: $tType,A: $tType] : aa(fun(product_prod(A,B),product_prod(B,B)),fun(product_prod(A,B),B),aa(fun(product_prod(B,B),B),fun(fun(product_prod(A,B),product_prod(B,B)),fun(product_prod(A,B),B)),comp(product_prod(B,B),B,product_prod(A,B)),product_fst(B,B)),aa(fun(product_prod(A,B),B),fun(product_prod(A,B),product_prod(B,B)),aa(fun(B,product_prod(B,B)),fun(fun(product_prod(A,B),B),fun(product_prod(A,B),product_prod(B,B))),comp(B,product_prod(B,B),product_prod(A,B)),aTP_Lamp_bm(B,product_prod(B,B))),product_snd(A,B))) = product_snd(A,B) ).
% fst_diag_snd
tff(fact_428_snd__diag__fst,axiom,
! [B: $tType,A: $tType] : aa(fun(product_prod(A,B),product_prod(A,A)),fun(product_prod(A,B),A),aa(fun(product_prod(A,A),A),fun(fun(product_prod(A,B),product_prod(A,A)),fun(product_prod(A,B),A)),comp(product_prod(A,A),A,product_prod(A,B)),product_snd(A,A)),aa(fun(product_prod(A,B),A),fun(product_prod(A,B),product_prod(A,A)),aa(fun(A,product_prod(A,A)),fun(fun(product_prod(A,B),A),fun(product_prod(A,B),product_prod(A,A))),comp(A,product_prod(A,A),product_prod(A,B)),aTP_Lamp_bi(A,product_prod(A,A))),product_fst(A,B))) = product_fst(A,B) ).
% snd_diag_fst
tff(fact_429_fun_Orel__Grp,axiom,
! [D: $tType,B: $tType,A: $tType,A6: set(A),F: fun(A,B)] : bNF_rel_fun(D,D,A,B,fequal(D),bNF_Grp(A,B,A6,F)) = bNF_Grp(fun(D,A),fun(D,B),aa(fun(fun(D,A),bool),set(fun(D,A)),collect(fun(D,A)),aTP_Lamp_bn(set(A),fun(fun(D,A),bool),A6)),aa(fun(A,B),fun(fun(D,A),fun(D,B)),comp(A,B,D),F)) ).
% fun.rel_Grp
tff(fact_430_Grp__fst__snd,axiom,
! [B: $tType,A: $tType,R3: fun(A,fun(B,bool))] : aa(fun(product_prod(A,B),fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(product_prod(A,B),bool)),fun(fun(product_prod(A,B),fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,product_prod(A,B),B),conversep(product_prod(A,B),A,bNF_Grp(product_prod(A,B),A,aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),R3)),product_fst(A,B)))),bNF_Grp(product_prod(A,B),B,aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),R3)),product_snd(A,B))) = R3 ).
% Grp_fst_snd
tff(fact_431_change__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [F: fun(A,A),R: ref(A)] : change(A,F,R) = heap_bind(A,A,lookup(A,R),aa(ref(A),fun(A,heap_Heap(A)),aTP_Lamp_bp(fun(A,A),fun(ref(A),fun(A,heap_Heap(A))),F),R)) ) ).
% change_def
tff(fact_432_type__copy__set__map0,axiom,
! [A: $tType,B: $tType,D: $tType,E: $tType,C: $tType,F2: $tType,Rep: fun(A,B),Abs: fun(B,A),S3: fun(B,set(D)),M: fun(C,B),F: fun(E,D),S4: fun(C,set(E)),G: fun(F2,C)] :
( type_definition(A,B,Rep,Abs,top_top(set(B)))
=> ( ( aa(fun(C,B),fun(C,set(D)),aa(fun(B,set(D)),fun(fun(C,B),fun(C,set(D))),comp(B,set(D),C),S3),M) = aa(fun(C,set(E)),fun(C,set(D)),aa(fun(set(E),set(D)),fun(fun(C,set(E)),fun(C,set(D))),comp(set(E),set(D),C),aa(fun(E,D),fun(set(E),set(D)),image2(E,D),F)),S4) )
=> ( aa(fun(F2,A),fun(F2,set(D)),aa(fun(A,set(D)),fun(fun(F2,A),fun(F2,set(D))),comp(A,set(D),F2),aa(fun(A,B),fun(A,set(D)),aa(fun(B,set(D)),fun(fun(A,B),fun(A,set(D))),comp(B,set(D),A),S3),Rep)),aa(fun(F2,C),fun(F2,A),aa(fun(C,A),fun(fun(F2,C),fun(F2,A)),comp(C,A,F2),aa(fun(C,B),fun(C,A),aa(fun(B,A),fun(fun(C,B),fun(C,A)),comp(B,A,C),Abs),M)),G)) = aa(fun(F2,set(E)),fun(F2,set(D)),aa(fun(set(E),set(D)),fun(fun(F2,set(E)),fun(F2,set(D))),comp(set(E),set(D),F2),aa(fun(E,D),fun(set(E),set(D)),image2(E,D),F)),aa(fun(F2,C),fun(F2,set(E)),aa(fun(C,set(E)),fun(fun(F2,C),fun(F2,set(E))),comp(C,set(E),F2),S4),G)) ) ) ) ).
% type_copy_set_map0
tff(fact_433_type__definition_ORep__range,axiom,
! [B: $tType,A: $tType,Rep: fun(B,A),Abs: fun(A,B),A6: set(A)] :
( type_definition(B,A,Rep,Abs,A6)
=> ( aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),Rep),top_top(set(B))) = A6 ) ) ).
% type_definition.Rep_range
tff(fact_434_type__definition_OAbs__image,axiom,
! [A: $tType,B: $tType,Rep: fun(B,A),Abs: fun(A,B),A6: set(A)] :
( type_definition(B,A,Rep,Abs,A6)
=> ( aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),Abs),A6) = top_top(set(B)) ) ) ).
% type_definition.Abs_image
tff(fact_435_Quotient__alt__def5,axiom,
! [B: $tType,A: $tType,R3: fun(A,fun(A,bool)),Abs: fun(A,B),Rep: fun(B,A),T3: fun(A,fun(B,bool))] :
( quotient(A,B,R3,Abs,Rep,T3)
<=> ( pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),T3),bNF_Grp(A,B,top_top(set(A)),Abs)))
& pp(aa(fun(B,fun(A,bool)),bool,aa(fun(B,fun(A,bool)),fun(fun(B,fun(A,bool)),bool),ord_less_eq(fun(B,fun(A,bool))),bNF_Grp(B,A,top_top(set(B)),Rep)),conversep(A,B,T3)))
& ( R3 = aa(fun(B,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(A,bool)),fun(A,fun(A,bool))),relcompp(A,B,A),T3),conversep(A,B,T3)) ) ) ) ).
% Quotient_alt_def5
tff(fact_436_K__record__comp,axiom,
! [A: $tType,C: $tType,B: $tType,C3: B,F: fun(A,C),X4: A] : aa(A,B,aa(fun(A,C),fun(A,B),aa(fun(C,B),fun(fun(A,C),fun(A,B)),comp(C,B,A),aTP_Lamp_bq(B,fun(C,B),C3)),F),X4) = C3 ).
% K_record_comp
tff(fact_437_image__paired__Times,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType,F: fun(C,A),G: fun(D,B),A6: set(C),B6: set(D)] : aa(set(product_prod(C,D)),set(product_prod(A,B)),aa(fun(product_prod(C,D),product_prod(A,B)),fun(set(product_prod(C,D)),set(product_prod(A,B))),image2(product_prod(C,D),product_prod(A,B)),aa(fun(C,fun(D,product_prod(A,B))),fun(product_prod(C,D),product_prod(A,B)),product_case_prod(C,D,product_prod(A,B)),aa(fun(D,B),fun(C,fun(D,product_prod(A,B))),aTP_Lamp_br(fun(C,A),fun(fun(D,B),fun(C,fun(D,product_prod(A,B)))),F),G))),product_Sigma(C,D,A6,aTP_Lamp_bs(set(D),fun(C,set(D)),B6))) = product_Sigma(A,B,aa(set(C),set(A),aa(fun(C,A),fun(set(C),set(A)),image2(C,A),F),A6),aa(set(D),fun(A,set(B)),aTP_Lamp_bt(fun(D,B),fun(set(D),fun(A,set(B))),G),B6)) ).
% image_paired_Times
tff(fact_438_Gr__incl,axiom,
! [A: $tType,B: $tType,A6: set(A),F: fun(A,B),B6: set(B)] :
( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),bNF_Gr(A,B,A6,F)),product_Sigma(A,B,A6,aTP_Lamp_bu(set(B),fun(A,set(B)),B6))))
<=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6)),B6)) ) ).
% Gr_incl
tff(fact_439_subset__fst__snd,axiom,
! [B: $tType,A: $tType,A6: set(product_prod(A,B))] : pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),A6),product_Sigma(A,B,aa(set(product_prod(A,B)),set(A),aa(fun(product_prod(A,B),A),fun(set(product_prod(A,B)),set(A)),image2(product_prod(A,B),A),product_fst(A,B)),A6),aTP_Lamp_bv(set(product_prod(A,B)),fun(A,set(B)),A6)))) ).
% subset_fst_snd
tff(fact_440_inv__image__def,axiom,
! [B: $tType,A: $tType,R: set(product_prod(B,B)),F: fun(A,B)] : inv_image(B,A,R,F) = aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(fun(A,B),fun(A,fun(A,bool)),aTP_Lamp_bw(set(product_prod(B,B)),fun(fun(A,B),fun(A,fun(A,bool))),R),F))) ).
% inv_image_def
tff(fact_441_The__case__prod,axiom,
! [B: $tType,A: $tType,P2: fun(A,fun(B,bool))] : the(product_prod(A,B),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),P2)) = the(product_prod(A,B),aTP_Lamp_bx(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),P2)) ).
% The_case_prod
tff(fact_442_Quotient__composition__ge__eq,axiom,
! [B: $tType,A: $tType,T3: fun(A,fun(B,bool)),R3: fun(B,fun(B,bool))] :
( left_total(A,B,T3)
=> ( pp(aa(fun(B,fun(B,bool)),bool,aa(fun(B,fun(B,bool)),fun(fun(B,fun(B,bool)),bool),ord_less_eq(fun(B,fun(B,bool))),fequal(B)),R3))
=> pp(aa(fun(A,fun(A,bool)),bool,aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),bool),ord_less_eq(fun(A,fun(A,bool))),fequal(A)),aa(fun(B,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(A,bool)),fun(A,fun(A,bool))),relcompp(A,B,A),T3),aa(fun(B,fun(A,bool)),fun(B,fun(A,bool)),aa(fun(B,fun(B,bool)),fun(fun(B,fun(A,bool)),fun(B,fun(A,bool))),relcompp(B,B,A),R3),conversep(A,B,T3))))) ) ) ).
% Quotient_composition_ge_eq
tff(fact_443_Quotient__composition__le__eq,axiom,
! [B: $tType,A: $tType,T3: fun(A,fun(B,bool)),R3: fun(B,fun(B,bool))] :
( left_unique(A,B,T3)
=> ( pp(aa(fun(B,fun(B,bool)),bool,aa(fun(B,fun(B,bool)),fun(fun(B,fun(B,bool)),bool),ord_less_eq(fun(B,fun(B,bool))),R3),fequal(B)))
=> pp(aa(fun(A,fun(A,bool)),bool,aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),bool),ord_less_eq(fun(A,fun(A,bool))),aa(fun(B,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(A,bool)),fun(A,fun(A,bool))),relcompp(A,B,A),T3),aa(fun(B,fun(A,bool)),fun(B,fun(A,bool)),aa(fun(B,fun(B,bool)),fun(fun(B,fun(A,bool)),fun(B,fun(A,bool))),relcompp(B,B,A),R3),conversep(A,B,T3)))),fequal(A))) ) ) ).
% Quotient_composition_le_eq
tff(fact_444_mem__Sigma__iff,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,A6: set(A),B6: fun(A,set(B))] :
( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),product_Sigma(A,B,A6,B6)))
<=> ( pp(aa(set(A),bool,member(A,A3),A6))
& pp(aa(set(B),bool,member(B,B2),aa(A,set(B),B6,A3))) ) ) ).
% mem_Sigma_iff
tff(fact_445_SigmaI,axiom,
! [B: $tType,A: $tType,A3: A,A6: set(A),B2: B,B6: fun(A,set(B))] :
( pp(aa(set(A),bool,member(A,A3),A6))
=> ( pp(aa(set(B),bool,member(B,B2),aa(A,set(B),B6,A3)))
=> pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),product_Sigma(A,B,A6,B6))) ) ) ).
% SigmaI
tff(fact_446_in__inv__image,axiom,
! [A: $tType,B: $tType,X2: A,Y2: A,R: set(product_prod(B,B)),F: fun(A,B)] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),inv_image(B,A,R,F)))
<=> pp(aa(set(product_prod(B,B)),bool,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,F,X2)),aa(A,B,F,Y2))),R)) ) ).
% in_inv_image
tff(fact_447_Collect__case__prod,axiom,
! [A: $tType,B: $tType,P2: fun(A,bool),Q2: fun(B,bool)] : aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aa(fun(B,bool),fun(A,fun(B,bool)),aTP_Lamp_by(fun(A,bool),fun(fun(B,bool),fun(A,fun(B,bool))),P2),Q2))) = product_Sigma(A,B,aa(fun(A,bool),set(A),collect(A),P2),aTP_Lamp_bz(fun(B,bool),fun(A,set(B)),Q2)) ).
% Collect_case_prod
tff(fact_448_UNIV__Times__UNIV,axiom,
! [B: $tType,A: $tType] : product_Sigma(A,B,top_top(set(A)),aTP_Lamp_ca(A,set(B))) = top_top(set(product_prod(A,B))) ).
% UNIV_Times_UNIV
tff(fact_449_The__split__eq,axiom,
! [A: $tType,B: $tType,X2: A,Y2: B] : the(product_prod(A,B),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aa(B,fun(A,fun(B,bool)),aTP_Lamp_cb(A,fun(B,fun(A,fun(B,bool))),X2),Y2))) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y2) ).
% The_split_eq
tff(fact_450_Times__eq__cancel2,axiom,
! [A: $tType,B: $tType,X2: A,C5: set(A),A6: set(B),B6: set(B)] :
( pp(aa(set(A),bool,member(A,X2),C5))
=> ( ( product_Sigma(B,A,A6,aTP_Lamp_cc(set(A),fun(B,set(A)),C5)) = product_Sigma(B,A,B6,aTP_Lamp_cc(set(A),fun(B,set(A)),C5)) )
<=> ( A6 = B6 ) ) ) ).
% Times_eq_cancel2
tff(fact_451_Sigma__cong,axiom,
! [B: $tType,A: $tType,A6: set(A),B6: set(A),C5: fun(A,set(B)),D3: fun(A,set(B))] :
( ( A6 = B6 )
=> ( ! [X: A] :
( pp(aa(set(A),bool,member(A,X),B6))
=> ( aa(A,set(B),C5,X) = aa(A,set(B),D3,X) ) )
=> ( product_Sigma(A,B,A6,C5) = product_Sigma(A,B,B6,D3) ) ) ) ).
% Sigma_cong
tff(fact_452_SigmaE2,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,A6: set(A),B6: fun(A,set(B))] :
( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),product_Sigma(A,B,A6,B6)))
=> ~ ( pp(aa(set(A),bool,member(A,A3),A6))
=> ~ pp(aa(set(B),bool,member(B,B2),aa(A,set(B),B6,A3))) ) ) ).
% SigmaE2
tff(fact_453_SigmaD2,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,A6: set(A),B6: fun(A,set(B))] :
( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),product_Sigma(A,B,A6,B6)))
=> pp(aa(set(B),bool,member(B,B2),aa(A,set(B),B6,A3))) ) ).
% SigmaD2
tff(fact_454_SigmaD1,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,A6: set(A),B6: fun(A,set(B))] :
( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),product_Sigma(A,B,A6,B6)))
=> pp(aa(set(A),bool,member(A,A3),A6)) ) ).
% SigmaD1
tff(fact_455_SigmaE,axiom,
! [A: $tType,B: $tType,C3: product_prod(A,B),A6: set(A),B6: fun(A,set(B))] :
( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),C3),product_Sigma(A,B,A6,B6)))
=> ~ ! [X: A] :
( pp(aa(set(A),bool,member(A,X),A6))
=> ! [Y: B] :
( pp(aa(set(B),bool,member(B,Y),aa(A,set(B),B6,X)))
=> ( C3 != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y) ) ) ) ) ).
% SigmaE
tff(fact_456_Collect__case__prod__Sigma,axiom,
! [B: $tType,A: $tType,P2: fun(A,bool),Q2: fun(A,fun(B,bool))] : aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aTP_Lamp_cd(fun(A,bool),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),P2),Q2))) = product_Sigma(A,B,aa(fun(A,bool),set(A),collect(A),P2),aTP_Lamp_ce(fun(A,fun(B,bool)),fun(A,set(B)),Q2)) ).
% Collect_case_prod_Sigma
tff(fact_457_Sigma__mono,axiom,
! [B: $tType,A: $tType,A6: set(A),C5: set(A),B6: fun(A,set(B)),D3: fun(A,set(B))] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),C5))
=> ( ! [X: A] :
( pp(aa(set(A),bool,member(A,X),A6))
=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),B6,X)),aa(A,set(B),D3,X))) )
=> pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),product_Sigma(A,B,A6,B6)),product_Sigma(A,B,C5,D3))) ) ) ).
% Sigma_mono
tff(fact_458_Quotient__alt__def4,axiom,
! [B: $tType,A: $tType,R3: fun(A,fun(A,bool)),Abs: fun(A,B),Rep: fun(B,A),T3: fun(A,fun(B,bool))] :
( quotient(A,B,R3,Abs,Rep,T3)
<=> ( ! [A7: A,B5: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),T3,A7),B5))
=> ( aa(A,B,Abs,A7) = B5 ) )
& ! [B5: B] : pp(aa(B,bool,aa(A,fun(B,bool),T3,aa(B,A,Rep,B5)),B5))
& ( R3 = aa(fun(B,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(A,bool)),fun(A,fun(A,bool))),relcompp(A,B,A),T3),conversep(A,B,T3)) ) ) ) ).
% Quotient_alt_def4
tff(fact_459_Times__subset__cancel2,axiom,
! [A: $tType,B: $tType,X2: A,C5: set(A),A6: set(B),B6: set(B)] :
( pp(aa(set(A),bool,member(A,X2),C5))
=> ( pp(aa(set(product_prod(B,A)),bool,aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),bool),ord_less_eq(set(product_prod(B,A))),product_Sigma(B,A,A6,aTP_Lamp_cc(set(A),fun(B,set(A)),C5))),product_Sigma(B,A,B6,aTP_Lamp_cc(set(A),fun(B,set(A)),C5))))
<=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A6),B6)) ) ) ).
% Times_subset_cancel2
tff(fact_460_mem__Times__iff,axiom,
! [A: $tType,B: $tType,X2: product_prod(A,B),A6: set(A),B6: set(B)] :
( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),X2),product_Sigma(A,B,A6,aTP_Lamp_bu(set(B),fun(A,set(B)),B6))))
<=> ( pp(aa(set(A),bool,member(A,aa(product_prod(A,B),A,product_fst(A,B),X2)),A6))
& pp(aa(set(B),bool,member(B,aa(product_prod(A,B),B,product_snd(A,B),X2)),B6)) ) ) ).
% mem_Times_iff
tff(fact_461_product__swap,axiom,
! [A: $tType,B: $tType,A6: set(B),B6: set(A)] : aa(set(product_prod(B,A)),set(product_prod(A,B)),aa(fun(product_prod(B,A),product_prod(A,B)),fun(set(product_prod(B,A)),set(product_prod(A,B))),image2(product_prod(B,A),product_prod(A,B)),product_swap(B,A)),product_Sigma(B,A,A6,aTP_Lamp_cc(set(A),fun(B,set(A)),B6))) = product_Sigma(A,B,B6,aTP_Lamp_bu(set(B),fun(A,set(B)),A6)) ).
% product_swap
tff(fact_462_Quotient__compose,axiom,
! [A: $tType,C: $tType,B: $tType,R1: fun(A,fun(A,bool)),Abs1: fun(A,B),Rep1: fun(B,A),T1: fun(A,fun(B,bool)),R22: fun(B,fun(B,bool)),Abs22: fun(B,C),Rep22: fun(C,B),T22: fun(B,fun(C,bool))] :
( quotient(A,B,R1,Abs1,Rep1,T1)
=> ( quotient(B,C,R22,Abs22,Rep22,T22)
=> quotient(A,C,aa(fun(B,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(A,bool)),fun(A,fun(A,bool))),relcompp(A,B,A),T1),aa(fun(B,fun(A,bool)),fun(B,fun(A,bool)),aa(fun(B,fun(B,bool)),fun(fun(B,fun(A,bool)),fun(B,fun(A,bool))),relcompp(B,B,A),R22),conversep(A,B,T1))),aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),Abs22),Abs1),aa(fun(C,B),fun(C,A),aa(fun(B,A),fun(fun(C,B),fun(C,A)),comp(B,A,C),Rep1),Rep22),aa(fun(B,fun(C,bool)),fun(A,fun(C,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(C,bool)),fun(A,fun(C,bool))),relcompp(A,B,C),T1),T22)) ) ) ).
% Quotient_compose
tff(fact_463_swap__product,axiom,
! [A: $tType,B: $tType,A6: set(B),B6: set(A)] : aa(set(product_prod(B,A)),set(product_prod(A,B)),aa(fun(product_prod(B,A),product_prod(A,B)),fun(set(product_prod(B,A)),set(product_prod(A,B))),image2(product_prod(B,A),product_prod(A,B)),aa(fun(B,fun(A,product_prod(A,B))),fun(product_prod(B,A),product_prod(A,B)),product_case_prod(B,A,product_prod(A,B)),aTP_Lamp_bk(B,fun(A,product_prod(A,B))))),product_Sigma(B,A,A6,aTP_Lamp_cc(set(A),fun(B,set(A)),B6))) = product_Sigma(A,B,B6,aTP_Lamp_bu(set(B),fun(A,set(B)),A6)) ).
% swap_product
tff(fact_464_map__prod__surj__on,axiom,
! [D: $tType,B: $tType,A: $tType,C: $tType,F: fun(B,A),A6: set(B),A8: set(A),G: fun(D,C),B6: set(D),B8: set(C)] :
( ( aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6) = A8 )
=> ( ( aa(set(D),set(C),aa(fun(D,C),fun(set(D),set(C)),image2(D,C),G),B6) = B8 )
=> ( aa(set(product_prod(B,D)),set(product_prod(A,C)),aa(fun(product_prod(B,D),product_prod(A,C)),fun(set(product_prod(B,D)),set(product_prod(A,C))),image2(product_prod(B,D),product_prod(A,C)),product_map_prod(B,A,D,C,F,G)),product_Sigma(B,D,A6,aTP_Lamp_cf(set(D),fun(B,set(D)),B6))) = product_Sigma(A,C,A8,aTP_Lamp_cg(set(C),fun(A,set(C)),B8)) ) ) ) ).
% map_prod_surj_on
tff(fact_465_old_Orec__prod__def,axiom,
! [T: $tType,B: $tType,A: $tType,X4: fun(A,fun(B,T)),Xa4: product_prod(A,B)] : product_rec_prod(A,B,T,X4,Xa4) = the(T,product_rec_set_prod(A,B,T,X4,Xa4)) ).
% old.rec_prod_def
tff(fact_466_left__total__alt__def,axiom,
! [B: $tType,A: $tType,R3: fun(A,fun(B,bool))] :
( left_total(A,B,R3)
<=> pp(aa(fun(A,fun(A,bool)),bool,aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),bool),ord_less_eq(fun(A,fun(A,bool))),fequal(A)),aa(fun(B,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(A,bool)),fun(A,fun(A,bool))),relcompp(A,B,A),R3),conversep(A,B,R3)))) ) ).
% left_total_alt_def
tff(fact_467_left__unique__alt__def,axiom,
! [B: $tType,A: $tType,R3: fun(A,fun(B,bool))] :
( left_unique(A,B,R3)
<=> pp(aa(fun(A,fun(A,bool)),bool,aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),bool),ord_less_eq(fun(A,fun(A,bool))),aa(fun(B,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(A,bool)),fun(A,fun(A,bool))),relcompp(A,B,A),R3),conversep(A,B,R3))),fequal(A))) ) ).
% left_unique_alt_def
tff(fact_468_rp__inv__image__def,axiom,
! [B: $tType,A: $tType] : fun_rp_inv_image(A,B) = aa(fun(set(product_prod(A,A)),fun(set(product_prod(A,A)),fun(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B)))))),fun(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),fun(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B))))),product_case_prod(set(product_prod(A,A)),set(product_prod(A,A)),fun(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B))))),aTP_Lamp_ch(set(product_prod(A,A)),fun(set(product_prod(A,A)),fun(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B))))))) ).
% rp_inv_image_def
tff(fact_469_left__total__OO,axiom,
! [A: $tType,C: $tType,B: $tType,R3: fun(A,fun(B,bool)),S3: fun(B,fun(C,bool))] :
( left_total(A,B,R3)
=> ( left_total(B,C,S3)
=> left_total(A,C,aa(fun(B,fun(C,bool)),fun(A,fun(C,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(C,bool)),fun(A,fun(C,bool))),relcompp(A,B,C),R3),S3)) ) ) ).
% left_total_OO
tff(fact_470_left__unique__OO,axiom,
! [A: $tType,C: $tType,B: $tType,R3: fun(A,fun(B,bool)),S3: fun(B,fun(C,bool))] :
( left_unique(A,B,R3)
=> ( left_unique(B,C,S3)
=> left_unique(A,C,aa(fun(B,fun(C,bool)),fun(A,fun(C,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(C,bool)),fun(A,fun(C,bool))),relcompp(A,B,C),R3),S3)) ) ) ).
% left_unique_OO
tff(fact_471_image__o__collect,axiom,
! [B: $tType,C: $tType,A: $tType,G: fun(C,B),F6: set(fun(A,set(C)))] : bNF_collect(A,B,aa(set(fun(A,set(C))),set(fun(A,set(B))),aa(fun(fun(A,set(C)),fun(A,set(B))),fun(set(fun(A,set(C))),set(fun(A,set(B)))),image2(fun(A,set(C)),fun(A,set(B))),aa(fun(set(C),set(B)),fun(fun(A,set(C)),fun(A,set(B))),comp(set(C),set(B),A),aa(fun(C,B),fun(set(C),set(B)),image2(C,B),G))),F6)) = aa(fun(A,set(C)),fun(A,set(B)),aa(fun(set(C),set(B)),fun(fun(A,set(C)),fun(A,set(B))),comp(set(C),set(B),A),aa(fun(C,B),fun(set(C),set(B)),image2(C,B),G)),bNF_collect(A,C,F6)) ).
% image_o_collect
tff(fact_472_Product__Type_Oproduct__def,axiom,
! [A: $tType,B: $tType,A6: set(A),B6: set(B)] : product_product(A,B,A6,B6) = product_Sigma(A,B,A6,aTP_Lamp_bu(set(B),fun(A,set(B)),B6)) ).
% Product_Type.product_def
tff(fact_473_member__product,axiom,
! [A: $tType,B: $tType,X2: product_prod(A,B),A6: set(A),B6: set(B)] :
( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),X2),product_product(A,B,A6,B6)))
<=> pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),X2),product_Sigma(A,B,A6,aTP_Lamp_bu(set(B),fun(A,set(B)),B6)))) ) ).
% member_product
tff(fact_474_snd__image__times,axiom,
! [B: $tType,A: $tType,A6: set(B),B6: set(A)] :
( ( ( A6 = bot_bot(set(B)) )
=> ( aa(set(product_prod(B,A)),set(A),aa(fun(product_prod(B,A),A),fun(set(product_prod(B,A)),set(A)),image2(product_prod(B,A),A),product_snd(B,A)),product_Sigma(B,A,A6,aTP_Lamp_cc(set(A),fun(B,set(A)),B6))) = bot_bot(set(A)) ) )
& ( ( A6 != bot_bot(set(B)) )
=> ( aa(set(product_prod(B,A)),set(A),aa(fun(product_prod(B,A),A),fun(set(product_prod(B,A)),set(A)),image2(product_prod(B,A),A),product_snd(B,A)),product_Sigma(B,A,A6,aTP_Lamp_cc(set(A),fun(B,set(A)),B6))) = B6 ) ) ) ).
% snd_image_times
tff(fact_475_fst__image__times,axiom,
! [B: $tType,A: $tType,B6: set(B),A6: set(A)] :
( ( ( B6 = bot_bot(set(B)) )
=> ( aa(set(product_prod(A,B)),set(A),aa(fun(product_prod(A,B),A),fun(set(product_prod(A,B)),set(A)),image2(product_prod(A,B),A),product_fst(A,B)),product_Sigma(A,B,A6,aTP_Lamp_bu(set(B),fun(A,set(B)),B6))) = bot_bot(set(A)) ) )
& ( ( B6 != bot_bot(set(B)) )
=> ( aa(set(product_prod(A,B)),set(A),aa(fun(product_prod(A,B),A),fun(set(product_prod(A,B)),set(A)),image2(product_prod(A,B),A),product_fst(A,B)),product_Sigma(A,B,A6,aTP_Lamp_bu(set(B),fun(A,set(B)),B6))) = A6 ) ) ) ).
% fst_image_times
tff(fact_476_image__is__empty,axiom,
! [A: $tType,B: $tType,F: fun(B,A),A6: set(B)] :
( ( aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6) = bot_bot(set(A)) )
<=> ( A6 = bot_bot(set(B)) ) ) ).
% image_is_empty
tff(fact_477_empty__is__image,axiom,
! [A: $tType,B: $tType,F: fun(B,A),A6: set(B)] :
( ( bot_bot(set(A)) = aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6) )
<=> ( A6 = bot_bot(set(B)) ) ) ).
% empty_is_image
tff(fact_478_image__empty,axiom,
! [B: $tType,A: $tType,F: fun(B,A)] : aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),bot_bot(set(B))) = bot_bot(set(A)) ).
% image_empty
tff(fact_479_Sigma__empty1,axiom,
! [B: $tType,A: $tType,B6: fun(A,set(B))] : product_Sigma(A,B,bot_bot(set(A)),B6) = bot_bot(set(product_prod(A,B))) ).
% Sigma_empty1
tff(fact_480_Sigma__empty2,axiom,
! [B: $tType,A: $tType,A6: set(A)] : product_Sigma(A,B,A6,aTP_Lamp_ci(A,set(B))) = bot_bot(set(product_prod(A,B))) ).
% Sigma_empty2
tff(fact_481_Times__empty,axiom,
! [A: $tType,B: $tType,A6: set(A),B6: set(B)] :
( ( product_Sigma(A,B,A6,aTP_Lamp_bu(set(B),fun(A,set(B)),B6)) = bot_bot(set(product_prod(A,B))) )
<=> ( ( A6 = bot_bot(set(A)) )
| ( B6 = bot_bot(set(B)) ) ) ) ).
% Times_empty
tff(fact_482_Collect__const__case__prod,axiom,
! [B: $tType,A: $tType,P2: bool] :
( ( pp(P2)
=> ( aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aTP_Lamp_cj(bool,fun(A,fun(B,bool)),P2))) = top_top(set(product_prod(A,B))) ) )
& ( ~ pp(P2)
=> ( aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aTP_Lamp_cj(bool,fun(A,fun(B,bool)),P2))) = bot_bot(set(product_prod(A,B))) ) ) ) ).
% Collect_const_case_prod
tff(fact_483_Sigma__empty__iff,axiom,
! [A: $tType,B: $tType,I4: set(A),X6: fun(A,set(B))] :
( ( product_Sigma(A,B,I4,X6) = bot_bot(set(product_prod(A,B))) )
<=> ! [X5: A] :
( pp(aa(set(A),bool,member(A,X5),I4))
=> ( aa(A,set(B),X6,X5) = bot_bot(set(B)) ) ) ) ).
% Sigma_empty_iff
tff(fact_484_subset__emptyI,axiom,
! [A: $tType,A6: set(A)] :
( ! [X: A] : ~ pp(aa(set(A),bool,member(A,X),A6))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),bot_bot(set(A)))) ) ).
% subset_emptyI
tff(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))) )
=> pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),A6),B6)) ) ).
% less_by_empty
tff(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,aTP_Lamp_bu(set(B),fun(A,set(B)),B6)) = product_Sigma(A,B,C5,aTP_Lamp_bu(set(B),fun(A,set(B)),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
tff(fact_487_times__subset__iff,axiom,
! [A: $tType,B: $tType,A6: set(A),C5: set(B),B6: set(A),D3: set(B)] :
( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),product_Sigma(A,B,A6,aTP_Lamp_bu(set(B),fun(A,set(B)),C5))),product_Sigma(A,B,B6,aTP_Lamp_bu(set(B),fun(A,set(B)),D3))))
<=> ( ( A6 = bot_bot(set(A)) )
| ( C5 = bot_bot(set(B)) )
| ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B6))
& pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),C5),D3)) ) ) ) ).
% times_subset_iff
tff(fact_488_fst__image__Sigma,axiom,
! [B: $tType,A: $tType,A6: set(A),B6: fun(A,set(B))] : aa(set(product_prod(A,B)),set(A),aa(fun(product_prod(A,B),A),fun(set(product_prod(A,B)),set(A)),image2(product_prod(A,B),A),product_fst(A,B)),product_Sigma(A,B,A6,B6)) = aa(fun(A,bool),set(A),collect(A),aa(fun(A,set(B)),fun(A,bool),aTP_Lamp_ck(set(A),fun(fun(A,set(B)),fun(A,bool)),A6),B6)) ).
% fst_image_Sigma
tff(fact_489_collect__comp,axiom,
! [A: $tType,B: $tType,C: $tType,F6: set(fun(C,set(B))),G: fun(A,C)] : aa(fun(A,C),fun(A,set(B)),aa(fun(C,set(B)),fun(fun(A,C),fun(A,set(B))),comp(C,set(B),A),bNF_collect(C,B,F6)),G) = bNF_collect(A,B,aa(set(fun(C,set(B))),set(fun(A,set(B))),aa(fun(fun(C,set(B)),fun(A,set(B))),fun(set(fun(C,set(B))),set(fun(A,set(B)))),image2(fun(C,set(B)),fun(A,set(B))),aTP_Lamp_cl(fun(A,C),fun(fun(C,set(B)),fun(A,set(B))),G)),F6)) ).
% collect_comp
tff(fact_490_empty__natural,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,F: fun(A,C),G: fun(D,B)] : aa(fun(A,C),fun(A,set(B)),aa(fun(C,set(B)),fun(fun(A,C),fun(A,set(B))),comp(C,set(B),A),aTP_Lamp_cm(C,set(B))),F) = aa(fun(A,set(D)),fun(A,set(B)),aa(fun(set(D),set(B)),fun(fun(A,set(D)),fun(A,set(B))),comp(set(D),set(B),A),aa(fun(D,B),fun(set(D),set(B)),image2(D,B),G)),aTP_Lamp_cn(A,set(D))) ).
% empty_natural
tff(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: fun(B,A)] :
( fun_reduction_pair(A,P2)
=> fun_reduction_pair(B,aa(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),fun(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),fun_rp_inv_image(A,B),P2),F)) ) ).
% rp_inv_image_rp
tff(fact_492_the__elem__image__unique,axiom,
! [B: $tType,A: $tType,A6: set(A),F: fun(A,B),X2: A] :
( ( A6 != bot_bot(set(A)) )
=> ( ! [Y: A] :
( pp(aa(set(A),bool,member(A,Y),A6))
=> ( aa(A,B,F,Y) = aa(A,B,F,X2) ) )
=> ( the_elem(B,aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6)) = aa(A,B,F,X2) ) ) ) ).
% the_elem_image_unique
tff(fact_493_snd__image__Sigma,axiom,
! [A: $tType,B: $tType,A6: set(B),B6: fun(B,set(A))] : aa(set(product_prod(B,A)),set(A),aa(fun(product_prod(B,A),A),fun(set(product_prod(B,A)),set(A)),image2(product_prod(B,A),A),product_snd(B,A)),product_Sigma(B,A,A6,B6)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),B6),A6)) ).
% snd_image_Sigma
tff(fact_494_old_Orec__unit__def,axiom,
! [T: $tType,X4: T,Xa4: product_unit] : product_rec_unit(T,X4,Xa4) = the(T,product_rec_set_unit(T,X4,Xa4)) ).
% old.rec_unit_def
tff(fact_495_range__constant,axiom,
! [B: $tType,A: $tType,X2: A] : aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),aa(A,fun(B,A),aTP_Lamp_at(A,fun(B,A)),X2)),top_top(set(B))) = aa(set(A),set(A),insert(A,X2),bot_bot(set(A))) ).
% range_constant
tff(fact_496_INT__constant,axiom,
! [B: $tType,A: $tType,A6: set(B),C3: set(A)] :
( ( ( A6 = bot_bot(set(B)) )
=> ( aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),aTP_Lamp_cc(set(A),fun(B,set(A)),C3)),A6)) = top_top(set(A)) ) )
& ( ( A6 != bot_bot(set(B)) )
=> ( aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),aTP_Lamp_cc(set(A),fun(B,set(A)),C3)),A6)) = C3 ) ) ) ).
% INT_constant
tff(fact_497_neg__fun__distr1,axiom,
! [D: $tType,A: $tType,B: $tType,C: $tType,E: $tType,F2: $tType,R3: fun(A,fun(B,bool)),R5: fun(B,fun(C,bool)),S3: fun(D,fun(F2,bool)),S4: fun(F2,fun(E,bool))] :
( left_unique(A,B,R3)
=> ( right_total(A,B,R3)
=> ( right_unique(B,C,R5)
=> ( left_total(B,C,R5)
=> pp(aa(fun(fun(A,D),fun(fun(C,E),bool)),bool,aa(fun(fun(A,D),fun(fun(C,E),bool)),fun(fun(fun(A,D),fun(fun(C,E),bool)),bool),ord_less_eq(fun(fun(A,D),fun(fun(C,E),bool))),bNF_rel_fun(A,C,D,E,aa(fun(B,fun(C,bool)),fun(A,fun(C,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(C,bool)),fun(A,fun(C,bool))),relcompp(A,B,C),R3),R5),aa(fun(F2,fun(E,bool)),fun(D,fun(E,bool)),aa(fun(D,fun(F2,bool)),fun(fun(F2,fun(E,bool)),fun(D,fun(E,bool))),relcompp(D,F2,E),S3),S4))),aa(fun(fun(B,F2),fun(fun(C,E),bool)),fun(fun(A,D),fun(fun(C,E),bool)),aa(fun(fun(A,D),fun(fun(B,F2),bool)),fun(fun(fun(B,F2),fun(fun(C,E),bool)),fun(fun(A,D),fun(fun(C,E),bool))),relcompp(fun(A,D),fun(B,F2),fun(C,E)),bNF_rel_fun(A,B,D,F2,R3,S3)),bNF_rel_fun(B,C,F2,E,R5,S4)))) ) ) ) ) ).
% neg_fun_distr1
tff(fact_498_neg__fun__distr2,axiom,
! [F2: $tType,E: $tType,A: $tType,B: $tType,D: $tType,C: $tType,R5: fun(A,fun(B,bool)),S4: fun(C,fun(D,bool)),R3: fun(E,fun(A,bool)),S3: fun(F2,fun(C,bool))] :
( right_unique(A,B,R5)
=> ( left_total(A,B,R5)
=> ( left_unique(C,D,S4)
=> ( right_total(C,D,S4)
=> pp(aa(fun(fun(E,F2),fun(fun(B,D),bool)),bool,aa(fun(fun(E,F2),fun(fun(B,D),bool)),fun(fun(fun(E,F2),fun(fun(B,D),bool)),bool),ord_less_eq(fun(fun(E,F2),fun(fun(B,D),bool))),bNF_rel_fun(E,B,F2,D,aa(fun(A,fun(B,bool)),fun(E,fun(B,bool)),aa(fun(E,fun(A,bool)),fun(fun(A,fun(B,bool)),fun(E,fun(B,bool))),relcompp(E,A,B),R3),R5),aa(fun(C,fun(D,bool)),fun(F2,fun(D,bool)),aa(fun(F2,fun(C,bool)),fun(fun(C,fun(D,bool)),fun(F2,fun(D,bool))),relcompp(F2,C,D),S3),S4))),aa(fun(fun(A,C),fun(fun(B,D),bool)),fun(fun(E,F2),fun(fun(B,D),bool)),aa(fun(fun(E,F2),fun(fun(A,C),bool)),fun(fun(fun(A,C),fun(fun(B,D),bool)),fun(fun(E,F2),fun(fun(B,D),bool))),relcompp(fun(E,F2),fun(A,C),fun(B,D)),bNF_rel_fun(E,A,F2,C,R3,S3)),bNF_rel_fun(A,B,C,D,R5,S4)))) ) ) ) ) ).
% neg_fun_distr2
tff(fact_499_refl__onI,axiom,
! [A: $tType,R: set(product_prod(A,A)),A6: set(A)] :
( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R),product_Sigma(A,A,A6,aTP_Lamp_co(set(A),fun(A,set(A)),A6))))
=> ( ! [X: A] :
( pp(aa(set(A),bool,member(A,X),A6))
=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X)),R)) )
=> refl_on(A,A6,R) ) ) ).
% refl_onI
tff(fact_500_refl__on__def,axiom,
! [A: $tType,A6: set(A),R: set(product_prod(A,A))] :
( refl_on(A,A6,R)
<=> ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R),product_Sigma(A,A,A6,aTP_Lamp_co(set(A),fun(A,set(A)),A6))))
& ! [X5: A] :
( pp(aa(set(A),bool,member(A,X5),A6))
=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),X5)),R)) ) ) ) ).
% refl_on_def
tff(fact_501_right__unique__alt__def,axiom,
! [A: $tType,B: $tType,R3: fun(A,fun(B,bool))] :
( right_unique(A,B,R3)
<=> pp(aa(fun(B,fun(B,bool)),bool,aa(fun(B,fun(B,bool)),fun(fun(B,fun(B,bool)),bool),ord_less_eq(fun(B,fun(B,bool))),aa(fun(A,fun(B,bool)),fun(B,fun(B,bool)),aa(fun(B,fun(A,bool)),fun(fun(A,fun(B,bool)),fun(B,fun(B,bool))),relcompp(B,A,B),conversep(A,B,R3)),R3)),fequal(B))) ) ).
% right_unique_alt_def
tff(fact_502_image__insert,axiom,
! [A: $tType,B: $tType,F: fun(B,A),A3: B,B6: set(B)] : aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),aa(set(B),set(B),insert(B,A3),B6)) = aa(set(A),set(A),insert(A,aa(B,A,F,A3)),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),B6)) ).
% image_insert
tff(fact_503_insert__image,axiom,
! [B: $tType,A: $tType,X2: A,A6: set(A),F: fun(A,B)] :
( pp(aa(set(A),bool,member(A,X2),A6))
=> ( aa(set(B),set(B),insert(B,aa(A,B,F,X2)),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6)) = aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6) ) ) ).
% insert_image
tff(fact_504_UN__ball__bex__simps_I4_J,axiom,
! [E: $tType,F2: $tType,B6: fun(E,set(F2)),A6: set(E),P2: fun(F2,bool)] :
( ? [X5: F2] :
( pp(aa(set(F2),bool,member(F2,X5),aa(set(set(F2)),set(F2),complete_Sup_Sup(set(F2)),aa(set(E),set(set(F2)),aa(fun(E,set(F2)),fun(set(E),set(set(F2))),image2(E,set(F2)),B6),A6))))
& pp(aa(F2,bool,P2,X5)) )
<=> ? [X5: E] :
( pp(aa(set(E),bool,member(E,X5),A6))
& ? [Xa3: F2] :
( pp(aa(set(F2),bool,member(F2,Xa3),aa(E,set(F2),B6,X5)))
& pp(aa(F2,bool,P2,Xa3)) ) ) ) ).
% UN_ball_bex_simps(4)
tff(fact_505_UN__ball__bex__simps_I2_J,axiom,
! [B: $tType,C: $tType,B6: fun(B,set(C)),A6: set(B),P2: fun(C,bool)] :
( ! [X5: C] :
( pp(aa(set(C),bool,member(C,X5),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),aa(fun(B,set(C)),fun(set(B),set(set(C))),image2(B,set(C)),B6),A6))))
=> pp(aa(C,bool,P2,X5)) )
<=> ! [X5: B] :
( pp(aa(set(B),bool,member(B,X5),A6))
=> ! [Xa3: C] :
( pp(aa(set(C),bool,member(C,Xa3),aa(B,set(C),B6,X5)))
=> pp(aa(C,bool,P2,Xa3)) ) ) ) ).
% UN_ball_bex_simps(2)
tff(fact_506_bex__UN,axiom,
! [B: $tType,A: $tType,B6: fun(B,set(A)),A6: set(B),P2: fun(A,bool)] :
( ? [X5: A] :
( pp(aa(set(A),bool,member(A,X5),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),B6),A6))))
& pp(aa(A,bool,P2,X5)) )
<=> ? [X5: B] :
( pp(aa(set(B),bool,member(B,X5),A6))
& ? [Xa3: A] :
( pp(aa(set(A),bool,member(A,Xa3),aa(B,set(A),B6,X5)))
& pp(aa(A,bool,P2,Xa3)) ) ) ) ).
% bex_UN
tff(fact_507_ball__UN,axiom,
! [B: $tType,A: $tType,B6: fun(B,set(A)),A6: set(B),P2: fun(A,bool)] :
( ! [X5: A] :
( pp(aa(set(A),bool,member(A,X5),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),B6),A6))))
=> pp(aa(A,bool,P2,X5)) )
<=> ! [X5: B] :
( pp(aa(set(B),bool,member(B,X5),A6))
=> ! [Xa3: A] :
( pp(aa(set(A),bool,member(A,Xa3),aa(B,set(A),B6,X5)))
=> pp(aa(A,bool,P2,Xa3)) ) ) ) ).
% ball_UN
tff(fact_508_SUP__identity__eq,axiom,
! [A: $tType] :
( complete_Sup(A)
=> ! [A6: set(A)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aTP_Lamp_cp(A,A)),A6)) = aa(set(A),A,complete_Sup_Sup(A),A6) ) ).
% SUP_identity_eq
tff(fact_509_INF__identity__eq,axiom,
! [A: $tType] :
( complete_Inf(A)
=> ! [A6: set(A)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aTP_Lamp_cq(A,A)),A6)) = aa(set(A),A,complete_Inf_Inf(A),A6) ) ).
% INF_identity_eq
tff(fact_510_UN__iff,axiom,
! [A: $tType,B: $tType,B2: A,B6: fun(B,set(A)),A6: set(B)] :
( pp(aa(set(A),bool,member(A,B2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),B6),A6))))
<=> ? [X5: B] :
( pp(aa(set(B),bool,member(B,X5),A6))
& pp(aa(set(A),bool,member(A,B2),aa(B,set(A),B6,X5))) ) ) ).
% UN_iff
tff(fact_511_UN__I,axiom,
! [B: $tType,A: $tType,A3: A,A6: set(A),B2: B,B6: fun(A,set(B))] :
( pp(aa(set(A),bool,member(A,A3),A6))
=> ( pp(aa(set(B),bool,member(B,B2),aa(A,set(B),B6,A3)))
=> pp(aa(set(B),bool,member(B,B2),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),aa(fun(A,set(B)),fun(set(A),set(set(B))),image2(A,set(B)),B6),A6)))) ) ) ).
% UN_I
tff(fact_512_INT__iff,axiom,
! [A: $tType,B: $tType,B2: A,B6: fun(B,set(A)),A6: set(B)] :
( pp(aa(set(A),bool,member(A,B2),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),B6),A6))))
<=> ! [X5: B] :
( pp(aa(set(B),bool,member(B,X5),A6))
=> pp(aa(set(A),bool,member(A,B2),aa(B,set(A),B6,X5))) ) ) ).
% INT_iff
tff(fact_513_INT__I,axiom,
! [B: $tType,A: $tType,A6: set(A),B2: B,B6: fun(A,set(B))] :
( ! [X: A] :
( pp(aa(set(A),bool,member(A,X),A6))
=> pp(aa(set(B),bool,member(B,B2),aa(A,set(B),B6,X))) )
=> pp(aa(set(B),bool,member(B,B2),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),aa(fun(A,set(B)),fun(set(A),set(set(B))),image2(A,set(B)),B6),A6)))) ) ).
% INT_I
tff(fact_514_Sup__apply,axiom,
! [B: $tType,A: $tType] :
( complete_Sup(B)
=> ! [A6: set(fun(A,B)),X2: A] : aa(A,B,aa(set(fun(A,B)),fun(A,B),complete_Sup_Sup(fun(A,B)),A6),X2) = aa(set(B),B,complete_Sup_Sup(B),aa(set(fun(A,B)),set(B),aa(fun(fun(A,B),B),fun(set(fun(A,B)),set(B)),image2(fun(A,B),B),aTP_Lamp_cr(A,fun(fun(A,B),B),X2)),A6)) ) ).
% Sup_apply
tff(fact_515_Inf__apply,axiom,
! [B: $tType,A: $tType] :
( complete_Inf(B)
=> ! [A6: set(fun(A,B)),X2: A] : aa(A,B,aa(set(fun(A,B)),fun(A,B),complete_Inf_Inf(fun(A,B)),A6),X2) = aa(set(B),B,complete_Inf_Inf(B),aa(set(fun(A,B)),set(B),aa(fun(fun(A,B),B),fun(set(fun(A,B)),set(B)),image2(fun(A,B),B),aTP_Lamp_cs(A,fun(fun(A,B),B),X2)),A6)) ) ).
% Inf_apply
tff(fact_516_SUP__id__eq,axiom,
! [A: $tType] :
( complete_Sup(A)
=> ! [A6: set(A)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),id(A)),A6)) = aa(set(A),A,complete_Sup_Sup(A),A6) ) ).
% SUP_id_eq
tff(fact_517_INF__id__eq,axiom,
! [A: $tType] :
( complete_Inf(A)
=> ! [A6: set(A)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),id(A)),A6)) = aa(set(A),A,complete_Inf_Inf(A),A6) ) ).
% INF_id_eq
tff(fact_518_SUP__bot,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A6: set(B)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),aTP_Lamp_ct(B,A)),A6)) = bot_bot(A) ) ).
% SUP_bot
tff(fact_519_SUP__bot__conv_I1_J,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [B6: fun(B,A),A6: set(B)] :
( ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),B6),A6)) = bot_bot(A) )
<=> ! [X5: B] :
( pp(aa(set(B),bool,member(B,X5),A6))
=> ( aa(B,A,B6,X5) = bot_bot(A) ) ) ) ) ).
% SUP_bot_conv(1)
tff(fact_520_SUP__bot__conv_I2_J,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [B6: fun(B,A),A6: set(B)] :
( ( bot_bot(A) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),B6),A6)) )
<=> ! [X5: B] :
( pp(aa(set(B),bool,member(B,X5),A6))
=> ( aa(B,A,B6,X5) = bot_bot(A) ) ) ) ) ).
% SUP_bot_conv(2)
tff(fact_521_SUP__const,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A6: set(B),F: A] :
( ( A6 != bot_bot(set(B)) )
=> ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),aTP_Lamp_cu(A,fun(B,A),F)),A6)) = F ) ) ) ).
% SUP_const
tff(fact_522_INF__const,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A6: set(B),F: A] :
( ( A6 != bot_bot(set(B)) )
=> ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),aTP_Lamp_cu(A,fun(B,A),F)),A6)) = F ) ) ) ).
% INF_const
tff(fact_523_INF__top__conv_I2_J,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [B6: fun(B,A),A6: set(B)] :
( ( top_top(A) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),B6),A6)) )
<=> ! [X5: B] :
( pp(aa(set(B),bool,member(B,X5),A6))
=> ( aa(B,A,B6,X5) = top_top(A) ) ) ) ) ).
% INF_top_conv(2)
tff(fact_524_INF__top__conv_I1_J,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [B6: fun(B,A),A6: set(B)] :
( ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),B6),A6)) = top_top(A) )
<=> ! [X5: B] :
( pp(aa(set(B),bool,member(B,X5),A6))
=> ( aa(B,A,B6,X5) = top_top(A) ) ) ) ) ).
% INF_top_conv(1)
tff(fact_525_INF__top,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A6: set(B)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),aTP_Lamp_cv(B,A)),A6)) = top_top(A) ) ).
% INF_top
tff(fact_526_UN__constant,axiom,
! [B: $tType,A: $tType,A6: set(B),C3: set(A)] :
( ( ( A6 = bot_bot(set(B)) )
=> ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),aTP_Lamp_cc(set(A),fun(B,set(A)),C3)),A6)) = bot_bot(set(A)) ) )
& ( ( A6 != bot_bot(set(B)) )
=> ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),aTP_Lamp_cc(set(A),fun(B,set(A)),C3)),A6)) = C3 ) ) ) ).
% UN_constant
tff(fact_527_SUP__apply,axiom,
! [B: $tType,A: $tType,C: $tType] :
( complete_Sup(A)
=> ! [F: fun(C,fun(B,A)),A6: set(C),X2: B] : aa(B,A,aa(set(fun(B,A)),fun(B,A),complete_Sup_Sup(fun(B,A)),aa(set(C),set(fun(B,A)),aa(fun(C,fun(B,A)),fun(set(C),set(fun(B,A))),image2(C,fun(B,A)),F),A6)),X2) = aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),aa(fun(C,A),fun(set(C),set(A)),image2(C,A),aa(B,fun(C,A),aTP_Lamp_cw(fun(C,fun(B,A)),fun(B,fun(C,A)),F),X2)),A6)) ) ).
% SUP_apply
tff(fact_528_INF__apply,axiom,
! [B: $tType,A: $tType,C: $tType] :
( complete_Inf(A)
=> ! [F: fun(C,fun(B,A)),A6: set(C),X2: B] : aa(B,A,aa(set(fun(B,A)),fun(B,A),complete_Inf_Inf(fun(B,A)),aa(set(C),set(fun(B,A)),aa(fun(C,fun(B,A)),fun(set(C),set(fun(B,A))),image2(C,fun(B,A)),F),A6)),X2) = aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),aa(fun(C,A),fun(set(C),set(A)),image2(C,A),aa(B,fun(C,A),aTP_Lamp_cx(fun(C,fun(B,A)),fun(B,fun(C,A)),F),X2)),A6)) ) ).
% INF_apply
tff(fact_529_UN__simps_I1_J,axiom,
! [A: $tType,B: $tType,C5: set(B),A3: A,B6: fun(B,set(A))] :
( ( ( C5 = bot_bot(set(B)) )
=> ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_cy(A,fun(fun(B,set(A)),fun(B,set(A))),A3),B6)),C5)) = bot_bot(set(A)) ) )
& ( ( C5 != bot_bot(set(B)) )
=> ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_cy(A,fun(fun(B,set(A)),fun(B,set(A))),A3),B6)),C5)) = aa(set(A),set(A),insert(A,A3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),B6),C5))) ) ) ) ).
% UN_simps(1)
tff(fact_530_UN__singleton,axiom,
! [A: $tType,A6: set(A)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(A),set(set(A)),aa(fun(A,set(A)),fun(set(A),set(set(A))),image2(A,set(A)),aTP_Lamp_cz(A,set(A))),A6)) = A6 ).
% UN_singleton
tff(fact_531_refl__on__UNION,axiom,
! [B: $tType,A: $tType,S3: set(A),A6: fun(A,set(B)),R: fun(A,set(product_prod(B,B)))] :
( ! [X: A] :
( pp(aa(set(A),bool,member(A,X),S3))
=> refl_on(B,aa(A,set(B),A6,X),aa(A,set(product_prod(B,B)),R,X)) )
=> refl_on(B,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),aa(fun(A,set(B)),fun(set(A),set(set(B))),image2(A,set(B)),A6),S3)),aa(set(set(product_prod(B,B))),set(product_prod(B,B)),complete_Sup_Sup(set(product_prod(B,B))),aa(set(A),set(set(product_prod(B,B))),aa(fun(A,set(product_prod(B,B))),fun(set(A),set(set(product_prod(B,B)))),image2(A,set(product_prod(B,B))),R),S3))) ) ).
% refl_on_UNION
tff(fact_532_refl__on__INTER,axiom,
! [B: $tType,A: $tType,S3: set(A),A6: fun(A,set(B)),R: fun(A,set(product_prod(B,B)))] :
( ! [X: A] :
( pp(aa(set(A),bool,member(A,X),S3))
=> refl_on(B,aa(A,set(B),A6,X),aa(A,set(product_prod(B,B)),R,X)) )
=> refl_on(B,aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),aa(fun(A,set(B)),fun(set(A),set(set(B))),image2(A,set(B)),A6),S3)),aa(set(set(product_prod(B,B))),set(product_prod(B,B)),complete_Inf_Inf(set(product_prod(B,B))),aa(set(A),set(set(product_prod(B,B))),aa(fun(A,set(product_prod(B,B))),fun(set(A),set(set(product_prod(B,B)))),image2(A,set(product_prod(B,B))),R),S3))) ) ).
% refl_on_INTER
tff(fact_533_refl__on__singleton,axiom,
! [A: $tType,X2: A] : refl_on(A,aa(set(A),set(A),insert(A,X2),bot_bot(set(A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),insert(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),X2)),bot_bot(set(product_prod(A,A))))) ).
% refl_on_singleton
tff(fact_534_INT__extend__simps_I9_J,axiom,
! [S5: $tType,R6: $tType,Q3: $tType,C5: fun(R6,set(S5)),B6: fun(Q3,set(R6)),A6: set(Q3)] : aa(set(set(S5)),set(S5),complete_Inf_Inf(set(S5)),aa(set(Q3),set(set(S5)),aa(fun(Q3,set(S5)),fun(set(Q3),set(set(S5))),image2(Q3,set(S5)),aa(fun(Q3,set(R6)),fun(Q3,set(S5)),aTP_Lamp_da(fun(R6,set(S5)),fun(fun(Q3,set(R6)),fun(Q3,set(S5))),C5),B6)),A6)) = aa(set(set(S5)),set(S5),complete_Inf_Inf(set(S5)),aa(set(R6),set(set(S5)),aa(fun(R6,set(S5)),fun(set(R6),set(set(S5))),image2(R6,set(S5)),C5),aa(set(set(R6)),set(R6),complete_Sup_Sup(set(R6)),aa(set(Q3),set(set(R6)),aa(fun(Q3,set(R6)),fun(set(Q3),set(set(R6))),image2(Q3,set(R6)),B6),A6)))) ).
% INT_extend_simps(9)
tff(fact_535_INT__insert__distrib,axiom,
! [B: $tType,A: $tType,U: A,A6: set(A),A3: B,B6: fun(A,set(B))] :
( pp(aa(set(A),bool,member(A,U),A6))
=> ( aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),aa(fun(A,set(B)),fun(set(A),set(set(B))),image2(A,set(B)),aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_db(B,fun(fun(A,set(B)),fun(A,set(B))),A3),B6)),A6)) = aa(set(B),set(B),insert(B,A3),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),aa(fun(A,set(B)),fun(set(A),set(set(B))),image2(A,set(B)),B6),A6))) ) ) ).
% INT_insert_distrib
tff(fact_536_INT__extend__simps_I5_J,axiom,
! [I5: $tType,J2: $tType,A3: I5,B6: fun(J2,set(I5)),C5: set(J2)] : aa(set(I5),set(I5),insert(I5,A3),aa(set(set(I5)),set(I5),complete_Inf_Inf(set(I5)),aa(set(J2),set(set(I5)),aa(fun(J2,set(I5)),fun(set(J2),set(set(I5))),image2(J2,set(I5)),B6),C5))) = aa(set(set(I5)),set(I5),complete_Inf_Inf(set(I5)),aa(set(J2),set(set(I5)),aa(fun(J2,set(I5)),fun(set(J2),set(set(I5))),image2(J2,set(I5)),aa(fun(J2,set(I5)),fun(J2,set(I5)),aTP_Lamp_dc(I5,fun(fun(J2,set(I5)),fun(J2,set(I5))),A3),B6)),C5)) ).
% INT_extend_simps(5)
tff(fact_537_UN__insert__distrib,axiom,
! [B: $tType,A: $tType,U: A,A6: set(A),A3: B,B6: fun(A,set(B))] :
( pp(aa(set(A),bool,member(A,U),A6))
=> ( aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),aa(fun(A,set(B)),fun(set(A),set(set(B))),image2(A,set(B)),aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_db(B,fun(fun(A,set(B)),fun(A,set(B))),A3),B6)),A6)) = aa(set(B),set(B),insert(B,A3),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),aa(fun(A,set(B)),fun(set(A),set(set(B))),image2(A,set(B)),B6),A6))) ) ) ).
% UN_insert_distrib
tff(fact_538_Sigma__Union,axiom,
! [B: $tType,A: $tType,X6: set(set(A)),B6: fun(A,set(B))] : product_Sigma(A,B,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),X6),B6) = aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(set(A)),set(set(product_prod(A,B))),aa(fun(set(A),set(product_prod(A,B))),fun(set(set(A)),set(set(product_prod(A,B)))),image2(set(A),set(product_prod(A,B))),aTP_Lamp_dd(fun(A,set(B)),fun(set(A),set(product_prod(A,B))),B6)),X6)) ).
% Sigma_Union
tff(fact_539_Inf__fun__def,axiom,
! [B: $tType,A: $tType] :
( complete_Inf(B)
=> ! [A6: set(fun(A,B)),X4: A] : aa(A,B,aa(set(fun(A,B)),fun(A,B),complete_Inf_Inf(fun(A,B)),A6),X4) = aa(set(B),B,complete_Inf_Inf(B),aa(set(fun(A,B)),set(B),aa(fun(fun(A,B),B),fun(set(fun(A,B)),set(B)),image2(fun(A,B),B),aTP_Lamp_cs(A,fun(fun(A,B),B),X4)),A6)) ) ).
% Inf_fun_def
tff(fact_540_Sup__fun__def,axiom,
! [B: $tType,A: $tType] :
( complete_Sup(B)
=> ! [A6: set(fun(A,B)),X4: A] : aa(A,B,aa(set(fun(A,B)),fun(A,B),complete_Sup_Sup(fun(A,B)),A6),X4) = aa(set(B),B,complete_Sup_Sup(B),aa(set(fun(A,B)),set(B),aa(fun(fun(A,B),B),fun(set(fun(A,B)),set(B)),image2(fun(A,B),B),aTP_Lamp_cr(A,fun(fun(A,B),B),X4)),A6)) ) ).
% Sup_fun_def
tff(fact_541_INF__cong,axiom,
! [A: $tType,B: $tType] :
( complete_Inf(A)
=> ! [A6: set(B),B6: set(B),C5: fun(B,A),D3: fun(B,A)] :
( ( A6 = B6 )
=> ( ! [X: B] :
( pp(aa(set(B),bool,member(B,X),B6))
=> ( aa(B,A,C5,X) = aa(B,A,D3,X) ) )
=> ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),C5),A6)) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),D3),B6)) ) ) ) ) ).
% INF_cong
tff(fact_542_SUP__cong,axiom,
! [A: $tType,B: $tType] :
( complete_Sup(A)
=> ! [A6: set(B),B6: set(B),C5: fun(B,A),D3: fun(B,A)] :
( ( A6 = B6 )
=> ( ! [X: B] :
( pp(aa(set(B),bool,member(B,X),B6))
=> ( aa(B,A,C5,X) = aa(B,A,D3,X) ) )
=> ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),C5),A6)) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),D3),B6)) ) ) ) ) ).
% SUP_cong
tff(fact_543_SUP__UNION,axiom,
! [B: $tType,A: $tType,C: $tType] :
( comple6319245703460814977attice(A)
=> ! [F: fun(B,A),G: fun(C,set(B)),A6: set(C)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),aa(fun(C,set(B)),fun(set(C),set(set(B))),image2(C,set(B)),G),A6)))) = aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),aa(fun(C,A),fun(set(C),set(A)),image2(C,A),aa(fun(C,set(B)),fun(C,A),aTP_Lamp_de(fun(B,A),fun(fun(C,set(B)),fun(C,A)),F),G)),A6)) ) ).
% SUP_UNION
tff(fact_544_insert__subsetI,axiom,
! [A: $tType,X2: A,A6: set(A),X6: set(A)] :
( pp(aa(set(A),bool,member(A,X2),A6))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X6),A6))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),insert(A,X2),X6)),A6)) ) ) ).
% insert_subsetI
tff(fact_545_UN__extend__simps_I1_J,axiom,
! [A: $tType,B: $tType,C5: set(B),A3: A,B6: fun(B,set(A))] :
( ( ( C5 = bot_bot(set(B)) )
=> ( aa(set(A),set(A),insert(A,A3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),B6),C5))) = aa(set(A),set(A),insert(A,A3),bot_bot(set(A))) ) )
& ( ( C5 != bot_bot(set(B)) )
=> ( aa(set(A),set(A),insert(A,A3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),B6),C5))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_cy(A,fun(fun(B,set(A)),fun(B,set(A))),A3),B6)),C5)) ) ) ) ).
% UN_extend_simps(1)
tff(fact_546_Sigma__def,axiom,
! [B: $tType,A: $tType,A6: set(A),B6: fun(A,set(B))] : product_Sigma(A,B,A6,B6) = aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(A),set(set(product_prod(A,B))),aa(fun(A,set(product_prod(A,B))),fun(set(A),set(set(product_prod(A,B)))),image2(A,set(product_prod(A,B))),aTP_Lamp_dg(fun(A,set(B)),fun(A,set(product_prod(A,B))),B6)),A6)) ).
% Sigma_def
tff(fact_547_UN__Times__distrib,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType,E4: fun(C,set(A)),F6: fun(D,set(B)),A6: set(C),B6: set(D)] : aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(product_prod(C,D)),set(set(product_prod(A,B))),aa(fun(product_prod(C,D),set(product_prod(A,B))),fun(set(product_prod(C,D)),set(set(product_prod(A,B)))),image2(product_prod(C,D),set(product_prod(A,B))),aa(fun(C,fun(D,set(product_prod(A,B)))),fun(product_prod(C,D),set(product_prod(A,B))),product_case_prod(C,D,set(product_prod(A,B))),aa(fun(D,set(B)),fun(C,fun(D,set(product_prod(A,B)))),aTP_Lamp_di(fun(C,set(A)),fun(fun(D,set(B)),fun(C,fun(D,set(product_prod(A,B))))),E4),F6))),product_Sigma(C,D,A6,aTP_Lamp_bs(set(D),fun(C,set(D)),B6)))) = product_Sigma(A,B,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),aa(fun(C,set(A)),fun(set(C),set(set(A))),image2(C,set(A)),E4),A6)),aa(set(D),fun(A,set(B)),aTP_Lamp_dj(fun(D,set(B)),fun(set(D),fun(A,set(B))),F6),B6)) ).
% UN_Times_distrib
tff(fact_548_Pow__INT__eq,axiom,
! [A: $tType,B: $tType,B6: fun(B,set(A)),A6: set(B)] : pow(A,aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),B6),A6))) = aa(set(set(set(A))),set(set(A)),complete_Inf_Inf(set(set(A))),aa(set(B),set(set(set(A))),aa(fun(B,set(set(A))),fun(set(B),set(set(set(A)))),image2(B,set(set(A))),aTP_Lamp_dk(fun(B,set(A)),fun(B,set(set(A))),B6)),A6)) ).
% Pow_INT_eq
tff(fact_549_right__total__OO,axiom,
! [A: $tType,C: $tType,B: $tType,A6: fun(A,fun(B,bool)),B6: fun(B,fun(C,bool))] :
( right_total(A,B,A6)
=> ( right_total(B,C,B6)
=> right_total(A,C,aa(fun(B,fun(C,bool)),fun(A,fun(C,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(C,bool)),fun(A,fun(C,bool))),relcompp(A,B,C),A6),B6)) ) ) ).
% right_total_OO
tff(fact_550_refl__onD,axiom,
! [A: $tType,A6: set(A),R: set(product_prod(A,A)),A3: A] :
( refl_on(A,A6,R)
=> ( pp(aa(set(A),bool,member(A,A3),A6))
=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A3)),R)) ) ) ).
% refl_onD
tff(fact_551_refl__onD1,axiom,
! [A: $tType,A6: set(A),R: set(product_prod(A,A)),X2: A,Y2: A] :
( refl_on(A,A6,R)
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),R))
=> pp(aa(set(A),bool,member(A,X2),A6)) ) ) ).
% refl_onD1
tff(fact_552_refl__onD2,axiom,
! [A: $tType,A6: set(A),R: set(product_prod(A,A)),X2: A,Y2: A] :
( refl_on(A,A6,R)
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),R))
=> pp(aa(set(A),bool,member(A,Y2),A6)) ) ) ).
% refl_onD2
tff(fact_553_INT__extend__simps_I8_J,axiom,
! [P3: $tType,O: $tType,B6: fun(O,set(P3)),A6: set(set(O))] : aa(set(set(P3)),set(P3),complete_Inf_Inf(set(P3)),aa(set(set(O)),set(set(P3)),aa(fun(set(O),set(P3)),fun(set(set(O)),set(set(P3))),image2(set(O),set(P3)),aTP_Lamp_dl(fun(O,set(P3)),fun(set(O),set(P3)),B6)),A6)) = aa(set(set(P3)),set(P3),complete_Inf_Inf(set(P3)),aa(set(O),set(set(P3)),aa(fun(O,set(P3)),fun(set(O),set(set(P3))),image2(O,set(P3)),B6),aa(set(set(O)),set(O),complete_Sup_Sup(set(O)),A6))) ).
% INT_extend_simps(8)
tff(fact_554_Id__on__def,axiom,
! [A: $tType,A6: set(A)] : id_on(A,A6) = aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),aa(set(A),set(set(product_prod(A,A))),aa(fun(A,set(product_prod(A,A))),fun(set(A),set(set(product_prod(A,A)))),image2(A,set(product_prod(A,A))),aTP_Lamp_dm(A,set(product_prod(A,A)))),A6)) ).
% Id_on_def
tff(fact_555_INF__le__SUP,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [A6: set(B),F: fun(B,A)] :
( ( A6 != bot_bot(set(B)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6)))) ) ) ).
% INF_le_SUP
tff(fact_556_right__unique__OO,axiom,
! [A: $tType,C: $tType,B: $tType,A6: fun(A,fun(B,bool)),B6: fun(B,fun(C,bool))] :
( right_unique(A,B,A6)
=> ( right_unique(B,C,B6)
=> right_unique(A,C,aa(fun(B,fun(C,bool)),fun(A,fun(C,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(C,bool)),fun(A,fun(C,bool))),relcompp(A,B,C),A6),B6)) ) ) ).
% right_unique_OO
tff(fact_557_INF__commute,axiom,
! [B: $tType,A: $tType,C: $tType] :
( comple6319245703460814977attice(A)
=> ! [F: fun(B,fun(C,A)),B6: set(C),A6: set(B)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),aa(set(C),fun(B,A),aTP_Lamp_dn(fun(B,fun(C,A)),fun(set(C),fun(B,A)),F),B6)),A6)) = aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),aa(fun(C,A),fun(set(C),set(A)),image2(C,A),aa(set(B),fun(C,A),aTP_Lamp_dp(fun(B,fun(C,A)),fun(set(B),fun(C,A)),F),A6)),B6)) ) ).
% INF_commute
tff(fact_558_SUP__commute,axiom,
! [B: $tType,A: $tType,C: $tType] :
( comple6319245703460814977attice(A)
=> ! [F: fun(B,fun(C,A)),B6: set(C),A6: set(B)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),aa(set(C),fun(B,A),aTP_Lamp_dq(fun(B,fun(C,A)),fun(set(C),fun(B,A)),F),B6)),A6)) = aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),aa(fun(C,A),fun(set(C),set(A)),image2(C,A),aa(set(B),fun(C,A),aTP_Lamp_dr(fun(B,fun(C,A)),fun(set(B),fun(C,A)),F),A6)),B6)) ) ).
% SUP_commute
tff(fact_559_INT__E,axiom,
! [A: $tType,B: $tType,B2: A,B6: fun(B,set(A)),A6: set(B),A3: B] :
( pp(aa(set(A),bool,member(A,B2),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),B6),A6))))
=> ( ~ pp(aa(set(A),bool,member(A,B2),aa(B,set(A),B6,A3)))
=> ~ pp(aa(set(B),bool,member(B,A3),A6)) ) ) ).
% INT_E
tff(fact_560_INT__D,axiom,
! [A: $tType,B: $tType,B2: A,B6: fun(B,set(A)),A6: set(B),A3: B] :
( pp(aa(set(A),bool,member(A,B2),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),B6),A6))))
=> ( pp(aa(set(B),bool,member(B,A3),A6))
=> pp(aa(set(A),bool,member(A,B2),aa(B,set(A),B6,A3))) ) ) ).
% INT_D
tff(fact_561_UN__UN__flatten,axiom,
! [B: $tType,A: $tType,C: $tType,C5: fun(B,set(A)),B6: fun(C,set(B)),A6: set(C)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),C5),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),aa(fun(C,set(B)),fun(set(C),set(set(B))),image2(C,set(B)),B6),A6)))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),aa(fun(C,set(A)),fun(set(C),set(set(A))),image2(C,set(A)),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_ds(fun(B,set(A)),fun(fun(C,set(B)),fun(C,set(A))),C5),B6)),A6)) ).
% UN_UN_flatten
tff(fact_562_UN__E,axiom,
! [A: $tType,B: $tType,B2: A,B6: fun(B,set(A)),A6: set(B)] :
( pp(aa(set(A),bool,member(A,B2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),B6),A6))))
=> ~ ! [X: B] :
( pp(aa(set(B),bool,member(B,X),A6))
=> ~ pp(aa(set(A),bool,member(A,B2),aa(B,set(A),B6,X))) ) ) ).
% UN_E
tff(fact_563_UN__extend__simps_I9_J,axiom,
! [S5: $tType,R6: $tType,Q3: $tType,C5: fun(R6,set(S5)),B6: fun(Q3,set(R6)),A6: set(Q3)] : aa(set(set(S5)),set(S5),complete_Sup_Sup(set(S5)),aa(set(Q3),set(set(S5)),aa(fun(Q3,set(S5)),fun(set(Q3),set(set(S5))),image2(Q3,set(S5)),aa(fun(Q3,set(R6)),fun(Q3,set(S5)),aTP_Lamp_dt(fun(R6,set(S5)),fun(fun(Q3,set(R6)),fun(Q3,set(S5))),C5),B6)),A6)) = aa(set(set(S5)),set(S5),complete_Sup_Sup(set(S5)),aa(set(R6),set(set(S5)),aa(fun(R6,set(S5)),fun(set(R6),set(set(S5))),image2(R6,set(S5)),C5),aa(set(set(R6)),set(R6),complete_Sup_Sup(set(R6)),aa(set(Q3),set(set(R6)),aa(fun(Q3,set(R6)),fun(set(Q3),set(set(R6))),image2(Q3,set(R6)),B6),A6)))) ).
% UN_extend_simps(9)
tff(fact_564_UNION__singleton__eq__range,axiom,
! [A: $tType,B: $tType,F: fun(B,A),A6: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),aTP_Lamp_du(fun(B,A),fun(B,set(A)),F)),A6)) = aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6) ).
% UNION_singleton_eq_range
tff(fact_565_in__Union__o__assoc,axiom,
! [B: $tType,A: $tType,C: $tType,X2: A,Gset: fun(B,set(set(A))),Gmap: fun(C,B),A6: C] :
( pp(aa(set(A),bool,member(A,X2),aa(C,set(A),aa(fun(C,B),fun(C,set(A)),aa(fun(B,set(A)),fun(fun(C,B),fun(C,set(A))),comp(B,set(A),C),aa(fun(B,set(set(A))),fun(B,set(A)),aa(fun(set(set(A)),set(A)),fun(fun(B,set(set(A))),fun(B,set(A))),comp(set(set(A)),set(A),B),complete_Sup_Sup(set(A))),Gset)),Gmap),A6)))
=> pp(aa(set(A),bool,member(A,X2),aa(C,set(A),aa(fun(C,set(set(A))),fun(C,set(A)),aa(fun(set(set(A)),set(A)),fun(fun(C,set(set(A))),fun(C,set(A))),comp(set(set(A)),set(A),C),complete_Sup_Sup(set(A))),aa(fun(C,B),fun(C,set(set(A))),aa(fun(B,set(set(A))),fun(fun(C,B),fun(C,set(set(A)))),comp(B,set(set(A)),C),Gset),Gmap)),A6))) ) ).
% in_Union_o_assoc
tff(fact_566_UN__Pow__subset,axiom,
! [A: $tType,B: $tType,B6: fun(B,set(A)),A6: set(B)] : pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),aa(set(set(set(A))),set(set(A)),complete_Sup_Sup(set(set(A))),aa(set(B),set(set(set(A))),aa(fun(B,set(set(A))),fun(set(B),set(set(set(A)))),image2(B,set(set(A))),aTP_Lamp_dk(fun(B,set(A)),fun(B,set(set(A))),B6)),A6))),pow(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),B6),A6))))) ).
% UN_Pow_subset
tff(fact_567_INF__eq,axiom,
! [B: $tType,A: $tType,C: $tType] :
( comple6319245703460814977attice(A)
=> ! [A6: set(B),B6: set(C),G: fun(C,A),F: fun(B,A)] :
( ! [I6: B] :
( pp(aa(set(B),bool,member(B,I6),A6))
=> ? [X4: C] :
( pp(aa(set(C),bool,member(C,X4),B6))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(C,A,G,X4)),aa(B,A,F,I6))) ) )
=> ( ! [J3: C] :
( pp(aa(set(C),bool,member(C,J3),B6))
=> ? [X4: B] :
( pp(aa(set(B),bool,member(B,X4),A6))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,X4)),aa(C,A,G,J3))) ) )
=> ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6)) = aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),aa(fun(C,A),fun(set(C),set(A)),image2(C,A),G),B6)) ) ) ) ) ).
% INF_eq
tff(fact_568_SUP__eq,axiom,
! [B: $tType,A: $tType,C: $tType] :
( comple6319245703460814977attice(A)
=> ! [A6: set(B),B6: set(C),F: fun(B,A),G: fun(C,A)] :
( ! [I6: B] :
( pp(aa(set(B),bool,member(B,I6),A6))
=> ? [X4: C] :
( pp(aa(set(C),bool,member(C,X4),B6))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,I6)),aa(C,A,G,X4))) ) )
=> ( ! [J3: C] :
( pp(aa(set(C),bool,member(C,J3),B6))
=> ? [X4: B] :
( pp(aa(set(B),bool,member(B,X4),A6))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(C,A,G,J3)),aa(B,A,F,X4))) ) )
=> ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6)) = aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),aa(fun(C,A),fun(set(C),set(A)),image2(C,A),G),B6)) ) ) ) ) ).
% SUP_eq
tff(fact_569_INF__eq__const,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [I4: set(B),F: fun(B,A),X2: A] :
( ( I4 != bot_bot(set(B)) )
=> ( ! [I6: B] :
( pp(aa(set(B),bool,member(B,I6),I4))
=> ( aa(B,A,F,I6) = X2 ) )
=> ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),I4)) = X2 ) ) ) ) ).
% INF_eq_const
tff(fact_570_SUP__eq__const,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [I4: set(B),F: fun(B,A),X2: A] :
( ( I4 != bot_bot(set(B)) )
=> ( ! [I6: B] :
( pp(aa(set(B),bool,member(B,I6),I4))
=> ( aa(B,A,F,I6) = X2 ) )
=> ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),I4)) = X2 ) ) ) ) ).
% SUP_eq_const
tff(fact_571_INF__image,axiom,
! [B: $tType,A: $tType,C: $tType] :
( complete_Inf(A)
=> ! [G: fun(B,A),F: fun(C,B),A6: set(C)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),G),aa(set(C),set(B),aa(fun(C,B),fun(set(C),set(B)),image2(C,B),F),A6))) = aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),aa(fun(C,A),fun(set(C),set(A)),image2(C,A),aa(fun(C,B),fun(C,A),aa(fun(B,A),fun(fun(C,B),fun(C,A)),comp(B,A,C),G),F)),A6)) ) ).
% INF_image
tff(fact_572_SUP__image,axiom,
! [B: $tType,A: $tType,C: $tType] :
( complete_Sup(A)
=> ! [G: fun(B,A),F: fun(C,B),A6: set(C)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),G),aa(set(C),set(B),aa(fun(C,B),fun(set(C),set(B)),image2(C,B),F),A6))) = aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),aa(fun(C,A),fun(set(C),set(A)),image2(C,A),aa(fun(C,B),fun(C,A),aa(fun(B,A),fun(fun(C,B),fun(C,A)),comp(B,A,C),G),F)),A6)) ) ).
% SUP_image
tff(fact_573_collect__def,axiom,
! [A: $tType,B: $tType,F6: set(fun(B,set(A))),X2: B] : aa(B,set(A),bNF_collect(B,A,F6),X2) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(fun(B,set(A))),set(set(A)),aa(fun(fun(B,set(A)),set(A)),fun(set(fun(B,set(A))),set(set(A))),image2(fun(B,set(A)),set(A)),aTP_Lamp_dv(B,fun(fun(B,set(A)),set(A)),X2)),F6)) ).
% collect_def
tff(fact_574_INF__greatest,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [A6: set(B),U: A,F: fun(B,A)] :
( ! [I6: B] :
( pp(aa(set(B),bool,member(B,I6),A6))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(B,A,F,I6))) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6)))) ) ) ).
% INF_greatest
tff(fact_575_le__INF__iff,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [U: A,F: fun(B,A),A6: set(B)] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))))
<=> ! [X5: B] :
( pp(aa(set(B),bool,member(B,X5),A6))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(B,A,F,X5))) ) ) ) ).
% le_INF_iff
tff(fact_576_INF__lower2,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [I: B,A6: set(B),F: fun(B,A),U: A] :
( pp(aa(set(B),bool,member(B,I),A6))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,I)),U))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))),U)) ) ) ) ).
% INF_lower2
tff(fact_577_INF__mono_H,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [F: fun(B,A),G: fun(B,A),A6: set(B)] :
( ! [X: B] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,X)),aa(B,A,G,X)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),G),A6)))) ) ) ).
% INF_mono'
tff(fact_578_INF__lower,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [I: B,A6: set(B),F: fun(B,A)] :
( pp(aa(set(B),bool,member(B,I),A6))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))),aa(B,A,F,I))) ) ) ).
% INF_lower
tff(fact_579_INF__mono,axiom,
! [C: $tType,A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [B6: set(B),A6: set(C),F: fun(C,A),G: fun(B,A)] :
( ! [M3: B] :
( pp(aa(set(B),bool,member(B,M3),B6))
=> ? [X4: C] :
( pp(aa(set(C),bool,member(C,X4),A6))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(C,A,F,X4)),aa(B,A,G,M3))) ) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),aa(fun(C,A),fun(set(C),set(A)),image2(C,A),F),A6))),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),G),B6)))) ) ) ).
% INF_mono
tff(fact_580_INF__eqI,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A6: set(B),X2: A,F: fun(B,A)] :
( ! [I6: B] :
( pp(aa(set(B),bool,member(B,I6),A6))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),aa(B,A,F,I6))) )
=> ( ! [Y: A] :
( ! [I7: B] :
( pp(aa(set(B),bool,member(B,I7),A6))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),aa(B,A,F,I7))) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X2)) )
=> ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6)) = X2 ) ) ) ) ).
% INF_eqI
tff(fact_581_SUP__upper2,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [I: B,A6: set(B),U: A,F: fun(B,A)] :
( pp(aa(set(B),bool,member(B,I),A6))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(B,A,F,I)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6)))) ) ) ) ).
% SUP_upper2
tff(fact_582_SUP__le__iff,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [F: fun(B,A),A6: set(B),U: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))),U))
<=> ! [X5: B] :
( pp(aa(set(B),bool,member(B,X5),A6))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,X5)),U)) ) ) ) ).
% SUP_le_iff
tff(fact_583_SUP__upper,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [I: B,A6: set(B),F: fun(B,A)] :
( pp(aa(set(B),bool,member(B,I),A6))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,I)),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6)))) ) ) ).
% SUP_upper
tff(fact_584_SUP__mono_H,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [F: fun(B,A),G: fun(B,A),A6: set(B)] :
( ! [X: B] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,X)),aa(B,A,G,X)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),G),A6)))) ) ) ).
% SUP_mono'
tff(fact_585_SUP__least,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A6: set(B),F: fun(B,A),U: A] :
( ! [I6: B] :
( pp(aa(set(B),bool,member(B,I6),A6))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,I6)),U)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))),U)) ) ) ).
% SUP_least
tff(fact_586_SUP__mono,axiom,
! [B: $tType,A: $tType,C: $tType] :
( comple6319245703460814977attice(A)
=> ! [A6: set(B),B6: set(C),F: fun(B,A),G: fun(C,A)] :
( ! [N2: B] :
( pp(aa(set(B),bool,member(B,N2),A6))
=> ? [X4: C] :
( pp(aa(set(C),bool,member(C,X4),B6))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,N2)),aa(C,A,G,X4))) ) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))),aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),aa(fun(C,A),fun(set(C),set(A)),image2(C,A),G),B6)))) ) ) ).
% SUP_mono
tff(fact_587_SUP__eqI,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A6: set(B),F: fun(B,A),X2: A] :
( ! [I6: B] :
( pp(aa(set(B),bool,member(B,I6),A6))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,I6)),X2)) )
=> ( ! [Y: A] :
( ! [I7: B] :
( pp(aa(set(B),bool,member(B,I7),A6))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,I7)),Y)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Y)) )
=> ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6)) = X2 ) ) ) ) ).
% SUP_eqI
tff(fact_588_INT__extend__simps_I10_J,axiom,
! [V4: $tType,U2: $tType,T: $tType,B6: fun(U2,set(V4)),F: fun(T,U2),A6: set(T)] : aa(set(set(V4)),set(V4),complete_Inf_Inf(set(V4)),aa(set(T),set(set(V4)),aa(fun(T,set(V4)),fun(set(T),set(set(V4))),image2(T,set(V4)),aa(fun(T,U2),fun(T,set(V4)),aTP_Lamp_dw(fun(U2,set(V4)),fun(fun(T,U2),fun(T,set(V4))),B6),F)),A6)) = aa(set(set(V4)),set(V4),complete_Inf_Inf(set(V4)),aa(set(U2),set(set(V4)),aa(fun(U2,set(V4)),fun(set(U2),set(set(V4))),image2(U2,set(V4)),B6),aa(set(T),set(U2),aa(fun(T,U2),fun(set(T),set(U2)),image2(T,U2),F),A6))) ).
% INT_extend_simps(10)
tff(fact_589_INT__lower,axiom,
! [B: $tType,A: $tType,A3: A,A6: set(A),B6: fun(A,set(B))] :
( pp(aa(set(A),bool,member(A,A3),A6))
=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),aa(fun(A,set(B)),fun(set(A),set(set(B))),image2(A,set(B)),B6),A6))),aa(A,set(B),B6,A3))) ) ).
% INT_lower
tff(fact_590_INT__greatest,axiom,
! [B: $tType,A: $tType,A6: set(A),C5: set(B),B6: fun(A,set(B))] :
( ! [X: A] :
( pp(aa(set(A),bool,member(A,X),A6))
=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),C5),aa(A,set(B),B6,X))) )
=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),C5),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),aa(fun(A,set(B)),fun(set(A),set(set(B))),image2(A,set(B)),B6),A6)))) ) ).
% INT_greatest
tff(fact_591_INT__anti__mono,axiom,
! [B: $tType,A: $tType,A6: set(A),B6: set(A),F: fun(A,set(B)),G: fun(A,set(B))] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B6))
=> ( ! [X: A] :
( pp(aa(set(A),bool,member(A,X),A6))
=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),F,X)),aa(A,set(B),G,X))) )
=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),aa(fun(A,set(B)),fun(set(A),set(set(B))),image2(A,set(B)),F),B6))),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),aa(fun(A,set(B)),fun(set(A),set(set(B))),image2(A,set(B)),G),A6)))) ) ) ).
% INT_anti_mono
tff(fact_592_INT__subset__iff,axiom,
! [A: $tType,B: $tType,B6: set(A),A6: fun(B,set(A)),I4: set(B)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B6),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),A6),I4))))
<=> ! [X5: B] :
( pp(aa(set(B),bool,member(B,X5),I4))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B6),aa(B,set(A),A6,X5))) ) ) ).
% INT_subset_iff
tff(fact_593_image__constant__conv,axiom,
! [B: $tType,A: $tType,A6: set(B),C3: A] :
( ( ( A6 = bot_bot(set(B)) )
=> ( aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),aa(A,fun(B,A),aTP_Lamp_at(A,fun(B,A)),C3)),A6) = bot_bot(set(A)) ) )
& ( ( A6 != bot_bot(set(B)) )
=> ( aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),aa(A,fun(B,A),aTP_Lamp_at(A,fun(B,A)),C3)),A6) = aa(set(A),set(A),insert(A,C3),bot_bot(set(A))) ) ) ) ).
% image_constant_conv
tff(fact_594_image__constant,axiom,
! [A: $tType,B: $tType,X2: A,A6: set(A),C3: B] :
( pp(aa(set(A),bool,member(A,X2),A6))
=> ( aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),aTP_Lamp_dx(B,fun(A,B),C3)),A6) = aa(set(B),set(B),insert(B,C3),bot_bot(set(B))) ) ) ).
% image_constant
tff(fact_595_INTER__UNIV__conv_I2_J,axiom,
! [B: $tType,A: $tType,B6: fun(B,set(A)),A6: set(B)] :
( ( aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),B6),A6)) = top_top(set(A)) )
<=> ! [X5: B] :
( pp(aa(set(B),bool,member(B,X5),A6))
=> ( aa(B,set(A),B6,X5) = top_top(set(A)) ) ) ) ).
% INTER_UNIV_conv(2)
tff(fact_596_INTER__UNIV__conv_I1_J,axiom,
! [B: $tType,A: $tType,B6: fun(B,set(A)),A6: set(B)] :
( ( top_top(set(A)) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),B6),A6)) )
<=> ! [X5: B] :
( pp(aa(set(B),bool,member(B,X5),A6))
=> ( aa(B,set(A),B6,X5) = top_top(set(A)) ) ) ) ).
% INTER_UNIV_conv(1)
tff(fact_597_UN__extend__simps_I10_J,axiom,
! [V4: $tType,U2: $tType,T: $tType,B6: fun(U2,set(V4)),F: fun(T,U2),A6: set(T)] : aa(set(set(V4)),set(V4),complete_Sup_Sup(set(V4)),aa(set(T),set(set(V4)),aa(fun(T,set(V4)),fun(set(T),set(set(V4))),image2(T,set(V4)),aa(fun(T,U2),fun(T,set(V4)),aTP_Lamp_dw(fun(U2,set(V4)),fun(fun(T,U2),fun(T,set(V4))),B6),F)),A6)) = aa(set(set(V4)),set(V4),complete_Sup_Sup(set(V4)),aa(set(U2),set(set(V4)),aa(fun(U2,set(V4)),fun(set(U2),set(set(V4))),image2(U2,set(V4)),B6),aa(set(T),set(U2),aa(fun(T,U2),fun(set(T),set(U2)),image2(T,U2),F),A6))) ).
% UN_extend_simps(10)
tff(fact_598_image__UN,axiom,
! [B: $tType,A: $tType,C: $tType,F: fun(B,A),B6: fun(C,set(B)),A6: set(C)] : aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),aa(fun(C,set(B)),fun(set(C),set(set(B))),image2(C,set(B)),B6),A6))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),aa(fun(C,set(A)),fun(set(C),set(set(A))),image2(C,set(A)),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_dy(fun(B,A),fun(fun(C,set(B)),fun(C,set(A))),F),B6)),A6)) ).
% image_UN
tff(fact_599_UN__empty2,axiom,
! [B: $tType,A: $tType,A6: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),aTP_Lamp_dz(B,set(A))),A6)) = bot_bot(set(A)) ).
% UN_empty2
tff(fact_600_UN__empty,axiom,
! [B: $tType,A: $tType,B6: fun(B,set(A))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),B6),bot_bot(set(B)))) = bot_bot(set(A)) ).
% UN_empty
tff(fact_601_UNION__empty__conv_I1_J,axiom,
! [B: $tType,A: $tType,B6: fun(B,set(A)),A6: set(B)] :
( ( bot_bot(set(A)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),B6),A6)) )
<=> ! [X5: B] :
( pp(aa(set(B),bool,member(B,X5),A6))
=> ( aa(B,set(A),B6,X5) = bot_bot(set(A)) ) ) ) ).
% UNION_empty_conv(1)
tff(fact_602_UNION__empty__conv_I2_J,axiom,
! [B: $tType,A: $tType,B6: fun(B,set(A)),A6: set(B)] :
( ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),B6),A6)) = bot_bot(set(A)) )
<=> ! [X5: B] :
( pp(aa(set(B),bool,member(B,X5),A6))
=> ( aa(B,set(A),B6,X5) = bot_bot(set(A)) ) ) ) ).
% UNION_empty_conv(2)
tff(fact_603_UN__mono,axiom,
! [B: $tType,A: $tType,A6: set(A),B6: set(A),F: fun(A,set(B)),G: fun(A,set(B))] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B6))
=> ( ! [X: A] :
( pp(aa(set(A),bool,member(A,X),A6))
=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),F,X)),aa(A,set(B),G,X))) )
=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),aa(fun(A,set(B)),fun(set(A),set(set(B))),image2(A,set(B)),F),A6))),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),aa(fun(A,set(B)),fun(set(A),set(set(B))),image2(A,set(B)),G),B6)))) ) ) ).
% UN_mono
tff(fact_604_UN__least,axiom,
! [A: $tType,B: $tType,A6: set(A),B6: fun(A,set(B)),C5: set(B)] :
( ! [X: A] :
( pp(aa(set(A),bool,member(A,X),A6))
=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),B6,X)),C5)) )
=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),aa(fun(A,set(B)),fun(set(A),set(set(B))),image2(A,set(B)),B6),A6))),C5)) ) ).
% UN_least
tff(fact_605_UN__upper,axiom,
! [B: $tType,A: $tType,A3: A,A6: set(A),B6: fun(A,set(B))] :
( pp(aa(set(A),bool,member(A,A3),A6))
=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),B6,A3)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),aa(fun(A,set(B)),fun(set(A),set(set(B))),image2(A,set(B)),B6),A6)))) ) ).
% UN_upper
tff(fact_606_UN__image__subset,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(B,set(A)),G: fun(C,set(B)),X2: C,X6: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),F),aa(C,set(B),G,X2)))),X6))
<=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(C,set(B),G,X2)),aa(fun(B,bool),set(B),collect(B),aa(set(A),fun(B,bool),aTP_Lamp_ea(fun(B,set(A)),fun(set(A),fun(B,bool)),F),X6)))) ) ).
% UN_image_subset
tff(fact_607_UN__subset__iff,axiom,
! [B: $tType,A: $tType,A6: fun(B,set(A)),I4: set(B),B6: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),A6),I4))),B6))
<=> ! [X5: B] :
( pp(aa(set(B),bool,member(B,X5),I4))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(B,set(A),A6,X5)),B6)) ) ) ).
% UN_subset_iff
tff(fact_608_Union__natural,axiom,
! [B: $tType,A: $tType,F: fun(A,B)] : aa(fun(set(set(A)),set(set(B))),fun(set(set(A)),set(B)),aa(fun(set(set(B)),set(B)),fun(fun(set(set(A)),set(set(B))),fun(set(set(A)),set(B))),comp(set(set(B)),set(B),set(set(A))),complete_Sup_Sup(set(B))),aa(fun(set(A),set(B)),fun(set(set(A)),set(set(B))),image2(set(A),set(B)),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F))) = aa(fun(set(set(A)),set(A)),fun(set(set(A)),set(B)),aa(fun(set(A),set(B)),fun(fun(set(set(A)),set(A)),fun(set(set(A)),set(B))),comp(set(A),set(B),set(set(A))),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F)),complete_Sup_Sup(set(A))) ).
% Union_natural
tff(fact_609_image__Union,axiom,
! [A: $tType,B: $tType,F: fun(B,A),S3: set(set(B))] : aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),S3)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(B)),set(set(A)),aa(fun(set(B),set(A)),fun(set(set(B)),set(set(A))),image2(set(B),set(A)),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F)),S3)) ).
% image_Union
tff(fact_610_UN__extend__simps_I8_J,axiom,
! [P3: $tType,O: $tType,B6: fun(O,set(P3)),A6: set(set(O))] : aa(set(set(P3)),set(P3),complete_Sup_Sup(set(P3)),aa(set(set(O)),set(set(P3)),aa(fun(set(O),set(P3)),fun(set(set(O)),set(set(P3))),image2(set(O),set(P3)),aTP_Lamp_eb(fun(O,set(P3)),fun(set(O),set(P3)),B6)),A6)) = aa(set(set(P3)),set(P3),complete_Sup_Sup(set(P3)),aa(set(O),set(set(P3)),aa(fun(O,set(P3)),fun(set(O),set(set(P3))),image2(O,set(P3)),B6),aa(set(set(O)),set(O),complete_Sup_Sup(set(O)),A6))) ).
% UN_extend_simps(8)
tff(fact_611_INF__eq__iff,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [I4: set(B),F: fun(B,A),C3: A] :
( ( I4 != bot_bot(set(B)) )
=> ( ! [I6: B] :
( pp(aa(set(B),bool,member(B,I6),I4))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,I6)),C3)) )
=> ( ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),I4)) = C3 )
<=> ! [X5: B] :
( pp(aa(set(B),bool,member(B,X5),I4))
=> ( aa(B,A,F,X5) = C3 ) ) ) ) ) ) ).
% INF_eq_iff
tff(fact_612_SUP__eq__iff,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [I4: set(B),C3: A,F: fun(B,A)] :
( ( I4 != bot_bot(set(B)) )
=> ( ! [I6: B] :
( pp(aa(set(B),bool,member(B,I6),I4))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C3),aa(B,A,F,I6))) )
=> ( ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),I4)) = C3 )
<=> ! [X5: B] :
( pp(aa(set(B),bool,member(B,X5),I4))
=> ( aa(B,A,F,X5) = C3 ) ) ) ) ) ) ).
% SUP_eq_iff
tff(fact_613_range__eq__singletonD,axiom,
! [B: $tType,A: $tType,F: fun(B,A),A3: A,X2: B] :
( ( aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),top_top(set(B))) = aa(set(A),set(A),insert(A,A3),bot_bot(set(A))) )
=> ( aa(B,A,F,X2) = A3 ) ) ).
% range_eq_singletonD
tff(fact_614_INF__superset__mono,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [B6: set(B),A6: set(B),F: fun(B,A),G: fun(B,A)] :
( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B6),A6))
=> ( ! [X: B] :
( pp(aa(set(B),bool,member(B,X),B6))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,X)),aa(B,A,G,X))) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),G),B6)))) ) ) ) ).
% INF_superset_mono
tff(fact_615_SUP__subset__mono,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [A6: set(B),B6: set(B),F: fun(B,A),G: fun(B,A)] :
( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A6),B6))
=> ( ! [X: B] :
( pp(aa(set(B),bool,member(B,X),A6))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,X)),aa(B,A,G,X))) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),G),B6)))) ) ) ) ).
% SUP_subset_mono
tff(fact_616_INF__constant,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A6: set(B),C3: A] :
( ( ( A6 = bot_bot(set(B)) )
=> ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),aTP_Lamp_cu(A,fun(B,A),C3)),A6)) = top_top(A) ) )
& ( ( A6 != bot_bot(set(B)) )
=> ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),aTP_Lamp_cu(A,fun(B,A),C3)),A6)) = C3 ) ) ) ) ).
% INF_constant
tff(fact_617_INF__empty,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [F: fun(B,A)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),bot_bot(set(B)))) = top_top(A) ) ).
% INF_empty
tff(fact_618_SUP__constant,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A6: set(B),C3: A] :
( ( ( A6 = bot_bot(set(B)) )
=> ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),aTP_Lamp_cu(A,fun(B,A),C3)),A6)) = bot_bot(A) ) )
& ( ( A6 != bot_bot(set(B)) )
=> ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),aTP_Lamp_cu(A,fun(B,A),C3)),A6)) = C3 ) ) ) ) ).
% SUP_constant
tff(fact_619_SUP__empty,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [F: fun(B,A)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),bot_bot(set(B)))) = bot_bot(A) ) ).
% SUP_empty
tff(fact_620_right__total__alt__def,axiom,
! [B: $tType,A: $tType,R3: fun(A,fun(B,bool))] :
( right_total(A,B,R3)
<=> pp(aa(fun(B,fun(B,bool)),bool,aa(fun(B,fun(B,bool)),fun(fun(B,fun(B,bool)),bool),ord_less_eq(fun(B,fun(B,bool))),fequal(B)),aa(fun(A,fun(B,bool)),fun(B,fun(B,bool)),aa(fun(B,fun(A,bool)),fun(fun(A,fun(B,bool)),fun(B,fun(B,bool))),relcompp(B,A,B),conversep(A,B,R3)),R3))) ) ).
% right_total_alt_def
tff(fact_621_INT__empty,axiom,
! [B: $tType,A: $tType,B6: fun(B,set(A))] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),B6),bot_bot(set(B)))) = top_top(set(A)) ).
% INT_empty
tff(fact_622_cINF__const,axiom,
! [B: $tType,A: $tType] :
( condit1219197933456340205attice(A)
=> ! [A6: set(B),C3: A] :
( ( A6 != bot_bot(set(B)) )
=> ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),aTP_Lamp_ec(A,fun(B,A),C3)),A6)) = C3 ) ) ) ).
% cINF_const
tff(fact_623_cSUP__const,axiom,
! [B: $tType,A: $tType] :
( condit1219197933456340205attice(A)
=> ! [A6: set(B),C3: A] :
( ( A6 != bot_bot(set(B)) )
=> ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),aTP_Lamp_ec(A,fun(B,A),C3)),A6)) = C3 ) ) ) ).
% cSUP_const
tff(fact_624_SUP__INF,axiom,
! [A: $tType,C: $tType,B: $tType] :
( comple592849572758109894attice(A)
=> ! [P2: fun(C,fun(B,A))] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),aTP_Lamp_ee(fun(C,fun(B,A)),fun(B,A),P2)),top_top(set(B)))) = aa(set(A),A,complete_Inf_Inf(A),aa(set(fun(B,C)),set(A),aa(fun(fun(B,C),A),fun(set(fun(B,C)),set(A)),image2(fun(B,C),A),aTP_Lamp_eg(fun(C,fun(B,A)),fun(fun(B,C),A),P2)),top_top(set(fun(B,C))))) ) ).
% SUP_INF
tff(fact_625_INF__SUP,axiom,
! [A: $tType,C: $tType,B: $tType] :
( comple592849572758109894attice(A)
=> ! [P2: fun(C,fun(B,A))] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),aTP_Lamp_eh(fun(C,fun(B,A)),fun(B,A),P2)),top_top(set(B)))) = aa(set(A),A,complete_Sup_Sup(A),aa(set(fun(B,C)),set(A),aa(fun(fun(B,C),A),fun(set(fun(B,C)),set(A)),image2(fun(B,C),A),aTP_Lamp_ei(fun(C,fun(B,A)),fun(fun(B,C),A),P2)),top_top(set(fun(B,C))))) ) ).
% INF_SUP
tff(fact_626_cINF__greatest,axiom,
! [A: $tType,B: $tType] :
( condit1219197933456340205attice(A)
=> ! [A6: set(B),M4: A,F: fun(B,A)] :
( ( A6 != bot_bot(set(B)) )
=> ( ! [X: B] :
( pp(aa(set(B),bool,member(B,X),A6))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M4),aa(B,A,F,X))) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M4),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6)))) ) ) ) ).
% cINF_greatest
tff(fact_627_cSUP__least,axiom,
! [B: $tType,A: $tType] :
( condit1219197933456340205attice(A)
=> ! [A6: set(B),F: fun(B,A),M: A] :
( ( A6 != bot_bot(set(B)) )
=> ( ! [X: B] :
( pp(aa(set(B),bool,member(B,X),A6))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,X)),M)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))),M)) ) ) ) ).
% cSUP_least
tff(fact_628_UN__UN__split__split__eq,axiom,
! [D: $tType,E: $tType,A: $tType,C: $tType,B: $tType,A6: fun(B,fun(C,fun(D,fun(E,set(A))))),Y3: set(product_prod(D,E)),X6: set(product_prod(B,C))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(product_prod(B,C)),set(set(A)),aa(fun(product_prod(B,C),set(A)),fun(set(product_prod(B,C)),set(set(A))),image2(product_prod(B,C),set(A)),aa(fun(B,fun(C,set(A))),fun(product_prod(B,C),set(A)),product_case_prod(B,C,set(A)),aa(set(product_prod(D,E)),fun(B,fun(C,set(A))),aTP_Lamp_ej(fun(B,fun(C,fun(D,fun(E,set(A))))),fun(set(product_prod(D,E)),fun(B,fun(C,set(A)))),A6),Y3))),X6)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(product_prod(B,C)),set(set(A)),aa(fun(product_prod(B,C),set(A)),fun(set(product_prod(B,C)),set(set(A))),image2(product_prod(B,C),set(A)),aa(set(product_prod(D,E)),fun(product_prod(B,C),set(A)),aTP_Lamp_em(fun(B,fun(C,fun(D,fun(E,set(A))))),fun(set(product_prod(D,E)),fun(product_prod(B,C),set(A))),A6),Y3)),X6)) ).
% UN_UN_split_split_eq
tff(fact_629_UN__constant__eq,axiom,
! [A: $tType,B: $tType,A3: A,A6: set(A),F: fun(A,set(B)),C3: set(B)] :
( pp(aa(set(A),bool,member(A,A3),A6))
=> ( ! [X: A] :
( pp(aa(set(A),bool,member(A,X),A6))
=> ( aa(A,set(B),F,X) = C3 ) )
=> ( aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),aa(fun(A,set(B)),fun(set(A),set(set(B))),image2(A,set(B)),F),A6)) = C3 ) ) ) ).
% UN_constant_eq
tff(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)
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R))
=> ( pp(aa(set(A),bool,member(A,A3),A6))
& pp(aa(set(A),bool,member(A,B2),A6)) ) ) ) ).
% refl_on_domain
tff(fact_631_INT__simps_I4_J,axiom,
! [G2: $tType,H5: $tType,C5: set(H5),A6: set(G2),B6: fun(H5,set(G2))] :
( ( ( C5 = bot_bot(set(H5)) )
=> ( aa(set(set(G2)),set(G2),complete_Inf_Inf(set(G2)),aa(set(H5),set(set(G2)),aa(fun(H5,set(G2)),fun(set(H5),set(set(G2))),image2(H5,set(G2)),aa(fun(H5,set(G2)),fun(H5,set(G2)),aTP_Lamp_en(set(G2),fun(fun(H5,set(G2)),fun(H5,set(G2))),A6),B6)),C5)) = top_top(set(G2)) ) )
& ( ( C5 != bot_bot(set(H5)) )
=> ( aa(set(set(G2)),set(G2),complete_Inf_Inf(set(G2)),aa(set(H5),set(set(G2)),aa(fun(H5,set(G2)),fun(set(H5),set(set(G2))),image2(H5,set(G2)),aa(fun(H5,set(G2)),fun(H5,set(G2)),aTP_Lamp_en(set(G2),fun(fun(H5,set(G2)),fun(H5,set(G2))),A6),B6)),C5)) = aa(set(G2),set(G2),aa(set(G2),fun(set(G2),set(G2)),minus_minus(set(G2)),A6),aa(set(set(G2)),set(G2),complete_Sup_Sup(set(G2)),aa(set(H5),set(set(G2)),aa(fun(H5,set(G2)),fun(set(H5),set(set(G2))),image2(H5,set(G2)),B6),C5))) ) ) ) ).
% INT_simps(4)
tff(fact_632_relcompp__bot2,axiom,
! [C: $tType,B: $tType,A: $tType,R3: fun(A,fun(C,bool))] : aa(fun(C,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(C,bool)),fun(fun(C,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,C,B),R3),bot_bot(fun(C,fun(B,bool)))) = bot_bot(fun(A,fun(B,bool))) ).
% relcompp_bot2
tff(fact_633_relcompp__bot1,axiom,
! [C: $tType,B: $tType,A: $tType,R3: fun(C,fun(B,bool))] : aa(fun(C,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(C,bool)),fun(fun(C,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,C,B),bot_bot(fun(A,fun(C,bool)))),R3) = bot_bot(fun(A,fun(B,bool))) ).
% relcompp_bot1
tff(fact_634_surj__diff__right,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [A3: A] : aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aTP_Lamp_eo(A,fun(A,A),A3)),top_top(set(A))) = top_top(set(A)) ) ).
% surj_diff_right
tff(fact_635_INT__simps_I3_J,axiom,
! [E: $tType,F2: $tType,C5: set(E),A6: fun(E,set(F2)),B6: set(F2)] :
( ( ( C5 = bot_bot(set(E)) )
=> ( aa(set(set(F2)),set(F2),complete_Inf_Inf(set(F2)),aa(set(E),set(set(F2)),aa(fun(E,set(F2)),fun(set(E),set(set(F2))),image2(E,set(F2)),aa(set(F2),fun(E,set(F2)),aTP_Lamp_ep(fun(E,set(F2)),fun(set(F2),fun(E,set(F2))),A6),B6)),C5)) = top_top(set(F2)) ) )
& ( ( C5 != bot_bot(set(E)) )
=> ( aa(set(set(F2)),set(F2),complete_Inf_Inf(set(F2)),aa(set(E),set(set(F2)),aa(fun(E,set(F2)),fun(set(E),set(set(F2))),image2(E,set(F2)),aa(set(F2),fun(E,set(F2)),aTP_Lamp_ep(fun(E,set(F2)),fun(set(F2),fun(E,set(F2))),A6),B6)),C5)) = aa(set(F2),set(F2),aa(set(F2),fun(set(F2),set(F2)),minus_minus(set(F2)),aa(set(set(F2)),set(F2),complete_Inf_Inf(set(F2)),aa(set(E),set(set(F2)),aa(fun(E,set(F2)),fun(set(E),set(set(F2))),image2(E,set(F2)),A6),C5))),B6) ) ) ) ).
% INT_simps(3)
tff(fact_636_Sigma__Diff__distrib1,axiom,
! [B: $tType,A: $tType,I4: set(A),J4: set(A),C5: fun(A,set(B))] : product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),I4),J4),C5) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),minus_minus(set(product_prod(A,B))),product_Sigma(A,B,I4,C5)),product_Sigma(A,B,J4,C5)) ).
% Sigma_Diff_distrib1
tff(fact_637_translation__subtract__diff,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [A3: A,S: set(A),T2: set(A)] : aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aTP_Lamp_eo(A,fun(A,A),A3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),T2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aTP_Lamp_eo(A,fun(A,A),A3)),S)),aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aTP_Lamp_eo(A,fun(A,A),A3)),T2)) ) ).
% translation_subtract_diff
tff(fact_638_Times__Diff__distrib1,axiom,
! [A: $tType,B: $tType,A6: set(A),B6: set(A),C5: set(B)] : product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),B6),aTP_Lamp_bu(set(B),fun(A,set(B)),C5)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),minus_minus(set(product_prod(A,B))),product_Sigma(A,B,A6,aTP_Lamp_bu(set(B),fun(A,set(B)),C5))),product_Sigma(A,B,B6,aTP_Lamp_bu(set(B),fun(A,set(B)),C5))) ).
% Times_Diff_distrib1
tff(fact_639_Sigma__Diff__distrib2,axiom,
! [B: $tType,A: $tType,I4: set(A),A6: fun(A,set(B)),B6: fun(A,set(B))] : product_Sigma(A,B,I4,aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_eq(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(B))),A6),B6)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),minus_minus(set(product_prod(A,B))),product_Sigma(A,B,I4,A6)),product_Sigma(A,B,I4,B6)) ).
% Sigma_Diff_distrib2
tff(fact_640_Sup__SUP__eq,axiom,
! [A: $tType,S3: set(fun(A,bool)),X4: A] :
( pp(aa(A,bool,aa(set(fun(A,bool)),fun(A,bool),complete_Sup_Sup(fun(A,bool)),S3),X4))
<=> pp(aa(set(A),bool,member(A,X4),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(fun(A,bool)),set(set(A)),aa(fun(fun(A,bool),set(A)),fun(set(fun(A,bool)),set(set(A))),image2(fun(A,bool),set(A)),collect(A)),S3)))) ) ).
% Sup_SUP_eq
tff(fact_641_Inf__INT__eq,axiom,
! [A: $tType,S3: set(fun(A,bool)),X4: A] :
( pp(aa(A,bool,aa(set(fun(A,bool)),fun(A,bool),complete_Inf_Inf(fun(A,bool)),S3),X4))
<=> pp(aa(set(A),bool,member(A,X4),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(fun(A,bool)),set(set(A)),aa(fun(fun(A,bool),set(A)),fun(set(fun(A,bool)),set(set(A))),image2(fun(A,bool),set(A)),collect(A)),S3)))) ) ).
% Inf_INT_eq
tff(fact_642_INF__INT__eq2,axiom,
! [A: $tType,B: $tType,C: $tType,R: fun(C,set(product_prod(A,B))),S3: set(C),X4: A,Xa4: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aa(set(fun(A,fun(B,bool))),fun(A,fun(B,bool)),complete_Inf_Inf(fun(A,fun(B,bool))),aa(set(C),set(fun(A,fun(B,bool))),aa(fun(C,fun(A,fun(B,bool))),fun(set(C),set(fun(A,fun(B,bool)))),image2(C,fun(A,fun(B,bool))),aTP_Lamp_er(fun(C,set(product_prod(A,B))),fun(C,fun(A,fun(B,bool))),R)),S3)),X4),Xa4))
<=> pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa4)),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Inf_Inf(set(product_prod(A,B))),aa(set(C),set(set(product_prod(A,B))),aa(fun(C,set(product_prod(A,B))),fun(set(C),set(set(product_prod(A,B)))),image2(C,set(product_prod(A,B))),R),S3)))) ) ).
% INF_INT_eq2
tff(fact_643_INF__Int__eq2,axiom,
! [B: $tType,A: $tType,S3: set(set(product_prod(A,B))),X4: A,Xa4: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aa(set(fun(A,fun(B,bool))),fun(A,fun(B,bool)),complete_Inf_Inf(fun(A,fun(B,bool))),aa(set(set(product_prod(A,B))),set(fun(A,fun(B,bool))),aa(fun(set(product_prod(A,B)),fun(A,fun(B,bool))),fun(set(set(product_prod(A,B))),set(fun(A,fun(B,bool)))),image2(set(product_prod(A,B)),fun(A,fun(B,bool))),aTP_Lamp_bg(set(product_prod(A,B)),fun(A,fun(B,bool)))),S3)),X4),Xa4))
<=> pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa4)),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Inf_Inf(set(product_prod(A,B))),S3))) ) ).
% INF_Int_eq2
tff(fact_644_SUP__UN__eq2,axiom,
! [A: $tType,B: $tType,C: $tType,R: fun(C,set(product_prod(A,B))),S3: set(C),X4: A,Xa4: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aa(set(fun(A,fun(B,bool))),fun(A,fun(B,bool)),complete_Sup_Sup(fun(A,fun(B,bool))),aa(set(C),set(fun(A,fun(B,bool))),aa(fun(C,fun(A,fun(B,bool))),fun(set(C),set(fun(A,fun(B,bool)))),image2(C,fun(A,fun(B,bool))),aTP_Lamp_er(fun(C,set(product_prod(A,B))),fun(C,fun(A,fun(B,bool))),R)),S3)),X4),Xa4))
<=> pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa4)),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(C),set(set(product_prod(A,B))),aa(fun(C,set(product_prod(A,B))),fun(set(C),set(set(product_prod(A,B)))),image2(C,set(product_prod(A,B))),R),S3)))) ) ).
% SUP_UN_eq2
tff(fact_645_SUP__Sup__eq2,axiom,
! [B: $tType,A: $tType,S3: set(set(product_prod(A,B))),X4: A,Xa4: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aa(set(fun(A,fun(B,bool))),fun(A,fun(B,bool)),complete_Sup_Sup(fun(A,fun(B,bool))),aa(set(set(product_prod(A,B))),set(fun(A,fun(B,bool))),aa(fun(set(product_prod(A,B)),fun(A,fun(B,bool))),fun(set(set(product_prod(A,B))),set(fun(A,fun(B,bool)))),image2(set(product_prod(A,B)),fun(A,fun(B,bool))),aTP_Lamp_bg(set(product_prod(A,B)),fun(A,fun(B,bool)))),S3)),X4),Xa4))
<=> pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa4)),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),S3))) ) ).
% SUP_Sup_eq2
tff(fact_646_SUP__Sup__eq,axiom,
! [A: $tType,S3: set(set(A)),X4: A] :
( pp(aa(A,bool,aa(set(fun(A,bool)),fun(A,bool),complete_Sup_Sup(fun(A,bool)),aa(set(set(A)),set(fun(A,bool)),aa(fun(set(A),fun(A,bool)),fun(set(set(A)),set(fun(A,bool))),image2(set(A),fun(A,bool)),aTP_Lamp_a(set(A),fun(A,bool))),S3)),X4))
<=> pp(aa(set(A),bool,member(A,X4),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),S3))) ) ).
% SUP_Sup_eq
tff(fact_647_INF__Int__eq,axiom,
! [A: $tType,S3: set(set(A)),X4: A] :
( pp(aa(A,bool,aa(set(fun(A,bool)),fun(A,bool),complete_Inf_Inf(fun(A,bool)),aa(set(set(A)),set(fun(A,bool)),aa(fun(set(A),fun(A,bool)),fun(set(set(A)),set(fun(A,bool))),image2(set(A),fun(A,bool)),aTP_Lamp_a(set(A),fun(A,bool))),S3)),X4))
<=> pp(aa(set(A),bool,member(A,X4),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),S3))) ) ).
% INF_Int_eq
tff(fact_648_relcompp__SUP__distrib,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType,S: fun(A,fun(C,bool)),R: fun(D,fun(C,fun(B,bool))),I4: set(D)] : aa(fun(C,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(C,bool)),fun(fun(C,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,C,B),S),aa(set(fun(C,fun(B,bool))),fun(C,fun(B,bool)),complete_Sup_Sup(fun(C,fun(B,bool))),aa(set(D),set(fun(C,fun(B,bool))),aa(fun(D,fun(C,fun(B,bool))),fun(set(D),set(fun(C,fun(B,bool)))),image2(D,fun(C,fun(B,bool))),R),I4))) = aa(set(fun(A,fun(B,bool))),fun(A,fun(B,bool)),complete_Sup_Sup(fun(A,fun(B,bool))),aa(set(D),set(fun(A,fun(B,bool))),aa(fun(D,fun(A,fun(B,bool))),fun(set(D),set(fun(A,fun(B,bool)))),image2(D,fun(A,fun(B,bool))),aa(fun(D,fun(C,fun(B,bool))),fun(D,fun(A,fun(B,bool))),aTP_Lamp_es(fun(A,fun(C,bool)),fun(fun(D,fun(C,fun(B,bool))),fun(D,fun(A,fun(B,bool)))),S),R)),I4)) ).
% relcompp_SUP_distrib
tff(fact_649_relcompp__SUP__distrib2,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType,R: fun(D,fun(A,fun(C,bool))),I4: set(D),S: fun(C,fun(B,bool))] : aa(fun(C,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(C,bool)),fun(fun(C,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,C,B),aa(set(fun(A,fun(C,bool))),fun(A,fun(C,bool)),complete_Sup_Sup(fun(A,fun(C,bool))),aa(set(D),set(fun(A,fun(C,bool))),aa(fun(D,fun(A,fun(C,bool))),fun(set(D),set(fun(A,fun(C,bool)))),image2(D,fun(A,fun(C,bool))),R),I4))),S) = aa(set(fun(A,fun(B,bool))),fun(A,fun(B,bool)),complete_Sup_Sup(fun(A,fun(B,bool))),aa(set(D),set(fun(A,fun(B,bool))),aa(fun(D,fun(A,fun(B,bool))),fun(set(D),set(fun(A,fun(B,bool)))),image2(D,fun(A,fun(B,bool))),aa(fun(C,fun(B,bool)),fun(D,fun(A,fun(B,bool))),aTP_Lamp_et(fun(D,fun(A,fun(C,bool))),fun(fun(C,fun(B,bool)),fun(D,fun(A,fun(B,bool)))),R),S)),I4)) ).
% relcompp_SUP_distrib2
tff(fact_650_Sup__set__def,axiom,
! [A: $tType,A6: set(set(A))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A6) = aa(fun(A,bool),set(A),collect(A),aTP_Lamp_eu(set(set(A)),fun(A,bool),A6)) ).
% Sup_set_def
tff(fact_651_Inf__set__def,axiom,
! [A: $tType,A6: set(set(A))] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),A6) = aa(fun(A,bool),set(A),collect(A),aTP_Lamp_ev(set(set(A)),fun(A,bool),A6)) ).
% Inf_set_def
tff(fact_652_image__diff__subset,axiom,
! [A: $tType,B: $tType,F: fun(B,A),A6: set(B),B6: set(B)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6)),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),B6))),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A6),B6)))) ).
% image_diff_subset
tff(fact_653_Inf__INT__eq2,axiom,
! [B: $tType,A: $tType,S3: set(fun(A,fun(B,bool))),X4: A,Xa4: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aa(set(fun(A,fun(B,bool))),fun(A,fun(B,bool)),complete_Inf_Inf(fun(A,fun(B,bool))),S3),X4),Xa4))
<=> pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa4)),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Inf_Inf(set(product_prod(A,B))),aa(set(fun(product_prod(A,B),bool)),set(set(product_prod(A,B))),aa(fun(fun(product_prod(A,B),bool),set(product_prod(A,B))),fun(set(fun(product_prod(A,B),bool)),set(set(product_prod(A,B)))),image2(fun(product_prod(A,B),bool),set(product_prod(A,B))),collect(product_prod(A,B))),aa(set(fun(A,fun(B,bool))),set(fun(product_prod(A,B),bool)),aa(fun(fun(A,fun(B,bool)),fun(product_prod(A,B),bool)),fun(set(fun(A,fun(B,bool))),set(fun(product_prod(A,B),bool))),image2(fun(A,fun(B,bool)),fun(product_prod(A,B),bool)),product_case_prod(A,B,bool)),S3))))) ) ).
% Inf_INT_eq2
tff(fact_654_Sup__SUP__eq2,axiom,
! [B: $tType,A: $tType,S3: set(fun(A,fun(B,bool))),X4: A,Xa4: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aa(set(fun(A,fun(B,bool))),fun(A,fun(B,bool)),complete_Sup_Sup(fun(A,fun(B,bool))),S3),X4),Xa4))
<=> pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa4)),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(fun(product_prod(A,B),bool)),set(set(product_prod(A,B))),aa(fun(fun(product_prod(A,B),bool),set(product_prod(A,B))),fun(set(fun(product_prod(A,B),bool)),set(set(product_prod(A,B)))),image2(fun(product_prod(A,B),bool),set(product_prod(A,B))),collect(product_prod(A,B))),aa(set(fun(A,fun(B,bool))),set(fun(product_prod(A,B),bool)),aa(fun(fun(A,fun(B,bool)),fun(product_prod(A,B),bool)),fun(set(fun(A,fun(B,bool))),set(fun(product_prod(A,B),bool))),image2(fun(A,fun(B,bool)),fun(product_prod(A,B),bool)),product_case_prod(A,B,bool)),S3))))) ) ).
% Sup_SUP_eq2
tff(fact_655_SUP__UN__eq,axiom,
! [A: $tType,B: $tType,R: fun(B,set(A)),S3: set(B),X4: A] :
( pp(aa(A,bool,aa(set(fun(A,bool)),fun(A,bool),complete_Sup_Sup(fun(A,bool)),aa(set(B),set(fun(A,bool)),aa(fun(B,fun(A,bool)),fun(set(B),set(fun(A,bool))),image2(B,fun(A,bool)),aTP_Lamp_ew(fun(B,set(A)),fun(B,fun(A,bool)),R)),S3)),X4))
<=> pp(aa(set(A),bool,member(A,X4),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),R),S3)))) ) ).
% SUP_UN_eq
tff(fact_656_bot__empty__eq2,axiom,
! [B: $tType,A: $tType,X4: A,Xa4: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),bot_bot(fun(A,fun(B,bool))),X4),Xa4))
<=> pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa4)),bot_bot(set(product_prod(A,B))))) ) ).
% bot_empty_eq2
tff(fact_657_INF__INT__eq,axiom,
! [A: $tType,B: $tType,R: fun(B,set(A)),S3: set(B),X4: A] :
( pp(aa(A,bool,aa(set(fun(A,bool)),fun(A,bool),complete_Inf_Inf(fun(A,bool)),aa(set(B),set(fun(A,bool)),aa(fun(B,fun(A,bool)),fun(set(B),set(fun(A,bool))),image2(B,fun(A,bool)),aTP_Lamp_ew(fun(B,set(A)),fun(B,fun(A,bool)),R)),S3)),X4))
<=> pp(aa(set(A),bool,member(A,X4),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),R),S3)))) ) ).
% INF_INT_eq
tff(fact_658_top__empty__eq2,axiom,
! [B: $tType,A: $tType,X4: A,Xa4: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),top_top(fun(A,fun(B,bool))),X4),Xa4))
<=> pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa4)),top_top(set(product_prod(A,B))))) ) ).
% top_empty_eq2
tff(fact_659_UN__extend__simps_I6_J,axiom,
! [L3: $tType,K2: $tType,A6: fun(K2,set(L3)),C5: set(K2),B6: set(L3)] : aa(set(L3),set(L3),aa(set(L3),fun(set(L3),set(L3)),minus_minus(set(L3)),aa(set(set(L3)),set(L3),complete_Sup_Sup(set(L3)),aa(set(K2),set(set(L3)),aa(fun(K2,set(L3)),fun(set(K2),set(set(L3))),image2(K2,set(L3)),A6),C5))),B6) = aa(set(set(L3)),set(L3),complete_Sup_Sup(set(L3)),aa(set(K2),set(set(L3)),aa(fun(K2,set(L3)),fun(set(K2),set(set(L3))),image2(K2,set(L3)),aa(set(L3),fun(K2,set(L3)),aTP_Lamp_ex(fun(K2,set(L3)),fun(set(L3),fun(K2,set(L3))),A6),B6)),C5)) ).
% UN_extend_simps(6)
tff(fact_660_in__image__insert__iff,axiom,
! [A: $tType,B6: set(set(A)),X2: A,A6: set(A)] :
( ! [C4: set(A)] :
( pp(aa(set(set(A)),bool,member(set(A),C4),B6))
=> ~ pp(aa(set(A),bool,member(A,X2),C4)) )
=> ( pp(aa(set(set(A)),bool,member(set(A),A6),aa(set(set(A)),set(set(A)),aa(fun(set(A),set(A)),fun(set(set(A)),set(set(A))),image2(set(A),set(A)),insert(A,X2)),B6)))
<=> ( pp(aa(set(A),bool,member(A,X2),A6))
& pp(aa(set(set(A)),bool,member(set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),insert(A,X2),bot_bot(set(A))))),B6)) ) ) ) ).
% in_image_insert_iff
tff(fact_661_UN__extend__simps_I7_J,axiom,
! [M5: $tType,N3: $tType,A6: set(M5),B6: fun(N3,set(M5)),C5: set(N3)] : aa(set(M5),set(M5),aa(set(M5),fun(set(M5),set(M5)),minus_minus(set(M5)),A6),aa(set(set(M5)),set(M5),complete_Inf_Inf(set(M5)),aa(set(N3),set(set(M5)),aa(fun(N3,set(M5)),fun(set(N3),set(set(M5))),image2(N3,set(M5)),B6),C5))) = aa(set(set(M5)),set(M5),complete_Sup_Sup(set(M5)),aa(set(N3),set(set(M5)),aa(fun(N3,set(M5)),fun(set(N3),set(set(M5))),image2(N3,set(M5)),aa(fun(N3,set(M5)),fun(N3,set(M5)),aTP_Lamp_ey(set(M5),fun(fun(N3,set(M5)),fun(N3,set(M5))),A6),B6)),C5)) ).
% UN_extend_simps(7)
tff(fact_662_INT__extend__simps_I3_J,axiom,
! [F2: $tType,E: $tType,C5: set(E),A6: fun(E,set(F2)),B6: set(F2)] :
( ( ( C5 = bot_bot(set(E)) )
=> ( aa(set(F2),set(F2),aa(set(F2),fun(set(F2),set(F2)),minus_minus(set(F2)),aa(set(set(F2)),set(F2),complete_Inf_Inf(set(F2)),aa(set(E),set(set(F2)),aa(fun(E,set(F2)),fun(set(E),set(set(F2))),image2(E,set(F2)),A6),C5))),B6) = aa(set(F2),set(F2),aa(set(F2),fun(set(F2),set(F2)),minus_minus(set(F2)),top_top(set(F2))),B6) ) )
& ( ( C5 != bot_bot(set(E)) )
=> ( aa(set(F2),set(F2),aa(set(F2),fun(set(F2),set(F2)),minus_minus(set(F2)),aa(set(set(F2)),set(F2),complete_Inf_Inf(set(F2)),aa(set(E),set(set(F2)),aa(fun(E,set(F2)),fun(set(E),set(set(F2))),image2(E,set(F2)),A6),C5))),B6) = aa(set(set(F2)),set(F2),complete_Inf_Inf(set(F2)),aa(set(E),set(set(F2)),aa(fun(E,set(F2)),fun(set(E),set(set(F2))),image2(E,set(F2)),aa(set(F2),fun(E,set(F2)),aTP_Lamp_ep(fun(E,set(F2)),fun(set(F2),fun(E,set(F2))),A6),B6)),C5)) ) ) ) ).
% INT_extend_simps(3)
tff(fact_663_INT__extend__simps_I4_J,axiom,
! [G2: $tType,H5: $tType,C5: set(H5),A6: set(G2),B6: fun(H5,set(G2))] :
( ( ( C5 = bot_bot(set(H5)) )
=> ( aa(set(G2),set(G2),aa(set(G2),fun(set(G2),set(G2)),minus_minus(set(G2)),A6),aa(set(set(G2)),set(G2),complete_Sup_Sup(set(G2)),aa(set(H5),set(set(G2)),aa(fun(H5,set(G2)),fun(set(H5),set(set(G2))),image2(H5,set(G2)),B6),C5))) = A6 ) )
& ( ( C5 != bot_bot(set(H5)) )
=> ( aa(set(G2),set(G2),aa(set(G2),fun(set(G2),set(G2)),minus_minus(set(G2)),A6),aa(set(set(G2)),set(G2),complete_Sup_Sup(set(G2)),aa(set(H5),set(set(G2)),aa(fun(H5,set(G2)),fun(set(H5),set(set(G2))),image2(H5,set(G2)),B6),C5))) = aa(set(set(G2)),set(G2),complete_Inf_Inf(set(G2)),aa(set(H5),set(set(G2)),aa(fun(H5,set(G2)),fun(set(H5),set(set(G2))),image2(H5,set(G2)),aa(fun(H5,set(G2)),fun(H5,set(G2)),aTP_Lamp_en(set(G2),fun(fun(H5,set(G2)),fun(H5,set(G2))),A6),B6)),C5)) ) ) ) ).
% INT_extend_simps(4)
tff(fact_664_linear__order__on__singleton,axiom,
! [A: $tType,X2: A] : order_679001287576687338der_on(A,aa(set(A),set(A),insert(A,X2),bot_bot(set(A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),insert(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),X2)),bot_bot(set(product_prod(A,A))))) ).
% linear_order_on_singleton
tff(fact_665_total__on__singleton,axiom,
! [A: $tType,X2: A] : total_on(A,aa(set(A),set(A),insert(A,X2),bot_bot(set(A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),insert(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),X2)),bot_bot(set(product_prod(A,A))))) ).
% total_on_singleton
tff(fact_666_relation__of__def,axiom,
! [A: $tType,P2: fun(A,fun(A,bool)),A6: set(A)] : order_relation_of(A,P2,A6) = aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(set(A),fun(A,fun(A,bool)),aTP_Lamp_ez(fun(A,fun(A,bool)),fun(set(A),fun(A,fun(A,bool))),P2),A6))) ).
% relation_of_def
tff(fact_667_insert__Times__insert,axiom,
! [A: $tType,B: $tType,A3: A,A6: set(A),B2: B,B6: set(B)] : product_Sigma(A,B,aa(set(A),set(A),insert(A,A3),A6),aa(set(B),fun(A,set(B)),aTP_Lamp_fa(B,fun(set(B),fun(A,set(B))),B2),B6)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),insert(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),product_Sigma(A,B,A6,aa(set(B),fun(A,set(B)),aTP_Lamp_fa(B,fun(set(B),fun(A,set(B))),B2),B6))),product_Sigma(A,B,aa(set(A),set(A),insert(A,A3),A6),aTP_Lamp_bu(set(B),fun(A,set(B)),B6)))) ).
% insert_Times_insert
tff(fact_668_INT__simps_I2_J,axiom,
! [C: $tType,D: $tType,C5: set(D),A6: set(C),B6: fun(D,set(C))] :
( ( ( C5 = bot_bot(set(D)) )
=> ( aa(set(set(C)),set(C),complete_Inf_Inf(set(C)),aa(set(D),set(set(C)),aa(fun(D,set(C)),fun(set(D),set(set(C))),image2(D,set(C)),aa(fun(D,set(C)),fun(D,set(C)),aTP_Lamp_fb(set(C),fun(fun(D,set(C)),fun(D,set(C))),A6),B6)),C5)) = top_top(set(C)) ) )
& ( ( C5 != bot_bot(set(D)) )
=> ( aa(set(set(C)),set(C),complete_Inf_Inf(set(C)),aa(set(D),set(set(C)),aa(fun(D,set(C)),fun(set(D),set(set(C))),image2(D,set(C)),aa(fun(D,set(C)),fun(D,set(C)),aTP_Lamp_fb(set(C),fun(fun(D,set(C)),fun(D,set(C))),A6),B6)),C5)) = aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),A6),aa(set(set(C)),set(C),complete_Inf_Inf(set(C)),aa(set(D),set(set(C)),aa(fun(D,set(C)),fun(set(D),set(set(C))),image2(D,set(C)),B6),C5))) ) ) ) ).
% INT_simps(2)
tff(fact_669_INT__simps_I1_J,axiom,
! [A: $tType,B: $tType,C5: set(A),A6: fun(A,set(B)),B6: set(B)] :
( ( ( C5 = bot_bot(set(A)) )
=> ( aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),aa(fun(A,set(B)),fun(set(A),set(set(B))),image2(A,set(B)),aa(set(B),fun(A,set(B)),aTP_Lamp_fc(fun(A,set(B)),fun(set(B),fun(A,set(B))),A6),B6)),C5)) = top_top(set(B)) ) )
& ( ( C5 != bot_bot(set(A)) )
=> ( aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),aa(fun(A,set(B)),fun(set(A),set(set(B))),image2(A,set(B)),aa(set(B),fun(A,set(B)),aTP_Lamp_fc(fun(A,set(B)),fun(set(B),fun(A,set(B))),A6),B6)),C5)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),aa(fun(A,set(B)),fun(set(A),set(set(B))),image2(A,set(B)),A6),C5))),B6) ) ) ) ).
% INT_simps(1)
tff(fact_670_cSUP__UNION,axiom,
! [D: $tType,B: $tType,C: $tType] :
( condit1219197933456340205attice(B)
=> ! [A6: set(C),B6: fun(C,set(D)),F: fun(D,B)] :
( ( A6 != bot_bot(set(C)) )
=> ( ! [X: C] :
( pp(aa(set(C),bool,member(C,X),A6))
=> ( aa(C,set(D),B6,X) != bot_bot(set(D)) ) )
=> ( condit941137186595557371_above(B,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),aa(fun(C,set(B)),fun(set(C),set(set(B))),image2(C,set(B)),aa(fun(D,B),fun(C,set(B)),aTP_Lamp_fd(fun(C,set(D)),fun(fun(D,B),fun(C,set(B))),B6),F)),A6)))
=> ( aa(set(B),B,complete_Sup_Sup(B),aa(set(D),set(B),aa(fun(D,B),fun(set(D),set(B)),image2(D,B),F),aa(set(set(D)),set(D),complete_Sup_Sup(set(D)),aa(set(C),set(set(D)),aa(fun(C,set(D)),fun(set(C),set(set(D))),image2(C,set(D)),B6),A6)))) = aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),aa(fun(C,B),fun(set(C),set(B)),image2(C,B),aa(fun(D,B),fun(C,B),aTP_Lamp_fe(fun(C,set(D)),fun(fun(D,B),fun(C,B)),B6),F)),A6)) ) ) ) ) ) ).
% cSUP_UNION
tff(fact_671_cINF__UNION,axiom,
! [D: $tType,B: $tType,C: $tType] :
( condit1219197933456340205attice(B)
=> ! [A6: set(C),B6: fun(C,set(D)),F: fun(D,B)] :
( ( A6 != bot_bot(set(C)) )
=> ( ! [X: C] :
( pp(aa(set(C),bool,member(C,X),A6))
=> ( aa(C,set(D),B6,X) != bot_bot(set(D)) ) )
=> ( condit1013018076250108175_below(B,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),aa(fun(C,set(B)),fun(set(C),set(set(B))),image2(C,set(B)),aa(fun(D,B),fun(C,set(B)),aTP_Lamp_fd(fun(C,set(D)),fun(fun(D,B),fun(C,set(B))),B6),F)),A6)))
=> ( aa(set(B),B,complete_Inf_Inf(B),aa(set(D),set(B),aa(fun(D,B),fun(set(D),set(B)),image2(D,B),F),aa(set(set(D)),set(D),complete_Sup_Sup(set(D)),aa(set(C),set(set(D)),aa(fun(C,set(D)),fun(set(C),set(set(D))),image2(C,set(D)),B6),A6)))) = aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),aa(fun(C,B),fun(set(C),set(B)),image2(C,B),aa(fun(D,B),fun(C,B),aTP_Lamp_ff(fun(C,set(D)),fun(fun(D,B),fun(C,B)),B6),F)),A6)) ) ) ) ) ) ).
% cINF_UNION
tff(fact_672_Ex__inj__on__UNION__Sigma,axiom,
! [A: $tType,B: $tType,A6: fun(B,set(A)),I4: set(B)] :
? [F3: fun(A,product_prod(B,A))] :
( inj_on(A,product_prod(B,A),F3,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),A6),I4)))
& pp(aa(set(product_prod(B,A)),bool,aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),bool),ord_less_eq(set(product_prod(B,A))),aa(set(A),set(product_prod(B,A)),aa(fun(A,product_prod(B,A)),fun(set(A),set(product_prod(B,A))),image2(A,product_prod(B,A)),F3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),A6),I4)))),product_Sigma(B,A,I4,A6))) ) ).
% Ex_inj_on_UNION_Sigma
tff(fact_673_prod__set__simps_I1_J,axiom,
! [B: $tType,A: $tType,X2: A,Y2: B] : aa(product_prod(A,B),set(A),basic_fsts(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y2)) = aa(set(A),set(A),insert(A,X2),bot_bot(set(A))) ).
% prod_set_simps(1)
tff(fact_674_SUP2__I,axiom,
! [B: $tType,A: $tType,C: $tType,A3: A,A6: set(A),B6: fun(A,fun(B,fun(C,bool))),B2: B,C3: C] :
( pp(aa(set(A),bool,member(A,A3),A6))
=> ( pp(aa(C,bool,aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),B6,A3),B2),C3))
=> pp(aa(C,bool,aa(B,fun(C,bool),aa(set(fun(B,fun(C,bool))),fun(B,fun(C,bool)),complete_Sup_Sup(fun(B,fun(C,bool))),aa(set(A),set(fun(B,fun(C,bool))),aa(fun(A,fun(B,fun(C,bool))),fun(set(A),set(fun(B,fun(C,bool)))),image2(A,fun(B,fun(C,bool))),B6),A6)),B2),C3)) ) ) ).
% SUP2_I
tff(fact_675_INF1__I,axiom,
! [A: $tType,B: $tType,A6: set(A),B6: fun(A,fun(B,bool)),B2: B] :
( ! [X: A] :
( pp(aa(set(A),bool,member(A,X),A6))
=> pp(aa(B,bool,aa(A,fun(B,bool),B6,X),B2)) )
=> pp(aa(B,bool,aa(set(fun(B,bool)),fun(B,bool),complete_Inf_Inf(fun(B,bool)),aa(set(A),set(fun(B,bool)),aa(fun(A,fun(B,bool)),fun(set(A),set(fun(B,bool))),image2(A,fun(B,bool)),B6),A6)),B2)) ) ).
% INF1_I
tff(fact_676_INF2__I,axiom,
! [B: $tType,A: $tType,C: $tType,A6: set(A),B6: fun(A,fun(B,fun(C,bool))),B2: B,C3: C] :
( ! [X: A] :
( pp(aa(set(A),bool,member(A,X),A6))
=> pp(aa(C,bool,aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),B6,X),B2),C3)) )
=> pp(aa(C,bool,aa(B,fun(C,bool),aa(set(fun(B,fun(C,bool))),fun(B,fun(C,bool)),complete_Inf_Inf(fun(B,fun(C,bool))),aa(set(A),set(fun(B,fun(C,bool))),aa(fun(A,fun(B,fun(C,bool))),fun(set(A),set(fun(B,fun(C,bool)))),image2(A,fun(B,fun(C,bool))),B6),A6)),B2),C3)) ) ).
% INF2_I
tff(fact_677_SUP1__I,axiom,
! [A: $tType,B: $tType,A3: A,A6: set(A),B6: fun(A,fun(B,bool)),B2: B] :
( pp(aa(set(A),bool,member(A,A3),A6))
=> ( pp(aa(B,bool,aa(A,fun(B,bool),B6,A3),B2))
=> pp(aa(B,bool,aa(set(fun(B,bool)),fun(B,bool),complete_Sup_Sup(fun(B,bool)),aa(set(A),set(fun(B,bool)),aa(fun(A,fun(B,bool)),fun(set(A),set(fun(B,bool))),image2(A,fun(B,bool)),B6),A6)),B2)) ) ) ).
% SUP1_I
tff(fact_678_if__image__distrib,axiom,
! [A: $tType,B: $tType,P2: fun(B,bool),F: fun(B,A),G: fun(B,A),S3: set(B)] : aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_fg(fun(B,bool),fun(fun(B,A),fun(fun(B,A),fun(B,A))),P2),F),G)),S3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),S3),aa(fun(B,bool),set(B),collect(B),P2)))),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),S3),aa(fun(B,bool),set(B),collect(B),aTP_Lamp_fh(fun(B,bool),fun(B,bool),P2))))) ).
% if_image_distrib
tff(fact_679_UN__Un,axiom,
! [A: $tType,B: $tType,M: fun(B,set(A)),A6: set(B),B6: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),M),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A6),B6))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),M),A6))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),M),B6))) ).
% UN_Un
tff(fact_680_bdd__below__image__inf,axiom,
! [A: $tType,B: $tType] :
( lattice(A)
=> ! [F: fun(B,A),G: fun(B,A),A6: set(B)] :
( condit1013018076250108175_below(A,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_fi(fun(B,A),fun(fun(B,A),fun(B,A)),F),G)),A6))
<=> ( condit1013018076250108175_below(A,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))
& condit1013018076250108175_below(A,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),G),A6)) ) ) ) ).
% bdd_below_image_inf
tff(fact_681_bdd__above__image__sup,axiom,
! [A: $tType,B: $tType] :
( lattice(A)
=> ! [F: fun(B,A),G: fun(B,A),A6: set(B)] :
( condit941137186595557371_above(A,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_fj(fun(B,A),fun(fun(B,A),fun(B,A)),F),G)),A6))
<=> ( condit941137186595557371_above(A,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))
& condit941137186595557371_above(A,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),G),A6)) ) ) ) ).
% bdd_above_image_sup
tff(fact_682_UN__simps_I3_J,axiom,
! [E: $tType,F2: $tType,C5: set(F2),A6: set(E),B6: fun(F2,set(E))] :
( ( ( C5 = bot_bot(set(F2)) )
=> ( aa(set(set(E)),set(E),complete_Sup_Sup(set(E)),aa(set(F2),set(set(E)),aa(fun(F2,set(E)),fun(set(F2),set(set(E))),image2(F2,set(E)),aa(fun(F2,set(E)),fun(F2,set(E)),aTP_Lamp_fk(set(E),fun(fun(F2,set(E)),fun(F2,set(E))),A6),B6)),C5)) = bot_bot(set(E)) ) )
& ( ( C5 != bot_bot(set(F2)) )
=> ( aa(set(set(E)),set(E),complete_Sup_Sup(set(E)),aa(set(F2),set(set(E)),aa(fun(F2,set(E)),fun(set(F2),set(set(E))),image2(F2,set(E)),aa(fun(F2,set(E)),fun(F2,set(E)),aTP_Lamp_fk(set(E),fun(fun(F2,set(E)),fun(F2,set(E))),A6),B6)),C5)) = aa(set(E),set(E),aa(set(E),fun(set(E),set(E)),sup_sup(set(E)),A6),aa(set(set(E)),set(E),complete_Sup_Sup(set(E)),aa(set(F2),set(set(E)),aa(fun(F2,set(E)),fun(set(F2),set(set(E))),image2(F2,set(E)),B6),C5))) ) ) ) ).
% UN_simps(3)
tff(fact_683_UN__simps_I2_J,axiom,
! [C: $tType,D: $tType,C5: set(C),A6: fun(C,set(D)),B6: set(D)] :
( ( ( C5 = bot_bot(set(C)) )
=> ( aa(set(set(D)),set(D),complete_Sup_Sup(set(D)),aa(set(C),set(set(D)),aa(fun(C,set(D)),fun(set(C),set(set(D))),image2(C,set(D)),aa(set(D),fun(C,set(D)),aTP_Lamp_fl(fun(C,set(D)),fun(set(D),fun(C,set(D))),A6),B6)),C5)) = bot_bot(set(D)) ) )
& ( ( C5 != bot_bot(set(C)) )
=> ( aa(set(set(D)),set(D),complete_Sup_Sup(set(D)),aa(set(C),set(set(D)),aa(fun(C,set(D)),fun(set(C),set(set(D))),image2(C,set(D)),aa(set(D),fun(C,set(D)),aTP_Lamp_fl(fun(C,set(D)),fun(set(D),fun(C,set(D))),A6),B6)),C5)) = aa(set(D),set(D),aa(set(D),fun(set(D),set(D)),sup_sup(set(D)),aa(set(set(D)),set(D),complete_Sup_Sup(set(D)),aa(set(C),set(set(D)),aa(fun(C,set(D)),fun(set(C),set(set(D))),image2(C,set(D)),A6),C5))),B6) ) ) ) ).
% UN_simps(2)
tff(fact_684_UN__insert,axiom,
! [A: $tType,B: $tType,B6: fun(B,set(A)),A3: B,A6: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),B6),aa(set(B),set(B),insert(B,A3),A6))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(B,set(A),B6,A3)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),B6),A6))) ).
% UN_insert
tff(fact_685_inj__apfst,axiom,
! [B: $tType,C: $tType,A: $tType,F: fun(A,C)] :
( inj_on(product_prod(A,B),product_prod(C,B),product_apfst(A,C,B,F),top_top(set(product_prod(A,B))))
<=> inj_on(A,C,F,top_top(set(A))) ) ).
% inj_apfst
tff(fact_686_inj__apsnd,axiom,
! [A: $tType,C: $tType,B: $tType,F: fun(B,C)] :
( inj_on(product_prod(A,B),product_prod(A,C),aa(fun(B,C),fun(product_prod(A,B),product_prod(A,C)),product_apsnd(B,C,A),F),top_top(set(product_prod(A,B))))
<=> inj_on(B,C,F,top_top(set(B))) ) ).
% inj_apsnd
tff(fact_687_INT__insert,axiom,
! [A: $tType,B: $tType,B6: fun(B,set(A)),A3: B,A6: set(B)] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),B6),aa(set(B),set(B),insert(B,A3),A6))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(B,set(A),B6,A3)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),B6),A6))) ).
% INT_insert
tff(fact_688_inj__on__apfst,axiom,
! [B: $tType,C: $tType,A: $tType,F: fun(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,aTP_Lamp_ca(A,set(B))))
<=> inj_on(A,C,F,A6) ) ).
% inj_on_apfst
tff(fact_689_inj__on__apsnd,axiom,
! [A: $tType,C: $tType,B: $tType,F: fun(B,C),A6: set(B)] :
( inj_on(product_prod(A,B),product_prod(A,C),aa(fun(B,C),fun(product_prod(A,B),product_prod(A,C)),product_apsnd(B,C,A),F),product_Sigma(A,B,top_top(set(A)),aTP_Lamp_bu(set(B),fun(A,set(B)),A6)))
<=> inj_on(B,C,F,A6) ) ).
% inj_on_apsnd
tff(fact_690_inj__on__insert,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A3: A,A6: set(A)] :
( inj_on(A,B,F,aa(set(A),set(A),insert(A,A3),A6))
<=> ( inj_on(A,B,F,A6)
& ~ pp(aa(set(B),bool,member(B,aa(A,B,F,A3)),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),aa(set(A),set(A),insert(A,A3),bot_bot(set(A))))))) ) ) ).
% inj_on_insert
tff(fact_691_inj__on__Un__image__eq__iff,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A6: set(A),B6: set(A)] :
( inj_on(A,B,F,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B6))
=> ( ( aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6) = aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),B6) )
<=> ( A6 = B6 ) ) ) ).
% inj_on_Un_image_eq_iff
tff(fact_692_INF2__D,axiom,
! [A: $tType,C: $tType,B: $tType,B6: fun(C,fun(A,fun(B,bool))),A6: set(C),B2: A,C3: B,A3: C] :
( pp(aa(B,bool,aa(A,fun(B,bool),aa(set(fun(A,fun(B,bool))),fun(A,fun(B,bool)),complete_Inf_Inf(fun(A,fun(B,bool))),aa(set(C),set(fun(A,fun(B,bool))),aa(fun(C,fun(A,fun(B,bool))),fun(set(C),set(fun(A,fun(B,bool)))),image2(C,fun(A,fun(B,bool))),B6),A6)),B2),C3))
=> ( pp(aa(set(C),bool,member(C,A3),A6))
=> pp(aa(B,bool,aa(A,fun(B,bool),aa(C,fun(A,fun(B,bool)),B6,A3),B2),C3)) ) ) ).
% INF2_D
tff(fact_693_INF2__E,axiom,
! [B: $tType,A: $tType,C: $tType,B6: fun(C,fun(A,fun(B,bool))),A6: set(C),B2: A,C3: B,A3: C] :
( pp(aa(B,bool,aa(A,fun(B,bool),aa(set(fun(A,fun(B,bool))),fun(A,fun(B,bool)),complete_Inf_Inf(fun(A,fun(B,bool))),aa(set(C),set(fun(A,fun(B,bool))),aa(fun(C,fun(A,fun(B,bool))),fun(set(C),set(fun(A,fun(B,bool)))),image2(C,fun(A,fun(B,bool))),B6),A6)),B2),C3))
=> ( ~ pp(aa(B,bool,aa(A,fun(B,bool),aa(C,fun(A,fun(B,bool)),B6,A3),B2),C3))
=> ~ pp(aa(set(C),bool,member(C,A3),A6)) ) ) ).
% INF2_E
tff(fact_694_INF1__D,axiom,
! [B: $tType,A: $tType,B6: fun(B,fun(A,bool)),A6: set(B),B2: A,A3: B] :
( pp(aa(A,bool,aa(set(fun(A,bool)),fun(A,bool),complete_Inf_Inf(fun(A,bool)),aa(set(B),set(fun(A,bool)),aa(fun(B,fun(A,bool)),fun(set(B),set(fun(A,bool))),image2(B,fun(A,bool)),B6),A6)),B2))
=> ( pp(aa(set(B),bool,member(B,A3),A6))
=> pp(aa(A,bool,aa(B,fun(A,bool),B6,A3),B2)) ) ) ).
% INF1_D
tff(fact_695_INF1__E,axiom,
! [A: $tType,B: $tType,B6: fun(B,fun(A,bool)),A6: set(B),B2: A,A3: B] :
( pp(aa(A,bool,aa(set(fun(A,bool)),fun(A,bool),complete_Inf_Inf(fun(A,bool)),aa(set(B),set(fun(A,bool)),aa(fun(B,fun(A,bool)),fun(set(B),set(fun(A,bool))),image2(B,fun(A,bool)),B6),A6)),B2))
=> ( ~ pp(aa(A,bool,aa(B,fun(A,bool),B6,A3),B2))
=> ~ pp(aa(set(B),bool,member(B,A3),A6)) ) ) ).
% INF1_E
tff(fact_696_SUP1__E,axiom,
! [B: $tType,A: $tType,B6: fun(B,fun(A,bool)),A6: set(B),B2: A] :
( pp(aa(A,bool,aa(set(fun(A,bool)),fun(A,bool),complete_Sup_Sup(fun(A,bool)),aa(set(B),set(fun(A,bool)),aa(fun(B,fun(A,bool)),fun(set(B),set(fun(A,bool))),image2(B,fun(A,bool)),B6),A6)),B2))
=> ~ ! [X: B] :
( pp(aa(set(B),bool,member(B,X),A6))
=> ~ pp(aa(A,bool,aa(B,fun(A,bool),B6,X),B2)) ) ) ).
% SUP1_E
tff(fact_697_SUP2__E,axiom,
! [A: $tType,C: $tType,B: $tType,B6: fun(C,fun(A,fun(B,bool))),A6: set(C),B2: A,C3: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aa(set(fun(A,fun(B,bool))),fun(A,fun(B,bool)),complete_Sup_Sup(fun(A,fun(B,bool))),aa(set(C),set(fun(A,fun(B,bool))),aa(fun(C,fun(A,fun(B,bool))),fun(set(C),set(fun(A,fun(B,bool)))),image2(C,fun(A,fun(B,bool))),B6),A6)),B2),C3))
=> ~ ! [X: C] :
( pp(aa(set(C),bool,member(C,X),A6))
=> ~ pp(aa(B,bool,aa(A,fun(B,bool),aa(C,fun(A,fun(B,bool)),B6,X),B2),C3)) ) ) ).
% SUP2_E
tff(fact_698_Sigma__Un__distrib1,axiom,
! [B: $tType,A: $tType,I4: set(A),J4: set(A),C5: fun(A,set(B))] : product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),I4),J4),C5) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),product_Sigma(A,B,I4,C5)),product_Sigma(A,B,J4,C5)) ).
% Sigma_Un_distrib1
tff(fact_699_Sigma__Int__distrib1,axiom,
! [B: $tType,A: $tType,I4: set(A),J4: set(A),C5: fun(A,set(B))] : product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),I4),J4),C5) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),inf_inf(set(product_prod(A,B))),product_Sigma(A,B,I4,C5)),product_Sigma(A,B,J4,C5)) ).
% Sigma_Int_distrib1
tff(fact_700_inj__on__disjoint__Un,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A6: set(A),G: fun(A,B),B6: set(A)] :
( inj_on(A,B,F,A6)
=> ( inj_on(A,B,G,B6)
=> ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6)),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),G),B6)) = bot_bot(set(B)) )
=> inj_on(A,B,aa(fun(A,B),fun(A,B),aa(set(A),fun(fun(A,B),fun(A,B)),aTP_Lamp_fm(fun(A,B),fun(set(A),fun(fun(A,B),fun(A,B))),F),A6),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B6)) ) ) ) ).
% inj_on_disjoint_Un
tff(fact_701_Sigma__Un__distrib2,axiom,
! [B: $tType,A: $tType,I4: set(A),A6: fun(A,set(B)),B6: fun(A,set(B))] : product_Sigma(A,B,I4,aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_fn(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(B))),A6),B6)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),product_Sigma(A,B,I4,A6)),product_Sigma(A,B,I4,B6)) ).
% Sigma_Un_distrib2
tff(fact_702_Times__Un__distrib1,axiom,
! [A: $tType,B: $tType,A6: set(A),B6: set(A),C5: set(B)] : product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B6),aTP_Lamp_bu(set(B),fun(A,set(B)),C5)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),product_Sigma(A,B,A6,aTP_Lamp_bu(set(B),fun(A,set(B)),C5))),product_Sigma(A,B,B6,aTP_Lamp_bu(set(B),fun(A,set(B)),C5))) ).
% Times_Un_distrib1
tff(fact_703_Times__Int__Times,axiom,
! [A: $tType,B: $tType,A6: set(A),B6: set(B),C5: set(A),D3: set(B)] : aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),inf_inf(set(product_prod(A,B))),product_Sigma(A,B,A6,aTP_Lamp_bu(set(B),fun(A,set(B)),B6))),product_Sigma(A,B,C5,aTP_Lamp_bu(set(B),fun(A,set(B)),D3))) = product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),C5),aa(set(B),fun(A,set(B)),aTP_Lamp_fo(set(B),fun(set(B),fun(A,set(B))),B6),D3)) ).
% Times_Int_Times
tff(fact_704_Sigma__Int__distrib2,axiom,
! [B: $tType,A: $tType,I4: set(A),A6: fun(A,set(B)),B6: fun(A,set(B))] : product_Sigma(A,B,I4,aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_fp(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(B))),A6),B6)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),inf_inf(set(product_prod(A,B))),product_Sigma(A,B,I4,A6)),product_Sigma(A,B,I4,B6)) ).
% Sigma_Int_distrib2
tff(fact_705_Times__Int__distrib1,axiom,
! [A: $tType,B: $tType,A6: set(A),B6: set(A),C5: set(B)] : product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B6),aTP_Lamp_bu(set(B),fun(A,set(B)),C5)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),inf_inf(set(product_prod(A,B))),product_Sigma(A,B,A6,aTP_Lamp_bu(set(B),fun(A,set(B)),C5))),product_Sigma(A,B,B6,aTP_Lamp_bu(set(B),fun(A,set(B)),C5))) ).
% Times_Int_distrib1
tff(fact_706_inj__on__Un,axiom,
! [A: $tType,B: $tType,F: fun(A,B),A6: set(A),B6: set(A)] :
( inj_on(A,B,F,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B6))
<=> ( inj_on(A,B,F,A6)
& inj_on(A,B,F,B6)
& ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),B6))),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),B6),A6))) = bot_bot(set(B)) ) ) ) ).
% inj_on_Un
tff(fact_707_inj__on__image,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A6: set(set(A))] :
( inj_on(A,B,F,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A6))
=> inj_on(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6) ) ).
% inj_on_image
tff(fact_708_cSUP__union,axiom,
! [A: $tType,B: $tType] :
( condit1219197933456340205attice(A)
=> ! [A6: set(B),F: fun(B,A),B6: set(B)] :
( ( A6 != bot_bot(set(B)) )
=> ( condit941137186595557371_above(A,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))
=> ( ( B6 != bot_bot(set(B)) )
=> ( condit941137186595557371_above(A,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),B6))
=> ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A6),B6))) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),B6))) ) ) ) ) ) ) ).
% cSUP_union
tff(fact_709_cINF__union,axiom,
! [A: $tType,B: $tType] :
( condit1219197933456340205attice(A)
=> ! [A6: set(B),F: fun(B,A),B6: set(B)] :
( ( A6 != bot_bot(set(B)) )
=> ( condit1013018076250108175_below(A,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))
=> ( ( B6 != bot_bot(set(B)) )
=> ( condit1013018076250108175_below(A,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),B6))
=> ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A6),B6))) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),B6))) ) ) ) ) ) ) ).
% cINF_union
tff(fact_710_inj__on__image__Int,axiom,
! [B: $tType,A: $tType,F: fun(A,B),C5: set(A),A6: set(A),B6: set(A)] :
( inj_on(A,B,F,C5)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),C5))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B6),C5))
=> ( aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B6)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6)),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),B6)) ) ) ) ) ).
% inj_on_image_Int
tff(fact_711_image__Int,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A6: set(A),B6: set(A)] :
( inj_on(A,B,F,top_top(set(A)))
=> ( aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B6)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6)),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),B6)) ) ) ).
% image_Int
tff(fact_712_inj__on__image__Pow,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A6: set(A)] :
( inj_on(A,B,F,A6)
=> inj_on(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),pow(A,A6)) ) ).
% inj_on_image_Pow
tff(fact_713_inj__on__image__iff,axiom,
! [B: $tType,A: $tType,A6: set(A),G: fun(A,B),F: fun(A,A)] :
( ! [X: A] :
( pp(aa(set(A),bool,member(A,X),A6))
=> ! [Xa: A] :
( pp(aa(set(A),bool,member(A,Xa),A6))
=> ( ( aa(A,B,G,aa(A,A,F,X)) = aa(A,B,G,aa(A,A,F,Xa)) )
<=> ( aa(A,B,G,X) = aa(A,B,G,Xa) ) ) ) )
=> ( inj_on(A,A,F,A6)
=> ( inj_on(A,B,G,aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),F),A6))
<=> inj_on(A,B,G,A6) ) ) ) ).
% inj_on_image_iff
tff(fact_714_inj__on__image__Fpow,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A6: set(A)] :
( inj_on(A,B,F,A6)
=> inj_on(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),finite_Fpow(A,A6)) ) ).
% inj_on_image_Fpow
tff(fact_715_inj__on__imageI2,axiom,
! [B: $tType,C: $tType,A: $tType,F4: fun(C,B),F: fun(A,C),A6: set(A)] :
( inj_on(A,B,aa(fun(A,C),fun(A,B),aa(fun(C,B),fun(fun(A,C),fun(A,B)),comp(C,B,A),F4),F),A6)
=> inj_on(A,C,F,A6) ) ).
% inj_on_imageI2
tff(fact_716_SUP__union,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [M: fun(B,A),A6: set(B),B6: set(B)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),M),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A6),B6))) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),M),A6))),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),M),B6))) ) ).
% SUP_union
tff(fact_717_map__prod__inj__on,axiom,
! [D: $tType,B: $tType,A: $tType,C: $tType,F: fun(A,B),A6: set(A),G: fun(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,aTP_Lamp_cg(set(C),fun(A,set(C)),B6))) ) ) ).
% map_prod_inj_on
tff(fact_718_INF__union,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [M: fun(B,A),A6: set(B),B6: set(B)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),M),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A6),B6))) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),M),A6))),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),M),B6))) ) ).
% INF_union
tff(fact_719_image__Un,axiom,
! [A: $tType,B: $tType,F: fun(B,A),A6: set(B),B6: set(B)] : aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A6),B6)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6)),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),B6)) ).
% image_Un
tff(fact_720_INT__Un,axiom,
! [A: $tType,B: $tType,M: fun(B,set(A)),A6: set(B),B6: set(B)] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),M),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A6),B6))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),M),A6))),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),M),B6))) ).
% INT_Un
tff(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
tff(fact_722_inj__on__convol__ident,axiom,
! [B: $tType,A: $tType,F: fun(A,B),X6: set(A)] : inj_on(A,product_prod(A,B),aTP_Lamp_fq(fun(A,B),fun(A,product_prod(A,B)),F),X6) ).
% inj_on_convol_ident
tff(fact_723_total__onI,axiom,
! [A: $tType,A6: set(A),R: set(product_prod(A,A))] :
( ! [X: A,Y: A] :
( pp(aa(set(A),bool,member(A,X),A6))
=> ( pp(aa(set(A),bool,member(A,Y),A6))
=> ( ( X != Y )
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R))
| pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),X)),R)) ) ) ) )
=> total_on(A,A6,R) ) ).
% total_onI
tff(fact_724_total__on__def,axiom,
! [A: $tType,A6: set(A),R: set(product_prod(A,A))] :
( total_on(A,A6,R)
<=> ! [X5: A] :
( pp(aa(set(A),bool,member(A,X5),A6))
=> ! [Xa3: A] :
( pp(aa(set(A),bool,member(A,Xa3),A6))
=> ( ( X5 != Xa3 )
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),Xa3)),R))
| pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa3),X5)),R)) ) ) ) ) ) ).
% total_on_def
tff(fact_725_SUP__UNIV__bool__expand,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A6: fun(bool,A)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(bool),set(A),aa(fun(bool,A),fun(set(bool),set(A)),image2(bool,A),A6),top_top(set(bool)))) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(bool,A,A6,fTrue)),aa(bool,A,A6,fFalse)) ) ).
% SUP_UNIV_bool_expand
tff(fact_726_INF__UNIV__bool__expand,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A6: fun(bool,A)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(bool),set(A),aa(fun(bool,A),fun(set(bool),set(A)),image2(bool,A),A6),top_top(set(bool)))) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(bool,A,A6,fTrue)),aa(bool,A,A6,fFalse)) ) ).
% INF_UNIV_bool_expand
tff(fact_727_fun_Oinj__map,axiom,
! [B: $tType,A: $tType,D: $tType,F: fun(A,B)] :
( inj_on(A,B,F,top_top(set(A)))
=> inj_on(fun(D,A),fun(D,B),aa(fun(A,B),fun(fun(D,A),fun(D,B)),comp(A,B,D),F),top_top(set(fun(D,A)))) ) ).
% fun.inj_map
tff(fact_728_Un__eq__UN,axiom,
! [A: $tType,A6: set(A),B6: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B6) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(bool),set(set(A)),aa(fun(bool,set(A)),fun(set(bool),set(set(A))),image2(bool,set(A)),aa(set(A),fun(bool,set(A)),aTP_Lamp_fr(set(A),fun(set(A),fun(bool,set(A))),A6),B6)),top_top(set(bool)))) ).
% Un_eq_UN
tff(fact_729_UN__bool__eq,axiom,
! [A: $tType,A6: fun(bool,set(A))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(bool),set(set(A)),aa(fun(bool,set(A)),fun(set(bool),set(set(A))),image2(bool,set(A)),A6),top_top(set(bool)))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(bool,set(A),A6,fTrue)),aa(bool,set(A),A6,fFalse)) ).
% UN_bool_eq
tff(fact_730_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
! [A: $tType,B: $tType] :
( condit1219197933456340205attice(A)
=> ! [A6: set(B),F: fun(B,A),G: fun(B,A)] :
( ( A6 != bot_bot(set(B)) )
=> ( condit941137186595557371_above(A,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))
=> ( condit941137186595557371_above(A,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),G),A6))
=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),G),A6))) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_fs(fun(B,A),fun(fun(B,A),fun(B,A)),F),G)),A6)) ) ) ) ) ) ).
% conditionally_complete_lattice_class.SUP_sup_distrib
tff(fact_731_cINF__inf__distrib,axiom,
! [A: $tType,B: $tType] :
( condit1219197933456340205attice(A)
=> ! [A6: set(B),F: fun(B,A),G: fun(B,A)] :
( ( A6 != bot_bot(set(B)) )
=> ( condit1013018076250108175_below(A,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))
=> ( condit1013018076250108175_below(A,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),G),A6))
=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),G),A6))) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_ft(fun(B,A),fun(fun(B,A),fun(B,A)),F),G)),A6)) ) ) ) ) ) ).
% cINF_inf_distrib
tff(fact_732_fsts_Ointros,axiom,
! [B: $tType,A: $tType,P: product_prod(A,B)] : pp(aa(set(A),bool,member(A,aa(product_prod(A,B),A,product_fst(A,B),P)),aa(product_prod(A,B),set(A),basic_fsts(A,B),P))) ).
% fsts.intros
tff(fact_733_fsts_Osimps,axiom,
! [B: $tType,A: $tType,A3: A,P: product_prod(A,B)] :
( pp(aa(set(A),bool,member(A,A3),aa(product_prod(A,B),set(A),basic_fsts(A,B),P)))
<=> ( A3 = aa(product_prod(A,B),A,product_fst(A,B),P) ) ) ).
% fsts.simps
tff(fact_734_fsts_Ocases,axiom,
! [B: $tType,A: $tType,A3: A,P: product_prod(A,B)] :
( pp(aa(set(A),bool,member(A,A3),aa(product_prod(A,B),set(A),basic_fsts(A,B),P)))
=> ( A3 = aa(product_prod(A,B),A,product_fst(A,B),P) ) ) ).
% fsts.cases
tff(fact_735_INT__bool__eq,axiom,
! [A: $tType,A6: fun(bool,set(A))] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(bool),set(set(A)),aa(fun(bool,set(A)),fun(set(bool),set(set(A))),image2(bool,set(A)),A6),top_top(set(bool)))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(bool,set(A),A6,fTrue)),aa(bool,set(A),A6,fFalse)) ).
% INT_bool_eq
tff(fact_736_cSUP__eq__cINF__D,axiom,
! [B: $tType,C: $tType] :
( condit1219197933456340205attice(B)
=> ! [F: fun(C,B),A6: set(C),A3: C] :
( ( aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),aa(fun(C,B),fun(set(C),set(B)),image2(C,B),F),A6)) = aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),aa(fun(C,B),fun(set(C),set(B)),image2(C,B),F),A6)) )
=> ( condit941137186595557371_above(B,aa(set(C),set(B),aa(fun(C,B),fun(set(C),set(B)),image2(C,B),F),A6))
=> ( condit1013018076250108175_below(B,aa(set(C),set(B),aa(fun(C,B),fun(set(C),set(B)),image2(C,B),F),A6))
=> ( pp(aa(set(C),bool,member(C,A3),A6))
=> ( aa(C,B,F,A3) = aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),aa(fun(C,B),fun(set(C),set(B)),image2(C,B),F),A6)) ) ) ) ) ) ) ).
% cSUP_eq_cINF_D
tff(fact_737_cSUP__insert,axiom,
! [A: $tType,B: $tType] :
( condit1219197933456340205attice(A)
=> ! [A6: set(B),F: fun(B,A),A3: B] :
( ( A6 != bot_bot(set(B)) )
=> ( condit941137186595557371_above(A,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))
=> ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),aa(set(B),set(B),insert(B,A3),A6))) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,F,A3)),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))) ) ) ) ) ).
% cSUP_insert
tff(fact_738_cINF__insert,axiom,
! [A: $tType,B: $tType] :
( condit1219197933456340205attice(A)
=> ! [A6: set(B),F: fun(B,A),A3: B] :
( ( A6 != bot_bot(set(B)) )
=> ( condit1013018076250108175_below(A,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))
=> ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),aa(set(B),set(B),insert(B,A3),A6))) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,F,A3)),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))) ) ) ) ) ).
% cINF_insert
tff(fact_739_inj__on__image__mem__iff,axiom,
! [B: $tType,A: $tType,F: fun(A,B),B6: set(A),A3: A,A6: set(A)] :
( inj_on(A,B,F,B6)
=> ( pp(aa(set(A),bool,member(A,A3),B6))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B6))
=> ( pp(aa(set(B),bool,member(B,aa(A,B,F,A3)),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6)))
<=> pp(aa(set(A),bool,member(A,A3),A6)) ) ) ) ) ).
% inj_on_image_mem_iff
tff(fact_740_inj__on__image__eq__iff,axiom,
! [B: $tType,A: $tType,F: fun(A,B),C5: set(A),A6: set(A),B6: set(A)] :
( inj_on(A,B,F,C5)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),C5))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B6),C5))
=> ( ( aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6) = aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),B6) )
<=> ( A6 = B6 ) ) ) ) ) ).
% inj_on_image_eq_iff
tff(fact_741_subset__image__inj,axiom,
! [A: $tType,B: $tType,S3: set(A),F: fun(B,A),T3: set(B)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S3),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),T3)))
<=> ? [U3: set(B)] :
( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),U3),T3))
& inj_on(B,A,F,U3)
& ( S3 = aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),U3) ) ) ) ).
% subset_image_inj
tff(fact_742_inj__img__insertE,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A6: set(A),X2: B,B6: set(B)] :
( inj_on(A,B,F,A6)
=> ( ~ pp(aa(set(B),bool,member(B,X2),B6))
=> ( ( aa(set(B),set(B),insert(B,X2),B6) = aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6) )
=> ~ ! [X7: A,A9: set(A)] :
( ~ pp(aa(set(A),bool,member(A,X7),A9))
=> ( ( A6 = aa(set(A),set(A),insert(A,X7),A9) )
=> ( ( X2 = aa(A,B,F,X7) )
=> ( B6 != aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A9) ) ) ) ) ) ) ) ).
% inj_img_insertE
tff(fact_743_inj__image__mem__iff,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A3: A,A6: set(A)] :
( inj_on(A,B,F,top_top(set(A)))
=> ( pp(aa(set(B),bool,member(B,aa(A,B,F,A3)),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6)))
<=> pp(aa(set(A),bool,member(A,A3),A6)) ) ) ).
% inj_image_mem_iff
tff(fact_744_inj__image__eq__iff,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A6: set(A),B6: set(A)] :
( inj_on(A,B,F,top_top(set(A)))
=> ( ( aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6) = aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),B6) )
<=> ( A6 = B6 ) ) ) ).
% inj_image_eq_iff
tff(fact_745_range__ex1__eq,axiom,
! [B: $tType,A: $tType,F: fun(A,B),B2: B] :
( inj_on(A,B,F,top_top(set(A)))
=> ( pp(aa(set(B),bool,member(B,B2),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),top_top(set(A)))))
<=> ? [X5: A] :
( ( B2 = aa(A,B,F,X5) )
& ! [Y4: A] :
( ( B2 = aa(A,B,F,Y4) )
=> ( Y4 = X5 ) ) ) ) ) ).
% range_ex1_eq
tff(fact_746_comp__inj__on,axiom,
! [B: $tType,C: $tType,A: $tType,F: fun(A,B),A6: set(A),G: fun(B,C)] :
( inj_on(A,B,F,A6)
=> ( inj_on(B,C,G,aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6))
=> inj_on(A,C,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),G),F),A6) ) ) ).
% comp_inj_on
tff(fact_747_inj__on__imageI,axiom,
! [B: $tType,C: $tType,A: $tType,G: fun(C,B),F: fun(A,C),A6: set(A)] :
( inj_on(A,B,aa(fun(A,C),fun(A,B),aa(fun(C,B),fun(fun(A,C),fun(A,B)),comp(C,B,A),G),F),A6)
=> inj_on(C,B,G,aa(set(A),set(C),aa(fun(A,C),fun(set(A),set(C)),image2(A,C),F),A6)) ) ).
% inj_on_imageI
tff(fact_748_comp__inj__on__iff,axiom,
! [B: $tType,C: $tType,A: $tType,F: fun(A,B),A6: set(A),F4: fun(B,C)] :
( inj_on(A,B,F,A6)
=> ( inj_on(B,C,F4,aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6))
<=> inj_on(A,C,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),F4),F),A6) ) ) ).
% comp_inj_on_iff
tff(fact_749_inj__compose,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(A,B),G: fun(C,A)] :
( inj_on(A,B,F,top_top(set(A)))
=> ( inj_on(C,A,G,top_top(set(C)))
=> inj_on(C,B,aa(fun(C,A),fun(C,B),aa(fun(A,B),fun(fun(C,A),fun(C,B)),comp(A,B,C),F),G),top_top(set(C))) ) ) ).
% inj_compose
tff(fact_750_inf__Sup,axiom,
! [A: $tType] :
( comple592849572758109894attice(A)
=> ! [A3: A,B6: set(A)] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(set(A),A,complete_Sup_Sup(A),B6)) = aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aa(A,fun(A,A),inf_inf(A),A3)),B6)) ) ).
% inf_Sup
tff(fact_751_sup__Inf,axiom,
! [A: $tType] :
( comple592849572758109894attice(A)
=> ! [A3: A,B6: set(A)] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(set(A),A,complete_Inf_Inf(A),B6)) = aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aa(A,fun(A,A),sup_sup(A),A3)),B6)) ) ).
% sup_Inf
tff(fact_752_bdd__below_OI2,axiom,
! [A: $tType,B: $tType] :
( preorder(A)
=> ! [A6: set(B),M: A,F: fun(B,A)] :
( ! [X: B] :
( pp(aa(set(B),bool,member(B,X),A6))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),aa(B,A,F,X))) )
=> condit1013018076250108175_below(A,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6)) ) ) ).
% bdd_below.I2
tff(fact_753_bdd__belowI2,axiom,
! [A: $tType,B: $tType] :
( preorder(A)
=> ! [A6: set(B),M4: A,F: fun(B,A)] :
( ! [X: B] :
( pp(aa(set(B),bool,member(B,X),A6))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M4),aa(B,A,F,X))) )
=> condit1013018076250108175_below(A,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6)) ) ) ).
% bdd_belowI2
tff(fact_754_bdd__above_OI2,axiom,
! [A: $tType,B: $tType] :
( preorder(A)
=> ! [A6: set(B),F: fun(B,A),M: A] :
( ! [X: B] :
( pp(aa(set(B),bool,member(B,X),A6))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,X)),M)) )
=> condit941137186595557371_above(A,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6)) ) ) ).
% bdd_above.I2
tff(fact_755_image__Int__subset,axiom,
! [A: $tType,B: $tType,F: fun(B,A),A6: set(B),B6: set(B)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A6),B6))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6)),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),B6)))) ).
% image_Int_subset
tff(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),aa(fun(A,fun(B,product_prod(B,A))),fun(product_prod(A,B),product_prod(B,A)),product_case_prod(A,B,product_prod(B,A)),aTP_Lamp_ba(A,fun(B,product_prod(B,A)))),A6) ).
% swap_inj_on
tff(fact_757_complete__lattice__class_OSUP__sup__distrib,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [F: fun(B,A),A6: set(B),G: fun(B,A)] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),G),A6))) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_fu(fun(B,A),fun(fun(B,A),fun(B,A)),F),G)),A6)) ) ).
% complete_lattice_class.SUP_sup_distrib
tff(fact_758_SUP__absorb,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [K: B,I4: set(B),A6: fun(B,A)] :
( pp(aa(set(B),bool,member(B,K),I4))
=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,A6,K)),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),A6),I4))) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),A6),I4)) ) ) ) ).
% SUP_absorb
tff(fact_759_SUP__inf,axiom,
! [A: $tType,B: $tType] :
( comple592849572758109894attice(A)
=> ! [F: fun(B,A),B6: set(B),A3: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),B6))),A3) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),aa(A,fun(B,A),aTP_Lamp_fv(fun(B,A),fun(A,fun(B,A)),F),A3)),B6)) ) ).
% SUP_inf
tff(fact_760_Sup__inf,axiom,
! [A: $tType] :
( comple592849572758109894attice(A)
=> ! [B6: set(A),A3: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),B6)),A3) = aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aTP_Lamp_fw(A,fun(A,A),A3)),B6)) ) ).
% Sup_inf
tff(fact_761_inf__SUP,axiom,
! [A: $tType,B: $tType] :
( comple592849572758109894attice(A)
=> ! [A3: A,F: fun(B,A),B6: set(B)] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),B6))) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_fx(A,fun(fun(B,A),fun(B,A)),A3),F)),B6)) ) ).
% inf_SUP
tff(fact_762_SUP__inf__distrib2,axiom,
! [C: $tType,A: $tType,B: $tType] :
( comple592849572758109894attice(A)
=> ! [F: fun(B,A),A6: set(B),G: fun(C,A),B6: set(C)] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))),aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),aa(fun(C,A),fun(set(C),set(A)),image2(C,A),G),B6))) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),aa(set(C),fun(B,A),aa(fun(C,A),fun(set(C),fun(B,A)),aTP_Lamp_fz(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),F),G),B6)),A6)) ) ).
% SUP_inf_distrib2
tff(fact_763_INF__sup,axiom,
! [A: $tType,B: $tType] :
( comple592849572758109894attice(A)
=> ! [F: fun(B,A),B6: set(B),A3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),B6))),A3) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),aa(A,fun(B,A),aTP_Lamp_ga(fun(B,A),fun(A,fun(B,A)),F),A3)),B6)) ) ).
% INF_sup
tff(fact_764_Inf__sup,axiom,
! [A: $tType] :
( comple592849572758109894attice(A)
=> ! [B6: set(A),A3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Inf_Inf(A),B6)),A3) = aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aTP_Lamp_gb(A,fun(A,A),A3)),B6)) ) ).
% Inf_sup
tff(fact_765_sup__INF,axiom,
! [A: $tType,B: $tType] :
( comple592849572758109894attice(A)
=> ! [A3: A,F: fun(B,A),B6: set(B)] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),B6))) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_gc(A,fun(fun(B,A),fun(B,A)),A3),F)),B6)) ) ).
% sup_INF
tff(fact_766_INF__sup__distrib2,axiom,
! [C: $tType,A: $tType,B: $tType] :
( comple592849572758109894attice(A)
=> ! [F: fun(B,A),A6: set(B),G: fun(C,A),B6: set(C)] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))),aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),aa(fun(C,A),fun(set(C),set(A)),image2(C,A),G),B6))) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),aa(set(C),fun(B,A),aa(fun(C,A),fun(set(C),fun(B,A)),aTP_Lamp_ge(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),F),G),B6)),A6)) ) ).
% INF_sup_distrib2
tff(fact_767_INF__inf__distrib,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [F: fun(B,A),A6: set(B),G: fun(B,A)] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),G),A6))) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_gf(fun(B,A),fun(fun(B,A),fun(B,A)),F),G)),A6)) ) ).
% INF_inf_distrib
tff(fact_768_INF__absorb,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [K: B,I4: set(B),A6: fun(B,A)] :
( pp(aa(set(B),bool,member(B,K),I4))
=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,A6,K)),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),A6),I4))) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),A6),I4)) ) ) ) ).
% INF_absorb
tff(fact_769_translation__subtract__Int,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [A3: A,S: set(A),T2: set(A)] : aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aTP_Lamp_eo(A,fun(A,A),A3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),T2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aTP_Lamp_eo(A,fun(A,A),A3)),S)),aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aTP_Lamp_eo(A,fun(A,A),A3)),T2)) ) ).
% translation_subtract_Int
tff(fact_770_Un__Union__image,axiom,
! [A: $tType,B: $tType,A6: fun(B,set(A)),B6: fun(B,set(A)),C5: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_gg(fun(B,set(A)),fun(fun(B,set(A)),fun(B,set(A))),A6),B6)),C5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),A6),C5))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),B6),C5))) ).
% Un_Union_image
tff(fact_771_UN__Un__distrib,axiom,
! [A: $tType,B: $tType,A6: fun(B,set(A)),B6: fun(B,set(A)),I4: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_gg(fun(B,set(A)),fun(fun(B,set(A)),fun(B,set(A))),A6),B6)),I4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),A6),I4))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),B6),I4))) ).
% UN_Un_distrib
tff(fact_772_UN__absorb,axiom,
! [B: $tType,A: $tType,K: A,I4: set(A),A6: fun(A,set(B))] :
( pp(aa(set(A),bool,member(A,K),I4))
=> ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(A,set(B),A6,K)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),aa(fun(A,set(B)),fun(set(A),set(set(B))),image2(A,set(B)),A6),I4))) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),aa(fun(A,set(B)),fun(set(A),set(set(B))),image2(A,set(B)),A6),I4)) ) ) ).
% UN_absorb
tff(fact_773_Int__UN__distrib2,axiom,
! [C: $tType,A: $tType,B: $tType,A6: fun(B,set(A)),I4: set(B),B6: fun(C,set(A)),J4: set(C)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),A6),I4))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),aa(fun(C,set(A)),fun(set(C),set(set(A))),image2(C,set(A)),B6),J4))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),aa(set(C),fun(B,set(A)),aa(fun(C,set(A)),fun(set(C),fun(B,set(A))),aTP_Lamp_gi(fun(B,set(A)),fun(fun(C,set(A)),fun(set(C),fun(B,set(A)))),A6),B6),J4)),I4)) ).
% Int_UN_distrib2
tff(fact_774_Int__UN__distrib,axiom,
! [A: $tType,B: $tType,B6: set(A),A6: fun(B,set(A)),I4: set(B)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B6),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),A6),I4))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_gj(set(A),fun(fun(B,set(A)),fun(B,set(A))),B6),A6)),I4)) ).
% Int_UN_distrib
tff(fact_775_UN__extend__simps_I4_J,axiom,
! [H5: $tType,G2: $tType,A6: fun(G2,set(H5)),C5: set(G2),B6: set(H5)] : aa(set(H5),set(H5),aa(set(H5),fun(set(H5),set(H5)),inf_inf(set(H5)),aa(set(set(H5)),set(H5),complete_Sup_Sup(set(H5)),aa(set(G2),set(set(H5)),aa(fun(G2,set(H5)),fun(set(G2),set(set(H5))),image2(G2,set(H5)),A6),C5))),B6) = aa(set(set(H5)),set(H5),complete_Sup_Sup(set(H5)),aa(set(G2),set(set(H5)),aa(fun(G2,set(H5)),fun(set(G2),set(set(H5))),image2(G2,set(H5)),aa(set(H5),fun(G2,set(H5)),aTP_Lamp_gk(fun(G2,set(H5)),fun(set(H5),fun(G2,set(H5))),A6),B6)),C5)) ).
% UN_extend_simps(4)
tff(fact_776_UN__extend__simps_I5_J,axiom,
! [I5: $tType,J2: $tType,A6: set(I5),B6: fun(J2,set(I5)),C5: set(J2)] : aa(set(I5),set(I5),aa(set(I5),fun(set(I5),set(I5)),inf_inf(set(I5)),A6),aa(set(set(I5)),set(I5),complete_Sup_Sup(set(I5)),aa(set(J2),set(set(I5)),aa(fun(J2,set(I5)),fun(set(J2),set(set(I5))),image2(J2,set(I5)),B6),C5))) = aa(set(set(I5)),set(I5),complete_Sup_Sup(set(I5)),aa(set(J2),set(set(I5)),aa(fun(J2,set(I5)),fun(set(J2),set(set(I5))),image2(J2,set(I5)),aa(fun(J2,set(I5)),fun(J2,set(I5)),aTP_Lamp_gl(set(I5),fun(fun(J2,set(I5)),fun(J2,set(I5))),A6),B6)),C5)) ).
% UN_extend_simps(5)
tff(fact_777_Un__INT__distrib2,axiom,
! [C: $tType,A: $tType,B: $tType,A6: fun(B,set(A)),I4: set(B),B6: fun(C,set(A)),J4: set(C)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),A6),I4))),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(C),set(set(A)),aa(fun(C,set(A)),fun(set(C),set(set(A))),image2(C,set(A)),B6),J4))) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),aa(set(C),fun(B,set(A)),aa(fun(C,set(A)),fun(set(C),fun(B,set(A))),aTP_Lamp_gn(fun(B,set(A)),fun(fun(C,set(A)),fun(set(C),fun(B,set(A)))),A6),B6),J4)),I4)) ).
% Un_INT_distrib2
tff(fact_778_Un__INT__distrib,axiom,
! [A: $tType,B: $tType,B6: set(A),A6: fun(B,set(A)),I4: set(B)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B6),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),A6),I4))) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_go(set(A),fun(fun(B,set(A)),fun(B,set(A))),B6),A6)),I4)) ).
% Un_INT_distrib
tff(fact_779_INT__extend__simps_I6_J,axiom,
! [L3: $tType,K2: $tType,A6: fun(K2,set(L3)),C5: set(K2),B6: set(L3)] : aa(set(L3),set(L3),aa(set(L3),fun(set(L3),set(L3)),sup_sup(set(L3)),aa(set(set(L3)),set(L3),complete_Inf_Inf(set(L3)),aa(set(K2),set(set(L3)),aa(fun(K2,set(L3)),fun(set(K2),set(set(L3))),image2(K2,set(L3)),A6),C5))),B6) = aa(set(set(L3)),set(L3),complete_Inf_Inf(set(L3)),aa(set(K2),set(set(L3)),aa(fun(K2,set(L3)),fun(set(K2),set(set(L3))),image2(K2,set(L3)),aa(set(L3),fun(K2,set(L3)),aTP_Lamp_gp(fun(K2,set(L3)),fun(set(L3),fun(K2,set(L3))),A6),B6)),C5)) ).
% INT_extend_simps(6)
tff(fact_780_INT__extend__simps_I7_J,axiom,
! [M5: $tType,N3: $tType,A6: set(M5),B6: fun(N3,set(M5)),C5: set(N3)] : aa(set(M5),set(M5),aa(set(M5),fun(set(M5),set(M5)),sup_sup(set(M5)),A6),aa(set(set(M5)),set(M5),complete_Inf_Inf(set(M5)),aa(set(N3),set(set(M5)),aa(fun(N3,set(M5)),fun(set(N3),set(set(M5))),image2(N3,set(M5)),B6),C5))) = aa(set(set(M5)),set(M5),complete_Inf_Inf(set(M5)),aa(set(N3),set(set(M5)),aa(fun(N3,set(M5)),fun(set(N3),set(set(M5))),image2(N3,set(M5)),aa(fun(N3,set(M5)),fun(N3,set(M5)),aTP_Lamp_gq(set(M5),fun(fun(N3,set(M5)),fun(N3,set(M5))),A6),B6)),C5)) ).
% INT_extend_simps(7)
tff(fact_781_Int__Inter__image,axiom,
! [A: $tType,B: $tType,A6: fun(B,set(A)),B6: fun(B,set(A)),C5: set(B)] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_gr(fun(B,set(A)),fun(fun(B,set(A)),fun(B,set(A))),A6),B6)),C5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),A6),C5))),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),B6),C5))) ).
% Int_Inter_image
tff(fact_782_INT__Int__distrib,axiom,
! [A: $tType,B: $tType,A6: fun(B,set(A)),B6: fun(B,set(A)),I4: set(B)] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_gr(fun(B,set(A)),fun(fun(B,set(A)),fun(B,set(A))),A6),B6)),I4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),A6),I4))),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),B6),I4))) ).
% INT_Int_distrib
tff(fact_783_INT__absorb,axiom,
! [B: $tType,A: $tType,K: A,I4: set(A),A6: fun(A,set(B))] :
( pp(aa(set(A),bool,member(A,K),I4))
=> ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(A,set(B),A6,K)),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),aa(fun(A,set(B)),fun(set(A),set(set(B))),image2(A,set(B)),A6),I4))) = aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),aa(fun(A,set(B)),fun(set(A),set(set(B))),image2(A,set(B)),A6),I4)) ) ) ).
% INT_absorb
tff(fact_784_Int__Union2,axiom,
! [A: $tType,B6: set(set(A)),A6: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B6)),A6) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(A)),set(set(A)),aa(fun(set(A),set(A)),fun(set(set(A)),set(set(A))),image2(set(A),set(A)),aTP_Lamp_gs(set(A),fun(set(A),set(A)),A6)),B6)) ).
% Int_Union2
tff(fact_785_Int__Union,axiom,
! [A: $tType,A6: set(A),B6: set(set(A))] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B6)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(A)),set(set(A)),aa(fun(set(A),set(A)),fun(set(set(A)),set(set(A))),image2(set(A),set(A)),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6)),B6)) ).
% Int_Union
tff(fact_786_Un__Inter,axiom,
! [A: $tType,A6: set(A),B6: set(set(A))] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B6)) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(set(A)),set(set(A)),aa(fun(set(A),set(A)),fun(set(set(A)),set(set(A))),image2(set(A),set(A)),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6)),B6)) ).
% Un_Inter
tff(fact_787_inj__on__iff__surj,axiom,
! [B: $tType,A: $tType,A6: set(A),A8: set(B)] :
( ( A6 != bot_bot(set(A)) )
=> ( ? [F7: fun(A,B)] :
( inj_on(A,B,F7,A6)
& pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F7),A6)),A8)) )
<=> ? [G5: fun(B,A)] : aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),G5),A8) = A6 ) ) ).
% inj_on_iff_surj
tff(fact_788_Int__Inter__eq_I1_J,axiom,
! [A: $tType,B9: set(set(A)),A6: set(A)] :
( ( ( B9 = bot_bot(set(set(A))) )
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B9)) = A6 ) )
& ( ( B9 != bot_bot(set(set(A))) )
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B9)) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(set(A)),set(set(A)),aa(fun(set(A),set(A)),fun(set(set(A)),set(set(A))),image2(set(A),set(A)),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6)),B9)) ) ) ) ).
% Int_Inter_eq(1)
tff(fact_789_Int__Inter__eq_I2_J,axiom,
! [A: $tType,B9: set(set(A)),A6: set(A)] :
( ( ( B9 = bot_bot(set(set(A))) )
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B9)),A6) = A6 ) )
& ( ( B9 != bot_bot(set(set(A))) )
=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B9)),A6) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(set(A)),set(set(A)),aa(fun(set(A),set(A)),fun(set(set(A)),set(set(A))),image2(set(A),set(A)),aTP_Lamp_gs(set(A),fun(set(A),set(A)),A6)),B9)) ) ) ) ).
% Int_Inter_eq(2)
tff(fact_790_inj__image__subset__iff,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A6: set(A),B6: set(A)] :
( inj_on(A,B,F,top_top(set(A)))
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6)),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),B6)))
<=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B6)) ) ) ).
% inj_image_subset_iff
tff(fact_791_inj__on__image__set__diff,axiom,
! [B: $tType,A: $tType,F: fun(A,B),C5: set(A),A6: set(A),B6: set(A)] :
( inj_on(A,B,F,C5)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),B6)),C5))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B6),C5))
=> ( aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),B6)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6)),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),B6)) ) ) ) ) ).
% inj_on_image_set_diff
tff(fact_792_image__set__diff,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A6: set(A),B6: set(A)] :
( inj_on(A,B,F,top_top(set(A)))
=> ( aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),A6),B6)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6)),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),B6)) ) ) ).
% image_set_diff
tff(fact_793_prod_Oset__map_I1_J,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,F1: fun(A,C),F22: fun(B,D),V: product_prod(A,B)] : aa(product_prod(C,D),set(C),basic_fsts(C,D),aa(product_prod(A,B),product_prod(C,D),product_map_prod(A,C,B,D,F1,F22),V)) = aa(set(A),set(C),aa(fun(A,C),fun(set(A),set(C)),image2(A,C),F1),aa(product_prod(A,B),set(A),basic_fsts(A,B),V)) ).
% prod.set_map(1)
tff(fact_794_cINF__lower2,axiom,
! [B: $tType,A: $tType] :
( condit1219197933456340205attice(A)
=> ! [F: fun(B,A),A6: set(B),X2: B,U: A] :
( condit1013018076250108175_below(A,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))
=> ( pp(aa(set(B),bool,member(B,X2),A6))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,X2)),U))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))),U)) ) ) ) ) ).
% cINF_lower2
tff(fact_795_cINF__lower,axiom,
! [A: $tType,B: $tType] :
( condit1219197933456340205attice(A)
=> ! [F: fun(B,A),A6: set(B),X2: B] :
( condit1013018076250108175_below(A,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))
=> ( pp(aa(set(B),bool,member(B,X2),A6))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))),aa(B,A,F,X2))) ) ) ) ).
% cINF_lower
tff(fact_796_cSUP__upper2,axiom,
! [A: $tType,B: $tType] :
( condit1219197933456340205attice(A)
=> ! [F: fun(B,A),A6: set(B),X2: B,U: A] :
( condit941137186595557371_above(A,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))
=> ( pp(aa(set(B),bool,member(B,X2),A6))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(B,A,F,X2)))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6)))) ) ) ) ) ).
% cSUP_upper2
tff(fact_797_cSUP__upper,axiom,
! [A: $tType,B: $tType] :
( condit1219197933456340205attice(A)
=> ! [X2: B,A6: set(B),F: fun(B,A)] :
( pp(aa(set(B),bool,member(B,X2),A6))
=> ( condit941137186595557371_above(A,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,X2)),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6)))) ) ) ) ).
% cSUP_upper
tff(fact_798_Union__image__empty,axiom,
! [B: $tType,A: $tType,A6: set(A),F: fun(B,set(A))] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),F),bot_bot(set(B))))) = A6 ).
% Union_image_empty
tff(fact_799_Union__image__insert,axiom,
! [A: $tType,B: $tType,F: fun(B,set(A)),A3: B,B6: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),F),aa(set(B),set(B),insert(B,A3),B6))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(B,set(A),F,A3)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),F),B6))) ).
% Union_image_insert
tff(fact_800_inj__on__UNION__chain,axiom,
! [C: $tType,B: $tType,A: $tType,I4: set(A),A6: fun(A,set(B)),F: fun(B,C)] :
( ! [I6: A,J3: A] :
( pp(aa(set(A),bool,member(A,I6),I4))
=> ( pp(aa(set(A),bool,member(A,J3),I4))
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),A6,I6)),aa(A,set(B),A6,J3)))
| pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),A6,J3)),aa(A,set(B),A6,I6))) ) ) )
=> ( ! [I6: A] :
( pp(aa(set(A),bool,member(A,I6),I4))
=> inj_on(B,C,F,aa(A,set(B),A6,I6)) )
=> inj_on(B,C,F,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),aa(fun(A,set(B)),fun(set(A),set(set(B))),image2(A,set(B)),A6),I4))) ) ) ).
% inj_on_UNION_chain
tff(fact_801_inj__on__INTER,axiom,
! [C: $tType,B: $tType,A: $tType,I4: set(A),F: fun(B,C),A6: fun(A,set(B))] :
( ( I4 != bot_bot(set(A)) )
=> ( ! [I6: A] :
( pp(aa(set(A),bool,member(A,I6),I4))
=> inj_on(B,C,F,aa(A,set(B),A6,I6)) )
=> inj_on(B,C,F,aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),aa(fun(A,set(B)),fun(set(A),set(set(B))),image2(A,set(B)),A6),I4))) ) ) ).
% inj_on_INTER
tff(fact_802_SUP__insert,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [F: fun(B,A),A3: B,A6: set(B)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),aa(set(B),set(B),insert(B,A3),A6))) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,F,A3)),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))) ) ).
% SUP_insert
tff(fact_803_INF__inf__const2,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [I4: set(B),F: fun(B,A),X2: A] :
( ( I4 != bot_bot(set(B)) )
=> ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),aa(A,fun(B,A),aTP_Lamp_gt(fun(B,A),fun(A,fun(B,A)),F),X2)),I4)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),I4))),X2) ) ) ) ).
% INF_inf_const2
tff(fact_804_INF__inf__const1,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [I4: set(B),X2: A,F: fun(B,A)] :
( ( I4 != bot_bot(set(B)) )
=> ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_gu(A,fun(fun(B,A),fun(B,A)),X2),F)),I4)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X2),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),I4))) ) ) ) ).
% INF_inf_const1
tff(fact_805_INF__insert,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [F: fun(B,A),A3: B,A6: set(B)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),aa(set(B),set(B),insert(B,A3),A6))) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,F,A3)),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))) ) ).
% INF_insert
tff(fact_806_UN__extend__simps_I2_J,axiom,
! [D: $tType,C: $tType,C5: set(C),A6: fun(C,set(D)),B6: set(D)] :
( ( ( C5 = bot_bot(set(C)) )
=> ( aa(set(D),set(D),aa(set(D),fun(set(D),set(D)),sup_sup(set(D)),aa(set(set(D)),set(D),complete_Sup_Sup(set(D)),aa(set(C),set(set(D)),aa(fun(C,set(D)),fun(set(C),set(set(D))),image2(C,set(D)),A6),C5))),B6) = B6 ) )
& ( ( C5 != bot_bot(set(C)) )
=> ( aa(set(D),set(D),aa(set(D),fun(set(D),set(D)),sup_sup(set(D)),aa(set(set(D)),set(D),complete_Sup_Sup(set(D)),aa(set(C),set(set(D)),aa(fun(C,set(D)),fun(set(C),set(set(D))),image2(C,set(D)),A6),C5))),B6) = aa(set(set(D)),set(D),complete_Sup_Sup(set(D)),aa(set(C),set(set(D)),aa(fun(C,set(D)),fun(set(C),set(set(D))),image2(C,set(D)),aa(set(D),fun(C,set(D)),aTP_Lamp_fl(fun(C,set(D)),fun(set(D),fun(C,set(D))),A6),B6)),C5)) ) ) ) ).
% UN_extend_simps(2)
tff(fact_807_UN__extend__simps_I3_J,axiom,
! [E: $tType,F2: $tType,C5: set(F2),A6: set(E),B6: fun(F2,set(E))] :
( ( ( C5 = bot_bot(set(F2)) )
=> ( aa(set(E),set(E),aa(set(E),fun(set(E),set(E)),sup_sup(set(E)),A6),aa(set(set(E)),set(E),complete_Sup_Sup(set(E)),aa(set(F2),set(set(E)),aa(fun(F2,set(E)),fun(set(F2),set(set(E))),image2(F2,set(E)),B6),C5))) = A6 ) )
& ( ( C5 != bot_bot(set(F2)) )
=> ( aa(set(E),set(E),aa(set(E),fun(set(E),set(E)),sup_sup(set(E)),A6),aa(set(set(E)),set(E),complete_Sup_Sup(set(E)),aa(set(F2),set(set(E)),aa(fun(F2,set(E)),fun(set(F2),set(set(E))),image2(F2,set(E)),B6),C5))) = aa(set(set(E)),set(E),complete_Sup_Sup(set(E)),aa(set(F2),set(set(E)),aa(fun(F2,set(E)),fun(set(F2),set(set(E))),image2(F2,set(E)),aa(fun(F2,set(E)),fun(F2,set(E)),aTP_Lamp_fk(set(E),fun(fun(F2,set(E)),fun(F2,set(E))),A6),B6)),C5)) ) ) ) ).
% UN_extend_simps(3)
tff(fact_808_INT__extend__simps_I1_J,axiom,
! [B: $tType,A: $tType,C5: set(A),A6: fun(A,set(B)),B6: set(B)] :
( ( ( C5 = bot_bot(set(A)) )
=> ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),aa(fun(A,set(B)),fun(set(A),set(set(B))),image2(A,set(B)),A6),C5))),B6) = B6 ) )
& ( ( C5 != bot_bot(set(A)) )
=> ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),aa(fun(A,set(B)),fun(set(A),set(set(B))),image2(A,set(B)),A6),C5))),B6) = aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),aa(fun(A,set(B)),fun(set(A),set(set(B))),image2(A,set(B)),aa(set(B),fun(A,set(B)),aTP_Lamp_fc(fun(A,set(B)),fun(set(B),fun(A,set(B))),A6),B6)),C5)) ) ) ) ).
% INT_extend_simps(1)
tff(fact_809_INT__extend__simps_I2_J,axiom,
! [C: $tType,D: $tType,C5: set(D),A6: set(C),B6: fun(D,set(C))] :
( ( ( C5 = bot_bot(set(D)) )
=> ( aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),A6),aa(set(set(C)),set(C),complete_Inf_Inf(set(C)),aa(set(D),set(set(C)),aa(fun(D,set(C)),fun(set(D),set(set(C))),image2(D,set(C)),B6),C5))) = A6 ) )
& ( ( C5 != bot_bot(set(D)) )
=> ( aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),A6),aa(set(set(C)),set(C),complete_Inf_Inf(set(C)),aa(set(D),set(set(C)),aa(fun(D,set(C)),fun(set(D),set(set(C))),image2(D,set(C)),B6),C5))) = aa(set(set(C)),set(C),complete_Inf_Inf(set(C)),aa(set(D),set(set(C)),aa(fun(D,set(C)),fun(set(D),set(set(C))),image2(D,set(C)),aa(fun(D,set(C)),fun(D,set(C)),aTP_Lamp_fb(set(C),fun(fun(D,set(C)),fun(D,set(C))),A6),B6)),C5)) ) ) ) ).
% INT_extend_simps(2)
tff(fact_810_le__cINF__iff,axiom,
! [A: $tType,B: $tType] :
( condit1219197933456340205attice(A)
=> ! [A6: set(B),F: fun(B,A),U: A] :
( ( A6 != bot_bot(set(B)) )
=> ( condit1013018076250108175_below(A,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))))
<=> ! [X5: B] :
( pp(aa(set(B),bool,member(B,X5),A6))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(B,A,F,X5))) ) ) ) ) ) ).
% le_cINF_iff
tff(fact_811_cINF__mono,axiom,
! [C: $tType,A: $tType,B: $tType] :
( condit1219197933456340205attice(A)
=> ! [B6: set(B),F: fun(C,A),A6: set(C),G: fun(B,A)] :
( ( B6 != bot_bot(set(B)) )
=> ( condit1013018076250108175_below(A,aa(set(C),set(A),aa(fun(C,A),fun(set(C),set(A)),image2(C,A),F),A6))
=> ( ! [M3: B] :
( pp(aa(set(B),bool,member(B,M3),B6))
=> ? [X4: C] :
( pp(aa(set(C),bool,member(C,X4),A6))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(C,A,F,X4)),aa(B,A,G,M3))) ) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),aa(fun(C,A),fun(set(C),set(A)),image2(C,A),F),A6))),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),G),B6)))) ) ) ) ) ).
% cINF_mono
tff(fact_812_cSUP__le__iff,axiom,
! [B: $tType,A: $tType] :
( condit1219197933456340205attice(A)
=> ! [A6: set(B),F: fun(B,A),U: A] :
( ( A6 != bot_bot(set(B)) )
=> ( condit941137186595557371_above(A,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))),U))
<=> ! [X5: B] :
( pp(aa(set(B),bool,member(B,X5),A6))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,X5)),U)) ) ) ) ) ) ).
% cSUP_le_iff
tff(fact_813_cSUP__mono,axiom,
! [B: $tType,A: $tType,C: $tType] :
( condit1219197933456340205attice(A)
=> ! [A6: set(B),G: fun(C,A),B6: set(C),F: fun(B,A)] :
( ( A6 != bot_bot(set(B)) )
=> ( condit941137186595557371_above(A,aa(set(C),set(A),aa(fun(C,A),fun(set(C),set(A)),image2(C,A),G),B6))
=> ( ! [N2: B] :
( pp(aa(set(B),bool,member(B,N2),A6))
=> ? [X4: C] :
( pp(aa(set(C),bool,member(C,X4),B6))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,N2)),aa(C,A,G,X4))) ) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))),aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),aa(fun(C,A),fun(set(C),set(A)),image2(C,A),G),B6)))) ) ) ) ) ).
% cSUP_mono
tff(fact_814_prod__set__defs_I1_J,axiom,
! [B: $tType,A: $tType,X4: product_prod(A,B)] : aa(product_prod(A,B),set(A),basic_fsts(A,B),X4) = aa(set(A),set(A),insert(A,aa(product_prod(A,B),A,product_fst(A,B),X4)),bot_bot(set(A))) ).
% prod_set_defs(1)
tff(fact_815_image__INT,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(A,B),C5: set(A),A6: set(C),B6: fun(C,set(A)),J5: C] :
( inj_on(A,B,F,C5)
=> ( ! [X: C] :
( pp(aa(set(C),bool,member(C,X),A6))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(C,set(A),B6,X)),C5)) )
=> ( pp(aa(set(C),bool,member(C,J5),A6))
=> ( aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(C),set(set(A)),aa(fun(C,set(A)),fun(set(C),set(set(A))),image2(C,set(A)),B6),A6))) = aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),aa(fun(C,set(B)),fun(set(C),set(set(B))),image2(C,set(B)),aa(fun(C,set(A)),fun(C,set(B)),aTP_Lamp_gv(fun(A,B),fun(fun(C,set(A)),fun(C,set(B))),F),B6)),A6)) ) ) ) ) ).
% image_INT
tff(fact_816_cINF__superset__mono,axiom,
! [A: $tType,B: $tType] :
( condit1219197933456340205attice(A)
=> ! [A6: set(B),G: fun(B,A),B6: set(B),F: fun(B,A)] :
( ( A6 != bot_bot(set(B)) )
=> ( condit1013018076250108175_below(A,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),G),B6))
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A6),B6))
=> ( ! [X: B] :
( pp(aa(set(B),bool,member(B,X),B6))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,G,X)),aa(B,A,F,X))) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),G),B6))),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6)))) ) ) ) ) ) ).
% cINF_superset_mono
tff(fact_817_cSUP__subset__mono,axiom,
! [A: $tType,B: $tType] :
( condit1219197933456340205attice(A)
=> ! [A6: set(B),G: fun(B,A),B6: set(B),F: fun(B,A)] :
( ( A6 != bot_bot(set(B)) )
=> ( condit941137186595557371_above(A,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),G),B6))
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A6),B6))
=> ( ! [X: B] :
( pp(aa(set(B),bool,member(B,X),A6))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F,X)),aa(B,A,G,X))) )
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),G),B6)))) ) ) ) ) ) ).
% cSUP_subset_mono
tff(fact_818_UNION__fun__upd,axiom,
! [B: $tType,A: $tType,A6: fun(B,set(A)),I: B,B6: set(A),J4: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),fun_upd(B,set(A),A6,I,B6)),J4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),A6),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),J4),aa(set(B),set(B),insert(B,I),bot_bot(set(B))))))),if(set(A),aa(set(B),bool,member(B,I),J4),B6,bot_bot(set(A)))) ).
% UNION_fun_upd
tff(fact_819_image__split__eq__Sigma,axiom,
! [B: $tType,A: $tType,C: $tType,F: fun(C,A),G: fun(C,B),A6: set(C)] : aa(set(C),set(product_prod(A,B)),aa(fun(C,product_prod(A,B)),fun(set(C),set(product_prod(A,B))),image2(C,product_prod(A,B)),aa(fun(C,B),fun(C,product_prod(A,B)),aTP_Lamp_gw(fun(C,A),fun(fun(C,B),fun(C,product_prod(A,B))),F),G)),A6) = product_Sigma(A,B,aa(set(C),set(A),aa(fun(C,A),fun(set(C),set(A)),image2(C,A),F),A6),aa(set(C),fun(A,set(B)),aa(fun(C,B),fun(set(C),fun(A,set(B))),aTP_Lamp_gx(fun(C,A),fun(fun(C,B),fun(set(C),fun(A,set(B)))),F),G),A6)) ).
% image_split_eq_Sigma
tff(fact_820_If__the__inv__into__f__f,axiom,
! [B: $tType,A: $tType,I: A,C5: set(A),G: fun(A,B),X2: A] :
( pp(aa(set(A),bool,member(A,I),C5))
=> ( inj_on(A,B,G,C5)
=> ( aa(A,A,aa(fun(A,B),fun(A,A),aa(fun(B,A),fun(fun(A,B),fun(A,A)),comp(B,A,A),aa(A,fun(B,A),aa(fun(A,B),fun(A,fun(B,A)),aTP_Lamp_gy(set(A),fun(fun(A,B),fun(A,fun(B,A))),C5),G),X2)),G),I) = aa(A,A,id(A),I) ) ) ) ).
% If_the_inv_into_f_f
tff(fact_821_Field__insert,axiom,
! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A))] : aa(set(product_prod(A,A)),set(A),field2(A),aa(set(product_prod(A,A)),set(product_prod(A,A)),insert(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),insert(A,A3),aa(set(A),set(A),insert(A,B2),bot_bot(set(A))))),aa(set(product_prod(A,A)),set(A),field2(A),R)) ).
% Field_insert
tff(fact_822_bdd__below__image__antimono,axiom,
! [B: $tType,A: $tType] :
( ( order(A)
& order(B) )
=> ! [F: fun(A,B),A6: set(A)] :
( order_antimono(A,B,F)
=> ( condit941137186595557371_above(A,A6)
=> condit1013018076250108175_below(B,aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6)) ) ) ) ).
% bdd_below_image_antimono
tff(fact_823_bdd__above__image__antimono,axiom,
! [B: $tType,A: $tType] :
( ( order(A)
& order(B) )
=> ! [F: fun(A,B),A6: set(A)] :
( order_antimono(A,B,F)
=> ( condit1013018076250108175_below(A,A6)
=> condit941137186595557371_above(B,aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6)) ) ) ) ).
% bdd_above_image_antimono
tff(fact_824_the__inv__f__o__f__id,axiom,
! [B: $tType,A: $tType,F: fun(A,B),Z2: A] :
( inj_on(A,B,F,top_top(set(A)))
=> ( aa(A,A,aa(fun(A,B),fun(A,A),aa(fun(B,A),fun(fun(A,B),fun(A,A)),comp(B,A,A),the_inv_into(A,B,top_top(set(A)),F)),F),Z2) = aa(A,A,id(A),Z2) ) ) ).
% the_inv_f_o_f_id
tff(fact_825_If__the__inv__into__in__Func,axiom,
! [B: $tType,A: $tType,G: fun(A,B),C5: set(A),B6: set(A),X2: A] :
( inj_on(A,B,G,C5)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B6),aa(set(A),set(A),insert(A,X2),bot_bot(set(A))))))
=> pp(aa(set(fun(B,A)),bool,member(fun(B,A),aa(A,fun(B,A),aa(set(A),fun(A,fun(B,A)),aTP_Lamp_gz(fun(A,B),fun(set(A),fun(A,fun(B,A))),G),C5),X2)),bNF_Wellorder_Func(B,A,top_top(set(B)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B6),aa(set(A),set(A),insert(A,X2),bot_bot(set(A))))))) ) ) ).
% If_the_inv_into_in_Func
tff(fact_826_mono__cInf,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice(A)
& condit1219197933456340205attice(B) )
=> ! [F: fun(A,B),A6: set(A)] :
( order_mono(A,B,F)
=> ( condit1013018076250108175_below(A,A6)
=> ( ( A6 != bot_bot(set(A)) )
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F,aa(set(A),A,complete_Inf_Inf(A),A6))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6)))) ) ) ) ) ).
% mono_cInf
tff(fact_827_mono__cINF,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( condit1219197933456340205attice(B)
& condit1219197933456340205attice(A) )
=> ! [F: fun(A,B),A6: fun(C,A),I4: set(C)] :
( order_mono(A,B,F)
=> ( condit1013018076250108175_below(A,aa(set(C),set(A),aa(fun(C,A),fun(set(C),set(A)),image2(C,A),A6),I4))
=> ( ( I4 != bot_bot(set(C)) )
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F,aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),aa(fun(C,A),fun(set(C),set(A)),image2(C,A),A6),I4)))),aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),aa(fun(C,B),fun(set(C),set(B)),image2(C,B),aa(fun(C,A),fun(C,B),aTP_Lamp_ha(fun(A,B),fun(fun(C,A),fun(C,B)),F),A6)),I4)))) ) ) ) ) ).
% mono_cINF
tff(fact_828_relcompp__distrib2,axiom,
! [A: $tType,B: $tType,C: $tType,S3: fun(A,fun(C,bool)),T3: fun(A,fun(C,bool)),R3: fun(C,fun(B,bool))] : aa(fun(C,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(C,bool)),fun(fun(C,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,C,B),aa(fun(A,fun(C,bool)),fun(A,fun(C,bool)),aa(fun(A,fun(C,bool)),fun(fun(A,fun(C,bool)),fun(A,fun(C,bool))),sup_sup(fun(A,fun(C,bool))),S3),T3)),R3) = aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),sup_sup(fun(A,fun(B,bool))),aa(fun(C,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(C,bool)),fun(fun(C,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,C,B),S3),R3)),aa(fun(C,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(C,bool)),fun(fun(C,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,C,B),T3),R3)) ).
% relcompp_distrib2
tff(fact_829_relcompp__distrib,axiom,
! [A: $tType,B: $tType,C: $tType,R3: fun(A,fun(C,bool)),S3: fun(C,fun(B,bool)),T3: fun(C,fun(B,bool))] : aa(fun(C,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(C,bool)),fun(fun(C,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,C,B),R3),aa(fun(C,fun(B,bool)),fun(C,fun(B,bool)),aa(fun(C,fun(B,bool)),fun(fun(C,fun(B,bool)),fun(C,fun(B,bool))),sup_sup(fun(C,fun(B,bool))),S3),T3)) = aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),sup_sup(fun(A,fun(B,bool))),aa(fun(C,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(C,bool)),fun(fun(C,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,C,B),R3),S3)),aa(fun(C,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(C,bool)),fun(fun(C,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,C,B),R3),T3)) ).
% relcompp_distrib
tff(fact_830_Field__Union,axiom,
! [A: $tType,R3: set(set(product_prod(A,A)))] : aa(set(product_prod(A,A)),set(A),field2(A),aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),R3)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(product_prod(A,A))),set(set(A)),aa(fun(set(product_prod(A,A)),set(A)),fun(set(set(product_prod(A,A))),set(set(A))),image2(set(product_prod(A,A)),set(A)),field2(A)),R3)) ).
% Field_Union
tff(fact_831_the__inv__into__onto,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A6: set(A)] :
( inj_on(A,B,F,A6)
=> ( aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),the_inv_into(A,B,A6,F)),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6)) = A6 ) ) ).
% the_inv_into_onto
tff(fact_832_image__vimage__eq,axiom,
! [A: $tType,B: $tType,F: fun(B,A),A6: set(A)] : aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),aa(set(A),set(B),vimage(B,A,F),A6)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),top_top(set(B)))) ).
% image_vimage_eq
tff(fact_833_vimage__comp,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(A,B),G: fun(B,C),X2: set(C)] : aa(set(B),set(A),vimage(A,B,F),aa(set(C),set(B),vimage(B,C,G),X2)) = aa(set(C),set(A),vimage(A,C,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),G),F)),X2) ).
% vimage_comp
tff(fact_834_set_Ocompositionality,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(C,B),G: fun(B,A),Set: set(A)] : aa(set(B),set(C),vimage(C,B,F),aa(set(A),set(B),vimage(B,A,G),Set)) = aa(set(A),set(C),vimage(C,A,aa(fun(C,B),fun(C,A),aa(fun(B,A),fun(fun(C,B),fun(C,A)),comp(B,A,C),G),F)),Set) ).
% set.compositionality
tff(fact_835_FieldI2,axiom,
! [A: $tType,I: A,J5: A,R3: set(product_prod(A,A))] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),I),J5)),R3))
=> pp(aa(set(A),bool,member(A,J5),aa(set(product_prod(A,A)),set(A),field2(A),R3))) ) ).
% FieldI2
tff(fact_836_FieldI1,axiom,
! [A: $tType,I: A,J5: A,R3: set(product_prod(A,A))] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),I),J5)),R3))
=> pp(aa(set(A),bool,member(A,I),aa(set(product_prod(A,A)),set(A),field2(A),R3))) ) ).
% FieldI1
tff(fact_837_inf__Int__eq2,axiom,
! [B: $tType,A: $tType,R3: set(product_prod(A,B)),S3: set(product_prod(A,B)),X4: A,Xa4: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),inf_inf(fun(A,fun(B,bool))),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_bg(set(product_prod(A,B)),fun(A,fun(B,bool))),R3)),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_bg(set(product_prod(A,B)),fun(A,fun(B,bool))),S3)),X4),Xa4))
<=> pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa4)),aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),inf_inf(set(product_prod(A,B))),R3),S3))) ) ).
% inf_Int_eq2
tff(fact_838_fun__upd__comp,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(C,B),G: fun(A,C),X2: A,Y2: C] : aa(fun(A,C),fun(A,B),aa(fun(C,B),fun(fun(A,C),fun(A,B)),comp(C,B,A),F),fun_upd(A,C,G,X2,Y2)) = fun_upd(A,B,aa(fun(A,C),fun(A,B),aa(fun(C,B),fun(fun(A,C),fun(A,B)),comp(C,B,A),F),G),X2,aa(C,B,F,Y2)) ).
% fun_upd_comp
tff(fact_839_image__vimage__subset,axiom,
! [B: $tType,A: $tType,F: fun(B,A),A6: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),aa(set(A),set(B),vimage(B,A,F),A6))),A6)) ).
% image_vimage_subset
tff(fact_840_image__subset__iff__subset__vimage,axiom,
! [B: $tType,A: $tType,F: fun(B,A),A6: set(B),B6: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6)),B6))
<=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A6),aa(set(A),set(B),vimage(B,A,F),B6))) ) ).
% image_subset_iff_subset_vimage
tff(fact_841_surj__image__vimage__eq,axiom,
! [B: $tType,A: $tType,F: fun(B,A),A6: set(A)] :
( ( aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),top_top(set(B))) = top_top(set(A)) )
=> ( aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),aa(set(A),set(B),vimage(B,A,F),A6)) = A6 ) ) ).
% surj_image_vimage_eq
tff(fact_842_bdd__below__image__mono,axiom,
! [B: $tType,A: $tType] :
( ( order(A)
& order(B) )
=> ! [F: fun(A,B),A6: set(A)] :
( order_mono(A,B,F)
=> ( condit1013018076250108175_below(A,A6)
=> condit1013018076250108175_below(B,aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6)) ) ) ) ).
% bdd_below_image_mono
tff(fact_843_bdd__above__image__mono,axiom,
! [B: $tType,A: $tType] :
( ( order(A)
& order(B) )
=> ! [F: fun(A,B),A6: set(A)] :
( order_mono(A,B,F)
=> ( condit941137186595557371_above(A,A6)
=> condit941137186595557371_above(B,aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6)) ) ) ) ).
% bdd_above_image_mono
tff(fact_844_vimage__UN,axiom,
! [B: $tType,A: $tType,C: $tType,F: fun(A,B),B6: fun(C,set(B)),A6: set(C)] : aa(set(B),set(A),vimage(A,B,F),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),aa(fun(C,set(B)),fun(set(C),set(set(B))),image2(C,set(B)),B6),A6))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),aa(fun(C,set(A)),fun(set(C),set(set(A))),image2(C,set(A)),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_hb(fun(A,B),fun(fun(C,set(B)),fun(C,set(A))),F),B6)),A6)) ).
% vimage_UN
tff(fact_845_vimage__INT,axiom,
! [B: $tType,A: $tType,C: $tType,F: fun(A,B),B6: fun(C,set(B)),A6: set(C)] : aa(set(B),set(A),vimage(A,B,F),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),aa(fun(C,set(B)),fun(set(C),set(set(B))),image2(C,set(B)),B6),A6))) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(C),set(set(A)),aa(fun(C,set(A)),fun(set(C),set(set(A))),image2(C,set(A)),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_hb(fun(A,B),fun(fun(C,set(B)),fun(C,set(A))),F),B6)),A6)) ).
% vimage_INT
tff(fact_846_sup__Un__eq2,axiom,
! [B: $tType,A: $tType,R3: set(product_prod(A,B)),S3: set(product_prod(A,B)),X4: A,Xa4: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),sup_sup(fun(A,fun(B,bool))),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_bg(set(product_prod(A,B)),fun(A,fun(B,bool))),R3)),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_bg(set(product_prod(A,B)),fun(A,fun(B,bool))),S3)),X4),Xa4))
<=> pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa4)),aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),R3),S3))) ) ).
% sup_Un_eq2
tff(fact_847_vimage__Union,axiom,
! [A: $tType,B: $tType,F: fun(A,B),A6: set(set(B))] : aa(set(B),set(A),vimage(A,B,F),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),A6)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(B)),set(set(A)),aa(fun(set(B),set(A)),fun(set(set(B)),set(set(A))),image2(set(B),set(A)),vimage(A,B,F)),A6)) ).
% vimage_Union
tff(fact_848_vimage__Times,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(A,product_prod(B,C)),A6: set(B),B6: set(C)] : aa(set(product_prod(B,C)),set(A),vimage(A,product_prod(B,C),F),product_Sigma(B,C,A6,aTP_Lamp_hc(set(C),fun(B,set(C)),B6))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(B),set(A),vimage(A,B,aa(fun(A,product_prod(B,C)),fun(A,B),aa(fun(product_prod(B,C),B),fun(fun(A,product_prod(B,C)),fun(A,B)),comp(product_prod(B,C),B,A),product_fst(B,C)),F)),A6)),aa(set(C),set(A),vimage(A,C,aa(fun(A,product_prod(B,C)),fun(A,C),aa(fun(product_prod(B,C),C),fun(fun(A,product_prod(B,C)),fun(A,C)),comp(product_prod(B,C),C,A),product_snd(B,C)),F)),B6)) ).
% vimage_Times
tff(fact_849_inj__on__fun__updI,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A6: set(A),Y2: B,X2: A] :
( inj_on(A,B,F,A6)
=> ( ~ pp(aa(set(B),bool,member(B,Y2),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6)))
=> inj_on(A,B,fun_upd(A,B,F,X2,Y2),A6) ) ) ).
% inj_on_fun_updI
tff(fact_850_set_Ocomp,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(C,B),G: fun(B,A)] : aa(fun(set(A),set(B)),fun(set(A),set(C)),aa(fun(set(B),set(C)),fun(fun(set(A),set(B)),fun(set(A),set(C))),comp(set(B),set(C),set(A)),vimage(C,B,F)),vimage(B,A,G)) = vimage(C,A,aa(fun(C,B),fun(C,A),aa(fun(B,A),fun(fun(C,B),fun(C,A)),comp(B,A,C),G),F)) ).
% set.comp
tff(fact_851_surj__vimage__empty,axiom,
! [B: $tType,A: $tType,F: fun(B,A),A6: set(A)] :
( ( aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),top_top(set(B))) = top_top(set(A)) )
=> ( ( aa(set(A),set(B),vimage(B,A,F),A6) = bot_bot(set(B)) )
<=> ( A6 = bot_bot(set(A)) ) ) ) ).
% surj_vimage_empty
tff(fact_852_vimage__subsetD,axiom,
! [A: $tType,B: $tType,F: fun(B,A),B6: set(A),A6: set(B)] :
( ( aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),top_top(set(B))) = top_top(set(A)) )
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),vimage(B,A,F),B6)),A6))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B6),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))) ) ) ).
% vimage_subsetD
tff(fact_853_Pair__vimage__Sigma,axiom,
! [B: $tType,A: $tType,X2: B,A6: set(B),F: fun(B,set(A))] :
( ( pp(aa(set(B),bool,member(B,X2),A6))
=> ( aa(set(product_prod(B,A)),set(A),vimage(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X2)),product_Sigma(B,A,A6,F)) = aa(B,set(A),F,X2) ) )
& ( ~ pp(aa(set(B),bool,member(B,X2),A6))
=> ( aa(set(product_prod(B,A)),set(A),vimage(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X2)),product_Sigma(B,A,A6,F)) = bot_bot(set(A)) ) ) ) ).
% Pair_vimage_Sigma
tff(fact_854_inj__vimage__image__eq,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A6: set(A)] :
( inj_on(A,B,F,top_top(set(A)))
=> ( aa(set(B),set(A),vimage(A,B,F),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6)) = A6 ) ) ).
% inj_vimage_image_eq
tff(fact_855_inj__graph,axiom,
! [B: $tType,A: $tType] : inj_on(fun(A,B),set(product_prod(A,B)),aTP_Lamp_he(fun(A,B),set(product_prod(A,B))),top_top(set(fun(A,B)))) ).
% inj_graph
tff(fact_856_f__the__inv__into__f,axiom,
! [A: $tType,B: $tType,F: fun(A,B),A6: set(A),Y2: B] :
( inj_on(A,B,F,A6)
=> ( pp(aa(set(B),bool,member(B,Y2),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6)))
=> ( aa(A,B,F,aa(B,A,the_inv_into(A,B,A6,F),Y2)) = Y2 ) ) ) ).
% f_the_inv_into_f
tff(fact_857_inj__on__the__inv__into,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A6: set(A)] :
( inj_on(A,B,F,A6)
=> inj_on(B,A,the_inv_into(A,B,A6,F),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6)) ) ).
% inj_on_the_inv_into
tff(fact_858_vimage__fst,axiom,
! [B: $tType,A: $tType,A6: set(A)] : aa(set(A),set(product_prod(A,B)),vimage(product_prod(A,B),A,product_fst(A,B)),A6) = product_Sigma(A,B,A6,aTP_Lamp_ca(A,set(B))) ).
% vimage_fst
tff(fact_859_vimage__snd,axiom,
! [A: $tType,B: $tType,A6: set(B)] : aa(set(B),set(product_prod(A,B)),vimage(product_prod(A,B),B,product_snd(A,B)),A6) = product_Sigma(A,B,top_top(set(A)),aTP_Lamp_bu(set(B),fun(A,set(B)),A6)) ).
% vimage_snd
tff(fact_860_mono__SUP,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comple6319245703460814977attice(A)
& comple6319245703460814977attice(B) )
=> ! [F: fun(A,B),A6: fun(C,A),I4: set(C)] :
( order_mono(A,B,F)
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),aa(fun(C,B),fun(set(C),set(B)),image2(C,B),aa(fun(C,A),fun(C,B),aTP_Lamp_hf(fun(A,B),fun(fun(C,A),fun(C,B)),F),A6)),I4))),aa(A,B,F,aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),aa(fun(C,A),fun(set(C),set(A)),image2(C,A),A6),I4))))) ) ) ).
% mono_SUP
tff(fact_861_mono__Sup,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice(A)
& comple6319245703460814977attice(B) )
=> ! [F: fun(A,B),A6: set(A)] :
( order_mono(A,B,F)
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6))),aa(A,B,F,aa(set(A),A,complete_Sup_Sup(A),A6)))) ) ) ).
% mono_Sup
tff(fact_862_mono__INF,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comple6319245703460814977attice(B)
& comple6319245703460814977attice(A) )
=> ! [F: fun(A,B),A6: fun(C,A),I4: set(C)] :
( order_mono(A,B,F)
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F,aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),aa(fun(C,A),fun(set(C),set(A)),image2(C,A),A6),I4)))),aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),aa(fun(C,B),fun(set(C),set(B)),image2(C,B),aa(fun(C,A),fun(C,B),aTP_Lamp_hf(fun(A,B),fun(fun(C,A),fun(C,B)),F),A6)),I4)))) ) ) ).
% mono_INF
tff(fact_863_mono__Inf,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice(A)
& comple6319245703460814977attice(B) )
=> ! [F: fun(A,B),A6: set(A)] :
( order_mono(A,B,F)
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F,aa(set(A),A,complete_Inf_Inf(A),A6))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6)))) ) ) ).
% mono_Inf
tff(fact_864_Pow__insert,axiom,
! [A: $tType,A3: A,A6: set(A)] : pow(A,aa(set(A),set(A),insert(A,A3),A6)) = aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),sup_sup(set(set(A))),pow(A,A6)),aa(set(set(A)),set(set(A)),aa(fun(set(A),set(A)),fun(set(set(A)),set(set(A))),image2(set(A),set(A)),insert(A,A3)),pow(A,A6))) ).
% Pow_insert
tff(fact_865_vimage__subsetI,axiom,
! [B: $tType,A: $tType,F: fun(A,B),B6: set(B),A6: set(A)] :
( inj_on(A,B,F,top_top(set(A)))
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B6),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6)))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),vimage(A,B,F),B6)),A6)) ) ) ).
% vimage_subsetI
tff(fact_866_vimage__eq__UN,axiom,
! [A: $tType,B: $tType,F: fun(A,B),B6: set(B)] : aa(set(B),set(A),vimage(A,B,F),B6) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),aTP_Lamp_hg(fun(A,B),fun(B,set(A)),F)),B6)) ).
% vimage_eq_UN
tff(fact_867_the__inv__into__into,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A6: set(A),X2: B,B6: set(A)] :
( inj_on(A,B,F,A6)
=> ( pp(aa(set(B),bool,member(B,X2),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6)))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),B6))
=> pp(aa(set(A),bool,member(A,aa(B,A,the_inv_into(A,B,A6,F),X2)),B6)) ) ) ) ).
% the_inv_into_into
tff(fact_868_the__inv__into__comp,axiom,
! [A: $tType,C: $tType,B: $tType,F: fun(A,B),G: fun(C,A),A6: set(C),X2: B] :
( inj_on(A,B,F,aa(set(C),set(A),aa(fun(C,A),fun(set(C),set(A)),image2(C,A),G),A6))
=> ( inj_on(C,A,G,A6)
=> ( pp(aa(set(B),bool,member(B,X2),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),aa(set(C),set(A),aa(fun(C,A),fun(set(C),set(A)),image2(C,A),G),A6))))
=> ( aa(B,C,the_inv_into(C,B,A6,aa(fun(C,A),fun(C,B),aa(fun(A,B),fun(fun(C,A),fun(C,B)),comp(A,B,C),F),G)),X2) = aa(B,C,aa(fun(B,A),fun(B,C),aa(fun(A,C),fun(fun(B,A),fun(B,C)),comp(A,C,B),the_inv_into(C,A,A6,G)),the_inv_into(A,B,aa(set(C),set(A),aa(fun(C,A),fun(set(C),set(A)),image2(C,A),G),A6),F)),X2) ) ) ) ) ).
% the_inv_into_comp
tff(fact_869_fun__upd__image,axiom,
! [A: $tType,B: $tType,X2: B,A6: set(B),F: fun(B,A),Y2: A] :
( ( pp(aa(set(B),bool,member(B,X2),A6))
=> ( aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),fun_upd(B,A,F,X2,Y2)),A6) = aa(set(A),set(A),insert(A,Y2),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),A6),aa(set(B),set(B),insert(B,X2),bot_bot(set(B)))))) ) )
& ( ~ pp(aa(set(B),bool,member(B,X2),A6))
=> ( aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),fun_upd(B,A,F,X2,Y2)),A6) = aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6) ) ) ) ).
% fun_upd_image
tff(fact_870_mono__cSUP,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( condit1219197933456340205attice(A)
& condit1219197933456340205attice(B) )
=> ! [F: fun(A,B),A6: fun(C,A),I4: set(C)] :
( order_mono(A,B,F)
=> ( condit941137186595557371_above(A,aa(set(C),set(A),aa(fun(C,A),fun(set(C),set(A)),image2(C,A),A6),I4))
=> ( ( I4 != bot_bot(set(C)) )
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),aa(fun(C,B),fun(set(C),set(B)),image2(C,B),aa(fun(C,A),fun(C,B),aTP_Lamp_ha(fun(A,B),fun(fun(C,A),fun(C,B)),F),A6)),I4))),aa(A,B,F,aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),aa(fun(C,A),fun(set(C),set(A)),image2(C,A),A6),I4))))) ) ) ) ) ).
% mono_cSUP
tff(fact_871_mono__cSup,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice(A)
& condit1219197933456340205attice(B) )
=> ! [F: fun(A,B),A6: set(A)] :
( order_mono(A,B,F)
=> ( condit941137186595557371_above(A,A6)
=> ( ( A6 != bot_bot(set(A)) )
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6))),aa(A,B,F,aa(set(A),A,complete_Sup_Sup(A),A6)))) ) ) ) ) ).
% mono_cSup
tff(fact_872_Func__map__surj,axiom,
! [C: $tType,A: $tType,D: $tType,B: $tType,F1: fun(B,A),A12: set(B),B1: set(A),F22: fun(C,D),B22: set(C),A23: set(D)] :
( ( aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F1),A12) = B1 )
=> ( inj_on(C,D,F22,B22)
=> ( pp(aa(set(D),bool,aa(set(D),fun(set(D),bool),ord_less_eq(set(D)),aa(set(C),set(D),aa(fun(C,D),fun(set(C),set(D)),image2(C,D),F22),B22)),A23))
=> ( ( ( B22 = bot_bot(set(C)) )
=> ( A23 = bot_bot(set(D)) ) )
=> ( bNF_Wellorder_Func(C,A,B22,B1) = aa(set(fun(D,B)),set(fun(C,A)),aa(fun(fun(D,B),fun(C,A)),fun(set(fun(D,B)),set(fun(C,A))),image2(fun(D,B),fun(C,A)),bNF_We4925052301507509544nc_map(C,B,A,D,B22,F1,F22)),bNF_Wellorder_Func(D,B,A23,A12)) ) ) ) ) ) ).
% Func_map_surj
tff(fact_873_cofinal__def,axiom,
! [A: $tType,A6: set(A),R: set(product_prod(A,A))] :
( bNF_Ca7293521722713021262ofinal(A,A6,R)
<=> ! [X5: A] :
( pp(aa(set(A),bool,member(A,X5),aa(set(product_prod(A,A)),set(A),field2(A),R)))
=> ? [Xa3: A] :
( pp(aa(set(A),bool,member(A,Xa3),A6))
& ( X5 != Xa3 )
& pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),Xa3)),R)) ) ) ) ).
% cofinal_def
tff(fact_874_Total__subset__Id,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( total_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R),R)
=> ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R),id2(A)))
=> ( ( R = bot_bot(set(product_prod(A,A))) )
| ? [A5: A] : R = aa(set(product_prod(A,A)),set(product_prod(A,A)),insert(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A5),A5)),bot_bot(set(product_prod(A,A)))) ) ) ) ).
% Total_subset_Id
tff(fact_875_inf__img__fin__dom_H,axiom,
! [A: $tType,B: $tType,F: fun(B,A),A6: set(B)] :
( finite_finite2(A,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))
=> ( ~ finite_finite2(B,A6)
=> ? [X: A] :
( pp(aa(set(A),bool,member(A,X),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6)))
& ~ finite_finite2(B,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),vimage(B,A,F),aa(set(A),set(A),insert(A,X),bot_bot(set(A))))),A6)) ) ) ) ).
% inf_img_fin_dom'
tff(fact_876_inf__img__fin__domE_H,axiom,
! [A: $tType,B: $tType,F: fun(B,A),A6: set(B)] :
( finite_finite2(A,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))
=> ( ~ finite_finite2(B,A6)
=> ~ ! [Y: A] :
( pp(aa(set(A),bool,member(A,Y),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6)))
=> finite_finite2(B,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),vimage(B,A,F),aa(set(A),set(A),insert(A,Y),bot_bot(set(A))))),A6)) ) ) ) ).
% inf_img_fin_domE'
tff(fact_877_prod__set__simps_I2_J,axiom,
! [A: $tType,B: $tType,X2: A,Y2: B] : basic_snds(A,B,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y2)) = aa(set(B),set(B),insert(B,Y2),bot_bot(set(B))) ).
% prod_set_simps(2)
tff(fact_878_Linear__order__in__diff__Id,axiom,
! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A] :
( order_679001287576687338der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R),R)
=> ( pp(aa(set(A),bool,member(A,A3),aa(set(product_prod(A,A)),set(A),field2(A),R)))
=> ( pp(aa(set(A),bool,member(A,B2),aa(set(product_prod(A,A)),set(A),field2(A),R)))
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R))
<=> ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A3)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),R),id2(A)))) ) ) ) ) ).
% Linear_order_in_diff_Id
tff(fact_879_aboveS__def,axiom,
! [A: $tType,R: set(product_prod(A,A)),A3: A] : order_aboveS(A,R,A3) = aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_hh(set(product_prod(A,A)),fun(A,fun(A,bool)),R),A3)) ).
% aboveS_def
tff(fact_880_card__vimage__inj,axiom,
! [A: $tType,B: $tType,F: fun(A,B),A6: set(B)] :
( inj_on(A,B,F,top_top(set(A)))
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A6),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),top_top(set(A)))))
=> ( aa(set(A),nat,finite_card(A),aa(set(B),set(A),vimage(A,B,F),A6)) = aa(set(B),nat,finite_card(B),A6) ) ) ) ).
% card_vimage_inj
tff(fact_881_Linear__order__Well__order__iff,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( order_679001287576687338der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R),R)
=> ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R),R)
<=> ! [A10: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A10),aa(set(product_prod(A,A)),set(A),field2(A),R)))
=> ( ( A10 != bot_bot(set(A)) )
=> ? [X5: A] :
( pp(aa(set(A),bool,member(A,X5),A10))
& ! [Xa3: A] :
( pp(aa(set(A),bool,member(A,Xa3),A10))
=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),Xa3)),R)) ) ) ) ) ) ) ).
% Linear_order_Well_order_iff
tff(fact_882_finite__imageI,axiom,
! [B: $tType,A: $tType,F6: set(A),H: fun(A,B)] :
( finite_finite2(A,F6)
=> finite_finite2(B,aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),H),F6)) ) ).
% finite_imageI
tff(fact_883_IdI,axiom,
! [A: $tType,A3: A] : pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A3)),id2(A))) ).
% IdI
tff(fact_884_pair__in__Id__conv,axiom,
! [A: $tType,A3: A,B2: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),id2(A)))
<=> ( A3 = B2 ) ) ).
% pair_in_Id_conv
tff(fact_885_finite__UN,axiom,
! [B: $tType,A: $tType,A6: set(A),B6: fun(A,set(B))] :
( finite_finite2(A,A6)
=> ( finite_finite2(B,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),aa(fun(A,set(B)),fun(set(A),set(set(B))),image2(A,set(B)),B6),A6)))
<=> ! [X5: A] :
( pp(aa(set(A),bool,member(A,X5),A6))
=> finite_finite2(B,aa(A,set(B),B6,X5)) ) ) ) ).
% finite_UN
tff(fact_886_finite__UN__I,axiom,
! [B: $tType,A: $tType,A6: set(A),B6: fun(A,set(B))] :
( finite_finite2(A,A6)
=> ( ! [A5: A] :
( pp(aa(set(A),bool,member(A,A5),A6))
=> finite_finite2(B,aa(A,set(B),B6,A5)) )
=> finite_finite2(B,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),aa(fun(A,set(B)),fun(set(A),set(set(B))),image2(A,set(B)),B6),A6))) ) ) ).
% finite_UN_I
tff(fact_887_finite__INT,axiom,
! [B: $tType,A: $tType,I4: set(A),A6: fun(A,set(B))] :
( ? [X4: A] :
( pp(aa(set(A),bool,member(A,X4),I4))
& finite_finite2(B,aa(A,set(B),A6,X4)) )
=> finite_finite2(B,aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),aa(fun(A,set(B)),fun(set(A),set(set(B))),image2(A,set(B)),A6),I4))) ) ).
% finite_INT
tff(fact_888_bdd__below__UN,axiom,
! [A: $tType,B: $tType] :
( lattice(A)
=> ! [I4: set(B),A6: fun(B,set(A))] :
( finite_finite2(B,I4)
=> ( condit1013018076250108175_below(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),A6),I4)))
<=> ! [X5: B] :
( pp(aa(set(B),bool,member(B,X5),I4))
=> condit1013018076250108175_below(A,aa(B,set(A),A6,X5)) ) ) ) ) ).
% bdd_below_UN
tff(fact_889_bdd__above__UN,axiom,
! [A: $tType,B: $tType] :
( lattice(A)
=> ! [I4: set(B),A6: fun(B,set(A))] :
( finite_finite2(B,I4)
=> ( condit941137186595557371_above(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),A6),I4)))
<=> ! [X5: B] :
( pp(aa(set(B),bool,member(B,X5),I4))
=> condit941137186595557371_above(A,aa(B,set(A),A6,X5)) ) ) ) ) ).
% bdd_above_UN
tff(fact_890_card__image__le,axiom,
! [B: $tType,A: $tType,A6: set(A),F: fun(A,B)] :
( finite_finite2(A,A6)
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(B),nat,finite_card(B),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6))),aa(set(A),nat,finite_card(A),A6))) ) ).
% card_image_le
tff(fact_891_surj__card__le,axiom,
! [B: $tType,A: $tType,A6: set(A),B6: set(B),F: fun(A,B)] :
( finite_finite2(A,A6)
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B6),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6)))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(B),nat,finite_card(B),B6)),aa(set(A),nat,finite_card(A),A6))) ) ) ).
% surj_card_le
tff(fact_892_eq__card__imp__inj__on,axiom,
! [B: $tType,A: $tType,A6: set(A),F: fun(A,B)] :
( finite_finite2(A,A6)
=> ( ( aa(set(B),nat,finite_card(B),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6)) = aa(set(A),nat,finite_card(A),A6) )
=> inj_on(A,B,F,A6) ) ) ).
% eq_card_imp_inj_on
tff(fact_893_inj__on__iff__eq__card,axiom,
! [B: $tType,A: $tType,A6: set(A),F: fun(A,B)] :
( finite_finite2(A,A6)
=> ( inj_on(A,B,F,A6)
<=> ( aa(set(B),nat,finite_card(B),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6)) = aa(set(A),nat,finite_card(A),A6) ) ) ) ).
% inj_on_iff_eq_card
tff(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)
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R))
=> ( pp(aa(set(A),bool,member(A,A3),A6))
& pp(aa(set(A),bool,member(A,B2),A6)) ) ) ) ).
% well_order_on_domain
tff(fact_895_infinite__countable__subset,axiom,
! [A: $tType,S3: set(A)] :
( ~ finite_finite2(A,S3)
=> ? [F3: fun(nat,A)] :
( inj_on(nat,A,F3,top_top(set(nat)))
& pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(nat),set(A),aa(fun(nat,A),fun(set(nat),set(A)),image2(nat,A),F3),top_top(set(nat)))),S3)) ) ) ).
% infinite_countable_subset
tff(fact_896_infinite__iff__countable__subset,axiom,
! [A: $tType,S3: set(A)] :
( ~ finite_finite2(A,S3)
<=> ? [F7: fun(nat,A)] :
( inj_on(nat,A,F7,top_top(set(nat)))
& pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(nat),set(A),aa(fun(nat,A),fun(set(nat),set(A)),image2(nat,A),F7),top_top(set(nat)))),S3)) ) ) ).
% infinite_iff_countable_subset
tff(fact_897_IdE,axiom,
! [A: $tType,P: product_prod(A,A)] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),P),id2(A)))
=> ~ ! [X: A] : P != aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X) ) ).
% IdE
tff(fact_898_IdD,axiom,
! [A: $tType,A3: A,B2: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),id2(A)))
=> ( A3 = B2 ) ) ).
% IdD
tff(fact_899_card__bij__eq,axiom,
! [A: $tType,B: $tType,F: fun(A,B),A6: set(A),B6: set(B),G: fun(B,A)] :
( inj_on(A,B,F,A6)
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6)),B6))
=> ( inj_on(B,A,G,B6)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),G),B6)),A6))
=> ( finite_finite2(A,A6)
=> ( finite_finite2(B,B6)
=> ( aa(set(A),nat,finite_card(A),A6) = aa(set(B),nat,finite_card(B),B6) ) ) ) ) ) ) ) ).
% card_bij_eq
tff(fact_900_card__le__inj,axiom,
! [B: $tType,A: $tType,A6: set(A),B6: set(B)] :
( finite_finite2(A,A6)
=> ( finite_finite2(B,B6)
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A6)),aa(set(B),nat,finite_card(B),B6)))
=> ? [F3: fun(A,B)] :
( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F3),A6)),B6))
& inj_on(A,B,F3,A6) ) ) ) ) ).
% card_le_inj
tff(fact_901_card__inj__on__le,axiom,
! [A: $tType,B: $tType,F: fun(A,B),A6: set(A),B6: set(B)] :
( inj_on(A,B,F,A6)
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6)),B6))
=> ( finite_finite2(B,B6)
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A6)),aa(set(B),nat,finite_card(B),B6))) ) ) ) ).
% card_inj_on_le
tff(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)
=> ( ? [F7: fun(A,B)] :
( inj_on(A,B,F7,A6)
& pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F7),A6)),B6)) )
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A6)),aa(set(B),nat,finite_card(B),B6))) ) ) ) ).
% inj_on_iff_card_le
tff(fact_903_surjective__iff__injective__gen,axiom,
! [B: $tType,A: $tType,S3: set(A),T3: set(B),F: fun(A,B)] :
( finite_finite2(A,S3)
=> ( finite_finite2(B,T3)
=> ( ( aa(set(A),nat,finite_card(A),S3) = aa(set(B),nat,finite_card(B),T3) )
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),S3)),T3))
=> ( ! [X5: B] :
( pp(aa(set(B),bool,member(B,X5),T3))
=> ? [Xa3: A] :
( pp(aa(set(A),bool,member(A,Xa3),S3))
& ( aa(A,B,F,Xa3) = X5 ) ) )
<=> inj_on(A,B,F,S3) ) ) ) ) ) ).
% surjective_iff_injective_gen
tff(fact_904_pigeonhole__infinite,axiom,
! [B: $tType,A: $tType,A6: set(A),F: fun(A,B)] :
( ~ finite_finite2(A,A6)
=> ( finite_finite2(B,aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6))
=> ? [X: A] :
( pp(aa(set(A),bool,member(A,X),A6))
& ~ finite_finite2(A,aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aa(fun(A,B),fun(A,fun(A,bool)),aTP_Lamp_hi(set(A),fun(fun(A,B),fun(A,fun(A,bool))),A6),F),X))) ) ) ) ).
% pigeonhole_infinite
tff(fact_905_snds_Ointros,axiom,
! [B: $tType,A: $tType,P: product_prod(A,B)] : pp(aa(set(B),bool,member(B,aa(product_prod(A,B),B,product_snd(A,B),P)),basic_snds(A,B,P))) ).
% snds.intros
tff(fact_906_snds_Osimps,axiom,
! [B: $tType,A: $tType,A3: B,P: product_prod(A,B)] :
( pp(aa(set(B),bool,member(B,A3),basic_snds(A,B,P)))
<=> ( A3 = aa(product_prod(A,B),B,product_snd(A,B),P) ) ) ).
% snds.simps
tff(fact_907_snds_Ocases,axiom,
! [B: $tType,A: $tType,A3: B,P: product_prod(A,B)] :
( pp(aa(set(B),bool,member(B,A3),basic_snds(A,B,P)))
=> ( A3 = aa(product_prod(A,B),B,product_snd(A,B),P) ) ) ).
% snds.cases
tff(fact_908_card__image,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A6: set(A)] :
( inj_on(A,B,F,A6)
=> ( aa(set(B),nat,finite_card(B),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6)) = aa(set(A),nat,finite_card(A),A6) ) ) ).
% card_image
tff(fact_909_all__finite__subset__image,axiom,
! [A: $tType,B: $tType,F: fun(B,A),A6: set(B),P2: fun(set(A),bool)] :
( ! [B7: set(A)] :
( ( finite_finite2(A,B7)
& pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B7),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))) )
=> pp(aa(set(A),bool,P2,B7)) )
<=> ! [B7: set(B)] :
( ( finite_finite2(B,B7)
& pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B7),A6)) )
=> pp(aa(set(A),bool,P2,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),B7))) ) ) ).
% all_finite_subset_image
tff(fact_910_ex__finite__subset__image,axiom,
! [A: $tType,B: $tType,F: fun(B,A),A6: set(B),P2: fun(set(A),bool)] :
( ? [B7: set(A)] :
( finite_finite2(A,B7)
& pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B7),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6)))
& pp(aa(set(A),bool,P2,B7)) )
<=> ? [B7: set(B)] :
( finite_finite2(B,B7)
& pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B7),A6))
& pp(aa(set(A),bool,P2,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),B7))) ) ) ).
% ex_finite_subset_image
tff(fact_911_finite__subset__image,axiom,
! [A: $tType,B: $tType,B6: set(A),F: fun(B,A),A6: set(B)] :
( finite_finite2(A,B6)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B6),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6)))
=> ? [C4: set(B)] :
( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),C4),A6))
& finite_finite2(B,C4)
& ( B6 = aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),C4) ) ) ) ) ).
% finite_subset_image
tff(fact_912_finite__surj,axiom,
! [A: $tType,B: $tType,A6: set(A),B6: set(B),F: fun(A,B)] :
( finite_finite2(A,A6)
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B6),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6)))
=> finite_finite2(B,B6) ) ) ).
% finite_surj
tff(fact_913_finite__image__iff,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A6: set(A)] :
( inj_on(A,B,F,A6)
=> ( finite_finite2(B,aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6))
<=> finite_finite2(A,A6) ) ) ).
% finite_image_iff
tff(fact_914_finite__imageD,axiom,
! [A: $tType,B: $tType,F: fun(B,A),A6: set(B)] :
( finite_finite2(A,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))
=> ( inj_on(B,A,F,A6)
=> finite_finite2(B,A6) ) ) ).
% finite_imageD
tff(fact_915_finite__range__imageI,axiom,
! [A: $tType,C: $tType,B: $tType,G: fun(B,A),F: fun(A,C)] :
( finite_finite2(A,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),G),top_top(set(B))))
=> finite_finite2(C,aa(set(B),set(C),aa(fun(B,C),fun(set(B),set(C)),image2(B,C),aa(fun(A,C),fun(B,C),aTP_Lamp_hj(fun(B,A),fun(fun(A,C),fun(B,C)),G),F)),top_top(set(B)))) ) ).
% finite_range_imageI
tff(fact_916_Id__fstsnd__eq,axiom,
! [A: $tType] : id2(A) = aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aTP_Lamp_hk(product_prod(A,A),bool)) ).
% Id_fstsnd_eq
tff(fact_917_prod_Oset__map_I2_J,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,F1: fun(A,C),F22: fun(B,D),V: product_prod(A,B)] : basic_snds(C,D,aa(product_prod(A,B),product_prod(C,D),product_map_prod(A,C,B,D,F1,F22),V)) = aa(set(B),set(D),aa(fun(B,D),fun(set(B),set(D)),image2(B,D),F22),basic_snds(A,B,V)) ).
% prod.set_map(2)
tff(fact_918_finite__surj__inj,axiom,
! [A: $tType,A6: set(A),F: fun(A,A)] :
( finite_finite2(A,A6)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),F),A6)))
=> inj_on(A,A,F,A6) ) ) ).
% finite_surj_inj
tff(fact_919_inj__on__finite,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A6: set(A),B6: set(B)] :
( inj_on(A,B,F,A6)
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6)),B6))
=> ( finite_finite2(B,B6)
=> finite_finite2(A,A6) ) ) ) ).
% inj_on_finite
tff(fact_920_endo__inj__surj,axiom,
! [A: $tType,A6: set(A),F: fun(A,A)] :
( finite_finite2(A,A6)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),F),A6)),A6))
=> ( inj_on(A,A,F,A6)
=> ( aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),F),A6) = A6 ) ) ) ) ).
% endo_inj_surj
tff(fact_921_finite__UNIV__surj__inj,axiom,
! [A: $tType,F: fun(A,A)] :
( finite_finite2(A,top_top(set(A)))
=> ( ( aa(set(A),set(A),aa(fun(A,A),fun(set(A),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
tff(fact_922_finite__UNIV__inj__surj,axiom,
! [A: $tType,F: fun(A,A)] :
( finite_finite2(A,top_top(set(A)))
=> ( inj_on(A,A,F,top_top(set(A)))
=> ( aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),F),top_top(set(A))) = top_top(set(A)) ) ) ) ).
% finite_UNIV_inj_surj
tff(fact_923_finite__vimageD,axiom,
! [A: $tType,B: $tType,H: fun(A,B),F6: set(B)] :
( finite_finite2(A,aa(set(B),set(A),vimage(A,B,H),F6))
=> ( ( aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),H),top_top(set(A))) = top_top(set(B)) )
=> finite_finite2(B,F6) ) ) ).
% finite_vimageD
tff(fact_924_prod__set__defs_I2_J,axiom,
! [C: $tType,D: $tType,X4: product_prod(C,D)] : basic_snds(C,D,X4) = aa(set(D),set(D),insert(D,aa(product_prod(C,D),D,product_snd(C,D),X4)),bot_bot(set(D))) ).
% prod_set_defs(2)
tff(fact_925_inf__img__fin__dom,axiom,
! [B: $tType,A: $tType,F: fun(B,A),A6: set(B)] :
( finite_finite2(A,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))
=> ( ~ finite_finite2(B,A6)
=> ? [X: A] :
( pp(aa(set(A),bool,member(A,X),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6)))
& ~ finite_finite2(B,aa(set(A),set(B),vimage(B,A,F),aa(set(A),set(A),insert(A,X),bot_bot(set(A))))) ) ) ) ).
% inf_img_fin_dom
tff(fact_926_inf__img__fin__domE,axiom,
! [B: $tType,A: $tType,F: fun(B,A),A6: set(B)] :
( finite_finite2(A,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))
=> ( ~ finite_finite2(B,A6)
=> ~ ! [Y: A] :
( pp(aa(set(A),bool,member(A,Y),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6)))
=> finite_finite2(B,aa(set(A),set(B),vimage(B,A,F),aa(set(A),set(A),insert(A,Y),bot_bot(set(A))))) ) ) ) ).
% inf_img_fin_domE
tff(fact_927_finite__vimageD_H,axiom,
! [A: $tType,B: $tType,F: fun(A,B),A6: set(B)] :
( finite_finite2(A,aa(set(B),set(A),vimage(A,B,F),A6))
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A6),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),top_top(set(A)))))
=> finite_finite2(B,A6) ) ) ).
% finite_vimageD'
tff(fact_928_Func__map,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,G: fun(A,B),A23: set(A),A12: set(B),F1: fun(B,C),B1: set(C),F22: fun(D,A),B22: set(D)] :
( pp(aa(set(fun(A,B)),bool,member(fun(A,B),G),bNF_Wellorder_Func(A,B,A23,A12)))
=> ( pp(aa(set(C),bool,aa(set(C),fun(set(C),bool),ord_less_eq(set(C)),aa(set(B),set(C),aa(fun(B,C),fun(set(B),set(C)),image2(B,C),F1),A12)),B1))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(D),set(A),aa(fun(D,A),fun(set(D),set(A)),image2(D,A),F22),B22)),A23))
=> pp(aa(set(fun(D,C)),bool,member(fun(D,C),aa(fun(A,B),fun(D,C),bNF_We4925052301507509544nc_map(D,B,C,A,B22,F1,F22),G)),bNF_Wellorder_Func(D,C,B22,B1))) ) ) ) ).
% Func_map
tff(fact_929_reflcl__set__eq,axiom,
! [A: $tType,R: set(product_prod(A,A)),X4: A,Xa4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),sup_sup(fun(A,fun(A,bool))),aTP_Lamp_hl(set(product_prod(A,A)),fun(A,fun(A,bool)),R)),fequal(A)),X4),Xa4))
<=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa4)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),R),id2(A)))) ) ).
% reflcl_set_eq
tff(fact_930_INF__filter__not__bot,axiom,
! [I5: $tType,A: $tType,B6: set(I5),F6: fun(I5,filter(A))] :
( ! [X8: set(I5)] :
( pp(aa(set(I5),bool,aa(set(I5),fun(set(I5),bool),ord_less_eq(set(I5)),X8),B6))
=> ( finite_finite2(I5,X8)
=> ( aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(I5),set(filter(A)),aa(fun(I5,filter(A)),fun(set(I5),set(filter(A))),image2(I5,filter(A)),F6),X8)) != bot_bot(filter(A)) ) ) )
=> ( aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(I5),set(filter(A)),aa(fun(I5,filter(A)),fun(set(I5),set(filter(A))),image2(I5,filter(A)),F6),B6)) != bot_bot(filter(A)) ) ) ).
% INF_filter_not_bot
tff(fact_931_Pow__fold,axiom,
! [A: $tType,A6: set(A)] :
( finite_finite2(A,A6)
=> ( pow(A,A6) = finite_fold(A,set(set(A)),aTP_Lamp_hm(A,fun(set(set(A)),set(set(A)))),aa(set(set(A)),set(set(A)),insert(set(A),bot_bot(set(A))),bot_bot(set(set(A)))),A6) ) ) ).
% Pow_fold
tff(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)
=> ( aa(set(product_prod(B,A)),set(product_prod(B,A)),aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),set(product_prod(B,A))),sup_sup(set(product_prod(B,A))),aa(set(set(product_prod(B,A))),set(product_prod(B,A)),complete_Sup_Sup(set(product_prod(B,A))),aa(set(A),set(set(product_prod(B,A))),aa(fun(A,set(product_prod(B,A))),fun(set(A),set(set(product_prod(B,A)))),image2(A,set(product_prod(B,A))),aTP_Lamp_hn(B,fun(A,set(product_prod(B,A))),X2)),B6))),A6) = finite_fold(A,set(product_prod(B,A)),aTP_Lamp_ho(B,fun(A,fun(set(product_prod(B,A)),set(product_prod(B,A)))),X2),A6,B6) ) ) ).
% fold_union_pair
tff(fact_933_relInvImage__Id__on,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A6: set(A),B6: set(B)] :
( ! [A13: A,A24: A] :
( ( aa(A,B,F,A13) = aa(A,B,F,A24) )
<=> ( A13 = A24 ) )
=> pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),bNF_Gr7122648621184425601vImage(A,B,A6,id_on(B,B6),F)),id2(A))) ) ).
% relInvImage_Id_on
tff(fact_934_Linear__order__wf__diff__Id,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( order_679001287576687338der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R),R)
=> ( wf(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),R),id2(A)))
<=> ! [A10: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A10),aa(set(product_prod(A,A)),set(A),field2(A),R)))
=> ( ( A10 != bot_bot(set(A)) )
=> ? [X5: A] :
( pp(aa(set(A),bool,member(A,X5),A10))
& ! [Xa3: A] :
( pp(aa(set(A),bool,member(A,Xa3),A10))
=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),Xa3)),R)) ) ) ) ) ) ) ).
% Linear_order_wf_diff_Id
tff(fact_935_bsqr__def,axiom,
! [A: $tType,R: set(product_prod(A,A))] : bNF_Wellorder_bsqr(A,R) = aa(fun(product_prod(product_prod(A,A),product_prod(A,A)),bool),set(product_prod(product_prod(A,A),product_prod(A,A))),collect(product_prod(product_prod(A,A),product_prod(A,A))),aa(fun(product_prod(A,A),fun(product_prod(A,A),bool)),fun(product_prod(product_prod(A,A),product_prod(A,A)),bool),product_case_prod(product_prod(A,A),product_prod(A,A),bool),aa(fun(A,fun(A,fun(product_prod(A,A),bool))),fun(product_prod(A,A),fun(product_prod(A,A),bool)),product_case_prod(A,A,fun(product_prod(A,A),bool)),aTP_Lamp_hq(set(product_prod(A,A)),fun(A,fun(A,fun(product_prod(A,A),bool))),R)))) ).
% bsqr_def
tff(fact_936_INF__fold__inf,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [A6: set(B),F: fun(B,A)] :
( finite_finite2(B,A6)
=> ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6)) = finite_fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),aa(fun(A,fun(A,A)),fun(fun(B,A),fun(B,fun(A,A))),comp(A,fun(A,A),B),inf_inf(A)),F),top_top(A),A6) ) ) ) ).
% INF_fold_inf
tff(fact_937_SUP__fold__sup,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [A6: set(B),F: fun(B,A)] :
( finite_finite2(B,A6)
=> ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6)) = finite_fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),aa(fun(A,fun(A,A)),fun(fun(B,A),fun(B,fun(A,A))),comp(A,fun(A,A),B),sup_sup(A)),F),bot_bot(A),A6) ) ) ) ).
% SUP_fold_sup
tff(fact_938_INF__filter__bot__base,axiom,
! [A: $tType,B: $tType,I4: set(A),F6: fun(A,filter(B))] :
( ! [I6: A] :
( pp(aa(set(A),bool,member(A,I6),I4))
=> ! [J3: A] :
( pp(aa(set(A),bool,member(A,J3),I4))
=> ? [X4: A] :
( pp(aa(set(A),bool,member(A,X4),I4))
& pp(aa(filter(B),bool,aa(filter(B),fun(filter(B),bool),ord_less_eq(filter(B)),aa(A,filter(B),F6,X4)),aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),inf_inf(filter(B)),aa(A,filter(B),F6,I6)),aa(A,filter(B),F6,J3)))) ) ) )
=> ( ( aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),aa(fun(A,filter(B)),fun(set(A),set(filter(B))),image2(A,filter(B)),F6),I4)) = bot_bot(filter(B)) )
<=> ? [X5: A] :
( pp(aa(set(A),bool,member(A,X5),I4))
& ( aa(A,filter(B),F6,X5) = bot_bot(filter(B)) ) ) ) ) ).
% INF_filter_bot_base
tff(fact_939_bsqr__max2,axiom,
! [A: $tType,R: set(product_prod(A,A)),A1: A,A22: A,B12: A,B23: A] :
( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R),R)
=> ( pp(aa(set(product_prod(product_prod(A,A),product_prod(A,A))),bool,member(product_prod(product_prod(A,A),product_prod(A,A)),aa(product_prod(A,A),product_prod(product_prod(A,A),product_prod(A,A)),aa(product_prod(A,A),fun(product_prod(A,A),product_prod(product_prod(A,A),product_prod(A,A))),product_Pair(product_prod(A,A),product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),A22)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B12),B23))),bNF_Wellorder_bsqr(A,R)))
=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),bNF_We1388413361240627857o_max2(A,R,A1,A22)),bNF_We1388413361240627857o_max2(A,R,B12,B23))),R)) ) ) ).
% bsqr_max2
tff(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: fun(B,A)] :
( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R1),R22))
=> pp(aa(set(product_prod(B,B)),bool,aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),bool),ord_less_eq(set(product_prod(B,B))),bNF_Gr7122648621184425601vImage(B,A,A6,R1,F)),bNF_Gr7122648621184425601vImage(B,A,A6,R22,F))) ) ).
% relInvImage_mono
tff(fact_941_fold__image,axiom,
! [C: $tType,B: $tType,A: $tType,G: fun(A,B),A6: set(A),F: fun(B,fun(C,C)),Z2: C] :
( inj_on(A,B,G,A6)
=> ( finite_fold(B,C,F,Z2,aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),G),A6)) = finite_fold(A,C,aa(fun(A,B),fun(A,fun(C,C)),aa(fun(B,fun(C,C)),fun(fun(A,B),fun(A,fun(C,C))),comp(B,fun(C,C),A),F),G),Z2,A6) ) ) ).
% fold_image
tff(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)),S3: set(product_prod(A,A))] :
( fun_reduction_pair(A,P2)
=> ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),set(product_prod(A,A)),product_fst(set(product_prod(A,A)),set(product_prod(A,A))),P2)))
=> ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),S3),aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),set(product_prod(A,A)),product_snd(set(product_prod(A,A)),set(product_prod(A,A))),P2)))
=> ( wf(A,S3)
=> wf(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),R3),S3)) ) ) ) ) ).
% reduction_pair_lemma
tff(fact_943_wf__eq__minimal2,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( wf(A,R)
<=> ! [A10: set(A)] :
( ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A10),aa(set(product_prod(A,A)),set(A),field2(A),R)))
& ( A10 != bot_bot(set(A)) ) )
=> ? [X5: A] :
( pp(aa(set(A),bool,member(A,X5),A10))
& ! [Xa3: A] :
( pp(aa(set(A),bool,member(A,Xa3),A10))
=> ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa3),X5)),R)) ) ) ) ) ).
% wf_eq_minimal2
tff(fact_944_image__fold__insert,axiom,
! [B: $tType,A: $tType,A6: set(A),F: fun(A,B)] :
( finite_finite2(A,A6)
=> ( aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6) = finite_fold(A,set(B),aTP_Lamp_hr(fun(A,B),fun(A,fun(set(B),set(B))),F),bot_bot(set(B)),A6) ) ) ).
% image_fold_insert
tff(fact_945_sup__SUP__fold__sup,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [A6: set(B),B6: A,F: fun(B,A)] :
( finite_finite2(B,A6)
=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),B6),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))) = finite_fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),aa(fun(A,fun(A,A)),fun(fun(B,A),fun(B,fun(A,A))),comp(A,fun(A,A),B),sup_sup(A)),F),B6,A6) ) ) ) ).
% sup_SUP_fold_sup
tff(fact_946_inf__INF__fold__inf,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [A6: set(B),B6: A,F: fun(B,A)] :
( finite_finite2(B,A6)
=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),B6),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))) = finite_fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),aa(fun(A,fun(A,A)),fun(fun(B,A),fun(B,fun(A,A))),comp(A,fun(A,A),B),inf_inf(A)),F),B6,A6) ) ) ) ).
% inf_INF_fold_inf
tff(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,aTP_Lamp_bu(set(B),fun(A,set(B)),B6)) = finite_fold(A,set(product_prod(A,B)),aTP_Lamp_ht(set(B),fun(A,fun(set(product_prod(A,B)),set(product_prod(A,B)))),B6),bot_bot(set(product_prod(A,B))),A6) ) ) ) ).
% product_fold
tff(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)),aTP_Lamp_hu(A,fun(set(product_prod(A,A)),set(product_prod(A,A)))),bot_bot(set(product_prod(A,A))),A6) ) ) ).
% Id_on_fold
tff(fact_949_wf__map__prod__image,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,A)),F: fun(A,B)] :
( wf(A,R)
=> ( inj_on(A,B,F,top_top(set(A)))
=> wf(B,aa(set(product_prod(A,A)),set(product_prod(B,B)),aa(fun(product_prod(A,A),product_prod(B,B)),fun(set(product_prod(A,A)),set(product_prod(B,B))),image2(product_prod(A,A),product_prod(B,B)),product_map_prod(A,B,A,B,F,F)),R)) ) ) ).
% wf_map_prod_image
tff(fact_950_wfI,axiom,
! [A: $tType,R: set(product_prod(A,A)),A6: set(A),B6: set(A)] :
( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R),product_Sigma(A,A,A6,aTP_Lamp_co(set(A),fun(A,set(A)),B6))))
=> ( ! [X: A,P4: fun(A,bool)] :
( ! [Xa4: A] :
( ! [Y: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Xa4)),R))
=> pp(aa(A,bool,P4,Y)) )
=> pp(aa(A,bool,P4,Xa4)) )
=> ( pp(aa(set(A),bool,member(A,X),A6))
=> ( pp(aa(set(A),bool,member(A,X),B6))
=> pp(aa(A,bool,P4,X)) ) ) )
=> wf(A,R) ) ) ).
% wfI
tff(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)) )
=> ~ ! [Z3: A] :
( pp(aa(set(A),bool,member(A,Z3),Q2))
=> ~ ! [Y5: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),Z3)),R3))
=> ~ pp(aa(set(A),bool,member(A,Y5),Q2)) ) ) ) ) ).
% wfE_min'
tff(fact_952_relImage__relInvImage,axiom,
! [B: $tType,A: $tType,R3: set(product_prod(A,A)),F: fun(B,A),A6: set(B)] :
( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),product_Sigma(A,A,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6),aa(set(B),fun(A,set(A)),aTP_Lamp_hv(fun(B,A),fun(set(B),fun(A,set(A))),F),A6))))
=> ( bNF_Gr4221423524335903396lImage(B,A,bNF_Gr7122648621184425601vImage(B,A,A6,R3,F),F) = R3 ) ) ).
% relImage_relInvImage
tff(fact_953_same__fst__def,axiom,
! [B: $tType,A: $tType,P2: fun(A,bool),R3: fun(A,set(product_prod(B,B)))] : same_fst(A,B,P2,R3) = aa(fun(product_prod(product_prod(A,B),product_prod(A,B)),bool),set(product_prod(product_prod(A,B),product_prod(A,B))),collect(product_prod(product_prod(A,B),product_prod(A,B))),aa(fun(product_prod(A,B),fun(product_prod(A,B),bool)),fun(product_prod(product_prod(A,B),product_prod(A,B)),bool),product_case_prod(product_prod(A,B),product_prod(A,B),bool),aa(fun(A,fun(B,fun(product_prod(A,B),bool))),fun(product_prod(A,B),fun(product_prod(A,B),bool)),product_case_prod(A,B,fun(product_prod(A,B),bool)),aa(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(product_prod(A,B),bool))),aTP_Lamp_hx(fun(A,bool),fun(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(product_prod(A,B),bool)))),P2),R3)))) ).
% same_fst_def
tff(fact_954_relInvImage__UNIV__relImage,axiom,
! [B: $tType,A: $tType,R3: set(product_prod(A,A)),F: fun(A,B)] : pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),bNF_Gr7122648621184425601vImage(A,B,top_top(set(A)),bNF_Gr4221423524335903396lImage(A,B,R3,F),F))) ).
% relInvImage_UNIV_relImage
tff(fact_955_comp__fun__commute__product__fold,axiom,
! [B: $tType,A: $tType,B6: set(A)] :
( finite_finite2(A,B6)
=> finite6289374366891150609ommute(B,set(product_prod(B,A)),aTP_Lamp_hy(set(A),fun(B,fun(set(product_prod(B,A)),set(product_prod(B,A)))),B6)) ) ).
% comp_fun_commute_product_fold
tff(fact_956_wf__def,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( wf(A,R)
<=> ! [P5: fun(A,bool)] :
( ! [X5: A] :
( ! [Y4: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),X5)),R))
=> pp(aa(A,bool,P5,Y4)) )
=> pp(aa(A,bool,P5,X5)) )
=> ! [X_1: A] : pp(aa(A,bool,P5,X_1)) ) ) ).
% wf_def
tff(fact_957_wfE__min,axiom,
! [A: $tType,R3: set(product_prod(A,A)),X2: A,Q2: set(A)] :
( wf(A,R3)
=> ( pp(aa(set(A),bool,member(A,X2),Q2))
=> ~ ! [Z3: A] :
( pp(aa(set(A),bool,member(A,Z3),Q2))
=> ~ ! [Y5: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),Z3)),R3))
=> ~ pp(aa(set(A),bool,member(A,Y5),Q2)) ) ) ) ) ).
% wfE_min
tff(fact_958_same__fstI,axiom,
! [B: $tType,A: $tType,P2: fun(A,bool),X2: A,Y6: B,Y2: B,R3: fun(A,set(product_prod(B,B)))] :
( pp(aa(A,bool,P2,X2))
=> ( pp(aa(set(product_prod(B,B)),bool,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Y6),Y2)),aa(A,set(product_prod(B,B)),R3,X2)))
=> pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y6)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y2))),same_fst(A,B,P2,R3))) ) ) ).
% same_fstI
tff(fact_959_comp__fun__commute_Ointro,axiom,
! [B: $tType,A: $tType,F: fun(A,fun(B,B))] :
( ! [Y: A,X: A] : aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,Y)),aa(A,fun(B,B),F,X)) = aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X)),aa(A,fun(B,B),F,Y))
=> finite6289374366891150609ommute(A,B,F) ) ).
% comp_fun_commute.intro
tff(fact_960_comp__fun__commute_Ocomp__fun__commute,axiom,
! [B: $tType,A: $tType,F: fun(A,fun(B,B)),Y2: A,X2: A] :
( finite6289374366891150609ommute(A,B,F)
=> ( aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,Y2)),aa(A,fun(B,B),F,X2)) = aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X2)),aa(A,fun(B,B),F,Y2)) ) ) ).
% comp_fun_commute.comp_fun_commute
tff(fact_961_comp__fun__commute__def,axiom,
! [B: $tType,A: $tType,F: fun(A,fun(B,B))] :
( finite6289374366891150609ommute(A,B,F)
<=> ! [Y4: A,X5: A] : aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,Y4)),aa(A,fun(B,B),F,X5)) = aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X5)),aa(A,fun(B,B),F,Y4)) ) ).
% comp_fun_commute_def
tff(fact_962_comp__fun__commute_Ocomp__comp__fun__commute,axiom,
! [B: $tType,A: $tType,C: $tType,F: fun(A,fun(B,B)),G: fun(C,A)] :
( finite6289374366891150609ommute(A,B,F)
=> finite6289374366891150609ommute(C,B,aa(fun(C,A),fun(C,fun(B,B)),aa(fun(A,fun(B,B)),fun(fun(C,A),fun(C,fun(B,B))),comp(A,fun(B,B),C),F),G)) ) ).
% comp_fun_commute.comp_comp_fun_commute
tff(fact_963_relImage__mono,axiom,
! [B: $tType,A: $tType,R1: set(product_prod(A,A)),R22: set(product_prod(A,A)),F: fun(A,B)] :
( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R1),R22))
=> pp(aa(set(product_prod(B,B)),bool,aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),bool),ord_less_eq(set(product_prod(B,B))),bNF_Gr4221423524335903396lImage(A,B,R1,F)),bNF_Gr4221423524335903396lImage(A,B,R22,F))) ) ).
% relImage_mono
tff(fact_964_comp__fun__commute__relcomp__fold,axiom,
! [C: $tType,B: $tType,A: $tType,S3: set(product_prod(A,B))] :
( finite_finite2(product_prod(A,B),S3)
=> finite6289374366891150609ommute(product_prod(C,A),set(product_prod(C,B)),aa(fun(C,fun(A,fun(set(product_prod(C,B)),set(product_prod(C,B))))),fun(product_prod(C,A),fun(set(product_prod(C,B)),set(product_prod(C,B)))),product_case_prod(C,A,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aTP_Lamp_ia(set(product_prod(A,B)),fun(C,fun(A,fun(set(product_prod(C,B)),set(product_prod(C,B))))),S3))) ) ).
% comp_fun_commute_relcomp_fold
tff(fact_965_wf__induct__rule,axiom,
! [A: $tType,R: set(product_prod(A,A)),P2: fun(A,bool),A3: A] :
( wf(A,R)
=> ( ! [X: A] :
( ! [Y5: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),X)),R))
=> pp(aa(A,bool,P2,Y5)) )
=> pp(aa(A,bool,P2,X)) )
=> pp(aa(A,bool,P2,A3)) ) ) ).
% wf_induct_rule
tff(fact_966_wf__eq__minimal,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( wf(A,R)
<=> ! [Q4: set(A)] :
( ? [X5: A] : pp(aa(set(A),bool,member(A,X5),Q4))
=> ? [X5: A] :
( pp(aa(set(A),bool,member(A,X5),Q4))
& ! [Y4: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),X5)),R))
=> ~ pp(aa(set(A),bool,member(A,Y4),Q4)) ) ) ) ) ).
% wf_eq_minimal
tff(fact_967_wf__not__refl,axiom,
! [A: $tType,R: set(product_prod(A,A)),A3: A] :
( wf(A,R)
=> ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A3)),R)) ) ).
% wf_not_refl
tff(fact_968_wf__not__sym,axiom,
! [A: $tType,R: set(product_prod(A,A)),A3: A,X2: A] :
( wf(A,R)
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),X2)),R))
=> ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),A3)),R)) ) ) ).
% wf_not_sym
tff(fact_969_wf__irrefl,axiom,
! [A: $tType,R: set(product_prod(A,A)),A3: A] :
( wf(A,R)
=> ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A3)),R)) ) ).
% wf_irrefl
tff(fact_970_wf__induct,axiom,
! [A: $tType,R: set(product_prod(A,A)),P2: fun(A,bool),A3: A] :
( wf(A,R)
=> ( ! [X: A] :
( ! [Y5: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),X)),R))
=> pp(aa(A,bool,P2,Y5)) )
=> pp(aa(A,bool,P2,X)) )
=> pp(aa(A,bool,P2,A3)) ) ) ).
% wf_induct
tff(fact_971_wf__asym,axiom,
! [A: $tType,R: set(product_prod(A,A)),A3: A,X2: A] :
( wf(A,R)
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),X2)),R))
=> ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),A3)),R)) ) ) ).
% wf_asym
tff(fact_972_wfUNIVI,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( ! [P4: fun(A,bool),X: A] :
( ! [Xa4: A] :
( ! [Y: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Xa4)),R))
=> pp(aa(A,bool,P4,Y)) )
=> pp(aa(A,bool,P4,Xa4)) )
=> pp(aa(A,bool,P4,X)) )
=> wf(A,R) ) ).
% wfUNIVI
tff(fact_973_wfI__min,axiom,
! [A: $tType,R3: set(product_prod(A,A))] :
( ! [X: A,Q5: set(A)] :
( pp(aa(set(A),bool,member(A,X),Q5))
=> ? [Xa4: A] :
( pp(aa(set(A),bool,member(A,Xa4),Q5))
& ! [Y: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Xa4)),R3))
=> ~ pp(aa(set(A),bool,member(A,Y),Q5)) ) ) )
=> wf(A,R3) ) ).
% wfI_min
tff(fact_974_lex__prod__def,axiom,
! [A: $tType,B: $tType,Ra2: set(product_prod(A,A)),Rb2: set(product_prod(B,B))] : lex_prod(A,B,Ra2,Rb2) = aa(fun(product_prod(product_prod(A,B),product_prod(A,B)),bool),set(product_prod(product_prod(A,B),product_prod(A,B))),collect(product_prod(product_prod(A,B),product_prod(A,B))),aa(fun(product_prod(A,B),fun(product_prod(A,B),bool)),fun(product_prod(product_prod(A,B),product_prod(A,B)),bool),product_case_prod(product_prod(A,B),product_prod(A,B),bool),aa(fun(A,fun(B,fun(product_prod(A,B),bool))),fun(product_prod(A,B),fun(product_prod(A,B),bool)),product_case_prod(A,B,fun(product_prod(A,B),bool)),aa(set(product_prod(B,B)),fun(A,fun(B,fun(product_prod(A,B),bool))),aTP_Lamp_ic(set(product_prod(A,A)),fun(set(product_prod(B,B)),fun(A,fun(B,fun(product_prod(A,B),bool)))),Ra2),Rb2)))) ).
% lex_prod_def
tff(fact_975_in__lex__prod,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,A4: A,B3: B,R: set(product_prod(A,A)),S: set(product_prod(B,B))] :
( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3))),lex_prod(A,B,R,S)))
<=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A4)),R))
| ( ( A3 = A4 )
& pp(aa(set(product_prod(B,B)),bool,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),B2),B3)),S)) ) ) ) ).
% in_lex_prod
tff(fact_976_dependent__wf__choice,axiom,
! [B: $tType,A: $tType,R3: set(product_prod(A,A)),P2: fun(fun(A,B),fun(A,fun(B,bool)))] :
( wf(A,R3)
=> ( ! [F3: fun(A,B),G3: fun(A,B),X: A,R4: B] :
( ! [Z6: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z6),X)),R3))
=> ( aa(A,B,F3,Z6) = aa(A,B,G3,Z6) ) )
=> ( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,B),fun(A,fun(B,bool)),P2,F3),X),R4))
<=> pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,B),fun(A,fun(B,bool)),P2,G3),X),R4)) ) )
=> ( ! [X: A,F3: fun(A,B)] :
( ! [Y5: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),X)),R3))
=> pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,B),fun(A,fun(B,bool)),P2,F3),Y5),aa(A,B,F3,Y5))) )
=> ? [X_12: B] : pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,B),fun(A,fun(B,bool)),P2,F3),X),X_12)) )
=> ? [F3: fun(A,B)] :
! [X4: A] : pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,B),fun(A,fun(B,bool)),P2,F3),X4),aa(A,B,F3,X4))) ) ) ) ).
% dependent_wf_choice
tff(fact_977_reduction__pair__def,axiom,
! [A: $tType,P2: product_prod(set(product_prod(A,A)),set(product_prod(A,A)))] :
( fun_reduction_pair(A,P2)
<=> ( wf(A,aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),set(product_prod(A,A)),product_fst(set(product_prod(A,A)),set(product_prod(A,A))),P2))
& pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),set(product_prod(A,A)),product_fst(set(product_prod(A,A)),set(product_prod(A,A))),P2),aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),set(product_prod(A,A)),product_snd(set(product_prod(A,A)),set(product_prod(A,A))),P2))),aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),set(product_prod(A,A)),product_fst(set(product_prod(A,A)),set(product_prod(A,A))),P2))) ) ) ).
% reduction_pair_def
tff(fact_978_INF__principal__finite,axiom,
! [B: $tType,A: $tType,X6: set(A),F: fun(A,set(B))] :
( finite_finite2(A,X6)
=> ( aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),aa(fun(A,filter(B)),fun(set(A),set(filter(B))),image2(A,filter(B)),aTP_Lamp_id(fun(A,set(B)),fun(A,filter(B)),F)),X6)) = principal(B,aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),aa(fun(A,set(B)),fun(set(A),set(set(B))),image2(A,set(B)),F),X6))) ) ) ).
% INF_principal_finite
tff(fact_979_insert__relcomp__union__fold,axiom,
! [C: $tType,B: $tType,A: $tType,S3: set(product_prod(A,B)),X2: product_prod(C,A),X6: set(product_prod(C,B))] :
( finite_finite2(product_prod(A,B),S3)
=> ( aa(set(product_prod(C,B)),set(product_prod(C,B)),aa(set(product_prod(C,B)),fun(set(product_prod(C,B)),set(product_prod(C,B))),sup_sup(set(product_prod(C,B))),relcomp(C,A,B,aa(set(product_prod(C,A)),set(product_prod(C,A)),insert(product_prod(C,A),X2),bot_bot(set(product_prod(C,A)))),S3)),X6) = finite_fold(product_prod(A,B),set(product_prod(C,B)),aa(fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),fun(product_prod(A,B),fun(set(product_prod(C,B)),set(product_prod(C,B)))),product_case_prod(A,B,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aTP_Lamp_ie(product_prod(C,A),fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),X2)),X6,S3) ) ) ).
% insert_relcomp_union_fold
tff(fact_980_max__ext_Osimps,axiom,
! [A: $tType,A1: set(A),A22: set(A),R3: set(product_prod(A,A))] :
( pp(aa(set(product_prod(set(A),set(A))),bool,member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),A1),A22)),max_ext(A,R3)))
<=> ( finite_finite2(A,A1)
& finite_finite2(A,A22)
& ( A22 != bot_bot(set(A)) )
& ! [X5: A] :
( pp(aa(set(A),bool,member(A,X5),A1))
=> ? [Xa3: A] :
( pp(aa(set(A),bool,member(A,Xa3),A22))
& pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),Xa3)),R3)) ) ) ) ) ).
% max_ext.simps
tff(fact_981_max__ext_Ocases,axiom,
! [A: $tType,A1: set(A),A22: set(A),R3: set(product_prod(A,A))] :
( pp(aa(set(product_prod(set(A),set(A))),bool,member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),A1),A22)),max_ext(A,R3)))
=> ~ ( finite_finite2(A,A1)
=> ( finite_finite2(A,A22)
=> ( ( A22 != bot_bot(set(A)) )
=> ~ ! [X4: A] :
( pp(aa(set(A),bool,member(A,X4),A1))
=> ? [Xa: A] :
( pp(aa(set(A),bool,member(A,Xa),A22))
& pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa)),R3)) ) ) ) ) ) ) ).
% max_ext.cases
tff(fact_982_SUP__principal,axiom,
! [A: $tType,B: $tType,A6: fun(B,set(A)),I4: set(B)] : aa(set(filter(A)),filter(A),complete_Sup_Sup(filter(A)),aa(set(B),set(filter(A)),aa(fun(B,filter(A)),fun(set(B),set(filter(A))),image2(B,filter(A)),aTP_Lamp_if(fun(B,set(A)),fun(B,filter(A)),A6)),I4)) = principal(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),A6),I4))) ).
% SUP_principal
tff(fact_983_reduction__pairI,axiom,
! [A: $tType,R3: set(product_prod(A,A)),S3: set(product_prod(A,A))] :
( wf(A,R3)
=> ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,R3,S3)),R3))
=> fun_reduction_pair(A,aa(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(A,A))),R3),S3)) ) ) ).
% reduction_pairI
tff(fact_984_max__ext__compat,axiom,
! [A: $tType,R3: set(product_prod(A,A)),S3: set(product_prod(A,A))] :
( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,R3,S3)),R3))
=> pp(aa(set(product_prod(set(A),set(A))),bool,aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(A),set(A))),bool),ord_less_eq(set(product_prod(set(A),set(A)))),relcomp(set(A),set(A),set(A),max_ext(A,R3),aa(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A))),aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A)))),sup_sup(set(product_prod(set(A),set(A)))),max_ext(A,S3)),aa(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A))),insert(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),bot_bot(set(A))),bot_bot(set(A)))),bot_bot(set(product_prod(set(A),set(A)))))))),max_ext(A,R3))) ) ).
% max_ext_compat
tff(fact_985_relcompEpair,axiom,
! [A: $tType,B: $tType,C: $tType,A3: A,C3: B,R: set(product_prod(A,C)),S: set(product_prod(C,B))] :
( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),C3)),relcomp(A,C,B,R,S)))
=> ~ ! [B4: C] :
( pp(aa(set(product_prod(A,C)),bool,member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A3),B4)),R))
=> ~ pp(aa(set(product_prod(C,B)),bool,member(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),B4),C3)),S)) ) ) ).
% relcompEpair
tff(fact_986_relcompE,axiom,
! [A: $tType,B: $tType,C: $tType,Xz: product_prod(A,B),R: set(product_prod(A,C)),S: set(product_prod(C,B))] :
( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),Xz),relcomp(A,C,B,R,S)))
=> ~ ! [X: A,Y: C,Z3: B] :
( ( Xz = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Z3) )
=> ( pp(aa(set(product_prod(A,C)),bool,member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),X),Y)),R))
=> ~ pp(aa(set(product_prod(C,B)),bool,member(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),Y),Z3)),S)) ) ) ) ).
% relcompE
tff(fact_987_relcomp_OrelcompI,axiom,
! [A: $tType,C: $tType,B: $tType,A3: A,B2: B,R: set(product_prod(A,B)),C3: C,S: set(product_prod(B,C))] :
( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),R))
=> ( pp(aa(set(product_prod(B,C)),bool,member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B2),C3)),S))
=> pp(aa(set(product_prod(A,C)),bool,member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A3),C3)),relcomp(A,B,C,R,S))) ) ) ).
% relcomp.relcompI
tff(fact_988_relcomp_Osimps,axiom,
! [A: $tType,C: $tType,B: $tType,A1: A,A22: C,R: set(product_prod(A,B)),S: set(product_prod(B,C))] :
( pp(aa(set(product_prod(A,C)),bool,member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A1),A22)),relcomp(A,B,C,R,S)))
<=> ? [A7: A,B5: B,C6: C] :
( ( A1 = A7 )
& ( A22 = C6 )
& pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A7),B5)),R))
& pp(aa(set(product_prod(B,C)),bool,member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B5),C6)),S)) ) ) ).
% relcomp.simps
tff(fact_989_relcomp_Ocases,axiom,
! [A: $tType,C: $tType,B: $tType,A1: A,A22: C,R: set(product_prod(A,B)),S: set(product_prod(B,C))] :
( pp(aa(set(product_prod(A,C)),bool,member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A1),A22)),relcomp(A,B,C,R,S)))
=> ~ ! [B4: B] :
( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A1),B4)),R))
=> ~ pp(aa(set(product_prod(B,C)),bool,member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B4),A22)),S)) ) ) ).
% relcomp.cases
tff(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
tff(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,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),B6),C5)) = bot_bot(set(product_prod(A,A))) ) ) ) ).
% union_comp_emptyR
tff(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,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),A6),B6),C5) = bot_bot(set(product_prod(A,A))) ) ) ) ).
% union_comp_emptyL
tff(fact_993_comp__fun__commute__Image__fold,axiom,
! [B: $tType,A: $tType,S3: set(A)] : finite6289374366891150609ommute(product_prod(A,B),set(B),aa(fun(A,fun(B,fun(set(B),set(B)))),fun(product_prod(A,B),fun(set(B),set(B))),product_case_prod(A,B,fun(set(B),set(B))),aTP_Lamp_ig(set(A),fun(A,fun(B,fun(set(B),set(B)))),S3))) ).
% comp_fun_commute_Image_fold
tff(fact_994_relcompp__relcomp__eq,axiom,
! [A: $tType,C: $tType,B: $tType,R: set(product_prod(A,B)),S: set(product_prod(B,C)),X4: A,Xa4: C] :
( pp(aa(C,bool,aa(A,fun(C,bool),aa(fun(B,fun(C,bool)),fun(A,fun(C,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(C,bool)),fun(A,fun(C,bool))),relcompp(A,B,C),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_bg(set(product_prod(A,B)),fun(A,fun(B,bool))),R)),aTP_Lamp_ih(set(product_prod(B,C)),fun(B,fun(C,bool)),S)),X4),Xa4))
<=> pp(aa(set(product_prod(A,C)),bool,member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),X4),Xa4)),relcomp(A,B,C,R,S))) ) ).
% relcompp_relcomp_eq
tff(fact_995_relcompp__in__rel,axiom,
! [A: $tType,B: $tType,C: $tType,R3: set(product_prod(A,C)),S3: set(product_prod(C,B))] : aa(fun(C,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(C,bool)),fun(fun(C,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,C,B),fun_in_rel(A,C,R3)),fun_in_rel(C,B,S3)) = fun_in_rel(A,B,relcomp(A,C,B,R3,S3)) ).
% relcompp_in_rel
tff(fact_996_relcomp__UNION__distrib,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType,S: set(product_prod(A,C)),R: fun(D,set(product_prod(C,B))),I4: set(D)] : relcomp(A,C,B,S,aa(set(set(product_prod(C,B))),set(product_prod(C,B)),complete_Sup_Sup(set(product_prod(C,B))),aa(set(D),set(set(product_prod(C,B))),aa(fun(D,set(product_prod(C,B))),fun(set(D),set(set(product_prod(C,B)))),image2(D,set(product_prod(C,B))),R),I4))) = aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(D),set(set(product_prod(A,B))),aa(fun(D,set(product_prod(A,B))),fun(set(D),set(set(product_prod(A,B)))),image2(D,set(product_prod(A,B))),aa(fun(D,set(product_prod(C,B))),fun(D,set(product_prod(A,B))),aTP_Lamp_ii(set(product_prod(A,C)),fun(fun(D,set(product_prod(C,B))),fun(D,set(product_prod(A,B)))),S),R)),I4)) ).
% relcomp_UNION_distrib
tff(fact_997_relcomp__UNION__distrib2,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType,R: fun(D,set(product_prod(A,C))),I4: set(D),S: set(product_prod(C,B))] : relcomp(A,C,B,aa(set(set(product_prod(A,C))),set(product_prod(A,C)),complete_Sup_Sup(set(product_prod(A,C))),aa(set(D),set(set(product_prod(A,C))),aa(fun(D,set(product_prod(A,C))),fun(set(D),set(set(product_prod(A,C)))),image2(D,set(product_prod(A,C))),R),I4)),S) = aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(D),set(set(product_prod(A,B))),aa(fun(D,set(product_prod(A,B))),fun(set(D),set(set(product_prod(A,B)))),image2(D,set(product_prod(A,B))),aa(set(product_prod(C,B)),fun(D,set(product_prod(A,B))),aTP_Lamp_ij(fun(D,set(product_prod(A,C))),fun(set(product_prod(C,B)),fun(D,set(product_prod(A,B)))),R),S)),I4)) ).
% relcomp_UNION_distrib2
tff(fact_998_relcomp__def,axiom,
! [A: $tType,C: $tType,B: $tType,X4: set(product_prod(A,B)),Xa4: set(product_prod(B,C))] : relcomp(A,B,C,X4,Xa4) = aa(fun(product_prod(A,C),bool),set(product_prod(A,C)),collect(product_prod(A,C)),aa(fun(A,fun(C,bool)),fun(product_prod(A,C),bool),product_case_prod(A,C,bool),aa(fun(B,fun(C,bool)),fun(A,fun(C,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(C,bool)),fun(A,fun(C,bool))),relcompp(A,B,C),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_bg(set(product_prod(A,B)),fun(A,fun(B,bool))),X4)),aTP_Lamp_ih(set(product_prod(B,C)),fun(B,fun(C,bool)),Xa4)))) ).
% relcomp_def
tff(fact_999_relcomp__fold,axiom,
! [C: $tType,B: $tType,A: $tType,R3: set(product_prod(A,B)),S3: set(product_prod(B,C))] :
( finite_finite2(product_prod(A,B),R3)
=> ( finite_finite2(product_prod(B,C),S3)
=> ( relcomp(A,B,C,R3,S3) = finite_fold(product_prod(A,B),set(product_prod(A,C)),aa(fun(A,fun(B,fun(set(product_prod(A,C)),set(product_prod(A,C))))),fun(product_prod(A,B),fun(set(product_prod(A,C)),set(product_prod(A,C)))),product_case_prod(A,B,fun(set(product_prod(A,C)),set(product_prod(A,C)))),aTP_Lamp_il(set(product_prod(B,C)),fun(A,fun(B,fun(set(product_prod(A,C)),set(product_prod(A,C))))),S3)),bot_bot(set(product_prod(A,C))),R3) ) ) ) ).
% relcomp_fold
tff(fact_1000_insert__relcomp__fold,axiom,
! [C: $tType,B: $tType,A: $tType,S3: set(product_prod(A,B)),X2: product_prod(C,A),R3: set(product_prod(C,A))] :
( finite_finite2(product_prod(A,B),S3)
=> ( relcomp(C,A,B,aa(set(product_prod(C,A)),set(product_prod(C,A)),insert(product_prod(C,A),X2),R3),S3) = finite_fold(product_prod(A,B),set(product_prod(C,B)),aa(fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),fun(product_prod(A,B),fun(set(product_prod(C,B)),set(product_prod(C,B)))),product_case_prod(A,B,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aTP_Lamp_ie(product_prod(C,A),fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),X2)),relcomp(C,A,B,R3,S3),S3) ) ) ).
% insert_relcomp_fold
tff(fact_1001_comp__fun__commute__Pow__fold,axiom,
! [A: $tType] : finite6289374366891150609ommute(A,set(set(A)),aTP_Lamp_hm(A,fun(set(set(A)),set(set(A))))) ).
% comp_fun_commute_Pow_fold
tff(fact_1002_max__ext_Omax__extI,axiom,
! [A: $tType,X6: set(A),Y3: set(A),R3: set(product_prod(A,A))] :
( finite_finite2(A,X6)
=> ( finite_finite2(A,Y3)
=> ( ( Y3 != bot_bot(set(A)) )
=> ( ! [X: A] :
( pp(aa(set(A),bool,member(A,X),X6))
=> ? [Xa4: A] :
( pp(aa(set(A),bool,member(A,Xa4),Y3))
& pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Xa4)),R3)) ) )
=> pp(aa(set(product_prod(set(A),set(A))),bool,member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),X6),Y3)),max_ext(A,R3))) ) ) ) ) ).
% max_ext.max_extI
tff(fact_1003_min__ext__compat,axiom,
! [A: $tType,R3: set(product_prod(A,A)),S3: set(product_prod(A,A))] :
( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,R3,S3)),R3))
=> pp(aa(set(product_prod(set(A),set(A))),bool,aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(A),set(A))),bool),ord_less_eq(set(product_prod(set(A),set(A)))),relcomp(set(A),set(A),set(A),min_ext(A,R3),aa(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A))),aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A)))),sup_sup(set(product_prod(set(A),set(A)))),min_ext(A,S3)),aa(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A))),insert(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),bot_bot(set(A))),bot_bot(set(A)))),bot_bot(set(product_prod(set(A),set(A)))))))),min_ext(A,R3))) ) ).
% min_ext_compat
tff(fact_1004_max__extp__max__ext__eq,axiom,
! [A: $tType,R3: set(product_prod(A,A)),X4: set(A),Xa4: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),max_extp(A,aTP_Lamp_hl(set(product_prod(A,A)),fun(A,fun(A,bool)),R3)),X4),Xa4))
<=> pp(aa(set(product_prod(set(A),set(A))),bool,member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),X4),Xa4)),max_ext(A,R3))) ) ).
% max_extp_max_ext_eq
tff(fact_1005_relInvImage__Gr,axiom,
! [A: $tType,B: $tType,R3: set(product_prod(A,A)),B6: set(A),A6: set(B),F: fun(B,A)] :
( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),product_Sigma(A,A,B6,aTP_Lamp_co(set(A),fun(A,set(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
tff(fact_1006_relImage__Gr,axiom,
! [B: $tType,A: $tType,R3: set(product_prod(A,A)),A6: set(A),F: fun(A,B)] :
( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),product_Sigma(A,A,A6,aTP_Lamp_co(set(A),fun(A,set(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
tff(fact_1007_Image__fold,axiom,
! [B: $tType,A: $tType,R3: set(product_prod(A,B)),S3: set(A)] :
( finite_finite2(product_prod(A,B),R3)
=> ( aa(set(A),set(B),image(A,B,R3),S3) = finite_fold(product_prod(A,B),set(B),aa(fun(A,fun(B,fun(set(B),set(B)))),fun(product_prod(A,B),fun(set(B),set(B))),product_case_prod(A,B,fun(set(B),set(B))),aTP_Lamp_ig(set(A),fun(A,fun(B,fun(set(B),set(B)))),S3)),bot_bot(set(B)),R3) ) ) ).
% Image_fold
tff(fact_1008_filterlim__base__iff,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,I4: set(A),F6: fun(A,set(B)),F: fun(B,C),G6: fun(D,set(C)),J4: set(D)] :
( ( I4 != bot_bot(set(A)) )
=> ( ! [I6: A] :
( pp(aa(set(A),bool,member(A,I6),I4))
=> ! [J3: A] :
( pp(aa(set(A),bool,member(A,J3),I4))
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),F6,I6)),aa(A,set(B),F6,J3)))
| pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),F6,J3)),aa(A,set(B),F6,I6))) ) ) )
=> ( filterlim(B,C,F,aa(set(filter(C)),filter(C),complete_Inf_Inf(filter(C)),aa(set(D),set(filter(C)),aa(fun(D,filter(C)),fun(set(D),set(filter(C))),image2(D,filter(C)),aTP_Lamp_im(fun(D,set(C)),fun(D,filter(C)),G6)),J4)),aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),aa(fun(A,filter(B)),fun(set(A),set(filter(B))),image2(A,filter(B)),aTP_Lamp_id(fun(A,set(B)),fun(A,filter(B)),F6)),I4)))
<=> ! [X5: D] :
( pp(aa(set(D),bool,member(D,X5),J4))
=> ? [Xa3: A] :
( pp(aa(set(A),bool,member(A,Xa3),I4))
& ! [Xb2: B] :
( pp(aa(set(B),bool,member(B,Xb2),aa(A,set(B),F6,Xa3)))
=> pp(aa(set(C),bool,member(C,aa(B,C,F,Xb2)),aa(D,set(C),G6,X5))) ) ) ) ) ) ) ).
% filterlim_base_iff
tff(fact_1009_wo__rel_Ocases__Total3,axiom,
! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A,Phi: fun(A,fun(A,bool))] :
( bNF_Wellorder_wo_rel(A,R)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),insert(A,A3),aa(set(A),set(A),insert(A,B2),bot_bot(set(A))))),aa(set(product_prod(A,A)),set(A),field2(A),R)))
=> ( ( ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),R),id2(A))))
| pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A3)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),R),id2(A)))) )
=> pp(aa(A,bool,aa(A,fun(A,bool),Phi,A3),B2)) )
=> ( ( ( A3 = B2 )
=> pp(aa(A,bool,aa(A,fun(A,bool),Phi,A3),B2)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),Phi,A3),B2)) ) ) ) ) ).
% wo_rel.cases_Total3
tff(fact_1010_eventually__INF__base,axiom,
! [B: $tType,A: $tType,B6: set(A),F6: fun(A,filter(B)),P2: fun(B,bool)] :
( ( B6 != bot_bot(set(A)) )
=> ( ! [A5: A] :
( pp(aa(set(A),bool,member(A,A5),B6))
=> ! [B4: A] :
( pp(aa(set(A),bool,member(A,B4),B6))
=> ? [X4: A] :
( pp(aa(set(A),bool,member(A,X4),B6))
& pp(aa(filter(B),bool,aa(filter(B),fun(filter(B),bool),ord_less_eq(filter(B)),aa(A,filter(B),F6,X4)),aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),inf_inf(filter(B)),aa(A,filter(B),F6,A5)),aa(A,filter(B),F6,B4)))) ) ) )
=> ( eventually(B,P2,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),aa(fun(A,filter(B)),fun(set(A),set(filter(B))),image2(A,filter(B)),F6),B6)))
<=> ? [X5: A] :
( pp(aa(set(A),bool,member(A,X5),B6))
& eventually(B,P2,aa(A,filter(B),F6,X5)) ) ) ) ) ).
% eventually_INF_base
tff(fact_1011_max__ext__def,axiom,
! [A: $tType,X4: set(product_prod(A,A))] : max_ext(A,X4) = aa(fun(product_prod(set(A),set(A)),bool),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aa(fun(set(A),fun(set(A),bool)),fun(product_prod(set(A),set(A)),bool),product_case_prod(set(A),set(A),bool),max_extp(A,aTP_Lamp_hl(set(product_prod(A,A)),fun(A,fun(A,bool)),X4)))) ).
% max_ext_def
tff(fact_1012_ImageI,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,R: set(product_prod(A,B)),A6: set(A)] :
( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),R))
=> ( pp(aa(set(A),bool,member(A,A3),A6))
=> pp(aa(set(B),bool,member(B,B2),aa(set(A),set(B),image(A,B,R),A6))) ) ) ).
% ImageI
tff(fact_1013_converse__iff,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,R: set(product_prod(B,A))] :
( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),converse(B,A,R)))
<=> pp(aa(set(product_prod(B,A)),bool,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),B2),A3)),R)) ) ).
% converse_iff
tff(fact_1014_Image__singleton__iff,axiom,
! [A: $tType,B: $tType,B2: A,R: set(product_prod(B,A)),A3: B] :
( pp(aa(set(A),bool,member(A,B2),aa(set(B),set(A),image(B,A,R),aa(set(B),set(B),insert(B,A3),bot_bot(set(B))))))
<=> pp(aa(set(product_prod(B,A)),bool,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A3),B2)),R)) ) ).
% Image_singleton_iff
tff(fact_1015_wo__rel_Owell__order__induct,axiom,
! [A: $tType,R: set(product_prod(A,A)),P2: fun(A,bool),A3: A] :
( bNF_Wellorder_wo_rel(A,R)
=> ( ! [X: A] :
( ! [Y5: A] :
( ( ( Y5 != X )
& pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),X)),R)) )
=> pp(aa(A,bool,P2,Y5)) )
=> pp(aa(A,bool,P2,X)) )
=> pp(aa(A,bool,P2,A3)) ) ) ).
% wo_rel.well_order_induct
tff(fact_1016_ImageE,axiom,
! [A: $tType,B: $tType,B2: A,R: set(product_prod(B,A)),A6: set(B)] :
( pp(aa(set(A),bool,member(A,B2),aa(set(B),set(A),image(B,A,R),A6)))
=> ~ ! [X: B] :
( pp(aa(set(product_prod(B,A)),bool,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X),B2)),R))
=> ~ pp(aa(set(B),bool,member(B,X),A6)) ) ) ).
% ImageE
tff(fact_1017_Image__iff,axiom,
! [A: $tType,B: $tType,B2: A,R: set(product_prod(B,A)),A6: set(B)] :
( pp(aa(set(A),bool,member(A,B2),aa(set(B),set(A),image(B,A,R),A6)))
<=> ? [X5: B] :
( pp(aa(set(B),bool,member(B,X5),A6))
& pp(aa(set(product_prod(B,A)),bool,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X5),B2)),R)) ) ) ).
% Image_iff
tff(fact_1018_rev__ImageI,axiom,
! [B: $tType,A: $tType,A3: A,A6: set(A),B2: B,R: set(product_prod(A,B))] :
( pp(aa(set(A),bool,member(A,A3),A6))
=> ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),R))
=> pp(aa(set(B),bool,member(B,B2),aa(set(A),set(B),image(A,B,R),A6))) ) ) ).
% rev_ImageI
tff(fact_1019_converse_Ocases,axiom,
! [B: $tType,A: $tType,A1: B,A22: A,R: set(product_prod(A,B))] :
( pp(aa(set(product_prod(B,A)),bool,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A1),A22)),converse(A,B,R)))
=> pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A22),A1)),R)) ) ).
% converse.cases
tff(fact_1020_converse_Osimps,axiom,
! [B: $tType,A: $tType,A1: B,A22: A,R: set(product_prod(A,B))] :
( pp(aa(set(product_prod(B,A)),bool,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A1),A22)),converse(A,B,R)))
<=> ? [A7: A,B5: B] :
( ( A1 = B5 )
& ( A22 = A7 )
& pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A7),B5)),R)) ) ) ).
% converse.simps
tff(fact_1021_converseD,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,R: set(product_prod(B,A))] :
( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),converse(B,A,R)))
=> pp(aa(set(product_prod(B,A)),bool,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),B2),A3)),R)) ) ).
% converseD
tff(fact_1022_converseE,axiom,
! [A: $tType,B: $tType,Yx: product_prod(A,B),R: set(product_prod(B,A))] :
( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),Yx),converse(B,A,R)))
=> ~ ! [X: B,Y: A] :
( ( Yx = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y),X) )
=> ~ pp(aa(set(product_prod(B,A)),bool,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X),Y)),R)) ) ) ).
% converseE
tff(fact_1023_converseI,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,R: set(product_prod(A,B))] :
( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),R))
=> pp(aa(set(product_prod(B,A)),bool,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),B2),A3)),converse(A,B,R))) ) ).
% converseI
tff(fact_1024_wo__rel_OTOTALS,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( bNF_Wellorder_wo_rel(A,R)
=> ! [X4: A] :
( pp(aa(set(A),bool,member(A,X4),aa(set(product_prod(A,A)),set(A),field2(A),R)))
=> ! [Xa4: A] :
( pp(aa(set(A),bool,member(A,Xa4),aa(set(product_prod(A,A)),set(A),field2(A),R)))
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa4)),R))
| pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa4),X4)),R)) ) ) ) ) ).
% wo_rel.TOTALS
tff(fact_1025_converse__Times,axiom,
! [A: $tType,B: $tType,A6: set(B),B6: set(A)] : converse(B,A,product_Sigma(B,A,A6,aTP_Lamp_cc(set(A),fun(B,set(A)),B6))) = product_Sigma(A,B,B6,aTP_Lamp_bu(set(B),fun(A,set(B)),A6)) ).
% converse_Times
tff(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)
=> ( ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R))
=> ( bNF_We1388413361240627857o_max2(A,R,A3,B2) = B2 ) )
& ( ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R))
=> ( bNF_We1388413361240627857o_max2(A,R,A3,B2) = A3 ) ) ) ) ).
% wo_rel.max2_def
tff(fact_1027_well__order__induct__imp,axiom,
! [A: $tType,R: set(product_prod(A,A)),P2: fun(A,bool),A3: A] :
( bNF_Wellorder_wo_rel(A,R)
=> ( ! [X: A] :
( ! [Y5: A] :
( ( ( Y5 != X )
& pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),X)),R)) )
=> ( pp(aa(set(A),bool,member(A,Y5),aa(set(product_prod(A,A)),set(A),field2(A),R)))
=> pp(aa(A,bool,P2,Y5)) ) )
=> ( pp(aa(set(A),bool,member(A,X),aa(set(product_prod(A,A)),set(A),field2(A),R)))
=> pp(aa(A,bool,P2,X)) ) )
=> ( pp(aa(set(A),bool,member(A,A3),aa(set(product_prod(A,A)),set(A),field2(A),R)))
=> pp(aa(A,bool,P2,A3)) ) ) ) ).
% well_order_induct_imp
tff(fact_1028_Image__UN,axiom,
! [B: $tType,A: $tType,C: $tType,R: set(product_prod(B,A)),B6: fun(C,set(B)),A6: set(C)] : aa(set(B),set(A),image(B,A,R),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),aa(fun(C,set(B)),fun(set(C),set(set(B))),image2(C,set(B)),B6),A6))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),aa(fun(C,set(A)),fun(set(C),set(set(A))),image2(C,set(A)),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_in(set(product_prod(B,A)),fun(fun(C,set(B)),fun(C,set(A))),R),B6)),A6)) ).
% Image_UN
tff(fact_1029_converse__unfold,axiom,
! [A: $tType,B: $tType,R: set(product_prod(B,A))] : converse(B,A,R) = aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aTP_Lamp_io(set(product_prod(B,A)),fun(A,fun(B,bool)),R))) ).
% converse_unfold
tff(fact_1030_conversep__converse__eq,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,B)),X4: B,Xa4: A] :
( pp(aa(A,bool,aa(B,fun(A,bool),conversep(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_bg(set(product_prod(A,B)),fun(A,fun(B,bool))),R)),X4),Xa4))
<=> pp(aa(set(product_prod(B,A)),bool,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X4),Xa4)),converse(A,B,R))) ) ).
% conversep_converse_eq
tff(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)
=> ( pp(aa(set(A),bool,member(A,A3),aa(set(product_prod(A,A)),set(A),field2(A),R)))
=> ( pp(aa(set(A),bool,member(A,B2),aa(set(product_prod(A,A)),set(A),field2(A),R)))
=> ( ( bNF_We1388413361240627857o_max2(A,R,A3,B2) = A3 )
<=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A3)),R)) ) ) ) ) ).
% wo_rel.max2_equals1
tff(fact_1032_wo__rel_Omax2__equals2,axiom,
! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A] :
( bNF_Wellorder_wo_rel(A,R)
=> ( pp(aa(set(A),bool,member(A,A3),aa(set(product_prod(A,A)),set(A),field2(A),R)))
=> ( pp(aa(set(A),bool,member(A,B2),aa(set(product_prod(A,A)),set(A),field2(A),R)))
=> ( ( bNF_We1388413361240627857o_max2(A,R,A3,B2) = B2 )
<=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R)) ) ) ) ) ).
% wo_rel.max2_equals2
tff(fact_1033_wo__rel_Omax2__greater,axiom,
! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A] :
( bNF_Wellorder_wo_rel(A,R)
=> ( pp(aa(set(A),bool,member(A,A3),aa(set(product_prod(A,A)),set(A),field2(A),R)))
=> ( pp(aa(set(A),bool,member(A,B2),aa(set(product_prod(A,A)),set(A),field2(A),R)))
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),bNF_We1388413361240627857o_max2(A,R,A3,B2))),R))
& pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),bNF_We1388413361240627857o_max2(A,R,A3,B2))),R)) ) ) ) ) ).
% wo_rel.max2_greater
tff(fact_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
tff(fact_1035_eventually__INF1,axiom,
! [B: $tType,A: $tType,I: A,I4: set(A),P2: fun(B,bool),F6: fun(A,filter(B))] :
( pp(aa(set(A),bool,member(A,I),I4))
=> ( eventually(B,P2,aa(A,filter(B),F6,I))
=> eventually(B,P2,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),aa(fun(A,filter(B)),fun(set(A),set(filter(B))),image2(A,filter(B)),F6),I4))) ) ) ).
% eventually_INF1
tff(fact_1036_converse__UNION,axiom,
! [B: $tType,A: $tType,C: $tType,R: fun(C,set(product_prod(B,A))),S3: set(C)] : converse(B,A,aa(set(set(product_prod(B,A))),set(product_prod(B,A)),complete_Sup_Sup(set(product_prod(B,A))),aa(set(C),set(set(product_prod(B,A))),aa(fun(C,set(product_prod(B,A))),fun(set(C),set(set(product_prod(B,A)))),image2(C,set(product_prod(B,A))),R),S3))) = aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(C),set(set(product_prod(A,B))),aa(fun(C,set(product_prod(A,B))),fun(set(C),set(set(product_prod(A,B)))),image2(C,set(product_prod(A,B))),aTP_Lamp_ip(fun(C,set(product_prod(B,A))),fun(C,set(product_prod(A,B))),R)),S3)) ).
% converse_UNION
tff(fact_1037_filterlim__INF_H,axiom,
! [C: $tType,B: $tType,A: $tType,X2: A,A6: set(A),F: fun(B,C),F6: filter(C),G6: fun(A,filter(B))] :
( pp(aa(set(A),bool,member(A,X2),A6))
=> ( filterlim(B,C,F,F6,aa(A,filter(B),G6,X2))
=> filterlim(B,C,F,F6,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),aa(fun(A,filter(B)),fun(set(A),set(filter(B))),image2(A,filter(B)),G6),A6))) ) ) ).
% filterlim_INF'
tff(fact_1038_filterlim__INF,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(A,B),G6: fun(C,filter(B)),B6: set(C),F6: filter(A)] :
( filterlim(A,B,F,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(C),set(filter(B)),aa(fun(C,filter(B)),fun(set(C),set(filter(B))),image2(C,filter(B)),G6),B6)),F6)
<=> ! [X5: C] :
( pp(aa(set(C),bool,member(C,X5),B6))
=> filterlim(A,B,F,aa(C,filter(B),G6,X5),F6) ) ) ).
% filterlim_INF
tff(fact_1039_converse__INTER,axiom,
! [B: $tType,A: $tType,C: $tType,R: fun(C,set(product_prod(B,A))),S3: set(C)] : converse(B,A,aa(set(set(product_prod(B,A))),set(product_prod(B,A)),complete_Inf_Inf(set(product_prod(B,A))),aa(set(C),set(set(product_prod(B,A))),aa(fun(C,set(product_prod(B,A))),fun(set(C),set(set(product_prod(B,A)))),image2(C,set(product_prod(B,A))),R),S3))) = aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Inf_Inf(set(product_prod(A,B))),aa(set(C),set(set(product_prod(A,B))),aa(fun(C,set(product_prod(A,B))),fun(set(C),set(set(product_prod(A,B)))),image2(C,set(product_prod(A,B))),aTP_Lamp_ip(fun(C,set(product_prod(B,A))),fun(C,set(product_prod(A,B))),R)),S3)) ).
% converse_INTER
tff(fact_1040_Image__singleton,axiom,
! [A: $tType,B: $tType,R: set(product_prod(B,A)),A3: B] : aa(set(B),set(A),image(B,A,R),aa(set(B),set(B),insert(B,A3),bot_bot(set(B)))) = aa(fun(A,bool),set(A),collect(A),aa(B,fun(A,bool),aTP_Lamp_iq(set(product_prod(B,A)),fun(B,fun(A,bool)),R),A3)) ).
% Image_singleton
tff(fact_1041_Image__INT__subset,axiom,
! [B: $tType,A: $tType,C: $tType,R: set(product_prod(B,A)),B6: fun(C,set(B)),A6: set(C)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,R),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),aa(fun(C,set(B)),fun(set(C),set(set(B))),image2(C,set(B)),B6),A6)))),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(C),set(set(A)),aa(fun(C,set(A)),fun(set(C),set(set(A))),image2(C,set(A)),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_in(set(product_prod(B,A)),fun(fun(C,set(B)),fun(C,set(A))),R),B6)),A6)))) ).
% Image_INT_subset
tff(fact_1042_converse__def,axiom,
! [B: $tType,A: $tType,X4: set(product_prod(A,B))] : converse(A,B,X4) = aa(fun(product_prod(B,A),bool),set(product_prod(B,A)),collect(product_prod(B,A)),aa(fun(B,fun(A,bool)),fun(product_prod(B,A),bool),product_case_prod(B,A,bool),conversep(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_bg(set(product_prod(A,B)),fun(A,fun(B,bool))),X4)))) ).
% converse_def
tff(fact_1043_eventually__INF__finite,axiom,
! [A: $tType,B: $tType,A6: set(A),P2: fun(B,bool),F6: fun(A,filter(B))] :
( finite_finite2(A,A6)
=> ( eventually(B,P2,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),aa(fun(A,filter(B)),fun(set(A),set(filter(B))),image2(A,filter(B)),F6),A6)))
<=> ? [Q4: fun(A,fun(B,bool))] :
( ! [X5: A] :
( pp(aa(set(A),bool,member(A,X5),A6))
=> eventually(B,aa(A,fun(B,bool),Q4,X5),aa(A,filter(B),F6,X5)) )
& ! [Y4: B] :
( ! [X5: A] :
( pp(aa(set(A),bool,member(A,X5),A6))
=> pp(aa(B,bool,aa(A,fun(B,bool),Q4,X5),Y4)) )
=> pp(aa(B,bool,P2,Y4)) ) ) ) ) ).
% eventually_INF_finite
tff(fact_1044_wo__rel_Ocases__Total,axiom,
! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A,Phi: fun(A,fun(A,bool))] :
( bNF_Wellorder_wo_rel(A,R)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),insert(A,A3),aa(set(A),set(A),insert(A,B2),bot_bot(set(A))))),aa(set(product_prod(A,A)),set(A),field2(A),R)))
=> ( ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R))
=> pp(aa(A,bool,aa(A,fun(A,bool),Phi,A3),B2)) )
=> ( ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A3)),R))
=> pp(aa(A,bool,aa(A,fun(A,bool),Phi,A3),B2)) )
=> pp(aa(A,bool,aa(A,fun(A,bool),Phi,A3),B2)) ) ) ) ) ).
% wo_rel.cases_Total
tff(fact_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)
=> ( pp(aa(set(A),bool,member(A,A3),aa(set(product_prod(A,A)),set(A),field2(A),R)))
=> ( pp(aa(set(A),bool,member(A,B2),aa(set(product_prod(A,A)),set(A),field2(A),R)))
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),bNF_We1388413361240627857o_max2(A,R,A3,B2))),R))
& pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),bNF_We1388413361240627857o_max2(A,R,A3,B2))),R))
& pp(aa(set(A),bool,member(A,bNF_We1388413361240627857o_max2(A,R,A3,B2)),aa(set(A),set(A),insert(A,A3),aa(set(A),set(A),insert(A,B2),bot_bot(set(A)))))) ) ) ) ) ).
% wo_rel.max2_greater_among
tff(fact_1046_filterlim__base,axiom,
! [B: $tType,A: $tType,E: $tType,D: $tType,C: $tType,J4: set(A),I: fun(A,C),I4: set(C),F6: fun(C,set(D)),F: fun(D,E),G6: fun(A,set(E))] :
( ! [M3: A,X: B] :
( pp(aa(set(A),bool,member(A,M3),J4))
=> pp(aa(set(C),bool,member(C,aa(A,C,I,M3)),I4)) )
=> ( ! [M3: A,X: D] :
( pp(aa(set(A),bool,member(A,M3),J4))
=> ( pp(aa(set(D),bool,member(D,X),aa(C,set(D),F6,aa(A,C,I,M3))))
=> pp(aa(set(E),bool,member(E,aa(D,E,F,X)),aa(A,set(E),G6,M3))) ) )
=> filterlim(D,E,F,aa(set(filter(E)),filter(E),complete_Inf_Inf(filter(E)),aa(set(A),set(filter(E)),aa(fun(A,filter(E)),fun(set(A),set(filter(E))),image2(A,filter(E)),aTP_Lamp_ir(fun(A,set(E)),fun(A,filter(E)),G6)),J4)),aa(set(filter(D)),filter(D),complete_Inf_Inf(filter(D)),aa(set(C),set(filter(D)),aa(fun(C,filter(D)),fun(set(C),set(filter(D))),image2(C,filter(D)),aTP_Lamp_is(fun(C,set(D)),fun(C,filter(D)),F6)),I4))) ) ) ).
% filterlim_base
tff(fact_1047_Image__eq__UN,axiom,
! [A: $tType,B: $tType,R: set(product_prod(B,A)),B6: set(B)] : aa(set(B),set(A),image(B,A,R),B6) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),aTP_Lamp_it(set(product_prod(B,A)),fun(B,set(A)),R)),B6)) ).
% Image_eq_UN
tff(fact_1048_image2__Gr,axiom,
! [A: $tType,B: $tType,C: $tType,A6: set(C),F: fun(C,A),G: fun(C,B)] : bNF_Greatest_image2(C,A,B,A6,F,G) = relcomp(A,C,B,converse(C,A,bNF_Gr(C,A,A6,F)),bNF_Gr(C,B,A6,G)) ).
% image2_Gr
tff(fact_1049_Sigma__Image,axiom,
! [A: $tType,B: $tType,A6: set(B),B6: fun(B,set(A)),X6: set(B)] : aa(set(B),set(A),image(B,A,product_Sigma(B,A,A6,B6)),X6) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),B6),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),X6),A6))) ).
% Sigma_Image
tff(fact_1050_eventually__INF,axiom,
! [A: $tType,B: $tType,P2: fun(A,bool),F6: fun(B,filter(A)),B6: set(B)] :
( eventually(A,P2,aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(B),set(filter(A)),aa(fun(B,filter(A)),fun(set(B),set(filter(A))),image2(B,filter(A)),F6),B6)))
<=> ? [X9: set(B)] :
( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),X9),B6))
& finite_finite2(B,X9)
& eventually(A,P2,aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(B),set(filter(A)),aa(fun(B,filter(A)),fun(set(B),set(filter(A))),image2(B,filter(A)),F6),X9))) ) ) ).
% eventually_INF
tff(fact_1051_UN__Image,axiom,
! [B: $tType,A: $tType,C: $tType,X6: fun(C,set(product_prod(B,A))),I4: set(C),S3: set(B)] : aa(set(B),set(A),image(B,A,aa(set(set(product_prod(B,A))),set(product_prod(B,A)),complete_Sup_Sup(set(product_prod(B,A))),aa(set(C),set(set(product_prod(B,A))),aa(fun(C,set(product_prod(B,A))),fun(set(C),set(set(product_prod(B,A)))),image2(C,set(product_prod(B,A))),X6),I4))),S3) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),aa(fun(C,set(A)),fun(set(C),set(set(A))),image2(C,set(A)),aa(set(B),fun(C,set(A)),aTP_Lamp_iu(fun(C,set(product_prod(B,A))),fun(set(B),fun(C,set(A))),X6),S3)),I4)) ).
% UN_Image
tff(fact_1052_max__extp__eq,axiom,
! [A: $tType,R: fun(A,fun(A,bool)),X2: set(A),Y2: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),max_extp(A,R),X2),Y2))
<=> pp(aa(set(product_prod(set(A),set(A))),bool,member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),X2),Y2)),max_ext(A,aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),R))))) ) ).
% max_extp_eq
tff(fact_1053_subset__Image1__Image1__iff,axiom,
! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A] :
( order_preorder_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R),R)
=> ( pp(aa(set(A),bool,member(A,A3),aa(set(product_prod(A,A)),set(A),field2(A),R)))
=> ( pp(aa(set(A),bool,member(A,B2),aa(set(product_prod(A,A)),set(A),field2(A),R)))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),image(A,A,R),aa(set(A),set(A),insert(A,A3),bot_bot(set(A))))),aa(set(A),set(A),image(A,A,R),aa(set(A),set(A),insert(A,B2),bot_bot(set(A))))))
<=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A3)),R)) ) ) ) ) ).
% subset_Image1_Image1_iff
tff(fact_1054_map__filter__on__comp,axiom,
! [A: $tType,C: $tType,B: $tType,G: fun(B,A),Y3: set(B),X6: set(A),F6: filter(B),F: fun(A,C)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),G),Y3)),X6))
=> ( eventually(B,aTP_Lamp_iv(set(B),fun(B,bool),Y3),F6)
=> ( map_filter_on(A,C,X6,F,map_filter_on(B,A,Y3,G,F6)) = map_filter_on(B,C,Y3,aa(fun(B,A),fun(B,C),aa(fun(A,C),fun(fun(B,A),fun(B,C)),comp(A,C,B),F),G),F6) ) ) ) ).
% map_filter_on_comp
tff(fact_1055_quotient__def,axiom,
! [A: $tType,A6: set(A),R: set(product_prod(A,A))] : equiv_quotient(A,A6,R) = aa(set(set(set(A))),set(set(A)),complete_Sup_Sup(set(set(A))),aa(set(A),set(set(set(A))),aa(fun(A,set(set(A))),fun(set(A),set(set(set(A)))),image2(A,set(set(A))),aTP_Lamp_iw(set(product_prod(A,A)),fun(A,set(set(A))),R)),A6)) ).
% quotient_def
tff(fact_1056_Image__INT__eq,axiom,
! [B: $tType,A: $tType,C: $tType,R: set(product_prod(B,A)),A6: set(C),B6: fun(C,set(B))] :
( single_valued(A,B,converse(B,A,R))
=> ( ( A6 != bot_bot(set(C)) )
=> ( aa(set(B),set(A),image(B,A,R),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),aa(fun(C,set(B)),fun(set(C),set(set(B))),image2(C,set(B)),B6),A6))) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(C),set(set(A)),aa(fun(C,set(A)),fun(set(C),set(set(A))),image2(C,set(A)),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_in(set(product_prod(B,A)),fun(fun(C,set(B)),fun(C,set(A))),R),B6)),A6)) ) ) ) ).
% Image_INT_eq
tff(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)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B6),aa(set(product_prod(A,A)),set(A),field2(A),R)))
=> ( pp(aa(set(A),bool,member(A,B2),B6))
=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),bNF_We6954850376910717587_minim(A,R,B6)),B2)),R)) ) ) ) ).
% wo_rel.minim_least
tff(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)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B6),aa(set(product_prod(A,A)),set(A),field2(A),R)))
=> ( pp(aa(set(A),bool,member(A,A3),B6))
=> ( ! [B4: A] :
( pp(aa(set(A),bool,member(A,B4),B6))
=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B4)),R)) )
=> ( A3 = bNF_We6954850376910717587_minim(A,R,B6) ) ) ) ) ) ).
% wo_rel.equals_minim
tff(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,aa(set(product_prod(A,A)),set(A),field2(A),R),R)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),aa(set(product_prod(A,A)),set(A),field2(A),R)))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B6),aa(set(product_prod(A,A)),set(A),field2(A),R)))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),image(A,A,R),A6)),aa(set(A),set(A),image(A,A,R),B6)))
<=> ! [X5: A] :
( pp(aa(set(A),bool,member(A,X5),A6))
=> ? [Xa3: A] :
( pp(aa(set(A),bool,member(A,Xa3),B6))
& pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa3),X5)),R)) ) ) ) ) ) ) ).
% subset_Image_Image_iff
tff(fact_1060_filterlim__INF__INF,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,J4: set(A),I4: set(B),F: fun(D,C),F6: fun(B,filter(D)),G6: fun(A,filter(C))] :
( ! [M3: A] :
( pp(aa(set(A),bool,member(A,M3),J4))
=> ? [X4: B] :
( pp(aa(set(B),bool,member(B,X4),I4))
& pp(aa(filter(C),bool,aa(filter(C),fun(filter(C),bool),ord_less_eq(filter(C)),filtermap(D,C,F,aa(B,filter(D),F6,X4))),aa(A,filter(C),G6,M3))) ) )
=> filterlim(D,C,F,aa(set(filter(C)),filter(C),complete_Inf_Inf(filter(C)),aa(set(A),set(filter(C)),aa(fun(A,filter(C)),fun(set(A),set(filter(C))),image2(A,filter(C)),G6),J4)),aa(set(filter(D)),filter(D),complete_Inf_Inf(filter(D)),aa(set(B),set(filter(D)),aa(fun(B,filter(D)),fun(set(B),set(filter(D))),image2(B,filter(D)),F6),I4))) ) ).
% filterlim_INF_INF
tff(fact_1061_filtermap__principal,axiom,
! [A: $tType,B: $tType,F: fun(B,A),A6: set(B)] : filtermap(B,A,F,principal(B,A6)) = principal(A,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6)) ).
% filtermap_principal
tff(fact_1062_single__valued__def,axiom,
! [A: $tType,B: $tType,R: set(product_prod(A,B))] :
( single_valued(A,B,R)
<=> ! [X5: A,Y4: B] :
( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X5),Y4)),R))
=> ! [Z4: B] :
( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X5),Z4)),R))
=> ( Y4 = Z4 ) ) ) ) ).
% single_valued_def
tff(fact_1063_single__valuedI,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,B))] :
( ! [X: A,Y: B,Z3: B] :
( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)),R))
=> ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Z3)),R))
=> ( Y = Z3 ) ) )
=> single_valued(A,B,R) ) ).
% single_valuedI
tff(fact_1064_single__valuedD,axiom,
! [A: $tType,B: $tType,R: set(product_prod(A,B)),X2: A,Y2: B,Z2: B] :
( single_valued(A,B,R)
=> ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y2)),R))
=> ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Z2)),R))
=> ( Y2 = Z2 ) ) ) ) ).
% single_valuedD
tff(fact_1065_filtermap__SUP,axiom,
! [B: $tType,A: $tType,C: $tType,F: fun(B,A),F6: fun(C,filter(B)),B6: set(C)] : filtermap(B,A,F,aa(set(filter(B)),filter(B),complete_Sup_Sup(filter(B)),aa(set(C),set(filter(B)),aa(fun(C,filter(B)),fun(set(C),set(filter(B))),image2(C,filter(B)),F6),B6))) = aa(set(filter(A)),filter(A),complete_Sup_Sup(filter(A)),aa(set(C),set(filter(A)),aa(fun(C,filter(A)),fun(set(C),set(filter(A))),image2(C,filter(A)),aa(fun(C,filter(B)),fun(C,filter(A)),aTP_Lamp_ix(fun(B,A),fun(fun(C,filter(B)),fun(C,filter(A))),F),F6)),B6)) ).
% filtermap_SUP
tff(fact_1066_filtermap__INF,axiom,
! [B: $tType,A: $tType,C: $tType,F: fun(B,A),F6: fun(C,filter(B)),B6: set(C)] : pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),filtermap(B,A,F,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(C),set(filter(B)),aa(fun(C,filter(B)),fun(set(C),set(filter(B))),image2(C,filter(B)),F6),B6)))),aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(C),set(filter(A)),aa(fun(C,filter(A)),fun(set(C),set(filter(A))),image2(C,filter(A)),aa(fun(C,filter(B)),fun(C,filter(A)),aTP_Lamp_ix(fun(B,A),fun(fun(C,filter(B)),fun(C,filter(A))),F),F6)),B6)))) ).
% filtermap_INF
tff(fact_1067_prod__filter__principal__singleton2,axiom,
! [B: $tType,A: $tType,F6: filter(A),X2: B] : prod_filter(A,B,F6,principal(B,aa(set(B),set(B),insert(B,X2),bot_bot(set(B))))) = filtermap(A,product_prod(A,B),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_bk(B,fun(A,product_prod(A,B))),X2),F6) ).
% prod_filter_principal_singleton2
tff(fact_1068_in__quotient__imp__in__rel,axiom,
! [A: $tType,A6: set(A),R: set(product_prod(A,A)),X6: set(A),X2: A,Y2: A] :
( equiv_equiv(A,A6,R)
=> ( pp(aa(set(set(A)),bool,member(set(A),X6),equiv_quotient(A,A6,R)))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),insert(A,X2),aa(set(A),set(A),insert(A,Y2),bot_bot(set(A))))),X6))
=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),R)) ) ) ) ).
% in_quotient_imp_in_rel
tff(fact_1069_relpow__finite__bounded,axiom,
! [A: $tType,R3: set(product_prod(A,A)),K: nat] :
( finite_finite2(product_prod(A,A),R3)
=> pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),compow(set(product_prod(A,A)),K,R3)),aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),aa(set(nat),set(set(product_prod(A,A))),aa(fun(nat,set(product_prod(A,A))),fun(set(nat),set(set(product_prod(A,A)))),image2(nat,set(product_prod(A,A))),aTP_Lamp_iy(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R3)),aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_iz(set(product_prod(A,A)),fun(nat,bool),R3)))))) ) ).
% relpow_finite_bounded
tff(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)
=> ( pp(aa(set(A),bool,member(A,X2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),image(A,A,R),aa(set(A),set(A),insert(A,A3),bot_bot(set(A))))),aa(set(A),set(A),image(A,A,R),aa(set(A),set(A),insert(A,B2),bot_bot(set(A)))))))
=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R)) ) ) ).
% equiv_class_nondisjoint
tff(fact_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)
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),image(A,A,R),aa(set(A),set(A),insert(A,A3),bot_bot(set(A))))),aa(set(A),set(A),image(A,A,R),aa(set(A),set(A),insert(A,B2),bot_bot(set(A)))))) ) ) ).
% equiv_class_subset
tff(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)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),image(A,A,R),aa(set(A),set(A),insert(A,B2),bot_bot(set(A))))),aa(set(A),set(A),image(A,A,R),aa(set(A),set(A),insert(A,A3),bot_bot(set(A))))))
=> ( pp(aa(set(A),bool,member(A,B2),A6))
=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R)) ) ) ) ).
% subset_equiv_class
tff(fact_1073_single__valuedp__single__valued__eq,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,B))] :
( single_valuedp(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_bg(set(product_prod(A,B)),fun(A,fun(B,bool))),R))
<=> single_valued(A,B,R) ) ).
% single_valuedp_single_valued_eq
tff(fact_1074_Range__Union,axiom,
! [A: $tType,B: $tType,S3: set(set(product_prod(B,A)))] : aa(set(product_prod(B,A)),set(A),range2(B,A),aa(set(set(product_prod(B,A))),set(product_prod(B,A)),complete_Sup_Sup(set(product_prod(B,A))),S3)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(product_prod(B,A))),set(set(A)),aa(fun(set(product_prod(B,A)),set(A)),fun(set(set(product_prod(B,A))),set(set(A))),image2(set(product_prod(B,A)),set(A)),range2(B,A)),S3)) ).
% Range_Union
tff(fact_1075_refines__equiv__image__eq,axiom,
! [A: $tType,R3: set(product_prod(A,A)),S3: set(product_prod(A,A)),A6: set(A)] :
( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),S3))
=> ( equiv_equiv(A,A6,R3)
=> ( equiv_equiv(A,A6,S3)
=> ( aa(set(set(A)),set(set(A)),aa(fun(set(A),set(A)),fun(set(set(A)),set(set(A))),image2(set(A),set(A)),image(A,A,S3)),equiv_quotient(A,A6,R3)) = equiv_quotient(A,A6,S3) ) ) ) ) ).
% refines_equiv_image_eq
tff(fact_1076_Range__insert,axiom,
! [A: $tType,B: $tType,A3: B,B2: A,R: set(product_prod(B,A))] : aa(set(product_prod(B,A)),set(A),range2(B,A),aa(set(product_prod(B,A)),set(product_prod(B,A)),insert(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A3),B2)),R)) = aa(set(A),set(A),insert(A,B2),aa(set(product_prod(B,A)),set(A),range2(B,A),R)) ).
% Range_insert
tff(fact_1077_prod__filter__assoc,axiom,
! [A: $tType,B: $tType,C: $tType,F6: filter(A),G6: filter(B),H6: filter(C)] : prod_filter(product_prod(A,B),C,prod_filter(A,B,F6,G6),H6) = filtermap(product_prod(A,product_prod(B,C)),product_prod(product_prod(A,B),C),aa(fun(A,fun(product_prod(B,C),product_prod(product_prod(A,B),C))),fun(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)),aTP_Lamp_jb(A,fun(product_prod(B,C),product_prod(product_prod(A,B),C)))),prod_filter(A,product_prod(B,C),F6,prod_filter(B,C,G6,H6))) ).
% prod_filter_assoc
tff(fact_1078_relpow__Suc__D2_H,axiom,
! [A: $tType,N4: nat,R3: set(product_prod(A,A)),X4: A,Y5: A,Z6: A] :
( ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Y5)),compow(set(product_prod(A,A)),N4,R3)))
& pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),Z6)),R3)) )
=> ? [W: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),W)),R3))
& pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),W),Z6)),compow(set(product_prod(A,A)),N4,R3))) ) ) ).
% relpow_Suc_D2'
tff(fact_1079_Range_Ocases,axiom,
! [B: $tType,A: $tType,A3: B,R: set(product_prod(A,B))] :
( pp(aa(set(B),bool,member(B,A3),aa(set(product_prod(A,B)),set(B),range2(A,B),R)))
=> ~ ! [A5: A] : ~ pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),A3)),R)) ) ).
% Range.cases
tff(fact_1080_Range_Osimps,axiom,
! [B: $tType,A: $tType,A3: B,R: set(product_prod(A,B))] :
( pp(aa(set(B),bool,member(B,A3),aa(set(product_prod(A,B)),set(B),range2(A,B),R)))
<=> ? [A7: A,B5: B] :
( ( A3 = B5 )
& pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A7),B5)),R)) ) ) ).
% Range.simps
tff(fact_1081_Range_Ointros,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,R: set(product_prod(A,B))] :
( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),R))
=> pp(aa(set(B),bool,member(B,B2),aa(set(product_prod(A,B)),set(B),range2(A,B),R))) ) ).
% Range.intros
tff(fact_1082_RangeE,axiom,
! [A: $tType,B: $tType,B2: A,R: set(product_prod(B,A))] :
( pp(aa(set(A),bool,member(A,B2),aa(set(product_prod(B,A)),set(A),range2(B,A),R)))
=> ~ ! [A5: B] : ~ pp(aa(set(product_prod(B,A)),bool,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A5),B2)),R)) ) ).
% RangeE
tff(fact_1083_Range__iff,axiom,
! [A: $tType,B: $tType,A3: A,R: set(product_prod(B,A))] :
( pp(aa(set(A),bool,member(A,A3),aa(set(product_prod(B,A)),set(A),range2(B,A),R)))
<=> ? [Y4: B] : pp(aa(set(product_prod(B,A)),bool,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Y4),A3)),R)) ) ).
% Range_iff
tff(fact_1084_prod__filter__commute,axiom,
! [B: $tType,A: $tType,F6: filter(A),G6: filter(B)] : prod_filter(A,B,F6,G6) = filtermap(product_prod(B,A),product_prod(A,B),product_swap(B,A),prod_filter(B,A,G6,F6)) ).
% prod_filter_commute
tff(fact_1085_in__quotient__imp__closed,axiom,
! [A: $tType,A6: set(A),R: set(product_prod(A,A)),X6: set(A),X2: A,Y2: A] :
( equiv_equiv(A,A6,R)
=> ( pp(aa(set(set(A)),bool,member(set(A),X6),equiv_quotient(A,A6,R)))
=> ( pp(aa(set(A),bool,member(A,X2),X6))
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),R))
=> pp(aa(set(A),bool,member(A,Y2),X6)) ) ) ) ) ).
% in_quotient_imp_closed
tff(fact_1086_quotient__eq__iff,axiom,
! [A: $tType,A6: set(A),R: set(product_prod(A,A)),X6: set(A),Y3: set(A),X2: A,Y2: A] :
( equiv_equiv(A,A6,R)
=> ( pp(aa(set(set(A)),bool,member(set(A),X6),equiv_quotient(A,A6,R)))
=> ( pp(aa(set(set(A)),bool,member(set(A),Y3),equiv_quotient(A,A6,R)))
=> ( pp(aa(set(A),bool,member(A,X2),X6))
=> ( pp(aa(set(A),bool,member(A,Y2),Y3))
=> ( ( X6 = Y3 )
<=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),R)) ) ) ) ) ) ) ).
% quotient_eq_iff
tff(fact_1087_quotient__eqI,axiom,
! [A: $tType,A6: set(A),R: set(product_prod(A,A)),X6: set(A),Y3: set(A),X2: A,Y2: A] :
( equiv_equiv(A,A6,R)
=> ( pp(aa(set(set(A)),bool,member(set(A),X6),equiv_quotient(A,A6,R)))
=> ( pp(aa(set(set(A)),bool,member(set(A),Y3),equiv_quotient(A,A6,R)))
=> ( pp(aa(set(A),bool,member(A,X2),X6))
=> ( pp(aa(set(A),bool,member(A,Y2),Y3))
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),R))
=> ( X6 = Y3 ) ) ) ) ) ) ) ).
% quotient_eqI
tff(fact_1088_eventually__prod__filter,axiom,
! [A: $tType,B: $tType,P2: fun(product_prod(A,B),bool),F6: filter(A),G6: filter(B)] :
( eventually(product_prod(A,B),P2,prod_filter(A,B,F6,G6))
<=> ? [Pf: fun(A,bool),Pg: fun(B,bool)] :
( eventually(A,Pf,F6)
& eventually(B,Pg,G6)
& ! [X5: A,Y4: B] :
( pp(aa(A,bool,Pf,X5))
=> ( pp(aa(B,bool,Pg,Y4))
=> pp(aa(product_prod(A,B),bool,P2,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X5),Y4))) ) ) ) ) ).
% eventually_prod_filter
tff(fact_1089_eventually__prod__same,axiom,
! [A: $tType,P2: fun(product_prod(A,A),bool),F6: filter(A)] :
( eventually(product_prod(A,A),P2,prod_filter(A,A,F6,F6))
<=> ? [Q4: fun(A,bool)] :
( eventually(A,Q4,F6)
& ! [X5: A,Y4: A] :
( pp(aa(A,bool,Q4,X5))
=> ( pp(aa(A,bool,Q4,Y4))
=> pp(aa(product_prod(A,A),bool,P2,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),Y4))) ) ) ) ) ).
% eventually_prod_same
tff(fact_1090_Range__snd,axiom,
! [A: $tType,B: $tType,R: set(product_prod(B,A))] : aa(set(product_prod(B,A)),set(A),range2(B,A),R) = aa(set(product_prod(B,A)),set(A),aa(fun(product_prod(B,A),A),fun(set(product_prod(B,A)),set(A)),image2(product_prod(B,A),A),product_snd(B,A)),R) ).
% Range_snd
tff(fact_1091_snd__eq__Range,axiom,
! [A: $tType,B: $tType,R3: set(product_prod(B,A))] : aa(set(product_prod(B,A)),set(A),aa(fun(product_prod(B,A),A),fun(set(product_prod(B,A)),set(A)),image2(product_prod(B,A),A),product_snd(B,A)),R3) = aa(set(product_prod(B,A)),set(A),range2(B,A),R3) ).
% snd_eq_Range
tff(fact_1092_filtermap__fst__prod__filter,axiom,
! [B: $tType,A: $tType,A6: filter(A),B6: filter(B)] : pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),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
tff(fact_1093_filtermap__snd__prod__filter,axiom,
! [B: $tType,A: $tType,A6: filter(B),B6: filter(A)] : pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),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
tff(fact_1094_filterlim__Pair,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(A,B),G6: filter(B),F6: filter(A),G: fun(A,C),H6: filter(C)] :
( filterlim(A,B,F,G6,F6)
=> ( filterlim(A,C,G,H6,F6)
=> filterlim(A,product_prod(B,C),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_jc(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),F),G),prod_filter(B,C,G6,H6),F6) ) ) ).
% filterlim_Pair
tff(fact_1095_filtermap__Pair,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(C,A),G: fun(C,B),F6: filter(C)] : pp(aa(filter(product_prod(A,B)),bool,aa(filter(product_prod(A,B)),fun(filter(product_prod(A,B)),bool),ord_less_eq(filter(product_prod(A,B))),filtermap(C,product_prod(A,B),aa(fun(C,B),fun(C,product_prod(A,B)),aTP_Lamp_gw(fun(C,A),fun(fun(C,B),fun(C,product_prod(A,B))),F),G),F6)),prod_filter(A,B,filtermap(C,A,F,F6),filtermap(C,B,G,F6)))) ).
% filtermap_Pair
tff(fact_1096_prod__filter__def,axiom,
! [A: $tType,B: $tType,F6: filter(A),G6: filter(B)] : prod_filter(A,B,F6,G6) = aa(set(filter(product_prod(A,B))),filter(product_prod(A,B)),complete_Inf_Inf(filter(product_prod(A,B))),aa(set(product_prod(fun(A,bool),fun(B,bool))),set(filter(product_prod(A,B))),aa(fun(product_prod(fun(A,bool),fun(B,bool)),filter(product_prod(A,B))),fun(set(product_prod(fun(A,bool),fun(B,bool))),set(filter(product_prod(A,B)))),image2(product_prod(fun(A,bool),fun(B,bool)),filter(product_prod(A,B))),aa(fun(fun(A,bool),fun(fun(B,bool),filter(product_prod(A,B)))),fun(product_prod(fun(A,bool),fun(B,bool)),filter(product_prod(A,B))),product_case_prod(fun(A,bool),fun(B,bool),filter(product_prod(A,B))),aTP_Lamp_jd(fun(A,bool),fun(fun(B,bool),filter(product_prod(A,B)))))),aa(fun(product_prod(fun(A,bool),fun(B,bool)),bool),set(product_prod(fun(A,bool),fun(B,bool))),collect(product_prod(fun(A,bool),fun(B,bool))),aa(fun(fun(A,bool),fun(fun(B,bool),bool)),fun(product_prod(fun(A,bool),fun(B,bool)),bool),product_case_prod(fun(A,bool),fun(B,bool),bool),aa(filter(B),fun(fun(A,bool),fun(fun(B,bool),bool)),aTP_Lamp_je(filter(A),fun(filter(B),fun(fun(A,bool),fun(fun(B,bool),bool))),F6),G6))))) ).
% prod_filter_def
tff(fact_1097_le__prod__filterI,axiom,
! [A: $tType,B: $tType,F6: filter(product_prod(A,B)),A6: filter(A),B6: filter(B)] :
( pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),filtermap(product_prod(A,B),A,product_fst(A,B),F6)),A6))
=> ( pp(aa(filter(B),bool,aa(filter(B),fun(filter(B),bool),ord_less_eq(filter(B)),filtermap(product_prod(A,B),B,product_snd(A,B),F6)),B6))
=> pp(aa(filter(product_prod(A,B)),bool,aa(filter(product_prod(A,B)),fun(filter(product_prod(A,B)),bool),ord_less_eq(filter(product_prod(A,B))),F6),prod_filter(A,B,A6,B6))) ) ) ).
% le_prod_filterI
tff(fact_1098_eq__equiv__class,axiom,
! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A,A6: set(A)] :
( ( aa(set(A),set(A),image(A,A,R),aa(set(A),set(A),insert(A,A3),bot_bot(set(A)))) = aa(set(A),set(A),image(A,A,R),aa(set(A),set(A),insert(A,B2),bot_bot(set(A)))) )
=> ( equiv_equiv(A,A6,R)
=> ( pp(aa(set(A),bool,member(A,B2),A6))
=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R)) ) ) ) ).
% eq_equiv_class
tff(fact_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)
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R))
=> ( aa(set(A),set(A),image(A,A,R),aa(set(A),set(A),insert(A,A3),bot_bot(set(A)))) = aa(set(A),set(A),image(A,A,R),aa(set(A),set(A),insert(A,B2),bot_bot(set(A)))) ) ) ) ).
% equiv_class_eq
tff(fact_1100_eq__equiv__class__iff,axiom,
! [A: $tType,A6: set(A),R: set(product_prod(A,A)),X2: A,Y2: A] :
( equiv_equiv(A,A6,R)
=> ( pp(aa(set(A),bool,member(A,X2),A6))
=> ( pp(aa(set(A),bool,member(A,Y2),A6))
=> ( ( aa(set(A),set(A),image(A,A,R),aa(set(A),set(A),insert(A,X2),bot_bot(set(A)))) = aa(set(A),set(A),image(A,A,R),aa(set(A),set(A),insert(A,Y2),bot_bot(set(A)))) )
<=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),R)) ) ) ) ) ).
% eq_equiv_class_iff
tff(fact_1101_equiv__class__eq__iff,axiom,
! [A: $tType,A6: set(A),R: set(product_prod(A,A)),X2: A,Y2: A] :
( equiv_equiv(A,A6,R)
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),R))
<=> ( ( aa(set(A),set(A),image(A,A,R),aa(set(A),set(A),insert(A,X2),bot_bot(set(A)))) = aa(set(A),set(A),image(A,A,R),aa(set(A),set(A),insert(A,Y2),bot_bot(set(A)))) )
& pp(aa(set(A),bool,member(A,X2),A6))
& pp(aa(set(A),bool,member(A,Y2),A6)) ) ) ) ).
% equiv_class_eq_iff
tff(fact_1102_eventually__prodI,axiom,
! [A: $tType,B: $tType,P2: fun(A,bool),F6: filter(A),Q2: fun(B,bool),G6: filter(B)] :
( eventually(A,P2,F6)
=> ( eventually(B,Q2,G6)
=> eventually(product_prod(A,B),aa(fun(B,bool),fun(product_prod(A,B),bool),aTP_Lamp_jf(fun(A,bool),fun(fun(B,bool),fun(product_prod(A,B),bool)),P2),Q2),prod_filter(A,B,F6,G6)) ) ) ).
% eventually_prodI
tff(fact_1103_eq__equiv__class__iff2,axiom,
! [A: $tType,A6: set(A),R: set(product_prod(A,A)),X2: A,Y2: A] :
( equiv_equiv(A,A6,R)
=> ( pp(aa(set(A),bool,member(A,X2),A6))
=> ( pp(aa(set(A),bool,member(A,Y2),A6))
=> ( ( equiv_quotient(A,aa(set(A),set(A),insert(A,X2),bot_bot(set(A))),R) = equiv_quotient(A,aa(set(A),set(A),insert(A,Y2),bot_bot(set(A))),R) )
<=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),R)) ) ) ) ) ).
% eq_equiv_class_iff2
tff(fact_1104_eventually__prod2,axiom,
! [A: $tType,B: $tType,A6: filter(A),P2: fun(B,bool),B6: filter(B)] :
( ( A6 != bot_bot(filter(A)) )
=> ( eventually(product_prod(A,B),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aTP_Lamp_jg(fun(B,bool),fun(A,fun(B,bool)),P2)),prod_filter(A,B,A6,B6))
<=> eventually(B,P2,B6) ) ) ).
% eventually_prod2
tff(fact_1105_eventually__prod1,axiom,
! [A: $tType,B: $tType,B6: filter(A),P2: fun(B,bool),A6: filter(B)] :
( ( B6 != bot_bot(filter(A)) )
=> ( eventually(product_prod(B,A),aa(fun(B,fun(A,bool)),fun(product_prod(B,A),bool),product_case_prod(B,A,bool),aTP_Lamp_jh(fun(B,bool),fun(B,fun(A,bool)),P2)),prod_filter(B,A,A6,B6))
<=> eventually(B,P2,A6) ) ) ).
% eventually_prod1
tff(fact_1106_prod__filter__INF2,axiom,
! [C: $tType,B: $tType,A: $tType,J4: set(A),A6: filter(B),B6: fun(A,filter(C))] :
( ( J4 != bot_bot(set(A)) )
=> ( prod_filter(B,C,A6,aa(set(filter(C)),filter(C),complete_Inf_Inf(filter(C)),aa(set(A),set(filter(C)),aa(fun(A,filter(C)),fun(set(A),set(filter(C))),image2(A,filter(C)),B6),J4))) = aa(set(filter(product_prod(B,C))),filter(product_prod(B,C)),complete_Inf_Inf(filter(product_prod(B,C))),aa(set(A),set(filter(product_prod(B,C))),aa(fun(A,filter(product_prod(B,C))),fun(set(A),set(filter(product_prod(B,C)))),image2(A,filter(product_prod(B,C))),aa(fun(A,filter(C)),fun(A,filter(product_prod(B,C))),aTP_Lamp_ji(filter(B),fun(fun(A,filter(C)),fun(A,filter(product_prod(B,C)))),A6),B6)),J4)) ) ) ).
% prod_filter_INF2
tff(fact_1107_prod__filter__INF1,axiom,
! [C: $tType,B: $tType,A: $tType,I4: set(A),A6: fun(A,filter(B)),B6: filter(C)] :
( ( I4 != bot_bot(set(A)) )
=> ( prod_filter(B,C,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),aa(fun(A,filter(B)),fun(set(A),set(filter(B))),image2(A,filter(B)),A6),I4)),B6) = aa(set(filter(product_prod(B,C))),filter(product_prod(B,C)),complete_Inf_Inf(filter(product_prod(B,C))),aa(set(A),set(filter(product_prod(B,C))),aa(fun(A,filter(product_prod(B,C))),fun(set(A),set(filter(product_prod(B,C)))),image2(A,filter(product_prod(B,C))),aa(filter(C),fun(A,filter(product_prod(B,C))),aTP_Lamp_jj(fun(A,filter(B)),fun(filter(C),fun(A,filter(product_prod(B,C)))),A6),B6)),I4)) ) ) ).
% prod_filter_INF1
tff(fact_1108_prod__filter__INF,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType,I4: set(A),J4: set(B),A6: fun(A,filter(C)),B6: fun(B,filter(D))] :
( ( I4 != bot_bot(set(A)) )
=> ( ( J4 != bot_bot(set(B)) )
=> ( prod_filter(C,D,aa(set(filter(C)),filter(C),complete_Inf_Inf(filter(C)),aa(set(A),set(filter(C)),aa(fun(A,filter(C)),fun(set(A),set(filter(C))),image2(A,filter(C)),A6),I4)),aa(set(filter(D)),filter(D),complete_Inf_Inf(filter(D)),aa(set(B),set(filter(D)),aa(fun(B,filter(D)),fun(set(B),set(filter(D))),image2(B,filter(D)),B6),J4))) = aa(set(filter(product_prod(C,D))),filter(product_prod(C,D)),complete_Inf_Inf(filter(product_prod(C,D))),aa(set(A),set(filter(product_prod(C,D))),aa(fun(A,filter(product_prod(C,D))),fun(set(A),set(filter(product_prod(C,D)))),image2(A,filter(product_prod(C,D))),aa(fun(B,filter(D)),fun(A,filter(product_prod(C,D))),aa(fun(A,filter(C)),fun(fun(B,filter(D)),fun(A,filter(product_prod(C,D)))),aTP_Lamp_jl(set(B),fun(fun(A,filter(C)),fun(fun(B,filter(D)),fun(A,filter(product_prod(C,D))))),J4),A6),B6)),I4)) ) ) ) ).
% prod_filter_INF
tff(fact_1109_prod__filter__principal__singleton,axiom,
! [A: $tType,B: $tType,X2: A,F6: filter(B)] : prod_filter(A,B,principal(A,aa(set(A),set(A),insert(A,X2),bot_bot(set(A)))),F6) = filtermap(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),F6) ).
% prod_filter_principal_singleton
tff(fact_1110_UN__equiv__class2,axiom,
! [A: $tType,C: $tType,B: $tType,A12: set(A),R12: set(product_prod(A,A)),A23: set(B),R23: set(product_prod(B,B)),F: fun(A,fun(B,set(C))),A1: A,A22: B] :
( equiv_equiv(A,A12,R12)
=> ( equiv_equiv(B,A23,R23)
=> ( equiv_congruent2(A,B,set(C),R12,R23,F)
=> ( pp(aa(set(A),bool,member(A,A1),A12))
=> ( pp(aa(set(B),bool,member(B,A22),A23))
=> ( aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(A),set(set(C)),aa(fun(A,set(C)),fun(set(A),set(set(C))),image2(A,set(C)),aa(B,fun(A,set(C)),aa(fun(A,fun(B,set(C))),fun(B,fun(A,set(C))),aTP_Lamp_jm(set(product_prod(B,B)),fun(fun(A,fun(B,set(C))),fun(B,fun(A,set(C)))),R23),F),A22)),aa(set(A),set(A),image(A,A,R12),aa(set(A),set(A),insert(A,A1),bot_bot(set(A)))))) = aa(B,set(C),aa(A,fun(B,set(C)),F,A1),A22) ) ) ) ) ) ) ).
% UN_equiv_class2
tff(fact_1111_UN__equiv__class,axiom,
! [B: $tType,A: $tType,A6: set(A),R: set(product_prod(A,A)),F: fun(A,set(B)),A3: A] :
( equiv_equiv(A,A6,R)
=> ( equiv_congruent(A,set(B),R,F)
=> ( pp(aa(set(A),bool,member(A,A3),A6))
=> ( aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),aa(fun(A,set(B)),fun(set(A),set(set(B))),image2(A,set(B)),F),aa(set(A),set(A),image(A,A,R),aa(set(A),set(A),insert(A,A3),bot_bot(set(A)))))) = aa(A,set(B),F,A3) ) ) ) ) ).
% UN_equiv_class
tff(fact_1112_UN__equiv__class__inject,axiom,
! [B: $tType,A: $tType,A6: set(A),R: set(product_prod(A,A)),F: fun(A,set(B)),X6: set(A),Y3: set(A)] :
( equiv_equiv(A,A6,R)
=> ( equiv_congruent(A,set(B),R,F)
=> ( ( aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),aa(fun(A,set(B)),fun(set(A),set(set(B))),image2(A,set(B)),F),X6)) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),aa(fun(A,set(B)),fun(set(A),set(set(B))),image2(A,set(B)),F),Y3)) )
=> ( pp(aa(set(set(A)),bool,member(set(A),X6),equiv_quotient(A,A6,R)))
=> ( pp(aa(set(set(A)),bool,member(set(A),Y3),equiv_quotient(A,A6,R)))
=> ( ! [X: A,Y: A] :
( pp(aa(set(A),bool,member(A,X),A6))
=> ( pp(aa(set(A),bool,member(A,Y),A6))
=> ( ( aa(A,set(B),F,X) = aa(A,set(B),F,Y) )
=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R)) ) ) )
=> ( X6 = Y3 ) ) ) ) ) ) ) ).
% UN_equiv_class_inject
tff(fact_1113_proj__iff,axiom,
! [A: $tType,A6: set(A),R: set(product_prod(A,A)),X2: A,Y2: A] :
( equiv_equiv(A,A6,R)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),insert(A,X2),aa(set(A),set(A),insert(A,Y2),bot_bot(set(A))))),A6))
=> ( ( aa(A,set(A),equiv_proj(A,A,R),X2) = aa(A,set(A),equiv_proj(A,A,R),Y2) )
<=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),R)) ) ) ) ).
% proj_iff
tff(fact_1114_UN__equiv__class__type2,axiom,
! [A: $tType,B: $tType,C: $tType,A12: set(A),R12: set(product_prod(A,A)),A23: set(B),R23: set(product_prod(B,B)),F: fun(A,fun(B,set(C))),X12: set(A),X23: set(B),B6: set(set(C))] :
( equiv_equiv(A,A12,R12)
=> ( equiv_equiv(B,A23,R23)
=> ( equiv_congruent2(A,B,set(C),R12,R23,F)
=> ( pp(aa(set(set(A)),bool,member(set(A),X12),equiv_quotient(A,A12,R12)))
=> ( pp(aa(set(set(B)),bool,member(set(B),X23),equiv_quotient(B,A23,R23)))
=> ( ! [X13: A,X24: B] :
( pp(aa(set(A),bool,member(A,X13),A12))
=> ( pp(aa(set(B),bool,member(B,X24),A23))
=> pp(aa(set(set(C)),bool,member(set(C),aa(B,set(C),aa(A,fun(B,set(C)),F,X13),X24)),B6)) ) )
=> pp(aa(set(set(C)),bool,member(set(C),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(A),set(set(C)),aa(fun(A,set(C)),fun(set(A),set(set(C))),image2(A,set(C)),aa(set(B),fun(A,set(C)),aTP_Lamp_jn(fun(A,fun(B,set(C))),fun(set(B),fun(A,set(C))),F),X23)),X12))),B6)) ) ) ) ) ) ) ).
% UN_equiv_class_type2
tff(fact_1115_UN__equiv__class__type,axiom,
! [A: $tType,B: $tType,A6: set(A),R: set(product_prod(A,A)),F: fun(A,set(B)),X6: set(A),B6: set(set(B))] :
( equiv_equiv(A,A6,R)
=> ( equiv_congruent(A,set(B),R,F)
=> ( pp(aa(set(set(A)),bool,member(set(A),X6),equiv_quotient(A,A6,R)))
=> ( ! [X: A] :
( pp(aa(set(A),bool,member(A,X),A6))
=> pp(aa(set(set(B)),bool,member(set(B),aa(A,set(B),F,X)),B6)) )
=> pp(aa(set(set(B)),bool,member(set(B),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),aa(fun(A,set(B)),fun(set(A),set(set(B))),image2(A,set(B)),F),X6))),B6)) ) ) ) ) ).
% UN_equiv_class_type
tff(fact_1116_wf__UN,axiom,
! [B: $tType,A: $tType,I4: set(A),R: fun(A,set(product_prod(B,B)))] :
( ! [I6: A] :
( pp(aa(set(A),bool,member(A,I6),I4))
=> wf(B,aa(A,set(product_prod(B,B)),R,I6)) )
=> ( ! [I6: A,J3: A] :
( pp(aa(set(A),bool,member(A,I6),I4))
=> ( pp(aa(set(A),bool,member(A,J3),I4))
=> ( ( aa(A,set(product_prod(B,B)),R,I6) != aa(A,set(product_prod(B,B)),R,J3) )
=> ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(product_prod(B,B)),set(B),domain(B,B),aa(A,set(product_prod(B,B)),R,I6))),aa(set(product_prod(B,B)),set(B),range2(B,B),aa(A,set(product_prod(B,B)),R,J3))) = bot_bot(set(B)) ) ) ) )
=> wf(B,aa(set(set(product_prod(B,B))),set(product_prod(B,B)),complete_Sup_Sup(set(product_prod(B,B))),aa(set(A),set(set(product_prod(B,B))),aa(fun(A,set(product_prod(B,B))),fun(set(A),set(set(product_prod(B,B)))),image2(A,set(product_prod(B,B))),R),I4))) ) ) ).
% wf_UN
tff(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) = aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),aa(set(nat),set(set(product_prod(A,A))),aa(fun(nat,set(product_prod(A,A))),fun(set(nat),set(set(product_prod(A,A)))),image2(nat,set(product_prod(A,A))),aTP_Lamp_iy(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R3)),aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_iz(set(product_prod(A,A)),fun(nat,bool),R3)))) ) ) ).
% rtrancl_finite_eq_relpow
tff(fact_1118_relImage__proj,axiom,
! [A: $tType,A6: set(A),R3: set(product_prod(A,A))] :
( equiv_equiv(A,A6,R3)
=> pp(aa(set(product_prod(set(A),set(A))),bool,aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(A),set(A))),bool),ord_less_eq(set(product_prod(set(A),set(A)))),bNF_Gr4221423524335903396lImage(A,set(A),R3,equiv_proj(A,A,R3))),id_on(set(A),equiv_quotient(A,A6,R3)))) ) ).
% relImage_proj
tff(fact_1119_wf__insert,axiom,
! [A: $tType,Y2: A,X2: A,R: set(product_prod(A,A))] :
( wf(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),insert(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),X2)),R))
<=> ( wf(A,R)
& ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),transitive_rtrancl(A,R))) ) ) ).
% wf_insert
tff(fact_1120_Domain__insert,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,R: set(product_prod(A,B))] : aa(set(product_prod(A,B)),set(A),domain(A,B),aa(set(product_prod(A,B)),set(product_prod(A,B)),insert(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),R)) = aa(set(A),set(A),insert(A,A3),aa(set(product_prod(A,B)),set(A),domain(A,B),R)) ).
% Domain_insert
tff(fact_1121_Not__Domain__rtrancl,axiom,
! [A: $tType,X2: A,R3: set(product_prod(A,A)),Y2: A] :
( ~ pp(aa(set(A),bool,member(A,X2),aa(set(product_prod(A,A)),set(A),domain(A,A),R3)))
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),transitive_rtrancl(A,R3)))
<=> ( X2 = Y2 ) ) ) ).
% Not_Domain_rtrancl
tff(fact_1122_rtrancl_Ocases,axiom,
! [A: $tType,A1: A,A22: A,R: set(product_prod(A,A))] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),A22)),transitive_rtrancl(A,R)))
=> ( ( A22 != A1 )
=> ~ ! [B4: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),B4)),transitive_rtrancl(A,R)))
=> ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B4),A22)),R)) ) ) ) ).
% rtrancl.cases
tff(fact_1123_rtrancl_Osimps,axiom,
! [A: $tType,A1: A,A22: A,R: set(product_prod(A,A))] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),A22)),transitive_rtrancl(A,R)))
<=> ( ? [A7: A] :
( ( A1 = A7 )
& ( A22 = A7 ) )
| ? [A7: A,B5: A,C6: A] :
( ( A1 = A7 )
& ( A22 = C6 )
& pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A7),B5)),transitive_rtrancl(A,R)))
& pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B5),C6)),R)) ) ) ) ).
% rtrancl.simps
tff(fact_1124_rtrancl_Ortrancl__refl,axiom,
! [A: $tType,A3: A,R: set(product_prod(A,A))] : pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A3)),transitive_rtrancl(A,R))) ).
% rtrancl.rtrancl_refl
tff(fact_1125_rtrancl_Ortrancl__into__rtrancl,axiom,
! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A)),C3: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_rtrancl(A,R)))
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C3)),R))
=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C3)),transitive_rtrancl(A,R))) ) ) ).
% rtrancl.rtrancl_into_rtrancl
tff(fact_1126_rtranclE,axiom,
! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A))] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_rtrancl(A,R)))
=> ( ( A3 != B2 )
=> ~ ! [Y: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),Y)),transitive_rtrancl(A,R)))
=> ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),B2)),R)) ) ) ) ).
% rtranclE
tff(fact_1127_rtrancl__trans,axiom,
! [A: $tType,X2: A,Y2: A,R: set(product_prod(A,A)),Z2: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),transitive_rtrancl(A,R)))
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z2)),transitive_rtrancl(A,R)))
=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Z2)),transitive_rtrancl(A,R))) ) ) ).
% rtrancl_trans
tff(fact_1128_rtrancl__induct,axiom,
! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A)),P2: fun(A,bool)] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_rtrancl(A,R)))
=> ( pp(aa(A,bool,P2,A3))
=> ( ! [Y: A,Z3: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),Y)),transitive_rtrancl(A,R)))
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z3)),R))
=> ( pp(aa(A,bool,P2,Y))
=> pp(aa(A,bool,P2,Z3)) ) ) )
=> pp(aa(A,bool,P2,B2)) ) ) ) ).
% rtrancl_induct
tff(fact_1129_converse__rtranclE,axiom,
! [A: $tType,X2: A,Z2: A,R: set(product_prod(A,A))] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Z2)),transitive_rtrancl(A,R)))
=> ( ( X2 != Z2 )
=> ~ ! [Y: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y)),R))
=> ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2)),transitive_rtrancl(A,R))) ) ) ) ).
% converse_rtranclE
tff(fact_1130_converse__rtrancl__induct,axiom,
! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A)),P2: fun(A,bool)] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_rtrancl(A,R)))
=> ( pp(aa(A,bool,P2,B2))
=> ( ! [Y: A,Z3: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z3)),R))
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z3),B2)),transitive_rtrancl(A,R)))
=> ( pp(aa(A,bool,P2,Z3))
=> pp(aa(A,bool,P2,Y)) ) ) )
=> pp(aa(A,bool,P2,A3)) ) ) ) ).
% converse_rtrancl_induct
tff(fact_1131_converse__rtrancl__into__rtrancl,axiom,
! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A)),C3: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R))
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C3)),transitive_rtrancl(A,R)))
=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C3)),transitive_rtrancl(A,R))) ) ) ).
% converse_rtrancl_into_rtrancl
tff(fact_1132_Domain__iff,axiom,
! [B: $tType,A: $tType,A3: A,R: set(product_prod(A,B))] :
( pp(aa(set(A),bool,member(A,A3),aa(set(product_prod(A,B)),set(A),domain(A,B),R)))
<=> ? [Y4: B] : pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),Y4)),R)) ) ).
% Domain_iff
tff(fact_1133_DomainE,axiom,
! [B: $tType,A: $tType,A3: A,R: set(product_prod(A,B))] :
( pp(aa(set(A),bool,member(A,A3),aa(set(product_prod(A,B)),set(A),domain(A,B),R)))
=> ~ ! [B4: B] : ~ pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B4)),R)) ) ).
% DomainE
tff(fact_1134_Domain_ODomainI,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,R: set(product_prod(A,B))] :
( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),R))
=> pp(aa(set(A),bool,member(A,A3),aa(set(product_prod(A,B)),set(A),domain(A,B),R))) ) ).
% Domain.DomainI
tff(fact_1135_Domain_Osimps,axiom,
! [B: $tType,A: $tType,A3: A,R: set(product_prod(A,B))] :
( pp(aa(set(A),bool,member(A,A3),aa(set(product_prod(A,B)),set(A),domain(A,B),R)))
<=> ? [A7: A,B5: B] :
( ( A3 = A7 )
& pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A7),B5)),R)) ) ) ).
% Domain.simps
tff(fact_1136_Domain_Ocases,axiom,
! [B: $tType,A: $tType,A3: A,R: set(product_prod(A,B))] :
( pp(aa(set(A),bool,member(A,A3),aa(set(product_prod(A,B)),set(A),domain(A,B),R)))
=> ~ ! [B4: B] : ~ pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B4)),R)) ) ).
% Domain.cases
tff(fact_1137_rtrancl__converseD,axiom,
! [A: $tType,X2: A,Y2: A,R: set(product_prod(A,A))] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),transitive_rtrancl(A,converse(A,A,R))))
=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),X2)),transitive_rtrancl(A,R))) ) ).
% rtrancl_converseD
tff(fact_1138_rtrancl__converseI,axiom,
! [A: $tType,Y2: A,X2: A,R: set(product_prod(A,A))] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),X2)),transitive_rtrancl(A,R)))
=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),transitive_rtrancl(A,converse(A,A,R)))) ) ).
% rtrancl_converseI
tff(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))] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),P2),Q2))))
=> ( ! [X: A,Y: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),B2)),transitive_rtrancl(A,P2)))
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),X)),Q2))
=> ( Y = X ) ) )
=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_rtrancl(A,P2))) ) ) ).
% rtrancl_Un_separator_converseE
tff(fact_1140_rtrancl__Un__separatorE,axiom,
! [A: $tType,A3: A,B2: A,P2: set(product_prod(A,A)),Q2: set(product_prod(A,A))] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),P2),Q2))))
=> ( ! [X: A,Y: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),X)),transitive_rtrancl(A,P2)))
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),Q2))
=> ( X = Y ) ) )
=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_rtrancl(A,P2))) ) ) ).
% rtrancl_Un_separatorE
tff(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: fun(A,fun(B,bool))] :
( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By))),transitive_rtrancl(product_prod(A,B),R)))
=> ( pp(aa(B,bool,aa(A,fun(B,bool),P2,Ax),Ay))
=> ( ! [A5: A,B4: B,Aa2: A,Ba: B] :
( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4))),transitive_rtrancl(product_prod(A,B),R)))
=> ( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba))),R))
=> ( pp(aa(B,bool,aa(A,fun(B,bool),P2,A5),B4))
=> pp(aa(B,bool,aa(A,fun(B,bool),P2,Aa2),Ba)) ) ) )
=> pp(aa(B,bool,aa(A,fun(B,bool),P2,Bx),By)) ) ) ) ).
% rtrancl_induct2
tff(fact_1142_converse__rtranclE2,axiom,
! [B: $tType,A: $tType,Xa2: A,Xb3: B,Za2: A,Zb: B,R: set(product_prod(product_prod(A,B),product_prod(A,B)))] :
( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa2),Xb3)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Za2),Zb))),transitive_rtrancl(product_prod(A,B),R)))
=> ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa2),Xb3) != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Za2),Zb) )
=> ~ ! [A5: A,B4: B] :
( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa2),Xb3)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4))),R))
=> ~ pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Za2),Zb))),transitive_rtrancl(product_prod(A,B),R))) ) ) ) ).
% converse_rtranclE2
tff(fact_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: fun(A,fun(B,bool))] :
( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By))),transitive_rtrancl(product_prod(A,B),R)))
=> ( pp(aa(B,bool,aa(A,fun(B,bool),P2,Bx),By))
=> ( ! [A5: A,B4: B,Aa2: A,Ba: B] :
( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba))),R))
=> ( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By))),transitive_rtrancl(product_prod(A,B),R)))
=> ( pp(aa(B,bool,aa(A,fun(B,bool),P2,Aa2),Ba))
=> pp(aa(B,bool,aa(A,fun(B,bool),P2,A5),B4)) ) ) )
=> pp(aa(B,bool,aa(A,fun(B,bool),P2,Ax),Ay)) ) ) ) ).
% converse_rtrancl_induct2
tff(fact_1144_single__valued__confluent,axiom,
! [A: $tType,R: set(product_prod(A,A)),X2: A,Y2: A,Z2: A] :
( single_valued(A,A,R)
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),transitive_rtrancl(A,R)))
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Z2)),transitive_rtrancl(A,R)))
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z2)),transitive_rtrancl(A,R)))
| pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z2),Y2)),transitive_rtrancl(A,R))) ) ) ) ) ).
% single_valued_confluent
tff(fact_1145_congruent2I_H,axiom,
! [C: $tType,B: $tType,A: $tType,R12: set(product_prod(A,A)),R23: set(product_prod(B,B)),F: fun(A,fun(B,C))] :
( ! [Y12: A,Z1: A,Y23: B,Z22: B] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y12),Z1)),R12))
=> ( pp(aa(set(product_prod(B,B)),bool,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Y23),Z22)),R23))
=> ( aa(B,C,aa(A,fun(B,C),F,Y12),Y23) = aa(B,C,aa(A,fun(B,C),F,Z1),Z22) ) ) )
=> equiv_congruent2(A,B,C,R12,R23,F) ) ).
% congruent2I'
tff(fact_1146_congruent2D,axiom,
! [A: $tType,C: $tType,B: $tType,R12: set(product_prod(A,A)),R23: set(product_prod(B,B)),F: fun(A,fun(B,C)),Y1: A,Z12: A,Y22: B,Z23: B] :
( equiv_congruent2(A,B,C,R12,R23,F)
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y1),Z12)),R12))
=> ( pp(aa(set(product_prod(B,B)),bool,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Y22),Z23)),R23))
=> ( aa(B,C,aa(A,fun(B,C),F,Y1),Y22) = aa(B,C,aa(A,fun(B,C),F,Z12),Z23) ) ) ) ) ).
% congruent2D
tff(fact_1147_congruentI,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,A)),F: fun(A,B)] :
( ! [Y: A,Z3: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z3)),R))
=> ( aa(A,B,F,Y) = aa(A,B,F,Z3) ) )
=> equiv_congruent(A,B,R,F) ) ).
% congruentI
tff(fact_1148_congruentD,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,A)),F: fun(A,B),Y2: A,Z2: A] :
( equiv_congruent(A,B,R,F)
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z2)),R))
=> ( aa(A,B,F,Y2) = aa(A,B,F,Z2) ) ) ) ).
% congruentD
tff(fact_1149_fst__eq__Domain,axiom,
! [B: $tType,A: $tType,R3: set(product_prod(A,B))] : aa(set(product_prod(A,B)),set(A),aa(fun(product_prod(A,B),A),fun(set(product_prod(A,B)),set(A)),image2(product_prod(A,B),A),product_fst(A,B)),R3) = aa(set(product_prod(A,B)),set(A),domain(A,B),R3) ).
% fst_eq_Domain
tff(fact_1150_Domain__fst,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,B))] : aa(set(product_prod(A,B)),set(A),domain(A,B),R) = aa(set(product_prod(A,B)),set(A),aa(fun(product_prod(A,B),A),fun(set(product_prod(A,B)),set(A)),image2(product_prod(A,B),A),product_fst(A,B)),R) ).
% Domain_fst
tff(fact_1151_trancl__subset__Sigma__aux,axiom,
! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A)),A6: set(A)] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_rtrancl(A,R)))
=> ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R),product_Sigma(A,A,A6,aTP_Lamp_co(set(A),fun(A,set(A)),A6))))
=> ( ( A3 = B2 )
| pp(aa(set(A),bool,member(A,A3),A6)) ) ) ) ).
% trancl_subset_Sigma_aux
tff(fact_1152_proj__image,axiom,
! [A: $tType,R: set(product_prod(A,A)),A6: set(A)] : aa(set(A),set(set(A)),aa(fun(A,set(A)),fun(set(A),set(set(A))),image2(A,set(A)),equiv_proj(A,A,R)),A6) = equiv_quotient(A,A6,R) ).
% proj_image
tff(fact_1153_rtrancl__insert,axiom,
! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A))] : transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),insert(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R)) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),transitive_rtrancl(A,R)),aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(set(product_prod(A,A)),fun(A,fun(A,bool)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool))),aTP_Lamp_jo(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool)))),A3),B2),R)))) ).
% rtrancl_insert
tff(fact_1154_congruent2__commuteI,axiom,
! [B: $tType,A: $tType,A6: set(A),R: set(product_prod(A,A)),F: fun(A,fun(A,B))] :
( equiv_equiv(A,A6,R)
=> ( ! [Y: A,Z3: A] :
( pp(aa(set(A),bool,member(A,Y),A6))
=> ( pp(aa(set(A),bool,member(A,Z3),A6))
=> ( aa(A,B,aa(A,fun(A,B),F,Y),Z3) = aa(A,B,aa(A,fun(A,B),F,Z3),Y) ) ) )
=> ( ! [Y: A,Z3: A,W: A] :
( pp(aa(set(A),bool,member(A,W),A6))
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z3)),R))
=> ( aa(A,B,aa(A,fun(A,B),F,W),Y) = aa(A,B,aa(A,fun(A,B),F,W),Z3) ) ) )
=> equiv_congruent2(A,A,B,R,R,F) ) ) ) ).
% congruent2_commuteI
tff(fact_1155_congruent2I,axiom,
! [C: $tType,B: $tType,A: $tType,A12: set(A),R12: set(product_prod(A,A)),A23: set(B),R23: set(product_prod(B,B)),F: fun(A,fun(B,C))] :
( equiv_equiv(A,A12,R12)
=> ( equiv_equiv(B,A23,R23)
=> ( ! [Y: A,Z3: A,W: B] :
( pp(aa(set(B),bool,member(B,W),A23))
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z3)),R12))
=> ( aa(B,C,aa(A,fun(B,C),F,Y),W) = aa(B,C,aa(A,fun(B,C),F,Z3),W) ) ) )
=> ( ! [Y: B,Z3: B,W: A] :
( pp(aa(set(A),bool,member(A,W),A12))
=> ( pp(aa(set(product_prod(B,B)),bool,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Y),Z3)),R23))
=> ( aa(B,C,aa(A,fun(B,C),F,W),Y) = aa(B,C,aa(A,fun(B,C),F,W),Z3) ) ) )
=> equiv_congruent2(A,B,C,R12,R23,F) ) ) ) ) ).
% congruent2I
tff(fact_1156_rtrancl__is__UN__relpow,axiom,
! [A: $tType,R3: set(product_prod(A,A))] : transitive_rtrancl(A,R3) = aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),aa(set(nat),set(set(product_prod(A,A))),aa(fun(nat,set(product_prod(A,A))),fun(set(nat),set(set(product_prod(A,A)))),image2(nat,set(product_prod(A,A))),aTP_Lamp_iy(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R3)),top_top(set(nat)))) ).
% rtrancl_is_UN_relpow
tff(fact_1157_rtrancl__imp__UN__relpow,axiom,
! [A: $tType,P: product_prod(A,A),R3: set(product_prod(A,A))] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),P),transitive_rtrancl(A,R3)))
=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),P),aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),aa(set(nat),set(set(product_prod(A,A))),aa(fun(nat,set(product_prod(A,A))),fun(set(nat),set(set(product_prod(A,A)))),image2(nat,set(product_prod(A,A))),aTP_Lamp_iy(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R3)),top_top(set(nat)))))) ) ).
% rtrancl_imp_UN_relpow
tff(fact_1158_Domain__Union,axiom,
! [B: $tType,A: $tType,S3: set(set(product_prod(A,B)))] : aa(set(product_prod(A,B)),set(A),domain(A,B),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),S3)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(product_prod(A,B))),set(set(A)),aa(fun(set(product_prod(A,B)),set(A)),fun(set(set(product_prod(A,B))),set(set(A))),image2(set(product_prod(A,B)),set(A)),domain(A,B)),S3)) ).
% Domain_Union
tff(fact_1159_wf__map__prod__image__Dom__Ran,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,A)),F: fun(A,B)] :
( wf(A,R)
=> ( ! [A5: A,A11: A] :
( pp(aa(set(A),bool,member(A,A5),aa(set(product_prod(A,A)),set(A),domain(A,A),R)))
=> ( pp(aa(set(A),bool,member(A,A11),aa(set(product_prod(A,A)),set(A),range2(A,A),R)))
=> ( ( aa(A,B,F,A5) = aa(A,B,F,A11) )
=> ( A5 = A11 ) ) ) )
=> wf(B,aa(set(product_prod(A,A)),set(product_prod(B,B)),aa(fun(product_prod(A,A),product_prod(B,B)),fun(set(product_prod(A,A)),set(product_prod(B,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
tff(fact_1160_congruent2__implies__congruent__UN,axiom,
! [A: $tType,C: $tType,B: $tType,A12: set(A),R12: set(product_prod(A,A)),A23: set(B),R23: set(product_prod(B,B)),F: fun(A,fun(B,set(C))),A3: B] :
( equiv_equiv(A,A12,R12)
=> ( equiv_equiv(B,A23,R23)
=> ( equiv_congruent2(A,B,set(C),R12,R23,F)
=> ( pp(aa(set(B),bool,member(B,A3),A23))
=> equiv_congruent(A,set(C),R12,aa(B,fun(A,set(C)),aa(fun(A,fun(B,set(C))),fun(B,fun(A,set(C))),aTP_Lamp_jm(set(product_prod(B,B)),fun(fun(A,fun(B,set(C))),fun(B,fun(A,set(C)))),R23),F),A3)) ) ) ) ) ).
% congruent2_implies_congruent_UN
tff(fact_1161_equiv__proj,axiom,
! [A: $tType,A6: set(A),R3: set(product_prod(A,A)),Z2: product_prod(A,A)] :
( equiv_equiv(A,A6,R3)
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),Z2),R3))
=> ( aa(product_prod(A,A),set(A),aa(fun(product_prod(A,A),A),fun(product_prod(A,A),set(A)),aa(fun(A,set(A)),fun(fun(product_prod(A,A),A),fun(product_prod(A,A),set(A))),comp(A,set(A),product_prod(A,A)),equiv_proj(A,A,R3)),product_fst(A,A)),Z2) = aa(product_prod(A,A),set(A),aa(fun(product_prod(A,A),A),fun(product_prod(A,A),set(A)),aa(fun(A,set(A)),fun(fun(product_prod(A,A),A),fun(product_prod(A,A),set(A))),comp(A,set(A),product_prod(A,A)),equiv_proj(A,A,R3)),product_snd(A,A)),Z2) ) ) ) ).
% equiv_proj
tff(fact_1162_univ__commute,axiom,
! [B: $tType,A: $tType,A6: set(A),R: set(product_prod(A,A)),F: fun(A,B),X2: A] :
( equiv_equiv(A,A6,R)
=> ( equiv_congruent(A,B,R,F)
=> ( pp(aa(set(A),bool,member(A,X2),A6))
=> ( bNF_Greatest_univ(A,B,F,aa(A,set(A),equiv_proj(A,A,R),X2)) = aa(A,B,F,X2) ) ) ) ) ).
% univ_commute
tff(fact_1163_univ__preserves,axiom,
! [A: $tType,B: $tType,A6: set(A),R: set(product_prod(A,A)),F: fun(A,B),B6: set(B)] :
( equiv_equiv(A,A6,R)
=> ( equiv_congruent(A,B,R,F)
=> ( ! [X: A] :
( pp(aa(set(A),bool,member(A,X),A6))
=> pp(aa(set(B),bool,member(B,aa(A,B,F,X)),B6)) )
=> ! [X4: set(A)] :
( pp(aa(set(set(A)),bool,member(set(A),X4),equiv_quotient(A,A6,R)))
=> pp(aa(set(B),bool,member(B,bNF_Greatest_univ(A,B,F,X4)),B6)) ) ) ) ) ).
% univ_preserves
tff(fact_1164_acyclic__insert,axiom,
! [A: $tType,Y2: A,X2: A,R: set(product_prod(A,A))] :
( transitive_acyclic(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),insert(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),X2)),R))
<=> ( transitive_acyclic(A,R)
& ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),transitive_rtrancl(A,R))) ) ) ).
% acyclic_insert
tff(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)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),aa(set(A),set(A),image(A,A,R),aa(set(A),set(A),insert(A,A3),bot_bot(set(A))))),aa(set(A),set(A),image(A,A,R),aa(set(A),set(A),insert(A,B2),bot_bot(set(A))))))
<=> ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R)) ) ) ).
% disjnt_equiv_class
tff(fact_1166_trancl__insert,axiom,
! [A: $tType,Y2: A,X2: A,R: set(product_prod(A,A))] : transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),insert(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),X2)),R)) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),transitive_trancl(A,R)),aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(set(product_prod(A,A)),fun(A,fun(A,bool)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool))),aTP_Lamp_jo(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool)))),Y2),X2),R)))) ).
% trancl_insert
tff(fact_1167_rel__filter_Ointros,axiom,
! [A: $tType,B: $tType,R3: fun(A,fun(B,bool)),Z5: filter(product_prod(A,B)),F6: filter(A),G6: filter(B)] :
( eventually(product_prod(A,B),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),R3),Z5)
=> ( ( map_filter_on(product_prod(A,B),A,aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),R3)),product_fst(A,B),Z5) = F6 )
=> ( ( map_filter_on(product_prod(A,B),B,aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),R3)),product_snd(A,B),Z5) = G6 )
=> pp(aa(filter(B),bool,aa(filter(A),fun(filter(B),bool),rel_filter(A,B,R3),F6),G6)) ) ) ) ).
% rel_filter.intros
tff(fact_1168_rel__filter_Osimps,axiom,
! [A: $tType,B: $tType,R3: fun(A,fun(B,bool)),F6: filter(A),G6: filter(B)] :
( pp(aa(filter(B),bool,aa(filter(A),fun(filter(B),bool),rel_filter(A,B,R3),F6),G6))
<=> ? [Z7: filter(product_prod(A,B))] :
( eventually(product_prod(A,B),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),R3),Z7)
& ( map_filter_on(product_prod(A,B),A,aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),R3)),product_fst(A,B),Z7) = F6 )
& ( map_filter_on(product_prod(A,B),B,aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),R3)),product_snd(A,B),Z7) = G6 ) ) ) ).
% rel_filter.simps
tff(fact_1169_rel__filter_Ocases,axiom,
! [A: $tType,B: $tType,R3: fun(A,fun(B,bool)),F6: filter(A),G6: filter(B)] :
( pp(aa(filter(B),bool,aa(filter(A),fun(filter(B),bool),rel_filter(A,B,R3),F6),G6))
=> ~ ! [Z8: filter(product_prod(A,B))] :
( eventually(product_prod(A,B),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),R3),Z8)
=> ( ( map_filter_on(product_prod(A,B),A,aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),R3)),product_fst(A,B),Z8) = F6 )
=> ( map_filter_on(product_prod(A,B),B,aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),R3)),product_snd(A,B),Z8) != G6 ) ) ) ) ).
% rel_filter.cases
tff(fact_1170_disjnt__Times1__iff,axiom,
! [A: $tType,B: $tType,C5: set(A),A6: set(B),B6: set(B)] :
( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),disjnt(product_prod(A,B)),product_Sigma(A,B,C5,aTP_Lamp_bu(set(B),fun(A,set(B)),A6))),product_Sigma(A,B,C5,aTP_Lamp_bu(set(B),fun(A,set(B)),B6))))
<=> ( ( C5 = bot_bot(set(A)) )
| pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),disjnt(B),A6),B6)) ) ) ).
% disjnt_Times1_iff
tff(fact_1171_disjnt__Times2__iff,axiom,
! [B: $tType,A: $tType,A6: set(A),C5: set(B),B6: set(A)] :
( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),disjnt(product_prod(A,B)),product_Sigma(A,B,A6,aTP_Lamp_bu(set(B),fun(A,set(B)),C5))),product_Sigma(A,B,B6,aTP_Lamp_bu(set(B),fun(A,set(B)),C5))))
<=> ( ( C5 = bot_bot(set(B)) )
| pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),A6),B6)) ) ) ).
% disjnt_Times2_iff
tff(fact_1172_acyclicP__converse,axiom,
! [A: $tType,R: fun(A,fun(A,bool))] :
( transitive_acyclic(A,aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),conversep(A,A,R))))
<=> transitive_acyclic(A,aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),R))) ) ).
% acyclicP_converse
tff(fact_1173_acyclicI,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( ! [X: A] : ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X)),transitive_trancl(A,R)))
=> transitive_acyclic(A,R) ) ).
% acyclicI
tff(fact_1174_acyclic__def,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( transitive_acyclic(A,R)
<=> ! [X5: A] : ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),X5)),transitive_trancl(A,R))) ) ).
% acyclic_def
tff(fact_1175_converse__trancl__induct,axiom,
! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A)),P2: fun(A,bool)] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_trancl(A,R)))
=> ( ! [Y: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),B2)),R))
=> pp(aa(A,bool,P2,Y)) )
=> ( ! [Y: A,Z3: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z3)),R))
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z3),B2)),transitive_trancl(A,R)))
=> ( pp(aa(A,bool,P2,Z3))
=> pp(aa(A,bool,P2,Y)) ) ) )
=> pp(aa(A,bool,P2,A3)) ) ) ) ).
% converse_trancl_induct
tff(fact_1176_trancl__trans__induct,axiom,
! [A: $tType,X2: A,Y2: A,R: set(product_prod(A,A)),P2: fun(A,fun(A,bool))] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),transitive_trancl(A,R)))
=> ( ! [X: A,Y: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R))
=> pp(aa(A,bool,aa(A,fun(A,bool),P2,X),Y)) )
=> ( ! [X: A,Y: A,Z3: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_trancl(A,R)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),P2,X),Y))
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z3)),transitive_trancl(A,R)))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),P2,Y),Z3))
=> pp(aa(A,bool,aa(A,fun(A,bool),P2,X),Z3)) ) ) ) )
=> pp(aa(A,bool,aa(A,fun(A,bool),P2,X2),Y2)) ) ) ) ).
% trancl_trans_induct
tff(fact_1177_trancl__into__trancl2,axiom,
! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A)),C3: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R))
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C3)),transitive_trancl(A,R)))
=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C3)),transitive_trancl(A,R))) ) ) ).
% trancl_into_trancl2
tff(fact_1178_Transitive__Closure_Otrancl__into__trancl,axiom,
! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A)),C3: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_trancl(A,R)))
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C3)),R))
=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C3)),transitive_trancl(A,R))) ) ) ).
% Transitive_Closure.trancl_into_trancl
tff(fact_1179_irrefl__trancl__rD,axiom,
! [A: $tType,R: set(product_prod(A,A)),X2: A,Y2: A] :
( ! [X: A] : ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X)),transitive_trancl(A,R)))
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),R))
=> ( X2 != Y2 ) ) ) ).
% irrefl_trancl_rD
tff(fact_1180_converse__tranclE,axiom,
! [A: $tType,X2: A,Z2: A,R: set(product_prod(A,A))] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Z2)),transitive_trancl(A,R)))
=> ( ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Z2)),R))
=> ~ ! [Y: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y)),R))
=> ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2)),transitive_trancl(A,R))) ) ) ) ).
% converse_tranclE
tff(fact_1181_r__r__into__trancl,axiom,
! [A: $tType,A3: A,B2: A,R3: set(product_prod(A,A)),C3: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R3))
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C3)),R3))
=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C3)),transitive_trancl(A,R3))) ) ) ).
% r_r_into_trancl
tff(fact_1182_trancl__induct,axiom,
! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A)),P2: fun(A,bool)] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_trancl(A,R)))
=> ( ! [Y: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),Y)),R))
=> pp(aa(A,bool,P2,Y)) )
=> ( ! [Y: A,Z3: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),Y)),transitive_trancl(A,R)))
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z3)),R))
=> ( pp(aa(A,bool,P2,Y))
=> pp(aa(A,bool,P2,Z3)) ) ) )
=> pp(aa(A,bool,P2,B2)) ) ) ) ).
% trancl_induct
tff(fact_1183_trancl__trans,axiom,
! [A: $tType,X2: A,Y2: A,R: set(product_prod(A,A)),Z2: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),transitive_trancl(A,R)))
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z2)),transitive_trancl(A,R)))
=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Z2)),transitive_trancl(A,R))) ) ) ).
% trancl_trans
tff(fact_1184_tranclE,axiom,
! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A))] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_trancl(A,R)))
=> ( ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R))
=> ~ ! [C2: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C2)),transitive_trancl(A,R)))
=> ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),C2),B2)),R)) ) ) ) ).
% tranclE
tff(fact_1185_trancl_Or__into__trancl,axiom,
! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A))] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R))
=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_trancl(A,R))) ) ).
% trancl.r_into_trancl
tff(fact_1186_trancl_Osimps,axiom,
! [A: $tType,A1: A,A22: A,R: set(product_prod(A,A))] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),A22)),transitive_trancl(A,R)))
<=> ( ? [A7: A,B5: A] :
( ( A1 = A7 )
& ( A22 = B5 )
& pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A7),B5)),R)) )
| ? [A7: A,B5: A,C6: A] :
( ( A1 = A7 )
& ( A22 = C6 )
& pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A7),B5)),transitive_trancl(A,R)))
& pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B5),C6)),R)) ) ) ) ).
% trancl.simps
tff(fact_1187_trancl_Ocases,axiom,
! [A: $tType,A1: A,A22: A,R: set(product_prod(A,A))] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),A22)),transitive_trancl(A,R)))
=> ( ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),A22)),R))
=> ~ ! [B4: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A1),B4)),transitive_trancl(A,R)))
=> ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B4),A22)),R)) ) ) ) ).
% trancl.cases
tff(fact_1188_rel__filter__distr,axiom,
! [A: $tType,B: $tType,C: $tType,A6: fun(A,fun(C,bool)),B6: fun(C,fun(B,bool))] : aa(fun(filter(C),fun(filter(B),bool)),fun(filter(A),fun(filter(B),bool)),aa(fun(filter(A),fun(filter(C),bool)),fun(fun(filter(C),fun(filter(B),bool)),fun(filter(A),fun(filter(B),bool))),relcompp(filter(A),filter(C),filter(B)),rel_filter(A,C,A6)),rel_filter(C,B,B6)) = rel_filter(A,B,aa(fun(C,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(C,bool)),fun(fun(C,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,C,B),A6),B6)) ).
% rel_filter_distr
tff(fact_1189_disjnt__Sigma__iff,axiom,
! [B: $tType,A: $tType,A6: set(A),C5: fun(A,set(B)),B6: set(A)] :
( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),disjnt(product_prod(A,B)),product_Sigma(A,B,A6,C5)),product_Sigma(A,B,B6,C5)))
<=> ( ! [X5: A] :
( pp(aa(set(A),bool,member(A,X5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A6),B6)))
=> ( aa(A,set(B),C5,X5) = bot_bot(set(B)) ) )
| pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),A6),B6)) ) ) ).
% disjnt_Sigma_iff
tff(fact_1190_trancl__rtrancl__trancl,axiom,
! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A)),C3: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_trancl(A,R)))
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C3)),transitive_rtrancl(A,R)))
=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C3)),transitive_trancl(A,R))) ) ) ).
% trancl_rtrancl_trancl
tff(fact_1191_rtrancl__trancl__trancl,axiom,
! [A: $tType,X2: A,Y2: A,R: set(product_prod(A,A)),Z2: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),transitive_rtrancl(A,R)))
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z2)),transitive_trancl(A,R)))
=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Z2)),transitive_trancl(A,R))) ) ) ).
% rtrancl_trancl_trancl
tff(fact_1192_rtrancl__into__trancl2,axiom,
! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A)),C3: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R))
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C3)),transitive_rtrancl(A,R)))
=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C3)),transitive_trancl(A,R))) ) ) ).
% rtrancl_into_trancl2
tff(fact_1193_rtrancl__into__trancl1,axiom,
! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A)),C3: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_rtrancl(A,R)))
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),C3)),R))
=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),C3)),transitive_trancl(A,R))) ) ) ).
% rtrancl_into_trancl1
tff(fact_1194_rtrancl__eq__or__trancl,axiom,
! [A: $tType,X2: A,Y2: A,R3: set(product_prod(A,A))] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),transitive_rtrancl(A,R3)))
<=> ( ( X2 = Y2 )
| ( ( X2 != Y2 )
& pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),transitive_trancl(A,R3))) ) ) ) ).
% rtrancl_eq_or_trancl
tff(fact_1195_trancl__into__rtrancl,axiom,
! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A))] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_trancl(A,R)))
=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_rtrancl(A,R))) ) ).
% trancl_into_rtrancl
tff(fact_1196_tranclD2,axiom,
! [A: $tType,X2: A,Y2: A,R3: set(product_prod(A,A))] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),transitive_trancl(A,R3)))
=> ? [Z3: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Z3)),transitive_rtrancl(A,R3)))
& pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z3),Y2)),R3)) ) ) ).
% tranclD2
tff(fact_1197_rtranclD,axiom,
! [A: $tType,A3: A,B2: A,R3: set(product_prod(A,A))] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_rtrancl(A,R3)))
=> ( ( A3 = B2 )
| ( ( A3 != B2 )
& pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_trancl(A,R3))) ) ) ) ).
% rtranclD
tff(fact_1198_tranclD,axiom,
! [A: $tType,X2: A,Y2: A,R3: set(product_prod(A,A))] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),transitive_trancl(A,R3)))
=> ? [Z3: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Z3)),R3))
& pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z3),Y2)),transitive_rtrancl(A,R3))) ) ) ).
% tranclD
tff(fact_1199_trancl__converseI,axiom,
! [A: $tType,X2: A,Y2: A,R: set(product_prod(A,A))] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),converse(A,A,transitive_trancl(A,R))))
=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),transitive_trancl(A,converse(A,A,R)))) ) ).
% trancl_converseI
tff(fact_1200_trancl__converseD,axiom,
! [A: $tType,X2: A,Y2: A,R: set(product_prod(A,A))] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),transitive_trancl(A,converse(A,A,R))))
=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),converse(A,A,transitive_trancl(A,R)))) ) ).
% trancl_converseD
tff(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: fun(A,fun(B,bool))] :
( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By))),transitive_trancl(product_prod(A,B),R)))
=> ( ! [A5: A,B4: B] :
( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4))),R))
=> pp(aa(B,bool,aa(A,fun(B,bool),P2,A5),B4)) )
=> ( ! [A5: A,B4: B,Aa2: A,Ba: B] :
( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4))),transitive_trancl(product_prod(A,B),R)))
=> ( pp(aa(set(product_prod(product_prod(A,B),product_prod(A,B))),bool,member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba))),R))
=> ( pp(aa(B,bool,aa(A,fun(B,bool),P2,A5),B4))
=> pp(aa(B,bool,aa(A,fun(B,bool),P2,Aa2),Ba)) ) ) )
=> pp(aa(B,bool,aa(A,fun(B,bool),P2,Bx),By)) ) ) ) ).
% trancl_induct2
tff(fact_1202_disjoint__UN__iff,axiom,
! [A: $tType,B: $tType,A6: set(A),B6: fun(B,set(A)),I4: set(B)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),A6),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),B6),I4))))
<=> ! [X5: B] :
( pp(aa(set(B),bool,member(B,X5),I4))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),A6),aa(B,set(A),B6,X5))) ) ) ).
% disjoint_UN_iff
tff(fact_1203_disjnt__inj__on__iff,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A14: set(set(A)),X6: set(A),Y3: set(A)] :
( inj_on(A,B,F,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A14))
=> ( pp(aa(set(set(A)),bool,member(set(A),X6),A14))
=> ( pp(aa(set(set(A)),bool,member(set(A),Y3),A14))
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),disjnt(B),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),X6)),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),Y3)))
<=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),disjnt(A),X6),Y3)) ) ) ) ) ).
% disjnt_inj_on_iff
tff(fact_1204_trancl__insert2,axiom,
! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A))] : transitive_trancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),insert(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R)) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),transitive_trancl(A,R)),aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(set(product_prod(A,A)),fun(A,fun(A,bool)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool))),aTP_Lamp_jp(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool)))),A3),B2),R)))) ).
% trancl_insert2
tff(fact_1205_irrefl__tranclI,axiom,
! [A: $tType,R: set(product_prod(A,A)),X2: A] :
( ( aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),converse(A,A,R)),transitive_rtrancl(A,R)) = bot_bot(set(product_prod(A,A))) )
=> ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),X2)),transitive_trancl(A,R))) ) ).
% irrefl_tranclI
tff(fact_1206_wf__linord__ex__has__least,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,A)),P2: fun(B,bool),K: B,M4: fun(B,A)] :
( wf(A,R)
=> ( ! [X: A,Y: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),transitive_trancl(A,R)))
<=> ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),X)),transitive_rtrancl(A,R))) )
=> ( pp(aa(B,bool,P2,K))
=> ? [X: B] :
( pp(aa(B,bool,P2,X))
& ! [Y5: B] :
( pp(aa(B,bool,P2,Y5))
=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(B,A,M4,X)),aa(B,A,M4,Y5))),transitive_rtrancl(A,R))) ) ) ) ) ) ).
% wf_linord_ex_has_least
tff(fact_1207_Range__def,axiom,
! [B: $tType,A: $tType,X4: set(product_prod(A,B))] : aa(set(product_prod(A,B)),set(B),range2(A,B),X4) = aa(fun(B,bool),set(B),collect(B),rangep(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_bg(set(product_prod(A,B)),fun(A,fun(B,bool))),X4))) ).
% Range_def
tff(fact_1208_Rangep__Range__eq,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,B)),X4: B] :
( pp(aa(B,bool,rangep(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_bg(set(product_prod(A,B)),fun(A,fun(B,bool))),R)),X4))
<=> pp(aa(set(B),bool,member(B,X4),aa(set(product_prod(A,B)),set(B),range2(A,B),R))) ) ).
% Rangep_Range_eq
tff(fact_1209_Least__mono,axiom,
! [B: $tType,A: $tType] :
( ( order(A)
& order(B) )
=> ! [F: fun(A,B),S3: set(A)] :
( order_mono(A,B,F)
=> ( ? [X4: A] :
( pp(aa(set(A),bool,member(A,X4),S3))
& ! [Xa: A] :
( pp(aa(set(A),bool,member(A,Xa),S3))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Xa)) ) )
=> ( ord_Least(B,aa(set(A),fun(B,bool),aTP_Lamp_jq(fun(A,B),fun(set(A),fun(B,bool)),F),S3)) = aa(A,B,F,ord_Least(A,aTP_Lamp_jr(set(A),fun(A,bool),S3))) ) ) ) ) ).
% Least_mono
tff(fact_1210_filtercomap__SUP,axiom,
! [A: $tType,C: $tType,B: $tType,F: fun(A,C),F6: fun(B,filter(C)),B6: set(B)] : pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),aa(set(filter(A)),filter(A),complete_Sup_Sup(filter(A)),aa(set(B),set(filter(A)),aa(fun(B,filter(A)),fun(set(B),set(filter(A))),image2(B,filter(A)),aa(fun(B,filter(C)),fun(B,filter(A)),aTP_Lamp_js(fun(A,C),fun(fun(B,filter(C)),fun(B,filter(A))),F),F6)),B6))),filtercomap(A,C,F,aa(set(filter(C)),filter(C),complete_Sup_Sup(filter(C)),aa(set(B),set(filter(C)),aa(fun(B,filter(C)),fun(set(B),set(filter(C))),image2(B,filter(C)),F6),B6))))) ).
% filtercomap_SUP
tff(fact_1211_bind__singleton__conv__image,axiom,
! [A: $tType,B: $tType,A6: set(B),F: fun(B,A)] : bind4(B,A,A6,aTP_Lamp_du(fun(B,A),fun(B,set(A)),F)) = aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6) ).
% bind_singleton_conv_image
tff(fact_1212_underS__incl__iff,axiom,
! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A] :
( order_679001287576687338der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R),R)
=> ( pp(aa(set(A),bool,member(A,A3),aa(set(product_prod(A,A)),set(A),field2(A),R)))
=> ( pp(aa(set(A),bool,member(A,B2),aa(set(product_prod(A,A)),set(A),field2(A),R)))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),order_underS(A,R,A3)),order_underS(A,R,B2)))
<=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R)) ) ) ) ) ).
% underS_incl_iff
tff(fact_1213_finite__subsets__at__top__def,axiom,
! [A: $tType,A6: set(A)] : finite5375528669736107172at_top(A,A6) = aa(set(filter(set(A))),filter(set(A)),complete_Inf_Inf(filter(set(A))),aa(set(set(A)),set(filter(set(A))),aa(fun(set(A),filter(set(A))),fun(set(set(A)),set(filter(set(A)))),image2(set(A),filter(set(A))),aTP_Lamp_ju(set(A),fun(set(A),filter(set(A))),A6)),aa(fun(set(A),bool),set(set(A)),collect(set(A)),aTP_Lamp_jv(set(A),fun(set(A),bool),A6)))) ).
% finite_subsets_at_top_def
tff(fact_1214_cINF__less__iff,axiom,
! [B: $tType,A: $tType] :
( condit6923001295902523014norder(A)
=> ! [A6: set(B),F: fun(B,A),A3: A] :
( ( A6 != bot_bot(set(B)) )
=> ( condit1013018076250108175_below(A,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))),A3))
<=> ? [X5: B] :
( pp(aa(set(B),bool,member(B,X5),A6))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F,X5)),A3)) ) ) ) ) ) ).
% cINF_less_iff
tff(fact_1215_member__bind,axiom,
! [A: $tType,B: $tType,X2: A,A6: set(B),F: fun(B,set(A))] :
( pp(aa(set(A),bool,member(A,X2),bind4(B,A,A6,F)))
<=> pp(aa(set(A),bool,member(A,X2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),F),A6)))) ) ).
% member_bind
tff(fact_1216_SUP__eq__top__iff,axiom,
! [A: $tType,B: $tType] :
( comple5582772986160207858norder(A)
=> ! [F: fun(B,A),A6: set(B)] :
( ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6)) = top_top(A) )
<=> ! [X5: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X5),top_top(A)))
=> ? [Xa3: B] :
( pp(aa(set(B),bool,member(B,Xa3),A6))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X5),aa(B,A,F,Xa3))) ) ) ) ) ).
% SUP_eq_top_iff
tff(fact_1217_INF__eq__bot__iff,axiom,
! [B: $tType,A: $tType] :
( comple5582772986160207858norder(A)
=> ! [F: fun(B,A),A6: set(B)] :
( ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6)) = bot_bot(A) )
<=> ! [X5: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),bot_bot(A)),X5))
=> ? [Xa3: B] :
( pp(aa(set(B),bool,member(B,Xa3),A6))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F,Xa3)),X5)) ) ) ) ) ).
% INF_eq_bot_iff
tff(fact_1218_nat__descend__induct,axiom,
! [N4: nat,P2: fun(nat,bool),M4: nat] :
( ! [K3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N4),K3))
=> pp(aa(nat,bool,P2,K3)) )
=> ( ! [K3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K3),N4))
=> ( ! [I7: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K3),I7))
=> pp(aa(nat,bool,P2,I7)) )
=> pp(aa(nat,bool,P2,K3)) ) )
=> pp(aa(nat,bool,P2,M4)) ) ) ).
% nat_descend_induct
tff(fact_1219_underS__I,axiom,
! [A: $tType,I: A,J5: A,R3: set(product_prod(A,A))] :
( ( I != J5 )
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),I),J5)),R3))
=> pp(aa(set(A),bool,member(A,I),order_underS(A,R3,J5))) ) ) ).
% underS_I
tff(fact_1220_underS__E,axiom,
! [A: $tType,I: A,R3: set(product_prod(A,A)),J5: A] :
( pp(aa(set(A),bool,member(A,I),order_underS(A,R3,J5)))
=> ( ( I != J5 )
& pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),I),J5)),R3)) ) ) ).
% underS_E
tff(fact_1221_BNF__Least__Fixpoint_OunderS__Field,axiom,
! [A: $tType,I: A,R3: set(product_prod(A,A)),J5: A] :
( pp(aa(set(A),bool,member(A,I),order_underS(A,R3,J5)))
=> pp(aa(set(A),bool,member(A,I),aa(set(product_prod(A,A)),set(A),field2(A),R3))) ) ).
% BNF_Least_Fixpoint.underS_Field
tff(fact_1222_finite__imp__inj__to__nat__seg,axiom,
! [A: $tType,A6: set(A)] :
( finite_finite2(A,A6)
=> ? [F3: fun(A,nat),N2: nat] :
( ( aa(set(A),set(nat),aa(fun(A,nat),fun(set(A),set(nat)),image2(A,nat),F3),A6) = aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_jw(nat,fun(nat,bool),N2)) )
& inj_on(A,nat,F3,A6) ) ) ).
% finite_imp_inj_to_nat_seg
tff(fact_1223_Field__natLeq__on,axiom,
! [N4: nat] : aa(set(product_prod(nat,nat)),set(nat),field2(nat),aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_jx(nat,fun(nat,fun(nat,bool)),N4)))) = aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_jw(nat,fun(nat,bool),N4)) ).
% Field_natLeq_on
tff(fact_1224_natLeq__on__well__order__on,axiom,
! [N4: nat] : order_well_order_on(nat,aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_jw(nat,fun(nat,bool),N4)),aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_jx(nat,fun(nat,fun(nat,bool)),N4)))) ).
% natLeq_on_well_order_on
tff(fact_1225_natLeq__on__Well__order,axiom,
! [N4: nat] : order_well_order_on(nat,aa(set(product_prod(nat,nat)),set(nat),field2(nat),aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_jx(nat,fun(nat,fun(nat,bool)),N4)))),aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_jx(nat,fun(nat,fun(nat,bool)),N4)))) ).
% natLeq_on_Well_order
tff(fact_1226_natLeq__on__wo__rel,axiom,
! [N4: nat] : bNF_Wellorder_wo_rel(nat,aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_jx(nat,fun(nat,fun(nat,bool)),N4)))) ).
% natLeq_on_wo_rel
tff(fact_1227_underS__def,axiom,
! [A: $tType,R: set(product_prod(A,A)),A3: A] : order_underS(A,R,A3) = aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_jy(set(product_prod(A,A)),fun(A,fun(A,bool)),R),A3)) ).
% underS_def
tff(fact_1228_inj__on__strict__subset,axiom,
! [B: $tType,A: $tType,F: fun(A,B),B6: set(A),A6: set(A)] :
( inj_on(A,B,F,B6)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A6),B6))
=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less(set(B)),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6)),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),B6))) ) ) ).
% inj_on_strict_subset
tff(fact_1229_acyclicI__order,axiom,
! [A: $tType,B: $tType] :
( preorder(A)
=> ! [R: set(product_prod(B,B)),F: fun(B,A)] :
( ! [A5: B,B4: B] :
( pp(aa(set(product_prod(B,B)),bool,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),A5),B4)),R))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F,B4)),aa(B,A,F,A5))) )
=> transitive_acyclic(B,R) ) ) ).
% acyclicI_order
tff(fact_1230_SUP__lessD,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [F: fun(B,A),A6: set(B),Y2: A,I: B] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))),Y2))
=> ( pp(aa(set(B),bool,member(B,I),A6))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F,I)),Y2)) ) ) ) ).
% SUP_lessD
tff(fact_1231_less__SUP__iff,axiom,
! [A: $tType,B: $tType] :
( comple5582772986160207858norder(A)
=> ! [A3: A,F: fun(B,A),A6: set(B)] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))))
<=> ? [X5: B] :
( pp(aa(set(B),bool,member(B,X5),A6))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(B,A,F,X5))) ) ) ) ).
% less_SUP_iff
tff(fact_1232_less__INF__D,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [Y2: A,F: fun(B,A),A6: set(B),I: B] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))))
=> ( pp(aa(set(B),bool,member(B,I),A6))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),aa(B,A,F,I))) ) ) ) ).
% less_INF_D
tff(fact_1233_INF__less__iff,axiom,
! [B: $tType,A: $tType] :
( comple5582772986160207858norder(A)
=> ! [F: fun(B,A),A6: set(B),A3: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))),A3))
<=> ? [X5: B] :
( pp(aa(set(B),bool,member(B,X5),A6))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F,X5)),A3)) ) ) ) ).
% INF_less_iff
tff(fact_1234_filterlim__filtercomap__iff,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(A,B),G: fun(B,C),G6: filter(C),F6: filter(A)] :
( filterlim(A,B,F,filtercomap(B,C,G,G6),F6)
<=> filterlim(A,C,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),G),F),G6,F6) ) ).
% filterlim_filtercomap_iff
tff(fact_1235_wf,axiom,
! [A: $tType] :
( wellorder(A)
=> wf(A,aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),ord_less(A)))) ) ).
% wf
tff(fact_1236_finite__conv__nat__seg__image,axiom,
! [A: $tType,A6: set(A)] :
( finite_finite2(A,A6)
<=> ? [N5: nat,F7: fun(nat,A)] : A6 = aa(set(nat),set(A),aa(fun(nat,A),fun(set(nat),set(A)),image2(nat,A),F7),aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_jw(nat,fun(nat,bool),N5))) ) ).
% finite_conv_nat_seg_image
tff(fact_1237_nat__seg__image__imp__finite,axiom,
! [A: $tType,A6: set(A),F: fun(nat,A),N4: nat] :
( ( A6 = aa(set(nat),set(A),aa(fun(nat,A),fun(set(nat),set(A)),image2(nat,A),F),aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_jw(nat,fun(nat,bool),N4))) )
=> finite_finite2(A,A6) ) ).
% nat_seg_image_imp_finite
tff(fact_1238_wf__if__measure,axiom,
! [A: $tType,P2: fun(A,bool),F: fun(A,nat),G: fun(A,A)] :
( ! [X: A] :
( pp(aa(A,bool,P2,X))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F,aa(A,A,G,X))),aa(A,nat,F,X))) )
=> wf(A,aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(fun(A,A),fun(A,fun(A,bool)),aTP_Lamp_jz(fun(A,bool),fun(fun(A,A),fun(A,fun(A,bool))),P2),G)))) ) ).
% wf_if_measure
tff(fact_1239_bind__UNION,axiom,
! [A: $tType,B: $tType,A6: set(B),F: fun(B,set(A))] : bind4(B,A,A6,F) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),F),A6)) ).
% bind_UNION
tff(fact_1240_le__SUP__iff,axiom,
! [A: $tType,B: $tType] :
( comple5582772986160207858norder(A)
=> ! [X2: A,F: fun(B,A),A6: set(B)] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))))
<=> ! [Y4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y4),X2))
=> ? [X5: B] :
( pp(aa(set(B),bool,member(B,X5),A6))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y4),aa(B,A,F,X5))) ) ) ) ) ).
% le_SUP_iff
tff(fact_1241_INF__le__iff,axiom,
! [B: $tType,A: $tType] :
( comple5582772986160207858norder(A)
=> ! [F: fun(B,A),A6: set(B),X2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))),X2))
<=> ! [Y4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Y4))
=> ? [X5: B] :
( pp(aa(set(B),bool,member(B,X5),A6))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F,X5)),Y4)) ) ) ) ) ).
% INF_le_iff
tff(fact_1242_bind__image,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(C,B),A6: set(C),G: fun(B,set(A))] : bind4(B,A,aa(set(C),set(B),aa(fun(C,B),fun(set(C),set(B)),image2(C,B),F),A6),G) = bind4(C,A,A6,aa(fun(C,B),fun(C,set(A)),aa(fun(B,set(A)),fun(fun(C,B),fun(C,set(A))),comp(B,set(A),C),G),F)) ).
% bind_image
tff(fact_1243_filtercomap__INF,axiom,
! [B: $tType,A: $tType,C: $tType,F: fun(A,B),F6: fun(C,filter(B)),B6: set(C)] : filtercomap(A,B,F,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(C),set(filter(B)),aa(fun(C,filter(B)),fun(set(C),set(filter(B))),image2(C,filter(B)),F6),B6))) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(C),set(filter(A)),aa(fun(C,filter(A)),fun(set(C),set(filter(A))),image2(C,filter(A)),aa(fun(C,filter(B)),fun(C,filter(A)),aTP_Lamp_ka(fun(A,B),fun(fun(C,filter(B)),fun(C,filter(A))),F),F6)),B6)) ).
% filtercomap_INF
tff(fact_1244_pigeonhole,axiom,
! [A: $tType,B: $tType,F: fun(B,A),A6: set(B)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(set(A),nat,finite_card(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))),aa(set(B),nat,finite_card(B),A6)))
=> ~ inj_on(B,A,F,A6) ) ).
% pigeonhole
tff(fact_1245_wf__bounded__measure,axiom,
! [A: $tType,R: set(product_prod(A,A)),Ub: fun(A,nat),F: fun(A,nat)] :
( ! [A5: A,B4: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B4),A5)),R))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,Ub,B4)),aa(A,nat,Ub,A5)))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,F,B4)),aa(A,nat,Ub,A5)))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F,A5)),aa(A,nat,F,B4))) ) )
=> wf(A,R) ) ).
% wf_bounded_measure
tff(fact_1246_image__bind,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(B,A),A6: set(C),G: fun(C,set(B))] : aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),bind4(C,B,A6,G)) = bind4(C,A,A6,aa(fun(C,set(B)),fun(C,set(A)),aa(fun(set(B),set(A)),fun(fun(C,set(B)),fun(C,set(A))),comp(set(B),set(A),C),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F)),G)) ).
% image_bind
tff(fact_1247_cSUP__lessD,axiom,
! [B: $tType,A: $tType] :
( condit1219197933456340205attice(A)
=> ! [F: fun(B,A),A6: set(B),Y2: A,I: B] :
( condit941137186595557371_above(A,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))),Y2))
=> ( pp(aa(set(B),bool,member(B,I),A6))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F,I)),Y2)) ) ) ) ) ).
% cSUP_lessD
tff(fact_1248_filtercomap__neq__bot__surj,axiom,
! [A: $tType,B: $tType,F6: filter(A),F: fun(B,A)] :
( ( F6 != bot_bot(filter(A)) )
=> ( ( aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),top_top(set(B))) = top_top(set(A)) )
=> ( filtercomap(B,A,F,F6) != bot_bot(filter(B)) ) ) ) ).
% filtercomap_neq_bot_surj
tff(fact_1249_less__cINF__D,axiom,
! [A: $tType,B: $tType] :
( condit1219197933456340205attice(A)
=> ! [F: fun(B,A),A6: set(B),Y2: A,I: B] :
( condit1013018076250108175_below(A,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))))
=> ( pp(aa(set(B),bool,member(B,I),A6))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y2),aa(B,A,F,I))) ) ) ) ) ).
% less_cINF_D
tff(fact_1250_finite__mono__strict__prefix__implies__finite__fixpoint,axiom,
! [A: $tType,F: fun(nat,set(A)),S3: set(A)] :
( ! [I6: nat] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(nat,set(A),F,I6)),S3))
=> ( finite_finite2(A,S3)
=> ( ? [N6: nat] :
( ! [N2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),N6))
=> ! [M3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M3),N6))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M3),N2))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),aa(nat,set(A),F,M3)),aa(nat,set(A),F,N2))) ) ) )
& ! [N2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N6),N2))
=> ( aa(nat,set(A),F,N6) = aa(nat,set(A),F,N2) ) ) )
=> ( aa(nat,set(A),F,aa(set(A),nat,finite_card(A),S3)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),aa(fun(nat,set(A)),fun(set(nat),set(set(A))),image2(nat,set(A)),F),top_top(set(nat)))) ) ) ) ) ).
% finite_mono_strict_prefix_implies_finite_fixpoint
tff(fact_1251_finite__subset__wf,axiom,
! [A: $tType,A6: set(A)] :
( finite_finite2(A,A6)
=> wf(set(A),aa(fun(product_prod(set(A),set(A)),bool),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aa(fun(set(A),fun(set(A),bool)),fun(product_prod(set(A),set(A)),bool),product_case_prod(set(A),set(A),bool),aTP_Lamp_kb(set(A),fun(set(A),fun(set(A),bool)),A6)))) ) ).
% finite_subset_wf
tff(fact_1252_finite__imp__nat__seg__image__inj__on,axiom,
! [A: $tType,A6: set(A)] :
( finite_finite2(A,A6)
=> ? [N2: nat,F3: fun(nat,A)] :
( ( A6 = aa(set(nat),set(A),aa(fun(nat,A),fun(set(nat),set(A)),image2(nat,A),F3),aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_jw(nat,fun(nat,bool),N2))) )
& inj_on(nat,A,F3,aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_jw(nat,fun(nat,bool),N2))) ) ) ).
% finite_imp_nat_seg_image_inj_on
tff(fact_1253_filtermap__image__finite__subsets__at__top,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A6: set(A)] :
( inj_on(A,B,F,A6)
=> ( filtermap(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),finite5375528669736107172at_top(A,A6)) = finite5375528669736107172at_top(B,aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6)) ) ) ).
% filtermap_image_finite_subsets_at_top
tff(fact_1254_wf__bounded__set,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,A)),Ub: fun(A,set(B)),F: fun(A,set(B))] :
( ! [A5: A,B4: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B4),A5)),R))
=> ( finite_finite2(B,aa(A,set(B),Ub,A5))
& pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),Ub,B4)),aa(A,set(B),Ub,A5)))
& pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),F,B4)),aa(A,set(B),Ub,A5)))
& pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less(set(B)),aa(A,set(B),F,A5)),aa(A,set(B),F,B4))) ) )
=> wf(A,R) ) ).
% wf_bounded_set
tff(fact_1255_less__cSUP__iff,axiom,
! [A: $tType,B: $tType] :
( condit6923001295902523014norder(A)
=> ! [A6: set(B),F: fun(B,A),A3: A] :
( ( A6 != bot_bot(set(B)) )
=> ( condit941137186595557371_above(A,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))))
<=> ? [X5: B] :
( pp(aa(set(B),bool,member(B,X5),A6))
& pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A3),aa(B,A,F,X5))) ) ) ) ) ) ).
% less_cSUP_iff
tff(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) = aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),aa(set(nat),set(set(product_prod(A,A))),aa(fun(nat,set(product_prod(A,A))),fun(set(nat),set(set(product_prod(A,A)))),image2(nat,set(product_prod(A,A))),aTP_Lamp_iy(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R3)),aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_kc(set(product_prod(A,A)),fun(nat,bool),R3)))) ) ) ).
% trancl_finite_eq_relpow
tff(fact_1257_relpow__finite__bounded1,axiom,
! [A: $tType,R3: set(product_prod(A,A)),K: nat] :
( finite_finite2(product_prod(A,A),R3)
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
=> pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),compow(set(product_prod(A,A)),K,R3)),aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),aa(set(nat),set(set(product_prod(A,A))),aa(fun(nat,set(product_prod(A,A))),fun(set(nat),set(set(product_prod(A,A)))),image2(nat,set(product_prod(A,A))),aTP_Lamp_iy(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R3)),aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_kc(set(product_prod(A,A)),fun(nat,bool),R3)))))) ) ) ).
% relpow_finite_bounded1
tff(fact_1258_finite__psubset__def,axiom,
! [A: $tType] : finite_psubset(A) = aa(fun(product_prod(set(A),set(A)),bool),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aa(fun(set(A),fun(set(A),bool)),fun(product_prod(set(A),set(A)),bool),product_case_prod(set(A),set(A),bool),aTP_Lamp_kd(set(A),fun(set(A),bool)))) ).
% finite_psubset_def
tff(fact_1259_mlex__eq,axiom,
! [A: $tType,F: fun(A,nat),R3: set(product_prod(A,A))] : mlex_prod(A,F,R3) = aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(set(product_prod(A,A)),fun(A,fun(A,bool)),aTP_Lamp_ke(fun(A,nat),fun(set(product_prod(A,A)),fun(A,fun(A,bool))),F),R3))) ).
% mlex_eq
tff(fact_1260_in__measure,axiom,
! [A: $tType,X2: A,Y2: A,F: fun(A,nat)] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),measure(A,F)))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F,X2)),aa(A,nat,F,Y2))) ) ).
% in_measure
tff(fact_1261_Ref_Opresent__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [H: heap_ext(product_unit),R: ref(A)] :
( pp(aa(ref(A),bool,present2(A,H),R))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(ref(A),nat,addr_of_ref(A),R)),lim(product_unit,H))) ) ) ).
% Ref.present_def
tff(fact_1262_Array_Opresent__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [H: heap_ext(product_unit),A3: array(A)] :
( pp(aa(array(A),bool,present(A,H),A3))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),addr_of_array(A,A3)),lim(product_unit,H))) ) ) ).
% Array.present_def
tff(fact_1263_success__swapI,axiom,
! [A: $tType] :
( heap(A)
=> ! [I: nat,H: heap_ext(product_unit),A3: array(A),X2: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(array(A),nat,length(A,H),A3)))
=> heap_success(A,swap(A,I,X2,A3),H) ) ) ).
% success_swapI
tff(fact_1264_upd__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [I: nat,X2: A,A3: array(A)] : upd(A,I,X2,A3) = heap_guard(array(A),aa(array(A),fun(heap_ext(product_unit),bool),aTP_Lamp_kf(nat,fun(array(A),fun(heap_ext(product_unit),bool)),I),A3),aa(array(A),fun(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit))),aa(A,fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit)))),aTP_Lamp_kg(nat,fun(A,fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit))))),I),X2),A3)) ) ).
% upd_def
tff(fact_1265_wf__less,axiom,
wf(nat,aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),ord_less(nat)))) ).
% wf_less
tff(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),aTP_Lamp_kh(array(A),fun(array(A),heap_Heap(array(A))),A3)) = upd(A,I,X2,A3) ) ).
% upd_return
tff(fact_1267_relpow__0__I,axiom,
! [A: $tType,X2: A,R3: set(product_prod(A,A))] : pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),X2)),compow(set(product_prod(A,A)),zero_zero(nat),R3))) ).
% relpow_0_I
tff(fact_1268_relpow__0__E,axiom,
! [A: $tType,X2: A,Y2: A,R3: set(product_prod(A,A))] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),compow(set(product_prod(A,A)),zero_zero(nat),R3)))
=> ( X2 = Y2 ) ) ).
% relpow_0_E
tff(fact_1269_mlex__iff,axiom,
! [A: $tType,X2: A,Y2: A,F: fun(A,nat),R3: set(product_prod(A,A))] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),mlex_prod(A,F,R3)))
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F,X2)),aa(A,nat,F,Y2)))
| ( ( aa(A,nat,F,X2) = aa(A,nat,F,Y2) )
& pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),R3)) ) ) ) ).
% mlex_iff
tff(fact_1270_mlex__less,axiom,
! [A: $tType,F: fun(A,nat),X2: A,Y2: A,R3: set(product_prod(A,A))] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F,X2)),aa(A,nat,F,Y2)))
=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),mlex_prod(A,F,R3))) ) ).
% mlex_less
tff(fact_1271_success__updI,axiom,
! [A: $tType] :
( heap(A)
=> ! [I: nat,H: heap_ext(product_unit),A3: array(A),X2: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(array(A),nat,length(A,H),A3)))
=> heap_success(array(A),upd(A,I,X2,A3),H) ) ) ).
% success_updI
tff(fact_1272_card__range__greater__zero,axiom,
! [A: $tType,B: $tType,F: fun(B,A)] :
( finite_finite2(A,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),top_top(set(B))))
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),top_top(set(B)))))) ) ).
% card_range_greater_zero
tff(fact_1273_mlex__leq,axiom,
! [A: $tType,F: fun(A,nat),X2: A,Y2: A,R3: set(product_prod(A,A))] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,F,X2)),aa(A,nat,F,Y2)))
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),R3))
=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),mlex_prod(A,F,R3))) ) ) ).
% mlex_leq
tff(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) )
=> ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(array(A),nat,length(A,H),A3))) ) ) ) ) ).
% effect_updE
tff(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] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(array(A),nat,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
tff(fact_1276_natLess__def,axiom,
bNF_Ca8459412986667044542atLess = aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),ord_less(nat))) ).
% natLess_def
tff(fact_1277_swap__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [I: nat,X2: A,A3: array(A)] : swap(A,I,X2,A3) = heap_guard(A,aa(array(A),fun(heap_ext(product_unit),bool),aTP_Lamp_kf(nat,fun(array(A),fun(heap_ext(product_unit),bool)),I),A3),aa(array(A),fun(heap_ext(product_unit),product_prod(A,heap_ext(product_unit))),aa(A,fun(array(A),fun(heap_ext(product_unit),product_prod(A,heap_ext(product_unit)))),aTP_Lamp_ki(nat,fun(A,fun(array(A),fun(heap_ext(product_unit),product_prod(A,heap_ext(product_unit))))),I),X2),A3)) ) ).
% swap_def
tff(fact_1278_success__nthI,axiom,
! [A: $tType] :
( heap(A)
=> ! [I: nat,H: heap_ext(product_unit),A3: array(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(array(A),nat,length(A,H),A3)))
=> heap_success(A,aa(nat,heap_Heap(A),nth(A,A3),I),H) ) ) ).
% success_nthI
tff(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 = aa(nat,A,nth3(A,aa(array(A),list(A),get(A,H),A3)),I) )
=> ( ( H2 = update(A,A3,I,X2,H) )
=> ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(array(A),nat,length(A,H),A3))) ) ) ) ) ).
% effect_swapE
tff(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] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(array(A),nat,length(A,H),A3)))
=> ( ( H2 = update(A,A3,I,X2,H) )
=> ( ( R = aa(nat,A,nth3(A,aa(array(A),list(A),get(A,H),A3)),I) )
=> heap_effect(A,swap(A,I,X2,A3),H,H2,R) ) ) ) ) ).
% effect_swapI
tff(fact_1281_finite__mono__remains__stable__implies__strict__prefix,axiom,
! [A: $tType] :
( order(A)
=> ! [F: fun(nat,A)] :
( finite_finite2(A,aa(set(nat),set(A),aa(fun(nat,A),fun(set(nat),set(A)),image2(nat,A),F),top_top(set(nat))))
=> ( order_mono(nat,A,F)
=> ( ! [N2: nat] :
( ( aa(nat,A,F,N2) = aa(nat,A,F,aa(nat,nat,suc,N2)) )
=> ( aa(nat,A,F,aa(nat,nat,suc,N2)) = aa(nat,A,F,aa(nat,nat,suc,aa(nat,nat,suc,N2))) ) )
=> ? [N7: nat] :
( ! [N8: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N8),N7))
=> ! [M6: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M6),N7))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M6),N8))
=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,F,M6)),aa(nat,A,F,N8))) ) ) )
& ! [N8: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N7),N8))
=> ( aa(nat,A,F,N7) = aa(nat,A,F,N8) ) ) ) ) ) ) ) ).
% finite_mono_remains_stable_implies_strict_prefix
tff(fact_1282_map__entry__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [I: nat,F: fun(A,A),A3: array(A)] : map_entry(A,I,F,A3) = heap_guard(array(A),aa(array(A),fun(heap_ext(product_unit),bool),aTP_Lamp_kf(nat,fun(array(A),fun(heap_ext(product_unit),bool)),I),A3),aa(array(A),fun(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit))),aa(fun(A,A),fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit)))),aTP_Lamp_kj(nat,fun(fun(A,A),fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit))))),I),F),A3)) ) ).
% map_entry_def
tff(fact_1283_success__map__entryI,axiom,
! [A: $tType] :
( heap(A)
=> ! [I: nat,H: heap_ext(product_unit),A3: array(A),F: fun(A,A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(array(A),nat,length(A,H),A3)))
=> heap_success(array(A),map_entry(A,I,F,A3),H) ) ) ).
% success_map_entryI
tff(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 )
=> ( aa(nat,A,nth3(A,aa(array(A),list(A),get(A,update(A,A3,J5,V,H)),A3)),I) = aa(nat,A,nth3(A,aa(array(A),list(A),get(A,H),A3)),I) ) ) ) ).
% get_update_elem_neqIndex
tff(fact_1285_card_Ocomp__fun__commute__on,axiom,
aa(fun(nat,nat),fun(nat,nat),aa(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),comp(nat,nat,nat),suc),suc) = aa(fun(nat,nat),fun(nat,nat),aa(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),comp(nat,nat,nat),suc),suc) ).
% card.comp_fun_commute_on
tff(fact_1286_wf__no__infinite__down__chainE,axiom,
! [A: $tType,R: set(product_prod(A,A)),F: fun(nat,A)] :
( wf(A,R)
=> ~ ! [K3: nat] : pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,F,aa(nat,nat,suc,K3))),aa(nat,A,F,K3))),R)) ) ).
% wf_no_infinite_down_chainE
tff(fact_1287_wf__iff__no__infinite__down__chain,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( wf(A,R)
<=> ~ ? [F7: fun(nat,A)] :
! [I3: nat] : pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,F7,aa(nat,nat,suc,I3))),aa(nat,A,F7,I3))),R)) ) ).
% wf_iff_no_infinite_down_chain
tff(fact_1288_relpow__Suc__E,axiom,
! [A: $tType,X2: A,Z2: A,N4: nat,R3: set(product_prod(A,A))] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Z2)),compow(set(product_prod(A,A)),aa(nat,nat,suc,N4),R3)))
=> ~ ! [Y: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y)),compow(set(product_prod(A,A)),N4,R3)))
=> ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2)),R3)) ) ) ).
% relpow_Suc_E
tff(fact_1289_relpow__Suc__I,axiom,
! [A: $tType,X2: A,Y2: A,N4: nat,R3: set(product_prod(A,A)),Z2: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),compow(set(product_prod(A,A)),N4,R3)))
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z2)),R3))
=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Z2)),compow(set(product_prod(A,A)),aa(nat,nat,suc,N4),R3))) ) ) ).
% relpow_Suc_I
tff(fact_1290_relpow__Suc__D2,axiom,
! [A: $tType,X2: A,Z2: A,N4: nat,R3: set(product_prod(A,A))] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Z2)),compow(set(product_prod(A,A)),aa(nat,nat,suc,N4),R3)))
=> ? [Y: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y)),R3))
& pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2)),compow(set(product_prod(A,A)),N4,R3))) ) ) ).
% relpow_Suc_D2
tff(fact_1291_relpow__Suc__E2,axiom,
! [A: $tType,X2: A,Z2: A,N4: nat,R3: set(product_prod(A,A))] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Z2)),compow(set(product_prod(A,A)),aa(nat,nat,suc,N4),R3)))
=> ~ ! [Y: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y)),R3))
=> ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2)),compow(set(product_prod(A,A)),N4,R3))) ) ) ).
% relpow_Suc_E2
tff(fact_1292_relpow__Suc__I2,axiom,
! [A: $tType,X2: A,Y2: A,R3: set(product_prod(A,A)),Z2: A,N4: nat] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),R3))
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z2)),compow(set(product_prod(A,A)),N4,R3)))
=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Z2)),compow(set(product_prod(A,A)),aa(nat,nat,suc,N4),R3))) ) ) ).
% relpow_Suc_I2
tff(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] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(array(A),nat,length(A,H),A3)))
=> ( ( H2 = H )
=> ( ( R = aa(nat,A,nth3(A,aa(array(A),list(A),get(A,H),A3)),I) )
=> heap_effect(A,aa(nat,heap_Heap(A),nth(A,A3),I),H,H2,R) ) ) ) ) ).
% effect_nthI
tff(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,aa(nat,heap_Heap(A),nth(A,A3),I),H,H2,R)
=> ~ ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(array(A),nat,length(A,H),A3)))
=> ( ( R = aa(nat,A,nth3(A,aa(array(A),list(A),get(A,H),A3)),I) )
=> ( H2 != H ) ) ) ) ) ).
% effect_nthE
tff(fact_1295_effect__map__entryE,axiom,
! [A: $tType] :
( heap(A)
=> ! [I: nat,F: fun(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,aa(A,A,F,aa(nat,A,nth3(A,aa(array(A),list(A),get(A,H),A3)),I)),H) )
=> ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(array(A),nat,length(A,H),A3))) ) ) ) ) ).
% effect_map_entryE
tff(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: fun(A,A),R: array(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(array(A),nat,length(A,H),A3)))
=> ( ( H2 = update(A,A3,I,aa(A,A,F,aa(nat,A,nth3(A,aa(array(A),list(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
tff(fact_1297_relpow__E,axiom,
! [A: $tType,X2: A,Z2: A,N4: nat,R3: set(product_prod(A,A))] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Z2)),compow(set(product_prod(A,A)),N4,R3)))
=> ( ( ( N4 = zero_zero(nat) )
=> ( X2 != Z2 ) )
=> ~ ! [Y: A,M3: nat] :
( ( N4 = aa(nat,nat,suc,M3) )
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y)),compow(set(product_prod(A,A)),M3,R3)))
=> ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2)),R3)) ) ) ) ) ).
% relpow_E
tff(fact_1298_relpow__E2,axiom,
! [A: $tType,X2: A,Z2: A,N4: nat,R3: set(product_prod(A,A))] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Z2)),compow(set(product_prod(A,A)),N4,R3)))
=> ( ( ( N4 = zero_zero(nat) )
=> ( X2 != Z2 ) )
=> ~ ! [Y: A,M3: nat] :
( ( N4 = aa(nat,nat,suc,M3) )
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y)),R3))
=> ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z2)),compow(set(product_prod(A,A)),M3,R3))) ) ) ) ) ).
% relpow_E2
tff(fact_1299_Array_Onth__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [A3: array(A),I: nat] : aa(nat,heap_Heap(A),nth(A,A3),I) = heap_guard(A,aa(nat,fun(heap_ext(product_unit),bool),aTP_Lamp_kk(array(A),fun(nat,fun(heap_ext(product_unit),bool)),A3),I),aa(nat,fun(heap_ext(product_unit),product_prod(A,heap_ext(product_unit))),aTP_Lamp_kl(array(A),fun(nat,fun(heap_ext(product_unit),product_prod(A,heap_ext(product_unit)))),A3),I)) ) ).
% Array.nth_def
tff(fact_1300_relpow__fun__conv,axiom,
! [A: $tType,A3: A,B2: A,N4: nat,R3: set(product_prod(A,A))] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),compow(set(product_prod(A,A)),N4,R3)))
<=> ? [F7: fun(nat,A)] :
( ( aa(nat,A,F7,zero_zero(nat)) = A3 )
& ( aa(nat,A,F7,N4) = B2 )
& ! [I3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),N4))
=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,F7,I3)),aa(nat,A,F7,aa(nat,nat,suc,I3)))),R3)) ) ) ) ).
% relpow_fun_conv
tff(fact_1301_ntrancl__def,axiom,
! [A: $tType,N4: nat,R3: set(product_prod(A,A))] : transitive_ntrancl(A,N4,R3) = aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),aa(set(nat),set(set(product_prod(A,A))),aa(fun(nat,set(product_prod(A,A))),fun(set(nat),set(set(product_prod(A,A)))),image2(nat,set(product_prod(A,A))),aTP_Lamp_iy(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R3)),aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_km(nat,fun(nat,bool),N4)))) ).
% ntrancl_def
tff(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)
=> ( aa(nat,A,nth3(A,aa(array(A),list(A),get(A,update(B,B2,J5,V,H)),A3)),I) = aa(nat,A,nth3(A,aa(array(A),list(A),get(A,H),A3)),I) ) ) ) ).
% nth_update_neq
tff(fact_1303_pred__nat__def,axiom,
pred_nat = aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_kn(nat,fun(nat,bool)))) ).
% pred_nat_def
tff(fact_1304_zero__notin__Suc__image,axiom,
! [A6: set(nat)] : ~ pp(aa(set(nat),bool,member(nat,zero_zero(nat)),aa(set(nat),set(nat),aa(fun(nat,nat),fun(set(nat),set(nat)),image2(nat,nat),suc),A6))) ).
% zero_notin_Suc_image
tff(fact_1305_size__prod__simp,axiom,
! [A: $tType,B: $tType,F: fun(A,nat),G: fun(B,nat),P: product_prod(A,B)] : aa(product_prod(A,B),nat,basic_BNF_size_prod(A,B,F,G),P) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,F,aa(product_prod(A,B),A,product_fst(A,B),P))),aa(B,nat,G,aa(product_prod(A,B),B,product_snd(A,B),P)))),aa(nat,nat,suc,zero_zero(nat))) ).
% size_prod_simp
tff(fact_1306_Restr__natLeq2,axiom,
! [N4: nat] : aa(set(product_prod(nat,nat)),set(product_prod(nat,nat)),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),set(product_prod(nat,nat))),inf_inf(set(product_prod(nat,nat))),bNF_Ca8665028551170535155natLeq),product_Sigma(nat,nat,order_underS(nat,bNF_Ca8665028551170535155natLeq,N4),aTP_Lamp_ko(nat,fun(nat,set(nat)),N4))) = aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_jx(nat,fun(nat,fun(nat,bool)),N4))) ).
% Restr_natLeq2
tff(fact_1307_Restr__natLeq,axiom,
! [N4: nat] : aa(set(product_prod(nat,nat)),set(product_prod(nat,nat)),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),set(product_prod(nat,nat))),inf_inf(set(product_prod(nat,nat))),bNF_Ca8665028551170535155natLeq),product_Sigma(nat,nat,aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_jw(nat,fun(nat,bool),N4)),aTP_Lamp_kp(nat,fun(nat,set(nat)),N4))) = aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_jx(nat,fun(nat,fun(nat,bool)),N4))) ).
% Restr_natLeq
tff(fact_1308_surj__from__nat,axiom,
! [A: $tType] :
( countable(A)
=> ( aa(set(nat),set(A),aa(fun(nat,A),fun(set(nat),set(A)),image2(nat,A),from_nat(A)),top_top(set(nat))) = top_top(set(A)) ) ) ).
% surj_from_nat
tff(fact_1309_Array_Ounequal,axiom,
! [A: $tType] :
( heap(A)
=> ! [A3: array(A),A4: array(A)] :
( ( A3 != A4 )
<=> noteq(A,A,A3,A4) ) ) ).
% Array.unequal
tff(fact_1310_image__add__0,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [S3: set(A)] : aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aa(A,fun(A,A),plus_plus(A),zero_zero(A))),S3) = S3 ) ).
% image_add_0
tff(fact_1311_surj__plus,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [A3: A] : aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aa(A,fun(A,A),plus_plus(A),A3)),top_top(set(A))) = top_top(set(A)) ) ).
% surj_plus
tff(fact_1312_Array_Onoteq__irrefl,axiom,
! [A: $tType] :
( heap(A)
=> ! [R: array(A)] : ~ noteq(A,A,R,R) ) ).
% Array.noteq_irrefl
tff(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
tff(fact_1314_translation__Int,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [A3: A,S: set(A),T2: set(A)] : aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aa(A,fun(A,A),plus_plus(A),A3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),T2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aa(A,fun(A,A),plus_plus(A),A3)),S)),aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aa(A,fun(A,A),plus_plus(A),A3)),T2)) ) ).
% translation_Int
tff(fact_1315_translation__diff,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [A3: A,S: set(A),T2: set(A)] : aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aa(A,fun(A,A),plus_plus(A),A3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),S),T2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),minus_minus(set(A)),aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aa(A,fun(A,A),plus_plus(A),A3)),S)),aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aa(A,fun(A,A),plus_plus(A),A3)),T2)) ) ).
% translation_diff
tff(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
tff(fact_1317_natLeq__def,axiom,
bNF_Ca8665028551170535155natLeq = aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),ord_less_eq(nat))) ).
% natLeq_def
tff(fact_1318_rat__denum,axiom,
? [F3: fun(nat,rat)] : aa(set(nat),set(rat),aa(fun(nat,rat),fun(set(nat),set(rat)),image2(nat,rat),F3),top_top(set(nat))) = top_top(set(rat)) ).
% rat_denum
tff(fact_1319_UN__finite2__subset,axiom,
! [A: $tType,A6: fun(nat,set(A)),B6: fun(nat,set(A)),K: nat] :
( ! [N2: nat] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),aa(fun(nat,set(A)),fun(set(nat),set(set(A))),image2(nat,set(A)),A6),set_or7035219750837199246ssThan(nat,zero_zero(nat),N2)))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),aa(fun(nat,set(A)),fun(set(nat),set(set(A))),image2(nat,set(A)),B6),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),K))))))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),aa(fun(nat,set(A)),fun(set(nat),set(set(A))),image2(nat,set(A)),A6),top_top(set(nat))))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),aa(fun(nat,set(A)),fun(set(nat),set(set(A))),image2(nat,set(A)),B6),top_top(set(nat)))))) ) ).
% UN_finite2_subset
tff(fact_1320_Array_Oget__set__neq,axiom,
! [B: $tType,A: $tType] :
( ( heap(A)
& heap(B) )
=> ! [R: array(A),S: array(B),X2: list(B),H: heap_ext(product_unit)] :
( noteq(A,B,R,S)
=> ( aa(array(A),list(A),get(A,aa(heap_ext(product_unit),heap_ext(product_unit),set2(B,S,X2),H)),R) = aa(array(A),list(A),get(A,H),R) ) ) ) ).
% Array.get_set_neq
tff(fact_1321_UN__finite2__eq,axiom,
! [A: $tType,A6: fun(nat,set(A)),B6: fun(nat,set(A)),K: nat] :
( ! [N2: nat] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),aa(fun(nat,set(A)),fun(set(nat),set(set(A))),image2(nat,set(A)),A6),set_or7035219750837199246ssThan(nat,zero_zero(nat),N2))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),aa(fun(nat,set(A)),fun(set(nat),set(set(A))),image2(nat,set(A)),B6),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),K))))
=> ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),aa(fun(nat,set(A)),fun(set(nat),set(set(A))),image2(nat,set(A)),A6),top_top(set(nat)))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),aa(fun(nat,set(A)),fun(set(nat),set(set(A))),image2(nat,set(A)),B6),top_top(set(nat)))) ) ) ).
% UN_finite2_eq
tff(fact_1322_UN__finite__subset,axiom,
! [A: $tType,A6: fun(nat,set(A)),C5: set(A)] :
( ! [N2: nat] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),aa(fun(nat,set(A)),fun(set(nat),set(set(A))),image2(nat,set(A)),A6),set_or7035219750837199246ssThan(nat,zero_zero(nat),N2)))),C5))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),aa(fun(nat,set(A)),fun(set(nat),set(set(A))),image2(nat,set(A)),A6),top_top(set(nat))))),C5)) ) ).
% UN_finite_subset
tff(fact_1323_greaterThan__0,axiom,
aa(nat,set(nat),set_ord_greaterThan(nat),zero_zero(nat)) = aa(set(nat),set(nat),aa(fun(nat,nat),fun(set(nat),set(nat)),image2(nat,nat),suc),top_top(set(nat))) ).
% greaterThan_0
tff(fact_1324_UN__UN__finite__eq,axiom,
! [A: $tType,A6: fun(nat,set(A))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),aa(fun(nat,set(A)),fun(set(nat),set(set(A))),image2(nat,set(A)),aTP_Lamp_kq(fun(nat,set(A)),fun(nat,set(A)),A6)),top_top(set(nat)))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),aa(fun(nat,set(A)),fun(set(nat),set(set(A))),image2(nat,set(A)),A6),top_top(set(nat)))) ).
% UN_UN_finite_eq
tff(fact_1325_INF__bool__eq,axiom,
! [A: $tType] : aTP_Lamp_kr(set(A),fun(fun(A,bool),bool)) = ball(A) ).
% INF_bool_eq
tff(fact_1326_Max__add__commute,axiom,
! [B: $tType,A: $tType] :
( linord4140545234300271783up_add(A)
=> ! [S3: set(B),F: fun(B,A),K: A] :
( finite_finite2(B,S3)
=> ( ( S3 != bot_bot(set(B)) )
=> ( lattic643756798349783984er_Max(A,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),aa(A,fun(B,A),aTP_Lamp_ks(fun(B,A),fun(A,fun(B,A)),F),K)),S3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),lattic643756798349783984er_Max(A,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),S3))),K) ) ) ) ) ).
% Max_add_commute
tff(fact_1327_Array_Oset__same,axiom,
! [A: $tType] :
( heap(A)
=> ! [R: array(A),X2: list(A),Y2: list(A),H: heap_ext(product_unit)] : aa(heap_ext(product_unit),heap_ext(product_unit),set2(A,R,X2),aa(heap_ext(product_unit),heap_ext(product_unit),set2(A,R,Y2),H)) = aa(heap_ext(product_unit),heap_ext(product_unit),set2(A,R,X2),H) ) ).
% Array.set_same
tff(fact_1328_not__False__in__image__Ball,axiom,
! [A: $tType,P2: fun(A,bool),A6: set(A)] :
( ~ pp(aa(set(bool),bool,member(bool,fFalse),aa(set(A),set(bool),aa(fun(A,bool),fun(set(A),set(bool)),image2(A,bool),P2),A6)))
<=> ! [X5: A] :
( pp(aa(set(A),bool,member(A,X5),A6))
=> pp(aa(A,bool,P2,X5)) ) ) ).
% not_False_in_image_Ball
tff(fact_1329_Array_Oget__set__eq,axiom,
! [A: $tType] :
( heap(A)
=> ! [R: array(A),X2: list(A),H: heap_ext(product_unit)] : aa(array(A),list(A),get(A,aa(heap_ext(product_unit),heap_ext(product_unit),set2(A,R,X2),H)),R) = X2 ) ).
% Array.get_set_eq
tff(fact_1330_image__add__atLeastLessThan,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [K: A,I: A,J5: A] : aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aa(A,fun(A,A),plus_plus(A),K)),set_or7035219750837199246ssThan(A,I,J5)) = set_or7035219750837199246ssThan(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),J5),K)) ) ).
% image_add_atLeastLessThan
tff(fact_1331_image__Suc__atLeastLessThan,axiom,
! [I: nat,J5: nat] : aa(set(nat),set(nat),aa(fun(nat,nat),fun(set(nat),set(nat)),image2(nat,nat),suc),set_or7035219750837199246ssThan(nat,I,J5)) = set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,I),aa(nat,nat,suc,J5)) ).
% image_Suc_atLeastLessThan
tff(fact_1332_image__add__atLeastLessThan_H,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [K: A,I: A,J5: A] : aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aTP_Lamp_kt(A,fun(A,A),K)),set_or7035219750837199246ssThan(A,I,J5)) = set_or7035219750837199246ssThan(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),J5),K)) ) ).
% image_add_atLeastLessThan'
tff(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,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),aTP_Lamp_ku(A,fun(B,A),C3)),A6)) = C3 ) ) ) ) ).
% Max_const
tff(fact_1334_Ball__image__comp,axiom,
! [A: $tType,B: $tType,F: fun(B,A),A6: set(B),G: fun(A,bool)] :
( ! [X5: A] :
( pp(aa(set(A),bool,member(A,X5),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6)))
=> pp(aa(A,bool,G,X5)) )
<=> ! [X5: B] :
( pp(aa(set(B),bool,member(B,X5),A6))
=> pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aa(fun(A,bool),fun(fun(B,A),fun(B,bool)),comp(A,bool,B),G),F),X5)) ) ) ).
% Ball_image_comp
tff(fact_1335_Ball__Collect,axiom,
! [A: $tType,A6: set(A),P2: fun(A,bool)] :
( ! [X5: A] :
( pp(aa(set(A),bool,member(A,X5),A6))
=> pp(aa(A,bool,P2,X5)) )
<=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A6),aa(fun(A,bool),set(A),collect(A),P2))) ) ).
% Ball_Collect
tff(fact_1336_Ball__comp__iff,axiom,
! [A: $tType,B: $tType,C: $tType,A6: fun(B,set(C)),F: fun(C,bool),G: fun(A,B),X4: A] :
( pp(aa(A,bool,aa(fun(A,B),fun(A,bool),aa(fun(B,bool),fun(fun(A,B),fun(A,bool)),comp(B,bool,A),aa(fun(C,bool),fun(B,bool),aTP_Lamp_kv(fun(B,set(C)),fun(fun(C,bool),fun(B,bool)),A6),F)),G),X4))
<=> ! [Xa3: C] :
( pp(aa(set(C),bool,member(C,Xa3),aa(A,set(C),aa(fun(A,B),fun(A,set(C)),aa(fun(B,set(C)),fun(fun(A,B),fun(A,set(C))),comp(B,set(C),A),A6),G),X4)))
=> pp(aa(C,bool,F,Xa3)) ) ) ).
% Ball_comp_iff
tff(fact_1337_INT__greaterThan__UNIV,axiom,
aa(set(set(nat)),set(nat),complete_Inf_Inf(set(nat)),aa(set(nat),set(set(nat)),aa(fun(nat,set(nat)),fun(set(nat),set(set(nat))),image2(nat,set(nat)),set_ord_greaterThan(nat)),top_top(set(nat)))) = bot_bot(set(nat)) ).
% INT_greaterThan_UNIV
tff(fact_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)] : aa(heap_ext(product_unit),heap_ext(product_unit),set2(A,A3,Xs),aa(heap_ext(product_unit),heap_ext(product_unit),set4(B,R,X3),H)) = aa(heap_ext(product_unit),heap_ext(product_unit),set4(B,R,X3),aa(heap_ext(product_unit),heap_ext(product_unit),set2(A,A3,Xs),H)) ) ).
% set_array_set_swap
tff(fact_1339_INTER__eq,axiom,
! [A: $tType,B: $tType,B6: fun(B,set(A)),A6: set(B)] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),B6),A6)) = aa(fun(A,bool),set(A),collect(A),aa(set(B),fun(A,bool),aTP_Lamp_kw(fun(B,set(A)),fun(set(B),fun(A,bool)),B6),A6)) ).
% INTER_eq
tff(fact_1340_Collect__ball__eq,axiom,
! [A: $tType,B: $tType,A6: set(B),P2: fun(A,fun(B,bool))] : aa(fun(A,bool),set(A),collect(A),aa(fun(A,fun(B,bool)),fun(A,bool),aTP_Lamp_kx(set(B),fun(fun(A,fun(B,bool)),fun(A,bool)),A6),P2)) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),aTP_Lamp_kz(fun(A,fun(B,bool)),fun(B,set(A)),P2)),A6)) ).
% Collect_ball_eq
tff(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)
=> ( aa(heap_ext(product_unit),heap_ext(product_unit),set2(A,R,X2),aa(heap_ext(product_unit),heap_ext(product_unit),set2(B,R2,X3),H)) = aa(heap_ext(product_unit),heap_ext(product_unit),set2(B,R2,X3),aa(heap_ext(product_unit),heap_ext(product_unit),set2(A,R,X2),H)) ) ) ) ).
% Array.set_set_swap
tff(fact_1342_congruent__def,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,A)),F: fun(A,B)] :
( equiv_congruent(A,B,R,F)
<=> ! [X5: product_prod(A,A)] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),X5),R))
=> pp(aa(product_prod(A,A),bool,aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aTP_Lamp_la(fun(A,B),fun(A,fun(A,bool)),F)),X5)) ) ) ).
% congruent_def
tff(fact_1343_congruent2__def,axiom,
! [B: $tType,C: $tType,A: $tType,R12: set(product_prod(A,A)),R23: set(product_prod(B,B)),F: fun(A,fun(B,C))] :
( equiv_congruent2(A,B,C,R12,R23,F)
<=> ! [X5: product_prod(A,A)] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),X5),R12))
=> pp(aa(product_prod(A,A),bool,aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(fun(A,fun(B,C)),fun(A,fun(A,bool)),aTP_Lamp_lc(set(product_prod(B,B)),fun(fun(A,fun(B,C)),fun(A,fun(A,bool))),R23),F)),X5)) ) ) ).
% congruent2_def
tff(fact_1344_mono__image__least,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [F: fun(A,B),M4: A,N4: A,M7: B,N9: B] :
( order_mono(A,B,F)
=> ( ( aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),set_or7035219750837199246ssThan(A,M4,N4)) = set_or7035219750837199246ssThan(B,M7,N9) )
=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M4),N4))
=> ( aa(A,B,F,M4) = M7 ) ) ) ) ) ).
% mono_image_least
tff(fact_1345_atLeast0__lessThan__Suc__eq__insert__0,axiom,
! [N4: nat] : set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N4)) = aa(set(nat),set(nat),insert(nat,zero_zero(nat)),aa(set(nat),set(nat),aa(fun(nat,nat),fun(set(nat),set(nat)),image2(nat,nat),suc),set_or7035219750837199246ssThan(nat,zero_zero(nat),N4))) ).
% atLeast0_lessThan_Suc_eq_insert_0
tff(fact_1346_refl__on__def_H,axiom,
! [A: $tType,A6: set(A),R: set(product_prod(A,A))] :
( refl_on(A,A6,R)
<=> ( ! [X5: product_prod(A,A)] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),X5),R))
=> pp(aa(product_prod(A,A),bool,aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aTP_Lamp_ld(set(A),fun(A,fun(A,bool)),A6)),X5)) )
& ! [X5: A] :
( pp(aa(set(A),bool,member(A,X5),A6))
=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),X5)),R)) ) ) ) ).
% refl_on_def'
tff(fact_1347_image__minus__const__atLeastLessThan__nat,axiom,
! [C3: nat,Y2: nat,X2: nat] :
( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),C3),Y2))
=> ( aa(set(nat),set(nat),aa(fun(nat,nat),fun(set(nat),set(nat)),image2(nat,nat),aTP_Lamp_le(nat,fun(nat,nat),C3)),set_or7035219750837199246ssThan(nat,X2,Y2)) = set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),X2),C3),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Y2),C3)) ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),C3),Y2))
=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X2),Y2))
=> ( aa(set(nat),set(nat),aa(fun(nat,nat),fun(set(nat),set(nat)),image2(nat,nat),aTP_Lamp_le(nat,fun(nat,nat),C3)),set_or7035219750837199246ssThan(nat,X2,Y2)) = aa(set(nat),set(nat),insert(nat,zero_zero(nat)),bot_bot(set(nat))) ) )
& ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X2),Y2))
=> ( aa(set(nat),set(nat),aa(fun(nat,nat),fun(set(nat),set(nat)),image2(nat,nat),aTP_Lamp_le(nat,fun(nat,nat),C3)),set_or7035219750837199246ssThan(nat,X2,Y2)) = bot_bot(set(nat)) ) ) ) ) ) ).
% image_minus_const_atLeastLessThan_nat
tff(fact_1348_mono__Max__commute,axiom,
! [B: $tType,A: $tType] :
( ( linorder(A)
& linorder(B) )
=> ! [F: fun(A,B),A6: set(A)] :
( order_mono(A,B,F)
=> ( finite_finite2(A,A6)
=> ( ( A6 != bot_bot(set(A)) )
=> ( aa(A,B,F,lattic643756798349783984er_Max(A,A6)) = lattic643756798349783984er_Max(B,aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6)) ) ) ) ) ) ).
% mono_Max_commute
tff(fact_1349_UnderS__def,axiom,
! [A: $tType,R: set(product_prod(A,A)),A6: set(A)] : order_UnderS(A,R,A6) = aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_lf(set(product_prod(A,A)),fun(set(A),fun(A,bool)),R),A6)) ).
% UnderS_def
tff(fact_1350_Under__def,axiom,
! [A: $tType,R: set(product_prod(A,A)),A6: set(A)] : order_Under(A,R,A6) = aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_lg(set(product_prod(A,A)),fun(set(A),fun(A,bool)),R),A6)) ).
% Under_def
tff(fact_1351_Above__def,axiom,
! [A: $tType,R: set(product_prod(A,A)),A6: set(A)] : order_Above(A,R,A6) = aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_lh(set(product_prod(A,A)),fun(set(A),fun(A,bool)),R),A6)) ).
% Above_def
tff(fact_1352_surj__nat__to__rat__surj,axiom,
aa(set(nat),set(rat),aa(fun(nat,rat),fun(set(nat),set(rat)),image2(nat,rat),nat_to_rat_surj),top_top(set(nat))) = top_top(set(rat)) ).
% surj_nat_to_rat_surj
tff(fact_1353_range__mod,axiom,
! [N4: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N4))
=> ( aa(set(nat),set(nat),aa(fun(nat,nat),fun(set(nat),set(nat)),image2(nat,nat),aTP_Lamp_li(nat,fun(nat,nat),N4)),top_top(set(nat))) = set_or7035219750837199246ssThan(nat,zero_zero(nat),N4) ) ) ).
% range_mod
tff(fact_1354_AboveS__def,axiom,
! [A: $tType,R: set(product_prod(A,A)),A6: set(A)] : order_AboveS(A,R,A6) = aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_lj(set(product_prod(A,A)),fun(set(A),fun(A,bool)),R),A6)) ).
% AboveS_def
tff(fact_1355_sum_OatLeast0__lessThan__Suc__shift,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(nat,A),N4: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N4))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),suc)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N4))) ) ).
% sum.atLeast0_lessThan_Suc_shift
tff(fact_1356_Chains__def,axiom,
! [A: $tType,R: set(product_prod(A,A))] : chains(A,R) = aa(fun(set(A),bool),set(set(A)),collect(set(A)),aTP_Lamp_lk(set(product_prod(A,A)),fun(set(A),bool),R)) ).
% Chains_def
tff(fact_1357_conj__comp__iff,axiom,
! [B: $tType,A: $tType,P2: fun(B,bool),Q2: fun(B,bool),G: fun(A,B),X4: A] :
( pp(aa(A,bool,aa(fun(A,B),fun(A,bool),aa(fun(B,bool),fun(fun(A,B),fun(A,bool)),comp(B,bool,A),aa(fun(B,bool),fun(B,bool),aTP_Lamp_ll(fun(B,bool),fun(fun(B,bool),fun(B,bool)),P2),Q2)),G),X4))
<=> ( pp(aa(A,bool,aa(fun(A,B),fun(A,bool),aa(fun(B,bool),fun(fun(A,B),fun(A,bool)),comp(B,bool,A),P2),G),X4))
& pp(aa(A,bool,aa(fun(A,B),fun(A,bool),aa(fun(B,bool),fun(fun(A,B),fun(A,bool)),comp(B,bool,A),Q2),G),X4)) ) ) ).
% conj_comp_iff
tff(fact_1358_sum_OatLeast__Suc__lessThan__Suc__shift,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(nat,A),M4: nat,N4: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M4),aa(nat,nat,suc,N4))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),suc)),set_or7035219750837199246ssThan(nat,M4,N4)) ) ).
% sum.atLeast_Suc_lessThan_Suc_shift
tff(fact_1359_sum_OatLeastLessThan__shift__bounds,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(nat,A),M4: nat,K: nat,N4: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M4),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N4),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),aa(nat,fun(nat,nat),plus_plus(nat),K))),set_or7035219750837199246ssThan(nat,M4,N4)) ) ).
% sum.atLeastLessThan_shift_bounds
tff(fact_1360_sum_OatLeastLessThan__shift__0,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(nat,A),M4: nat,N4: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M4,N4)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),aa(nat,fun(nat,nat),plus_plus(nat),M4))),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N4),M4))) ) ).
% sum.atLeastLessThan_shift_0
tff(fact_1361_sum_OatLeast__lessThan__pred__shift,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(nat,A),M4: nat,N4: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),aTP_Lamp_lm(nat,nat))),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M4),aa(nat,nat,suc,N4))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M4,N4)) ) ).
% sum.atLeast_lessThan_pred_shift
tff(fact_1362_sum_OUNION__disjoint,axiom,
! [C: $tType,A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [I4: set(B),A6: fun(B,set(C)),G: fun(C,A)] :
( finite_finite2(B,I4)
=> ( ! [X: B] :
( pp(aa(set(B),bool,member(B,X),I4))
=> finite_finite2(C,aa(B,set(C),A6,X)) )
=> ( ! [X: B] :
( pp(aa(set(B),bool,member(B,X),I4))
=> ! [Xa: B] :
( pp(aa(set(B),bool,member(B,Xa),I4))
=> ( ( X != Xa )
=> ( aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),aa(B,set(C),A6,X)),aa(B,set(C),A6,Xa)) = bot_bot(set(C)) ) ) ) )
=> ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),G),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),aa(fun(B,set(C)),fun(set(B),set(set(C))),image2(B,set(C)),A6),I4))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(C,A),fun(B,A),aTP_Lamp_ln(fun(B,set(C)),fun(fun(C,A),fun(B,A)),A6),G)),I4) ) ) ) ) ) ).
% sum.UNION_disjoint
tff(fact_1363_sum__image__le,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ordere6911136660526730532id_add(B)
=> ! [I4: set(C),G: fun(A,B),F: fun(C,A)] :
( finite_finite2(C,I4)
=> ( ! [I6: C] :
( pp(aa(set(C),bool,member(C,I6),I4))
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),aa(A,B,G,aa(C,A,F,I6)))) )
=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(C),set(A),aa(fun(C,A),fun(set(C),set(A)),image2(C,A),F),I4))),aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),aa(fun(C,A),fun(C,B),aa(fun(A,B),fun(fun(C,A),fun(C,B)),comp(A,B,C),G),F)),I4))) ) ) ) ).
% sum_image_le
tff(fact_1364_sum_Oeq__fold,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(B,A),A6: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A6) = finite_fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),aa(fun(A,fun(A,A)),fun(fun(B,A),fun(B,fun(A,A))),comp(A,fun(A,A),B),plus_plus(A)),G),zero_zero(A),A6) ) ).
% sum.eq_fold
tff(fact_1365_sum_OUnion__disjoint,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [C5: set(set(B)),G: fun(B,A)] :
( ! [X: set(B)] :
( pp(aa(set(set(B)),bool,member(set(B),X),C5))
=> finite_finite2(B,X) )
=> ( ! [X: set(B)] :
( pp(aa(set(set(B)),bool,member(set(B),X),C5))
=> ! [Xa: set(B)] :
( pp(aa(set(set(B)),bool,member(set(B),Xa),C5))
=> ( ( X != Xa )
=> ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),X),Xa) = bot_bot(set(B)) ) ) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),C5)) = aa(set(set(B)),A,aa(fun(B,A),fun(set(set(B)),A),aa(fun(fun(B,A),fun(set(B),A)),fun(fun(B,A),fun(set(set(B)),A)),aa(fun(fun(set(B),A),fun(set(set(B)),A)),fun(fun(fun(B,A),fun(set(B),A)),fun(fun(B,A),fun(set(set(B)),A))),comp(fun(set(B),A),fun(set(set(B)),A),fun(B,A)),groups7311177749621191930dd_sum(set(B),A)),groups7311177749621191930dd_sum(B,A)),G),C5) ) ) ) ) ).
% sum.Union_disjoint
tff(fact_1366_sum_Oreindex__nontrivial,axiom,
! [C: $tType,A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [A6: set(B),H: fun(B,C),G: fun(C,A)] :
( finite_finite2(B,A6)
=> ( ! [X: B,Y: B] :
( pp(aa(set(B),bool,member(B,X),A6))
=> ( pp(aa(set(B),bool,member(B,Y),A6))
=> ( ( X != Y )
=> ( ( aa(B,C,H,X) = aa(B,C,H,Y) )
=> ( aa(C,A,G,aa(B,C,H,X)) = zero_zero(A) ) ) ) ) )
=> ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),G),aa(set(B),set(C),aa(fun(B,C),fun(set(B),set(C)),image2(B,C),H),A6)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,C),fun(B,A),aa(fun(C,A),fun(fun(B,C),fun(B,A)),comp(C,A,B),G),H)),A6) ) ) ) ) ).
% sum.reindex_nontrivial
tff(fact_1367_sum_OSigma,axiom,
! [A: $tType,C: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [A6: set(B),B6: fun(B,set(C)),G: fun(B,fun(C,A))] :
( finite_finite2(B,A6)
=> ( ! [X: B] :
( pp(aa(set(B),bool,member(B,X),A6))
=> finite_finite2(C,aa(B,set(C),B6,X)) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,fun(C,A)),fun(B,A),aTP_Lamp_lo(fun(B,set(C)),fun(fun(B,fun(C,A)),fun(B,A)),B6),G)),A6) = aa(set(product_prod(B,C)),A,aa(fun(product_prod(B,C),A),fun(set(product_prod(B,C)),A),groups7311177749621191930dd_sum(product_prod(B,C),A),aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),G)),product_Sigma(B,C,A6,B6)) ) ) ) ) ).
% sum.Sigma
tff(fact_1368_sum_Ogroup,axiom,
! [C: $tType,A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [S3: set(B),T3: set(C),G: fun(B,C),H: fun(B,A)] :
( finite_finite2(B,S3)
=> ( finite_finite2(C,T3)
=> ( pp(aa(set(C),bool,aa(set(C),fun(set(C),bool),ord_less_eq(set(C)),aa(set(B),set(C),aa(fun(B,C),fun(set(B),set(C)),image2(B,C),G),S3)),T3))
=> ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),aa(fun(B,A),fun(C,A),aa(fun(B,C),fun(fun(B,A),fun(C,A)),aTP_Lamp_lq(set(B),fun(fun(B,C),fun(fun(B,A),fun(C,A))),S3),G),H)),T3) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),S3) ) ) ) ) ) ).
% sum.group
tff(fact_1369_sum_Oreindex,axiom,
! [C: $tType,A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [H: fun(B,C),A6: set(B),G: fun(C,A)] :
( inj_on(B,C,H,A6)
=> ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),G),aa(set(B),set(C),aa(fun(B,C),fun(set(B),set(C)),image2(B,C),H),A6)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,C),fun(B,A),aa(fun(C,A),fun(fun(B,C),fun(B,A)),comp(C,A,B),G),H)),A6) ) ) ) ).
% sum.reindex
tff(fact_1370_card__UN__le,axiom,
! [B: $tType,A: $tType,I4: set(A),A6: fun(A,set(B))] :
( finite_finite2(A,I4)
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(B),nat,finite_card(B),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),aa(fun(A,set(B)),fun(set(A),set(set(B))),image2(A,set(B)),A6),I4)))),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_lr(fun(A,set(B)),fun(A,nat),A6)),I4))) ) ).
% card_UN_le
tff(fact_1371_card__UN__disjoint,axiom,
! [B: $tType,A: $tType,I4: set(A),A6: fun(A,set(B))] :
( finite_finite2(A,I4)
=> ( ! [X: A] :
( pp(aa(set(A),bool,member(A,X),I4))
=> finite_finite2(B,aa(A,set(B),A6,X)) )
=> ( ! [X: A] :
( pp(aa(set(A),bool,member(A,X),I4))
=> ! [Xa: A] :
( pp(aa(set(A),bool,member(A,Xa),I4))
=> ( ( X != Xa )
=> ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(A,set(B),A6,X)),aa(A,set(B),A6,Xa)) = bot_bot(set(B)) ) ) ) )
=> ( aa(set(B),nat,finite_card(B),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),aa(fun(A,set(B)),fun(set(A),set(set(B))),image2(A,set(B)),A6),I4))) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_lr(fun(A,set(B)),fun(A,nat),A6)),I4) ) ) ) ) ).
% card_UN_disjoint
tff(fact_1372_sum_Oreindex__cong,axiom,
! [C: $tType,A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [L: fun(B,C),B6: set(B),A6: set(C),G: fun(C,A),H: fun(B,A)] :
( inj_on(B,C,L,B6)
=> ( ( A6 = aa(set(B),set(C),aa(fun(B,C),fun(set(B),set(C)),image2(B,C),L),B6) )
=> ( ! [X: B] :
( pp(aa(set(B),bool,member(B,X),B6))
=> ( aa(C,A,G,aa(B,C,L,X)) = aa(B,A,H,X) ) )
=> ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),G),A6) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),B6) ) ) ) ) ) ).
% sum.reindex_cong
tff(fact_1373_sum_Oimage__gen,axiom,
! [A: $tType,C: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [S3: set(B),H: fun(B,A),G: fun(B,C)] :
( finite_finite2(B,S3)
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),H),S3) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),aa(fun(B,C),fun(C,A),aa(fun(B,A),fun(fun(B,C),fun(C,A)),aTP_Lamp_ls(set(B),fun(fun(B,A),fun(fun(B,C),fun(C,A))),S3),H),G)),aa(set(B),set(C),aa(fun(B,C),fun(set(B),set(C)),image2(B,C),G),S3)) ) ) ) ).
% sum.image_gen
tff(fact_1374_sum_Oimage__eq,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(B,A),A6: set(B)] :
( inj_on(B,A,G,A6)
=> ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_lt(A,A)),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),G),A6)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A6) ) ) ) ).
% sum.image_eq
tff(fact_1375_sum_Ocartesian__product,axiom,
! [A: $tType,B: $tType,C: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(B,fun(C,A)),B6: set(C),A6: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(set(C),fun(B,A),aTP_Lamp_lu(fun(B,fun(C,A)),fun(set(C),fun(B,A)),G),B6)),A6) = aa(set(product_prod(B,C)),A,aa(fun(product_prod(B,C),A),fun(set(product_prod(B,C)),A),groups7311177749621191930dd_sum(product_prod(B,C),A),aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),G)),product_Sigma(B,C,A6,aTP_Lamp_hc(set(C),fun(B,set(C)),B6))) ) ).
% sum.cartesian_product
tff(fact_1376_sum__comp__morphism,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comm_monoid_add(B)
& comm_monoid_add(A) )
=> ! [H: fun(B,A),G: fun(C,B),A6: set(C)] :
( ( aa(B,A,H,zero_zero(B)) = zero_zero(A) )
=> ( ! [X: B,Y: B] : aa(B,A,H,aa(B,B,aa(B,fun(B,B),plus_plus(B),X),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,H,X)),aa(B,A,H,Y))
=> ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),aa(fun(C,B),fun(C,A),aa(fun(B,A),fun(fun(C,B),fun(C,A)),comp(B,A,C),H),G)),A6) = aa(B,A,H,aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),G),A6)) ) ) ) ) ).
% sum_comp_morphism
tff(fact_1377_sum_OUnion__comp,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [B6: set(set(B)),G: fun(B,A)] :
( ! [X: set(B)] :
( pp(aa(set(set(B)),bool,member(set(B),X),B6))
=> finite_finite2(B,X) )
=> ( ! [A15: set(B)] :
( pp(aa(set(set(B)),bool,member(set(B),A15),B6))
=> ! [A25: set(B)] :
( pp(aa(set(set(B)),bool,member(set(B),A25),B6))
=> ( ( A15 != A25 )
=> ! [X: B] :
( pp(aa(set(B),bool,member(B,X),A15))
=> ( pp(aa(set(B),bool,member(B,X),A25))
=> ( aa(B,A,G,X) = zero_zero(A) ) ) ) ) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),B6)) = aa(set(set(B)),A,aa(fun(B,A),fun(set(set(B)),A),aa(fun(fun(B,A),fun(set(B),A)),fun(fun(B,A),fun(set(set(B)),A)),aa(fun(fun(set(B),A),fun(set(set(B)),A)),fun(fun(fun(B,A),fun(set(B),A)),fun(fun(B,A),fun(set(set(B)),A))),comp(fun(set(B),A),fun(set(set(B)),A),fun(B,A)),groups7311177749621191930dd_sum(set(B),A)),groups7311177749621191930dd_sum(B,A)),G),B6) ) ) ) ) ).
% sum.Union_comp
tff(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)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B6),aa(set(product_prod(A,A)),set(A),field2(A),R)))
=> ( ( order_AboveS(A,R,B6) != bot_bot(set(A)) )
=> ( pp(aa(set(A),bool,member(A,B2),B6))
=> ( ( bNF_Wellorder_wo_suc(A,R,B6) != B2 )
& pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),bNF_Wellorder_wo_suc(A,R,B6))),R)) ) ) ) ) ) ).
% wo_rel.suc_greater
tff(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)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B6),aa(set(product_prod(A,A)),set(A),field2(A),R)))
=> ( pp(aa(set(A),bool,member(A,A3),order_AboveS(A,R,B6)))
=> ( ! [A11: A] :
( pp(aa(set(A),bool,member(A,A11),order_AboveS(A,R,B6)))
=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A11)),R)) )
=> ( A3 = bNF_Wellorder_wo_suc(A,R,B6) ) ) ) ) ) ).
% wo_rel.equals_suc_AboveS
tff(fact_1380_Chains__subset_H,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( refl_on(A,top_top(set(A)),R)
=> pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),aa(fun(set(A),bool),set(set(A)),collect(set(A)),pred_chain(A,top_top(set(A)),aTP_Lamp_hl(set(product_prod(A,A)),fun(A,fun(A,bool)),R)))),chains(A,R))) ) ).
% Chains_subset'
tff(fact_1381_sum_Otriangle__reindex__eq,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(nat,fun(nat,A)),N4: nat] : aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups7311177749621191930dd_sum(product_prod(nat,nat),A),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G)),aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_lv(nat,fun(nat,fun(nat,bool)),N4)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_lx(fun(nat,fun(nat,A)),fun(nat,A),G)),aa(nat,set(nat),set_ord_atMost(nat),N4)) ) ).
% sum.triangle_reindex_eq
tff(fact_1382_sum__card__image,axiom,
! [B: $tType,A: $tType,A6: set(A),F: fun(A,set(B))] :
( finite_finite2(A,A6)
=> ( pairwise(A,aTP_Lamp_ly(fun(A,set(B)),fun(A,fun(A,bool)),F),A6)
=> ( aa(set(set(B)),nat,aa(fun(set(B),nat),fun(set(set(B)),nat),groups7311177749621191930dd_sum(set(B),nat),finite_card(B)),aa(set(A),set(set(B)),aa(fun(A,set(B)),fun(set(A),set(set(B))),image2(A,set(B)),F),A6)) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_lr(fun(A,set(B)),fun(A,nat),F)),A6) ) ) ) ).
% sum_card_image
tff(fact_1383_sum_OatLeastAtMost__shift__0,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [M4: nat,N4: nat,G: fun(nat,A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M4),N4))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M4,N4)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),aa(nat,fun(nat,nat),plus_plus(nat),M4))),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N4),M4))) ) ) ) ).
% sum.atLeastAtMost_shift_0
tff(fact_1384_divmod__nat__if,axiom,
! [N4: nat,M4: nat] :
( ( ( ( N4 = zero_zero(nat) )
| pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M4),N4)) )
=> ( divmod_nat(M4,N4) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),zero_zero(nat)),M4) ) )
& ( ~ ( ( N4 = zero_zero(nat) )
| pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M4),N4)) )
=> ( divmod_nat(M4,N4) = aa(product_prod(nat,nat),product_prod(nat,nat),aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aTP_Lamp_lz(nat,fun(nat,product_prod(nat,nat)))),divmod_nat(aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),M4),N4),N4)) ) ) ) ).
% divmod_nat_if
tff(fact_1385_Rats__eq__range__nat__to__rat__surj,axiom,
field_char_0_Rats(rat) = aa(set(nat),set(rat),aa(fun(nat,rat),fun(set(nat),set(rat)),image2(nat,rat),nat_to_rat_surj),top_top(set(nat))) ).
% Rats_eq_range_nat_to_rat_surj
tff(fact_1386_image__add__atLeastAtMost,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [K: A,I: A,J5: A] : aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aa(A,fun(A,A),plus_plus(A),K)),set_or1337092689740270186AtMost(A,I,J5)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),J5),K)) ) ).
% image_add_atLeastAtMost
tff(fact_1387_image__diff__atLeastAtMost,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [D4: A,A3: A,B2: A] : aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aa(A,fun(A,A),minus_minus(A),D4)),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),D4),B2),aa(A,A,aa(A,fun(A,A),minus_minus(A),D4),A3)) ) ).
% image_diff_atLeastAtMost
tff(fact_1388_image__add__atMost,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [C3: A,A3: A] : aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aa(A,fun(A,A),plus_plus(A),C3)),aa(A,set(A),set_ord_atMost(A),A3)) = aa(A,set(A),set_ord_atMost(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A3)) ) ).
% image_add_atMost
tff(fact_1389_image__add__atLeastAtMost_H,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [K: A,I: A,J5: A] : aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aTP_Lamp_kt(A,fun(A,A),K)),set_or1337092689740270186AtMost(A,I,J5)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),J5),K)) ) ).
% image_add_atLeastAtMost'
tff(fact_1390_image__Suc__atLeastAtMost,axiom,
! [I: nat,J5: nat] : aa(set(nat),set(nat),aa(fun(nat,nat),fun(set(nat),set(nat)),image2(nat,nat),suc),set_or1337092689740270186AtMost(nat,I,J5)) = set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,I),aa(nat,nat,suc,J5)) ).
% image_Suc_atLeastAtMost
tff(fact_1391_image__minus__const__atLeastAtMost_H,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [D4: A,A3: A,B2: A] : aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aTP_Lamp_ma(A,fun(A,A),D4)),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),A3),D4),aa(A,A,aa(A,fun(A,A),minus_minus(A),B2),D4)) ) ).
% image_minus_const_atLeastAtMost'
tff(fact_1392_pairwise__imageI,axiom,
! [B: $tType,A: $tType,A6: set(A),F: fun(A,B),P2: fun(B,fun(B,bool))] :
( ! [X: A,Y: A] :
( pp(aa(set(A),bool,member(A,X),A6))
=> ( pp(aa(set(A),bool,member(A,Y),A6))
=> ( ( X != Y )
=> ( ( aa(A,B,F,X) != aa(A,B,F,Y) )
=> pp(aa(B,bool,aa(B,fun(B,bool),P2,aa(A,B,F,X)),aa(A,B,F,Y))) ) ) ) )
=> pairwise(B,P2,aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6)) ) ).
% pairwise_imageI
tff(fact_1393_UN__le__add__shift,axiom,
! [A: $tType,M: fun(nat,set(A)),K: nat,N4: nat] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),aa(fun(nat,set(A)),fun(set(nat),set(set(A))),image2(nat,set(A)),aa(nat,fun(nat,set(A)),aTP_Lamp_mb(fun(nat,set(A)),fun(nat,fun(nat,set(A))),M),K)),aa(nat,set(nat),set_ord_atMost(nat),N4))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),aa(fun(nat,set(A)),fun(set(nat),set(set(A))),image2(nat,set(A)),M),set_or1337092689740270186AtMost(nat,K,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N4),K)))) ).
% UN_le_add_shift
tff(fact_1394_pairwise__image,axiom,
! [A: $tType,B: $tType,R: fun(A,fun(A,bool)),F: fun(B,A),S: set(B)] :
( pairwise(A,R,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),S))
<=> pairwise(B,aa(fun(B,A),fun(B,fun(B,bool)),aTP_Lamp_mc(fun(A,fun(A,bool)),fun(fun(B,A),fun(B,fun(B,bool))),R),F),S) ) ).
% pairwise_image
tff(fact_1395_UN__atMost__UNIV,axiom,
aa(set(set(nat)),set(nat),complete_Sup_Sup(set(nat)),aa(set(nat),set(set(nat)),aa(fun(nat,set(nat)),fun(set(nat),set(set(nat))),image2(nat,set(nat)),set_ord_atMost(nat)),top_top(set(nat)))) = top_top(set(nat)) ).
% UN_atMost_UNIV
tff(fact_1396_sum_OatLeast__Suc__atMost__Suc__shift,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(nat,A),M4: nat,N4: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M4),aa(nat,nat,suc,N4))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),suc)),set_or1337092689740270186AtMost(nat,M4,N4)) ) ).
% sum.atLeast_Suc_atMost_Suc_shift
tff(fact_1397_sum_OatLeastAtMost__shift__bounds,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(nat,A),M4: nat,K: nat,N4: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M4),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N4),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),aa(nat,fun(nat,nat),plus_plus(nat),K))),set_or1337092689740270186AtMost(nat,M4,N4)) ) ).
% sum.atLeastAtMost_shift_bounds
tff(fact_1398_atMost__Suc__eq__insert__0,axiom,
! [N4: nat] : aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,N4)) = aa(set(nat),set(nat),insert(nat,zero_zero(nat)),aa(set(nat),set(nat),aa(fun(nat,nat),fun(set(nat),set(nat)),image2(nat,nat),suc),aa(nat,set(nat),set_ord_atMost(nat),N4))) ).
% atMost_Suc_eq_insert_0
tff(fact_1399_disjoint__image__subset,axiom,
! [A: $tType,A14: set(set(A)),F: fun(set(A),set(A))] :
( pairwise(set(A),disjnt(A),A14)
=> ( ! [X8: set(A)] :
( pp(aa(set(set(A)),bool,member(set(A),X8),A14))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),F,X8)),X8)) )
=> pairwise(set(A),disjnt(A),aa(set(set(A)),set(set(A)),aa(fun(set(A),set(A)),fun(set(set(A)),set(set(A))),image2(set(A),set(A)),F),A14)) ) ) ).
% disjoint_image_subset
tff(fact_1400_atLeast0__atMost__Suc__eq__insert__0,axiom,
! [N4: nat] : set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N4)) = aa(set(nat),set(nat),insert(nat,zero_zero(nat)),aa(set(nat),set(nat),aa(fun(nat,nat),fun(set(nat),set(nat)),image2(nat,nat),suc),set_or1337092689740270186AtMost(nat,zero_zero(nat),N4))) ).
% atLeast0_atMost_Suc_eq_insert_0
tff(fact_1401_sum_OatLeast0__atMost__Suc__shift,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(nat,A),N4: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N4))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),suc)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N4))) ) ).
% sum.atLeast0_atMost_Suc_shift
tff(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)
=> ( pp(aa(set(A),bool,member(A,A3),order_AboveS(A,R,B6)))
=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),bNF_Wellorder_wo_suc(A,R,B6)),A3)),R)) ) ) ).
% wo_rel.suc_least_AboveS
tff(fact_1403_sum_OatLeast__atMost__pred__shift,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(nat,A),M4: nat,N4: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),aTP_Lamp_lm(nat,nat))),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M4),aa(nat,nat,suc,N4))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M4,N4)) ) ).
% sum.atLeast_atMost_pred_shift
tff(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) = aa(fun(set(A),bool),set(set(A)),collect(set(A)),pred_chain(A,top_top(set(A)),aTP_Lamp_hl(set(product_prod(A,A)),fun(A,fun(A,bool)),R))) ) ) ).
% Chains_alt_def
tff(fact_1405_Chains__subset,axiom,
! [A: $tType,R: set(product_prod(A,A))] : pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),chains(A,R)),aa(fun(set(A),bool),set(set(A)),collect(set(A)),pred_chain(A,top_top(set(A)),aTP_Lamp_hl(set(product_prod(A,A)),fun(A,fun(A,bool)),R))))) ).
% Chains_subset
tff(fact_1406_divides__aux__eq,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [Q: A,R: A] :
( unique5940410009612947441es_aux(A,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q),R))
<=> ( R = zero_zero(A) ) ) ) ).
% divides_aux_eq
tff(fact_1407_at__bot__def,axiom,
! [A: $tType] :
( order(A)
=> ( at_bot(A) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(A),set(filter(A)),aa(fun(A,filter(A)),fun(set(A),set(filter(A))),image2(A,filter(A)),aTP_Lamp_md(A,filter(A))),top_top(set(A)))) ) ) ).
% at_bot_def
tff(fact_1408_sum_Otriangle__reindex,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(nat,fun(nat,A)),N4: nat] : aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups7311177749621191930dd_sum(product_prod(nat,nat),A),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G)),aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_me(nat,fun(nat,fun(nat,bool)),N4)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_lx(fun(nat,fun(nat,A)),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),N4)) ) ).
% sum.triangle_reindex
tff(fact_1409_at__bot__sub,axiom,
! [A: $tType] :
( linorder(A)
=> ! [C3: A] : at_bot(A) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(A),set(filter(A)),aa(fun(A,filter(A)),fun(set(A),set(filter(A))),image2(A,filter(A)),aTP_Lamp_mf(A,filter(A))),aa(A,set(A),set_ord_atMost(A),C3))) ) ).
% at_bot_sub
tff(fact_1410_UN__le__eq__Un0,axiom,
! [A: $tType,M: fun(nat,set(A)),N4: nat] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),aa(fun(nat,set(A)),fun(set(nat),set(set(A))),image2(nat,set(A)),M),aa(nat,set(nat),set_ord_atMost(nat),N4))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),aa(fun(nat,set(A)),fun(set(nat),set(set(A))),image2(nat,set(A)),M),set_or1337092689740270186AtMost(nat,one_one(nat),N4)))),aa(nat,set(A),M,zero_zero(nat))) ).
% UN_le_eq_Un0
tff(fact_1411_divides__aux__def,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [Qr: product_prod(A,A)] :
( unique5940410009612947441es_aux(A,Qr)
<=> ( aa(product_prod(A,A),A,product_snd(A,A),Qr) = zero_zero(A) ) ) ) ).
% divides_aux_def
tff(fact_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)) )
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),bNF_Wellorder_wo_suc(A,R,A6))),R))
=> ( ( B2 != bNF_Wellorder_wo_suc(A,R,A6) )
=> pp(aa(set(A),bool,member(A,B2),A6)) ) ) ) ) ) ).
% wo_rel.suc_ofilter_in
tff(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)) )
=> ( aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aa(A,fun(A,A),aTP_Lamp_mg(A,fun(A,fun(A,A)),M4),C3)),set_or1337092689740270186AtMost(A,A3,B2)) = bot_bot(set(A)) ) )
& ( ( set_or1337092689740270186AtMost(A,A3,B2) != bot_bot(set(A)) )
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M4))
=> ( aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aa(A,fun(A,A),aTP_Lamp_mg(A,fun(A,fun(A,A)),M4),C3)),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M4),A3)),C3),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M4),B2)),C3)) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M4))
=> ( aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aa(A,fun(A,A),aTP_Lamp_mg(A,fun(A,fun(A,A)),M4),C3)),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M4),B2)),C3),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M4),A3)),C3)) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_diff
tff(fact_1414_image__mult__atLeastAtMost,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [D4: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),D4))
=> ( aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aa(A,fun(A,A),times_times(A),D4)),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),D4),A3),aa(A,A,aa(A,fun(A,A),times_times(A),D4),B2)) ) ) ) ).
% image_mult_atLeastAtMost
tff(fact_1415_image__Suc__lessThan,axiom,
! [N4: nat] : aa(set(nat),set(nat),aa(fun(nat,nat),fun(set(nat),set(nat)),image2(nat,nat),suc),aa(nat,set(nat),set_ord_lessThan(nat),N4)) = set_or1337092689740270186AtMost(nat,one_one(nat),N4) ).
% image_Suc_lessThan
tff(fact_1416_UN__lessThan__UNIV,axiom,
aa(set(set(nat)),set(nat),complete_Sup_Sup(set(nat)),aa(set(nat),set(set(nat)),aa(fun(nat,set(nat)),fun(set(nat),set(set(nat))),image2(nat,set(nat)),set_ord_lessThan(nat)),top_top(set(nat)))) = top_top(set(nat)) ).
% UN_lessThan_UNIV
tff(fact_1417_wo__rel_Oofilter__UNION,axiom,
! [A: $tType,B: $tType,R: set(product_prod(A,A)),I4: set(B),A6: fun(B,set(A))] :
( bNF_Wellorder_wo_rel(A,R)
=> ( ! [I6: B] :
( pp(aa(set(B),bool,member(B,I6),I4))
=> order_ofilter(A,R,aa(B,set(A),A6,I6)) )
=> order_ofilter(A,R,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),A6),I4))) ) ) ).
% wo_rel.ofilter_UNION
tff(fact_1418_lessThan__Suc__eq__insert__0,axiom,
! [N4: nat] : aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,N4)) = aa(set(nat),set(nat),insert(nat,zero_zero(nat)),aa(set(nat),set(nat),aa(fun(nat,nat),fun(set(nat),set(nat)),image2(nat,nat),suc),aa(nat,set(nat),set_ord_lessThan(nat),N4))) ).
% lessThan_Suc_eq_insert_0
tff(fact_1419_UN__le__add__shift__strict,axiom,
! [A: $tType,M: fun(nat,set(A)),K: nat,N4: nat] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),aa(fun(nat,set(A)),fun(set(nat),set(set(A))),image2(nat,set(A)),aa(nat,fun(nat,set(A)),aTP_Lamp_mb(fun(nat,set(A)),fun(nat,fun(nat,set(A))),M),K)),aa(nat,set(nat),set_ord_lessThan(nat),N4))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),aa(fun(nat,set(A)),fun(set(nat),set(set(A))),image2(nat,set(A)),M),set_or7035219750837199246ssThan(nat,K,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N4),K)))) ).
% UN_le_add_shift_strict
tff(fact_1420_image__Suc__atMost,axiom,
! [N4: nat] : aa(set(nat),set(nat),aa(fun(nat,nat),fun(set(nat),set(nat)),image2(nat,nat),suc),aa(nat,set(nat),set_ord_atMost(nat),N4)) = set_or1337092689740270186AtMost(nat,one_one(nat),aa(nat,nat,suc,N4)) ).
% image_Suc_atMost
tff(fact_1421_sum__le__card__Max,axiom,
! [A: $tType,A6: set(A),F: fun(A,nat)] :
( finite_finite2(A,A6)
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F),A6)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(A),nat,finite_card(A),A6)),lattic643756798349783984er_Max(nat,aa(set(A),set(nat),aa(fun(A,nat),fun(set(A),set(nat)),image2(A,nat),F),A6))))) ) ).
% sum_le_card_Max
tff(fact_1422_image__mult__atLeastAtMost__if,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [C3: A,X2: A,Y2: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> ( aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aa(A,fun(A,A),times_times(A),C3)),set_or1337092689740270186AtMost(A,X2,Y2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),X2),aa(A,A,aa(A,fun(A,A),times_times(A),C3),Y2)) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Y2))
=> ( aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aa(A,fun(A,A),times_times(A),C3)),set_or1337092689740270186AtMost(A,X2,Y2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),Y2),aa(A,A,aa(A,fun(A,A),times_times(A),C3),X2)) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Y2))
=> ( aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aa(A,fun(A,A),times_times(A),C3)),set_or1337092689740270186AtMost(A,X2,Y2)) = bot_bot(set(A)) ) ) ) ) ) ) ).
% image_mult_atLeastAtMost_if
tff(fact_1423_image__mult__atLeastAtMost__if_H,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [X2: A,Y2: A,C3: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Y2))
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> ( aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aTP_Lamp_mh(A,fun(A,A),C3)),set_or1337092689740270186AtMost(A,X2,Y2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),X2),C3),aa(A,A,aa(A,fun(A,A),times_times(A),Y2),C3)) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C3))
=> ( aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aTP_Lamp_mh(A,fun(A,A),C3)),set_or1337092689740270186AtMost(A,X2,Y2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),Y2),C3),aa(A,A,aa(A,fun(A,A),times_times(A),X2),C3)) ) ) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Y2))
=> ( aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aTP_Lamp_mh(A,fun(A,A),C3)),set_or1337092689740270186AtMost(A,X2,Y2)) = bot_bot(set(A)) ) ) ) ) ).
% image_mult_atLeastAtMost_if'
tff(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)) )
=> ( aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aa(A,fun(A,A),aTP_Lamp_mi(A,fun(A,fun(A,A)),M4),C3)),set_or1337092689740270186AtMost(A,A3,B2)) = bot_bot(set(A)) ) )
& ( ( set_or1337092689740270186AtMost(A,A3,B2) != bot_bot(set(A)) )
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M4))
=> ( aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aa(A,fun(A,A),aTP_Lamp_mi(A,fun(A,fun(A,A)),M4),C3)),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M4),A3)),C3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M4),B2)),C3)) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M4))
=> ( aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aa(A,fun(A,A),aTP_Lamp_mi(A,fun(A,fun(A,A)),M4),C3)),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M4),B2)),C3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M4),A3)),C3)) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost
tff(fact_1425_UNION__inj__on__ofilter,axiom,
! [C: $tType,A: $tType,B: $tType,R: set(product_prod(A,A)),I4: set(B),A6: fun(B,set(A)),F: fun(A,C)] :
( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R),R)
=> ( ! [I6: B] :
( pp(aa(set(B),bool,member(B,I6),I4))
=> order_ofilter(A,R,aa(B,set(A),A6,I6)) )
=> ( ! [I6: B] :
( pp(aa(set(B),bool,member(B,I6),I4))
=> inj_on(A,C,F,aa(B,set(A),A6,I6)) )
=> inj_on(A,C,F,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),A6),I4))) ) ) ) ).
% UNION_inj_on_ofilter
tff(fact_1426_ofilterIncl__def,axiom,
! [A: $tType,R: set(product_prod(A,A))] : bNF_We413866401316099525erIncl(A,R) = aa(fun(product_prod(set(A),set(A)),bool),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aa(fun(set(A),fun(set(A),bool)),fun(product_prod(set(A),set(A)),bool),product_case_prod(set(A),set(A),bool),aTP_Lamp_mj(set(product_prod(A,A)),fun(set(A),fun(set(A),bool)),R))) ).
% ofilterIncl_def
tff(fact_1427_sum__mult__sum__if__inj,axiom,
! [A: $tType,B: $tType,C: $tType] :
( semiring_0(B)
=> ! [F: fun(A,B),G: fun(C,B),A6: set(A),B6: set(C)] :
( inj_on(product_prod(A,C),B,aa(fun(A,fun(C,B)),fun(product_prod(A,C),B),product_case_prod(A,C,B),aa(fun(C,B),fun(A,fun(C,B)),aTP_Lamp_mk(fun(A,B),fun(fun(C,B),fun(A,fun(C,B))),F),G)),product_Sigma(A,C,A6,aTP_Lamp_cg(set(C),fun(A,set(C)),B6)))
=> ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F),A6)),aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),G),B6)) = aa(set(B),B,aa(fun(B,B),fun(set(B),B),groups7311177749621191930dd_sum(B,B),id(B)),aa(fun(B,bool),set(B),collect(B),aa(set(C),fun(B,bool),aa(set(A),fun(set(C),fun(B,bool)),aa(fun(C,B),fun(set(A),fun(set(C),fun(B,bool))),aTP_Lamp_ml(fun(A,B),fun(fun(C,B),fun(set(A),fun(set(C),fun(B,bool)))),F),G),A6),B6))) ) ) ) ).
% sum_mult_sum_if_inj
tff(fact_1428_Ref_Oalloc__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [X2: A,H: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(ref(A),heap_ext(product_unit)),alloc2(A,X2),H) = aa(heap_ext(product_unit),product_prod(ref(A),heap_ext(product_unit)),aa(ref(A),fun(heap_ext(product_unit),product_prod(ref(A),heap_ext(product_unit))),product_Pair(ref(A),heap_ext(product_unit)),ref2(A,lim(product_unit,H))),aa(heap_ext(product_unit),heap_ext(product_unit),set4(A,ref2(A,lim(product_unit,H)),X2),lim_update(product_unit,aTP_Lamp_mm(heap_ext(product_unit),fun(nat,nat),H),H))) ) ).
% Ref.alloc_def
tff(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)) )
=> ( aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aa(A,fun(A,A),aTP_Lamp_mn(A,fun(A,fun(A,A)),M4),C3)),set_or1337092689740270186AtMost(A,A3,B2)) = bot_bot(set(A)) ) )
& ( ( set_or1337092689740270186AtMost(A,A3,B2) != bot_bot(set(A)) )
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M4))
=> ( aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aa(A,fun(A,A),aTP_Lamp_mn(A,fun(A,fun(A,A)),M4),C3)),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),M4)),C3),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),M4)),C3)) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M4))
=> ( aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aa(A,fun(A,A),aTP_Lamp_mn(A,fun(A,fun(A,A)),M4),C3)),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),M4)),C3),aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),M4)),C3)) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_div_diff
tff(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)) )
=> ( aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aa(A,fun(A,A),aTP_Lamp_mo(A,fun(A,fun(A,A)),M4),C3)),set_or1337092689740270186AtMost(A,A3,B2)) = bot_bot(set(A)) ) )
& ( ( set_or1337092689740270186AtMost(A,A3,B2) != bot_bot(set(A)) )
=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M4))
=> ( aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aa(A,fun(A,A),aTP_Lamp_mo(A,fun(A,fun(A,A)),M4),C3)),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),M4)),C3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),M4)),C3)) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M4))
=> ( aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aa(A,fun(A,A),aTP_Lamp_mo(A,fun(A,fun(A,A)),M4),C3)),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),M4)),C3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),M4)),C3)) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_div
tff(fact_1431_sum__fun__comp,axiom,
! [C: $tType,A: $tType,B: $tType] :
( semiring_1(C)
=> ! [S3: set(A),R3: set(B),G: fun(A,B),F: fun(B,C)] :
( finite_finite2(A,S3)
=> ( finite_finite2(B,R3)
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),G),S3)),R3))
=> ( aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),aa(fun(B,C),fun(A,C),aTP_Lamp_mp(fun(A,B),fun(fun(B,C),fun(A,C)),G),F)),S3) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),aa(fun(B,C),fun(B,C),aa(fun(A,B),fun(fun(B,C),fun(B,C)),aTP_Lamp_mr(set(A),fun(fun(A,B),fun(fun(B,C),fun(B,C))),S3),G),F)),R3) ) ) ) ) ) ).
% sum_fun_comp
tff(fact_1432_prod_OatLeastAtMost__shift__0,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [M4: nat,N4: nat,G: fun(nat,A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M4),N4))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M4,N4)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),aa(nat,fun(nat,nat),plus_plus(nat),M4))),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N4),M4))) ) ) ) ).
% prod.atLeastAtMost_shift_0
tff(fact_1433_Domain__Collect__case__prod,axiom,
! [B: $tType,A: $tType,P2: fun(A,fun(B,bool))] : aa(set(product_prod(A,B)),set(A),domain(A,B),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),P2))) = aa(fun(A,bool),set(A),collect(A),aTP_Lamp_ms(fun(A,fun(B,bool)),fun(A,bool),P2)) ).
% Domain_Collect_case_prod
tff(fact_1434_Range__Collect__case__prod,axiom,
! [A: $tType,B: $tType,P2: fun(B,fun(A,bool))] : aa(set(product_prod(B,A)),set(A),range2(B,A),aa(fun(product_prod(B,A),bool),set(product_prod(B,A)),collect(product_prod(B,A)),aa(fun(B,fun(A,bool)),fun(product_prod(B,A),bool),product_case_prod(B,A,bool),P2))) = aa(fun(A,bool),set(A),collect(A),aTP_Lamp_mt(fun(B,fun(A,bool)),fun(A,bool),P2)) ).
% Range_Collect_case_prod
tff(fact_1435_image__divide__atLeastAtMost,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [D4: A,A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),D4))
=> ( aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aTP_Lamp_mu(A,fun(A,A),D4)),set_or1337092689740270186AtMost(A,A3,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),D4),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),D4)) ) ) ) ).
% image_divide_atLeastAtMost
tff(fact_1436_prod_Ocartesian__product,axiom,
! [A: $tType,B: $tType,C: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(B,fun(C,A)),B6: set(C),A6: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(set(C),fun(B,A),aTP_Lamp_mv(fun(B,fun(C,A)),fun(set(C),fun(B,A)),G),B6)),A6) = aa(set(product_prod(B,C)),A,aa(fun(product_prod(B,C),A),fun(set(product_prod(B,C)),A),groups7121269368397514597t_prod(product_prod(B,C),A),aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),G)),product_Sigma(B,C,A6,aTP_Lamp_hc(set(C),fun(B,set(C)),B6))) ) ).
% prod.cartesian_product
tff(fact_1437_image__int__atLeastAtMost,axiom,
! [A3: nat,B2: nat] : aa(set(nat),set(int),aa(fun(nat,int),fun(set(nat),set(int)),image2(nat,int),semiring_1_of_nat(int)),set_or1337092689740270186AtMost(nat,A3,B2)) = set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),A3),aa(nat,int,semiring_1_of_nat(int),B2)) ).
% image_int_atLeastAtMost
tff(fact_1438_image__int__atLeastLessThan,axiom,
! [A3: nat,B2: nat] : aa(set(nat),set(int),aa(fun(nat,int),fun(set(nat),set(int)),image2(nat,int),semiring_1_of_nat(int)),set_or7035219750837199246ssThan(nat,A3,B2)) = set_or7035219750837199246ssThan(int,aa(nat,int,semiring_1_of_nat(int),A3),aa(nat,int,semiring_1_of_nat(int),B2)) ).
% image_int_atLeastLessThan
tff(fact_1439_setcompr__eq__image,axiom,
! [A: $tType,B: $tType,F: fun(B,A),P2: fun(B,bool)] : aa(fun(A,bool),set(A),collect(A),aa(fun(B,bool),fun(A,bool),aTP_Lamp_mw(fun(B,A),fun(fun(B,bool),fun(A,bool)),F),P2)) = aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),aa(fun(B,bool),set(B),collect(B),P2)) ).
% setcompr_eq_image
tff(fact_1440_Setcompr__eq__image,axiom,
! [A: $tType,B: $tType,F: fun(B,A),A6: set(B)] : aa(fun(A,bool),set(A),collect(A),aa(set(B),fun(A,bool),aTP_Lamp_mx(fun(B,A),fun(set(B),fun(A,bool)),F),A6)) = aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6) ).
% Setcompr_eq_image
tff(fact_1441_prod_OSigma,axiom,
! [A: $tType,C: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [A6: set(B),B6: fun(B,set(C)),G: fun(B,fun(C,A))] :
( finite_finite2(B,A6)
=> ( ! [X: B] :
( pp(aa(set(B),bool,member(B,X),A6))
=> finite_finite2(C,aa(B,set(C),B6,X)) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,fun(C,A)),fun(B,A),aTP_Lamp_my(fun(B,set(C)),fun(fun(B,fun(C,A)),fun(B,A)),B6),G)),A6) = aa(set(product_prod(B,C)),A,aa(fun(product_prod(B,C),A),fun(set(product_prod(B,C)),A),groups7121269368397514597t_prod(product_prod(B,C),A),aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),G)),product_Sigma(B,C,A6,B6)) ) ) ) ) ).
% prod.Sigma
tff(fact_1442_OO__def,axiom,
! [A: $tType,C: $tType,B: $tType,R3: fun(A,fun(C,bool)),S3: fun(C,fun(B,bool)),X4: A,Xa4: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(C,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(C,bool)),fun(fun(C,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,C,B),R3),S3),X4),Xa4))
<=> ? [Y4: C] :
( pp(aa(C,bool,aa(A,fun(C,bool),R3,X4),Y4))
& pp(aa(B,bool,aa(C,fun(B,bool),S3,Y4),Xa4)) ) ) ).
% OO_def
tff(fact_1443_prod_Oreindex__cong,axiom,
! [C: $tType,A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [L: fun(B,C),B6: set(B),A6: set(C),G: fun(C,A),H: fun(B,A)] :
( inj_on(B,C,L,B6)
=> ( ( A6 = aa(set(B),set(C),aa(fun(B,C),fun(set(B),set(C)),image2(B,C),L),B6) )
=> ( ! [X: B] :
( pp(aa(set(B),bool,member(B,X),B6))
=> ( aa(C,A,G,aa(B,C,L,X)) = aa(B,A,H,X) ) )
=> ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),G),A6) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),B6) ) ) ) ) ) ).
% prod.reindex_cong
tff(fact_1444_prod_OUnion__comp,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [B6: set(set(B)),G: fun(B,A)] :
( ! [X: set(B)] :
( pp(aa(set(set(B)),bool,member(set(B),X),B6))
=> finite_finite2(B,X) )
=> ( ! [A15: set(B)] :
( pp(aa(set(set(B)),bool,member(set(B),A15),B6))
=> ! [A25: set(B)] :
( pp(aa(set(set(B)),bool,member(set(B),A25),B6))
=> ( ( A15 != A25 )
=> ! [X: B] :
( pp(aa(set(B),bool,member(B,X),A15))
=> ( pp(aa(set(B),bool,member(B,X),A25))
=> ( aa(B,A,G,X) = one_one(A) ) ) ) ) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),B6)) = aa(set(set(B)),A,aa(fun(B,A),fun(set(set(B)),A),aa(fun(fun(B,A),fun(set(B),A)),fun(fun(B,A),fun(set(set(B)),A)),aa(fun(fun(set(B),A),fun(set(set(B)),A)),fun(fun(fun(B,A),fun(set(B),A)),fun(fun(B,A),fun(set(set(B)),A))),comp(fun(set(B),A),fun(set(set(B)),A),fun(B,A)),groups7121269368397514597t_prod(set(B),A)),groups7121269368397514597t_prod(B,A)),G),B6) ) ) ) ) ).
% prod.Union_comp
tff(fact_1445_image2p__def,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F: fun(C,A),G: fun(D,B),R3: fun(C,fun(D,bool)),X4: A,Xa4: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),bNF_Greatest_image2p(C,A,D,B,F,G,R3),X4),Xa4))
<=> ? [X10: C,Y7: D] :
( pp(aa(D,bool,aa(C,fun(D,bool),R3,X10),Y7))
& ( aa(C,A,F,X10) = X4 )
& ( aa(D,B,G,Y7) = Xa4 ) ) ) ).
% image2p_def
tff(fact_1446_prod_Oimage__gen,axiom,
! [A: $tType,C: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [S3: set(B),H: fun(B,A),G: fun(B,C)] :
( finite_finite2(B,S3)
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),S3) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),aa(fun(B,C),fun(C,A),aa(fun(B,A),fun(fun(B,C),fun(C,A)),aTP_Lamp_mz(set(B),fun(fun(B,A),fun(fun(B,C),fun(C,A))),S3),H),G)),aa(set(B),set(C),aa(fun(B,C),fun(set(B),set(C)),image2(B,C),G),S3)) ) ) ) ).
% prod.image_gen
tff(fact_1447_sum_OatLeast__int__atMost__int__shift,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(int,A),M4: nat,N4: nat] : aa(set(int),A,aa(fun(int,A),fun(set(int),A),groups7311177749621191930dd_sum(int,A),G),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),M4),aa(nat,int,semiring_1_of_nat(int),N4))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,int),fun(nat,A),aa(fun(int,A),fun(fun(nat,int),fun(nat,A)),comp(int,A,nat),G),semiring_1_of_nat(int))),set_or1337092689740270186AtMost(nat,M4,N4)) ) ).
% sum.atLeast_int_atMost_int_shift
tff(fact_1448_sum_OatLeast__int__lessThan__int__shift,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [G: fun(int,A),M4: nat,N4: nat] : aa(set(int),A,aa(fun(int,A),fun(set(int),A),groups7311177749621191930dd_sum(int,A),G),set_or7035219750837199246ssThan(int,aa(nat,int,semiring_1_of_nat(int),M4),aa(nat,int,semiring_1_of_nat(int),N4))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,int),fun(nat,A),aa(fun(int,A),fun(fun(nat,int),fun(nat,A)),comp(int,A,nat),G),semiring_1_of_nat(int))),set_or7035219750837199246ssThan(nat,M4,N4)) ) ).
% sum.atLeast_int_lessThan_int_shift
tff(fact_1449_prod_Oimage__eq,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(B,A),A6: set(B)] :
( inj_on(B,A,G,A6)
=> ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7121269368397514597t_prod(A,A),aTP_Lamp_na(A,A)),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),G),A6)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A6) ) ) ) ).
% prod.image_eq
tff(fact_1450_full__SetCompr__eq,axiom,
! [A: $tType,B: $tType,F: fun(B,A)] : aa(fun(A,bool),set(A),collect(A),aTP_Lamp_nb(fun(B,A),fun(A,bool),F)) = aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),top_top(set(B))) ).
% full_SetCompr_eq
tff(fact_1451_prod_OUnion__disjoint,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [C5: set(set(B)),G: fun(B,A)] :
( ! [X: set(B)] :
( pp(aa(set(set(B)),bool,member(set(B),X),C5))
=> finite_finite2(B,X) )
=> ( ! [X: set(B)] :
( pp(aa(set(set(B)),bool,member(set(B),X),C5))
=> ! [Xa: set(B)] :
( pp(aa(set(set(B)),bool,member(set(B),Xa),C5))
=> ( ( X != Xa )
=> ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),X),Xa) = bot_bot(set(B)) ) ) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),C5)) = aa(set(set(B)),A,aa(fun(B,A),fun(set(set(B)),A),aa(fun(fun(B,A),fun(set(B),A)),fun(fun(B,A),fun(set(set(B)),A)),aa(fun(fun(set(B),A),fun(set(set(B)),A)),fun(fun(fun(B,A),fun(set(B),A)),fun(fun(B,A),fun(set(set(B)),A))),comp(fun(set(B),A),fun(set(set(B)),A),fun(B,A)),groups7121269368397514597t_prod(set(B),A)),groups7121269368397514597t_prod(B,A)),G),C5) ) ) ) ) ).
% prod.Union_disjoint
tff(fact_1452_Id__def,axiom,
! [A: $tType] : id2(A) = aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aTP_Lamp_nc(product_prod(A,A),bool)) ).
% Id_def
tff(fact_1453_Domain__unfold,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,B))] : aa(set(product_prod(A,B)),set(A),domain(A,B),R) = aa(fun(A,bool),set(A),collect(A),aTP_Lamp_nd(set(product_prod(A,B)),fun(A,bool),R)) ).
% Domain_unfold
tff(fact_1454_Gr__def,axiom,
! [B: $tType,A: $tType,A6: set(A),F: fun(A,B)] : bNF_Gr(A,B,A6,F) = aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,B),fun(product_prod(A,B),bool),aTP_Lamp_ne(set(A),fun(fun(A,B),fun(product_prod(A,B),bool)),A6),F)) ).
% Gr_def
tff(fact_1455_relImage__def,axiom,
! [A: $tType,B: $tType,R3: set(product_prod(B,B)),F: fun(B,A)] : bNF_Gr4221423524335903396lImage(B,A,R3,F) = aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(B,A),fun(product_prod(A,A),bool),aTP_Lamp_nf(set(product_prod(B,B)),fun(fun(B,A),fun(product_prod(A,A),bool)),R3),F)) ).
% relImage_def
tff(fact_1456_relInvImage__def,axiom,
! [B: $tType,A: $tType,A6: set(A),R3: set(product_prod(B,B)),F: fun(A,B)] : bNF_Gr7122648621184425601vImage(A,B,A6,R3,F) = aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,B),fun(product_prod(A,A),bool),aa(set(product_prod(B,B)),fun(fun(A,B),fun(product_prod(A,A),bool)),aTP_Lamp_ng(set(A),fun(set(product_prod(B,B)),fun(fun(A,B),fun(product_prod(A,A),bool))),A6),R3),F)) ).
% relInvImage_def
tff(fact_1457_prod_Oreindex,axiom,
! [C: $tType,A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [H: fun(B,C),A6: set(B),G: fun(C,A)] :
( inj_on(B,C,H,A6)
=> ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),G),aa(set(B),set(C),aa(fun(B,C),fun(set(B),set(C)),image2(B,C),H),A6)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,C),fun(B,A),aa(fun(C,A),fun(fun(B,C),fun(B,A)),comp(C,A,B),G),H)),A6) ) ) ) ).
% prod.reindex
tff(fact_1458_prod_OatLeast__Suc__atMost__Suc__shift,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),M4: nat,N4: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M4),aa(nat,nat,suc,N4))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),suc)),set_or1337092689740270186AtMost(nat,M4,N4)) ) ).
% prod.atLeast_Suc_atMost_Suc_shift
tff(fact_1459_prod_OatLeast__Suc__lessThan__Suc__shift,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),M4: nat,N4: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M4),aa(nat,nat,suc,N4))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),suc)),set_or7035219750837199246ssThan(nat,M4,N4)) ) ).
% prod.atLeast_Suc_lessThan_Suc_shift
tff(fact_1460_prod_Ogroup,axiom,
! [C: $tType,A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [S3: set(B),T3: set(C),G: fun(B,C),H: fun(B,A)] :
( finite_finite2(B,S3)
=> ( finite_finite2(C,T3)
=> ( pp(aa(set(C),bool,aa(set(C),fun(set(C),bool),ord_less_eq(set(C)),aa(set(B),set(C),aa(fun(B,C),fun(set(B),set(C)),image2(B,C),G),S3)),T3))
=> ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),aa(fun(B,A),fun(C,A),aa(fun(B,C),fun(fun(B,A),fun(C,A)),aTP_Lamp_nh(set(B),fun(fun(B,C),fun(fun(B,A),fun(C,A))),S3),G),H)),T3) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),S3) ) ) ) ) ) ).
% prod.group
tff(fact_1461_prod_OatLeastAtMost__shift__bounds,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),M4: nat,K: nat,N4: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M4),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N4),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),aa(nat,fun(nat,nat),plus_plus(nat),K))),set_or1337092689740270186AtMost(nat,M4,N4)) ) ).
% prod.atLeastAtMost_shift_bounds
tff(fact_1462_prod_OatLeastLessThan__shift__bounds,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),M4: nat,K: nat,N4: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M4),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N4),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),aa(nat,fun(nat,nat),plus_plus(nat),K))),set_or7035219750837199246ssThan(nat,M4,N4)) ) ).
% prod.atLeastLessThan_shift_bounds
tff(fact_1463_image2__def,axiom,
! [A: $tType,B: $tType,C: $tType,A6: set(C),F: fun(C,A),G: fun(C,B)] : bNF_Greatest_image2(C,A,B,A6,F,G) = aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(C,B),fun(product_prod(A,B),bool),aa(fun(C,A),fun(fun(C,B),fun(product_prod(A,B),bool)),aTP_Lamp_ni(set(C),fun(fun(C,A),fun(fun(C,B),fun(product_prod(A,B),bool))),A6),F),G)) ).
% image2_def
tff(fact_1464_Collect__ex__eq,axiom,
! [A: $tType,B: $tType,P2: fun(A,fun(B,bool))] : aa(fun(A,bool),set(A),collect(A),aTP_Lamp_ms(fun(A,fun(B,bool)),fun(A,bool),P2)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),aTP_Lamp_kz(fun(A,fun(B,bool)),fun(B,set(A)),P2)),top_top(set(B)))) ).
% Collect_ex_eq
tff(fact_1465_relcomp__unfold,axiom,
! [A: $tType,B: $tType,C: $tType,R: set(product_prod(A,C)),S: set(product_prod(C,B))] : relcomp(A,C,B,R,S) = aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aa(set(product_prod(C,B)),fun(A,fun(B,bool)),aTP_Lamp_nj(set(product_prod(A,C)),fun(set(product_prod(C,B)),fun(A,fun(B,bool))),R),S))) ).
% relcomp_unfold
tff(fact_1466_OO__Grp__alt,axiom,
! [A: $tType,C: $tType,B: $tType,A6: set(C),F: fun(C,A),G: fun(C,B),X4: A,Xa4: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(C,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(C,bool)),fun(fun(C,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,C,B),conversep(C,A,bNF_Grp(C,A,A6,F))),bNF_Grp(C,B,A6,G)),X4),Xa4))
<=> ? [Z4: C] :
( pp(aa(set(C),bool,member(C,Z4),A6))
& ( aa(C,A,F,Z4) = X4 )
& ( aa(C,B,G,Z4) = Xa4 ) ) ) ).
% OO_Grp_alt
tff(fact_1467_Inf__Sup,axiom,
! [A: $tType] :
( comple592849572758109894attice(A)
=> ! [A6: set(set(A))] : aa(set(A),A,complete_Inf_Inf(A),aa(set(set(A)),set(A),aa(fun(set(A),A),fun(set(set(A)),set(A)),image2(set(A),A),complete_Sup_Sup(A)),A6)) = aa(set(A),A,complete_Sup_Sup(A),aa(set(set(A)),set(A),aa(fun(set(A),A),fun(set(set(A)),set(A)),image2(set(A),A),complete_Inf_Inf(A)),aa(fun(set(A),bool),set(set(A)),collect(set(A)),aTP_Lamp_nk(set(set(A)),fun(set(A),bool),A6)))) ) ).
% Inf_Sup
tff(fact_1468_Sup__Inf,axiom,
! [A: $tType] :
( comple592849572758109894attice(A)
=> ! [A6: set(set(A))] : aa(set(A),A,complete_Sup_Sup(A),aa(set(set(A)),set(A),aa(fun(set(A),A),fun(set(set(A)),set(A)),image2(set(A),A),complete_Inf_Inf(A)),A6)) = aa(set(A),A,complete_Inf_Inf(A),aa(set(set(A)),set(A),aa(fun(set(A),A),fun(set(set(A)),set(A)),image2(set(A),A),complete_Sup_Sup(A)),aa(fun(set(A),bool),set(set(A)),collect(set(A)),aTP_Lamp_nk(set(set(A)),fun(set(A),bool),A6)))) ) ).
% Sup_Inf
tff(fact_1469_prod_Oreindex__nontrivial,axiom,
! [C: $tType,A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [A6: set(B),H: fun(B,C),G: fun(C,A)] :
( finite_finite2(B,A6)
=> ( ! [X: B,Y: B] :
( pp(aa(set(B),bool,member(B,X),A6))
=> ( pp(aa(set(B),bool,member(B,Y),A6))
=> ( ( X != Y )
=> ( ( aa(B,C,H,X) = aa(B,C,H,Y) )
=> ( aa(C,A,G,aa(B,C,H,X)) = one_one(A) ) ) ) ) )
=> ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),G),aa(set(B),set(C),aa(fun(B,C),fun(set(B),set(C)),image2(B,C),H),A6)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,C),fun(B,A),aa(fun(C,A),fun(fun(B,C),fun(B,A)),comp(C,A,B),G),H)),A6) ) ) ) ) ).
% prod.reindex_nontrivial
tff(fact_1470_prod_Oeq__fold,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(B,A),A6: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A6) = finite_fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),aa(fun(A,fun(A,A)),fun(fun(B,A),fun(B,fun(A,A))),comp(A,fun(A,A),B),times_times(A)),G),one_one(A),A6) ) ).
% prod.eq_fold
tff(fact_1471_Inf__Sup__le,axiom,
! [A: $tType] :
( comple592849572758109894attice(A)
=> ! [A6: set(set(A))] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(set(A)),set(A),aa(fun(set(A),A),fun(set(set(A)),set(A)),image2(set(A),A),complete_Sup_Sup(A)),A6))),aa(set(A),A,complete_Sup_Sup(A),aa(set(set(A)),set(A),aa(fun(set(A),A),fun(set(set(A)),set(A)),image2(set(A),A),complete_Inf_Inf(A)),aa(fun(set(A),bool),set(set(A)),collect(set(A)),aTP_Lamp_nk(set(set(A)),fun(set(A),bool),A6)))))) ) ).
% Inf_Sup_le
tff(fact_1472_Sup__Inf__le,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [A6: set(set(A))] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(set(A)),set(A),aa(fun(set(A),A),fun(set(set(A)),set(A)),image2(set(A),A),complete_Inf_Inf(A)),aa(fun(set(A),bool),set(set(A)),collect(set(A)),aTP_Lamp_nl(set(set(A)),fun(set(A),bool),A6))))),aa(set(A),A,complete_Inf_Inf(A),aa(set(set(A)),set(A),aa(fun(set(A),A),fun(set(set(A)),set(A)),image2(set(A),A),complete_Sup_Sup(A)),A6)))) ) ).
% Sup_Inf_le
tff(fact_1473_finite__Inf__Sup,axiom,
! [A: $tType] :
( finite8700451911770168679attice(A)
=> ! [A6: set(set(A))] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(set(A)),set(A),aa(fun(set(A),A),fun(set(set(A)),set(A)),image2(set(A),A),complete_Sup_Sup(A)),A6))),aa(set(A),A,complete_Sup_Sup(A),aa(set(set(A)),set(A),aa(fun(set(A),A),fun(set(set(A)),set(A)),image2(set(A),A),complete_Inf_Inf(A)),aa(fun(set(A),bool),set(set(A)),collect(set(A)),aTP_Lamp_nm(set(set(A)),fun(set(A),bool),A6)))))) ) ).
% finite_Inf_Sup
tff(fact_1474_INF__SUP__set,axiom,
! [A: $tType,B: $tType] :
( comple592849572758109894attice(A)
=> ! [G: fun(B,A),A6: set(set(B))] : aa(set(A),A,complete_Inf_Inf(A),aa(set(set(B)),set(A),aa(fun(set(B),A),fun(set(set(B)),set(A)),image2(set(B),A),aTP_Lamp_nn(fun(B,A),fun(set(B),A),G)),A6)) = aa(set(A),A,complete_Sup_Sup(A),aa(set(set(B)),set(A),aa(fun(set(B),A),fun(set(set(B)),set(A)),image2(set(B),A),aTP_Lamp_no(fun(B,A),fun(set(B),A),G)),aa(fun(set(B),bool),set(set(B)),collect(set(B)),aTP_Lamp_np(set(set(B)),fun(set(B),bool),A6)))) ) ).
% INF_SUP_set
tff(fact_1475_SUP__INF__set,axiom,
! [A: $tType,B: $tType] :
( comple592849572758109894attice(A)
=> ! [G: fun(B,A),A6: set(set(B))] : aa(set(A),A,complete_Sup_Sup(A),aa(set(set(B)),set(A),aa(fun(set(B),A),fun(set(set(B)),set(A)),image2(set(B),A),aTP_Lamp_no(fun(B,A),fun(set(B),A),G)),A6)) = aa(set(A),A,complete_Inf_Inf(A),aa(set(set(B)),set(A),aa(fun(set(B),A),fun(set(set(B)),set(A)),image2(set(B),A),aTP_Lamp_nn(fun(B,A),fun(set(B),A),G)),aa(fun(set(B),bool),set(set(B)),collect(set(B)),aTP_Lamp_np(set(set(B)),fun(set(B),bool),A6)))) ) ).
% SUP_INF_set
tff(fact_1476_prod_OatLeast0__atMost__Suc__shift,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),N4: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N4))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),suc)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N4))) ) ).
% prod.atLeast0_atMost_Suc_shift
tff(fact_1477_prod_OatLeast0__lessThan__Suc__shift,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),N4: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N4))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),suc)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N4))) ) ).
% prod.atLeast0_lessThan_Suc_shift
tff(fact_1478_prod_OatLeastLessThan__shift__0,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),M4: nat,N4: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M4,N4)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),aa(nat,fun(nat,nat),plus_plus(nat),M4))),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N4),M4))) ) ).
% prod.atLeastLessThan_shift_0
tff(fact_1479_prod_OUNION__disjoint,axiom,
! [C: $tType,A: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [I4: set(B),A6: fun(B,set(C)),G: fun(C,A)] :
( finite_finite2(B,I4)
=> ( ! [X: B] :
( pp(aa(set(B),bool,member(B,X),I4))
=> finite_finite2(C,aa(B,set(C),A6,X)) )
=> ( ! [X: B] :
( pp(aa(set(B),bool,member(B,X),I4))
=> ! [Xa: B] :
( pp(aa(set(B),bool,member(B,Xa),I4))
=> ( ( X != Xa )
=> ( aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),aa(B,set(C),A6,X)),aa(B,set(C),A6,Xa)) = bot_bot(set(C)) ) ) ) )
=> ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),G),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),aa(fun(B,set(C)),fun(set(B),set(set(C))),image2(B,set(C)),A6),I4))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(C,A),fun(B,A),aTP_Lamp_nq(fun(B,set(C)),fun(fun(C,A),fun(B,A)),A6),G)),I4) ) ) ) ) ) ).
% prod.UNION_disjoint
tff(fact_1480_prod_OatLeast__atMost__pred__shift,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),M4: nat,N4: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),aTP_Lamp_lm(nat,nat))),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M4),aa(nat,nat,suc,N4))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M4,N4)) ) ).
% prod.atLeast_atMost_pred_shift
tff(fact_1481_prod_OatLeast__lessThan__pred__shift,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,A),M4: nat,N4: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),aTP_Lamp_lm(nat,nat))),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M4),aa(nat,nat,suc,N4))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M4,N4)) ) ).
% prod.atLeast_lessThan_pred_shift
tff(fact_1482_len_H__def,axiom,
! [B: $tType,A: $tType] :
( ( semiring_1(A)
& heap(B) )
=> ! [A3: array(B)] : len2(B,A,A3) = heap_bind(nat,A,len(B,A3),aTP_Lamp_nr(nat,heap_Heap(A))) ) ).
% len'_def
tff(fact_1483_fold__congs_I3_J,axiom,
! [Z9: $tType,R: heap_ext(Z9),R2: heap_ext(Z9),V2: nat,F: fun(nat,nat),F4: fun(nat,nat)] :
( ( R = R2 )
=> ( ( lim(Z9,R2) = V2 )
=> ( ! [V5: nat] :
( ( V2 = V5 )
=> ( aa(nat,nat,F,V5) = aa(nat,nat,F4,V5) ) )
=> ( lim_update(Z9,F,R) = lim_update(Z9,F4,R2) ) ) ) ) ).
% fold_congs(3)
tff(fact_1484_unfold__congs_I3_J,axiom,
! [Z9: $tType,R: heap_ext(Z9),R2: heap_ext(Z9),V2: nat,F: fun(nat,nat),F4: fun(nat,nat)] :
( ( R = R2 )
=> ( ( lim(Z9,R2) = V2 )
=> ( ! [V5: nat] :
( ( V5 = V2 )
=> ( aa(nat,nat,F,V5) = aa(nat,nat,F4,V5) ) )
=> ( lim_update(Z9,F,R) = lim_update(Z9,F4,R2) ) ) ) ) ).
% unfold_congs(3)
tff(fact_1485_prod_Otriangle__reindex,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,fun(nat,A)),N4: nat] : aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups7121269368397514597t_prod(product_prod(nat,nat),A),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G)),aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_me(nat,fun(nat,fun(nat,bool)),N4)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_nt(fun(nat,fun(nat,A)),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),N4)) ) ).
% prod.triangle_reindex
tff(fact_1486_prod_Otriangle__reindex__eq,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(nat,fun(nat,A)),N4: nat] : aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups7121269368397514597t_prod(product_prod(nat,nat),A),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G)),aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_lv(nat,fun(nat,fun(nat,bool)),N4)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_nt(fun(nat,fun(nat,A)),fun(nat,A),G)),aa(nat,set(nat),set_ord_atMost(nat),N4)) ) ).
% prod.triangle_reindex_eq
tff(fact_1487_card__Min__le__sum,axiom,
! [A: $tType,A6: set(A),F: fun(A,nat)] :
( finite_finite2(A,A6)
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(A),nat,finite_card(A),A6)),lattic643756798350308766er_Min(nat,aa(set(A),set(nat),aa(fun(A,nat),fun(set(A),set(nat)),image2(A,nat),F),A6)))),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F),A6))) ) ).
% card_Min_le_sum
tff(fact_1488_embedS__Field,axiom,
! [A: $tType,B: $tType,R: set(product_prod(A,A)),R2: set(product_prod(B,B)),F: fun(A,B)] :
( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R),R)
=> ( bNF_Wellorder_embedS(A,B,R,R2,F)
=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less(set(B)),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),aa(set(product_prod(A,A)),set(A),field2(A),R))),aa(set(product_prod(B,B)),set(B),field2(B),R2))) ) ) ).
% embedS_Field
tff(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 = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(A),set(set(A)),aa(fun(A,set(A)),fun(set(A),set(set(A))),image2(A,set(A)),order_under(A,R)),A6)) ) ) ) ).
% wo_rel.ofilter_under_UNION
tff(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,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),aTP_Lamp_ku(A,fun(B,A),C3)),A6)) = C3 ) ) ) ) ).
% Min_const
tff(fact_1491_image__add__int__atLeastLessThan,axiom,
! [L: int,U: int] : aa(set(int),set(int),aa(fun(int,int),fun(set(int),set(int)),image2(int,int),aTP_Lamp_nu(int,fun(int,int),L)),set_or7035219750837199246ssThan(int,zero_zero(int),aa(int,int,aa(int,fun(int,int),minus_minus(int),U),L))) = set_or7035219750837199246ssThan(int,L,U) ).
% image_add_int_atLeastLessThan
tff(fact_1492_under__def,axiom,
! [A: $tType,R: set(product_prod(A,A)),A3: A] : aa(A,set(A),order_under(A,R),A3) = aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_nv(set(product_prod(A,A)),fun(A,fun(A,bool)),R),A3)) ).
% under_def
tff(fact_1493_prod_OatLeast__int__atMost__int__shift,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(int,A),M4: nat,N4: nat] : aa(set(int),A,aa(fun(int,A),fun(set(int),A),groups7121269368397514597t_prod(int,A),G),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),M4),aa(nat,int,semiring_1_of_nat(int),N4))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,int),fun(nat,A),aa(fun(int,A),fun(fun(nat,int),fun(nat,A)),comp(int,A,nat),G),semiring_1_of_nat(int))),set_or1337092689740270186AtMost(nat,M4,N4)) ) ).
% prod.atLeast_int_atMost_int_shift
tff(fact_1494_prod_OatLeast__int__lessThan__int__shift,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [G: fun(int,A),M4: nat,N4: nat] : aa(set(int),A,aa(fun(int,A),fun(set(int),A),groups7121269368397514597t_prod(int,A),G),set_or7035219750837199246ssThan(int,aa(nat,int,semiring_1_of_nat(int),M4),aa(nat,int,semiring_1_of_nat(int),N4))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,int),fun(nat,A),aa(fun(int,A),fun(fun(nat,int),fun(nat,A)),comp(int,A,nat),G),semiring_1_of_nat(int))),set_or7035219750837199246ssThan(nat,M4,N4)) ) ).
% prod.atLeast_int_lessThan_int_shift
tff(fact_1495_mono__Min__commute,axiom,
! [B: $tType,A: $tType] :
( ( linorder(A)
& linorder(B) )
=> ! [F: fun(A,B),A6: set(A)] :
( order_mono(A,B,F)
=> ( finite_finite2(A,A6)
=> ( ( A6 != bot_bot(set(A)) )
=> ( aa(A,B,F,lattic643756798350308766er_Min(A,A6)) = lattic643756798350308766er_Min(B,aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6)) ) ) ) ) ) ).
% mono_Min_commute
tff(fact_1496_Min__add__commute,axiom,
! [B: $tType,A: $tType] :
( linord4140545234300271783up_add(A)
=> ! [S3: set(B),F: fun(B,A),K: A] :
( finite_finite2(B,S3)
=> ( ( S3 != bot_bot(set(B)) )
=> ( lattic643756798350308766er_Min(A,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),aa(A,fun(B,A),aTP_Lamp_ks(fun(B,A),fun(A,fun(B,A)),F),K)),S3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),lattic643756798350308766er_Min(A,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),S3))),K) ) ) ) ) ).
% Min_add_commute
tff(fact_1497_minus__Min__eq__Max,axiom,
! [A: $tType] :
( linord5086331880401160121up_add(A)
=> ! [S3: set(A)] :
( finite_finite2(A,S3)
=> ( ( S3 != bot_bot(set(A)) )
=> ( aa(A,A,uminus_uminus(A),lattic643756798350308766er_Min(A,S3)) = lattic643756798349783984er_Max(A,aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),uminus_uminus(A)),S3)) ) ) ) ) ).
% minus_Min_eq_Max
tff(fact_1498_minus__Max__eq__Min,axiom,
! [A: $tType] :
( linord5086331880401160121up_add(A)
=> ! [S3: set(A)] :
( finite_finite2(A,S3)
=> ( ( S3 != bot_bot(set(A)) )
=> ( aa(A,A,uminus_uminus(A),lattic643756798349783984er_Max(A,S3)) = lattic643756798350308766er_Min(A,aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),uminus_uminus(A)),S3)) ) ) ) ) ).
% minus_Max_eq_Min
tff(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: fun(A,A2)] :
( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R),R)
=> ( order_well_order_on(A2,aa(set(product_prod(A2,A2)),set(A2),field2(A2),R2),R2)
=> ( pp(aa(set(A),bool,member(A,A3),aa(set(product_prod(A,A)),set(A),field2(A),R)))
=> ( ! [X: A] :
( pp(aa(set(A),bool,member(A,X),order_underS(A,R,A3)))
=> bij_betw(A,A2,F,aa(A,set(A),order_under(A,R),X),aa(A2,set(A2),order_under(A2,R2),aa(A,A2,F,X))) )
=> ( ( aa(set(A),set(A2),aa(fun(A,A2),fun(set(A),set(A2)),image2(A,A2),F),order_underS(A,R,A3)) != aa(set(product_prod(A2,A2)),set(A2),field2(A2),R2) )
=> ( ( aa(A,A2,F,A3) = bNF_Wellorder_wo_suc(A2,R2,aa(set(A),set(A2),aa(fun(A,A2),fun(set(A),set(A2)),image2(A,A2),F),order_underS(A,R,A3))) )
=> bij_betw(A,A2,F,aa(A,set(A),order_under(A,R),A3),aa(A2,set(A2),order_under(A2,R2),aa(A,A2,F,A3))) ) ) ) ) ) ) ).
% wellorders_totally_ordered_aux
tff(fact_1500_ordLess__def,axiom,
! [A2: $tType,A: $tType] : bNF_We4044943003108391690rdLess(A,A2) = aa(fun(product_prod(set(product_prod(A,A)),set(product_prod(A2,A2))),bool),set(product_prod(set(product_prod(A,A)),set(product_prod(A2,A2)))),collect(product_prod(set(product_prod(A,A)),set(product_prod(A2,A2)))),aa(fun(set(product_prod(A,A)),fun(set(product_prod(A2,A2)),bool)),fun(product_prod(set(product_prod(A,A)),set(product_prod(A2,A2))),bool),product_case_prod(set(product_prod(A,A)),set(product_prod(A2,A2)),bool),aTP_Lamp_nw(set(product_prod(A,A)),fun(set(product_prod(A2,A2)),bool)))) ).
% ordLess_def
tff(fact_1501_pair__lessI2,axiom,
! [A3: nat,B2: nat,S: nat,T2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),B2))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),S),T2))
=> pp(aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool,member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A3),S)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),B2),T2))),fun_pair_less)) ) ) ).
% pair_lessI2
tff(fact_1502_pair__less__iff1,axiom,
! [X2: nat,Y2: nat,Z2: nat] :
( pp(aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool,member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X2),Y2)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),X2),Z2))),fun_pair_less))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Y2),Z2)) ) ).
% pair_less_iff1
tff(fact_1503_Array_Oalloc__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [Xs: list(A),H: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit)),alloc(A,Xs),H) = aa(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit)),aa(array(A),fun(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit))),product_Pair(array(A),heap_ext(product_unit)),array2(A,lim(product_unit,H))),aa(heap_ext(product_unit),heap_ext(product_unit),set2(A,array2(A,lim(product_unit,H)),Xs),lim_update(product_unit,aTP_Lamp_mm(heap_ext(product_unit),fun(nat,nat),H),H))) ) ).
% Array.alloc_def
tff(fact_1504_embed__determined,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,A)),R2: set(product_prod(B,B)),F: fun(A,B),A3: A] :
( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R),R)
=> ( order_well_order_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R2),R2)
=> ( pp(aa(fun(A,B),bool,bNF_Wellorder_embed(A,B,R,R2),F))
=> ( pp(aa(set(A),bool,member(A,A3),aa(set(product_prod(A,A)),set(A),field2(A),R)))
=> ( aa(A,B,F,A3) = bNF_Wellorder_wo_suc(B,R2,aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),order_underS(A,R,A3))) ) ) ) ) ) ).
% embed_determined
tff(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
tff(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)
<=> ( aa(set(nat),set(A),aa(fun(nat,A),fun(set(nat),set(A)),image2(nat,A),semiring_1_of_nat(A)),N) = A6 ) ) ) ).
% bij_betw_of_nat
tff(fact_1507_Compl__Times__UNIV2,axiom,
! [B: $tType,A: $tType,A6: set(A)] : aa(set(product_prod(A,B)),set(product_prod(A,B)),uminus_uminus(set(product_prod(A,B))),product_Sigma(A,B,A6,aTP_Lamp_ca(A,set(B)))) = product_Sigma(A,B,aa(set(A),set(A),uminus_uminus(set(A)),A6),aTP_Lamp_ca(A,set(B))) ).
% Compl_Times_UNIV2
tff(fact_1508_Compl__Times__UNIV1,axiom,
! [A: $tType,B: $tType,A6: set(B)] : aa(set(product_prod(A,B)),set(product_prod(A,B)),uminus_uminus(set(product_prod(A,B))),product_Sigma(A,B,top_top(set(A)),aTP_Lamp_bu(set(B),fun(A,set(B)),A6))) = product_Sigma(A,B,top_top(set(A)),aTP_Lamp_nx(set(B),fun(A,set(B)),A6)) ).
% Compl_Times_UNIV1
tff(fact_1509_image__uminus__atLeastAtMost,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [X2: A,Y2: A] : aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),uminus_uminus(A)),set_or1337092689740270186AtMost(A,X2,Y2)) = set_or1337092689740270186AtMost(A,aa(A,A,uminus_uminus(A),Y2),aa(A,A,uminus_uminus(A),X2)) ) ).
% image_uminus_atLeastAtMost
tff(fact_1510_bij__betw__add,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [A3: A,A6: set(A),B6: set(A)] :
( bij_betw(A,A,aa(A,fun(A,A),plus_plus(A),A3),A6,B6)
<=> ( aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aa(A,fun(A,A),plus_plus(A),A3)),A6) = B6 ) ) ) ).
% bij_betw_add
tff(fact_1511_boolean__algebra__class_Ominus__comp__minus,axiom,
! [A: $tType] :
( boolea8198339166811842893lgebra(A)
=> ( aa(fun(A,A),fun(A,A),aa(fun(A,A),fun(fun(A,A),fun(A,A)),comp(A,A,A),uminus_uminus(A)),uminus_uminus(A)) = id(A) ) ) ).
% boolean_algebra_class.minus_comp_minus
tff(fact_1512_group__add__class_Ominus__comp__minus,axiom,
! [A: $tType] :
( group_add(A)
=> ( aa(fun(A,A),fun(A,A),aa(fun(A,A),fun(fun(A,A),fun(A,A)),comp(A,A,A),uminus_uminus(A)),uminus_uminus(A)) = id(A) ) ) ).
% group_add_class.minus_comp_minus
tff(fact_1513_Compl__UN,axiom,
! [A: $tType,B: $tType,B6: fun(B,set(A)),A6: set(B)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),B6),A6))) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),aTP_Lamp_ny(fun(B,set(A)),fun(B,set(A)),B6)),A6)) ).
% Compl_UN
tff(fact_1514_Compl__INT,axiom,
! [A: $tType,B: $tType,B6: fun(B,set(A)),A6: set(B)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),B6),A6))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),aTP_Lamp_ny(fun(B,set(A)),fun(B,set(A)),B6)),A6)) ).
% Compl_INT
tff(fact_1515_bdd__below__uminus,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [X6: set(A)] :
( condit1013018076250108175_below(A,aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),uminus_uminus(A)),X6))
<=> condit941137186595557371_above(A,X6) ) ) ).
% bdd_below_uminus
tff(fact_1516_bdd__above__uminus,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [X6: set(A)] :
( condit941137186595557371_above(A,aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),uminus_uminus(A)),X6))
<=> condit1013018076250108175_below(A,X6) ) ) ).
% bdd_above_uminus
tff(fact_1517_image__uminus__lessThan,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [X2: A] : aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),uminus_uminus(A)),aa(A,set(A),set_ord_lessThan(A),X2)) = aa(A,set(A),set_ord_greaterThan(A),aa(A,A,uminus_uminus(A),X2)) ) ).
% image_uminus_lessThan
tff(fact_1518_image__uminus__greaterThan,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [X2: A] : aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),uminus_uminus(A)),aa(A,set(A),set_ord_greaterThan(A),X2)) = aa(A,set(A),set_ord_lessThan(A),aa(A,A,uminus_uminus(A),X2)) ) ).
% image_uminus_greaterThan
tff(fact_1519_Inf__int__def,axiom,
! [X6: set(int)] : aa(set(int),int,complete_Inf_Inf(int),X6) = aa(int,int,uminus_uminus(int),aa(set(int),int,complete_Sup_Sup(int),aa(set(int),set(int),aa(fun(int,int),fun(set(int),set(int)),image2(int,int),uminus_uminus(int)),X6))) ).
% Inf_int_def
tff(fact_1520_wf__pair__less,axiom,
wf(product_prod(nat,nat),fun_pair_less) ).
% wf_pair_less
tff(fact_1521_bij__betwI_H,axiom,
! [A: $tType,B: $tType,X6: set(A),F: fun(A,B),Y3: set(B)] :
( ! [X: A] :
( pp(aa(set(A),bool,member(A,X),X6))
=> ! [Y: A] :
( pp(aa(set(A),bool,member(A,Y),X6))
=> ( ( aa(A,B,F,X) = aa(A,B,F,Y) )
<=> ( X = Y ) ) ) )
=> ( ! [X: A] :
( pp(aa(set(A),bool,member(A,X),X6))
=> pp(aa(set(B),bool,member(B,aa(A,B,F,X)),Y3)) )
=> ( ! [Y: B] :
( pp(aa(set(B),bool,member(B,Y),Y3))
=> ? [X4: A] :
( pp(aa(set(A),bool,member(A,X4),X6))
& ( Y = aa(A,B,F,X4) ) ) )
=> bij_betw(A,B,F,X6,Y3) ) ) ) ).
% bij_betwI'
tff(fact_1522_bij__betw__Pow,axiom,
! [A: $tType,B: $tType,F: fun(A,B),A6: set(A),B6: set(B)] :
( bij_betw(A,B,F,A6,B6)
=> bij_betw(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),pow(A,A6),pow(B,B6)) ) ).
% bij_betw_Pow
tff(fact_1523_bij__image__Compl__eq,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A6: set(A)] :
( bij_betw(A,B,F,top_top(set(A)),top_top(set(B)))
=> ( aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),aa(set(A),set(A),uminus_uminus(set(A)),A6)) = aa(set(B),set(B),uminus_uminus(set(B)),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6)) ) ) ).
% bij_image_Compl_eq
tff(fact_1524_bij__betw__imp__surj__on,axiom,
! [A: $tType,B: $tType,F: fun(A,B),A6: set(A),B6: set(B)] :
( bij_betw(A,B,F,A6,B6)
=> ( aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6) = B6 ) ) ).
% bij_betw_imp_surj_on
tff(fact_1525_bijI_H,axiom,
! [A: $tType,B: $tType,F: fun(A,B)] :
( ! [X: A,Y: A] :
( ( aa(A,B,F,X) = aa(A,B,F,Y) )
<=> ( X = Y ) )
=> ( ! [Y: B] :
? [X4: A] : Y = aa(A,B,F,X4)
=> bij_betw(A,B,F,top_top(set(A)),top_top(set(B))) ) ) ).
% bijI'
tff(fact_1526_bij__betw__trans,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(A,B),A6: set(A),B6: set(B),G: fun(B,C),C5: set(C)] :
( bij_betw(A,B,F,A6,B6)
=> ( bij_betw(B,C,G,B6,C5)
=> bij_betw(A,C,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),G),F),A6,C5) ) ) ).
% bij_betw_trans
tff(fact_1527_bij__betw__comp__iff,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(A,B),A6: set(A),A8: set(B),F4: fun(B,C),A16: set(C)] :
( bij_betw(A,B,F,A6,A8)
=> ( bij_betw(B,C,F4,A8,A16)
<=> bij_betw(A,C,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),F4),F),A6,A16) ) ) ).
% bij_betw_comp_iff
tff(fact_1528_comp__embed,axiom,
! [C: $tType,B: $tType,A: $tType,R: set(product_prod(A,A)),R2: set(product_prod(B,B)),F: fun(A,B),R7: set(product_prod(C,C)),F4: fun(B,C)] :
( pp(aa(fun(A,B),bool,bNF_Wellorder_embed(A,B,R,R2),F))
=> ( pp(aa(fun(B,C),bool,bNF_Wellorder_embed(B,C,R2,R7),F4))
=> pp(aa(fun(A,C),bool,bNF_Wellorder_embed(A,C,R,R7),aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),F4),F))) ) ) ).
% comp_embed
tff(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: fun(A,B)] :
( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R12),R23)),bNF_We4044943003108391690rdLess(A,B)))
=> ( pp(aa(fun(A,B),bool,bNF_Wellorder_embed(A,B,R12,R23),F))
=> ( aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),aa(set(product_prod(A,A)),set(A),field2(A),R12)) != aa(set(product_prod(B,B)),set(B),field2(B),R23) ) ) ) ).
% BNF_Wellorder_Constructions.ordLess_Field
tff(fact_1530_Array_Oget__alloc,axiom,
! [A: $tType] :
( heap(A)
=> ! [Xs: list(A),H: heap_ext(product_unit),Ys: list(A)] : aa(array(A),list(A),get(A,aa(product_prod(array(A),heap_ext(product_unit)),heap_ext(product_unit),product_snd(array(A),heap_ext(product_unit)),aa(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit)),alloc(A,Xs),H))),aa(product_prod(array(A),heap_ext(product_unit)),array(A),product_fst(array(A),heap_ext(product_unit)),aa(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit)),alloc(A,Ys),H))) = Xs ) ).
% Array.get_alloc
tff(fact_1531_bij__betw__byWitness,axiom,
! [A: $tType,B: $tType,A6: set(A),F4: fun(B,A),F: fun(A,B),A8: set(B)] :
( ! [X: A] :
( pp(aa(set(A),bool,member(A,X),A6))
=> ( aa(B,A,F4,aa(A,B,F,X)) = X ) )
=> ( ! [X: B] :
( pp(aa(set(B),bool,member(B,X),A8))
=> ( aa(A,B,F,aa(B,A,F4,X)) = X ) )
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6)),A8))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F4),A8)),A6))
=> bij_betw(A,B,F,A6,A8) ) ) ) ) ).
% bij_betw_byWitness
tff(fact_1532_bij__betw__subset,axiom,
! [A: $tType,B: $tType,F: fun(A,B),A6: set(A),A8: set(B),B6: set(A),B8: set(B)] :
( bij_betw(A,B,F,A6,A8)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B6),A6))
=> ( ( aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),B6) = B8 )
=> bij_betw(A,B,F,B6,B8) ) ) ) ).
% bij_betw_subset
tff(fact_1533_bij__betw__imp__surj,axiom,
! [A: $tType,B: $tType,F: fun(A,B),A6: set(A)] :
( bij_betw(A,B,F,A6,top_top(set(B)))
=> ( aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),top_top(set(A))) = top_top(set(B)) ) ) ).
% bij_betw_imp_surj
tff(fact_1534_bij__is__surj,axiom,
! [A: $tType,B: $tType,F: fun(A,B)] :
( bij_betw(A,B,F,top_top(set(A)),top_top(set(B)))
=> ( aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),top_top(set(A))) = top_top(set(B)) ) ) ).
% bij_is_surj
tff(fact_1535_bij__comp,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(A,B),G: fun(B,C)] :
( bij_betw(A,B,F,top_top(set(A)),top_top(set(B)))
=> ( bij_betw(B,C,G,top_top(set(B)),top_top(set(C)))
=> bij_betw(A,C,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),G),F),top_top(set(A)),top_top(set(C))) ) ) ).
% bij_comp
tff(fact_1536_inj__on__imp__bij__betw,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A6: set(A)] :
( inj_on(A,B,F,A6)
=> bij_betw(A,B,F,A6,aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6)) ) ).
% inj_on_imp_bij_betw
tff(fact_1537_bij__betw__imageI,axiom,
! [A: $tType,B: $tType,F: fun(A,B),A6: set(A),B6: set(B)] :
( inj_on(A,B,F,A6)
=> ( ( aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6) = B6 )
=> bij_betw(A,B,F,A6,B6) ) ) ).
% bij_betw_imageI
tff(fact_1538_bij__betw__def,axiom,
! [A: $tType,B: $tType,F: fun(A,B),A6: set(A),B6: set(B)] :
( bij_betw(A,B,F,A6,B6)
<=> ( inj_on(A,B,F,A6)
& ( aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6) = B6 ) ) ) ).
% bij_betw_def
tff(fact_1539_Array_Opresent__alloc,axiom,
! [A: $tType] :
( heap(A)
=> ! [Xs: list(A),H: heap_ext(product_unit)] : pp(aa(array(A),bool,present(A,aa(product_prod(array(A),heap_ext(product_unit)),heap_ext(product_unit),product_snd(array(A),heap_ext(product_unit)),aa(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit)),alloc(A,Xs),H))),aa(product_prod(array(A),heap_ext(product_unit)),array(A),product_fst(array(A),heap_ext(product_unit)),aa(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit)),alloc(A,Xs),H)))) ) ).
% Array.present_alloc
tff(fact_1540_Array_Onot__present__alloc,axiom,
! [A: $tType] :
( heap(A)
=> ! [H: heap_ext(product_unit),Xs: list(A)] : ~ pp(aa(array(A),bool,present(A,H),aa(product_prod(array(A),heap_ext(product_unit)),array(A),product_fst(array(A),heap_ext(product_unit)),aa(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit)),alloc(A,Xs),H)))) ) ).
% Array.not_present_alloc
tff(fact_1541_set,axiom,
! [A: $tType] :
( heap(A)
=> ! [Ls: list(A),H: heap_ext(product_unit),New_ls: list(A)] : aa(heap_ext(product_unit),heap_ext(product_unit),set2(A,aa(product_prod(array(A),heap_ext(product_unit)),array(A),product_fst(array(A),heap_ext(product_unit)),aa(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit)),alloc(A,Ls),H)),New_ls),aa(product_prod(array(A),heap_ext(product_unit)),heap_ext(product_unit),product_snd(array(A),heap_ext(product_unit)),aa(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit)),alloc(A,Ls),H))) = aa(product_prod(array(A),heap_ext(product_unit)),heap_ext(product_unit),product_snd(array(A),heap_ext(product_unit)),aa(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit)),alloc(A,New_ls),H)) ) ).
% set
tff(fact_1542_translation__Compl,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [A3: A,T2: set(A)] : aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aa(A,fun(A,A),plus_plus(A),A3)),aa(set(A),set(A),uminus_uminus(set(A)),T2)) = aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aa(A,fun(A,A),plus_plus(A),A3)),T2)) ) ).
% translation_Compl
tff(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
tff(fact_1544_translation__subtract__Compl,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [A3: A,T2: set(A)] : aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aTP_Lamp_eo(A,fun(A,A),A3)),aa(set(A),set(A),uminus_uminus(set(A)),T2)) = aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aTP_Lamp_eo(A,fun(A,A),A3)),T2)) ) ).
% translation_subtract_Compl
tff(fact_1545_bij__betw__comp__iff2,axiom,
! [C: $tType,A: $tType,B: $tType,F4: fun(A,B),A8: set(A),A16: set(B),F: fun(C,A),A6: set(C)] :
( bij_betw(A,B,F4,A8,A16)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(C),set(A),aa(fun(C,A),fun(set(C),set(A)),image2(C,A),F),A6)),A8))
=> ( bij_betw(C,A,F,A6,A8)
<=> bij_betw(C,B,aa(fun(C,A),fun(C,B),aa(fun(A,B),fun(fun(C,A),fun(C,B)),comp(A,B,C),F4),F),A6,A16) ) ) ) ).
% bij_betw_comp_iff2
tff(fact_1546_bijI,axiom,
! [A: $tType,B: $tType,F: fun(A,B)] :
( inj_on(A,B,F,top_top(set(A)))
=> ( ( aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),top_top(set(A))) = top_top(set(B)) )
=> bij_betw(A,B,F,top_top(set(A)),top_top(set(B))) ) ) ).
% bijI
tff(fact_1547_bij__def,axiom,
! [A: $tType,B: $tType,F: fun(A,B)] :
( bij_betw(A,B,F,top_top(set(A)),top_top(set(B)))
<=> ( inj_on(A,B,F,top_top(set(A)))
& ( aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),top_top(set(A))) = top_top(set(B)) ) ) ) ).
% bij_def
tff(fact_1548_o__bij,axiom,
! [A: $tType,B: $tType,G: fun(B,A),F: fun(A,B)] :
( ( aa(fun(A,B),fun(A,A),aa(fun(B,A),fun(fun(A,B),fun(A,A)),comp(B,A,A),G),F) = id(A) )
=> ( ( aa(fun(B,A),fun(B,B),aa(fun(A,B),fun(fun(B,A),fun(B,B)),comp(A,B,B),F),G) = id(B) )
=> bij_betw(A,B,F,top_top(set(A)),top_top(set(B))) ) ) ).
% o_bij
tff(fact_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: fun(A,C),F23: fun(B,C)] :
( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),R12),R23)),bNF_We4044943003108391690rdLess(A,B)))
=> ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(B,B)),fun(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),product_Pair(set(product_prod(B,B)),set(product_prod(C,C))),R23),R33)),bNF_We4044943003108391690rdLess(B,C)))
=> ( pp(aa(fun(A,C),bool,bNF_Wellorder_embed(A,C,R12,R33),F13))
=> ( pp(aa(fun(B,C),bool,bNF_Wellorder_embed(B,C,R23,R33),F23))
=> pp(aa(set(product_prod(set(C),set(C))),bool,member(product_prod(set(C),set(C)),aa(set(C),product_prod(set(C),set(C)),aa(set(C),fun(set(C),product_prod(set(C),set(C))),product_Pair(set(C),set(C)),aa(set(A),set(C),aa(fun(A,C),fun(set(A),set(C)),image2(A,C),F13),aa(set(product_prod(A,A)),set(A),field2(A),R12))),aa(set(B),set(C),aa(fun(B,C),fun(set(B),set(C)),image2(B,C),F23),aa(set(product_prod(B,B)),set(B),field2(B),R23)))),bNF_We413866401316099525erIncl(C,R33))) ) ) ) ) ).
% embed_ordLess_ofilterIncl
tff(fact_1550_embed__Field,axiom,
! [A: $tType,B: $tType,R: set(product_prod(A,A)),R2: set(product_prod(B,B)),F: fun(A,B)] :
( pp(aa(fun(A,B),bool,bNF_Wellorder_embed(A,B,R,R2),F))
=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),aa(set(product_prod(A,A)),set(A),field2(A),R))),aa(set(product_prod(B,B)),set(B),field2(B),R2))) ) ).
% embed_Field
tff(fact_1551_surj__Compl__image__subset,axiom,
! [A: $tType,B: $tType,F: fun(B,A),A6: set(B)] :
( ( aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),top_top(set(B))) = top_top(set(A)) )
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),aa(set(B),set(B),uminus_uminus(set(B)),A6)))) ) ).
% surj_Compl_image_subset
tff(fact_1552_bij__betw__UNION__chain,axiom,
! [B: $tType,C: $tType,A: $tType,I4: set(A),A6: fun(A,set(B)),F: fun(B,C),A8: fun(A,set(C))] :
( ! [I6: A,J3: A] :
( pp(aa(set(A),bool,member(A,I6),I4))
=> ( pp(aa(set(A),bool,member(A,J3),I4))
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),A6,I6)),aa(A,set(B),A6,J3)))
| pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),A6,J3)),aa(A,set(B),A6,I6))) ) ) )
=> ( ! [I6: A] :
( pp(aa(set(A),bool,member(A,I6),I4))
=> bij_betw(B,C,F,aa(A,set(B),A6,I6),aa(A,set(C),A8,I6)) )
=> bij_betw(B,C,F,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),aa(fun(A,set(B)),fun(set(A),set(set(B))),image2(A,set(B)),A6),I4)),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(A),set(set(C)),aa(fun(A,set(C)),fun(set(A),set(set(C))),image2(A,set(C)),A8),I4))) ) ) ).
% bij_betw_UNION_chain
tff(fact_1553_uminus__Inf,axiom,
! [A: $tType] :
( comple489889107523837845lgebra(A)
=> ! [A6: set(A)] : aa(A,A,uminus_uminus(A),aa(set(A),A,complete_Inf_Inf(A),A6)) = aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),uminus_uminus(A)),A6)) ) ).
% uminus_Inf
tff(fact_1554_uminus__Sup,axiom,
! [A: $tType] :
( comple489889107523837845lgebra(A)
=> ! [A6: set(A)] : aa(A,A,uminus_uminus(A),aa(set(A),A,complete_Sup_Sup(A),A6)) = aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),uminus_uminus(A)),A6)) ) ).
% uminus_Sup
tff(fact_1555_uminus__INF,axiom,
! [A: $tType,B: $tType] :
( comple489889107523837845lgebra(A)
=> ! [B6: fun(B,A),A6: set(B)] : aa(A,A,uminus_uminus(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),B6),A6))) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),aTP_Lamp_nz(fun(B,A),fun(B,A),B6)),A6)) ) ).
% uminus_INF
tff(fact_1556_uminus__SUP,axiom,
! [A: $tType,B: $tType] :
( comple489889107523837845lgebra(A)
=> ! [B6: fun(B,A),A6: set(B)] : aa(A,A,uminus_uminus(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),B6),A6))) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),aTP_Lamp_nz(fun(B,A),fun(B,A),B6)),A6)) ) ).
% uminus_SUP
tff(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)] :
( pp(aa(array(A),bool,present(A,H),A3))
=> noteq(A,B,A3,aa(product_prod(array(B),heap_ext(product_unit)),array(B),product_fst(array(B),heap_ext(product_unit)),aa(heap_ext(product_unit),product_prod(array(B),heap_ext(product_unit)),alloc(B,Xs),H))) ) ) ).
% present_alloc_noteq
tff(fact_1558_vimage__subset__eq,axiom,
! [B: $tType,A: $tType,F: fun(A,B),B6: set(B),A6: set(A)] :
( bij_betw(A,B,F,top_top(set(A)),top_top(set(B)))
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),vimage(A,B,F),B6)),A6))
<=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B6),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6))) ) ) ).
% vimage_subset_eq
tff(fact_1559_mono__bij__Inf,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder(A)
& comple5582772986160207858norder(B) )
=> ! [F: fun(A,B),A6: set(A)] :
( order_mono(A,B,F)
=> ( bij_betw(A,B,F,top_top(set(A)),top_top(set(B)))
=> ( aa(A,B,F,aa(set(A),A,complete_Inf_Inf(A),A6)) = aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6)) ) ) ) ) ).
% mono_bij_Inf
tff(fact_1560_wellorders__totally__ordered__aux2,axiom,
! [A: $tType,A2: $tType,R: set(product_prod(A,A)),R2: set(product_prod(A2,A2)),G: fun(A,bool),F: fun(A,A2),A3: A] :
( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R),R)
=> ( order_well_order_on(A2,aa(set(product_prod(A2,A2)),set(A2),field2(A2),R2),R2)
=> ( ! [A5: A] :
( ( ( ~ pp(aa(set(bool),bool,member(bool,fFalse),aa(set(A),set(bool),aa(fun(A,bool),fun(set(A),set(bool)),image2(A,bool),G),order_underS(A,R,A5))))
& ( aa(set(A),set(A2),aa(fun(A,A2),fun(set(A),set(A2)),image2(A,A2),F),order_underS(A,R,A5)) != aa(set(product_prod(A2,A2)),set(A2),field2(A2),R2) ) )
=> ( ( aa(A,A2,F,A5) = bNF_Wellorder_wo_suc(A2,R2,aa(set(A),set(A2),aa(fun(A,A2),fun(set(A),set(A2)),image2(A,A2),F),order_underS(A,R,A5))) )
& pp(aa(A,bool,G,A5)) ) )
& ( ~ ( ~ pp(aa(set(bool),bool,member(bool,fFalse),aa(set(A),set(bool),aa(fun(A,bool),fun(set(A),set(bool)),image2(A,bool),G),order_underS(A,R,A5))))
& ( aa(set(A),set(A2),aa(fun(A,A2),fun(set(A),set(A2)),image2(A,A2),F),order_underS(A,R,A5)) != aa(set(product_prod(A2,A2)),set(A2),field2(A2),R2) ) )
=> ~ pp(aa(A,bool,G,A5)) ) )
=> ( ! [A5: A] :
( ( pp(aa(set(A),bool,member(A,A5),aa(set(product_prod(A,A)),set(A),field2(A),R)))
& ~ pp(aa(set(bool),bool,member(bool,fFalse),aa(set(A),set(bool),aa(fun(A,bool),fun(set(A),set(bool)),image2(A,bool),G),aa(A,set(A),order_under(A,R),A5)))) )
=> bij_betw(A,A2,F,aa(A,set(A),order_under(A,R),A5),aa(A2,set(A2),order_under(A2,R2),aa(A,A2,F,A5))) )
=> ( ( pp(aa(set(A),bool,member(A,A3),aa(set(product_prod(A,A)),set(A),field2(A),R)))
& pp(aa(set(bool),bool,member(bool,fFalse),aa(set(A),set(bool),aa(fun(A,bool),fun(set(A),set(bool)),image2(A,bool),G),aa(A,set(A),order_under(A,R),A3)))) )
=> ? [X_13: fun(A2,A)] : pp(aa(fun(A2,A),bool,bNF_Wellorder_embed(A2,A,R2,R),X_13)) ) ) ) ) ) ).
% wellorders_totally_ordered_aux2
tff(fact_1561_bij__image__INT,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(A,B),B6: fun(C,set(A)),A6: set(C)] :
( bij_betw(A,B,F,top_top(set(A)),top_top(set(B)))
=> ( aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(C),set(set(A)),aa(fun(C,set(A)),fun(set(C),set(set(A))),image2(C,set(A)),B6),A6))) = aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),aa(fun(C,set(B)),fun(set(C),set(set(B))),image2(C,set(B)),aa(fun(C,set(A)),fun(C,set(B)),aTP_Lamp_gv(fun(A,B),fun(fun(C,set(A)),fun(C,set(B))),F),B6)),A6)) ) ) ).
% bij_image_INT
tff(fact_1562_pair__lessI1,axiom,
! [A3: nat,B2: nat,S: nat,T2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),B2))
=> pp(aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool,member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A3),S)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),B2),T2))),fun_pair_less)) ) ).
% pair_lessI1
tff(fact_1563_inj__image__Compl__subset,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A6: set(A)] :
( inj_on(A,B,F,top_top(set(A)))
=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),aa(set(A),set(A),uminus_uminus(set(A)),A6))),aa(set(B),set(B),uminus_uminus(set(B)),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6)))) ) ).
% inj_image_Compl_subset
tff(fact_1564_embed__preserves__ofilter,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,A)),R2: set(product_prod(B,B)),F: fun(A,B),A6: set(A)] :
( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R),R)
=> ( order_well_order_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R2),R2)
=> ( pp(aa(fun(A,B),bool,bNF_Wellorder_embed(A,B,R,R2),F))
=> ( order_ofilter(A,R,A6)
=> order_ofilter(B,R2,aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6)) ) ) ) ) ).
% embed_preserves_ofilter
tff(fact_1565_embed__Field__ofilter,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,A)),R2: set(product_prod(B,B)),F: fun(A,B)] :
( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R),R)
=> ( order_well_order_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R2),R2)
=> ( pp(aa(fun(A,B),bool,bNF_Wellorder_embed(A,B,R,R2),F))
=> order_ofilter(B,R2,aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),aa(set(product_prod(A,A)),set(A),field2(A),R))) ) ) ) ).
% embed_Field_ofilter
tff(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: fun(A,B),R7: set(product_prod(C,C)),F4: fun(B,C)] :
( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R),R)
=> ( order_well_order_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R2),R2)
=> ( bNF_Wellorder_embedS(A,B,R,R2,F)
=> ( pp(aa(fun(B,C),bool,bNF_Wellorder_embed(B,C,R2,R7),F4))
=> bNF_Wellorder_embedS(A,C,R,R7,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),F4),F)) ) ) ) ) ).
% embedS_comp_embed
tff(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: fun(A,B),R7: set(product_prod(C,C)),F4: fun(B,C)] :
( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R),R)
=> ( order_well_order_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R2),R2)
=> ( pp(aa(fun(A,B),bool,bNF_Wellorder_embed(A,B,R,R2),F))
=> ( bNF_Wellorder_embedS(B,C,R2,R7,F4)
=> bNF_Wellorder_embedS(A,C,R,R7,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),F4),F)) ) ) ) ) ).
% embed_comp_embedS
tff(fact_1568_sum_OatLeastAtMost__reindex,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add(A)
& ord(B) )
=> ! [H: fun(nat,B),M4: nat,N4: nat,G: fun(B,A)] :
( bij_betw(nat,B,H,set_or1337092689740270186AtMost(nat,M4,N4),set_or1337092689740270186AtMost(B,aa(nat,B,H,M4),aa(nat,B,H,N4)))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),set_or1337092689740270186AtMost(B,aa(nat,B,H,M4),aa(nat,B,H,N4))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,B),fun(nat,A),aa(fun(B,A),fun(fun(nat,B),fun(nat,A)),comp(B,A,nat),G),H)),set_or1337092689740270186AtMost(nat,M4,N4)) ) ) ) ).
% sum.atLeastAtMost_reindex
tff(fact_1569_sum_OatLeastLessThan__reindex,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add(A)
& ord(B) )
=> ! [H: fun(nat,B),M4: nat,N4: nat,G: fun(B,A)] :
( bij_betw(nat,B,H,set_or7035219750837199246ssThan(nat,M4,N4),set_or7035219750837199246ssThan(B,aa(nat,B,H,M4),aa(nat,B,H,N4)))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),set_or7035219750837199246ssThan(B,aa(nat,B,H,M4),aa(nat,B,H,N4))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(fun(nat,B),fun(nat,A),aa(fun(B,A),fun(fun(nat,B),fun(nat,A)),comp(B,A,nat),G),H)),set_or7035219750837199246ssThan(nat,M4,N4)) ) ) ) ).
% sum.atLeastLessThan_reindex
tff(fact_1570_prod_OatLeastAtMost__reindex,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult(A)
& ord(B) )
=> ! [H: fun(nat,B),M4: nat,N4: nat,G: fun(B,A)] :
( bij_betw(nat,B,H,set_or1337092689740270186AtMost(nat,M4,N4),set_or1337092689740270186AtMost(B,aa(nat,B,H,M4),aa(nat,B,H,N4)))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),set_or1337092689740270186AtMost(B,aa(nat,B,H,M4),aa(nat,B,H,N4))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,B),fun(nat,A),aa(fun(B,A),fun(fun(nat,B),fun(nat,A)),comp(B,A,nat),G),H)),set_or1337092689740270186AtMost(nat,M4,N4)) ) ) ) ).
% prod.atLeastAtMost_reindex
tff(fact_1571_prod_OatLeastLessThan__reindex,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult(A)
& ord(B) )
=> ! [H: fun(nat,B),M4: nat,N4: nat,G: fun(B,A)] :
( bij_betw(nat,B,H,set_or7035219750837199246ssThan(nat,M4,N4),set_or7035219750837199246ssThan(B,aa(nat,B,H,M4),aa(nat,B,H,N4)))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),set_or7035219750837199246ssThan(B,aa(nat,B,H,M4),aa(nat,B,H,N4))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,B),fun(nat,A),aa(fun(B,A),fun(fun(nat,B),fun(nat,A)),comp(B,A,nat),G),H)),set_or7035219750837199246ssThan(nat,M4,N4)) ) ) ) ).
% prod.atLeastLessThan_reindex
tff(fact_1572_embedS__iff,axiom,
! [A: $tType,B: $tType,R: set(product_prod(A,A)),R2: set(product_prod(B,B)),F: fun(A,B)] :
( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R),R)
=> ( pp(aa(fun(A,B),bool,bNF_Wellorder_embed(A,B,R,R2),F))
=> ( bNF_Wellorder_embedS(A,B,R,R2,F)
<=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less(set(B)),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),aa(set(product_prod(A,A)),set(A),field2(A),R))),aa(set(product_prod(B,B)),set(B),field2(B),R2))) ) ) ) ).
% embedS_iff
tff(fact_1573_bij__betw__Suc,axiom,
! [M: set(nat),N: set(nat)] :
( bij_betw(nat,nat,suc,M,N)
<=> ( aa(set(nat),set(nat),aa(fun(nat,nat),fun(set(nat),set(nat)),image2(nat,nat),suc),M) = N ) ) ).
% bij_betw_Suc
tff(fact_1574_Schroeder__Bernstein,axiom,
! [A: $tType,B: $tType,F: fun(A,B),A6: set(A),B6: set(B),G: fun(B,A)] :
( inj_on(A,B,F,A6)
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6)),B6))
=> ( inj_on(B,A,G,B6)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),G),B6)),A6))
=> ? [H7: fun(A,B)] : bij_betw(A,B,H7,A6,B6) ) ) ) ) ).
% Schroeder_Bernstein
tff(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: fun(A,B)] :
( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R),R)
=> ( order_well_order_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R2),R2)
=> ( pp(aa(fun(A,B),bool,bNF_Wellorder_embed(A,B,R,R2),F))
<=> ( bNF_Wellorder_compat(A,B,R,R2,F)
& inj_on(A,B,F,aa(set(product_prod(A,A)),set(A),field2(A),R))
& order_ofilter(B,R2,aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),aa(set(product_prod(A,A)),set(A),field2(A),R))) ) ) ) ) ).
% embed_iff_compat_inj_on_ofilter
tff(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 = aa(product_prod(array(A),heap_ext(product_unit)),array(A),product_fst(array(A),heap_ext(product_unit)),aa(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit)),alloc(A,Xs),H)) )
=> ( ( H2 = aa(product_prod(array(A),heap_ext(product_unit)),heap_ext(product_unit),product_snd(array(A),heap_ext(product_unit)),aa(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit)),alloc(A,Xs),H)) )
=> ( ( aa(array(A),list(A),get(A,H2),R) = Xs )
=> ( pp(aa(array(A),bool,present(A,H2),R))
=> pp(aa(array(A),bool,present(A,H),R)) ) ) ) ) ) ) ).
% effect_of_listE
tff(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)] :
( ( aa(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit)),aa(array(A),fun(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit))),product_Pair(array(A),heap_ext(product_unit)),A3),H2) = aa(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit)),alloc(A,Xs),H) )
=> heap_effect(array(A),of_list(A,Xs),H,H2,A3) ) ) ).
% effect_of_listI
tff(fact_1578_embed__implies__iso__Restr,axiom,
! [A: $tType,B: $tType,R: set(product_prod(A,A)),R2: set(product_prod(B,B)),F: fun(B,A)] :
( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R),R)
=> ( order_well_order_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R2),R2)
=> ( pp(aa(fun(B,A),bool,bNF_Wellorder_embed(B,A,R2,R),F))
=> bNF_Wellorder_iso(B,A,R2,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R),product_Sigma(A,A,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),aa(set(product_prod(B,B)),set(B),field2(B),R2)),aa(fun(B,A),fun(A,set(A)),aTP_Lamp_oa(set(product_prod(B,B)),fun(fun(B,A),fun(A,set(A))),R2),F))),F) ) ) ) ).
% embed_implies_iso_Restr
tff(fact_1579_pair__leqI2,axiom,
! [A3: nat,B2: nat,S: nat,T2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A3),B2))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),S),T2))
=> pp(aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool,member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A3),S)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),B2),T2))),fun_pair_leq)) ) ) ).
% pair_leqI2
tff(fact_1580_pair__leqI1,axiom,
! [A3: nat,B2: nat,S: nat,T2: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),B2))
=> pp(aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool,member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),A3),S)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),B2),T2))),fun_pair_leq)) ) ).
% pair_leqI1
tff(fact_1581_iso__forward,axiom,
! [A: $tType,B: $tType,X2: A,Y2: A,R: set(product_prod(A,A)),R2: set(product_prod(B,B)),F: fun(A,B)] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),R))
=> ( bNF_Wellorder_iso(A,B,R,R2,F)
=> pp(aa(set(product_prod(B,B)),bool,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,F,X2)),aa(A,B,F,Y2))),R2)) ) ) ).
% iso_forward
tff(fact_1582_comp__iso,axiom,
! [C: $tType,B: $tType,A: $tType,R: set(product_prod(A,A)),R2: set(product_prod(B,B)),F: fun(A,B),R7: set(product_prod(C,C)),F4: fun(B,C)] :
( bNF_Wellorder_iso(A,B,R,R2,F)
=> ( bNF_Wellorder_iso(B,C,R2,R7,F4)
=> bNF_Wellorder_iso(A,C,R,R7,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),F4),F)) ) ) ).
% comp_iso
tff(fact_1583_compat__def,axiom,
! [A: $tType,A2: $tType,R: set(product_prod(A,A)),R2: set(product_prod(A2,A2)),F: fun(A,A2)] :
( bNF_Wellorder_compat(A,A2,R,R2,F)
<=> ! [A7: A,B5: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A7),B5)),R))
=> pp(aa(set(product_prod(A2,A2)),bool,member(product_prod(A2,A2),aa(A2,product_prod(A2,A2),aa(A2,fun(A2,product_prod(A2,A2)),product_Pair(A2,A2),aa(A,A2,F,A7)),aa(A,A2,F,B5))),R2)) ) ) ).
% compat_def
tff(fact_1584_iso__Field,axiom,
! [A: $tType,B: $tType,R: set(product_prod(A,A)),R2: set(product_prod(B,B)),F: fun(A,B)] :
( bNF_Wellorder_iso(A,B,R,R2,F)
=> ( aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),aa(set(product_prod(A,A)),set(A),field2(A),R)) = aa(set(product_prod(B,B)),set(B),field2(B),R2) ) ) ).
% iso_Field
tff(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: fun(A,B),R7: set(product_prod(C,C)),F4: fun(B,C)] :
( bNF_Wellorder_iso(A,B,R,R2,F)
=> ( pp(aa(fun(B,C),bool,bNF_Wellorder_embed(B,C,R2,R7),F4))
=> pp(aa(fun(A,C),bool,bNF_Wellorder_embed(A,C,R,R7),aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),F4),F))) ) ) ).
% iso_comp_embed
tff(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: fun(A,B),R7: set(product_prod(C,C)),F4: fun(B,C)] :
( pp(aa(fun(A,B),bool,bNF_Wellorder_embed(A,B,R,R2),F))
=> ( bNF_Wellorder_iso(B,C,R2,R7,F4)
=> pp(aa(fun(A,C),bool,bNF_Wellorder_embed(A,C,R,R7),aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),F4),F))) ) ) ).
% embed_comp_iso
tff(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: fun(A,B),R7: set(product_prod(C,C)),F4: fun(B,C)] :
( bNF_Wellorder_embedS(A,B,R,R2,F)
=> ( bNF_Wellorder_iso(B,C,R2,R7,F4)
=> bNF_Wellorder_embedS(A,C,R,R7,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),F4),F)) ) ) ).
% embedS_comp_iso
tff(fact_1588_of__list__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [Xs: list(A)] : of_list(A,Xs) = heap_heap(array(A),alloc(A,Xs)) ) ).
% of_list_def
tff(fact_1589_pair__leq__def,axiom,
fun_pair_leq = aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),fun(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),set(product_prod(product_prod(nat,nat),product_prod(nat,nat)))),sup_sup(set(product_prod(product_prod(nat,nat),product_prod(nat,nat)))),fun_pair_less),id2(product_prod(nat,nat))) ).
% pair_leq_def
tff(fact_1590_iso__iff2,axiom,
! [A: $tType,B: $tType,R: set(product_prod(A,A)),R2: set(product_prod(B,B)),F: fun(A,B)] :
( bNF_Wellorder_iso(A,B,R,R2,F)
<=> ( bij_betw(A,B,F,aa(set(product_prod(A,A)),set(A),field2(A),R),aa(set(product_prod(B,B)),set(B),field2(B),R2))
& ! [X5: A] :
( pp(aa(set(A),bool,member(A,X5),aa(set(product_prod(A,A)),set(A),field2(A),R)))
=> ! [Xa3: A] :
( pp(aa(set(A),bool,member(A,Xa3),aa(set(product_prod(A,A)),set(A),field2(A),R)))
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),Xa3)),R))
<=> pp(aa(set(product_prod(B,B)),bool,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,F,X5)),aa(A,B,F,Xa3))),R2)) ) ) ) ) ) ).
% iso_iff2
tff(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
tff(fact_1592_iso__iff,axiom,
! [A: $tType,B: $tType,R: set(product_prod(A,A)),R2: set(product_prod(B,B)),F: fun(A,B)] :
( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R),R)
=> ( bNF_Wellorder_iso(A,B,R,R2,F)
<=> ( pp(aa(fun(A,B),bool,bNF_Wellorder_embed(A,B,R,R2),F))
& ( aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),aa(set(product_prod(A,A)),set(A),field2(A),R)) = aa(set(product_prod(B,B)),set(B),field2(B),R2) ) ) ) ) ).
% iso_iff
tff(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: fun(A,B),R7: set(product_prod(C,C)),F4: fun(B,C)] :
( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R),R)
=> ( order_well_order_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R2),R2)
=> ( bNF_Wellorder_iso(A,B,R,R2,F)
=> ( bNF_Wellorder_embedS(B,C,R2,R7,F4)
=> bNF_Wellorder_embedS(A,C,R,R7,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),F4),F)) ) ) ) ) ).
% iso_comp_embedS
tff(fact_1594_wmax__insertI,axiom,
! [Y2: product_prod(nat,nat),YS: set(product_prod(nat,nat)),X2: product_prod(nat,nat),XS: set(product_prod(nat,nat))] :
( pp(aa(set(product_prod(nat,nat)),bool,member(product_prod(nat,nat),Y2),YS))
=> ( pp(aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool,member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),X2),Y2)),fun_pair_leq))
=> ( pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),XS),YS)),fun_max_weak))
=> pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),set(product_prod(nat,nat)),insert(product_prod(nat,nat),X2),XS)),YS)),fun_max_weak)) ) ) ) ).
% wmax_insertI
tff(fact_1595_wmin__insertI,axiom,
! [X2: product_prod(nat,nat),XS: set(product_prod(nat,nat)),Y2: product_prod(nat,nat),YS: set(product_prod(nat,nat))] :
( pp(aa(set(product_prod(nat,nat)),bool,member(product_prod(nat,nat),X2),XS))
=> ( pp(aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool,member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),X2),Y2)),fun_pair_leq))
=> ( pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),XS),YS)),fun_min_weak))
=> pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),XS),aa(set(product_prod(nat,nat)),set(product_prod(nat,nat)),insert(product_prod(nat,nat),Y2),YS))),fun_min_weak)) ) ) ) ).
% wmin_insertI
tff(fact_1596_smax__insertI,axiom,
! [Y2: product_prod(nat,nat),Y3: set(product_prod(nat,nat)),X2: product_prod(nat,nat),X6: set(product_prod(nat,nat))] :
( pp(aa(set(product_prod(nat,nat)),bool,member(product_prod(nat,nat),Y2),Y3))
=> ( pp(aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool,member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),X2),Y2)),fun_pair_less))
=> ( pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),X6),Y3)),fun_max_strict))
=> pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),set(product_prod(nat,nat)),insert(product_prod(nat,nat),X2),X6)),Y3)),fun_max_strict)) ) ) ) ).
% smax_insertI
tff(fact_1597_smin__insertI,axiom,
! [X2: product_prod(nat,nat),XS: set(product_prod(nat,nat)),Y2: product_prod(nat,nat),YS: set(product_prod(nat,nat))] :
( pp(aa(set(product_prod(nat,nat)),bool,member(product_prod(nat,nat),X2),XS))
=> ( pp(aa(set(product_prod(product_prod(nat,nat),product_prod(nat,nat))),bool,member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat))),product_Pair(product_prod(nat,nat),product_prod(nat,nat)),X2),Y2)),fun_pair_less))
=> ( pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),XS),YS)),fun_min_strict))
=> pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),XS),aa(set(product_prod(nat,nat)),set(product_prod(nat,nat)),insert(product_prod(nat,nat),Y2),YS))),fun_min_strict)) ) ) ) ).
% smin_insertI
tff(fact_1598_inv__into__ofilter__embed,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,A)),A6: set(A),F: fun(A,B),R2: set(product_prod(B,B))] :
( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R),R)
=> ( order_ofilter(A,R,A6)
=> ( ! [X: A] :
( pp(aa(set(A),bool,member(A,X),A6))
=> bij_betw(A,B,F,aa(A,set(A),order_under(A,R),X),aa(B,set(B),order_under(B,R2),aa(A,B,F,X))) )
=> ( ( aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6) = aa(set(product_prod(B,B)),set(B),field2(B),R2) )
=> pp(aa(fun(B,A),bool,bNF_Wellorder_embed(B,A,R2,R),hilbert_inv_into(A,B,A6,F))) ) ) ) ) ).
% inv_into_ofilter_embed
tff(fact_1599_inv__into__underS__embed,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,A)),A3: A,F: fun(A,B),R2: set(product_prod(B,B))] :
( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R),R)
=> ( ! [X: A] :
( pp(aa(set(A),bool,member(A,X),order_underS(A,R,A3)))
=> bij_betw(A,B,F,aa(A,set(A),order_under(A,R),X),aa(B,set(B),order_under(B,R2),aa(A,B,F,X))) )
=> ( pp(aa(set(A),bool,member(A,A3),aa(set(product_prod(A,A)),set(A),field2(A),R)))
=> ( ( aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),order_underS(A,R,A3)) = aa(set(product_prod(B,B)),set(B),field2(B),R2) )
=> pp(aa(fun(B,A),bool,bNF_Wellorder_embed(B,A,R2,R),hilbert_inv_into(A,B,order_underS(A,R,A3),F))) ) ) ) ) ).
% inv_into_underS_embed
tff(fact_1600_ordIso__def,axiom,
! [A2: $tType,A: $tType] : bNF_Wellorder_ordIso(A,A2) = aa(fun(product_prod(set(product_prod(A,A)),set(product_prod(A2,A2))),bool),set(product_prod(set(product_prod(A,A)),set(product_prod(A2,A2)))),collect(product_prod(set(product_prod(A,A)),set(product_prod(A2,A2)))),aa(fun(set(product_prod(A,A)),fun(set(product_prod(A2,A2)),bool)),fun(product_prod(set(product_prod(A,A)),set(product_prod(A2,A2))),bool),product_case_prod(set(product_prod(A,A)),set(product_prod(A2,A2)),bool),aTP_Lamp_ob(set(product_prod(A,A)),fun(set(product_prod(A2,A2)),bool)))) ).
% ordIso_def
tff(fact_1601_ordLeq__def,axiom,
! [A2: $tType,A: $tType] : bNF_Wellorder_ordLeq(A,A2) = aa(fun(product_prod(set(product_prod(A,A)),set(product_prod(A2,A2))),bool),set(product_prod(set(product_prod(A,A)),set(product_prod(A2,A2)))),collect(product_prod(set(product_prod(A,A)),set(product_prod(A2,A2)))),aa(fun(set(product_prod(A,A)),fun(set(product_prod(A2,A2)),bool)),fun(product_prod(set(product_prod(A,A)),set(product_prod(A2,A2))),bool),product_case_prod(set(product_prod(A,A)),set(product_prod(A2,A2)),bool),aTP_Lamp_oc(set(product_prod(A,A)),fun(set(product_prod(A2,A2)),bool)))) ).
% ordLeq_def
tff(fact_1602_inv__into__image__cancel,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A6: set(A),S3: set(A)] :
( inj_on(A,B,F,A6)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S3),A6))
=> ( aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),hilbert_inv_into(A,B,A6,F)),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),S3)) = S3 ) ) ) ).
% inv_into_image_cancel
tff(fact_1603_o__inv__o__cancel,axiom,
! [B: $tType,C: $tType,A: $tType,F: fun(A,B),G: fun(A,C)] :
( inj_on(A,B,F,top_top(set(A)))
=> ( aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),aa(fun(B,A),fun(B,C),aa(fun(A,C),fun(fun(B,A),fun(B,C)),comp(A,C,B),G),hilbert_inv_into(A,B,top_top(set(A)),F))),F) = G ) ) ).
% o_inv_o_cancel
tff(fact_1604_inv__o__cancel,axiom,
! [B: $tType,A: $tType,F: fun(A,B)] :
( inj_on(A,B,F,top_top(set(A)))
=> ( aa(fun(A,B),fun(A,A),aa(fun(B,A),fun(fun(A,B),fun(A,A)),comp(B,A,A),hilbert_inv_into(A,B,top_top(set(A)),F)),F) = id(A) ) ) ).
% inv_o_cancel
tff(fact_1605_max__rpair__set,axiom,
fun_reduction_pair(set(product_prod(nat,nat)),aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_prod(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))),aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),fun(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_prod(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))))),product_Pair(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))),fun_max_strict),fun_max_weak)) ).
% max_rpair_set
tff(fact_1606_min__rpair__set,axiom,
fun_reduction_pair(set(product_prod(nat,nat)),aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_prod(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))),aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),fun(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_prod(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))))),product_Pair(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))),fun_min_strict),fun_min_weak)) ).
% min_rpair_set
tff(fact_1607_wmax__emptyI,axiom,
! [X6: set(product_prod(nat,nat))] :
( finite_finite2(product_prod(nat,nat),X6)
=> pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),bot_bot(set(product_prod(nat,nat)))),X6)),fun_max_weak)) ) ).
% wmax_emptyI
tff(fact_1608_smax__emptyI,axiom,
! [Y3: set(product_prod(nat,nat))] :
( finite_finite2(product_prod(nat,nat),Y3)
=> ( ( Y3 != bot_bot(set(product_prod(nat,nat))) )
=> pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),bot_bot(set(product_prod(nat,nat)))),Y3)),fun_max_strict)) ) ) ).
% smax_emptyI
tff(fact_1609_smin__emptyI,axiom,
! [X6: set(product_prod(nat,nat))] :
( ( X6 != bot_bot(set(product_prod(nat,nat))) )
=> pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),X6),bot_bot(set(product_prod(nat,nat))))),fun_min_strict)) ) ).
% smin_emptyI
tff(fact_1610_wmin__emptyI,axiom,
! [X6: set(product_prod(nat,nat))] : pp(aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),bool,member(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),X6),bot_bot(set(product_prod(nat,nat))))),fun_min_weak)) ).
% wmin_emptyI
tff(fact_1611_f__inv__into__f,axiom,
! [B: $tType,A: $tType,Y2: A,F: fun(B,A),A6: set(B)] :
( pp(aa(set(A),bool,member(A,Y2),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6)))
=> ( aa(B,A,F,aa(A,B,hilbert_inv_into(B,A,A6,F),Y2)) = Y2 ) ) ).
% f_inv_into_f
tff(fact_1612_inv__into__into,axiom,
! [A: $tType,B: $tType,X2: A,F: fun(B,A),A6: set(B)] :
( pp(aa(set(A),bool,member(A,X2),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6)))
=> pp(aa(set(B),bool,member(B,aa(A,B,hilbert_inv_into(B,A,A6,F),X2)),A6)) ) ).
% inv_into_into
tff(fact_1613_inv__into__injective,axiom,
! [A: $tType,B: $tType,A6: set(A),F: fun(A,B),X2: B,Y2: B] :
( ( aa(B,A,hilbert_inv_into(A,B,A6,F),X2) = aa(B,A,hilbert_inv_into(A,B,A6,F),Y2) )
=> ( pp(aa(set(B),bool,member(B,X2),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6)))
=> ( pp(aa(set(B),bool,member(B,Y2),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6)))
=> ( X2 = Y2 ) ) ) ) ).
% inv_into_injective
tff(fact_1614_image__inv__into__cancel,axiom,
! [B: $tType,A: $tType,F: fun(B,A),A6: set(B),A8: set(A),B8: set(A)] :
( ( aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6) = A8 )
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B8),A8))
=> ( aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),hilbert_inv_into(B,A,A6,F)),B8)) = B8 ) ) ) ).
% image_inv_into_cancel
tff(fact_1615_surj__f__inv__f,axiom,
! [B: $tType,A: $tType,F: fun(B,A),Y2: A] :
( ( aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),top_top(set(B))) = top_top(set(A)) )
=> ( aa(B,A,F,aa(A,B,hilbert_inv_into(B,A,top_top(set(B)),F),Y2)) = Y2 ) ) ).
% surj_f_inv_f
tff(fact_1616_surj__iff__all,axiom,
! [B: $tType,A: $tType,F: fun(B,A)] :
( ( aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),top_top(set(B))) = top_top(set(A)) )
<=> ! [X5: A] : aa(B,A,F,aa(A,B,hilbert_inv_into(B,A,top_top(set(B)),F),X5)) = X5 ) ).
% surj_iff_all
tff(fact_1617_image__f__inv__f,axiom,
! [B: $tType,A: $tType,F: fun(B,A),A6: set(A)] :
( ( aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),top_top(set(B))) = top_top(set(A)) )
=> ( aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),hilbert_inv_into(B,A,top_top(set(B)),F)),A6)) = A6 ) ) ).
% image_f_inv_f
tff(fact_1618_surj__imp__inv__eq,axiom,
! [B: $tType,A: $tType,F: fun(B,A),G: fun(A,B)] :
( ( aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),top_top(set(B))) = top_top(set(A)) )
=> ( ! [X: B] : aa(A,B,G,aa(B,A,F,X)) = X
=> ( hilbert_inv_into(B,A,top_top(set(B)),F) = G ) ) ) ).
% surj_imp_inv_eq
tff(fact_1619_inj__on__inv__into,axiom,
! [B: $tType,A: $tType,B6: set(A),F: fun(B,A),A6: set(B)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B6),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6)))
=> inj_on(A,B,hilbert_inv_into(B,A,A6,F),B6) ) ).
% inj_on_inv_into
tff(fact_1620_inj__transfer,axiom,
! [B: $tType,A: $tType,F: fun(A,B),P2: fun(A,bool),X2: A] :
( inj_on(A,B,F,top_top(set(A)))
=> ( ! [Y: B] :
( pp(aa(set(B),bool,member(B,Y),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),top_top(set(A)))))
=> pp(aa(A,bool,P2,aa(B,A,hilbert_inv_into(A,B,top_top(set(A)),F),Y))) )
=> pp(aa(A,bool,P2,X2)) ) ) ).
% inj_transfer
tff(fact_1621_image__inv__f__f,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A6: set(A)] :
( inj_on(A,B,F,top_top(set(A)))
=> ( aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),hilbert_inv_into(A,B,top_top(set(A)),F)),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6)) = A6 ) ) ).
% image_inv_f_f
tff(fact_1622_inj__imp__surj__inv,axiom,
! [B: $tType,A: $tType,F: fun(A,B)] :
( inj_on(A,B,F,top_top(set(A)))
=> ( aa(set(B),set(A),aa(fun(B,A),fun(set(B),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
tff(fact_1623_surj__imp__inj__inv,axiom,
! [B: $tType,A: $tType,F: fun(B,A)] :
( ( aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(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
tff(fact_1624_inv__into__comp,axiom,
! [A: $tType,C: $tType,B: $tType,F: fun(A,B),G: fun(C,A),A6: set(C),X2: B] :
( inj_on(A,B,F,aa(set(C),set(A),aa(fun(C,A),fun(set(C),set(A)),image2(C,A),G),A6))
=> ( inj_on(C,A,G,A6)
=> ( pp(aa(set(B),bool,member(B,X2),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),aa(set(C),set(A),aa(fun(C,A),fun(set(C),set(A)),image2(C,A),G),A6))))
=> ( aa(B,C,hilbert_inv_into(C,B,A6,aa(fun(C,A),fun(C,B),aa(fun(A,B),fun(fun(C,A),fun(C,B)),comp(A,B,C),F),G)),X2) = aa(B,C,aa(fun(B,A),fun(B,C),aa(fun(A,C),fun(fun(B,A),fun(B,C)),comp(A,C,B),hilbert_inv_into(C,A,A6,G)),hilbert_inv_into(A,B,aa(set(C),set(A),aa(fun(C,A),fun(set(C),set(A)),image2(C,A),G),A6),F)),X2) ) ) ) ) ).
% inv_into_comp
tff(fact_1625_bij__betw__inv__into__subset,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A6: set(A),A8: set(B),B6: set(A),B8: set(B)] :
( bij_betw(A,B,F,A6,A8)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B6),A6))
=> ( ( aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),B6) = B8 )
=> bij_betw(B,A,hilbert_inv_into(A,B,A6,F),B8,B6) ) ) ) ).
% bij_betw_inv_into_subset
tff(fact_1626_inv__unique__comp,axiom,
! [B: $tType,A: $tType,F: fun(B,A),G: fun(A,B)] :
( ( aa(fun(A,B),fun(A,A),aa(fun(B,A),fun(fun(A,B),fun(A,A)),comp(B,A,A),F),G) = id(A) )
=> ( ( aa(fun(B,A),fun(B,B),aa(fun(A,B),fun(fun(B,A),fun(B,B)),comp(A,B,B),G),F) = id(B) )
=> ( hilbert_inv_into(B,A,top_top(set(B)),F) = G ) ) ) ).
% inv_unique_comp
tff(fact_1627_o__inv__distrib,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(A,B),G: fun(C,A)] :
( bij_betw(A,B,F,top_top(set(A)),top_top(set(B)))
=> ( bij_betw(C,A,G,top_top(set(C)),top_top(set(A)))
=> ( hilbert_inv_into(C,B,top_top(set(C)),aa(fun(C,A),fun(C,B),aa(fun(A,B),fun(fun(C,A),fun(C,B)),comp(A,B,C),F),G)) = aa(fun(B,A),fun(B,C),aa(fun(A,C),fun(fun(B,A),fun(B,C)),comp(A,C,B),hilbert_inv_into(C,A,top_top(set(C)),G)),hilbert_inv_into(A,B,top_top(set(A)),F)) ) ) ) ).
% o_inv_distrib
tff(fact_1628_iso__backward,axiom,
! [A: $tType,B: $tType,X2: A,Y2: A,R2: set(product_prod(A,A)),R: set(product_prod(B,B)),F: fun(B,A)] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),R2))
=> ( bNF_Wellorder_iso(B,A,R,R2,F)
=> pp(aa(set(product_prod(B,B)),bool,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,hilbert_inv_into(B,A,aa(set(product_prod(B,B)),set(B),field2(B),R),F),X2)),aa(A,B,hilbert_inv_into(B,A,aa(set(product_prod(B,B)),set(B),field2(B),R),F),Y2))),R)) ) ) ).
% iso_backward
tff(fact_1629_bij__image__Collect__eq,axiom,
! [B: $tType,A: $tType,F: fun(A,B),P2: fun(A,bool)] :
( bij_betw(A,B,F,top_top(set(A)),top_top(set(B)))
=> ( aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),aa(fun(A,bool),set(A),collect(A),P2)) = aa(fun(B,bool),set(B),collect(B),aa(fun(A,bool),fun(B,bool),aTP_Lamp_od(fun(A,B),fun(fun(A,bool),fun(B,bool)),F),P2)) ) ) ).
% bij_image_Collect_eq
tff(fact_1630_surj__iff,axiom,
! [B: $tType,A: $tType,F: fun(B,A)] :
( ( aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),top_top(set(B))) = top_top(set(A)) )
<=> ( aa(fun(A,B),fun(A,A),aa(fun(B,A),fun(fun(A,B),fun(A,A)),comp(B,A,A),F),hilbert_inv_into(B,A,top_top(set(B)),F)) = id(A) ) ) ).
% surj_iff
tff(fact_1631_inj__imp__bij__betw__inv,axiom,
! [B: $tType,A: $tType,F: fun(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),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),M),M) ) ).
% inj_imp_bij_betw_inv
tff(fact_1632_inj__iff,axiom,
! [B: $tType,A: $tType,F: fun(A,B)] :
( inj_on(A,B,F,top_top(set(A)))
<=> ( aa(fun(A,B),fun(A,A),aa(fun(B,A),fun(fun(A,B),fun(A,A)),comp(B,A,A),hilbert_inv_into(A,B,top_top(set(A)),F)),F) = id(A) ) ) ).
% inj_iff
tff(fact_1633_bij__vimage__eq__inv__image,axiom,
! [A: $tType,B: $tType,F: fun(A,B),A6: set(B)] :
( bij_betw(A,B,F,top_top(set(A)),top_top(set(B)))
=> ( aa(set(B),set(A),vimage(A,B,F),A6) = aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),hilbert_inv_into(A,B,top_top(set(A)),F)),A6) ) ) ).
% bij_vimage_eq_inv_image
tff(fact_1634_max__strict__def,axiom,
fun_max_strict = max_ext(product_prod(nat,nat),fun_pair_less) ).
% max_strict_def
tff(fact_1635_min__strict__def,axiom,
fun_min_strict = min_ext(product_prod(nat,nat),fun_pair_less) ).
% min_strict_def
tff(fact_1636_min__weak__def,axiom,
fun_min_weak = aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),fun(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))),sup_sup(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))),min_ext(product_prod(nat,nat),fun_pair_leq)),aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),insert(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),bot_bot(set(product_prod(nat,nat)))),bot_bot(set(product_prod(nat,nat))))),bot_bot(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))))) ).
% min_weak_def
tff(fact_1637_max__weak__def,axiom,
fun_max_weak = aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),fun(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))),sup_sup(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))),max_ext(product_prod(nat,nat),fun_pair_leq)),aa(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),insert(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat)))),product_Pair(set(product_prod(nat,nat)),set(product_prod(nat,nat))),bot_bot(set(product_prod(nat,nat)))),bot_bot(set(product_prod(nat,nat))))),bot_bot(set(product_prod(set(product_prod(nat,nat)),set(product_prod(nat,nat))))))) ).
% max_weak_def
tff(fact_1638_inv__into__Field__embed,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,A)),R2: set(product_prod(B,B)),F: fun(A,B)] :
( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R),R)
=> ( pp(aa(fun(A,B),bool,bNF_Wellorder_embed(A,B,R,R2),F))
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(product_prod(B,B)),set(B),field2(B),R2)),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),aa(set(product_prod(A,A)),set(A),field2(A),R))))
=> pp(aa(fun(B,A),bool,bNF_Wellorder_embed(B,A,R2,R),hilbert_inv_into(A,B,aa(set(product_prod(A,A)),set(A),field2(A),R),F))) ) ) ) ).
% inv_into_Field_embed
tff(fact_1639_strict__mono__inv__on__range,axiom,
! [B: $tType,A: $tType] :
( ( linorder(A)
& order(B) )
=> ! [F: fun(A,B)] :
( order_strict_mono(A,B,F)
=> strict_mono_on(B,A,hilbert_inv_into(A,B,top_top(set(A)),F),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),top_top(set(A)))) ) ) ).
% strict_mono_inv_on_range
tff(fact_1640_comp__set__bd__Union__o__collect,axiom,
! [C: $tType,B: $tType,A: $tType,X2: C,X6: set(fun(C,set(set(A)))),Hbd: set(product_prod(B,B))] :
( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(set(A))),set(set(A)),complete_Sup_Sup(set(set(A))),aa(set(fun(C,set(set(A)))),set(set(set(A))),aa(fun(fun(C,set(set(A))),set(set(A))),fun(set(fun(C,set(set(A)))),set(set(set(A)))),image2(fun(C,set(set(A))),set(set(A))),aTP_Lamp_oe(C,fun(fun(C,set(set(A))),set(set(A))),X2)),X6))))),Hbd)),bNF_Wellorder_ordLeq(A,B)))
=> pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,aa(C,set(A),aa(fun(C,set(set(A))),fun(C,set(A)),aa(fun(set(set(A)),set(A)),fun(fun(C,set(set(A))),fun(C,set(A))),comp(set(set(A)),set(A),C),complete_Sup_Sup(set(A))),bNF_collect(C,set(A),X6)),X2))),Hbd)),bNF_Wellorder_ordLeq(A,B))) ) ).
% comp_set_bd_Union_o_collect
tff(fact_1641_bijection_Oinv__comp__left,axiom,
! [A: $tType,F: fun(A,A)] :
( hilbert_bijection(A,F)
=> ( aa(fun(A,A),fun(A,A),aa(fun(A,A),fun(fun(A,A),fun(A,A)),comp(A,A,A),hilbert_inv_into(A,A,top_top(set(A)),F)),F) = id(A) ) ) ).
% bijection.inv_comp_left
tff(fact_1642_bijection_Oinv__comp__right,axiom,
! [A: $tType,F: fun(A,A)] :
( hilbert_bijection(A,F)
=> ( aa(fun(A,A),fun(A,A),aa(fun(A,A),fun(fun(A,A),fun(A,A)),comp(A,A,A),F),hilbert_inv_into(A,A,top_top(set(A)),F)) = id(A) ) ) ).
% bijection.inv_comp_right
tff(fact_1643_image__atLeastZeroLessThan__int,axiom,
! [U: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),U))
=> ( set_or7035219750837199246ssThan(int,zero_zero(int),U) = aa(set(nat),set(int),aa(fun(nat,int),fun(set(nat),set(int)),image2(nat,int),semiring_1_of_nat(int)),aa(nat,set(nat),set_ord_lessThan(nat),aa(int,nat,nat2,U))) ) ) ).
% image_atLeastZeroLessThan_int
tff(fact_1644_UN__atLeast__UNIV,axiom,
aa(set(set(nat)),set(nat),complete_Sup_Sup(set(nat)),aa(set(nat),set(set(nat)),aa(fun(nat,set(nat)),fun(set(nat),set(set(nat))),image2(nat,set(nat)),set_ord_atLeast(nat)),top_top(set(nat)))) = top_top(set(nat)) ).
% UN_atLeast_UNIV
tff(fact_1645_Array_Olength__alloc,axiom,
! [A: $tType] :
( heap(A)
=> ! [Xs: list(A),H: heap_ext(product_unit),Ys: list(A)] : aa(array(A),nat,length(A,aa(product_prod(array(A),heap_ext(product_unit)),heap_ext(product_unit),product_snd(array(A),heap_ext(product_unit)),aa(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit)),alloc(A,Xs),H))),aa(product_prod(array(A),heap_ext(product_unit)),array(A),product_fst(array(A),heap_ext(product_unit)),aa(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit)),alloc(A,Ys),H))) = aa(list(A),nat,size_size(list(A)),Xs) ) ).
% Array.length_alloc
tff(fact_1646_min__ext__def,axiom,
! [A: $tType,R: set(product_prod(A,A))] : min_ext(A,R) = aa(fun(product_prod(set(A),set(A)),bool),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aTP_Lamp_of(set(product_prod(A,A)),fun(product_prod(set(A),set(A)),bool),R)) ).
% min_ext_def
tff(fact_1647_True__in__image__Bex,axiom,
! [A: $tType,P2: fun(A,bool),A6: set(A)] :
( pp(aa(set(bool),bool,member(bool,fTrue),aa(set(A),set(bool),aa(fun(A,bool),fun(set(A),set(bool)),image2(A,bool),P2),A6)))
<=> ? [X5: A] :
( pp(aa(set(A),bool,member(A,X5),A6))
& pp(aa(A,bool,P2,X5)) ) ) ).
% True_in_image_Bex
tff(fact_1648_image__add__atLeast,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [K: A,I: A] : aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aa(A,fun(A,A),plus_plus(A),K)),aa(A,set(A),set_ord_atLeast(A),I)) = aa(A,set(A),set_ord_atLeast(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),K),I)) ) ).
% image_add_atLeast
tff(fact_1649_Image__Collect__case__prod,axiom,
! [A: $tType,B: $tType,P2: fun(B,fun(A,bool)),A6: set(B)] : aa(set(B),set(A),image(B,A,aa(fun(product_prod(B,A),bool),set(product_prod(B,A)),collect(product_prod(B,A)),aa(fun(B,fun(A,bool)),fun(product_prod(B,A),bool),product_case_prod(B,A,bool),P2))),A6) = aa(fun(A,bool),set(A),collect(A),aa(set(B),fun(A,bool),aTP_Lamp_og(fun(B,fun(A,bool)),fun(set(B),fun(A,bool)),P2),A6)) ).
% Image_Collect_case_prod
tff(fact_1650_image__minus__const__atLeast,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [C3: A,A3: A] : aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aa(A,fun(A,A),minus_minus(A),C3)),aa(A,set(A),set_ord_atLeast(A),A3)) = aa(A,set(A),set_ord_atMost(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),A3)) ) ).
% image_minus_const_atLeast
tff(fact_1651_image__minus__const__AtMost,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [C3: A,B2: A] : aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aa(A,fun(A,A),minus_minus(A),C3)),aa(A,set(A),set_ord_atMost(A),B2)) = aa(A,set(A),set_ord_atLeast(A),aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),B2)) ) ).
% image_minus_const_AtMost
tff(fact_1652_image__uminus__atLeast,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [X2: A] : aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),uminus_uminus(A)),aa(A,set(A),set_ord_atLeast(A),X2)) = aa(A,set(A),set_ord_atMost(A),aa(A,A,uminus_uminus(A),X2)) ) ).
% image_uminus_atLeast
tff(fact_1653_image__uminus__atMost,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [X2: A] : aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),uminus_uminus(A)),aa(A,set(A),set_ord_atMost(A),X2)) = aa(A,set(A),set_ord_atLeast(A),aa(A,A,uminus_uminus(A),X2)) ) ).
% image_uminus_atMost
tff(fact_1654_SUP__bool__eq,axiom,
! [A: $tType] : aTP_Lamp_oh(set(A),fun(fun(A,bool),bool)) = bex(A) ).
% SUP_bool_eq
tff(fact_1655_image__def,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A6: set(A)] : aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6) = aa(fun(B,bool),set(B),collect(B),aa(set(A),fun(B,bool),aTP_Lamp_oi(fun(A,B),fun(set(A),fun(B,bool)),F),A6)) ).
% image_def
tff(fact_1656_card__of__UNION__Sigma,axiom,
! [B: $tType,A: $tType,A6: fun(B,set(A)),I4: set(B)] : pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A))))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A)))),aa(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A)))),aa(set(product_prod(A,A)),fun(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A))))),product_Pair(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A)))),bNF_Ca6860139660246222851ard_of(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(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
tff(fact_1657_UNION__eq,axiom,
! [A: $tType,B: $tType,B6: fun(B,set(A)),A6: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),B6),A6)) = aa(fun(A,bool),set(A),collect(A),aa(set(B),fun(A,bool),aTP_Lamp_oj(fun(B,set(A)),fun(set(B),fun(A,bool)),B6),A6)) ).
% UNION_eq
tff(fact_1658_Collect__bex__eq,axiom,
! [A: $tType,B: $tType,A6: set(B),P2: fun(A,fun(B,bool))] : aa(fun(A,bool),set(A),collect(A),aa(fun(A,fun(B,bool)),fun(A,bool),aTP_Lamp_ok(set(B),fun(fun(A,fun(B,bool)),fun(A,bool)),A6),P2)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),aTP_Lamp_kz(fun(A,fun(B,bool)),fun(B,set(A)),P2)),A6)) ).
% Collect_bex_eq
tff(fact_1659_Image__def,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,B)),S: set(A)] : aa(set(A),set(B),image(A,B,R),S) = aa(fun(B,bool),set(B),collect(B),aa(set(A),fun(B,bool),aTP_Lamp_ol(set(product_prod(A,B)),fun(set(A),fun(B,bool)),R),S)) ).
% Image_def
tff(fact_1660_vimage__image__eq,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A6: set(A)] : aa(set(B),set(A),vimage(A,B,F),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6)) = aa(fun(A,bool),set(A),collect(A),aa(set(A),fun(A,bool),aTP_Lamp_om(fun(A,B),fun(set(A),fun(A,bool)),F),A6)) ).
% vimage_image_eq
tff(fact_1661_length__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [H: heap_ext(product_unit),A3: array(A)] : aa(array(A),nat,length(A,H),A3) = aa(list(A),nat,size_size(list(A)),aa(array(A),list(A),get(A,H),A3)) ) ).
% length_def
tff(fact_1662_Set_Obind__def,axiom,
! [B: $tType,A: $tType,A6: set(A),F: fun(A,set(B))] : bind4(A,B,A6,F) = aa(fun(B,bool),set(B),collect(B),aa(fun(A,set(B)),fun(B,bool),aTP_Lamp_on(set(A),fun(fun(A,set(B)),fun(B,bool)),A6),F)) ).
% Set.bind_def
tff(fact_1663_bijection_Osurj,axiom,
! [A: $tType,F: fun(A,A)] :
( hilbert_bijection(A,F)
=> ( aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),F),top_top(set(A))) = top_top(set(A)) ) ) ).
% bijection.surj
tff(fact_1664_card__of__image,axiom,
! [B: $tType,A: $tType,F: fun(B,A),A6: set(B)] : pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6))),bNF_Ca6860139660246222851ard_of(B,A6))),bNF_Wellorder_ordLeq(A,B))) ).
% card_of_image
tff(fact_1665_ex__bij__betw,axiom,
! [B: $tType,A: $tType,A6: set(A),R: set(product_prod(B,B))] :
( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A6)),R)),bNF_Wellorder_ordLeq(A,B)))
=> ? [F3: fun(B,A),B10: set(B)] : bij_betw(B,A,F3,B10,A6) ) ).
% ex_bij_betw
tff(fact_1666_type__copy__set__bd,axiom,
! [A: $tType,D: $tType,C: $tType,B: $tType,S3: fun(A,set(B)),Bd: set(product_prod(C,C)),Rep: fun(D,A),X2: D] :
( ! [Y: A] : pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(B,B)),fun(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),product_Pair(set(product_prod(B,B)),set(product_prod(C,C))),bNF_Ca6860139660246222851ard_of(B,aa(A,set(B),S3,Y))),Bd)),bNF_Wellorder_ordLeq(B,C)))
=> pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C))),aa(set(product_prod(B,B)),fun(set(product_prod(C,C)),product_prod(set(product_prod(B,B)),set(product_prod(C,C)))),product_Pair(set(product_prod(B,B)),set(product_prod(C,C))),bNF_Ca6860139660246222851ard_of(B,aa(D,set(B),aa(fun(D,A),fun(D,set(B)),aa(fun(A,set(B)),fun(fun(D,A),fun(D,set(B))),comp(A,set(B),D),S3),Rep),X2))),Bd)),bNF_Wellorder_ordLeq(B,C))) ) ).
% type_copy_set_bd
tff(fact_1667_card__of__ordLeq2,axiom,
! [B: $tType,A: $tType,A6: set(A),B6: set(B)] :
( ( A6 != bot_bot(set(A)) )
=> ( ? [G5: fun(B,A)] : aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),G5),B6) = A6
<=> pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(B,B6))),bNF_Wellorder_ordLeq(A,B))) ) ) ).
% card_of_ordLeq2
tff(fact_1668_surj__imp__ordLeq,axiom,
! [B: $tType,A: $tType,B6: set(A),F: fun(B,A),A6: set(B)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B6),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6)))
=> pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,B6)),bNF_Ca6860139660246222851ard_of(B,A6))),bNF_Wellorder_ordLeq(A,B))) ) ).
% surj_imp_ordLeq
tff(fact_1669_card__of__ordLess2,axiom,
! [A: $tType,B: $tType,B6: set(A),A6: set(B)] :
( ( B6 != bot_bot(set(A)) )
=> ( ~ ? [F7: fun(B,A)] : aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F7),A6) = B6
<=> pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,A6)),bNF_Ca6860139660246222851ard_of(A,B6))),bNF_We4044943003108391690rdLess(B,A))) ) ) ).
% card_of_ordLess2
tff(fact_1670_card__of__Func__UNIV,axiom,
! [B: $tType,A: $tType,B6: set(B)] : pp(aa(set(product_prod(set(product_prod(fun(A,B),fun(A,B))),set(product_prod(fun(A,B),fun(A,B))))),bool,member(product_prod(set(product_prod(fun(A,B),fun(A,B))),set(product_prod(fun(A,B),fun(A,B)))),aa(set(product_prod(fun(A,B),fun(A,B))),product_prod(set(product_prod(fun(A,B),fun(A,B))),set(product_prod(fun(A,B),fun(A,B)))),aa(set(product_prod(fun(A,B),fun(A,B))),fun(set(product_prod(fun(A,B),fun(A,B))),product_prod(set(product_prod(fun(A,B),fun(A,B))),set(product_prod(fun(A,B),fun(A,B))))),product_Pair(set(product_prod(fun(A,B),fun(A,B))),set(product_prod(fun(A,B),fun(A,B)))),bNF_Ca6860139660246222851ard_of(fun(A,B),bNF_Wellorder_Func(A,B,top_top(set(A)),B6))),bNF_Ca6860139660246222851ard_of(fun(A,B),aa(fun(fun(A,B),bool),set(fun(A,B)),collect(fun(A,B)),aTP_Lamp_oo(set(B),fun(fun(A,B),bool),B6))))),bNF_Wellorder_ordIso(fun(A,B),fun(A,B)))) ).
% card_of_Func_UNIV
tff(fact_1671_bijection_Osurj__inv,axiom,
! [A: $tType,F: fun(A,A)] :
( hilbert_bijection(A,F)
=> ( aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),hilbert_inv_into(A,A,top_top(set(A)),F)),top_top(set(A))) = top_top(set(A)) ) ) ).
% bijection.surj_inv
tff(fact_1672_card__of__ordLeq,axiom,
! [B: $tType,A: $tType,A6: set(A),B6: set(B)] :
( ? [F7: fun(A,B)] :
( inj_on(A,B,F7,A6)
& pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F7),A6)),B6)) )
<=> pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A6)),bNF_Ca6860139660246222851ard_of(B,B6))),bNF_Wellorder_ordLeq(A,B))) ) ).
% card_of_ordLeq
tff(fact_1673_card__of__ordLess,axiom,
! [A: $tType,B: $tType,A6: set(A),B6: set(B)] :
( ~ ? [F7: fun(A,B)] :
( inj_on(A,B,F7,A6)
& pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F7),A6)),B6)) )
<=> pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B6)),bNF_Ca6860139660246222851ard_of(A,A6))),bNF_We4044943003108391690rdLess(B,A))) ) ).
% card_of_ordLess
tff(fact_1674_card__of__UNION__ordLeq__infinite,axiom,
! [B: $tType,A: $tType,C: $tType,B6: set(A),I4: set(B),A6: fun(B,set(C))] :
( ~ finite_finite2(A,B6)
=> ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,I4)),bNF_Ca6860139660246222851ard_of(A,B6))),bNF_Wellorder_ordLeq(B,A)))
=> ( ! [X: B] :
( pp(aa(set(B),bool,member(B,X),I4))
=> pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,aa(B,set(C),A6,X))),bNF_Ca6860139660246222851ard_of(A,B6))),bNF_Wellorder_ordLeq(C,A))) )
=> pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),aa(fun(B,set(C)),fun(set(B),set(set(C))),image2(B,set(C)),A6),I4)))),bNF_Ca6860139660246222851ard_of(A,B6))),bNF_Wellorder_ordLeq(C,A))) ) ) ) ).
% card_of_UNION_ordLeq_infinite
tff(fact_1675_at__top__def,axiom,
! [A: $tType] :
( order(A)
=> ( at_top(A) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(A),set(filter(A)),aa(fun(A,filter(A)),fun(set(A),set(filter(A))),image2(A,filter(A)),aTP_Lamp_op(A,filter(A))),top_top(set(A)))) ) ) ).
% at_top_def
tff(fact_1676_array__of__list__make,axiom,
! [A: $tType] :
( heap(A)
=> ! [Xs: list(A)] : of_list(A,Xs) = make(A,aa(list(A),nat,size_size(list(A)),Xs),nth3(A,Xs)) ) ).
% array_of_list_make
tff(fact_1677_at__top__sub,axiom,
! [A: $tType] :
( linorder(A)
=> ! [C3: A] : at_top(A) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(A),set(filter(A)),aa(fun(A,filter(A)),fun(set(A),set(filter(A))),image2(A,filter(A)),aTP_Lamp_oq(A,filter(A))),aa(A,set(A),set_ord_atLeast(A),C3))) ) ).
% at_top_sub
tff(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: fun(B,set(C))] :
( ~ finite_finite2(A,aa(set(product_prod(A,A)),set(A),field2(A),R))
=> ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R),R)
=> ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,I4)),R)),bNF_Wellorder_ordLeq(B,A)))
=> ( ! [X: B] :
( pp(aa(set(B),bool,member(B,X),I4))
=> pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,aa(B,set(C),A6,X))),R)),bNF_Wellorder_ordLeq(C,A))) )
=> pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),aa(fun(B,set(C)),fun(set(B),set(set(C))),image2(B,set(C)),A6),I4)))),R)),bNF_Wellorder_ordLeq(C,A))) ) ) ) ) ).
% card_of_UNION_ordLeq_infinite_Field
tff(fact_1679_product__nth,axiom,
! [A: $tType,B: $tType,N4: nat,Xs: list(A),Ys: list(B)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N4),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(B),nat,size_size(list(B)),Ys))))
=> ( aa(nat,product_prod(A,B),nth3(product_prod(A,B),product(A,B,Xs,Ys)),N4) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(nat,A,nth3(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N4),aa(list(B),nat,size_size(list(B)),Ys)))),aa(nat,B,nth3(B,Ys),modulo_modulo(nat,N4,aa(list(B),nat,size_size(list(B)),Ys)))) ) ) ).
% product_nth
tff(fact_1680_max__ext__eq,axiom,
! [A: $tType,R3: set(product_prod(A,A))] : max_ext(A,R3) = aa(fun(product_prod(set(A),set(A)),bool),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aa(fun(set(A),fun(set(A),bool)),fun(product_prod(set(A),set(A)),bool),product_case_prod(set(A),set(A),bool),aTP_Lamp_or(set(product_prod(A,A)),fun(set(A),fun(set(A),bool)),R3))) ).
% max_ext_eq
tff(fact_1681_remdups__adj__altdef,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] :
( ( remdups_adj(A,Xs) = Ys )
<=> ? [F7: fun(nat,nat)] :
( order_mono(nat,nat,F7)
& ( aa(set(nat),set(nat),aa(fun(nat,nat),fun(set(nat),set(nat)),image2(nat,nat),F7),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))) = set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Ys)) )
& ! [I3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( aa(nat,A,nth3(A,Xs),I3) = aa(nat,A,nth3(A,Ys),aa(nat,nat,F7,I3)) ) )
& ! [I3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I3),one_one(nat))),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( ( aa(nat,A,nth3(A,Xs),I3) = aa(nat,A,nth3(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I3),one_one(nat))) )
<=> ( aa(nat,nat,F7,I3) = aa(nat,nat,F7,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I3),one_one(nat))) ) ) ) ) ) ).
% remdups_adj_altdef
tff(fact_1682_right__total__relcompp__transfer,axiom,
! [A: $tType,E: $tType,C: $tType,F2: $tType,B: $tType,D: $tType,B6: fun(A,fun(B,bool)),A6: fun(C,fun(D,bool)),C5: fun(E,fun(F2,bool))] :
( right_total(A,B,B6)
=> pp(aa(fun(fun(D,fun(B,bool)),fun(fun(B,fun(F2,bool)),fun(D,fun(F2,bool)))),bool,aa(fun(fun(C,fun(A,bool)),fun(fun(A,fun(E,bool)),fun(C,fun(E,bool)))),fun(fun(fun(D,fun(B,bool)),fun(fun(B,fun(F2,bool)),fun(D,fun(F2,bool)))),bool),bNF_rel_fun(fun(C,fun(A,bool)),fun(D,fun(B,bool)),fun(fun(A,fun(E,bool)),fun(C,fun(E,bool))),fun(fun(B,fun(F2,bool)),fun(D,fun(F2,bool))),bNF_rel_fun(C,D,fun(A,bool),fun(B,bool),A6,bNF_rel_fun(A,B,bool,bool,B6,fequal(bool))),bNF_rel_fun(fun(A,fun(E,bool)),fun(B,fun(F2,bool)),fun(C,fun(E,bool)),fun(D,fun(F2,bool)),bNF_rel_fun(A,B,fun(E,bool),fun(F2,bool),B6,bNF_rel_fun(E,F2,bool,bool,C5,fequal(bool))),bNF_rel_fun(C,D,fun(E,bool),fun(F2,bool),A6,bNF_rel_fun(E,F2,bool,bool,C5,fequal(bool))))),aTP_Lamp_os(fun(A,fun(B,bool)),fun(fun(C,fun(A,bool)),fun(fun(A,fun(E,bool)),fun(C,fun(E,bool)))),B6)),relcompp(D,B,F2))) ) ).
% right_total_relcompp_transfer
tff(fact_1683_rel__fun__def,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,A6: fun(A,fun(C,bool)),B6: fun(B,fun(D,bool)),X4: fun(A,B),Xa4: fun(C,D)] :
( pp(aa(fun(C,D),bool,aa(fun(A,B),fun(fun(C,D),bool),bNF_rel_fun(A,C,B,D,A6,B6),X4),Xa4))
<=> ! [Xb2: A,Y4: C] :
( pp(aa(C,bool,aa(A,fun(C,bool),A6,Xb2),Y4))
=> pp(aa(D,bool,aa(B,fun(D,bool),B6,aa(A,B,X4,Xb2)),aa(C,D,Xa4,Y4))) ) ) ).
% rel_fun_def
tff(fact_1684_Card__order__trans,axiom,
! [A: $tType,R: set(product_prod(A,A)),X2: A,Y2: A,Z2: A] :
( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R),R)
=> ( ( X2 != Y2 )
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),R))
=> ( ( Y2 != Z2 )
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z2)),R))
=> ( ( X2 != Z2 )
& pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Z2)),R)) ) ) ) ) ) ) ).
% Card_order_trans
tff(fact_1685_Card__order__wo__rel,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R),R)
=> bNF_Wellorder_wo_rel(A,R) ) ).
% Card_order_wo_rel
tff(fact_1686_pcr__Domainp__par,axiom,
! [A: $tType,B: $tType,C: $tType,B6: fun(A,fun(B,bool)),P23: fun(A,bool),A6: fun(C,fun(A,bool)),P12: fun(C,bool),P24: fun(C,bool)] :
( ( domainp(A,B,B6) = P23 )
=> ( ( domainp(C,A,A6) = P12 )
=> ( pp(aa(fun(A,bool),bool,aa(fun(C,bool),fun(fun(A,bool),bool),bNF_rel_fun(C,A,bool,bool,A6,fequal(bool)),P24),P23))
=> ( domainp(C,B,aa(fun(A,fun(B,bool)),fun(C,fun(B,bool)),aa(fun(C,fun(A,bool)),fun(fun(A,fun(B,bool)),fun(C,fun(B,bool))),relcompp(C,A,B),A6),B6)) = aa(fun(C,bool),fun(C,bool),aa(fun(C,bool),fun(fun(C,bool),fun(C,bool)),inf_inf(fun(C,bool)),P12),P24) ) ) ) ) ).
% pcr_Domainp_par
tff(fact_1687_pcr__Domainp__total,axiom,
! [A: $tType,B: $tType,C: $tType,B6: fun(A,fun(B,bool)),A6: fun(C,fun(A,bool)),P2: fun(C,bool)] :
( left_total(A,B,B6)
=> ( ( domainp(C,A,A6) = P2 )
=> ( domainp(C,B,aa(fun(A,fun(B,bool)),fun(C,fun(B,bool)),aa(fun(C,fun(A,bool)),fun(fun(A,fun(B,bool)),fun(C,fun(B,bool))),relcompp(C,A,B),A6),B6)) = P2 ) ) ) ).
% pcr_Domainp_total
tff(fact_1688_pcr__Domainp,axiom,
! [C: $tType,B: $tType,A: $tType,B6: fun(A,fun(B,bool)),P2: fun(A,bool),A6: fun(C,fun(A,bool))] :
( ( domainp(A,B,B6) = P2 )
=> ! [X4: C] :
( pp(aa(C,bool,domainp(C,B,aa(fun(A,fun(B,bool)),fun(C,fun(B,bool)),aa(fun(C,fun(A,bool)),fun(fun(A,fun(B,bool)),fun(C,fun(B,bool))),relcompp(C,A,B),A6),B6)),X4))
<=> ? [Y4: A] :
( pp(aa(A,bool,aa(C,fun(A,bool),A6,X4),Y4))
& pp(aa(A,bool,P2,Y4)) ) ) ) ).
% pcr_Domainp
tff(fact_1689_Domain__def,axiom,
! [B: $tType,A: $tType,X4: set(product_prod(A,B))] : aa(set(product_prod(A,B)),set(A),domain(A,B),X4) = aa(fun(A,bool),set(A),collect(A),domainp(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_bg(set(product_prod(A,B)),fun(A,fun(B,bool))),X4))) ).
% Domain_def
tff(fact_1690_Domainp__Domain__eq,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,B)),X4: A] :
( pp(aa(A,bool,domainp(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_bg(set(product_prod(A,B)),fun(A,fun(B,bool))),R)),X4))
<=> pp(aa(set(A),bool,member(A,X4),aa(set(product_prod(A,B)),set(A),domain(A,B),R))) ) ).
% Domainp_Domain_eq
tff(fact_1691_infinite__Card__order__limit,axiom,
! [A: $tType,R: set(product_prod(A,A)),A3: A] :
( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R),R)
=> ( ~ finite_finite2(A,aa(set(product_prod(A,A)),set(A),field2(A),R))
=> ( pp(aa(set(A),bool,member(A,A3),aa(set(product_prod(A,A)),set(A),field2(A),R)))
=> ? [X: A] :
( pp(aa(set(A),bool,member(A,X),aa(set(product_prod(A,A)),set(A),field2(A),R)))
& ( A3 != X )
& pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),X)),R)) ) ) ) ) ).
% infinite_Card_order_limit
tff(fact_1692_Collect__all__eq,axiom,
! [A: $tType,B: $tType,P2: fun(A,fun(B,bool))] : aa(fun(A,bool),set(A),collect(A),aTP_Lamp_ot(fun(A,fun(B,bool)),fun(A,bool),P2)) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),aTP_Lamp_kz(fun(A,fun(B,bool)),fun(B,set(A)),P2)),top_top(set(B)))) ).
% Collect_all_eq
tff(fact_1693_success__makeI,axiom,
! [A: $tType] :
( heap(A)
=> ! [N4: nat,F: fun(nat,A),H: heap_ext(product_unit)] : heap_success(array(A),make(A,N4,F),H) ) ).
% success_makeI
tff(fact_1694_pcr__Domainp__par__left__total,axiom,
! [A: $tType,B: $tType,C: $tType,B6: fun(A,fun(B,bool)),P2: fun(A,bool),A6: fun(C,fun(A,bool)),P6: fun(C,bool)] :
( ( domainp(A,B,B6) = P2 )
=> ( left_total(C,A,A6)
=> ( pp(aa(fun(A,bool),bool,aa(fun(C,bool),fun(fun(A,bool),bool),bNF_rel_fun(C,A,bool,bool,A6,fequal(bool)),P6),P2))
=> ( domainp(C,B,aa(fun(A,fun(B,bool)),fun(C,fun(B,bool)),aa(fun(C,fun(A,bool)),fun(fun(A,fun(B,bool)),fun(C,fun(B,bool))),relcompp(C,A,B),A6),B6)) = P6 ) ) ) ) ).
% pcr_Domainp_par_left_total
tff(fact_1695_regularCard__UNION,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,A)),As: fun(A,set(B)),B6: set(B)] :
( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R),R)
=> ( bNF_Ca7133664381575040944arCard(A,R)
=> ( bNF_Ca3754400796208372196lChain(A,set(B),R,As)
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B6),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),aa(fun(A,set(B)),fun(set(A),set(set(B))),image2(A,set(B)),As),aa(set(product_prod(A,A)),set(A),field2(A),R)))))
=> ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B6)),R)),bNF_We4044943003108391690rdLess(B,A)))
=> ? [X: A] :
( pp(aa(set(A),bool,member(A,X),aa(set(product_prod(A,A)),set(A),field2(A),R)))
& pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B6),aa(A,set(B),As,X))) ) ) ) ) ) ) ).
% regularCard_UNION
tff(fact_1696_ex__toCard__pred,axiom,
! [B: $tType,A: $tType,A6: set(A),R: set(product_prod(B,B))] :
( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A6)),R)),bNF_Wellorder_ordLeq(A,B)))
=> ( bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R),R)
=> ? [X_13: fun(A,B)] : pp(aa(fun(A,B),bool,bNF_Gr1419584066657907630d_pred(A,B,A6,R),X_13)) ) ) ).
% ex_toCard_pred
tff(fact_1697_toCard__pred__def,axiom,
! [A: $tType,B: $tType,A6: set(A),R: set(product_prod(B,B)),F: fun(A,B)] :
( pp(aa(fun(A,B),bool,bNF_Gr1419584066657907630d_pred(A,B,A6,R),F))
<=> ( inj_on(A,B,F,A6)
& pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),A6)),aa(set(product_prod(B,B)),set(B),field2(B),R)))
& bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R),R) ) ) ).
% toCard_pred_def
tff(fact_1698_cardSuc__UNION,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,A)),As: fun(set(A),set(B)),B6: set(B)] :
( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R),R)
=> ( ~ finite_finite2(A,aa(set(product_prod(A,A)),set(A),field2(A),R))
=> ( bNF_Ca3754400796208372196lChain(set(A),set(B),bNF_Ca8387033319878233205ardSuc(A,R),As)
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B6),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(set(A)),set(set(B)),aa(fun(set(A),set(B)),fun(set(set(A)),set(set(B))),image2(set(A),set(B)),As),aa(set(product_prod(set(A),set(A))),set(set(A)),field2(set(A)),bNF_Ca8387033319878233205ardSuc(A,R))))))
=> ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B6)),R)),bNF_Wellorder_ordLeq(B,A)))
=> ? [X: set(A)] :
( pp(aa(set(set(A)),bool,member(set(A),X),aa(set(product_prod(set(A),set(A))),set(set(A)),field2(set(A)),bNF_Ca8387033319878233205ardSuc(A,R))))
& pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B6),aa(set(A),set(B),As,X))) ) ) ) ) ) ) ).
% cardSuc_UNION
tff(fact_1699_toCard__inj,axiom,
! [B: $tType,A: $tType,A6: set(A),R: set(product_prod(B,B)),X2: A,Y2: A] :
( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A6)),R)),bNF_Wellorder_ordLeq(A,B)))
=> ( bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R),R)
=> ( pp(aa(set(A),bool,member(A,X2),A6))
=> ( pp(aa(set(A),bool,member(A,Y2),A6))
=> ( ( aa(A,B,bNF_Greatest_toCard(A,B,A6,R),X2) = aa(A,B,bNF_Greatest_toCard(A,B,A6,R),Y2) )
<=> ( X2 = Y2 ) ) ) ) ) ) ).
% toCard_inj
tff(fact_1700_comp__single__set__bd,axiom,
! [B: $tType,D: $tType,A: $tType,E: $tType,C: $tType,Fbd: set(product_prod(A,A)),Fset: fun(B,set(C)),Gset: fun(D,set(B)),Gbd: set(product_prod(E,E)),X2: D] :
( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),Fbd),Fbd)
=> ( ! [X: B] : pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(C,C)),fun(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A)))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(C,aa(B,set(C),Fset,X))),Fbd)),bNF_Wellorder_ordLeq(C,A)))
=> ( ! [X: D] : pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(E,E)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(E,E))),aa(set(product_prod(E,E)),product_prod(set(product_prod(B,B)),set(product_prod(E,E))),aa(set(product_prod(B,B)),fun(set(product_prod(E,E)),product_prod(set(product_prod(B,B)),set(product_prod(E,E)))),product_Pair(set(product_prod(B,B)),set(product_prod(E,E))),bNF_Ca6860139660246222851ard_of(B,aa(D,set(B),Gset,X))),Gbd)),bNF_Wellorder_ordLeq(B,E)))
=> pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(product_prod(E,A),product_prod(E,A))))),bool,member(product_prod(set(product_prod(C,C)),set(product_prod(product_prod(E,A),product_prod(E,A)))),aa(set(product_prod(product_prod(E,A),product_prod(E,A))),product_prod(set(product_prod(C,C)),set(product_prod(product_prod(E,A),product_prod(E,A)))),aa(set(product_prod(C,C)),fun(set(product_prod(product_prod(E,A),product_prod(E,A))),product_prod(set(product_prod(C,C)),set(product_prod(product_prod(E,A),product_prod(E,A))))),product_Pair(set(product_prod(C,C)),set(product_prod(product_prod(E,A),product_prod(E,A)))),bNF_Ca6860139660246222851ard_of(C,aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),aa(fun(B,set(C)),fun(set(B),set(set(C))),image2(B,set(C)),Fset),aa(D,set(B),Gset,X2))))),bNF_Cardinal_cprod(E,A,Gbd,Fbd))),bNF_Wellorder_ordLeq(C,product_prod(E,A)))) ) ) ) ).
% comp_single_set_bd
tff(fact_1701_init__seg__of__def,axiom,
! [A: $tType] : init_seg_of(A) = aa(fun(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),bool),set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),collect(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),aa(fun(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool)),fun(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),bool),product_case_prod(set(product_prod(A,A)),set(product_prod(A,A)),bool),aTP_Lamp_ou(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool)))) ).
% init_seg_of_def
tff(fact_1702_wfP__SUP,axiom,
! [B: $tType,A: $tType,R: fun(A,fun(B,fun(B,bool)))] :
( ! [I6: A] : wfP(B,aa(A,fun(B,fun(B,bool)),R,I6))
=> ( ! [I6: A,J3: A] :
( ( aa(A,fun(B,fun(B,bool)),R,I6) != aa(A,fun(B,fun(B,bool)),R,J3) )
=> ( aa(fun(B,bool),fun(B,bool),aa(fun(B,bool),fun(fun(B,bool),fun(B,bool)),inf_inf(fun(B,bool)),domainp(B,B,aa(A,fun(B,fun(B,bool)),R,I6))),rangep(B,B,aa(A,fun(B,fun(B,bool)),R,J3))) = bot_bot(fun(B,bool)) ) )
=> wfP(B,aa(set(fun(B,fun(B,bool))),fun(B,fun(B,bool)),complete_Sup_Sup(fun(B,fun(B,bool))),aa(set(A),set(fun(B,fun(B,bool))),aa(fun(A,fun(B,fun(B,bool))),fun(set(A),set(fun(B,fun(B,bool)))),image2(A,fun(B,fun(B,bool))),R),top_top(set(A))))) ) ) ).
% wfP_SUP
tff(fact_1703_wfP__wf__eq,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( wfP(A,aTP_Lamp_hl(set(product_prod(A,A)),fun(A,fun(A,bool)),R))
<=> wf(A,R) ) ).
% wfP_wf_eq
tff(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
tff(fact_1705_wfP__def,axiom,
! [A: $tType,R: fun(A,fun(A,bool))] :
( wfP(A,R)
<=> wf(A,aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),R))) ) ).
% wfP_def
tff(fact_1706_wfP__acyclicP,axiom,
! [A: $tType,R: fun(A,fun(A,bool))] :
( wfP(A,R)
=> transitive_acyclic(A,aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),R))) ) ).
% wfP_acyclicP
tff(fact_1707_toCard__pred__toCard,axiom,
! [A: $tType,B: $tType,A6: set(A),R: set(product_prod(B,B))] :
( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A6)),R)),bNF_Wellorder_ordLeq(A,B)))
=> ( bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R),R)
=> pp(aa(fun(A,B),bool,bNF_Gr1419584066657907630d_pred(A,B,A6,R),bNF_Greatest_toCard(A,B,A6,R))) ) ) ).
% toCard_pred_toCard
tff(fact_1708_fromCard__toCard,axiom,
! [B: $tType,A: $tType,A6: set(A),R: set(product_prod(B,B)),B2: A] :
( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,A6)),R)),bNF_Wellorder_ordLeq(A,B)))
=> ( bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R),R)
=> ( pp(aa(set(A),bool,member(A,B2),A6))
=> ( bNF_Gr5436034075474128252omCard(A,B,A6,R,aa(A,B,bNF_Greatest_toCard(A,B,A6,R),B2)) = B2 ) ) ) ) ).
% fromCard_toCard
tff(fact_1709_cardSuc__UNION__Cinfinite,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,A)),As: fun(set(A),set(B)),B6: set(B)] :
( ( bNF_Ca4139267488887388095finite(A,R)
& bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R),R) )
=> ( bNF_Ca3754400796208372196lChain(set(A),set(B),bNF_Ca8387033319878233205ardSuc(A,R),As)
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B6),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(set(A)),set(set(B)),aa(fun(set(A),set(B)),fun(set(set(A)),set(set(B))),image2(set(A),set(B)),As),aa(set(product_prod(set(A),set(A))),set(set(A)),field2(set(A)),bNF_Ca8387033319878233205ardSuc(A,R))))))
=> ( pp(aa(set(product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),bool,member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(B,B)),fun(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A))),bNF_Ca6860139660246222851ard_of(B,B6)),R)),bNF_Wellorder_ordLeq(B,A)))
=> ? [X: set(A)] :
( pp(aa(set(set(A)),bool,member(set(A),X),aa(set(product_prod(set(A),set(A))),set(set(A)),field2(set(A)),bNF_Ca8387033319878233205ardSuc(A,R))))
& pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B6),aa(set(A),set(B),As,X))) ) ) ) ) ) ).
% cardSuc_UNION_Cinfinite
tff(fact_1710_composed__equiv__rel__eq__onp,axiom,
! [B: $tType,A: $tType,R3: fun(A,fun(B,bool)),P2: fun(A,bool),P6: fun(B,bool),P7: fun(A,bool)] :
( left_unique(A,B,R3)
=> ( pp(aa(fun(B,bool),bool,aa(fun(A,bool),fun(fun(B,bool),bool),bNF_rel_fun(A,B,bool,bool,R3,fequal(bool)),P2),P6))
=> ( ( domainp(A,B,R3) = P7 )
=> ( aa(fun(B,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(A,bool)),fun(A,fun(A,bool))),relcompp(A,B,A),R3),aa(fun(B,fun(A,bool)),fun(B,fun(A,bool)),aa(fun(B,fun(B,bool)),fun(fun(B,fun(A,bool)),fun(B,fun(A,bool))),relcompp(B,B,A),bNF_eq_onp(B,P6)),conversep(A,B,R3))) = bNF_eq_onp(A,aa(fun(A,bool),fun(A,bool),aa(fun(A,bool),fun(fun(A,bool),fun(A,bool)),inf_inf(fun(A,bool)),P7),P2)) ) ) ) ) ).
% composed_equiv_rel_eq_onp
tff(fact_1711_nth__zip,axiom,
! [A: $tType,B: $tType,I: nat,Xs: list(A),Ys: list(B)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(list(B),nat,size_size(list(B)),Ys)))
=> ( aa(nat,product_prod(A,B),nth3(product_prod(A,B),zip(A,B,Xs,Ys)),I) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(nat,A,nth3(A,Xs),I)),aa(nat,B,nth3(B,Ys),I)) ) ) ) ).
% nth_zip
tff(fact_1712_relcompp__transfer,axiom,
! [C: $tType,A: $tType,E: $tType,F2: $tType,B: $tType,D: $tType,B6: fun(A,fun(B,bool)),A6: fun(C,fun(D,bool)),C5: fun(E,fun(F2,bool))] :
( bi_total(A,B,B6)
=> pp(aa(fun(fun(D,fun(B,bool)),fun(fun(B,fun(F2,bool)),fun(D,fun(F2,bool)))),bool,aa(fun(fun(C,fun(A,bool)),fun(fun(A,fun(E,bool)),fun(C,fun(E,bool)))),fun(fun(fun(D,fun(B,bool)),fun(fun(B,fun(F2,bool)),fun(D,fun(F2,bool)))),bool),bNF_rel_fun(fun(C,fun(A,bool)),fun(D,fun(B,bool)),fun(fun(A,fun(E,bool)),fun(C,fun(E,bool))),fun(fun(B,fun(F2,bool)),fun(D,fun(F2,bool))),bNF_rel_fun(C,D,fun(A,bool),fun(B,bool),A6,bNF_rel_fun(A,B,bool,bool,B6,fequal(bool))),bNF_rel_fun(fun(A,fun(E,bool)),fun(B,fun(F2,bool)),fun(C,fun(E,bool)),fun(D,fun(F2,bool)),bNF_rel_fun(A,B,fun(E,bool),fun(F2,bool),B6,bNF_rel_fun(E,F2,bool,bool,C5,fequal(bool))),bNF_rel_fun(C,D,fun(E,bool),fun(F2,bool),A6,bNF_rel_fun(E,F2,bool,bool,C5,fequal(bool))))),relcompp(C,A,E)),relcompp(D,B,F2))) ) ).
% relcompp_transfer
tff(fact_1713_UNION__Cinfinite__bound,axiom,
! [A: $tType,B: $tType,C: $tType,I4: set(A),R: set(product_prod(B,B)),A6: fun(A,set(C))] :
( pp(aa(set(product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(A,I4)),R)),bNF_Wellorder_ordLeq(A,B)))
=> ( ! [X: A] :
( pp(aa(set(A),bool,member(A,X),I4))
=> pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(C,C)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(C,C)),set(product_prod(B,B))),aa(set(product_prod(C,C)),fun(set(product_prod(B,B)),product_prod(set(product_prod(C,C)),set(product_prod(B,B)))),product_Pair(set(product_prod(C,C)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(C,aa(A,set(C),A6,X))),R)),bNF_Wellorder_ordLeq(C,B))) )
=> ( ( bNF_Ca4139267488887388095finite(B,R)
& bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R),R) )
=> pp(aa(set(product_prod(set(product_prod(C,C)),set(product_prod(B,B)))),bool,member(product_prod(set(product_prod(C,C)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(C,C)),set(product_prod(B,B))),aa(set(product_prod(C,C)),fun(set(product_prod(B,B)),product_prod(set(product_prod(C,C)),set(product_prod(B,B)))),product_Pair(set(product_prod(C,C)),set(product_prod(B,B))),bNF_Ca6860139660246222851ard_of(C,aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(A),set(set(C)),aa(fun(A,set(C)),fun(set(A),set(set(C))),image2(A,set(C)),A6),I4)))),R)),bNF_Wellorder_ordLeq(C,B))) ) ) ) ).
% UNION_Cinfinite_bound
tff(fact_1714_Cinfinite__limit__finite,axiom,
! [A: $tType,X6: set(A),R: set(product_prod(A,A))] :
( finite_finite2(A,X6)
=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X6),aa(set(product_prod(A,A)),set(A),field2(A),R)))
=> ( ( bNF_Ca4139267488887388095finite(A,R)
& bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R),R) )
=> ? [X: A] :
( pp(aa(set(A),bool,member(A,X),aa(set(product_prod(A,A)),set(A),field2(A),R)))
& ! [Xa4: A] :
( pp(aa(set(A),bool,member(A,Xa4),X6))
=> ( ( Xa4 != X )
& pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa4),X)),R)) ) ) ) ) ) ) ).
% Cinfinite_limit_finite
tff(fact_1715_eq__onp__top__eq__eq,axiom,
! [A: $tType] : bNF_eq_onp(A,top_top(fun(A,bool))) = fequal(A) ).
% eq_onp_top_eq_eq
tff(fact_1716_eq__onp__def,axiom,
! [A: $tType,R3: fun(A,bool),X4: A,Xa4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),bNF_eq_onp(A,R3),X4),Xa4))
<=> ( pp(aa(A,bool,R3,X4))
& ( X4 = Xa4 ) ) ) ).
% eq_onp_def
tff(fact_1717_eq__onp__True,axiom,
! [A: $tType] : bNF_eq_onp(A,aTP_Lamp_ov(A,bool)) = fequal(A) ).
% eq_onp_True
tff(fact_1718_eq__onp__same__args,axiom,
! [A: $tType,P2: fun(A,bool),X2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),bNF_eq_onp(A,P2),X2),X2))
<=> pp(aa(A,bool,P2,X2)) ) ).
% eq_onp_same_args
tff(fact_1719_eq__onp__to__eq,axiom,
! [A: $tType,P2: fun(A,bool),X2: A,Y2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),bNF_eq_onp(A,P2),X2),Y2))
=> ( X2 = Y2 ) ) ).
% eq_onp_to_eq
tff(fact_1720_eq__onp__mono0,axiom,
! [A: $tType,A6: set(A),P2: fun(A,bool),Q2: fun(A,bool)] :
( ! [X: A] :
( pp(aa(set(A),bool,member(A,X),A6))
=> ( pp(aa(A,bool,P2,X))
=> pp(aa(A,bool,Q2,X)) ) )
=> ! [X4: A] :
( pp(aa(set(A),bool,member(A,X4),A6))
=> ! [Xa4: A] :
( pp(aa(set(A),bool,member(A,Xa4),A6))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),bNF_eq_onp(A,P2),X4),Xa4))
=> pp(aa(A,bool,aa(A,fun(A,bool),bNF_eq_onp(A,Q2),X4),Xa4)) ) ) ) ) ).
% eq_onp_mono0
tff(fact_1721_eq__onp__eqD,axiom,
! [A: $tType,P2: fun(A,bool),Q2: fun(A,fun(A,bool)),X2: A] :
( ( bNF_eq_onp(A,P2) = Q2 )
=> ( pp(aa(A,bool,P2,X2))
<=> pp(aa(A,bool,aa(A,fun(A,bool),Q2,X2),X2)) ) ) ).
% eq_onp_eqD
tff(fact_1722_bi__total__OO,axiom,
! [A: $tType,C: $tType,B: $tType,A6: fun(A,fun(B,bool)),B6: fun(B,fun(C,bool))] :
( bi_total(A,B,A6)
=> ( bi_total(B,C,B6)
=> bi_total(A,C,aa(fun(B,fun(C,bool)),fun(A,fun(C,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(C,bool)),fun(A,fun(C,bool))),relcompp(A,B,C),A6),B6)) ) ) ).
% bi_total_OO
tff(fact_1723_eq__onp__mono__iff,axiom,
! [A: $tType,P2: fun(A,bool),Q2: fun(A,bool)] :
( pp(aa(fun(A,fun(A,bool)),bool,aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),bool),ord_less_eq(fun(A,fun(A,bool))),bNF_eq_onp(A,P2)),bNF_eq_onp(A,Q2)))
<=> pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),P2),Q2)) ) ).
% eq_onp_mono_iff
tff(fact_1724_eq__onp__Grp,axiom,
! [A: $tType,P2: fun(A,bool)] : bNF_eq_onp(A,P2) = bNF_Grp(A,A,aa(fun(A,bool),set(A),collect(A),P2),id(A)) ).
% eq_onp_Grp
tff(fact_1725_Cinfinite__limit,axiom,
! [A: $tType,X2: A,R: set(product_prod(A,A))] :
( pp(aa(set(A),bool,member(A,X2),aa(set(product_prod(A,A)),set(A),field2(A),R)))
=> ( ( bNF_Ca4139267488887388095finite(A,R)
& bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R),R) )
=> ? [X: A] :
( pp(aa(set(A),bool,member(A,X),aa(set(product_prod(A,A)),set(A),field2(A),R)))
& ( X2 != X )
& pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),X)),R)) ) ) ) ).
% Cinfinite_limit
tff(fact_1726_Cinfinite__limit2,axiom,
! [A: $tType,X1: A,R: set(product_prod(A,A)),X22: A] :
( pp(aa(set(A),bool,member(A,X1),aa(set(product_prod(A,A)),set(A),field2(A),R)))
=> ( pp(aa(set(A),bool,member(A,X22),aa(set(product_prod(A,A)),set(A),field2(A),R)))
=> ( ( bNF_Ca4139267488887388095finite(A,R)
& bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R),R) )
=> ? [X: A] :
( pp(aa(set(A),bool,member(A,X),aa(set(product_prod(A,A)),set(A),field2(A),R)))
& ( X1 != X )
& pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X1),X)),R))
& ( X22 != X )
& pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X22),X)),R)) ) ) ) ) ).
% Cinfinite_limit2
tff(fact_1727_composed__equiv__rel__eq__eq__onp,axiom,
! [B: $tType,A: $tType,R3: fun(A,fun(B,bool)),P2: fun(A,bool)] :
( left_unique(A,B,R3)
=> ( ( domainp(A,B,R3) = P2 )
=> ( aa(fun(B,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(A,bool)),fun(A,fun(A,bool))),relcompp(A,B,A),R3),aa(fun(B,fun(A,bool)),fun(B,fun(A,bool)),aa(fun(B,fun(B,bool)),fun(fun(B,fun(A,bool)),fun(B,fun(A,bool))),relcompp(B,B,A),fequal(B)),conversep(A,B,R3))) = bNF_eq_onp(A,P2) ) ) ) ).
% composed_equiv_rel_eq_eq_onp
tff(fact_1728_in__set__zip,axiom,
! [A: $tType,B: $tType,P: product_prod(A,B),Xs: list(A),Ys: list(B)] :
( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),P),aa(list(product_prod(A,B)),set(product_prod(A,B)),set3(product_prod(A,B)),zip(A,B,Xs,Ys))))
<=> ? [N5: nat] :
( ( aa(nat,A,nth3(A,Xs),N5) = aa(product_prod(A,B),A,product_fst(A,B),P) )
& ( aa(nat,B,nth3(B,Ys),N5) = aa(product_prod(A,B),B,product_snd(A,B),P) )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N5),aa(list(A),nat,size_size(list(A)),Xs)))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N5),aa(list(B),nat,size_size(list(B)),Ys))) ) ) ).
% in_set_zip
tff(fact_1729_effect__newE,axiom,
! [A: $tType] :
( heap(A)
=> ! [N4: nat,X2: A,H: heap_ext(product_unit),H2: heap_ext(product_unit),R: array(A)] :
( heap_effect(array(A),aa(A,heap_Heap(array(A)),aa(nat,fun(A,heap_Heap(array(A))),new(A),N4),X2),H,H2,R)
=> ~ ( ( R = aa(product_prod(array(A),heap_ext(product_unit)),array(A),product_fst(array(A),heap_ext(product_unit)),aa(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit)),alloc(A,replicate(A,N4,X2)),H)) )
=> ( ( H2 = aa(product_prod(array(A),heap_ext(product_unit)),heap_ext(product_unit),product_snd(array(A),heap_ext(product_unit)),aa(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit)),alloc(A,replicate(A,N4,X2)),H)) )
=> ( ( aa(array(A),list(A),get(A,H2),R) = replicate(A,N4,X2) )
=> ( pp(aa(array(A),bool,present(A,H2),R))
=> pp(aa(array(A),bool,present(A,H),R)) ) ) ) ) ) ) ).
% effect_newE
tff(fact_1730_Rats__eq__range__of__rat__o__nat__to__rat__surj,axiom,
! [A: $tType] :
( field_char_0(A)
=> ( field_char_0_Rats(A) = aa(set(nat),set(A),aa(fun(nat,A),fun(set(nat),set(A)),image2(nat,A),aa(fun(nat,rat),fun(nat,A),aa(fun(rat,A),fun(fun(nat,rat),fun(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
tff(fact_1731_list__eq__iff__zip__eq,axiom,
! [A: $tType,Xs: list(A),Ys: list(A)] :
( ( Xs = Ys )
<=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) )
& ! [X5: product_prod(A,A)] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),X5),aa(list(product_prod(A,A)),set(product_prod(A,A)),set3(product_prod(A,A)),zip(A,A,Xs,Ys))))
=> pp(aa(product_prod(A,A),bool,aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),fequal(A)),X5)) ) ) ) ).
% list_eq_iff_zip_eq
tff(fact_1732_Nats__def,axiom,
! [A: $tType] :
( semiring_1(A)
=> ( semiring_1_Nats(A) = aa(set(nat),set(A),aa(fun(nat,A),fun(set(nat),set(A)),image2(nat,A),semiring_1_of_nat(A)),top_top(set(nat))) ) ) ).
% Nats_def
tff(fact_1733_effect__makeE,axiom,
! [A: $tType] :
( heap(A)
=> ! [N4: nat,F: fun(nat,A),H: heap_ext(product_unit),H2: heap_ext(product_unit),R: array(A)] :
( heap_effect(array(A),make(A,N4,F),H,H2,R)
=> ~ ( ( R = aa(product_prod(array(A),heap_ext(product_unit)),array(A),product_fst(array(A),heap_ext(product_unit)),aa(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit)),alloc(A,aa(list(nat),list(A),map(nat,A,F),upt(zero_zero(nat),N4))),H)) )
=> ( ( H2 = aa(product_prod(array(A),heap_ext(product_unit)),heap_ext(product_unit),product_snd(array(A),heap_ext(product_unit)),aa(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit)),alloc(A,aa(list(nat),list(A),map(nat,A,F),upt(zero_zero(nat),N4))),H)) )
=> ( ( aa(array(A),list(A),get(A,H2),R) = aa(list(nat),list(A),map(nat,A,F),upt(zero_zero(nat),N4)) )
=> ( pp(aa(array(A),bool,present(A,H2),R))
=> pp(aa(array(A),bool,present(A,H),R)) ) ) ) ) ) ) ).
% effect_makeE
tff(fact_1734_Zorns__po__lemma,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( order_7125193373082350890der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R),R)
=> ( ! [C4: set(A)] :
( pp(aa(set(set(A)),bool,member(set(A),C4),chains(A,R)))
=> ? [X4: A] :
( pp(aa(set(A),bool,member(A,X4),aa(set(product_prod(A,A)),set(A),field2(A),R)))
& ! [Xa: A] :
( pp(aa(set(A),bool,member(A,Xa),C4))
=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),X4)),R)) ) ) )
=> ? [X: A] :
( pp(aa(set(A),bool,member(A,X),aa(set(product_prod(A,A)),set(A),field2(A),R)))
& ! [Xa4: A] :
( pp(aa(set(A),bool,member(A,Xa4),aa(set(product_prod(A,A)),set(A),field2(A),R)))
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Xa4)),R))
=> ( Xa4 = X ) ) ) ) ) ) ).
% Zorns_po_lemma
tff(fact_1735_map__comp__map,axiom,
! [B: $tType,C: $tType,A: $tType,F: fun(C,B),G: fun(A,C)] : aa(fun(list(A),list(C)),fun(list(A),list(B)),aa(fun(list(C),list(B)),fun(fun(list(A),list(C)),fun(list(A),list(B))),comp(list(C),list(B),list(A)),map(C,B,F)),map(A,C,G)) = map(A,B,aa(fun(A,C),fun(A,B),aa(fun(C,B),fun(fun(A,C),fun(A,B)),comp(C,B,A),F),G)) ).
% map_comp_map
tff(fact_1736_map__map,axiom,
! [B: $tType,A: $tType,C: $tType,F: fun(B,A),G: fun(C,B),Xs: list(C)] : aa(list(B),list(A),map(B,A,F),aa(list(C),list(B),map(C,B,G),Xs)) = aa(list(C),list(A),map(C,A,aa(fun(C,B),fun(C,A),aa(fun(B,A),fun(fun(C,B),fun(C,A)),comp(B,A,C),F),G)),Xs) ).
% map_map
tff(fact_1737_List_Omap_Ocompositionality,axiom,
! [B: $tType,C: $tType,A: $tType,F: fun(B,C),G: fun(A,B),List: list(A)] : aa(list(B),list(C),map(B,C,F),aa(list(A),list(B),map(A,B,G),List)) = aa(list(A),list(C),map(A,C,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),F),G)),List) ).
% List.map.compositionality
tff(fact_1738_list_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,G: fun(B,C),F: fun(A,B),V: list(A)] : aa(list(B),list(C),map(B,C,G),aa(list(A),list(B),map(A,B,F),V)) = aa(list(A),list(C),map(A,C,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),G),F)),V) ).
% list.map_comp
tff(fact_1739_List_Omap_Ocomp,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(B,C),G: fun(A,B)] : aa(fun(list(A),list(B)),fun(list(A),list(C)),aa(fun(list(B),list(C)),fun(fun(list(A),list(B)),fun(list(A),list(C))),comp(list(B),list(C),list(A)),map(B,C,F)),map(A,B,G)) = map(A,C,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),F),G)) ).
% List.map.comp
tff(fact_1740_list_Oset__map,axiom,
! [B: $tType,A: $tType,F: fun(A,B),V: list(A)] : aa(list(B),set(B),set3(B),aa(list(A),list(B),map(A,B,F),V)) = aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),aa(list(A),set(A),set3(A),V)) ).
% list.set_map
tff(fact_1741_image__set,axiom,
! [A: $tType,B: $tType,F: fun(B,A),Xs: list(B)] : aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),aa(list(B),set(B),set3(B),Xs)) = aa(list(A),set(A),set3(A),aa(list(B),list(A),map(B,A,F),Xs)) ).
% image_set
tff(fact_1742_inj__on__mapI,axiom,
! [B: $tType,A: $tType,F: fun(A,B),A6: set(list(A))] :
( inj_on(A,B,F,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(list(A)),set(set(A)),aa(fun(list(A),set(A)),fun(set(list(A)),set(set(A))),image2(list(A),set(A)),set3(A)),A6)))
=> inj_on(list(A),list(B),map(A,B,F),A6) ) ).
% inj_on_mapI
tff(fact_1743_zip__same,axiom,
! [A: $tType,A3: A,B2: A,Xs: list(A)] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),aa(list(product_prod(A,A)),set(product_prod(A,A)),set3(product_prod(A,A)),zip(A,A,Xs,Xs))))
<=> ( pp(aa(set(A),bool,member(A,A3),aa(list(A),set(A),set3(A),Xs)))
& ( A3 = B2 ) ) ) ).
% zip_same
tff(fact_1744_in__set__zipE,axiom,
! [A: $tType,B: $tType,X2: A,Y2: B,Xs: list(A),Ys: list(B)] :
( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y2)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set3(product_prod(A,B)),zip(A,B,Xs,Ys))))
=> ~ ( pp(aa(set(A),bool,member(A,X2),aa(list(A),set(A),set3(A),Xs)))
=> ~ pp(aa(set(B),bool,member(B,Y2),aa(list(B),set(B),set3(B),Ys))) ) ) ).
% in_set_zipE
tff(fact_1745_set__zip__leftD,axiom,
! [B: $tType,A: $tType,X2: A,Y2: B,Xs: list(A),Ys: list(B)] :
( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y2)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set3(product_prod(A,B)),zip(A,B,Xs,Ys))))
=> pp(aa(set(A),bool,member(A,X2),aa(list(A),set(A),set3(A),Xs))) ) ).
% set_zip_leftD
tff(fact_1746_set__zip__rightD,axiom,
! [A: $tType,B: $tType,X2: A,Y2: B,Xs: list(A),Ys: list(B)] :
( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y2)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set3(product_prod(A,B)),zip(A,B,Xs,Ys))))
=> pp(aa(set(B),bool,member(B,Y2),aa(list(B),set(B),set3(B),Ys))) ) ).
% set_zip_rightD
tff(fact_1747_map2__map__map,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,H: fun(B,fun(C,A)),F: fun(D,B),Xs: list(D),G: fun(D,C)] : aa(list(product_prod(B,C)),list(A),map(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),H)),zip(B,C,aa(list(D),list(B),map(D,B,F),Xs),aa(list(D),list(C),map(D,C,G),Xs))) = aa(list(D),list(A),map(D,A,aa(fun(D,C),fun(D,A),aa(fun(D,B),fun(fun(D,C),fun(D,A)),aTP_Lamp_ow(fun(B,fun(C,A)),fun(fun(D,B),fun(fun(D,C),fun(D,A))),H),F),G)),Xs) ).
% map2_map_map
tff(fact_1748_in__set__impl__in__set__zip1,axiom,
! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),X2: A] :
( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
=> ( pp(aa(set(A),bool,member(A,X2),aa(list(A),set(A),set3(A),Xs)))
=> ~ ! [Y: B] : ~ pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set3(product_prod(A,B)),zip(A,B,Xs,Ys)))) ) ) ).
% in_set_impl_in_set_zip1
tff(fact_1749_in__set__impl__in__set__zip2,axiom,
! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),Y2: B] :
( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
=> ( pp(aa(set(B),bool,member(B,Y2),aa(list(B),set(B),set3(B),Ys)))
=> ~ ! [X: A] : ~ pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y2)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set3(product_prod(A,B)),zip(A,B,Xs,Ys)))) ) ) ).
% in_set_impl_in_set_zip2
tff(fact_1750_map__zip__map,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F: fun(product_prod(B,C),A),G: fun(D,B),Xs: list(D),Ys: list(C)] : aa(list(product_prod(B,C)),list(A),map(product_prod(B,C),A,F),zip(B,C,aa(list(D),list(B),map(D,B,G),Xs),Ys)) = aa(list(product_prod(D,C)),list(A),map(product_prod(D,C),A,aa(fun(D,fun(C,A)),fun(product_prod(D,C),A),product_case_prod(D,C,A),aa(fun(D,B),fun(D,fun(C,A)),aTP_Lamp_ox(fun(product_prod(B,C),A),fun(fun(D,B),fun(D,fun(C,A))),F),G))),zip(D,C,Xs,Ys)) ).
% map_zip_map
tff(fact_1751_map__zip__map2,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F: fun(product_prod(B,C),A),Xs: list(B),G: fun(D,C),Ys: list(D)] : aa(list(product_prod(B,C)),list(A),map(product_prod(B,C),A,F),zip(B,C,Xs,aa(list(D),list(C),map(D,C,G),Ys))) = aa(list(product_prod(B,D)),list(A),map(product_prod(B,D),A,aa(fun(B,fun(D,A)),fun(product_prod(B,D),A),product_case_prod(B,D,A),aa(fun(D,C),fun(B,fun(D,A)),aTP_Lamp_oy(fun(product_prod(B,C),A),fun(fun(D,C),fun(B,fun(D,A))),F),G))),zip(B,D,Xs,Ys)) ).
% map_zip_map2
tff(fact_1752_zip__map__map,axiom,
! [B: $tType,A: $tType,C: $tType,D: $tType,F: fun(C,A),Xs: list(C),G: fun(D,B),Ys: list(D)] : zip(A,B,aa(list(C),list(A),map(C,A,F),Xs),aa(list(D),list(B),map(D,B,G),Ys)) = aa(list(product_prod(C,D)),list(product_prod(A,B)),map(product_prod(C,D),product_prod(A,B),aa(fun(C,fun(D,product_prod(A,B))),fun(product_prod(C,D),product_prod(A,B)),product_case_prod(C,D,product_prod(A,B)),aa(fun(D,B),fun(C,fun(D,product_prod(A,B))),aTP_Lamp_br(fun(C,A),fun(fun(D,B),fun(C,fun(D,product_prod(A,B)))),F),G))),zip(C,D,Xs,Ys)) ).
% zip_map_map
tff(fact_1753_zip__map2,axiom,
! [B: $tType,A: $tType,C: $tType,Xs: list(A),F: fun(C,B),Ys: list(C)] : zip(A,B,Xs,aa(list(C),list(B),map(C,B,F),Ys)) = aa(list(product_prod(A,C)),list(product_prod(A,B)),map(product_prod(A,C),product_prod(A,B),aa(fun(A,fun(C,product_prod(A,B))),fun(product_prod(A,C),product_prod(A,B)),product_case_prod(A,C,product_prod(A,B)),aTP_Lamp_oz(fun(C,B),fun(A,fun(C,product_prod(A,B))),F))),zip(A,C,Xs,Ys)) ).
% zip_map2
tff(fact_1754_zip__map1,axiom,
! [A: $tType,C: $tType,B: $tType,F: fun(C,A),Xs: list(C),Ys: list(B)] : zip(A,B,aa(list(C),list(A),map(C,A,F),Xs),Ys) = aa(list(product_prod(C,B)),list(product_prod(A,B)),map(product_prod(C,B),product_prod(A,B),aa(fun(C,fun(B,product_prod(A,B))),fun(product_prod(C,B),product_prod(A,B)),product_case_prod(C,B,product_prod(A,B)),aTP_Lamp_pa(fun(C,A),fun(C,fun(B,product_prod(A,B))),F))),zip(C,B,Xs,Ys)) ).
% zip_map1
tff(fact_1755_new__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [N4: nat,X2: A] : aa(A,heap_Heap(array(A)),aa(nat,fun(A,heap_Heap(array(A))),new(A),N4),X2) = heap_heap(array(A),alloc(A,replicate(A,N4,X2))) ) ).
% new_def
tff(fact_1756_Array_Omake__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [N4: nat,F: fun(nat,A)] : make(A,N4,F) = heap_heap(array(A),alloc(A,aa(list(nat),list(A),map(nat,A,F),upt(zero_zero(nat),N4)))) ) ).
% Array.make_def
tff(fact_1757_effect__newI,axiom,
! [A: $tType] :
( heap(A)
=> ! [A3: array(A),H2: heap_ext(product_unit),N4: nat,X2: A,H: heap_ext(product_unit)] :
( ( aa(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit)),aa(array(A),fun(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit))),product_Pair(array(A),heap_ext(product_unit)),A3),H2) = aa(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit)),alloc(A,replicate(A,N4,X2)),H) )
=> heap_effect(array(A),aa(A,heap_Heap(array(A)),aa(nat,fun(A,heap_Heap(array(A))),new(A),N4),X2),H,H2,A3) ) ) ).
% effect_newI
tff(fact_1758_effect__makeI,axiom,
! [A: $tType] :
( heap(A)
=> ! [A3: array(A),H2: heap_ext(product_unit),F: fun(nat,A),N4: nat,H: heap_ext(product_unit)] :
( ( aa(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit)),aa(array(A),fun(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit))),product_Pair(array(A),heap_ext(product_unit)),A3),H2) = aa(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit)),alloc(A,aa(list(nat),list(A),map(nat,A,F),upt(zero_zero(nat),N4))),H) )
=> heap_effect(array(A),make(A,N4,F),H,H2,A3) ) ) ).
% effect_makeI
tff(fact_1759_success__newI,axiom,
! [A: $tType] :
( heap(A)
=> ! [N4: nat,X2: A,H: heap_ext(product_unit)] : heap_success(array(A),aa(A,heap_Heap(array(A)),aa(nat,fun(A,heap_Heap(array(A))),new(A),N4),X2),H) ) ).
% success_newI
tff(fact_1760_array__make,axiom,
! [A: $tType] :
( heap(A)
=> ! [N4: nat,X2: A] : aa(A,heap_Heap(array(A)),aa(nat,fun(A,heap_Heap(array(A))),new(A),N4),X2) = make(A,N4,aTP_Lamp_pb(A,fun(nat,A),X2)) ) ).
% array_make
tff(fact_1761_Rats__def,axiom,
! [A: $tType] :
( field_char_0(A)
=> ( field_char_0_Rats(A) = aa(set(rat),set(A),aa(fun(rat,A),fun(set(rat),set(A)),image2(rat,A),field_char_0_of_rat(A)),top_top(set(rat))) ) ) ).
% Rats_def
tff(fact_1762_listrel__iff__zip,axiom,
! [B: $tType,A: $tType,Xs: list(A),Ys: list(B),R: set(product_prod(A,B))] :
( pp(aa(set(product_prod(list(A),list(B))),bool,member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs),Ys)),listrel(A,B,R)))
<=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
& ! [X5: product_prod(A,B)] :
( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),X5),aa(list(product_prod(A,B)),set(product_prod(A,B)),set3(product_prod(A,B)),zip(A,B,Xs,Ys))))
=> pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_bg(set(product_prod(A,B)),fun(A,fun(B,bool))),R)),X5)) ) ) ) ).
% listrel_iff_zip
tff(fact_1763_set__list__bind,axiom,
! [A: $tType,B: $tType,Xs: list(B),F: fun(B,list(A))] : aa(list(A),set(A),set3(A),bind(B,A,Xs,F)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),aTP_Lamp_pc(fun(B,list(A)),fun(B,set(A)),F)),aa(list(B),set(B),set3(B),Xs))) ).
% set_list_bind
tff(fact_1764_nth__image,axiom,
! [A: $tType,L: nat,Xs: list(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),L),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( aa(set(nat),set(A),aa(fun(nat,A),fun(set(nat),set(A)),image2(nat,A),nth3(A,Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),L)) = aa(list(A),set(A),set3(A),take(A,L,Xs)) ) ) ).
% nth_image
tff(fact_1765_INF__set__fold,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [F: fun(B,A),Xs: list(B)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),aa(list(B),set(B),set3(B),Xs))) = aa(A,A,fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),aa(fun(A,fun(A,A)),fun(fun(B,A),fun(B,fun(A,A))),comp(A,fun(A,A),B),inf_inf(A)),F),Xs),top_top(A)) ) ).
% INF_set_fold
tff(fact_1766_SUP__set__fold,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [F: fun(B,A),Xs: list(B)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),aa(list(B),set(B),set3(B),Xs))) = aa(A,A,fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),aa(fun(A,fun(A,A)),fun(fun(B,A),fun(B,fun(A,A))),comp(A,fun(A,A),B),sup_sup(A)),F),Xs),bot_bot(A)) ) ).
% SUP_set_fold
tff(fact_1767_set__zip,axiom,
! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] : aa(list(product_prod(A,B)),set(product_prod(A,B)),set3(product_prod(A,B)),zip(A,B,Xs,Ys)) = aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(list(B),fun(product_prod(A,B),bool),aTP_Lamp_pd(list(A),fun(list(B),fun(product_prod(A,B),bool)),Xs),Ys)) ).
% set_zip
tff(fact_1768_map__fst__zip,axiom,
! [B: $tType,A: $tType,Xs: list(A),Ys: list(B)] :
( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
=> ( aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),zip(A,B,Xs,Ys)) = Xs ) ) ).
% map_fst_zip
tff(fact_1769_map__snd__zip,axiom,
! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] :
( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
=> ( aa(list(product_prod(A,B)),list(B),map(product_prod(A,B),B,product_snd(A,B)),zip(A,B,Xs,Ys)) = Ys ) ) ).
% map_snd_zip
tff(fact_1770_zip__replicate,axiom,
! [A: $tType,B: $tType,I: nat,X2: A,J5: nat,Y2: B] : zip(A,B,replicate(A,I,X2),replicate(B,J5,Y2)) = replicate(product_prod(A,B),ord_min(nat,I,J5),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y2)) ).
% zip_replicate
tff(fact_1771_map__fst__zip__take,axiom,
! [B: $tType,A: $tType,Xs: list(A),Ys: list(B)] : aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),zip(A,B,Xs,Ys)) = take(A,ord_min(nat,aa(list(A),nat,size_size(list(A)),Xs),aa(list(B),nat,size_size(list(B)),Ys)),Xs) ).
% map_fst_zip_take
tff(fact_1772_map__snd__zip__take,axiom,
! [B: $tType,A: $tType,Xs: list(B),Ys: list(A)] : aa(list(product_prod(B,A)),list(A),map(product_prod(B,A),A,product_snd(B,A)),zip(B,A,Xs,Ys)) = take(A,ord_min(nat,aa(list(B),nat,size_size(list(B)),Xs),aa(list(A),nat,size_size(list(A)),Ys)),Ys) ).
% map_snd_zip_take
tff(fact_1773_zip__replicate1,axiom,
! [A: $tType,B: $tType,N4: nat,X2: A,Ys: list(B)] : zip(A,B,replicate(A,N4,X2),Ys) = aa(list(B),list(product_prod(A,B)),map(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2)),take(B,N4,Ys)) ).
% zip_replicate1
tff(fact_1774_zip__replicate2,axiom,
! [B: $tType,A: $tType,Xs: list(A),N4: nat,Y2: B] : zip(A,B,Xs,replicate(B,N4,Y2)) = aa(list(A),list(product_prod(A,B)),map(A,product_prod(A,B),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_bk(B,fun(A,product_prod(A,B))),Y2)),take(A,N4,Xs)) ).
% zip_replicate2
tff(fact_1775_pair__list__eqI,axiom,
! [B: $tType,A: $tType,Xs: list(product_prod(A,B)),Ys: list(product_prod(A,B))] :
( ( aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs) = aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Ys) )
=> ( ( aa(list(product_prod(A,B)),list(B),map(product_prod(A,B),B,product_snd(A,B)),Xs) = aa(list(product_prod(A,B)),list(B),map(product_prod(A,B),B,product_snd(A,B)),Ys) )
=> ( Xs = Ys ) ) ) ).
% pair_list_eqI
tff(fact_1776_fold__commute__apply,axiom,
! [A: $tType,C: $tType,B: $tType,Xs: list(A),H: fun(B,C),G: fun(A,fun(B,B)),F: fun(A,fun(C,C)),S: B] :
( ! [X: A] :
( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set3(A),Xs)))
=> ( aa(fun(B,B),fun(B,C),aa(fun(B,C),fun(fun(B,B),fun(B,C)),comp(B,C,B),H),aa(A,fun(B,B),G,X)) = aa(fun(B,C),fun(B,C),aa(fun(C,C),fun(fun(B,C),fun(B,C)),comp(C,C,B),aa(A,fun(C,C),F,X)),H) ) )
=> ( aa(B,C,H,aa(B,B,fold(A,B,G,Xs),S)) = aa(C,C,fold(A,C,F,Xs),aa(B,C,H,S)) ) ) ).
% fold_commute_apply
tff(fact_1777_fold__commute,axiom,
! [A: $tType,C: $tType,B: $tType,Xs: list(A),H: fun(B,C),G: fun(A,fun(B,B)),F: fun(A,fun(C,C))] :
( ! [X: A] :
( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set3(A),Xs)))
=> ( aa(fun(B,B),fun(B,C),aa(fun(B,C),fun(fun(B,B),fun(B,C)),comp(B,C,B),H),aa(A,fun(B,B),G,X)) = aa(fun(B,C),fun(B,C),aa(fun(C,C),fun(fun(B,C),fun(B,C)),comp(C,C,B),aa(A,fun(C,C),F,X)),H) ) )
=> ( aa(fun(B,B),fun(B,C),aa(fun(B,C),fun(fun(B,B),fun(B,C)),comp(B,C,B),H),fold(A,B,G,Xs)) = aa(fun(B,C),fun(B,C),aa(fun(C,C),fun(fun(B,C),fun(B,C)),comp(C,C,B),fold(A,C,F,Xs)),H) ) ) ).
% fold_commute
tff(fact_1778_zip__same__conv__map,axiom,
! [A: $tType,Xs: list(A)] : zip(A,A,Xs,Xs) = aa(list(A),list(product_prod(A,A)),map(A,product_prod(A,A),aTP_Lamp_bi(A,product_prod(A,A))),Xs) ).
% zip_same_conv_map
tff(fact_1779_fold__map,axiom,
! [B: $tType,A: $tType,C: $tType,G: fun(B,fun(A,A)),F: fun(C,B),Xs: list(C)] : fold(B,A,G,aa(list(C),list(B),map(C,B,F),Xs)) = fold(C,A,aa(fun(C,B),fun(C,fun(A,A)),aa(fun(B,fun(A,A)),fun(fun(C,B),fun(C,fun(A,A))),comp(B,fun(A,A),C),G),F),Xs) ).
% fold_map
tff(fact_1780_zip__map__fst__snd,axiom,
! [B: $tType,A: $tType,Zs: list(product_prod(A,B))] : zip(A,B,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Zs),aa(list(product_prod(A,B)),list(B),map(product_prod(A,B),B,product_snd(A,B)),Zs)) = Zs ).
% zip_map_fst_snd
tff(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)) = aa(list(product_prod(B,product_prod(A,C))),list(product_prod(A,product_prod(B,C))),map(product_prod(B,product_prod(A,C)),product_prod(A,product_prod(B,C)),aa(fun(B,fun(product_prod(A,C),product_prod(A,product_prod(B,C)))),fun(product_prod(B,product_prod(A,C)),product_prod(A,product_prod(B,C))),product_case_prod(B,product_prod(A,C),product_prod(A,product_prod(B,C))),aTP_Lamp_pf(B,fun(product_prod(A,C),product_prod(A,product_prod(B,C)))))),zip(B,product_prod(A,C),Ys,zip(A,C,Xs,Zs))) ).
% zip_left_commute
tff(fact_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)) = aa(list(product_prod(product_prod(A,B),C)),list(product_prod(A,product_prod(B,C))),map(product_prod(product_prod(A,B),C),product_prod(A,product_prod(B,C)),aa(fun(product_prod(A,B),fun(C,product_prod(A,product_prod(B,C)))),fun(product_prod(product_prod(A,B),C),product_prod(A,product_prod(B,C))),product_case_prod(product_prod(A,B),C,product_prod(A,product_prod(B,C))),aa(fun(A,fun(B,fun(C,product_prod(A,product_prod(B,C))))),fun(product_prod(A,B),fun(C,product_prod(A,product_prod(B,C)))),product_case_prod(A,B,fun(C,product_prod(A,product_prod(B,C)))),aTP_Lamp_pg(A,fun(B,fun(C,product_prod(A,product_prod(B,C)))))))),zip(product_prod(A,B),C,zip(A,B,Xs,Ys),Zs)) ).
% zip_assoc
tff(fact_1783_zip__commute,axiom,
! [B: $tType,A: $tType,Xs: list(A),Ys: list(B)] : zip(A,B,Xs,Ys) = aa(list(product_prod(B,A)),list(product_prod(A,B)),map(product_prod(B,A),product_prod(A,B),aa(fun(B,fun(A,product_prod(A,B))),fun(product_prod(B,A),product_prod(A,B)),product_case_prod(B,A,product_prod(A,B)),aTP_Lamp_bk(B,fun(A,product_prod(A,B))))),zip(B,A,Ys,Xs)) ).
% zip_commute
tff(fact_1784_hom__Min__commute,axiom,
! [A: $tType] :
( linorder(A)
=> ! [H: fun(A,A),N: set(A)] :
( ! [X: A,Y: A] : aa(A,A,H,ord_min(A,X,Y)) = ord_min(A,aa(A,A,H,X),aa(A,A,H,Y))
=> ( finite_finite2(A,N)
=> ( ( N != bot_bot(set(A)) )
=> ( aa(A,A,H,lattic643756798350308766er_Min(A,N)) = lattic643756798350308766er_Min(A,aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),H),N)) ) ) ) ) ) ).
% hom_Min_commute
tff(fact_1785_zip__eq__conv,axiom,
! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),Zs: list(product_prod(A,B))] :
( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
=> ( ( zip(A,B,Xs,Ys) = Zs )
<=> ( ( aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Zs) = Xs )
& ( aa(list(product_prod(A,B)),list(B),map(product_prod(A,B),B,product_snd(A,B)),Zs) = Ys ) ) ) ) ).
% zip_eq_conv
tff(fact_1786_Id__on__set,axiom,
! [A: $tType,Xs: list(A)] : id_on(A,aa(list(A),set(A),set3(A),Xs)) = aa(list(product_prod(A,A)),set(product_prod(A,A)),set3(product_prod(A,A)),aa(list(A),list(product_prod(A,A)),map(A,product_prod(A,A),aTP_Lamp_bi(A,product_prod(A,A))),Xs)) ).
% Id_on_set
tff(fact_1787_listrel__iff__nth,axiom,
! [B: $tType,A: $tType,Xs: list(A),Ys: list(B),R: set(product_prod(A,B))] :
( pp(aa(set(product_prod(list(A),list(B))),bool,member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs),Ys)),listrel(A,B,R)))
<=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
& ! [N5: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N5),aa(list(A),nat,size_size(list(A)),Xs)))
=> pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(nat,A,nth3(A,Xs),N5)),aa(nat,B,nth3(B,Ys),N5))),R)) ) ) ) ).
% listrel_iff_nth
tff(fact_1788_lexord__take__index__conv,axiom,
! [A: $tType,X2: list(A),Y2: list(A),R: set(product_prod(A,A))] :
( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X2),Y2)),lexord(A,R)))
<=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),X2)),aa(list(A),nat,size_size(list(A)),Y2)))
& ( take(A,aa(list(A),nat,size_size(list(A)),X2),Y2) = X2 ) )
| ? [I3: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),ord_min(nat,aa(list(A),nat,size_size(list(A)),X2),aa(list(A),nat,size_size(list(A)),Y2))))
& ( take(A,I3,X2) = take(A,I3,Y2) )
& pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,nth3(A,X2),I3)),aa(nat,A,nth3(A,Y2),I3))),R)) ) ) ) ).
% lexord_take_index_conv
tff(fact_1789_lex__take__index,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),R: set(product_prod(A,A))] :
( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),lex(A,R)))
=> ~ ! [I6: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I6),aa(list(A),nat,size_size(list(A)),Xs)))
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I6),aa(list(A),nat,size_size(list(A)),Ys)))
=> ( ( take(A,I6,Xs) = take(A,I6,Ys) )
=> ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,nth3(A,Xs),I6)),aa(nat,A,nth3(A,Ys),I6))),R)) ) ) ) ) ).
% lex_take_index
tff(fact_1790_product__code,axiom,
! [B: $tType,A: $tType,Xs: list(A),Ys: list(B)] : product_product(A,B,aa(list(A),set(A),set3(A),Xs),aa(list(B),set(B),set3(B),Ys)) = aa(list(product_prod(A,B)),set(product_prod(A,B)),set3(product_prod(A,B)),concat(product_prod(A,B),aa(list(A),list(list(product_prod(A,B))),map(A,list(product_prod(A,B)),aTP_Lamp_ph(list(B),fun(A,list(product_prod(A,B))),Ys)),Xs))) ).
% product_code
tff(fact_1791_subseqs__powset,axiom,
! [A: $tType,Xs: list(A)] : aa(set(list(A)),set(set(A)),aa(fun(list(A),set(A)),fun(set(list(A)),set(set(A))),image2(list(A),set(A)),set3(A)),aa(list(list(A)),set(list(A)),set3(list(A)),subseqs(A,Xs))) = pow(A,aa(list(A),set(A),set3(A),Xs)) ).
% subseqs_powset
tff(fact_1792_list_Oin__rel,axiom,
! [A: $tType,B: $tType,R3: fun(A,fun(B,bool)),A3: list(A),B2: list(B)] :
( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,R3),A3),B2))
<=> ? [Z4: list(product_prod(A,B))] :
( pp(aa(set(list(product_prod(A,B))),bool,member(list(product_prod(A,B)),Z4),aa(fun(list(product_prod(A,B)),bool),set(list(product_prod(A,B))),collect(list(product_prod(A,B))),aTP_Lamp_pi(fun(A,fun(B,bool)),fun(list(product_prod(A,B)),bool),R3))))
& ( aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Z4) = A3 )
& ( aa(list(product_prod(A,B)),list(B),map(product_prod(A,B),B,product_snd(A,B)),Z4) = B2 ) ) ) ).
% list.in_rel
tff(fact_1793_list_Orel__compp__Grp,axiom,
! [B: $tType,A: $tType,R3: fun(A,fun(B,bool))] : list_all2(A,B,R3) = aa(fun(list(product_prod(A,B)),fun(list(B),bool)),fun(list(A),fun(list(B),bool)),aa(fun(list(A),fun(list(product_prod(A,B)),bool)),fun(fun(list(product_prod(A,B)),fun(list(B),bool)),fun(list(A),fun(list(B),bool))),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),aa(fun(list(product_prod(A,B)),bool),set(list(product_prod(A,B))),collect(list(product_prod(A,B))),aTP_Lamp_pi(fun(A,fun(B,bool)),fun(list(product_prod(A,B)),bool),R3)),map(product_prod(A,B),A,product_fst(A,B))))),bNF_Grp(list(product_prod(A,B)),list(B),aa(fun(list(product_prod(A,B)),bool),set(list(product_prod(A,B))),collect(list(product_prod(A,B))),aTP_Lamp_pi(fun(A,fun(B,bool)),fun(list(product_prod(A,B)),bool),R3)),map(product_prod(A,B),B,product_snd(A,B)))) ).
% list.rel_compp_Grp
tff(fact_1794_Pow__set_I2_J,axiom,
! [B: $tType,X2: B,Xs: list(B)] : pow(B,aa(list(B),set(B),set3(B),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X2),Xs))) = aa(set(set(B)),set(set(B)),aa(set(set(B)),fun(set(set(B)),set(set(B))),sup_sup(set(set(B))),pow(B,aa(list(B),set(B),set3(B),Xs))),aa(set(set(B)),set(set(B)),aa(fun(set(B),set(B)),fun(set(set(B)),set(set(B))),image2(set(B),set(B)),insert(B,X2)),pow(B,aa(list(B),set(B),set3(B),Xs)))) ).
% Pow_set(2)
tff(fact_1795_zip__Cons__Cons,axiom,
! [A: $tType,B: $tType,X2: A,Xs: list(A),Y2: B,Ys: list(B)] : zip(A,B,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Ys)) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y2)),zip(A,B,Xs,Ys)) ).
% zip_Cons_Cons
tff(fact_1796_lexord__cons__cons,axiom,
! [A: $tType,A3: A,X2: list(A),B2: A,Y2: list(A),R: set(product_prod(A,A))] :
( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),X2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B2),Y2))),lexord(A,R)))
<=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R))
| ( ( A3 = B2 )
& pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X2),Y2)),lexord(A,R))) ) ) ) ).
% lexord_cons_cons
tff(fact_1797_Cons__in__lex,axiom,
! [A: $tType,X2: A,Xs: list(A),Y2: A,Ys: list(A),R: set(product_prod(A,A))] :
( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys))),lex(A,R)))
<=> ( ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),R))
& ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) ) )
| ( ( X2 = Y2 )
& pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),lex(A,R))) ) ) ) ).
% Cons_in_lex
tff(fact_1798_set__concat,axiom,
! [A: $tType,Xs: list(list(A))] : aa(list(A),set(A),set3(A),concat(A,Xs)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(list(A)),set(set(A)),aa(fun(list(A),set(A)),fun(set(list(A)),set(set(A))),image2(list(A),set(A)),set3(A)),aa(list(list(A)),set(list(A)),set3(list(A)),Xs))) ).
% set_concat
tff(fact_1799_inj__split__Cons,axiom,
! [A: $tType,X6: set(product_prod(list(A),A))] : inj_on(product_prod(list(A),A),list(A),aa(fun(list(A),fun(A,list(A))),fun(product_prod(list(A),A),list(A)),product_case_prod(list(A),A,list(A)),aTP_Lamp_pj(list(A),fun(A,list(A)))),X6) ).
% inj_split_Cons
tff(fact_1800_zip__eq__ConsE,axiom,
! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),Xy: product_prod(A,B),Xys: list(product_prod(A,B))] :
( ( zip(A,B,Xs,Ys) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),Xy),Xys) )
=> ~ ! [X: A,Xs2: list(A)] :
( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs2) )
=> ! [Y: B,Ys2: list(B)] :
( ( Ys = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys2) )
=> ( ( Xy = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y) )
=> ( Xys != zip(A,B,Xs2,Ys2) ) ) ) ) ) ).
% zip_eq_ConsE
tff(fact_1801_list_Orel__compp,axiom,
! [A: $tType,C: $tType,B: $tType,R3: fun(A,fun(B,bool)),S3: fun(B,fun(C,bool))] : list_all2(A,C,aa(fun(B,fun(C,bool)),fun(A,fun(C,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(C,bool)),fun(A,fun(C,bool))),relcompp(A,B,C),R3),S3)) = aa(fun(list(B),fun(list(C),bool)),fun(list(A),fun(list(C),bool)),aa(fun(list(A),fun(list(B),bool)),fun(fun(list(B),fun(list(C),bool)),fun(list(A),fun(list(C),bool))),relcompp(list(A),list(B),list(C)),list_all2(A,B,R3)),list_all2(B,C,S3)) ).
% list.rel_compp
tff(fact_1802_concat__injective,axiom,
! [A: $tType,Xs: list(list(A)),Ys: list(list(A))] :
( ( concat(A,Xs) = concat(A,Ys) )
=> ( ( aa(list(list(A)),nat,size_size(list(list(A))),Xs) = aa(list(list(A)),nat,size_size(list(list(A))),Ys) )
=> ( ! [X: product_prod(list(A),list(A))] :
( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),X),aa(list(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),set3(product_prod(list(A),list(A))),zip(list(A),list(A),Xs,Ys))))
=> pp(aa(product_prod(list(A),list(A)),bool,aa(fun(list(A),fun(list(A),bool)),fun(product_prod(list(A),list(A)),bool),product_case_prod(list(A),list(A),bool),aTP_Lamp_pk(list(A),fun(list(A),bool))),X)) )
=> ( Xs = Ys ) ) ) ) ).
% concat_injective
tff(fact_1803_concat__eq__concat__iff,axiom,
! [A: $tType,Xs: list(list(A)),Ys: list(list(A))] :
( ! [X: product_prod(list(A),list(A))] :
( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),X),aa(list(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),set3(product_prod(list(A),list(A))),zip(list(A),list(A),Xs,Ys))))
=> pp(aa(product_prod(list(A),list(A)),bool,aa(fun(list(A),fun(list(A),bool)),fun(product_prod(list(A),list(A)),bool),product_case_prod(list(A),list(A),bool),aTP_Lamp_pk(list(A),fun(list(A),bool))),X)) )
=> ( ( aa(list(list(A)),nat,size_size(list(list(A))),Xs) = aa(list(list(A)),nat,size_size(list(list(A))),Ys) )
=> ( ( concat(A,Xs) = concat(A,Ys) )
<=> ( Xs = Ys ) ) ) ) ).
% concat_eq_concat_iff
tff(fact_1804_length__Cons,axiom,
! [A: $tType,X2: A,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs)) ).
% length_Cons
tff(fact_1805_fold__Cons,axiom,
! [B: $tType,A: $tType,F: fun(A,fun(B,B)),X2: A,Xs: list(A)] : fold(A,B,F,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)) = aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),fold(A,B,F,Xs)),aa(A,fun(B,B),F,X2)) ).
% fold_Cons
tff(fact_1806_lexord__linear,axiom,
! [A: $tType,R: set(product_prod(A,A)),X2: list(A),Y2: list(A)] :
( ! [A5: A,B4: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A5),B4)),R))
| ( A5 = B4 )
| pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B4),A5)),R)) )
=> ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X2),Y2)),lexord(A,R)))
| ( X2 = Y2 )
| pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Y2),X2)),lexord(A,R))) ) ) ).
% lexord_linear
tff(fact_1807_lexord__irreflexive,axiom,
! [A: $tType,R: set(product_prod(A,A)),Xs: list(A)] :
( ! [X: A] : ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X)),R))
=> ~ pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Xs)),lexord(A,R))) ) ).
% lexord_irreflexive
tff(fact_1808_listrel_OCons,axiom,
! [B: $tType,A: $tType,X2: A,Y2: B,R: set(product_prod(A,B)),Xs: list(A),Ys: list(B)] :
( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y2)),R))
=> ( pp(aa(set(product_prod(list(A),list(B))),bool,member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs),Ys)),listrel(A,B,R)))
=> pp(aa(set(product_prod(list(A),list(B))),bool,member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Ys))),listrel(A,B,R))) ) ) ).
% listrel.Cons
tff(fact_1809_listrel__Cons1,axiom,
! [B: $tType,A: $tType,Y2: A,Ys: list(A),Xs: list(B),R: set(product_prod(A,B))] :
( pp(aa(set(product_prod(list(A),list(B))),bool,member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys)),Xs)),listrel(A,B,R)))
=> ~ ! [Y: B,Ys3: list(B)] :
( ( Xs = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys3) )
=> ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y2),Y)),R))
=> ~ pp(aa(set(product_prod(list(A),list(B))),bool,member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Ys),Ys3)),listrel(A,B,R))) ) ) ) ).
% listrel_Cons1
tff(fact_1810_listrel__Cons2,axiom,
! [B: $tType,A: $tType,Xs: list(A),Y2: B,Ys: list(B),R: set(product_prod(A,B))] :
( pp(aa(set(product_prod(list(A),list(B))),bool,member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Ys))),listrel(A,B,R)))
=> ~ ! [X: A,Xs3: list(A)] :
( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs3) )
=> ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y2)),R))
=> ~ pp(aa(set(product_prod(list(A),list(B))),bool,member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs3),Ys)),listrel(A,B,R))) ) ) ) ).
% listrel_Cons2
tff(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,Z3: A] :
( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set3(A),Xs)))
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R))
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z3)),R))
=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z3)),R)) ) ) )
=> ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),lexord(A,R)))
=> ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys),Zs)),lexord(A,R)))
=> pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Zs)),lexord(A,R))) ) ) ) ).
% lexord_partial_trans
tff(fact_1812_list__all2I,axiom,
! [A: $tType,B: $tType,A3: list(A),B2: list(B),P2: fun(A,fun(B,bool))] :
( ! [X: product_prod(A,B)] :
( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),X),aa(list(product_prod(A,B)),set(product_prod(A,B)),set3(product_prod(A,B)),zip(A,B,A3,B2))))
=> pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),P2),X)) )
=> ( ( aa(list(A),nat,size_size(list(A)),A3) = aa(list(B),nat,size_size(list(B)),B2) )
=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P2),A3),B2)) ) ) ).
% list_all2I
tff(fact_1813_product__concat__map,axiom,
! [B: $tType,A: $tType,Xs: list(A),Ys: list(B)] : product(A,B,Xs,Ys) = concat(product_prod(A,B),aa(list(A),list(list(product_prod(A,B))),map(A,list(product_prod(A,B)),aTP_Lamp_ph(list(B),fun(A,list(product_prod(A,B))),Ys)),Xs)) ).
% product_concat_map
tff(fact_1814_lists__length__Suc__eq,axiom,
! [A: $tType,A6: set(A),N4: nat] : aa(fun(list(A),bool),set(list(A)),collect(list(A)),aa(nat,fun(list(A),bool),aTP_Lamp_pl(set(A),fun(nat,fun(list(A),bool)),A6),N4)) = aa(set(product_prod(list(A),A)),set(list(A)),aa(fun(product_prod(list(A),A),list(A)),fun(set(product_prod(list(A),A)),set(list(A))),image2(product_prod(list(A),A),list(A)),aa(fun(list(A),fun(A,list(A))),fun(product_prod(list(A),A),list(A)),product_case_prod(list(A),A,list(A)),aTP_Lamp_pj(list(A),fun(A,list(A))))),product_Sigma(list(A),A,aa(fun(list(A),bool),set(list(A)),collect(list(A)),aa(nat,fun(list(A),bool),aTP_Lamp_pm(set(A),fun(nat,fun(list(A),bool)),A6),N4)),aTP_Lamp_pn(set(A),fun(list(A),set(A)),A6))) ).
% lists_length_Suc_eq
tff(fact_1815_list__all2__iff,axiom,
! [B: $tType,A: $tType,P2: fun(A,fun(B,bool)),Xs: list(A),Ys: list(B)] :
( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P2),Xs),Ys))
<=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
& ! [X5: product_prod(A,B)] :
( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),X5),aa(list(product_prod(A,B)),set(product_prod(A,B)),set3(product_prod(A,B)),zip(A,B,Xs,Ys))))
=> pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),P2),X5)) ) ) ) ).
% list_all2_iff
tff(fact_1816_subset__subseqs,axiom,
! [A: $tType,X6: set(A),Xs: list(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X6),aa(list(A),set(A),set3(A),Xs)))
=> pp(aa(set(set(A)),bool,member(set(A),X6),aa(set(list(A)),set(set(A)),aa(fun(list(A),set(A)),fun(set(list(A)),set(set(A))),image2(list(A),set(A)),set3(A)),aa(list(list(A)),set(list(A)),set3(list(A)),subseqs(A,Xs))))) ) ).
% subset_subseqs
tff(fact_1817_List_Olexordp__def,axiom,
! [A: $tType,R: fun(A,fun(A,bool)),Xs: list(A),Ys: list(A)] :
( lexordp(A,R,Xs,Ys)
<=> pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),lexord(A,aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),R))))) ) ).
% List.lexordp_def
tff(fact_1818_length__remdups__concat,axiom,
! [A: $tType,Xss: list(list(A))] : aa(list(A),nat,size_size(list(A)),remdups(A,concat(A,Xss))) = aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(list(A)),set(set(A)),aa(fun(list(A),set(A)),fun(set(list(A)),set(set(A))),image2(list(A),set(A)),set3(A)),aa(list(list(A)),set(list(A)),set3(list(A)),Xss)))) ).
% length_remdups_concat
tff(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,aa(list(product_prod(A,C)),set(product_prod(A,C)),set3(product_prod(A,C)),Xys),aa(list(product_prod(C,B)),set(product_prod(C,B)),set3(product_prod(C,B)),Yzs)) = aa(list(product_prod(A,B)),set(product_prod(A,B)),set3(product_prod(A,B)),concat(product_prod(A,B),aa(list(product_prod(A,C)),list(list(product_prod(A,B))),map(product_prod(A,C),list(product_prod(A,B)),aTP_Lamp_pp(list(product_prod(C,B)),fun(product_prod(A,C),list(product_prod(A,B))),Yzs)),Xys))) ).
% set_relcomp
tff(fact_1820_shift__def,axiom,
! [B: $tType,A: $tType,Lab: fun(list(A),B),K: A,X4: list(A)] : bNF_Greatest_shift(A,B,Lab,K,X4) = aa(list(A),B,Lab,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),K),X4)) ).
% shift_def
tff(fact_1821_Cons__lenlex__iff,axiom,
! [A: $tType,M4: A,Ms: list(A),N4: A,Ns: list(A),R: set(product_prod(A,A))] :
( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),M4),Ms)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),N4),Ns))),lenlex(A,R)))
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Ms)),aa(list(A),nat,size_size(list(A)),Ns)))
| ( ( aa(list(A),nat,size_size(list(A)),Ms) = aa(list(A),nat,size_size(list(A)),Ns) )
& pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),M4),N4)),R)) )
| ( ( M4 = N4 )
& pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ms),Ns)),lenlex(A,R))) ) ) ) ).
% Cons_lenlex_iff
tff(fact_1822_lexn_Osimps_I2_J,axiom,
! [A: $tType,R: set(product_prod(A,A)),N4: nat] : aa(nat,set(product_prod(list(A),list(A))),lexn(A,R),aa(nat,nat,suc,N4)) = aa(set(product_prod(list(A),list(A))),set(product_prod(list(A),list(A))),aa(set(product_prod(list(A),list(A))),fun(set(product_prod(list(A),list(A))),set(product_prod(list(A),list(A)))),inf_inf(set(product_prod(list(A),list(A)))),aa(set(product_prod(product_prod(A,list(A)),product_prod(A,list(A)))),set(product_prod(list(A),list(A))),aa(fun(product_prod(product_prod(A,list(A)),product_prod(A,list(A))),product_prod(list(A),list(A))),fun(set(product_prod(product_prod(A,list(A)),product_prod(A,list(A)))),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),aa(fun(A,fun(list(A),list(A))),fun(product_prod(A,list(A)),list(A)),product_case_prod(A,list(A),list(A)),cons(A)),aa(fun(A,fun(list(A),list(A))),fun(product_prod(A,list(A)),list(A)),product_case_prod(A,list(A),list(A)),cons(A)))),lex_prod(A,list(A),R,aa(nat,set(product_prod(list(A),list(A))),lexn(A,R),N4)))),aa(fun(product_prod(list(A),list(A)),bool),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),bool)),fun(product_prod(list(A),list(A)),bool),product_case_prod(list(A),list(A),bool),aTP_Lamp_pq(nat,fun(list(A),fun(list(A),bool)),N4)))) ).
% lexn.simps(2)
tff(fact_1823_lenlex__conv,axiom,
! [A: $tType,R: set(product_prod(A,A))] : lenlex(A,R) = aa(fun(product_prod(list(A),list(A)),bool),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),bool)),fun(product_prod(list(A),list(A)),bool),product_case_prod(list(A),list(A),bool),aTP_Lamp_pr(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),R))) ).
% lenlex_conv
tff(fact_1824_lenlex__irreflexive,axiom,
! [A: $tType,R: set(product_prod(A,A)),Xs: list(A)] :
( ! [X: A] : ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X)),R))
=> ~ pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Xs)),lenlex(A,R))) ) ).
% lenlex_irreflexive
tff(fact_1825_listrel_Osimps,axiom,
! [B: $tType,A: $tType,A1: list(A),A22: list(B),R: set(product_prod(A,B))] :
( pp(aa(set(product_prod(list(A),list(B))),bool,member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),A1),A22)),listrel(A,B,R)))
<=> ( ( ( A1 = nil(A) )
& ( A22 = nil(B) ) )
| ? [X5: A,Y4: B,Xs4: list(A),Ys4: list(B)] :
( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X5),Xs4) )
& ( A22 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y4),Ys4) )
& pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X5),Y4)),R))
& pp(aa(set(product_prod(list(A),list(B))),bool,member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs4),Ys4)),listrel(A,B,R))) ) ) ) ).
% listrel.simps
tff(fact_1826_listrel_Ocases,axiom,
! [B: $tType,A: $tType,A1: list(A),A22: list(B),R: set(product_prod(A,B))] :
( pp(aa(set(product_prod(list(A),list(B))),bool,member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),A1),A22)),listrel(A,B,R)))
=> ( ( ( A1 = nil(A) )
=> ( A22 != nil(B) ) )
=> ~ ! [X: A,Y: B,Xs3: list(A)] :
( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs3) )
=> ! [Ys3: list(B)] :
( ( A22 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys3) )
=> ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y)),R))
=> ~ pp(aa(set(product_prod(list(A),list(B))),bool,member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xs3),Ys3)),listrel(A,B,R))) ) ) ) ) ) ).
% listrel.cases
tff(fact_1827_lex__def,axiom,
! [A: $tType,R: set(product_prod(A,A))] : lex(A,R) = aa(set(set(product_prod(list(A),list(A)))),set(product_prod(list(A),list(A))),complete_Sup_Sup(set(product_prod(list(A),list(A)))),aa(set(nat),set(set(product_prod(list(A),list(A)))),aa(fun(nat,set(product_prod(list(A),list(A)))),fun(set(nat),set(set(product_prod(list(A),list(A))))),image2(nat,set(product_prod(list(A),list(A)))),lexn(A,R)),top_top(set(nat)))) ).
% lex_def
tff(fact_1828_set__Cons__sing__Nil,axiom,
! [A: $tType,A6: set(A)] : set_Cons(A,A6,aa(set(list(A)),set(list(A)),insert(list(A),nil(A)),bot_bot(set(list(A))))) = aa(set(A),set(list(A)),aa(fun(A,list(A)),fun(set(A),set(list(A))),image2(A,list(A)),aTP_Lamp_ps(A,list(A))),A6) ).
% set_Cons_sing_Nil
tff(fact_1829_zip__Cons1,axiom,
! [A: $tType,B: $tType,X2: A,Xs: list(A),Ys: list(B)] : zip(A,B,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs),Ys) = case_list(list(product_prod(A,B)),B,nil(product_prod(A,B)),aa(list(A),fun(B,fun(list(B),list(product_prod(A,B)))),aTP_Lamp_pt(A,fun(list(A),fun(B,fun(list(B),list(product_prod(A,B))))),X2),Xs),Ys) ).
% zip_Cons1
tff(fact_1830_zip__Cons,axiom,
! [B: $tType,A: $tType,Xs: list(A),Y2: B,Ys: list(B)] : zip(A,B,Xs,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y2),Ys)) = case_list(list(product_prod(A,B)),A,nil(product_prod(A,B)),aa(list(B),fun(A,fun(list(A),list(product_prod(A,B)))),aTP_Lamp_pu(B,fun(list(B),fun(A,fun(list(A),list(product_prod(A,B))))),Y2),Ys),Xs) ).
% zip_Cons
tff(fact_1831_f__arg__min__list__f,axiom,
! [B: $tType,A: $tType] :
( linorder(B)
=> ! [Xs: list(A),F: fun(A,B)] :
( ( Xs != nil(A) )
=> ( aa(A,B,F,arg_min_list(A,B,F,Xs)) = lattic643756798350308766er_Min(B,aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),aa(list(A),set(A),set3(A),Xs))) ) ) ) ).
% f_arg_min_list_f
tff(fact_1832_lexn__conv,axiom,
! [A: $tType,R: set(product_prod(A,A)),N4: nat] : aa(nat,set(product_prod(list(A),list(A))),lexn(A,R),N4) = aa(fun(product_prod(list(A),list(A)),bool),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),bool)),fun(product_prod(list(A),list(A)),bool),product_case_prod(list(A),list(A),bool),aa(nat,fun(list(A),fun(list(A),bool)),aTP_Lamp_pv(set(product_prod(A,A)),fun(nat,fun(list(A),fun(list(A),bool))),R),N4))) ).
% lexn_conv
tff(fact_1833_lex__conv,axiom,
! [A: $tType,R: set(product_prod(A,A))] : lex(A,R) = aa(fun(product_prod(list(A),list(A)),bool),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),bool)),fun(product_prod(list(A),list(A)),bool),product_case_prod(list(A),list(A),bool),aTP_Lamp_pw(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),R))) ).
% lex_conv
tff(fact_1834_listrel__def,axiom,
! [B: $tType,A: $tType,X4: set(product_prod(A,B))] : listrel(A,B,X4) = aa(fun(product_prod(list(A),list(B)),bool),set(product_prod(list(A),list(B))),collect(product_prod(list(A),list(B))),aa(fun(list(A),fun(list(B),bool)),fun(product_prod(list(A),list(B)),bool),product_case_prod(list(A),list(B),bool),listrelp(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_bg(set(product_prod(A,B)),fun(A,fun(B,bool))),X4)))) ).
% listrel_def
tff(fact_1835_fold__append,axiom,
! [A: $tType,B: $tType,F: fun(B,fun(A,A)),Xs: list(B),Ys: list(B)] : fold(B,A,F,append(B,Xs,Ys)) = aa(fun(A,A),fun(A,A),aa(fun(A,A),fun(fun(A,A),fun(A,A)),comp(A,A,A),fold(B,A,F,Ys)),fold(B,A,F,Xs)) ).
% fold_append
tff(fact_1836_lexord__append__leftD,axiom,
! [A: $tType,X2: list(A),U: list(A),V: list(A),R: set(product_prod(A,A))] :
( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),append(A,X2,U)),append(A,X2,V))),lexord(A,R)))
=> ( ! [A5: A] : ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A5),A5)),R))
=> pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),U),V)),lexord(A,R))) ) ) ).
% lexord_append_leftD
tff(fact_1837_lex__append__leftD,axiom,
! [A: $tType,R: set(product_prod(A,A)),Xs: list(A),Ys: list(A),Zs: list(A)] :
( ! [X: A] : ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X)),R))
=> ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),append(A,Xs,Ys)),append(A,Xs,Zs))),lex(A,R)))
=> pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys),Zs)),lex(A,R))) ) ) ).
% lex_append_leftD
tff(fact_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] : ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X)),R))
=> ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),append(A,Xs,Ys)),append(A,Xs,Zs))),lex(A,R)))
<=> pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys),Zs)),lex(A,R))) ) ) ).
% lex_append_left_iff
tff(fact_1839_length__append__singleton,axiom,
! [A: $tType,Xs: list(A),X2: A] : aa(list(A),nat,size_size(list(A)),append(A,Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A)))) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs)) ).
% length_append_singleton
tff(fact_1840_lexord__append__left__rightI,axiom,
! [A: $tType,A3: A,B2: A,R: set(product_prod(A,A)),U: list(A),X2: list(A),Y2: list(A)] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R))
=> pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),append(A,U,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A3),X2))),append(A,U,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B2),Y2)))),lexord(A,R))) ) ).
% lexord_append_left_rightI
tff(fact_1841_lexord__same__pref__iff,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A),R: set(product_prod(A,A))] :
( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),append(A,Xs,Ys)),append(A,Xs,Zs))),lexord(A,R)))
<=> ( ? [X5: A] :
( pp(aa(set(A),bool,member(A,X5),aa(list(A),set(A),set3(A),Xs)))
& pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),X5)),R)) )
| pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys),Zs)),lexord(A,R))) ) ) ).
% lexord_same_pref_iff
tff(fact_1842_product_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,X2: A,Xs: list(A),Ys: list(B)] : product(A,B,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs),Ys) = append(product_prod(A,B),aa(list(B),list(product_prod(A,B)),map(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2)),Ys),product(A,B,Xs,Ys)) ).
% product.simps(2)
tff(fact_1843_lexord__def,axiom,
! [A: $tType,R: set(product_prod(A,A))] : lexord(A,R) = aa(fun(product_prod(list(A),list(A)),bool),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),bool)),fun(product_prod(list(A),list(A)),bool),product_case_prod(list(A),list(A),bool),aTP_Lamp_px(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),R))) ).
% lexord_def
tff(fact_1844_listrelp__listrel__eq,axiom,
! [B: $tType,A: $tType,R: set(product_prod(A,B)),X4: list(A),Xa4: list(B)] :
( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),listrelp(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_bg(set(product_prod(A,B)),fun(A,fun(B,bool))),R)),X4),Xa4))
<=> pp(aa(set(product_prod(list(A),list(B))),bool,member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),X4),Xa4)),listrel(A,B,R))) ) ).
% listrelp_listrel_eq
tff(fact_1845_Succ__def,axiom,
! [A: $tType,Kl: set(list(A)),Kl2: list(A)] : bNF_Greatest_Succ(A,Kl,Kl2) = aa(fun(A,bool),set(A),collect(A),aa(list(A),fun(A,bool),aTP_Lamp_py(set(list(A)),fun(list(A),fun(A,bool)),Kl),Kl2)) ).
% Succ_def
tff(fact_1846_listrel1__def,axiom,
! [A: $tType,R: set(product_prod(A,A))] : listrel1(A,R) = aa(fun(product_prod(list(A),list(A)),bool),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),bool)),fun(product_prod(list(A),list(A)),bool),product_case_prod(list(A),list(A),bool),aTP_Lamp_pz(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),R))) ).
% listrel1_def
tff(fact_1847_SuccI,axiom,
! [A: $tType,Kl2: list(A),K: A,Kl: set(list(A))] :
( pp(aa(set(list(A)),bool,member(list(A),append(A,Kl2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),K),nil(A)))),Kl))
=> pp(aa(set(A),bool,member(A,K),bNF_Greatest_Succ(A,Kl,Kl2))) ) ).
% SuccI
tff(fact_1848_SuccD,axiom,
! [A: $tType,K: A,Kl: set(list(A)),Kl2: list(A)] :
( pp(aa(set(A),bool,member(A,K),bNF_Greatest_Succ(A,Kl,Kl2)))
=> pp(aa(set(list(A)),bool,member(list(A),append(A,Kl2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),K),nil(A)))),Kl)) ) ).
% SuccD
tff(fact_1849_prod_Orel__compp__Grp,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,R1: fun(A,fun(C,bool)),R22: fun(B,fun(D,bool))] : basic_rel_prod(A,C,B,D,R1,R22) = aa(fun(product_prod(product_prod(A,C),product_prod(B,D)),fun(product_prod(C,D),bool)),fun(product_prod(A,B),fun(product_prod(C,D),bool)),aa(fun(product_prod(A,B),fun(product_prod(product_prod(A,C),product_prod(B,D)),bool)),fun(fun(product_prod(product_prod(A,C),product_prod(B,D)),fun(product_prod(C,D),bool)),fun(product_prod(A,B),fun(product_prod(C,D),bool))),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),aa(fun(product_prod(product_prod(A,C),product_prod(B,D)),bool),set(product_prod(product_prod(A,C),product_prod(B,D))),collect(product_prod(product_prod(A,C),product_prod(B,D))),aa(fun(B,fun(D,bool)),fun(product_prod(product_prod(A,C),product_prod(B,D)),bool),aTP_Lamp_qa(fun(A,fun(C,bool)),fun(fun(B,fun(D,bool)),fun(product_prod(product_prod(A,C),product_prod(B,D)),bool)),R1),R22)),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),aa(fun(product_prod(product_prod(A,C),product_prod(B,D)),bool),set(product_prod(product_prod(A,C),product_prod(B,D))),collect(product_prod(product_prod(A,C),product_prod(B,D))),aa(fun(B,fun(D,bool)),fun(product_prod(product_prod(A,C),product_prod(B,D)),bool),aTP_Lamp_qa(fun(A,fun(C,bool)),fun(fun(B,fun(D,bool)),fun(product_prod(product_prod(A,C),product_prod(B,D)),bool)),R1),R22)),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
tff(fact_1850_prod_Oin__rel,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,R1: fun(A,fun(C,bool)),R22: fun(B,fun(D,bool)),A3: product_prod(A,B),B2: product_prod(C,D)] :
( pp(aa(product_prod(C,D),bool,aa(product_prod(A,B),fun(product_prod(C,D),bool),basic_rel_prod(A,C,B,D,R1,R22),A3),B2))
<=> ? [Z4: product_prod(product_prod(A,C),product_prod(B,D))] :
( pp(aa(set(product_prod(product_prod(A,C),product_prod(B,D))),bool,member(product_prod(product_prod(A,C),product_prod(B,D)),Z4),aa(fun(product_prod(product_prod(A,C),product_prod(B,D)),bool),set(product_prod(product_prod(A,C),product_prod(B,D))),collect(product_prod(product_prod(A,C),product_prod(B,D))),aa(fun(B,fun(D,bool)),fun(product_prod(product_prod(A,C),product_prod(B,D)),bool),aTP_Lamp_qa(fun(A,fun(C,bool)),fun(fun(B,fun(D,bool)),fun(product_prod(product_prod(A,C),product_prod(B,D)),bool)),R1),R22))))
& ( aa(product_prod(product_prod(A,C),product_prod(B,D)),product_prod(A,B),product_map_prod(product_prod(A,C),A,product_prod(B,D),B,product_fst(A,C),product_fst(B,D)),Z4) = A3 )
& ( aa(product_prod(product_prod(A,C),product_prod(B,D)),product_prod(C,D),product_map_prod(product_prod(A,C),C,product_prod(B,D),D,product_snd(A,C),product_snd(B,D)),Z4) = B2 ) ) ) ).
% prod.in_rel
tff(fact_1851_in__measures_I2_J,axiom,
! [A: $tType,X2: A,Y2: A,F: fun(A,nat),Fs: list(fun(A,nat))] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),measures(A,aa(list(fun(A,nat)),list(fun(A,nat)),aa(fun(A,nat),fun(list(fun(A,nat)),list(fun(A,nat))),cons(fun(A,nat)),F),Fs))))
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F,X2)),aa(A,nat,F,Y2)))
| ( ( aa(A,nat,F,X2) = aa(A,nat,F,Y2) )
& pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),measures(A,Fs))) ) ) ) ).
% in_measures(2)
tff(fact_1852_rel__prod__inject,axiom,
! [B: $tType,A: $tType,C: $tType,D: $tType,R1: fun(A,fun(B,bool)),R22: fun(C,fun(D,bool)),A3: A,B2: C,C3: B,D4: D] :
( pp(aa(product_prod(B,D),bool,aa(product_prod(A,C),fun(product_prod(B,D),bool),basic_rel_prod(A,B,C,D,R1,R22),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A3),B2)),aa(D,product_prod(B,D),aa(B,fun(D,product_prod(B,D)),product_Pair(B,D),C3),D4)))
<=> ( pp(aa(B,bool,aa(A,fun(B,bool),R1,A3),C3))
& pp(aa(D,bool,aa(C,fun(D,bool),R22,B2),D4)) ) ) ).
% rel_prod_inject
tff(fact_1853_in__measures_I1_J,axiom,
! [A: $tType,X2: A,Y2: A] : ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),measures(A,nil(fun(A,nat))))) ).
% in_measures(1)
tff(fact_1854_Cons__listrel1__Cons,axiom,
! [A: $tType,X2: A,Xs: list(A),Y2: A,Ys: list(A),R: set(product_prod(A,A))] :
( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys))),listrel1(A,R)))
<=> ( ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),R))
& ( Xs = Ys ) )
| ( ( X2 = Y2 )
& pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),listrel1(A,R))) ) ) ) ).
% Cons_listrel1_Cons
tff(fact_1855_rel__prod_Ocases,axiom,
! [B: $tType,A: $tType,C: $tType,D: $tType,R1: fun(A,fun(B,bool)),R22: fun(C,fun(D,bool)),A1: product_prod(A,C),A22: product_prod(B,D)] :
( pp(aa(product_prod(B,D),bool,aa(product_prod(A,C),fun(product_prod(B,D),bool),basic_rel_prod(A,B,C,D,R1,R22),A1),A22))
=> ~ ! [A5: A,B4: B,C2: C] :
( ( A1 = aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A5),C2) )
=> ! [D2: D] :
( ( A22 = aa(D,product_prod(B,D),aa(B,fun(D,product_prod(B,D)),product_Pair(B,D),B4),D2) )
=> ( pp(aa(B,bool,aa(A,fun(B,bool),R1,A5),B4))
=> ~ pp(aa(D,bool,aa(C,fun(D,bool),R22,C2),D2)) ) ) ) ) ).
% rel_prod.cases
tff(fact_1856_rel__prod_Osimps,axiom,
! [B: $tType,A: $tType,C: $tType,D: $tType,R1: fun(A,fun(B,bool)),R22: fun(C,fun(D,bool)),A1: product_prod(A,C),A22: product_prod(B,D)] :
( pp(aa(product_prod(B,D),bool,aa(product_prod(A,C),fun(product_prod(B,D),bool),basic_rel_prod(A,B,C,D,R1,R22),A1),A22))
<=> ? [A7: A,B5: B,C6: C,D5: D] :
( ( A1 = aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A7),C6) )
& ( A22 = aa(D,product_prod(B,D),aa(B,fun(D,product_prod(B,D)),product_Pair(B,D),B5),D5) )
& pp(aa(B,bool,aa(A,fun(B,bool),R1,A7),B5))
& pp(aa(D,bool,aa(C,fun(D,bool),R22,C6),D5)) ) ) ).
% rel_prod.simps
tff(fact_1857_rel__prod_Ointros,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,R1: fun(A,fun(B,bool)),A3: A,B2: B,R22: fun(C,fun(D,bool)),C3: C,D4: D] :
( pp(aa(B,bool,aa(A,fun(B,bool),R1,A3),B2))
=> ( pp(aa(D,bool,aa(C,fun(D,bool),R22,C3),D4))
=> pp(aa(product_prod(B,D),bool,aa(product_prod(A,C),fun(product_prod(B,D),bool),basic_rel_prod(A,B,C,D,R1,R22),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),A3),C3)),aa(D,product_prod(B,D),aa(B,fun(D,product_prod(B,D)),product_Pair(B,D),B2),D4))) ) ) ).
% rel_prod.intros
tff(fact_1858_Pair__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A6: fun(A,fun(B,bool)),B6: fun(C,fun(D,bool))] : pp(aa(fun(B,fun(D,product_prod(B,D))),bool,aa(fun(A,fun(C,product_prod(A,C))),fun(fun(B,fun(D,product_prod(B,D))),bool),bNF_rel_fun(A,B,fun(C,product_prod(A,C)),fun(D,product_prod(B,D)),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
tff(fact_1859_fst__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,A6: fun(A,fun(C,bool)),B6: fun(B,fun(D,bool))] : pp(aa(fun(product_prod(C,D),C),bool,aa(fun(product_prod(A,B),A),fun(fun(product_prod(C,D),C),bool),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
tff(fact_1860_snd__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,A6: fun(A,fun(C,bool)),B6: fun(B,fun(D,bool))] : pp(aa(fun(product_prod(C,D),D),bool,aa(fun(product_prod(A,B),B),fun(fun(product_prod(C,D),D),bool),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
tff(fact_1861_prod_Orel__compp,axiom,
! [B: $tType,A: $tType,C: $tType,E: $tType,F2: $tType,D: $tType,R1: fun(A,fun(C,bool)),S1: fun(C,fun(E,bool)),R22: fun(B,fun(D,bool)),S22: fun(D,fun(F2,bool))] : basic_rel_prod(A,E,B,F2,aa(fun(C,fun(E,bool)),fun(A,fun(E,bool)),aa(fun(A,fun(C,bool)),fun(fun(C,fun(E,bool)),fun(A,fun(E,bool))),relcompp(A,C,E),R1),S1),aa(fun(D,fun(F2,bool)),fun(B,fun(F2,bool)),aa(fun(B,fun(D,bool)),fun(fun(D,fun(F2,bool)),fun(B,fun(F2,bool))),relcompp(B,D,F2),R22),S22)) = aa(fun(product_prod(C,D),fun(product_prod(E,F2),bool)),fun(product_prod(A,B),fun(product_prod(E,F2),bool)),aa(fun(product_prod(A,B),fun(product_prod(C,D),bool)),fun(fun(product_prod(C,D),fun(product_prod(E,F2),bool)),fun(product_prod(A,B),fun(product_prod(E,F2),bool))),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
tff(fact_1862_rel__prod__sel,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,R1: fun(A,fun(B,bool)),R22: fun(C,fun(D,bool)),P: product_prod(A,C),Q: product_prod(B,D)] :
( pp(aa(product_prod(B,D),bool,aa(product_prod(A,C),fun(product_prod(B,D),bool),basic_rel_prod(A,B,C,D,R1,R22),P),Q))
<=> ( pp(aa(B,bool,aa(A,fun(B,bool),R1,aa(product_prod(A,C),A,product_fst(A,C),P)),aa(product_prod(B,D),B,product_fst(B,D),Q)))
& pp(aa(D,bool,aa(C,fun(D,bool),R22,aa(product_prod(A,C),C,product_snd(A,C),P)),aa(product_prod(B,D),D,product_snd(B,D),Q))) ) ) ).
% rel_prod_sel
tff(fact_1863_rel__prod__conv,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,R1: fun(A,fun(C,bool)),R22: fun(B,fun(D,bool))] : basic_rel_prod(A,C,B,D,R1,R22) = aa(fun(A,fun(B,fun(product_prod(C,D),bool))),fun(product_prod(A,B),fun(product_prod(C,D),bool)),product_case_prod(A,B,fun(product_prod(C,D),bool)),aa(fun(B,fun(D,bool)),fun(A,fun(B,fun(product_prod(C,D),bool))),aTP_Lamp_qc(fun(A,fun(C,bool)),fun(fun(B,fun(D,bool)),fun(A,fun(B,fun(product_prod(C,D),bool)))),R1),R22)) ).
% rel_prod_conv
tff(fact_1864_case__prod__transfer,axiom,
! [A: $tType,B: $tType,C: $tType,F2: $tType,E: $tType,D: $tType,A6: fun(A,fun(D,bool)),B6: fun(B,fun(E,bool)),C5: fun(C,fun(F2,bool))] : pp(aa(fun(fun(D,fun(E,F2)),fun(product_prod(D,E),F2)),bool,aa(fun(fun(A,fun(B,C)),fun(product_prod(A,B),C)),fun(fun(fun(D,fun(E,F2)),fun(product_prod(D,E),F2)),bool),bNF_rel_fun(fun(A,fun(B,C)),fun(D,fun(E,F2)),fun(product_prod(A,B),C),fun(product_prod(D,E),F2),bNF_rel_fun(A,D,fun(B,C),fun(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
tff(fact_1865_curry__transfer,axiom,
! [A: $tType,B: $tType,C: $tType,F2: $tType,E: $tType,D: $tType,A6: fun(A,fun(D,bool)),B6: fun(B,fun(E,bool)),C5: fun(C,fun(F2,bool))] : pp(aa(fun(fun(product_prod(D,E),F2),fun(D,fun(E,F2))),bool,aa(fun(fun(product_prod(A,B),C),fun(A,fun(B,C))),fun(fun(fun(product_prod(D,E),F2),fun(D,fun(E,F2))),bool),bNF_rel_fun(fun(product_prod(A,B),C),fun(product_prod(D,E),F2),fun(A,fun(B,C)),fun(D,fun(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,fun(B,C),fun(E,F2),A6,bNF_rel_fun(B,E,C,F2,B6,C5))),product_curry(A,B,C)),product_curry(D,E,F2))) ).
% curry_transfer
tff(fact_1866_Cons__listrel1E2,axiom,
! [A: $tType,Xs: list(A),Y2: A,Ys: list(A),R: set(product_prod(A,A))] :
( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys))),listrel1(A,R)))
=> ( ! [X: A] :
( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys) )
=> ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y2)),R)) )
=> ~ ! [Zs2: list(A)] :
( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Zs2) )
=> ~ pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Zs2),Ys)),listrel1(A,R))) ) ) ) ).
% Cons_listrel1E2
tff(fact_1867_Cons__listrel1E1,axiom,
! [A: $tType,X2: A,Xs: list(A),Ys: list(A),R: set(product_prod(A,A))] :
( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)),Ys)),listrel1(A,R)))
=> ( ! [Y: A] :
( ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xs) )
=> ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y)),R)) )
=> ~ ! [Zs2: list(A)] :
( ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Zs2) )
=> ~ pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Zs2)),listrel1(A,R))) ) ) ) ).
% Cons_listrel1E1
tff(fact_1868_listrel1I1,axiom,
! [A: $tType,X2: A,Y2: A,R: set(product_prod(A,A)),Xs: list(A)] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),R))
=> pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Xs))),listrel1(A,R))) ) ).
% listrel1I1
tff(fact_1869_measures__less,axiom,
! [A: $tType,F: fun(A,nat),X2: A,Y2: A,Fs: list(fun(A,nat))] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F,X2)),aa(A,nat,F,Y2)))
=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),measures(A,aa(list(fun(A,nat)),list(fun(A,nat)),aa(fun(A,nat),fun(list(fun(A,nat)),list(fun(A,nat))),cons(fun(A,nat)),F),Fs)))) ) ).
% measures_less
tff(fact_1870_measures__lesseq,axiom,
! [A: $tType,F: fun(A,nat),X2: A,Y2: A,Fs: list(fun(A,nat))] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,F,X2)),aa(A,nat,F,Y2)))
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),measures(A,Fs)))
=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),measures(A,aa(list(fun(A,nat)),list(fun(A,nat)),aa(fun(A,nat),fun(list(fun(A,nat)),list(fun(A,nat))),cons(fun(A,nat)),F),Fs)))) ) ) ).
% measures_lesseq
tff(fact_1871_listrel1E,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),R: set(product_prod(A,A))] :
( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),listrel1(A,R)))
=> ~ ! [X: A,Y: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R))
=> ! [Us: list(A),Vs: list(A)] :
( ( Xs = append(A,Us,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Vs)) )
=> ( Ys != append(A,Us,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Vs)) ) ) ) ) ).
% listrel1E
tff(fact_1872_listrel1I,axiom,
! [A: $tType,X2: A,Y2: A,R: set(product_prod(A,A)),Xs: list(A),Us2: list(A),Vs2: list(A),Ys: list(A)] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),R))
=> ( ( Xs = append(A,Us2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Vs2)) )
=> ( ( Ys = append(A,Us2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Vs2)) )
=> pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),listrel1(A,R))) ) ) ) ).
% listrel1I
tff(fact_1873_rtrancl__listrel1__ConsI2,axiom,
! [A: $tType,X2: A,Y2: A,R: set(product_prod(A,A)),Xs: list(A),Ys: list(A)] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),transitive_rtrancl(A,R)))
=> ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),transitive_rtrancl(list(A),listrel1(A,R))))
=> pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys))),transitive_rtrancl(list(A),listrel1(A,R)))) ) ) ).
% rtrancl_listrel1_ConsI2
tff(fact_1874_snoc__listrel1__snoc__iff,axiom,
! [A: $tType,Xs: list(A),X2: A,Ys: list(A),Y2: A,R: set(product_prod(A,A))] :
( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),append(A,Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A)))),append(A,Ys,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),nil(A))))),listrel1(A,R)))
<=> ( ( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),listrel1(A,R)))
& ( X2 = Y2 ) )
| ( ( Xs = Ys )
& pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),R)) ) ) ) ).
% snoc_listrel1_snoc_iff
tff(fact_1875_listrel1p__def,axiom,
! [A: $tType,R: fun(A,fun(A,bool)),Xs: list(A),Ys: list(A)] :
( listrel1p(A,R,Xs,Ys)
<=> pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),listrel1(A,aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),R))))) ) ).
% listrel1p_def
tff(fact_1876_empty__Shift,axiom,
! [A: $tType,Kl: set(list(A)),K: A] :
( pp(aa(set(list(A)),bool,member(list(A),nil(A)),Kl))
=> ( pp(aa(set(A),bool,member(A,K),bNF_Greatest_Succ(A,Kl,nil(A))))
=> pp(aa(set(list(A)),bool,member(list(A),nil(A)),bNF_Greatest_Shift(A,Kl,K))) ) ) ).
% empty_Shift
tff(fact_1877_Succ__Shift,axiom,
! [A: $tType,Kl: set(list(A)),K: A,Kl2: list(A)] : bNF_Greatest_Succ(A,bNF_Greatest_Shift(A,Kl,K),Kl2) = bNF_Greatest_Succ(A,Kl,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),K),Kl2)) ).
% Succ_Shift
tff(fact_1878_listrel1__iff__update,axiom,
! [A: $tType,Xs: list(A),Ys: list(A),R: set(product_prod(A,A))] :
( pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),listrel1(A,R)))
<=> ? [Y4: A,N5: nat] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(nat,A,nth3(A,Xs),N5)),Y4)),R))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N5),aa(list(A),nat,size_size(list(A)),Xs)))
& ( Ys = list_update(A,Xs,N5,Y4) ) ) ) ).
% listrel1_iff_update
tff(fact_1879_lexordp__conv__lexord,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Xs: list(A),Ys: list(A)] :
( ord_lexordp(A,Xs,Ys)
<=> pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys)),lexord(A,aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),ord_less(A)))))) ) ) ).
% lexordp_conv_lexord
tff(fact_1880_list_Orec__o__map,axiom,
! [C: $tType,B: $tType,A: $tType,G: C,Ga: fun(B,fun(list(B),fun(C,C))),F: fun(A,B)] : aa(fun(list(A),list(B)),fun(list(A),C),aa(fun(list(B),C),fun(fun(list(A),list(B)),fun(list(A),C)),comp(list(B),C,list(A)),rec_list(C,B,G,Ga)),map(A,B,F)) = rec_list(C,A,G,aa(fun(A,B),fun(A,fun(list(A),fun(C,C))),aTP_Lamp_qd(fun(B,fun(list(B),fun(C,C))),fun(fun(A,B),fun(A,fun(list(A),fun(C,C)))),Ga),F)) ).
% list.rec_o_map
tff(fact_1881_shuffles_Oelims,axiom,
! [A: $tType,X2: list(A),Xa2: list(A),Y2: set(list(A))] :
( ( shuffles(A,X2,Xa2) = Y2 )
=> ( ( ( X2 = nil(A) )
=> ( Y2 != aa(set(list(A)),set(list(A)),insert(list(A),Xa2),bot_bot(set(list(A)))) ) )
=> ( ( ( Xa2 = nil(A) )
=> ( Y2 != aa(set(list(A)),set(list(A)),insert(list(A),X2),bot_bot(set(list(A)))) ) )
=> ~ ! [X: A,Xs3: list(A)] :
( ( X2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs3) )
=> ! [Y: A,Ys3: list(A)] :
( ( Xa2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys3) )
=> ( Y2 != aa(set(list(A)),set(list(A)),aa(set(list(A)),fun(set(list(A)),set(list(A))),sup_sup(set(list(A))),aa(set(list(A)),set(list(A)),aa(fun(list(A),list(A)),fun(set(list(A)),set(list(A))),image2(list(A),list(A)),aa(A,fun(list(A),list(A)),cons(A),X)),shuffles(A,Xs3,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys3)))),aa(set(list(A)),set(list(A)),aa(fun(list(A),list(A)),fun(set(list(A)),set(list(A))),image2(list(A),list(A)),aa(A,fun(list(A),list(A)),cons(A),Y)),shuffles(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs3),Ys3))) ) ) ) ) ) ) ).
% shuffles.elims
tff(fact_1882_get__update__eq,axiom,
! [A: $tType] :
( heap(A)
=> ! [A3: array(A),I: nat,V: A,H: heap_ext(product_unit)] : aa(array(A),list(A),get(A,update(A,A3,I,V,H)),A3) = list_update(A,aa(array(A),list(A),get(A,H),A3),I,V) ) ).
% get_update_eq
tff(fact_1883_zip__update,axiom,
! [A: $tType,B: $tType,Xs: list(A),I: nat,X2: A,Ys: list(B),Y2: B] : zip(A,B,list_update(A,Xs,I,X2),list_update(B,Ys,I,Y2)) = list_update(product_prod(A,B),zip(A,B,Xs,Ys),I,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y2)) ).
% zip_update
tff(fact_1884_Cons__shuffles__subset2,axiom,
! [A: $tType,Y2: A,Xs: list(A),Ys: list(A)] : pp(aa(set(list(A)),bool,aa(set(list(A)),fun(set(list(A)),bool),ord_less_eq(set(list(A))),aa(set(list(A)),set(list(A)),aa(fun(list(A),list(A)),fun(set(list(A)),set(list(A))),image2(list(A),list(A)),aa(A,fun(list(A),list(A)),cons(A),Y2)),shuffles(A,Xs,Ys))),shuffles(A,Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys)))) ).
% Cons_shuffles_subset2
tff(fact_1885_Cons__shuffles__subset1,axiom,
! [A: $tType,X2: A,Xs: list(A),Ys: list(A)] : pp(aa(set(list(A)),bool,aa(set(list(A)),fun(set(list(A)),bool),ord_less_eq(set(list(A))),aa(set(list(A)),set(list(A)),aa(fun(list(A),list(A)),fun(set(list(A)),set(list(A))),image2(list(A),list(A)),aa(A,fun(list(A),list(A)),cons(A),X2)),shuffles(A,Xs,Ys))),shuffles(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs),Ys))) ).
% Cons_shuffles_subset1
tff(fact_1886_shuffles_Osimps_I3_J,axiom,
! [A: $tType,X2: A,Xs: list(A),Y2: A,Ys: list(A)] : shuffles(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys)) = aa(set(list(A)),set(list(A)),aa(set(list(A)),fun(set(list(A)),set(list(A))),sup_sup(set(list(A))),aa(set(list(A)),set(list(A)),aa(fun(list(A),list(A)),fun(set(list(A)),set(list(A))),image2(list(A),list(A)),aa(A,fun(list(A),list(A)),cons(A),X2)),shuffles(A,Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys)))),aa(set(list(A)),set(list(A)),aa(fun(list(A),list(A)),fun(set(list(A)),set(list(A))),image2(list(A),list(A)),aa(A,fun(list(A),list(A)),cons(A),Y2)),shuffles(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs),Ys))) ).
% shuffles.simps(3)
tff(fact_1887_update__zip,axiom,
! [B: $tType,A: $tType,Xs: list(A),Ys: list(B),I: nat,Xy: product_prod(A,B)] : list_update(product_prod(A,B),zip(A,B,Xs,Ys),I,Xy) = zip(A,B,list_update(A,Xs,I,aa(product_prod(A,B),A,product_fst(A,B),Xy)),list_update(B,Ys,I,aa(product_prod(A,B),B,product_snd(A,B),Xy))) ).
% update_zip
tff(fact_1888_rec__list__Cons__imp,axiom,
! [B: $tType,A: $tType,F: fun(list(A),B),F1: B,F22: fun(A,fun(list(A),fun(B,B))),X2: A,Xs: list(A)] :
( ( F = rec_list(B,A,F1,F22) )
=> ( aa(list(A),B,F,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)) = aa(B,B,aa(list(A),fun(B,B),aa(A,fun(list(A),fun(B,B)),F22,X2),Xs),aa(list(A),B,F,Xs)) ) ) ).
% rec_list_Cons_imp
tff(fact_1889_rec__list__Nil__imp,axiom,
! [A: $tType,B: $tType,F: fun(list(A),B),F1: B,F22: fun(A,fun(list(A),fun(B,B)))] :
( ( F = rec_list(B,A,F1,F22) )
=> ( aa(list(A),B,F,nil(A)) = F1 ) ) ).
% rec_list_Nil_imp
tff(fact_1890_ShiftD,axiom,
! [A: $tType,Kl2: list(A),Kl: set(list(A)),K: A] :
( pp(aa(set(list(A)),bool,member(list(A),Kl2),bNF_Greatest_Shift(A,Kl,K)))
=> pp(aa(set(list(A)),bool,member(list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),K),Kl2)),Kl)) ) ).
% ShiftD
tff(fact_1891_Array_Oupdate__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [A3: array(A),I: nat,X2: A,H: heap_ext(product_unit)] : update(A,A3,I,X2,H) = aa(heap_ext(product_unit),heap_ext(product_unit),set2(A,A3,list_update(A,aa(array(A),list(A),get(A,H),A3),I,X2)),H) ) ).
% Array.update_def
tff(fact_1892_Shift__def,axiom,
! [A: $tType,Kl: set(list(A)),K: A] : bNF_Greatest_Shift(A,Kl,K) = aa(fun(list(A),bool),set(list(A)),collect(list(A)),aa(A,fun(list(A),bool),aTP_Lamp_qe(set(list(A)),fun(A,fun(list(A),bool)),Kl),K)) ).
% Shift_def
tff(fact_1893_shuffles_Opelims,axiom,
! [A: $tType,X2: list(A),Xa2: list(A),Y2: set(list(A))] :
( ( shuffles(A,X2,Xa2) = Y2 )
=> ( pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X2),Xa2)))
=> ( ( ( X2 = nil(A) )
=> ( ( Y2 = aa(set(list(A)),set(list(A)),insert(list(A),Xa2),bot_bot(set(list(A)))) )
=> ~ pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Xa2))) ) )
=> ( ( ( Xa2 = nil(A) )
=> ( ( Y2 = aa(set(list(A)),set(list(A)),insert(list(A),X2),bot_bot(set(list(A)))) )
=> ~ pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),X2),nil(A)))) ) )
=> ~ ! [X: A,Xs3: list(A)] :
( ( X2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs3) )
=> ! [Y: A,Ys3: list(A)] :
( ( Xa2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys3) )
=> ( ( Y2 = aa(set(list(A)),set(list(A)),aa(set(list(A)),fun(set(list(A)),set(list(A))),sup_sup(set(list(A))),aa(set(list(A)),set(list(A)),aa(fun(list(A),list(A)),fun(set(list(A)),set(list(A))),image2(list(A),list(A)),aa(A,fun(list(A),list(A)),cons(A),X)),shuffles(A,Xs3,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys3)))),aa(set(list(A)),set(list(A)),aa(fun(list(A),list(A)),fun(set(list(A)),set(list(A))),image2(list(A),list(A)),aa(A,fun(list(A),list(A)),cons(A),Y)),shuffles(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs3),Ys3))) )
=> ~ pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs3)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys3)))) ) ) ) ) ) ) ) ).
% shuffles.pelims
tff(fact_1894_fun_Opred__set,axiom,
! [D: $tType,A: $tType,P2: fun(A,bool),X4: fun(D,A)] :
( basic_pred_fun(D,A,aTP_Lamp_qf(D,bool),P2,X4)
<=> ! [Xa3: A] :
( pp(aa(set(A),bool,member(A,Xa3),aa(set(D),set(A),aa(fun(D,A),fun(set(D),set(A)),image2(D,A),X4),top_top(set(D)))))
=> pp(aa(A,bool,P2,Xa3)) ) ) ).
% fun.pred_set
tff(fact_1895_make_H__def,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1(B)
& heap(A) )
=> ! [I: code_integer,F: fun(B,A)] : make2(B,A,I,F) = make(A,aa(code_integer,nat,code_nat_of_integer,I),aa(fun(nat,B),fun(nat,A),aa(fun(B,A),fun(fun(nat,B),fun(nat,A)),comp(B,A,nat),F),semiring_1_of_nat(B))) ) ).
% make'_def
tff(fact_1896_Sup__fin_Ohom__commute,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [H: fun(A,A),N: set(A)] :
( ! [X: A,Y: A] : aa(A,A,H,aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,H,X)),aa(A,A,H,Y))
=> ( finite_finite2(A,N)
=> ( ( N != bot_bot(set(A)) )
=> ( aa(A,A,H,lattic5882676163264333800up_fin(A,N)) = lattic5882676163264333800up_fin(A,aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),H),N)) ) ) ) ) ) ).
% Sup_fin.hom_commute
tff(fact_1897_Inf__fin_Ohom__commute,axiom,
! [A: $tType] :
( semilattice_inf(A)
=> ! [H: fun(A,A),N: set(A)] :
( ! [X: A,Y: A] : aa(A,A,H,aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,H,X)),aa(A,A,H,Y))
=> ( finite_finite2(A,N)
=> ( ( N != bot_bot(set(A)) )
=> ( aa(A,A,H,lattic7752659483105999362nf_fin(A,N)) = lattic7752659483105999362nf_fin(A,aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),H),N)) ) ) ) ) ) ).
% Inf_fin.hom_commute
tff(fact_1898_irrefl__distinct,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( irrefl(A,R)
<=> ! [X5: product_prod(A,A)] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),X5),R))
=> pp(aa(product_prod(A,A),bool,aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aTP_Lamp_qg(A,fun(A,bool))),X5)) ) ) ).
% irrefl_distinct
tff(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)
<=> ( pp(aa(set(A),bool,member(A,B2),A6))
& ! [X5: A] :
( pp(aa(set(A),bool,member(A,X5),A6))
=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),X5)),R)) ) ) ) ) ).
% wo_rel.isMinim_def
tff(fact_1900_irreflI,axiom,
! [A: $tType,R3: set(product_prod(A,A))] :
( ! [A5: A] : ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A5),A5)),R3))
=> irrefl(A,R3) ) ).
% irreflI
tff(fact_1901_irrefl__def,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( irrefl(A,R)
<=> ! [A7: A] : ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A7),A7)),R)) ) ).
% irrefl_def
tff(fact_1902_fun_Omap__cong__pred,axiom,
! [B: $tType,A: $tType,D: $tType,X2: fun(D,A),Ya: fun(D,A),F: fun(A,B),G: fun(A,B)] :
( ( X2 = Ya )
=> ( basic_pred_fun(D,A,aTP_Lamp_qf(D,bool),aa(fun(A,B),fun(A,bool),aTP_Lamp_qh(fun(A,B),fun(fun(A,B),fun(A,bool)),F),G),Ya)
=> ( aa(fun(D,A),fun(D,B),aa(fun(A,B),fun(fun(D,A),fun(D,B)),comp(A,B,D),F),X2) = aa(fun(D,A),fun(D,B),aa(fun(A,B),fun(fun(D,A),fun(D,B)),comp(A,B,D),G),Ya) ) ) ) ).
% fun.map_cong_pred
tff(fact_1903_fun_Opred__mono__strong,axiom,
! [A: $tType,D: $tType,P2: fun(A,bool),X2: fun(D,A),Pa: fun(A,bool)] :
( basic_pred_fun(D,A,aTP_Lamp_qf(D,bool),P2,X2)
=> ( ! [Z3: A] :
( pp(aa(set(A),bool,member(A,Z3),aa(set(D),set(A),aa(fun(D,A),fun(set(D),set(A)),image2(D,A),X2),top_top(set(D)))))
=> ( pp(aa(A,bool,P2,Z3))
=> pp(aa(A,bool,Pa,Z3)) ) )
=> basic_pred_fun(D,A,aTP_Lamp_qf(D,bool),Pa,X2) ) ) ).
% fun.pred_mono_strong
tff(fact_1904_fun_Opred__cong,axiom,
! [A: $tType,D: $tType,X2: fun(D,A),Ya: fun(D,A),P2: fun(A,bool),Pa: fun(A,bool)] :
( ( X2 = Ya )
=> ( ! [Z3: A] :
( pp(aa(set(A),bool,member(A,Z3),aa(set(D),set(A),aa(fun(D,A),fun(set(D),set(A)),image2(D,A),Ya),top_top(set(D)))))
=> ( pp(aa(A,bool,P2,Z3))
<=> pp(aa(A,bool,Pa,Z3)) ) )
=> ( basic_pred_fun(D,A,aTP_Lamp_qf(D,bool),P2,X2)
<=> basic_pred_fun(D,A,aTP_Lamp_qf(D,bool),Pa,Ya) ) ) ) ).
% fun.pred_cong
tff(fact_1905_fun_Opred__map,axiom,
! [B: $tType,A: $tType,D: $tType,Q2: fun(B,bool),F: fun(A,B),X2: fun(D,A)] :
( basic_pred_fun(D,B,aTP_Lamp_qf(D,bool),Q2,aa(fun(D,A),fun(D,B),aa(fun(A,B),fun(fun(D,A),fun(D,B)),comp(A,B,D),F),X2))
<=> basic_pred_fun(D,A,aTP_Lamp_qf(D,bool),aa(fun(A,B),fun(A,bool),aa(fun(B,bool),fun(fun(A,B),fun(A,bool)),comp(B,bool,A),Q2),F),X2) ) ).
% fun.pred_map
tff(fact_1906_shuffles_Opsimps_I3_J,axiom,
! [A: $tType,X2: A,Xs: list(A),Y2: A,Ys: list(A)] :
( pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys))))
=> ( shuffles(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys)) = aa(set(list(A)),set(list(A)),aa(set(list(A)),fun(set(list(A)),set(list(A))),sup_sup(set(list(A))),aa(set(list(A)),set(list(A)),aa(fun(list(A),list(A)),fun(set(list(A)),set(list(A))),image2(list(A),list(A)),aa(A,fun(list(A),list(A)),cons(A),X2)),shuffles(A,Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y2),Ys)))),aa(set(list(A)),set(list(A)),aa(fun(list(A),list(A)),fun(set(list(A)),set(list(A))),image2(list(A),list(A)),aa(A,fun(list(A),list(A)),cons(A),Y2)),shuffles(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs),Ys))) ) ) ).
% shuffles.psimps(3)
tff(fact_1907_nth_H__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [A3: array(A)] : nth2(A,A3) = aa(fun(code_integer,nat),fun(code_integer,heap_Heap(A)),aa(fun(nat,heap_Heap(A)),fun(fun(code_integer,nat),fun(code_integer,heap_Heap(A))),comp(nat,heap_Heap(A),code_integer),nth(A,A3)),code_nat_of_integer) ) ).
% nth'_def
tff(fact_1908_new_H__def,axiom,
! [A: $tType] :
( heap(A)
=> ( new2(A) = aa(fun(code_integer,nat),fun(code_integer,fun(A,heap_Heap(array(A)))),aa(fun(nat,fun(A,heap_Heap(array(A)))),fun(fun(code_integer,nat),fun(code_integer,fun(A,heap_Heap(array(A))))),comp(nat,fun(A,heap_Heap(array(A))),code_integer),new(A)),code_nat_of_integer) ) ) ).
% new'_def
tff(fact_1909_pred__fun__True__id,axiom,
! [A: $tType,B: $tType,C: $tType,P: fun(B,bool),F: fun(C,B)] :
( nO_MATCH(fun(A,A),fun(B,bool),id(A),P)
=> ( basic_pred_fun(C,B,aTP_Lamp_qi(C,bool),P,F)
<=> basic_pred_fun(C,bool,aTP_Lamp_qi(C,bool),id(bool),aa(fun(C,B),fun(C,bool),aa(fun(B,bool),fun(fun(C,B),fun(C,bool)),comp(B,bool,C),P),F)) ) ) ).
% pred_fun_True_id
tff(fact_1910_irreflp__irrefl__eq,axiom,
! [A: $tType,R3: set(product_prod(A,A))] :
( irreflp(A,aTP_Lamp_hl(set(product_prod(A,A)),fun(A,fun(A,bool)),R3))
<=> irrefl(A,R3) ) ).
% irreflp_irrefl_eq
tff(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,aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_nv(set(product_prod(A,A)),fun(A,fun(A,bool)),R),X)))
=> wf(A,R) ) ) ) ).
% wf_finite_segments
tff(fact_1912_size__list__map,axiom,
! [A: $tType,B: $tType,F: fun(A,nat),G: fun(B,A),Xs: list(B)] : aa(list(A),nat,size_list(A,F),aa(list(B),list(A),map(B,A,G),Xs)) = aa(list(B),nat,size_list(B,aa(fun(B,A),fun(B,nat),aa(fun(A,nat),fun(fun(B,A),fun(B,nat)),comp(A,nat,B),F),G)),Xs) ).
% size_list_map
tff(fact_1913_fromCard__def,axiom,
! [A: $tType,B: $tType,A6: set(A),R: set(product_prod(B,B)),K: B] : bNF_Gr5436034075474128252omCard(A,B,A6,R,K) = fChoice(A,aa(B,fun(A,bool),aa(set(product_prod(B,B)),fun(B,fun(A,bool)),aTP_Lamp_qj(set(A),fun(set(product_prod(B,B)),fun(B,fun(A,bool))),A6),R),K)) ).
% fromCard_def
tff(fact_1914_trans__pair__less,axiom,
trans(product_prod(nat,nat),fun_pair_less) ).
% trans_pair_less
tff(fact_1915_Eps__case__prod__eq,axiom,
! [A: $tType,B: $tType,X2: A,Y2: B] : fChoice(product_prod(A,B),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aa(B,fun(A,fun(B,bool)),aTP_Lamp_cb(A,fun(B,fun(A,fun(B,bool))),X2),Y2))) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y2) ).
% Eps_case_prod_eq
tff(fact_1916_trans__def,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( trans(A,R)
<=> ! [X5: A,Y4: A,Z4: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),Y4)),R))
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),Z4)),R))
=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),Z4)),R)) ) ) ) ).
% trans_def
tff(fact_1917_transI,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( ! [X: A,Y: A,Z3: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R))
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),Z3)),R))
=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Z3)),R)) ) )
=> trans(A,R) ) ).
% transI
tff(fact_1918_transE,axiom,
! [A: $tType,R: set(product_prod(A,A)),X2: A,Y2: A,Z2: A] :
( trans(A,R)
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),R))
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z2)),R))
=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Z2)),R)) ) ) ) ).
% transE
tff(fact_1919_transD,axiom,
! [A: $tType,R: set(product_prod(A,A)),X2: A,Y2: A,Z2: A] :
( trans(A,R)
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),R))
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),Z2)),R))
=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Z2)),R)) ) ) ) ).
% transD
tff(fact_1920_pick__middlep__def,axiom,
! [B: $tType,A: $tType,C: $tType,P2: fun(B,fun(A,bool)),Q2: fun(A,fun(C,bool)),A3: B,C3: C] : bNF_pick_middlep(B,A,C,P2,Q2,A3,C3) = fChoice(A,aa(C,fun(A,bool),aa(B,fun(C,fun(A,bool)),aa(fun(A,fun(C,bool)),fun(B,fun(C,fun(A,bool))),aTP_Lamp_qk(fun(B,fun(A,bool)),fun(fun(A,fun(C,bool)),fun(B,fun(C,fun(A,bool)))),P2),Q2),A3),C3)) ).
% pick_middlep_def
tff(fact_1921_univ__def,axiom,
! [A: $tType,B: $tType,F: fun(B,A),X6: set(B)] : bNF_Greatest_univ(B,A,F,X6) = aa(B,A,F,fChoice(B,aTP_Lamp_iv(set(B),fun(B,bool),X6))) ).
% univ_def
tff(fact_1922_split__paired__Eps,axiom,
! [B: $tType,A: $tType,P2: fun(product_prod(A,B),bool)] : fChoice(product_prod(A,B),P2) = fChoice(product_prod(A,B),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aTP_Lamp_ql(fun(product_prod(A,B),bool),fun(A,fun(B,bool)),P2))) ).
% split_paired_Eps
tff(fact_1923_trans__INTER,axiom,
! [B: $tType,A: $tType,S3: set(A),R: fun(A,set(product_prod(B,B)))] :
( ! [X: A] :
( pp(aa(set(A),bool,member(A,X),S3))
=> trans(B,aa(A,set(product_prod(B,B)),R,X)) )
=> trans(B,aa(set(set(product_prod(B,B))),set(product_prod(B,B)),complete_Inf_Inf(set(product_prod(B,B))),aa(set(A),set(set(product_prod(B,B))),aa(fun(A,set(product_prod(B,B))),fun(set(A),set(set(product_prod(B,B)))),image2(A,set(product_prod(B,B))),R),S3))) ) ).
% trans_INTER
tff(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)
=> ( pp(aa(set(set(A)),bool,member(set(A),X6),equiv_quotient(A,A6,R)))
=> pp(aa(set(A),bool,member(A,fChoice(A,aa(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool)),X6))),X6)) ) ) ).
% equiv_Eps_in
tff(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)
=> ( pp(aa(set(set(A)),bool,member(set(A),X6),equiv_quotient(A,A6,R)))
=> pp(aa(set(A),bool,member(A,fChoice(A,aa(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool)),X6))),A6)) ) ) ).
% equiv_Eps_preserves
tff(fact_1926_under__incr,axiom,
! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A] :
( trans(A,R)
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),order_under(A,R),A3)),aa(A,set(A),order_under(A,R),B2))) ) ) ).
% under_incr
tff(fact_1927_list_Osize__gen__o__map,axiom,
! [B: $tType,A: $tType,F: fun(B,nat),G: fun(A,B)] : aa(fun(list(A),list(B)),fun(list(A),nat),aa(fun(list(B),nat),fun(fun(list(A),list(B)),fun(list(A),nat)),comp(list(B),nat,list(A)),size_list(B,F)),map(A,B,G)) = size_list(A,aa(fun(A,B),fun(A,nat),aa(fun(B,nat),fun(fun(A,B),fun(A,nat)),comp(B,nat,A),F),G)) ).
% list.size_gen_o_map
tff(fact_1928_trans__singleton,axiom,
! [A: $tType,A3: A] : trans(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),insert(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),A3)),bot_bot(set(product_prod(A,A))))) ).
% trans_singleton
tff(fact_1929_Eps__case__prod,axiom,
! [B: $tType,A: $tType,P2: fun(A,fun(B,bool))] : fChoice(product_prod(A,B),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),P2)) = fChoice(product_prod(A,B),aTP_Lamp_bx(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),P2)) ).
% Eps_case_prod
tff(fact_1930_toCard__def,axiom,
! [A: $tType,B: $tType,A6: set(A),R: set(product_prod(B,B))] : bNF_Greatest_toCard(A,B,A6,R) = fChoice(fun(A,B),bNF_Gr1419584066657907630d_pred(A,B,A6,R)) ).
% toCard_def
tff(fact_1931_trans__join,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( trans(A,R)
<=> ! [X5: product_prod(A,A)] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),X5),R))
=> pp(aa(product_prod(A,A),bool,aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aTP_Lamp_qn(set(product_prod(A,A)),fun(A,fun(A,bool)),R)),X5)) ) ) ).
% trans_join
tff(fact_1932_proj__Eps,axiom,
! [A: $tType,A6: set(A),R: set(product_prod(A,A)),X6: set(A)] :
( equiv_equiv(A,A6,R)
=> ( pp(aa(set(set(A)),bool,member(set(A),X6),equiv_quotient(A,A6,R)))
=> ( aa(A,set(A),equiv_proj(A,A,R),fChoice(A,aa(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool)),X6))) = X6 ) ) ) ).
% proj_Eps
tff(fact_1933_ord__to__filter__def,axiom,
! [A: $tType,R0: set(product_prod(A,A)),R: set(product_prod(A,A))] : bNF_We8469521843155493636filter(A,R0,R) = aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),fChoice(fun(A,A),bNF_Wellorder_embed(A,A,R,R0))),aa(set(product_prod(A,A)),set(A),field2(A),R)) ).
% ord_to_filter_def
tff(fact_1934_arg__min__SOME__Min,axiom,
! [B: $tType,A: $tType] :
( linorder(B)
=> ! [S3: set(A),F: fun(A,B)] :
( finite_finite2(A,S3)
=> ( lattic7623131987881927897min_on(A,B,F,S3) = fChoice(A,aa(fun(A,B),fun(A,bool),aTP_Lamp_qo(set(A),fun(fun(A,B),fun(A,bool)),S3),F)) ) ) ) ).
% arg_min_SOME_Min
tff(fact_1935_underS__incr,axiom,
! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A] :
( trans(A,R)
=> ( antisym(A,R)
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R))
=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),order_underS(A,R,A3)),order_underS(A,R,B2))) ) ) ) ).
% underS_incr
tff(fact_1936_equivp__equiv,axiom,
! [A: $tType,A6: set(product_prod(A,A))] :
( equiv_equiv(A,top_top(set(A)),A6)
<=> equiv_equivp(A,aTP_Lamp_hl(set(product_prod(A,A)),fun(A,fun(A,bool)),A6)) ) ).
% equivp_equiv
tff(fact_1937_partition_Osimps_I2_J,axiom,
! [A: $tType,P2: fun(A,bool),X2: A,Xs: list(A)] : partition(A,P2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)) = aa(product_prod(list(A),list(A)),product_prod(list(A),list(A)),aa(fun(list(A),fun(list(A),product_prod(list(A),list(A)))),fun(product_prod(list(A),list(A)),product_prod(list(A),list(A))),product_case_prod(list(A),list(A),product_prod(list(A),list(A))),aa(A,fun(list(A),fun(list(A),product_prod(list(A),list(A)))),aTP_Lamp_qp(fun(A,bool),fun(A,fun(list(A),fun(list(A),product_prod(list(A),list(A))))),P2),X2)),partition(A,P2,Xs)) ).
% partition.simps(2)
tff(fact_1938_less__eq__int_Orep__eq,axiom,
! [X2: int,Xa2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X2),Xa2))
<=> pp(aa(product_prod(nat,nat),bool,aa(product_prod(nat,nat),fun(product_prod(nat,nat),bool),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_qr(nat,fun(nat,fun(product_prod(nat,nat),bool)))),aa(int,product_prod(nat,nat),rep_Integ,X2)),aa(int,product_prod(nat,nat),rep_Integ,Xa2))) ) ).
% less_eq_int.rep_eq
tff(fact_1939_above__def,axiom,
! [A: $tType,R: set(product_prod(A,A)),A3: A] : order_above(A,R,A3) = aa(fun(A,bool),set(A),collect(A),aa(A,fun(A,bool),aTP_Lamp_hl(set(product_prod(A,A)),fun(A,fun(A,bool)),R),A3)) ).
% above_def
tff(fact_1940_antisymD,axiom,
! [A: $tType,R: set(product_prod(A,A)),A3: A,B2: A] :
( antisym(A,R)
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R))
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A3)),R))
=> ( A3 = B2 ) ) ) ) ).
% antisymD
tff(fact_1941_antisymI,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( ! [X: A,Y: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y)),R))
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),X)),R))
=> ( X = Y ) ) )
=> antisym(A,R) ) ).
% antisymI
tff(fact_1942_antisym__def,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( antisym(A,R)
<=> ! [X5: A,Y4: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),Y4)),R))
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),X5)),R))
=> ( X5 = Y4 ) ) ) ) ).
% antisym_def
tff(fact_1943_nat_Orep__eq,axiom,
! [X2: int] : aa(int,nat,nat2,X2) = aa(product_prod(nat,nat),nat,aa(fun(nat,fun(nat,nat)),fun(product_prod(nat,nat),nat),product_case_prod(nat,nat,nat),minus_minus(nat)),aa(int,product_prod(nat,nat),rep_Integ,X2)) ).
% nat.rep_eq
tff(fact_1944_less__int_Orep__eq,axiom,
! [X2: int,Xa2: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X2),Xa2))
<=> pp(aa(product_prod(nat,nat),bool,aa(product_prod(nat,nat),fun(product_prod(nat,nat),bool),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_qt(nat,fun(nat,fun(product_prod(nat,nat),bool)))),aa(int,product_prod(nat,nat),rep_Integ,X2)),aa(int,product_prod(nat,nat),rep_Integ,Xa2))) ) ).
% less_int.rep_eq
tff(fact_1945_less__eq__int_Otransfer,axiom,
pp(aa(fun(int,fun(int,bool)),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),fun(fun(int,fun(int,bool)),bool),bNF_rel_fun(product_prod(nat,nat),int,fun(product_prod(nat,nat),bool),fun(int,bool),pcr_int,bNF_rel_fun(product_prod(nat,nat),int,bool,bool,pcr_int,fequal(bool))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_qr(nat,fun(nat,fun(product_prod(nat,nat),bool))))),ord_less_eq(int))) ).
% less_eq_int.transfer
tff(fact_1946_less__eq__int_Oabs__eq,axiom,
! [Xa2: product_prod(nat,nat),X2: product_prod(nat,nat)] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(product_prod(nat,nat),int,abs_Integ,Xa2)),aa(product_prod(nat,nat),int,abs_Integ,X2)))
<=> pp(aa(product_prod(nat,nat),bool,aa(product_prod(nat,nat),fun(product_prod(nat,nat),bool),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_qr(nat,fun(nat,fun(product_prod(nat,nat),bool)))),Xa2),X2)) ) ).
% less_eq_int.abs_eq
tff(fact_1947_int__ge__less__than2__def,axiom,
! [D4: int] : int_ge_less_than2(D4) = aa(fun(product_prod(int,int),bool),set(product_prod(int,int)),collect(product_prod(int,int)),aa(fun(int,fun(int,bool)),fun(product_prod(int,int),bool),product_case_prod(int,int,bool),aTP_Lamp_qu(int,fun(int,fun(int,bool)),D4))) ).
% int_ge_less_than2_def
tff(fact_1948_int__ge__less__than__def,axiom,
! [D4: int] : int_ge_less_than(D4) = aa(fun(product_prod(int,int),bool),set(product_prod(int,int)),collect(product_prod(int,int)),aa(fun(int,fun(int,bool)),fun(product_prod(int,int),bool),product_case_prod(int,int,bool),aTP_Lamp_qv(int,fun(int,fun(int,bool)),D4))) ).
% int_ge_less_than_def
tff(fact_1949_less__int_Otransfer,axiom,
pp(aa(fun(int,fun(int,bool)),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),fun(fun(int,fun(int,bool)),bool),bNF_rel_fun(product_prod(nat,nat),int,fun(product_prod(nat,nat),bool),fun(int,bool),pcr_int,bNF_rel_fun(product_prod(nat,nat),int,bool,bool,pcr_int,fequal(bool))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_qt(nat,fun(nat,fun(product_prod(nat,nat),bool))))),ord_less(int))) ).
% less_int.transfer
tff(fact_1950_minus__int_Otransfer,axiom,
pp(aa(fun(int,fun(int,int)),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(int,fun(int,int)),bool),bNF_rel_fun(product_prod(nat,nat),int,fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),pcr_int,bNF_rel_fun(product_prod(nat,nat),int,product_prod(nat,nat),int,pcr_int,pcr_int)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_qx(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),minus_minus(int))) ).
% minus_int.transfer
tff(fact_1951_plus__int_Otransfer,axiom,
pp(aa(fun(int,fun(int,int)),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(int,fun(int,int)),bool),bNF_rel_fun(product_prod(nat,nat),int,fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),pcr_int,bNF_rel_fun(product_prod(nat,nat),int,product_prod(nat,nat),int,pcr_int,pcr_int)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_qz(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),plus_plus(int))) ).
% plus_int.transfer
tff(fact_1952_nat_Oabs__eq,axiom,
! [X2: product_prod(nat,nat)] : aa(int,nat,nat2,aa(product_prod(nat,nat),int,abs_Integ,X2)) = aa(product_prod(nat,nat),nat,aa(fun(nat,fun(nat,nat)),fun(product_prod(nat,nat),nat),product_case_prod(nat,nat,nat),minus_minus(nat)),X2) ).
% nat.abs_eq
tff(fact_1953_nat_Otransfer,axiom,
pp(aa(fun(int,nat),bool,aa(fun(product_prod(nat,nat),nat),fun(fun(int,nat),bool),bNF_rel_fun(product_prod(nat,nat),int,nat,nat,pcr_int,fequal(nat)),aa(fun(nat,fun(nat,nat)),fun(product_prod(nat,nat),nat),product_case_prod(nat,nat,nat),minus_minus(nat))),nat2)) ).
% nat.transfer
tff(fact_1954_uminus__int_Oabs__eq,axiom,
! [X2: product_prod(nat,nat)] : aa(int,int,uminus_uminus(int),aa(product_prod(nat,nat),int,abs_Integ,X2)) = aa(product_prod(nat,nat),int,abs_Integ,aa(product_prod(nat,nat),product_prod(nat,nat),aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aTP_Lamp_ra(nat,fun(nat,product_prod(nat,nat)))),X2)) ).
% uminus_int.abs_eq
tff(fact_1955_uminus__int_Otransfer,axiom,
pp(aa(fun(int,int),bool,aa(fun(product_prod(nat,nat),product_prod(nat,nat)),fun(fun(int,int),bool),bNF_rel_fun(product_prod(nat,nat),int,product_prod(nat,nat),int,pcr_int,pcr_int),aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aTP_Lamp_ra(nat,fun(nat,product_prod(nat,nat))))),uminus_uminus(int))) ).
% uminus_int.transfer
tff(fact_1956_times__int_Oabs__eq,axiom,
! [Xa2: product_prod(nat,nat),X2: product_prod(nat,nat)] : aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(nat,nat),int,abs_Integ,Xa2)),aa(product_prod(nat,nat),int,abs_Integ,X2)) = aa(product_prod(nat,nat),int,abs_Integ,aa(product_prod(nat,nat),product_prod(nat,nat),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_rc(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xa2),X2)) ).
% times_int.abs_eq
tff(fact_1957_plus__int_Oabs__eq,axiom,
! [Xa2: product_prod(nat,nat),X2: product_prod(nat,nat)] : aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(product_prod(nat,nat),int,abs_Integ,Xa2)),aa(product_prod(nat,nat),int,abs_Integ,X2)) = aa(product_prod(nat,nat),int,abs_Integ,aa(product_prod(nat,nat),product_prod(nat,nat),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_qz(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xa2),X2)) ).
% plus_int.abs_eq
tff(fact_1958_minus__int_Oabs__eq,axiom,
! [Xa2: product_prod(nat,nat),X2: product_prod(nat,nat)] : aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(product_prod(nat,nat),int,abs_Integ,Xa2)),aa(product_prod(nat,nat),int,abs_Integ,X2)) = aa(product_prod(nat,nat),int,abs_Integ,aa(product_prod(nat,nat),product_prod(nat,nat),aa(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_qx(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xa2),X2)) ).
% minus_int.abs_eq
tff(fact_1959_times__int_Otransfer,axiom,
pp(aa(fun(int,fun(int,int)),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(int,fun(int,int)),bool),bNF_rel_fun(product_prod(nat,nat),int,fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),pcr_int,bNF_rel_fun(product_prod(nat,nat),int,product_prod(nat,nat),int,pcr_int,pcr_int)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_rc(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),times_times(int))) ).
% times_int.transfer
tff(fact_1960_less__int_Oabs__eq,axiom,
! [Xa2: product_prod(nat,nat),X2: product_prod(nat,nat)] :
( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(product_prod(nat,nat),int,abs_Integ,Xa2)),aa(product_prod(nat,nat),int,abs_Integ,X2)))
<=> pp(aa(product_prod(nat,nat),bool,aa(product_prod(nat,nat),fun(product_prod(nat,nat),bool),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_qt(nat,fun(nat,fun(product_prod(nat,nat),bool)))),Xa2),X2)) ) ).
% less_int.abs_eq
tff(fact_1961_uminus__int__def,axiom,
uminus_uminus(int) = aa(fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),map_fun(int,product_prod(nat,nat),product_prod(nat,nat),int,rep_Integ,abs_Integ),aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aTP_Lamp_ra(nat,fun(nat,product_prod(nat,nat))))) ).
% uminus_int_def
tff(fact_1962_times__int_Orsp,axiom,
pp(aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),bool),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),fun(product_prod(nat,nat),product_prod(nat,nat)),intrel,bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),intrel,intrel)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_rc(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_rc(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))))) ).
% times_int.rsp
tff(fact_1963_nat__def,axiom,
nat2 = aa(fun(product_prod(nat,nat),nat),fun(int,nat),map_fun(int,product_prod(nat,nat),nat,nat,rep_Integ,id(nat)),aa(fun(nat,fun(nat,nat)),fun(product_prod(nat,nat),nat),product_case_prod(nat,nat,nat),minus_minus(nat))) ).
% nat_def
tff(fact_1964_of__int_Otransfer,axiom,
! [A: $tType] :
( ring_1(A)
=> pp(aa(fun(int,A),bool,aa(fun(product_prod(nat,nat),A),fun(fun(int,A),bool),bNF_rel_fun(product_prod(nat,nat),int,A,A,pcr_int,fequal(A)),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),aTP_Lamp_rd(nat,fun(nat,A)))),ring_1_of_int(A))) ) ).
% of_int.transfer
tff(fact_1965_minus__int_Orsp,axiom,
pp(aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),bool),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),fun(product_prod(nat,nat),product_prod(nat,nat)),intrel,bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),intrel,intrel)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_qx(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_qx(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))))) ).
% minus_int.rsp
tff(fact_1966_plus__int_Orsp,axiom,
pp(aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),bool),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),fun(product_prod(nat,nat),product_prod(nat,nat)),intrel,bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),intrel,intrel)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_qz(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_qz(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))))) ).
% plus_int.rsp
tff(fact_1967_less__eq__int_Orsp,axiom,
pp(aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),fun(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),bool),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),fun(product_prod(nat,nat),bool),fun(product_prod(nat,nat),bool),intrel,bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),bool,bool,intrel,fequal(bool))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_qr(nat,fun(nat,fun(product_prod(nat,nat),bool))))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_qr(nat,fun(nat,fun(product_prod(nat,nat),bool)))))) ).
% less_eq_int.rsp
tff(fact_1968_map__fun_Ocompositionality,axiom,
! [D: $tType,F2: $tType,C: $tType,E: $tType,B: $tType,A: $tType,F: fun(E,C),G: fun(D,F2),H: fun(C,A),I: fun(B,D),Fun: fun(A,B)] : aa(fun(C,D),fun(E,F2),map_fun(E,C,D,F2,F,G),aa(fun(A,B),fun(C,D),map_fun(C,A,B,D,H,I),Fun)) = aa(fun(A,B),fun(E,F2),map_fun(E,A,B,F2,aa(fun(E,C),fun(E,A),aa(fun(C,A),fun(fun(E,C),fun(E,A)),comp(C,A,E),H),F),aa(fun(B,D),fun(B,F2),aa(fun(D,F2),fun(fun(B,D),fun(B,F2)),comp(D,F2,B),G),I)),Fun) ).
% map_fun.compositionality
tff(fact_1969_map__fun__def,axiom,
! [B: $tType,D: $tType,A: $tType,C: $tType,F: fun(C,A),G: fun(B,D),H: fun(A,B)] : aa(fun(A,B),fun(C,D),map_fun(C,A,B,D,F,G),H) = aa(fun(C,A),fun(C,D),aa(fun(A,D),fun(fun(C,A),fun(C,D)),comp(A,D,C),aa(fun(A,B),fun(A,D),aa(fun(B,D),fun(fun(A,B),fun(A,D)),comp(B,D,A),G),H)),F) ).
% map_fun_def
tff(fact_1970_map__fun_Ocomp,axiom,
! [E: $tType,C: $tType,A: $tType,F2: $tType,D: $tType,B: $tType,F: fun(E,C),G: fun(D,F2),H: fun(C,A),I: fun(B,D)] : aa(fun(fun(A,B),fun(C,D)),fun(fun(A,B),fun(E,F2)),aa(fun(fun(C,D),fun(E,F2)),fun(fun(fun(A,B),fun(C,D)),fun(fun(A,B),fun(E,F2))),comp(fun(C,D),fun(E,F2),fun(A,B)),map_fun(E,C,D,F2,F,G)),map_fun(C,A,B,D,H,I)) = map_fun(E,A,B,F2,aa(fun(E,C),fun(E,A),aa(fun(C,A),fun(fun(E,C),fun(E,A)),comp(C,A,E),H),F),aa(fun(B,D),fun(B,F2),aa(fun(D,F2),fun(fun(B,D),fun(B,F2)),comp(D,F2,B),G),I)) ).
% map_fun.comp
tff(fact_1971_ring__1__class_Oof__int__def,axiom,
! [A: $tType] :
( ring_1(A)
=> ( ring_1_of_int(A) = aa(fun(product_prod(nat,nat),A),fun(int,A),map_fun(int,product_prod(nat,nat),A,A,rep_Integ,id(A)),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),aTP_Lamp_rd(nat,fun(nat,A)))) ) ) ).
% ring_1_class.of_int_def
tff(fact_1972_uminus__int_Orsp,axiom,
pp(aa(fun(product_prod(nat,nat),product_prod(nat,nat)),bool,aa(fun(product_prod(nat,nat),product_prod(nat,nat)),fun(fun(product_prod(nat,nat),product_prod(nat,nat)),bool),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),product_prod(nat,nat),intrel,intrel),aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aTP_Lamp_ra(nat,fun(nat,product_prod(nat,nat))))),aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aTP_Lamp_ra(nat,fun(nat,product_prod(nat,nat)))))) ).
% uminus_int.rsp
tff(fact_1973_nat_Orsp,axiom,
pp(aa(fun(product_prod(nat,nat),nat),bool,aa(fun(product_prod(nat,nat),nat),fun(fun(product_prod(nat,nat),nat),bool),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),nat,nat,intrel,fequal(nat)),aa(fun(nat,fun(nat,nat)),fun(product_prod(nat,nat),nat),product_case_prod(nat,nat,nat),minus_minus(nat))),aa(fun(nat,fun(nat,nat)),fun(product_prod(nat,nat),nat),product_case_prod(nat,nat,nat),minus_minus(nat)))) ).
% nat.rsp
tff(fact_1974_of__int_Orsp,axiom,
! [A: $tType] :
( ring_1(A)
=> pp(aa(fun(product_prod(nat,nat),A),bool,aa(fun(product_prod(nat,nat),A),fun(fun(product_prod(nat,nat),A),bool),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),A,A,intrel,fequal(A)),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),aTP_Lamp_rd(nat,fun(nat,A)))),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),aTP_Lamp_rd(nat,fun(nat,A))))) ) ).
% of_int.rsp
tff(fact_1975_intrel__def,axiom,
intrel = aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_rf(nat,fun(nat,fun(product_prod(nat,nat),bool)))) ).
% intrel_def
tff(fact_1976_of__int_Oabs__eq,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [X2: product_prod(nat,nat)] : aa(int,A,ring_1_of_int(A),aa(product_prod(nat,nat),int,abs_Integ,X2)) = aa(product_prod(nat,nat),A,aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),aTP_Lamp_rd(nat,fun(nat,A))),X2) ) ).
% of_int.abs_eq
tff(fact_1977_of__int_Orep__eq,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [X2: int] : aa(int,A,ring_1_of_int(A),X2) = aa(product_prod(nat,nat),A,aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),aTP_Lamp_rd(nat,fun(nat,A))),aa(int,product_prod(nat,nat),rep_Integ,X2)) ) ).
% of_int.rep_eq
tff(fact_1978_less__int_Orsp,axiom,
pp(aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),bool,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),fun(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),bool),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),fun(product_prod(nat,nat),bool),fun(product_prod(nat,nat),bool),intrel,bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),bool,bool,intrel,fequal(bool))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_qt(nat,fun(nat,fun(product_prod(nat,nat),bool))))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_qt(nat,fun(nat,fun(product_prod(nat,nat),bool)))))) ).
% less_int.rsp
tff(fact_1979_times__int__def,axiom,
times_times(int) = aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(int,fun(int,int)),map_fun(int,product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),rep_Integ,map_fun(int,product_prod(nat,nat),product_prod(nat,nat),int,rep_Integ,abs_Integ)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_rc(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).
% times_int_def
tff(fact_1980_minus__int__def,axiom,
minus_minus(int) = aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(int,fun(int,int)),map_fun(int,product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),rep_Integ,map_fun(int,product_prod(nat,nat),product_prod(nat,nat),int,rep_Integ,abs_Integ)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_qx(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).
% minus_int_def
tff(fact_1981_plus__int__def,axiom,
plus_plus(int) = aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),fun(int,fun(int,int)),map_fun(int,product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat)),fun(int,int),rep_Integ,map_fun(int,product_prod(nat,nat),product_prod(nat,nat),int,rep_Integ,abs_Integ)),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),product_prod(nat,nat))),product_case_prod(nat,nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_qz(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))))) ).
% plus_int_def
tff(fact_1982_antisymp__antisym__eq,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( antisymp(A,aTP_Lamp_hl(set(product_prod(A,A)),fun(A,fun(A,bool)),R))
<=> antisym(A,R) ) ).
% antisymp_antisym_eq
tff(fact_1983_o__prs_I2_J,axiom,
! [F2: $tType,E: $tType,C: $tType,D: $tType,A: $tType,B: $tType,H5: $tType,G2: $tType,R1: fun(A,fun(A,bool)),Abs1: fun(A,B),Rep1: fun(B,A),R22: fun(C,fun(C,bool)),Abs22: fun(C,D),Rep22: fun(D,C),R32: fun(E,fun(E,bool)),Abs32: fun(E,F2),Rep32: fun(F2,E)] :
( quotient3(A,B,R1,Abs1,Rep1)
=> ( quotient3(C,D,R22,Abs22,Rep22)
=> ( quotient3(E,F2,R32,Abs32,Rep32)
=> ( aa(fun(fun(G2,H5),fun(fun(A,G2),fun(A,H5))),fun(fun(G2,H5),fun(fun(B,G2),fun(B,H5))),map_fun(fun(G2,H5),fun(G2,H5),fun(fun(A,G2),fun(A,H5)),fun(fun(B,G2),fun(B,H5)),id(fun(G2,H5)),map_fun(fun(B,G2),fun(A,G2),fun(A,H5),fun(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)
tff(fact_1984_rat__less__code,axiom,
! [P: rat,Q: rat] :
( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),P),Q))
<=> pp(aa(product_prod(int,int),bool,aa(fun(int,fun(int,bool)),fun(product_prod(int,int),bool),product_case_prod(int,int,bool),aTP_Lamp_rh(rat,fun(int,fun(int,bool)),Q)),quotient_of(P))) ) ).
% rat_less_code
tff(fact_1985_rat__less__eq__code,axiom,
! [P: rat,Q: rat] :
( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),P),Q))
<=> pp(aa(product_prod(int,int),bool,aa(fun(int,fun(int,bool)),fun(product_prod(int,int),bool),product_case_prod(int,int,bool),aTP_Lamp_rj(rat,fun(int,fun(int,bool)),Q)),quotient_of(P))) ) ).
% rat_less_eq_code
tff(fact_1986_rat__uminus__code,axiom,
! [P: rat] : quotient_of(aa(rat,rat,uminus_uminus(rat),P)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_rk(int,fun(int,product_prod(int,int)))),quotient_of(P)) ).
% rat_uminus_code
tff(fact_1987_abs__o__rep,axiom,
! [A: $tType,B: $tType,R3: fun(A,fun(A,bool)),Abs: fun(A,B),Rep: fun(B,A)] :
( quotient3(A,B,R3,Abs,Rep)
=> ( aa(fun(B,A),fun(B,B),aa(fun(A,B),fun(fun(B,A),fun(B,B)),comp(A,B,B),Abs),Rep) = id(B) ) ) ).
% abs_o_rep
tff(fact_1988_o__prs_I1_J,axiom,
! [C: $tType,E: $tType,A: $tType,B: $tType,F2: $tType,D: $tType,R1: fun(A,fun(A,bool)),Abs1: fun(A,B),Rep1: fun(B,A),R22: fun(C,fun(C,bool)),Abs22: fun(C,D),Rep22: fun(D,C),R32: fun(E,fun(E,bool)),Abs32: fun(E,F2),Rep32: fun(F2,E)] :
( quotient3(A,B,R1,Abs1,Rep1)
=> ( quotient3(C,D,R22,Abs22,Rep22)
=> ( quotient3(E,F2,R32,Abs32,Rep32)
=> ( aa(fun(fun(C,E),fun(fun(A,C),fun(A,E))),fun(fun(D,F2),fun(fun(B,D),fun(B,F2))),map_fun(fun(D,F2),fun(C,E),fun(fun(A,C),fun(A,E)),fun(fun(B,D),fun(B,F2)),map_fun(C,D,F2,E,Abs22,Rep32),map_fun(fun(B,D),fun(A,C),fun(A,E),fun(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)
tff(fact_1989_OOO__eq__quotient3,axiom,
! [A: $tType,B: $tType,C: $tType,R1: fun(A,fun(A,bool)),Abs1: fun(A,B),Rep1: fun(B,A),Abs22: fun(B,C),Rep22: fun(C,B)] :
( quotient3(A,B,R1,Abs1,Rep1)
=> ( quotient3(B,C,fequal(B),Abs22,Rep22)
=> quotient3(A,C,aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),relcompp(A,A,A),R1),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),relcompp(A,A,A),fequal(A)),R1)),aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),Abs22),Abs1),aa(fun(C,B),fun(C,A),aa(fun(B,A),fun(fun(C,B),fun(C,A)),comp(B,A,C),Rep1),Rep22)) ) ) ).
% OOO_eq_quotient3
tff(fact_1990_OOO__quotient3,axiom,
! [A: $tType,B: $tType,C: $tType,R1: fun(A,fun(A,bool)),Abs1: fun(A,B),Rep1: fun(B,A),R22: fun(B,fun(B,bool)),Abs22: fun(B,C),Rep22: fun(C,B),R24: fun(A,fun(A,bool))] :
( quotient3(A,B,R1,Abs1,Rep1)
=> ( quotient3(B,C,R22,Abs22,Rep22)
=> ( ! [X: A,Y: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),R24,X),Y))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),R1,X),X))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),R1,Y),Y))
=> pp(aa(B,bool,aa(B,fun(B,bool),R22,aa(A,B,Abs1,X)),aa(A,B,Abs1,Y))) ) ) )
=> ( ! [X: B,Y: B] :
( pp(aa(B,bool,aa(B,fun(B,bool),R22,X),Y))
=> pp(aa(A,bool,aa(A,fun(A,bool),R24,aa(B,A,Rep1,X)),aa(B,A,Rep1,Y))) )
=> quotient3(A,C,aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),relcompp(A,A,A),R1),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),relcompp(A,A,A),R24),R1)),aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),Abs22),Abs1),aa(fun(C,B),fun(C,A),aa(fun(B,A),fun(fun(C,B),fun(C,A)),comp(B,A,C),Rep1),Rep22)) ) ) ) ) ).
% OOO_quotient3
tff(fact_1991_less__int__def,axiom,
ord_less(int) = aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),fun(int,fun(int,bool)),map_fun(int,product_prod(nat,nat),fun(product_prod(nat,nat),bool),fun(int,bool),rep_Integ,map_fun(int,product_prod(nat,nat),bool,bool,rep_Integ,id(bool))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_qt(nat,fun(nat,fun(product_prod(nat,nat),bool))))) ).
% less_int_def
tff(fact_1992_less__eq__int__def,axiom,
ord_less_eq(int) = aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),fun(int,fun(int,bool)),map_fun(int,product_prod(nat,nat),fun(product_prod(nat,nat),bool),fun(int,bool),rep_Integ,map_fun(int,product_prod(nat,nat),bool,bool,rep_Integ,id(bool))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),bool))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_qr(nat,fun(nat,fun(product_prod(nat,nat),bool))))) ).
% less_eq_int_def
tff(fact_1993_rat__minus__code,axiom,
! [P: rat,Q: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),minus_minus(rat),P),Q)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_rm(rat,fun(int,fun(int,product_prod(int,int))),Q)),quotient_of(P)) ).
% rat_minus_code
tff(fact_1994_rat__plus__code,axiom,
! [P: rat,Q: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),P),Q)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_ro(rat,fun(int,fun(int,product_prod(int,int))),Q)),quotient_of(P)) ).
% rat_plus_code
tff(fact_1995_rat__divide__code,axiom,
! [P: rat,Q: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),divide_divide(rat),P),Q)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_rq(rat,fun(int,fun(int,product_prod(int,int))),Q)),quotient_of(P)) ).
% rat_divide_code
tff(fact_1996_rat__times__code,axiom,
! [P: rat,Q: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),times_times(rat),P),Q)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_rs(rat,fun(int,fun(int,product_prod(int,int))),Q)),quotient_of(P)) ).
% rat_times_code
tff(fact_1997_ID_Oin__rel,axiom,
! [A: $tType,B: $tType,R3: fun(A,fun(B,bool)),A3: A,B2: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),bNF_id_bnf(fun(A,fun(B,bool))),R3),A3),B2))
<=> ? [Z4: product_prod(A,B)] :
( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),Z4),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aTP_Lamp_rt(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),R3))))
& ( aa(product_prod(A,B),A,aa(fun(product_prod(A,B),A),fun(product_prod(A,B),A),bNF_id_bnf(fun(product_prod(A,B),A)),product_fst(A,B)),Z4) = A3 )
& ( aa(product_prod(A,B),B,aa(fun(product_prod(A,B),B),fun(product_prod(A,B),B),bNF_id_bnf(fun(product_prod(A,B),B)),product_snd(A,B)),Z4) = B2 ) ) ) ).
% ID.in_rel
tff(fact_1998_execute__map__entry_I1_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [I: nat,H: heap_ext(product_unit),A3: array(A),F: fun(A,A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(array(A),nat,length(A,H),A3)))
=> ( aa(heap_ext(product_unit),option(product_prod(array(A),heap_ext(product_unit))),aa(heap_Heap(array(A)),fun(heap_ext(product_unit),option(product_prod(array(A),heap_ext(product_unit)))),heap_execute(array(A)),map_entry(A,I,F,A3)),H) = aa(product_prod(array(A),heap_ext(product_unit)),option(product_prod(array(A),heap_ext(product_unit))),some(product_prod(array(A),heap_ext(product_unit))),aa(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit)),aa(array(A),fun(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit))),product_Pair(array(A),heap_ext(product_unit)),A3),update(A,A3,I,aa(A,A,F,aa(nat,A,nth3(A,aa(array(A),list(A),get(A,H),A3)),I)),H))) ) ) ) ).
% execute_map_entry(1)
tff(fact_1999_foldr__fold,axiom,
! [B: $tType,A: $tType,Xs: list(A),F: fun(A,fun(B,B))] :
( ! [X: A,Y: A] :
( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set3(A),Xs)))
=> ( pp(aa(set(A),bool,member(A,Y),aa(list(A),set(A),set3(A),Xs)))
=> ( aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,Y)),aa(A,fun(B,B),F,X)) = aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X)),aa(A,fun(B,B),F,Y)) ) ) )
=> ( foldr(A,B,F,Xs) = fold(A,B,F,Xs) ) ) ).
% foldr_fold
tff(fact_2000_ID_Opred__map,axiom,
! [B: $tType,A: $tType,Q2: fun(B,bool),F: fun(A,B),X2: A] :
( pp(aa(B,bool,aa(fun(B,bool),fun(B,bool),bNF_id_bnf(fun(B,bool)),Q2),aa(A,B,aa(fun(A,B),fun(A,B),bNF_id_bnf(fun(A,B)),F),X2)))
<=> pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),bNF_id_bnf(fun(A,bool)),aa(fun(A,B),fun(A,bool),aa(fun(B,bool),fun(fun(A,B),fun(A,bool)),comp(B,bool,A),Q2),F)),X2)) ) ).
% ID.pred_map
tff(fact_2001_execute__heap,axiom,
! [A: $tType,F: fun(heap_ext(product_unit),product_prod(A,heap_ext(product_unit)))] : aa(heap_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_execute(A),heap_heap(A,F)) = aa(fun(heap_ext(product_unit),product_prod(A,heap_ext(product_unit))),fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),aa(fun(product_prod(A,heap_ext(product_unit)),option(product_prod(A,heap_ext(product_unit)))),fun(fun(heap_ext(product_unit),product_prod(A,heap_ext(product_unit))),fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))))),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
tff(fact_2002_execute__guard_I2_J,axiom,
! [A: $tType,P2: fun(heap_ext(product_unit),bool),H: heap_ext(product_unit),F: fun(heap_ext(product_unit),product_prod(A,heap_ext(product_unit)))] :
( pp(aa(heap_ext(product_unit),bool,P2,H))
=> ( aa(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))),aa(heap_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_execute(A),heap_guard(A,P2,F)),H) = aa(product_prod(A,heap_ext(product_unit)),option(product_prod(A,heap_ext(product_unit))),some(product_prod(A,heap_ext(product_unit))),aa(heap_ext(product_unit),product_prod(A,heap_ext(product_unit)),F,H)) ) ) ).
% execute_guard(2)
tff(fact_2003_Heap__eqI,axiom,
! [A: $tType,F: heap_Heap(A),G: heap_Heap(A)] :
( ! [H7: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))),aa(heap_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_execute(A),F),H7) = aa(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))),aa(heap_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_execute(A),G),H7)
=> ( F = G ) ) ).
% Heap_eqI
tff(fact_2004_effect__def,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)
<=> ( aa(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))),aa(heap_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_execute(A),C3),H) = aa(product_prod(A,heap_ext(product_unit)),option(product_prod(A,heap_ext(product_unit))),some(product_prod(A,heap_ext(product_unit))),aa(heap_ext(product_unit),product_prod(A,heap_ext(product_unit)),aa(A,fun(heap_ext(product_unit),product_prod(A,heap_ext(product_unit))),product_Pair(A,heap_ext(product_unit)),R),H2)) ) ) ).
% effect_def
tff(fact_2005_effectI,axiom,
! [A: $tType,C3: heap_Heap(A),H: heap_ext(product_unit),R: A,H2: heap_ext(product_unit)] :
( ( aa(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))),aa(heap_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_execute(A),C3),H) = aa(product_prod(A,heap_ext(product_unit)),option(product_prod(A,heap_ext(product_unit))),some(product_prod(A,heap_ext(product_unit))),aa(heap_ext(product_unit),product_prod(A,heap_ext(product_unit)),aa(A,fun(heap_ext(product_unit),product_prod(A,heap_ext(product_unit))),product_Pair(A,heap_ext(product_unit)),R),H2)) )
=> heap_effect(A,C3,H,H2,R) ) ).
% effectI
tff(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)] : aa(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))),aa(heap_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_execute(A),F),H) != aa(product_prod(A,heap_ext(product_unit)),option(product_prod(A,heap_ext(product_unit))),some(product_prod(A,heap_ext(product_unit))),aa(heap_ext(product_unit),product_prod(A,heap_ext(product_unit)),aa(A,fun(heap_ext(product_unit),product_prod(A,heap_ext(product_unit))),product_Pair(A,heap_ext(product_unit)),R4),H4)) ) ).
% successE
tff(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: fun(A,heap_Heap(B)),Y2: B,H3: heap_ext(product_unit)] :
( ( aa(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))),aa(heap_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_execute(A),F),H) = aa(product_prod(A,heap_ext(product_unit)),option(product_prod(A,heap_ext(product_unit))),some(product_prod(A,heap_ext(product_unit))),aa(heap_ext(product_unit),product_prod(A,heap_ext(product_unit)),aa(A,fun(heap_ext(product_unit),product_prod(A,heap_ext(product_unit))),product_Pair(A,heap_ext(product_unit)),X2),H2)) )
=> ( ( aa(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit))),aa(heap_Heap(B),fun(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit)))),heap_execute(B),aa(A,heap_Heap(B),G,X2)),H2) = aa(product_prod(B,heap_ext(product_unit)),option(product_prod(B,heap_ext(product_unit))),some(product_prod(B,heap_ext(product_unit))),aa(heap_ext(product_unit),product_prod(B,heap_ext(product_unit)),aa(B,fun(heap_ext(product_unit),product_prod(B,heap_ext(product_unit))),product_Pair(B,heap_ext(product_unit)),Y2),H3)) )
=> ( aa(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit))),aa(heap_Heap(B),fun(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit)))),heap_execute(B),heap_bind(A,B,F,G)),H) = aa(product_prod(B,heap_ext(product_unit)),option(product_prod(B,heap_ext(product_unit))),some(product_prod(B,heap_ext(product_unit))),aa(heap_ext(product_unit),product_prod(B,heap_ext(product_unit)),aa(B,fun(heap_ext(product_unit),product_prod(B,heap_ext(product_unit))),product_Pair(B,heap_ext(product_unit)),Y2),H3)) ) ) ) ).
% execute_bind_eq_SomeI
tff(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: fun(A,heap_Heap(B))] :
( ( aa(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))),aa(heap_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_execute(A),F),H) = aa(product_prod(A,heap_ext(product_unit)),option(product_prod(A,heap_ext(product_unit))),some(product_prod(A,heap_ext(product_unit))),aa(heap_ext(product_unit),product_prod(A,heap_ext(product_unit)),aa(A,fun(heap_ext(product_unit),product_prod(A,heap_ext(product_unit))),product_Pair(A,heap_ext(product_unit)),X2),H2)) )
=> ( aa(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit))),aa(heap_Heap(B),fun(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit)))),heap_execute(B),heap_bind(A,B,F,G)),H) = aa(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit))),aa(heap_Heap(B),fun(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit)))),heap_execute(B),aa(A,heap_Heap(B),G,X2)),H2) ) ) ).
% execute_bind(1)
tff(fact_2009_execute__return,axiom,
! [A: $tType,X2: A] : aa(heap_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_execute(A),aa(A,heap_Heap(A),heap_return(A),X2)) = aa(fun(heap_ext(product_unit),product_prod(A,heap_ext(product_unit))),fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),aa(fun(product_prod(A,heap_ext(product_unit)),option(product_prod(A,heap_ext(product_unit)))),fun(fun(heap_ext(product_unit),product_prod(A,heap_ext(product_unit))),fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))))),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)))),aa(A,fun(heap_ext(product_unit),product_prod(A,heap_ext(product_unit))),product_Pair(A,heap_ext(product_unit)),X2)) ).
% execute_return
tff(fact_2010_execute__tap,axiom,
! [A: $tType,F: fun(heap_ext(product_unit),A),H: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))),aa(heap_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_execute(A),heap_tap(A,F)),H) = aa(product_prod(A,heap_ext(product_unit)),option(product_prod(A,heap_ext(product_unit))),some(product_prod(A,heap_ext(product_unit))),aa(heap_ext(product_unit),product_prod(A,heap_ext(product_unit)),aa(A,fun(heap_ext(product_unit),product_prod(A,heap_ext(product_unit))),product_Pair(A,heap_ext(product_unit)),aa(heap_ext(product_unit),A,F,H)),H)) ).
% execute_tap
tff(fact_2011_execute__of__list,axiom,
! [A: $tType] :
( heap(A)
=> ! [Xs: list(A),H: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(array(A),heap_ext(product_unit))),aa(heap_Heap(array(A)),fun(heap_ext(product_unit),option(product_prod(array(A),heap_ext(product_unit)))),heap_execute(array(A)),of_list(A,Xs)),H) = aa(product_prod(array(A),heap_ext(product_unit)),option(product_prod(array(A),heap_ext(product_unit))),some(product_prod(array(A),heap_ext(product_unit))),aa(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit)),alloc(A,Xs),H)) ) ).
% execute_of_list
tff(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: fun(A,heap_Heap(B))] :
( ( aa(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))),aa(heap_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_execute(A),F),H) = aa(product_prod(A,heap_ext(product_unit)),option(product_prod(A,heap_ext(product_unit))),some(product_prod(A,heap_ext(product_unit))),aa(heap_ext(product_unit),product_prod(A,heap_ext(product_unit)),aa(A,fun(heap_ext(product_unit),product_prod(A,heap_ext(product_unit))),product_Pair(A,heap_ext(product_unit)),X2),H2)) )
=> ( heap_success(B,aa(A,heap_Heap(B),G,X2),H2)
=> heap_success(B,heap_bind(A,B,F,G),H) ) ) ).
% success_bind_executeI
tff(fact_2013_execute__new,axiom,
! [A: $tType] :
( heap(A)
=> ! [N4: nat,X2: A,H: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(array(A),heap_ext(product_unit))),aa(heap_Heap(array(A)),fun(heap_ext(product_unit),option(product_prod(array(A),heap_ext(product_unit)))),heap_execute(array(A)),aa(A,heap_Heap(array(A)),aa(nat,fun(A,heap_Heap(array(A))),new(A),N4),X2)),H) = aa(product_prod(array(A),heap_ext(product_unit)),option(product_prod(array(A),heap_ext(product_unit))),some(product_prod(array(A),heap_ext(product_unit))),aa(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit)),alloc(A,replicate(A,N4,X2)),H)) ) ).
% execute_new
tff(fact_2014_execute__lookup,axiom,
! [A: $tType] :
( heap(A)
=> ! [R: ref(A),H: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))),aa(heap_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_execute(A),lookup(A,R)),H) = aa(product_prod(A,heap_ext(product_unit)),option(product_prod(A,heap_ext(product_unit))),some(product_prod(A,heap_ext(product_unit))),aa(heap_ext(product_unit),product_prod(A,heap_ext(product_unit)),aa(A,fun(heap_ext(product_unit),product_prod(A,heap_ext(product_unit))),product_Pair(A,heap_ext(product_unit)),aa(ref(A),A,get2(A,H),R)),H)) ) ).
% execute_lookup
tff(fact_2015_execute__change,axiom,
! [A: $tType] :
( heap(A)
=> ! [F: fun(A,A),R: ref(A),H: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))),aa(heap_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_execute(A),change(A,F,R)),H) = aa(product_prod(A,heap_ext(product_unit)),option(product_prod(A,heap_ext(product_unit))),some(product_prod(A,heap_ext(product_unit))),aa(heap_ext(product_unit),product_prod(A,heap_ext(product_unit)),aa(A,fun(heap_ext(product_unit),product_prod(A,heap_ext(product_unit))),product_Pair(A,heap_ext(product_unit)),aa(A,A,F,aa(ref(A),A,get2(A,H),R))),aa(heap_ext(product_unit),heap_ext(product_unit),set4(A,R,aa(A,A,F,aa(ref(A),A,get2(A,H),R))),H))) ) ).
% execute_change
tff(fact_2016_foldr__Cons,axiom,
! [B: $tType,A: $tType,F: fun(A,fun(B,B)),X2: A,Xs: list(A)] : foldr(A,B,F,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)) = aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X2)),foldr(A,B,F,Xs)) ).
% foldr_Cons
tff(fact_2017_execute__make,axiom,
! [A: $tType] :
( heap(A)
=> ! [N4: nat,F: fun(nat,A),H: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(array(A),heap_ext(product_unit))),aa(heap_Heap(array(A)),fun(heap_ext(product_unit),option(product_prod(array(A),heap_ext(product_unit)))),heap_execute(array(A)),make(A,N4,F)),H) = aa(product_prod(array(A),heap_ext(product_unit)),option(product_prod(array(A),heap_ext(product_unit))),some(product_prod(array(A),heap_ext(product_unit))),aa(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit)),alloc(A,aa(list(nat),list(A),map(nat,A,F),upt(zero_zero(nat),N4))),H)) ) ).
% execute_make
tff(fact_2018_foldr__map,axiom,
! [C: $tType,B: $tType,A: $tType,G: fun(B,fun(A,A)),F: fun(C,B),Xs: list(C),A3: A] : aa(A,A,foldr(B,A,G,aa(list(C),list(B),map(C,B,F),Xs)),A3) = aa(A,A,foldr(C,A,aa(fun(C,B),fun(C,fun(A,A)),aa(fun(B,fun(A,A)),fun(fun(C,B),fun(C,fun(A,A))),comp(B,fun(A,A),C),G),F),Xs),A3) ).
% foldr_map
tff(fact_2019_execute__upd_I1_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [I: nat,H: heap_ext(product_unit),A3: array(A),X2: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(array(A),nat,length(A,H),A3)))
=> ( aa(heap_ext(product_unit),option(product_prod(array(A),heap_ext(product_unit))),aa(heap_Heap(array(A)),fun(heap_ext(product_unit),option(product_prod(array(A),heap_ext(product_unit)))),heap_execute(array(A)),upd(A,I,X2,A3)),H) = aa(product_prod(array(A),heap_ext(product_unit)),option(product_prod(array(A),heap_ext(product_unit))),some(product_prod(array(A),heap_ext(product_unit))),aa(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit)),aa(array(A),fun(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit))),product_Pair(array(A),heap_ext(product_unit)),A3),update(A,A3,I,X2,H))) ) ) ) ).
% execute_upd(1)
tff(fact_2020_execute__nth_I1_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [I: nat,H: heap_ext(product_unit),A3: array(A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(array(A),nat,length(A,H),A3)))
=> ( aa(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))),aa(heap_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_execute(A),aa(nat,heap_Heap(A),nth(A,A3),I)),H) = aa(product_prod(A,heap_ext(product_unit)),option(product_prod(A,heap_ext(product_unit))),some(product_prod(A,heap_ext(product_unit))),aa(heap_ext(product_unit),product_prod(A,heap_ext(product_unit)),aa(A,fun(heap_ext(product_unit),product_prod(A,heap_ext(product_unit))),product_Pair(A,heap_ext(product_unit)),aa(nat,A,nth3(A,aa(array(A),list(A),get(A,H),A3)),I)),H)) ) ) ) ).
% execute_nth(1)
tff(fact_2021_execute__swap_I1_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [I: nat,H: heap_ext(product_unit),A3: array(A),X2: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(array(A),nat,length(A,H),A3)))
=> ( aa(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))),aa(heap_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_execute(A),swap(A,I,X2,A3)),H) = aa(product_prod(A,heap_ext(product_unit)),option(product_prod(A,heap_ext(product_unit))),some(product_prod(A,heap_ext(product_unit))),aa(heap_ext(product_unit),product_prod(A,heap_ext(product_unit)),aa(A,fun(heap_ext(product_unit),product_prod(A,heap_ext(product_unit))),product_Pair(A,heap_ext(product_unit)),aa(nat,A,nth3(A,aa(array(A),list(A),get(A,H),A3)),I)),update(A,A3,I,X2,H))) ) ) ) ).
% execute_swap(1)
tff(fact_2022_finite__range__Some,axiom,
! [A: $tType] :
( finite_finite2(option(A),aa(set(A),set(option(A)),aa(fun(A,option(A)),fun(set(A),set(option(A))),image2(A,option(A)),some(A)),top_top(set(A))))
<=> finite_finite2(A,top_top(set(A))) ) ).
% finite_range_Some
tff(fact_2023_image__map__upd,axiom,
! [B: $tType,A: $tType,X2: A,A6: set(A),M4: fun(A,option(B)),Y2: B] :
( ~ pp(aa(set(A),bool,member(A,X2),A6))
=> ( aa(set(A),set(option(B)),aa(fun(A,option(B)),fun(set(A),set(option(B))),image2(A,option(B)),fun_upd(A,option(B),M4,X2,aa(B,option(B),some(B),Y2))),A6) = aa(set(A),set(option(B)),aa(fun(A,option(B)),fun(set(A),set(option(B))),image2(A,option(B)),M4),A6) ) ) ).
% image_map_upd
tff(fact_2024_ctor__rec__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,R3: fun(A,fun(C,bool)),S3: fun(B,fun(D,bool))] : pp(aa(fun(fun(C,D),fun(C,D)),bool,aa(fun(fun(A,B),fun(A,B)),fun(fun(fun(C,D),fun(C,D)),bool),bNF_rel_fun(fun(A,B),fun(C,D),fun(A,B),fun(C,D),bNF_rel_fun(A,C,B,D,aa(fun(A,fun(C,bool)),fun(A,fun(C,bool)),bNF_vimage2p(A,A,C,C,bool,bNF_id_bnf(A),bNF_id_bnf(C)),R3),S3),bNF_rel_fun(A,C,B,D,R3,S3)),basic_BNF_ctor_rec(fun(A,B))),basic_BNF_ctor_rec(fun(C,D)))) ).
% ctor_rec_transfer
tff(fact_2025_finite__range__updI,axiom,
! [A: $tType,B: $tType,F: fun(B,option(A)),A3: B,B2: A] :
( finite_finite2(option(A),aa(set(B),set(option(A)),aa(fun(B,option(A)),fun(set(B),set(option(A))),image2(B,option(A)),F),top_top(set(B))))
=> finite_finite2(option(A),aa(set(B),set(option(A)),aa(fun(B,option(A)),fun(set(B),set(option(A))),image2(B,option(A)),fun_upd(B,option(A),F,A3,aa(A,option(A),some(A),B2))),top_top(set(B)))) ) ).
% finite_range_updI
tff(fact_2026_execute__assert_I1_J,axiom,
! [A: $tType,P2: fun(A,bool),X2: A,H: heap_ext(product_unit)] :
( pp(aa(A,bool,P2,X2))
=> ( aa(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))),aa(heap_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_execute(A),heap_assert(A,P2,X2)),H) = aa(product_prod(A,heap_ext(product_unit)),option(product_prod(A,heap_ext(product_unit))),some(product_prod(A,heap_ext(product_unit))),aa(heap_ext(product_unit),product_prod(A,heap_ext(product_unit)),aa(A,fun(heap_ext(product_unit),product_prod(A,heap_ext(product_unit))),product_Pair(A,heap_ext(product_unit)),X2),H)) ) ) ).
% execute_assert(1)
tff(fact_2027_execute__bind__success,axiom,
! [B: $tType,A: $tType,F: heap_Heap(A),H: heap_ext(product_unit),G: fun(A,heap_Heap(B))] :
( heap_success(A,F,H)
=> ( aa(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit))),aa(heap_Heap(B),fun(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit)))),heap_execute(B),heap_bind(A,B,F,G)),H) = aa(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit))),aa(heap_Heap(B),fun(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit)))),heap_execute(B),aa(A,heap_Heap(B),G,aa(product_prod(A,heap_ext(product_unit)),A,product_fst(A,heap_ext(product_unit)),aa(option(product_prod(A,heap_ext(product_unit))),product_prod(A,heap_ext(product_unit)),the2(product_prod(A,heap_ext(product_unit))),aa(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))),aa(heap_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_execute(A),F),H))))),aa(product_prod(A,heap_ext(product_unit)),heap_ext(product_unit),product_snd(A,heap_ext(product_unit)),aa(option(product_prod(A,heap_ext(product_unit))),product_prod(A,heap_ext(product_unit)),the2(product_prod(A,heap_ext(product_unit))),aa(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))),aa(heap_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_execute(A),F),H)))) ) ) ).
% execute_bind_success
tff(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 = aa(product_prod(A,heap_ext(product_unit)),A,product_fst(A,heap_ext(product_unit)),aa(option(product_prod(A,heap_ext(product_unit))),product_prod(A,heap_ext(product_unit)),the2(product_prod(A,heap_ext(product_unit))),aa(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))),aa(heap_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_execute(A),C3),H))) )
=> ( ( H2 = aa(product_prod(A,heap_ext(product_unit)),heap_ext(product_unit),product_snd(A,heap_ext(product_unit)),aa(option(product_prod(A,heap_ext(product_unit))),product_prod(A,heap_ext(product_unit)),the2(product_prod(A,heap_ext(product_unit))),aa(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))),aa(heap_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_execute(A),C3),H))) )
=> ~ heap_success(A,C3,H) ) ) ) ).
% effectE
tff(fact_2029_comp__the__Some,axiom,
! [A: $tType] : aa(fun(A,option(A)),fun(A,A),aa(fun(option(A),A),fun(fun(A,option(A)),fun(A,A)),comp(option(A),A,A),the2(A)),some(A)) = id(A) ).
% comp_the_Some
tff(fact_2030_execute__ref,axiom,
! [A: $tType] :
( heap(A)
=> ! [V: A,H: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(ref(A),heap_ext(product_unit))),aa(heap_Heap(ref(A)),fun(heap_ext(product_unit),option(product_prod(ref(A),heap_ext(product_unit)))),heap_execute(ref(A)),aa(A,heap_Heap(ref(A)),ref3(A),V)),H) = aa(product_prod(ref(A),heap_ext(product_unit)),option(product_prod(ref(A),heap_ext(product_unit))),some(product_prod(ref(A),heap_ext(product_unit))),aa(heap_ext(product_unit),product_prod(ref(A),heap_ext(product_unit)),alloc2(A,V),H)) ) ).
% execute_ref
tff(fact_2031_execute__len,axiom,
! [A: $tType] :
( heap(A)
=> ! [A3: array(A),H: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(nat,heap_ext(product_unit))),aa(heap_Heap(nat),fun(heap_ext(product_unit),option(product_prod(nat,heap_ext(product_unit)))),heap_execute(nat),len(A,A3)),H) = aa(product_prod(nat,heap_ext(product_unit)),option(product_prod(nat,heap_ext(product_unit))),some(product_prod(nat,heap_ext(product_unit))),aa(heap_ext(product_unit),product_prod(nat,heap_ext(product_unit)),aa(nat,fun(heap_ext(product_unit),product_prod(nat,heap_ext(product_unit))),product_Pair(nat,heap_ext(product_unit)),aa(array(A),nat,length(A,H),A3)),H)) ) ).
% execute_len
tff(fact_2032_execute__freeze,axiom,
! [A: $tType] :
( heap(A)
=> ! [A3: array(A),H: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(list(A),heap_ext(product_unit))),aa(heap_Heap(list(A)),fun(heap_ext(product_unit),option(product_prod(list(A),heap_ext(product_unit)))),heap_execute(list(A)),freeze(A,A3)),H) = aa(product_prod(list(A),heap_ext(product_unit)),option(product_prod(list(A),heap_ext(product_unit))),some(product_prod(list(A),heap_ext(product_unit))),aa(heap_ext(product_unit),product_prod(list(A),heap_ext(product_unit)),aa(list(A),fun(heap_ext(product_unit),product_prod(list(A),heap_ext(product_unit))),product_Pair(list(A),heap_ext(product_unit)),aa(array(A),list(A),get(A,H),A3)),H)) ) ).
% execute_freeze
tff(fact_2033_Basic__BNF__LFPs_Octor__rec__def,axiom,
! [A: $tType,X2: A] : aa(A,A,basic_BNF_ctor_rec(A),X2) = X2 ).
% Basic_BNF_LFPs.ctor_rec_def
tff(fact_2034_effect__assertI,axiom,
! [A: $tType,P2: fun(A,bool),X2: A,H2: heap_ext(product_unit),H: heap_ext(product_unit),R: A] :
( pp(aa(A,bool,P2,X2))
=> ( ( H2 = H )
=> ( ( R = X2 )
=> heap_effect(A,heap_assert(A,P2,X2),H,H2,R) ) ) ) ).
% effect_assertI
tff(fact_2035_effect__assertE,axiom,
! [A: $tType,P2: fun(A,bool),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)
=> ~ ( pp(aa(A,bool,P2,X2))
=> ( ( R = X2 )
=> ( H2 != H ) ) ) ) ).
% effect_assertE
tff(fact_2036_success__assertI,axiom,
! [A: $tType,P2: fun(A,bool),X2: A,H: heap_ext(product_unit)] :
( pp(aa(A,bool,P2,X2))
=> heap_success(A,heap_assert(A,P2,X2),H) ) ).
% success_assertI
tff(fact_2037_assert__cong,axiom,
! [B: $tType,A: $tType,P2: fun(A,bool),P6: fun(A,bool),F: fun(A,heap_Heap(B)),F4: fun(A,heap_Heap(B)),X2: A] :
( ( P2 = P6 )
=> ( ! [X: A] :
( pp(aa(A,bool,P6,X))
=> ( aa(A,heap_Heap(B),F,X) = aa(A,heap_Heap(B),F4,X) ) )
=> ( heap_bind(A,B,heap_assert(A,P2,X2),F) = heap_bind(A,B,heap_assert(A,P6,X2),F4) ) ) ) ).
% assert_cong
tff(fact_2038_Basic__BNF__LFPs_Octor__rec__o__map,axiom,
! [B: $tType,C: $tType,A: $tType,F: fun(C,B),G: fun(A,C)] : aa(fun(A,C),fun(A,B),aa(fun(C,B),fun(fun(A,C),fun(A,B)),comp(C,B,A),aa(fun(C,B),fun(C,B),basic_BNF_ctor_rec(fun(C,B)),F)),G) = aa(fun(A,B),fun(A,B),basic_BNF_ctor_rec(fun(A,B)),aa(fun(A,C),fun(A,B),aa(fun(C,B),fun(fun(A,C),fun(A,B)),comp(C,B,A),F),aa(fun(A,A),fun(A,C),aa(fun(A,C),fun(fun(A,A),fun(A,C)),comp(A,C,A),aa(fun(A,C),fun(A,C),aa(fun(C,C),fun(fun(A,C),fun(A,C)),comp(C,C,A),bNF_id_bnf(C)),G)),bNF_id_bnf(A)))) ).
% Basic_BNF_LFPs.ctor_rec_o_map
tff(fact_2039_Basic__BNF__LFPs_Octor__rec__def__alt,axiom,
! [B: $tType,A: $tType,F: fun(A,B)] : F = aa(fun(A,B),fun(A,B),basic_BNF_ctor_rec(fun(A,B)),aa(fun(A,A),fun(A,B),aa(fun(A,B),fun(fun(A,A),fun(A,B)),comp(A,B,A),F),bNF_id_bnf(A))) ).
% Basic_BNF_LFPs.ctor_rec_def_alt
tff(fact_2040_execute__fold__map__unchanged__heap,axiom,
! [A: $tType,B: $tType,Xs: list(A),F: fun(A,heap_Heap(B)),H: heap_ext(product_unit)] :
( ! [X: A] :
( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set3(A),Xs)))
=> ? [Y5: B] : aa(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit))),aa(heap_Heap(B),fun(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit)))),heap_execute(B),aa(A,heap_Heap(B),F,X)),H) = aa(product_prod(B,heap_ext(product_unit)),option(product_prod(B,heap_ext(product_unit))),some(product_prod(B,heap_ext(product_unit))),aa(heap_ext(product_unit),product_prod(B,heap_ext(product_unit)),aa(B,fun(heap_ext(product_unit),product_prod(B,heap_ext(product_unit))),product_Pair(B,heap_ext(product_unit)),Y5),H)) )
=> ( aa(heap_ext(product_unit),option(product_prod(list(B),heap_ext(product_unit))),aa(heap_Heap(list(B)),fun(heap_ext(product_unit),option(product_prod(list(B),heap_ext(product_unit)))),heap_execute(list(B)),heap_fold_map(A,B,F,Xs)),H) = aa(product_prod(list(B),heap_ext(product_unit)),option(product_prod(list(B),heap_ext(product_unit))),some(product_prod(list(B),heap_ext(product_unit))),aa(heap_ext(product_unit),product_prod(list(B),heap_ext(product_unit)),aa(list(B),fun(heap_ext(product_unit),product_prod(list(B),heap_ext(product_unit))),product_Pair(list(B),heap_ext(product_unit)),aa(list(A),list(B),map(A,B,aa(heap_ext(product_unit),fun(A,B),aTP_Lamp_ru(fun(A,heap_Heap(B)),fun(heap_ext(product_unit),fun(A,B)),F),H)),Xs)),H)) ) ) ).
% execute_fold_map_unchanged_heap
tff(fact_2041_map__of__mapk__SomeI,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(A,B),T2: list(product_prod(A,C)),K: A,X2: C] :
( inj_on(A,B,F,top_top(set(A)))
=> ( ( aa(A,option(C),map_of(A,C,T2),K) = aa(C,option(C),some(C),X2) )
=> ( aa(B,option(C),map_of(B,C,aa(list(product_prod(A,C)),list(product_prod(B,C)),map(product_prod(A,C),product_prod(B,C),aa(fun(A,fun(C,product_prod(B,C))),fun(product_prod(A,C),product_prod(B,C)),product_case_prod(A,C,product_prod(B,C)),aTP_Lamp_rv(fun(A,B),fun(A,fun(C,product_prod(B,C))),F))),T2)),aa(A,B,F,K)) = aa(C,option(C),some(C),X2) ) ) ) ).
% map_of_mapk_SomeI
tff(fact_2042_Basic__BNF__LFPs_Octor__rec,axiom,
! [B: $tType,A: $tType,G: fun(A,A),F: fun(A,B),X2: A] :
( ( G = id(A) )
=> ( aa(A,B,aa(fun(A,B),fun(A,B),basic_BNF_ctor_rec(fun(A,B)),F),aa(A,A,basic_BNF_xtor(A),X2)) = aa(A,B,F,aa(A,A,aa(fun(A,A),fun(A,A),aa(fun(A,A),fun(fun(A,A),fun(A,A)),comp(A,A,A),aa(fun(A,A),fun(A,A),aa(fun(A,A),fun(fun(A,A),fun(A,A)),comp(A,A,A),bNF_id_bnf(A)),G)),bNF_id_bnf(A)),X2)) ) ) ).
% Basic_BNF_LFPs.ctor_rec
tff(fact_2043_ctor__rec__unique,axiom,
! [B: $tType,A: $tType,G: fun(A,A),F: fun(A,B),S: fun(A,B)] :
( ( G = id(A) )
=> ( ( aa(fun(A,A),fun(A,B),aa(fun(A,B),fun(fun(A,A),fun(A,B)),comp(A,B,A),F),basic_BNF_xtor(A)) = aa(fun(A,A),fun(A,B),aa(fun(A,B),fun(fun(A,A),fun(A,B)),comp(A,B,A),S),aa(fun(A,A),fun(A,A),aa(fun(A,A),fun(fun(A,A),fun(A,A)),comp(A,A,A),aa(fun(A,A),fun(A,A),aa(fun(A,A),fun(fun(A,A),fun(A,A)),comp(A,A,A),bNF_id_bnf(A)),G)),bNF_id_bnf(A))) )
=> ( F = aa(fun(A,B),fun(A,B),basic_BNF_ctor_rec(fun(A,B)),S) ) ) ) ).
% ctor_rec_unique
tff(fact_2044_these__image__Some__eq,axiom,
! [A: $tType,A6: set(A)] : these(A,aa(set(A),set(option(A)),aa(fun(A,option(A)),fun(set(A),set(option(A))),image2(A,option(A)),some(A)),A6)) = A6 ).
% these_image_Some_eq
tff(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,aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),P),Ps)) = fun_upd(A,option(B),map_of(A,B,Ps),aa(product_prod(A,B),A,product_fst(A,B),P),aa(B,option(B),some(B),aa(product_prod(A,B),B,product_snd(A,B),P))) ).
% map_of.simps(2)
tff(fact_2046_execute__map__entry_I2_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [H: heap_ext(product_unit),A3: array(A),I: nat,F: fun(A,A)] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(array(A),nat,length(A,H),A3)),I))
=> ( aa(heap_ext(product_unit),option(product_prod(array(A),heap_ext(product_unit))),aa(heap_Heap(array(A)),fun(heap_ext(product_unit),option(product_prod(array(A),heap_ext(product_unit)))),heap_execute(array(A)),map_entry(A,I,F,A3)),H) = none(product_prod(array(A),heap_ext(product_unit))) ) ) ) ).
% execute_map_entry(2)
tff(fact_2047_xtor__iff__xtor,axiom,
! [A: $tType,U: A,W2: A] :
( ( U = aa(A,A,basic_BNF_xtor(A),W2) )
<=> ( aa(A,A,basic_BNF_xtor(A),U) = W2 ) ) ).
% xtor_iff_xtor
tff(fact_2048_Basic__BNF__LFPs_Oxtor__inject,axiom,
! [A: $tType,X2: A,Y2: A] :
( ( aa(A,A,basic_BNF_xtor(A),X2) = aa(A,A,basic_BNF_xtor(A),Y2) )
<=> ( X2 = Y2 ) ) ).
% Basic_BNF_LFPs.xtor_inject
tff(fact_2049_Basic__BNF__LFPs_Oxtor__induct,axiom,
! [A: $tType,P2: fun(A,bool),Z2: A] :
( ! [X: A] : pp(aa(A,bool,P2,aa(A,A,basic_BNF_xtor(A),X)))
=> pp(aa(A,bool,P2,Z2)) ) ).
% Basic_BNF_LFPs.xtor_induct
tff(fact_2050_Basic__BNF__LFPs_Oxtor__xtor,axiom,
! [A: $tType,X2: A] : aa(A,A,basic_BNF_xtor(A),aa(A,A,basic_BNF_xtor(A),X2)) = X2 ).
% Basic_BNF_LFPs.xtor_xtor
tff(fact_2051_Basic__BNF__LFPs_Oxtor__set,axiom,
! [A: $tType,B: $tType,F: fun(B,A),X2: B] : aa(B,A,F,aa(B,B,basic_BNF_xtor(B),X2)) = aa(B,A,F,X2) ).
% Basic_BNF_LFPs.xtor_set
tff(fact_2052_Basic__BNF__LFPs_Oxtor__rel,axiom,
! [B: $tType,A: $tType,C: $tType,R3: fun(B,fun(C,A)),X2: B,Y2: C] : aa(C,A,aa(B,fun(C,A),R3,aa(B,B,basic_BNF_xtor(B),X2)),aa(C,C,basic_BNF_xtor(C),Y2)) = aa(C,A,aa(B,fun(C,A),R3,X2),Y2) ).
% Basic_BNF_LFPs.xtor_rel
tff(fact_2053_Basic__BNF__LFPs_Oxtor__map,axiom,
! [A: $tType,B: $tType,F: fun(B,A),X2: B] : aa(B,A,F,aa(B,B,basic_BNF_xtor(B),X2)) = aa(A,A,basic_BNF_xtor(A),aa(B,A,F,X2)) ).
% Basic_BNF_LFPs.xtor_map
tff(fact_2054_Basic__BNF__LFPs_Oxtor__def,axiom,
! [A: $tType,X2: A] : aa(A,A,basic_BNF_xtor(A),X2) = X2 ).
% Basic_BNF_LFPs.xtor_def
tff(fact_2055_Option_Othese__def,axiom,
! [A: $tType,A6: set(option(A))] : these(A,A6) = aa(set(option(A)),set(A),aa(fun(option(A),A),fun(set(option(A)),set(A)),image2(option(A),A),the2(A)),aa(fun(option(A),bool),set(option(A)),collect(option(A)),aTP_Lamp_rw(set(option(A)),fun(option(A),bool),A6))) ).
% Option.these_def
tff(fact_2056_map__of__eq__None__iff,axiom,
! [A: $tType,B: $tType,Xys: list(product_prod(B,A)),X2: B] :
( ( aa(B,option(A),map_of(B,A,Xys),X2) = none(A) )
<=> ~ pp(aa(set(B),bool,member(B,X2),aa(set(product_prod(B,A)),set(B),aa(fun(product_prod(B,A),B),fun(set(product_prod(B,A)),set(B)),image2(product_prod(B,A),B),product_fst(B,A)),aa(list(product_prod(B,A)),set(product_prod(B,A)),set3(product_prod(B,A)),Xys)))) ) ).
% map_of_eq_None_iff
tff(fact_2057_xtor__map__unique,axiom,
! [B: $tType,A: $tType,U: fun(A,B),F: fun(A,B)] :
( ( aa(fun(A,A),fun(A,B),aa(fun(A,B),fun(fun(A,A),fun(A,B)),comp(A,B,A),U),basic_BNF_xtor(A)) = aa(fun(A,B),fun(A,B),aa(fun(B,B),fun(fun(A,B),fun(A,B)),comp(B,B,A),basic_BNF_xtor(B)),F) )
=> ( U = F ) ) ).
% xtor_map_unique
tff(fact_2058_Some__image__these__eq,axiom,
! [A: $tType,A6: set(option(A))] : aa(set(A),set(option(A)),aa(fun(A,option(A)),fun(set(A),set(option(A))),image2(A,option(A)),some(A)),these(A,A6)) = aa(fun(option(A),bool),set(option(A)),collect(option(A)),aTP_Lamp_rw(set(option(A)),fun(option(A),bool),A6)) ).
% Some_image_these_eq
tff(fact_2059_Basic__BNF__LFPs_OPair__def__alt,axiom,
! [A: $tType,B: $tType,X4: A,Xa4: B] : aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa4) = aa(product_prod(A,B),product_prod(A,B),basic_BNF_xtor(product_prod(A,B)),aa(product_prod(A,B),product_prod(A,B),bNF_id_bnf(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa4))) ).
% Basic_BNF_LFPs.Pair_def_alt
tff(fact_2060_fold__map_Osimps_I1_J,axiom,
! [A: $tType,B: $tType,F: fun(A,heap_Heap(B))] : heap_fold_map(A,B,F,nil(A)) = aa(list(B),heap_Heap(list(B)),heap_return(list(B)),nil(B)) ).
% fold_map.simps(1)
tff(fact_2061_success__def,axiom,
! [A: $tType,F: heap_Heap(A),H: heap_ext(product_unit)] :
( heap_success(A,F,H)
<=> ( aa(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))),aa(heap_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_execute(A),F),H) != none(product_prod(A,heap_ext(product_unit))) ) ) ).
% success_def
tff(fact_2062_successI,axiom,
! [A: $tType,F: heap_Heap(A),H: heap_ext(product_unit)] :
( ( aa(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))),aa(heap_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_execute(A),F),H) != none(product_prod(A,heap_ext(product_unit))) )
=> heap_success(A,F,H) ) ).
% successI
tff(fact_2063_execute__bind_I2_J,axiom,
! [A: $tType,B: $tType,F: heap_Heap(A),H: heap_ext(product_unit),G: fun(A,heap_Heap(B))] :
( ( aa(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))),aa(heap_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_execute(A),F),H) = none(product_prod(A,heap_ext(product_unit))) )
=> ( aa(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit))),aa(heap_Heap(B),fun(heap_ext(product_unit),option(product_prod(B,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)
tff(fact_2064_None__notin__image__Some,axiom,
! [A: $tType,A6: set(A)] : ~ pp(aa(set(option(A)),bool,member(option(A),none(A)),aa(set(A),set(option(A)),aa(fun(A,option(A)),fun(set(A),set(option(A))),image2(A,option(A)),some(A)),A6))) ).
% None_notin_image_Some
tff(fact_2065_execute__guard_I1_J,axiom,
! [A: $tType,P2: fun(heap_ext(product_unit),bool),H: heap_ext(product_unit),F: fun(heap_ext(product_unit),product_prod(A,heap_ext(product_unit)))] :
( ~ pp(aa(heap_ext(product_unit),bool,P2,H))
=> ( aa(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))),aa(heap_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_execute(A),heap_guard(A,P2,F)),H) = none(product_prod(A,heap_ext(product_unit))) ) ) ).
% execute_guard(1)
tff(fact_2066_execute__assert_I2_J,axiom,
! [A: $tType,P2: fun(A,bool),X2: A,H: heap_ext(product_unit)] :
( ~ pp(aa(A,bool,P2,X2))
=> ( aa(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))),aa(heap_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_execute(A),heap_assert(A,P2,X2)),H) = none(product_prod(A,heap_ext(product_unit))) ) ) ).
% execute_assert(2)
tff(fact_2067_execute__raise,axiom,
! [A: $tType,S: literal,X4: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))),aa(heap_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_execute(A),heap_raise(A,S)),X4) = none(product_prod(A,heap_ext(product_unit))) ).
% execute_raise
tff(fact_2068_fold__map_Osimps_I2_J,axiom,
! [B: $tType,A: $tType,F: fun(A,heap_Heap(B)),X2: A,Xs: list(A)] : heap_fold_map(A,B,F,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)) = heap_bind(B,list(B),aa(A,heap_Heap(B),F,X2),aa(list(A),fun(B,heap_Heap(list(B))),aTP_Lamp_ry(fun(A,heap_Heap(B)),fun(list(A),fun(B,heap_Heap(list(B)))),F),Xs)) ).
% fold_map.simps(2)
tff(fact_2069_fold__map__append,axiom,
! [A: $tType,B: $tType,F: fun(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),aa(list(B),fun(list(A),heap_Heap(list(A))),aTP_Lamp_sa(fun(B,heap_Heap(A)),fun(list(B),fun(list(A),heap_Heap(list(A)))),F),Ys)) ).
% fold_map_append
tff(fact_2070_weak__map__of__SomeI,axiom,
! [A: $tType,B: $tType,K: A,X2: B,L: list(product_prod(A,B))] :
( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),X2)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set3(product_prod(A,B)),L)))
=> ? [X: B] : aa(A,option(B),map_of(A,B,L),K) = aa(B,option(B),some(B),X) ) ).
% weak_map_of_SomeI
tff(fact_2071_map__of__SomeD,axiom,
! [A: $tType,B: $tType,Xs: list(product_prod(B,A)),K: B,Y2: A] :
( ( aa(B,option(A),map_of(B,A,Xs),K) = aa(A,option(A),some(A),Y2) )
=> pp(aa(set(product_prod(B,A)),bool,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),K),Y2)),aa(list(product_prod(B,A)),set(product_prod(B,A)),set3(product_prod(B,A)),Xs))) ) ).
% map_of_SomeD
tff(fact_2072_map__of__eqI,axiom,
! [B: $tType,A: $tType,Xs: list(product_prod(A,B)),Ys: list(product_prod(A,B))] :
( ( aa(list(A),set(A),set3(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs)) = aa(list(A),set(A),set3(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Ys)) )
=> ( ! [X: A] :
( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set3(A),aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs))))
=> ( aa(A,option(B),map_of(A,B,Xs),X) = aa(A,option(B),map_of(A,B,Ys),X) ) )
=> ( map_of(A,B,Xs) = map_of(A,B,Ys) ) ) ) ).
% map_of_eqI
tff(fact_2073_notin__range__Some,axiom,
! [A: $tType,X2: option(A)] :
( ~ pp(aa(set(option(A)),bool,member(option(A),X2),aa(set(A),set(option(A)),aa(fun(A,option(A)),fun(set(A),set(option(A))),image2(A,option(A)),some(A)),top_top(set(A)))))
<=> ( X2 = none(A) ) ) ).
% notin_range_Some
tff(fact_2074_UNIV__option__conv,axiom,
! [A: $tType] : top_top(set(option(A))) = aa(set(option(A)),set(option(A)),insert(option(A),none(A)),aa(set(A),set(option(A)),aa(fun(A,option(A)),fun(set(A),set(option(A))),image2(A,option(A)),some(A)),top_top(set(A)))) ).
% UNIV_option_conv
tff(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) )
=> ( aa(set(product_prod(A,B)),set(A),aa(fun(product_prod(A,B),A),fun(set(product_prod(A,B)),set(A)),image2(product_prod(A,B),A),product_fst(A,B)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set3(product_prod(A,B)),Xs)) = aa(set(product_prod(A,B)),set(A),aa(fun(product_prod(A,B),A),fun(set(product_prod(A,B)),set(A)),image2(product_prod(A,B),A),product_fst(A,B)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set3(product_prod(A,B)),Ys)) ) ) ).
% map_of_eq_dom
tff(fact_2076_Heap__cases,axiom,
! [A: $tType,F: heap_Heap(A),H: heap_ext(product_unit)] :
( ! [X: A,H4: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))),aa(heap_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_execute(A),F),H) != aa(product_prod(A,heap_ext(product_unit)),option(product_prod(A,heap_ext(product_unit))),some(product_prod(A,heap_ext(product_unit))),aa(heap_ext(product_unit),product_prod(A,heap_ext(product_unit)),aa(A,fun(heap_ext(product_unit),product_prod(A,heap_ext(product_unit))),product_Pair(A,heap_ext(product_unit)),X),H4))
=> ( aa(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))),aa(heap_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_execute(A),F),H) = none(product_prod(A,heap_ext(product_unit))) ) ) ).
% Heap_cases
tff(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 )
=> ( aa(B,option(C),map_of(B,C,aa(list(product_prod(B,C)),list(product_prod(B,C)),aa(product_prod(B,C),fun(list(product_prod(B,C)),list(product_prod(B,C))),cons(product_prod(B,C)),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),L),V)),Ps)),K) = aa(C,option(C),some(C),V) ) )
& ( ( L != K )
=> ( aa(B,option(C),map_of(B,C,aa(list(product_prod(B,C)),list(product_prod(B,C)),aa(product_prod(B,C),fun(list(product_prod(B,C)),list(product_prod(B,C))),cons(product_prod(B,C)),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),L),V)),Ps)),K) = aa(B,option(C),map_of(B,C,Ps),K) ) ) ) ).
% map_of_Cons_code(2)
tff(fact_2078_finite__range__map__of,axiom,
! [A: $tType,B: $tType,Xys: list(product_prod(B,A))] : finite_finite2(option(A),aa(set(B),set(option(A)),aa(fun(B,option(A)),fun(set(B),set(option(A))),image2(B,option(A)),map_of(B,A,Xys)),top_top(set(B)))) ).
% finite_range_map_of
tff(fact_2079_execute__nth_I2_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [H: heap_ext(product_unit),A3: array(A),I: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(array(A),nat,length(A,H),A3)),I))
=> ( aa(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))),aa(heap_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_execute(A),aa(nat,heap_Heap(A),nth(A,A3),I)),H) = none(product_prod(A,heap_ext(product_unit))) ) ) ) ).
% execute_nth(2)
tff(fact_2080_execute__swap_I2_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [H: heap_ext(product_unit),A3: array(A),I: nat,X2: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(array(A),nat,length(A,H),A3)),I))
=> ( aa(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))),aa(heap_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_execute(A),swap(A,I,X2,A3)),H) = none(product_prod(A,heap_ext(product_unit))) ) ) ) ).
% execute_swap(2)
tff(fact_2081_Basic__BNF__LFPs_Oxtor__rel__induct,axiom,
! [B: $tType,A: $tType,R3: fun(A,fun(B,bool)),IR: fun(A,fun(B,bool))] :
( ! [X: A,Y: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),bNF_vimage2p(A,A,B,B,bool,bNF_id_bnf(A),bNF_id_bnf(B)),R3),X),Y))
=> pp(aa(B,bool,aa(A,fun(B,bool),IR,aa(A,A,basic_BNF_xtor(A),X)),aa(B,B,basic_BNF_xtor(B),Y))) )
=> pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),R3),IR)) ) ).
% Basic_BNF_LFPs.xtor_rel_induct
tff(fact_2082_execute__upd_I2_J,axiom,
! [A: $tType] :
( heap(A)
=> ! [H: heap_ext(product_unit),A3: array(A),I: nat,X2: A] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(array(A),nat,length(A,H),A3)),I))
=> ( aa(heap_ext(product_unit),option(product_prod(array(A),heap_ext(product_unit))),aa(heap_Heap(array(A)),fun(heap_ext(product_unit),option(product_prod(array(A),heap_ext(product_unit)))),heap_execute(array(A)),upd(A,I,X2,A3)),H) = none(product_prod(array(A),heap_ext(product_unit))) ) ) ) ).
% execute_upd(2)
tff(fact_2083_graph__map__upd,axiom,
! [A: $tType,B: $tType,M4: fun(A,option(B)),K: A,V: B] : graph(A,B,fun_upd(A,option(B),M4,K,aa(B,option(B),some(B),V))) = aa(set(product_prod(A,B)),set(product_prod(A,B)),insert(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),V)),graph(A,B,fun_upd(A,option(B),M4,K,none(B)))) ).
% graph_map_upd
tff(fact_2084_map__of__map__restrict,axiom,
! [B: $tType,A: $tType,F: fun(A,B),Ks: list(A)] : map_of(A,B,aa(list(A),list(product_prod(A,B)),map(A,product_prod(A,B),aTP_Lamp_fq(fun(A,B),fun(A,product_prod(A,B)),F)),Ks)) = restrict_map(A,B,aa(fun(A,B),fun(A,option(B)),aa(fun(B,option(B)),fun(fun(A,B),fun(A,option(B))),comp(B,option(B),A),some(B)),F),aa(list(A),set(A),set3(A),Ks)) ).
% map_of_map_restrict
tff(fact_2085_map__of__is__SomeI,axiom,
! [A: $tType,B: $tType,Xys: list(product_prod(A,B)),X2: A,Y2: B] :
( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xys))
=> ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y2)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set3(product_prod(A,B)),Xys)))
=> ( aa(A,option(B),map_of(A,B,Xys),X2) = aa(B,option(B),some(B),Y2) ) ) ) ).
% map_of_is_SomeI
tff(fact_2086_Some__eq__map__of__iff,axiom,
! [B: $tType,A: $tType,Xys: list(product_prod(A,B)),Y2: B,X2: A] :
( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xys))
=> ( ( aa(B,option(B),some(B),Y2) = aa(A,option(B),map_of(A,B,Xys),X2) )
<=> pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y2)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set3(product_prod(A,B)),Xys))) ) ) ).
% Some_eq_map_of_iff
tff(fact_2087_map__of__eq__Some__iff,axiom,
! [B: $tType,A: $tType,Xys: list(product_prod(A,B)),X2: A,Y2: B] :
( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xys))
=> ( ( aa(A,option(B),map_of(A,B,Xys),X2) = aa(B,option(B),some(B),Y2) )
<=> pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y2)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set3(product_prod(A,B)),Xys))) ) ) ).
% map_of_eq_Some_iff
tff(fact_2088_set__map__of__compr,axiom,
! [B: $tType,A: $tType,Xs: list(product_prod(A,B))] :
( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs))
=> ( aa(list(product_prod(A,B)),set(product_prod(A,B)),set3(product_prod(A,B)),Xs) = aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aTP_Lamp_sb(list(product_prod(A,B)),fun(A,fun(B,bool)),Xs))) ) ) ).
% set_map_of_compr
tff(fact_2089_map__of__map__keys,axiom,
! [B: $tType,A: $tType,Xs: list(A),M4: fun(A,option(B))] :
( ( aa(list(A),set(A),set3(A),Xs) = dom(A,B,M4) )
=> ( map_of(A,B,aa(list(A),list(product_prod(A,B)),map(A,product_prod(A,B),aTP_Lamp_sc(fun(A,option(B)),fun(A,product_prod(A,B)),M4)),Xs)) = M4 ) ) ).
% map_of_map_keys
tff(fact_2090_graph__restrictD_I1_J,axiom,
! [B: $tType,A: $tType,K: A,V: B,M4: fun(A,option(B)),A6: set(A)] :
( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),V)),graph(A,B,restrict_map(A,B,M4,A6))))
=> pp(aa(set(A),bool,member(A,K),A6)) ) ).
% graph_restrictD(1)
tff(fact_2091_graph__domD,axiom,
! [B: $tType,A: $tType,X2: product_prod(A,B),M4: fun(A,option(B))] :
( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),X2),graph(A,B,M4)))
=> pp(aa(set(A),bool,member(A,aa(product_prod(A,B),A,product_fst(A,B),X2)),dom(A,B,M4))) ) ).
% graph_domD
tff(fact_2092_fst__graph__eq__dom,axiom,
! [B: $tType,A: $tType,M4: fun(A,option(B))] : aa(set(product_prod(A,B)),set(A),aa(fun(product_prod(A,B),A),fun(set(product_prod(A,B)),set(A)),image2(product_prod(A,B),A),product_fst(A,B)),graph(A,B,M4)) = dom(A,B,M4) ).
% fst_graph_eq_dom
tff(fact_2093_graph__restrictD_I2_J,axiom,
! [A: $tType,B: $tType,K: A,V: B,M4: fun(A,option(B)),A6: set(A)] :
( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),V)),graph(A,B,restrict_map(A,B,M4,A6))))
=> ( aa(A,option(B),M4,K) = aa(B,option(B),some(B),V) ) ) ).
% graph_restrictD(2)
tff(fact_2094_graph__eq__to__snd__dom,axiom,
! [B: $tType,A: $tType,M4: fun(A,option(B))] : graph(A,B,M4) = aa(set(A),set(product_prod(A,B)),aa(fun(A,product_prod(A,B)),fun(set(A),set(product_prod(A,B))),image2(A,product_prod(A,B)),aTP_Lamp_sc(fun(A,option(B)),fun(A,product_prod(A,B)),M4)),dom(A,B,M4)) ).
% graph_eq_to_snd_dom
tff(fact_2095_graph__map__of__if__distinct__dom,axiom,
! [B: $tType,A: $tType,Al: list(product_prod(A,B))] :
( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Al))
=> ( graph(A,B,map_of(A,B,Al)) = aa(list(product_prod(A,B)),set(product_prod(A,B)),set3(product_prod(A,B)),Al) ) ) ).
% graph_map_of_if_distinct_dom
tff(fact_2096_in__graphI,axiom,
! [A: $tType,B: $tType,M4: fun(B,option(A)),K: B,V: A] :
( ( aa(B,option(A),M4,K) = aa(A,option(A),some(A),V) )
=> pp(aa(set(product_prod(B,A)),bool,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),K),V)),graph(B,A,M4))) ) ).
% in_graphI
tff(fact_2097_in__graphD,axiom,
! [A: $tType,B: $tType,K: A,V: B,M4: fun(A,option(B))] :
( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),V)),graph(A,B,M4)))
=> ( aa(A,option(B),M4,K) = aa(B,option(B),some(B),V) ) ) ).
% in_graphD
tff(fact_2098_inj__on__fst__graph,axiom,
! [B: $tType,A: $tType,M4: fun(A,option(B))] : inj_on(product_prod(A,B),A,product_fst(A,B),graph(A,B,M4)) ).
% inj_on_fst_graph
tff(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,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs))
=> ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),V1)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set3(product_prod(A,B)),Xs)))
=> ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),V22)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set3(product_prod(A,B)),Xs)))
=> ( V1 = V22 ) ) ) ) ).
% eq_key_imp_eq_value
tff(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,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs))
=> ( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Ys))
=> ( ( map_of(A,B,Xs) = map_of(A,B,Ys) )
<=> ( aa(list(product_prod(A,B)),set(product_prod(A,B)),set3(product_prod(A,B)),Xs) = aa(list(product_prod(A,B)),set(product_prod(A,B)),set3(product_prod(A,B)),Ys) ) ) ) ) ).
% map_of_inject_set
tff(fact_2101_graph__def,axiom,
! [B: $tType,A: $tType,M4: fun(A,option(B))] : graph(A,B,M4) = aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aTP_Lamp_sd(fun(A,option(B)),fun(product_prod(A,B),bool),M4)) ).
% graph_def
tff(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)) = aa(set(product_prod(A,B)),set(A),aa(fun(product_prod(A,B),A),fun(set(product_prod(A,B)),set(A)),image2(product_prod(A,B),A),product_fst(A,B)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set3(product_prod(A,B)),Xys)) ).
% dom_map_of_conv_image_fst
tff(fact_2103_graph__fun__upd__None,axiom,
! [B: $tType,A: $tType,M4: fun(A,option(B)),K: A] : graph(A,B,fun_upd(A,option(B),M4,K,none(B))) = aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(A,fun(product_prod(A,B),bool),aTP_Lamp_se(fun(A,option(B)),fun(A,fun(product_prod(A,B),bool)),M4),K)) ).
% graph_fun_upd_None
tff(fact_2104_ran__distinct,axiom,
! [B: $tType,A: $tType,Al: list(product_prod(A,B))] :
( distinct(A,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Al))
=> ( ran(A,B,map_of(A,B,Al)) = aa(set(product_prod(A,B)),set(B),aa(fun(product_prod(A,B),B),fun(set(product_prod(A,B)),set(B)),image2(product_prod(A,B),B),product_snd(A,B)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set3(product_prod(A,B)),Al)) ) ) ).
% ran_distinct
tff(fact_2105_finite__range__map__of__map__add,axiom,
! [A: $tType,B: $tType,F: fun(B,option(A)),L: list(product_prod(B,A))] :
( finite_finite2(option(A),aa(set(B),set(option(A)),aa(fun(B,option(A)),fun(set(B),set(option(A))),image2(B,option(A)),F),top_top(set(B))))
=> finite_finite2(option(A),aa(set(B),set(option(A)),aa(fun(B,option(A)),fun(set(B),set(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
tff(fact_2106_heap__def,axiom,
! [A: $tType,F: fun(heap_ext(product_unit),product_prod(A,heap_ext(product_unit)))] : heap_heap(A,F) = aa(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_Heap(A),heap_Heap2(A),aa(fun(heap_ext(product_unit),product_prod(A,heap_ext(product_unit))),fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),aa(fun(product_prod(A,heap_ext(product_unit)),option(product_prod(A,heap_ext(product_unit)))),fun(fun(heap_ext(product_unit),product_prod(A,heap_ext(product_unit))),fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))))),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)) ).
% heap_def
tff(fact_2107_execute__bind__case,axiom,
! [A: $tType,B: $tType,F: heap_Heap(B),G: fun(B,heap_Heap(A)),H: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))),aa(heap_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,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))),aa(fun(B,fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))))),fun(product_prod(B,heap_ext(product_unit)),option(product_prod(A,heap_ext(product_unit)))),product_case_prod(B,heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),aTP_Lamp_sf(fun(B,heap_Heap(A)),fun(B,fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))))),G)),aa(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit))),aa(heap_Heap(B),fun(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit)))),heap_execute(B),F),H)) ).
% execute_bind_case
tff(fact_2108_map__of__map,axiom,
! [B: $tType,C: $tType,A: $tType,F: fun(C,B),Xs: list(product_prod(A,C))] : map_of(A,B,aa(list(product_prod(A,C)),list(product_prod(A,B)),map(product_prod(A,C),product_prod(A,B),aa(fun(A,fun(C,product_prod(A,B))),fun(product_prod(A,C),product_prod(A,B)),product_case_prod(A,C,product_prod(A,B)),aTP_Lamp_oz(fun(C,B),fun(A,fun(C,product_prod(A,B))),F))),Xs)) = aa(fun(A,option(C)),fun(A,option(B)),aa(fun(option(C),option(B)),fun(fun(A,option(C)),fun(A,option(B))),comp(option(C),option(B),A),map_option(C,B,F)),map_of(A,C,Xs)) ).
% map_of_map
tff(fact_2109_guard__def,axiom,
! [A: $tType,P2: fun(heap_ext(product_unit),bool),F: fun(heap_ext(product_unit),product_prod(A,heap_ext(product_unit)))] : heap_guard(A,P2,F) = aa(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_Heap(A),heap_Heap2(A),aa(fun(heap_ext(product_unit),product_prod(A,heap_ext(product_unit))),fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),aTP_Lamp_sg(fun(heap_ext(product_unit),bool),fun(fun(heap_ext(product_unit),product_prod(A,heap_ext(product_unit))),fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))))),P2),F)) ).
% guard_def
tff(fact_2110_tap__def,axiom,
! [A: $tType,F: fun(heap_ext(product_unit),A)] : heap_tap(A,F) = aa(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_Heap(A),heap_Heap2(A),aTP_Lamp_sh(fun(heap_ext(product_unit),A),fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),F)) ).
% tap_def
tff(fact_2111_Heap_Oinject,axiom,
! [A: $tType,X2: fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),Ya: fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))))] :
( ( aa(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_Heap(A),heap_Heap2(A),X2) = aa(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_Heap(A),heap_Heap2(A),Ya) )
<=> ( X2 = Ya ) ) ).
% Heap.inject
tff(fact_2112_Heap__execute,axiom,
! [A: $tType,F: heap_Heap(A)] : aa(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_Heap(A),heap_Heap2(A),aa(heap_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_execute(A),F)) = F ).
% Heap_execute
tff(fact_2113_case__map__option,axiom,
! [B: $tType,A: $tType,C: $tType,G: A,H: fun(B,A),F: fun(C,B),X2: option(C)] : case_option(A,B,G,H,aa(option(C),option(B),map_option(C,B,F),X2)) = case_option(A,C,G,aa(fun(C,B),fun(C,A),aa(fun(B,A),fun(fun(C,B),fun(C,A)),comp(B,A,C),H),F),X2) ).
% case_map_option
tff(fact_2114_dom__map__option__comp,axiom,
! [B: $tType,C: $tType,A: $tType,G: fun(C,B),M4: fun(A,option(C))] : dom(A,B,aa(fun(A,option(C)),fun(A,option(B)),aa(fun(option(C),option(B)),fun(fun(A,option(C)),fun(A,option(B))),comp(option(C),option(B),A),map_option(C,B,G)),M4)) = dom(A,C,M4) ).
% dom_map_option_comp
tff(fact_2115_map__option__o__empty,axiom,
! [A: $tType,C: $tType,B: $tType,F: fun(C,B),X4: A] : aa(A,option(B),aa(fun(A,option(C)),fun(A,option(B)),aa(fun(option(C),option(B)),fun(fun(A,option(C)),fun(A,option(B))),comp(option(C),option(B),A),map_option(C,B,F)),aTP_Lamp_si(A,option(C))),X4) = none(B) ).
% map_option_o_empty
tff(fact_2116_map__option__o__map__upd,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(C,B),M4: fun(A,option(C)),A3: A,B2: C] : aa(fun(A,option(C)),fun(A,option(B)),aa(fun(option(C),option(B)),fun(fun(A,option(C)),fun(A,option(B))),comp(option(C),option(B),A),map_option(C,B,F)),fun_upd(A,option(C),M4,A3,aa(C,option(C),some(C),B2))) = fun_upd(A,option(B),aa(fun(A,option(C)),fun(A,option(B)),aa(fun(option(C),option(B)),fun(fun(A,option(C)),fun(A,option(B))),comp(option(C),option(B),A),map_option(C,B,F)),M4),A3,aa(B,option(B),some(B),aa(C,B,F,B2))) ).
% map_option_o_map_upd
tff(fact_2117_ran__map__option,axiom,
! [A: $tType,C: $tType,B: $tType,F: fun(C,A),M4: fun(B,option(C))] : ran(B,A,aa(fun(B,option(C)),fun(B,option(A)),aTP_Lamp_sj(fun(C,A),fun(fun(B,option(C)),fun(B,option(A))),F),M4)) = aa(set(C),set(A),aa(fun(C,A),fun(set(C),set(A)),image2(C,A),F),ran(B,C,M4)) ).
% ran_map_option
tff(fact_2118_Heap_Oexhaust,axiom,
! [A: $tType,Y2: heap_Heap(A)] :
~ ! [X: fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))))] : Y2 != aa(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_Heap(A),heap_Heap2(A),X) ).
% Heap.exhaust
tff(fact_2119_disjE__realizer2,axiom,
! [B: $tType,A: $tType,P2: bool,Q2: fun(A,bool),X2: option(A),R3: fun(B,bool),F: B,G: fun(A,B)] :
( pp(case_option(bool,A,P2,Q2,X2))
=> ( ( pp(P2)
=> pp(aa(B,bool,R3,F)) )
=> ( ! [Q6: A] :
( pp(aa(A,bool,Q2,Q6))
=> pp(aa(B,bool,R3,aa(A,B,G,Q6))) )
=> pp(aa(B,bool,R3,case_option(B,A,F,G,X2))) ) ) ) ).
% disjE_realizer2
tff(fact_2120_map__option_Ocompositionality,axiom,
! [B: $tType,C: $tType,A: $tType,F: fun(B,C),G: fun(A,B),Option: option(A)] : aa(option(B),option(C),map_option(B,C,F),aa(option(A),option(B),map_option(A,B,G),Option)) = aa(option(A),option(C),map_option(A,C,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),F),G)),Option) ).
% map_option.compositionality
tff(fact_2121_option_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,G: fun(B,C),F: fun(A,B),V: option(A)] : aa(option(B),option(C),map_option(B,C,G),aa(option(A),option(B),map_option(A,B,F),V)) = aa(option(A),option(C),map_option(A,C,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),G),F)),V) ).
% option.map_comp
tff(fact_2122_map__option_Ocomp,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(B,C),G: fun(A,B)] : aa(fun(option(A),option(B)),fun(option(A),option(C)),aa(fun(option(B),option(C)),fun(fun(option(A),option(B)),fun(option(A),option(C))),comp(option(B),option(C),option(A)),map_option(B,C,F)),map_option(A,B,G)) = map_option(A,C,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),F),G)) ).
% map_option.comp
tff(fact_2123_execute_Osimps,axiom,
! [A: $tType,F: fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))))] : aa(heap_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_execute(A),aa(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_Heap(A),heap_Heap2(A),F)) = F ).
% execute.simps
tff(fact_2124_Heap__Monad_Obind__def,axiom,
! [B: $tType,A: $tType,F: heap_Heap(A),G: fun(A,heap_Heap(B))] : heap_bind(A,B,F,G) = aa(fun(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit)))),heap_Heap(B),heap_Heap2(B),aa(fun(A,heap_Heap(B)),fun(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit)))),aTP_Lamp_sl(heap_Heap(A),fun(fun(A,heap_Heap(B)),fun(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit))))),F),G)) ).
% Heap_Monad.bind_def
tff(fact_2125_graph__ranD,axiom,
! [B: $tType,A: $tType,X2: product_prod(A,B),M4: fun(A,option(B))] :
( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),X2),graph(A,B,M4)))
=> pp(aa(set(B),bool,member(B,aa(product_prod(A,B),B,product_snd(A,B),X2)),ran(A,B,M4))) ) ).
% graph_ranD
tff(fact_2126_raise__def,axiom,
! [A: $tType,S: literal] : heap_raise(A,S) = aa(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_Heap(A),heap_Heap2(A),aTP_Lamp_sm(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))))) ).
% raise_def
tff(fact_2127_map__add__map__of__foldr,axiom,
! [B: $tType,A: $tType,M4: fun(A,option(B)),Ps: list(product_prod(A,B))] : map_add(A,B,M4,map_of(A,B,Ps)) = aa(fun(A,option(B)),fun(A,option(B)),foldr(product_prod(A,B),fun(A,option(B)),aa(fun(A,fun(B,fun(fun(A,option(B)),fun(A,option(B))))),fun(product_prod(A,B),fun(fun(A,option(B)),fun(A,option(B)))),product_case_prod(A,B,fun(fun(A,option(B)),fun(A,option(B)))),aTP_Lamp_sn(A,fun(B,fun(fun(A,option(B)),fun(A,option(B)))))),Ps),M4) ).
% map_add_map_of_foldr
tff(fact_2128_snd__graph__ran,axiom,
! [A: $tType,B: $tType,M4: fun(B,option(A))] : aa(set(product_prod(B,A)),set(A),aa(fun(product_prod(B,A),A),fun(set(product_prod(B,A)),set(A)),image2(product_prod(B,A),A),product_snd(B,A)),graph(B,A,M4)) = ran(B,A,M4) ).
% snd_graph_ran
tff(fact_2129_Heap_Osize__gen,axiom,
! [A: $tType,Xa2: fun(A,nat),X2: fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))))] : aa(heap_Heap(A),nat,heap_size_Heap(A,Xa2),aa(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_Heap(A),heap_Heap2(A),X2)) = aa(nat,nat,suc,zero_zero(nat)) ).
% Heap.size_gen
tff(fact_2130_Heap_Osize_I2_J,axiom,
! [A: $tType,X2: fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))))] : aa(heap_Heap(A),nat,size_size(heap_Heap(A)),aa(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_Heap(A),heap_Heap2(A),X2)) = aa(nat,nat,suc,zero_zero(nat)) ).
% Heap.size(2)
tff(fact_2131_option_Osize__gen__o__map,axiom,
! [B: $tType,A: $tType,F: fun(B,nat),G: fun(A,B)] : aa(fun(option(A),option(B)),fun(option(A),nat),aa(fun(option(B),nat),fun(fun(option(A),option(B)),fun(option(A),nat)),comp(option(B),nat,option(A)),size_option(B,F)),map_option(A,B,G)) = size_option(A,aa(fun(A,B),fun(A,nat),aa(fun(B,nat),fun(fun(A,B),fun(A,nat)),comp(B,nat,A),F),G)) ).
% option.size_gen_o_map
tff(fact_2132_Heap__lub__empty,axiom,
! [A: $tType] : aa(set(heap_Heap(A)),heap_Heap(A),heap_Heap_lub(A),bot_bot(set(heap_Heap(A)))) = aa(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_Heap(A),heap_Heap2(A),aTP_Lamp_sm(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))))) ).
% Heap_lub_empty
tff(fact_2133_distinct__insort__key,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F: fun(B,A),X2: B,Xs: list(B)] :
( distinct(A,aa(list(B),list(A),map(B,A,F),aa(list(B),list(B),linorder_insort_key(B,A,F,X2),Xs)))
<=> ( ~ pp(aa(set(A),bool,member(A,aa(B,A,F,X2)),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),aa(list(B),set(B),set3(B),Xs))))
& distinct(A,aa(list(B),list(A),map(B,A,F),Xs)) ) ) ) ).
% distinct_insort_key
tff(fact_2134_map__conv__bind__option,axiom,
! [A: $tType,B: $tType,F: fun(B,A),X2: option(B)] : aa(option(B),option(A),map_option(B,A,F),X2) = bind2(B,A,X2,aa(fun(B,A),fun(B,option(A)),aa(fun(A,option(A)),fun(fun(B,A),fun(B,option(A))),comp(A,option(A),B),some(A)),F)) ).
% map_conv_bind_option
tff(fact_2135_assert__def,axiom,
! [A: $tType,P2: fun(A,bool),X2: A] :
( ( pp(aa(A,bool,P2,X2))
=> ( heap_assert(A,P2,X2) = aa(A,heap_Heap(A),heap_return(A),X2) ) )
& ( ~ pp(aa(A,bool,P2,X2))
=> ( heap_assert(A,P2,X2) = heap_raise(A,literal2(fTrue,fFalse,fFalse,fFalse,fFalse,fTrue,fTrue,literal2(fTrue,fTrue,fFalse,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fTrue,fFalse,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,literal2(fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,fTrue,literal2(fFalse,fFalse,fTrue,fFalse,fTrue,fTrue,fTrue,zero_zero(literal)))))))) ) ) ) ).
% assert_def
tff(fact_2136_sorted__list__of__set_Ofold__insort__key_Ocomp__fun__commute__on,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y2: A,X2: A] : aa(fun(list(A),list(A)),fun(list(A),list(A)),aa(fun(list(A),list(A)),fun(fun(list(A),list(A)),fun(list(A),list(A))),comp(list(A),list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_so(A,A),Y2)),linorder_insort_key(A,A,aTP_Lamp_so(A,A),X2)) = aa(fun(list(A),list(A)),fun(list(A),list(A)),aa(fun(list(A),list(A)),fun(fun(list(A),list(A)),fun(list(A),list(A))),comp(list(A),list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_so(A,A),X2)),linorder_insort_key(A,A,aTP_Lamp_so(A,A),Y2)) ) ).
% sorted_list_of_set.fold_insort_key.comp_fun_commute_on
tff(fact_2137_Heap_Osize__neq,axiom,
! [A: $tType,X2: heap_Heap(A)] : aa(heap_Heap(A),nat,size_size(heap_Heap(A)),X2) != zero_zero(nat) ).
% Heap.size_neq
tff(fact_2138_bind__map__option,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(C,B),X2: option(C),G: fun(B,option(A))] : bind2(B,A,aa(option(C),option(B),map_option(C,B,F),X2),G) = bind2(C,A,X2,aa(fun(C,B),fun(C,option(A)),aa(fun(B,option(A)),fun(fun(C,B),fun(C,option(A))),comp(B,option(A),C),G),F)) ).
% bind_map_option
tff(fact_2139_map__option__bind,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(B,A),X2: option(C),G: fun(C,option(B))] : aa(option(B),option(A),map_option(B,A,F),bind2(C,B,X2,G)) = bind2(C,A,X2,aa(fun(C,option(B)),fun(C,option(A)),aa(fun(option(B),option(A)),fun(fun(C,option(B)),fun(C,option(A))),comp(option(B),option(A),C),map_option(B,A,F)),G)) ).
% map_option_bind
tff(fact_2140_option_Orec__o__map,axiom,
! [C: $tType,B: $tType,A: $tType,G: C,Ga: fun(B,C),F: fun(A,B)] : aa(fun(option(A),option(B)),fun(option(A),C),aa(fun(option(B),C),fun(fun(option(A),option(B)),fun(option(A),C)),comp(option(B),C,option(A)),rec_option(C,B,G,Ga)),map_option(A,B,F)) = rec_option(C,A,G,aa(fun(A,B),fun(A,C),aTP_Lamp_sp(fun(B,C),fun(fun(A,B),fun(A,C)),Ga),F)) ).
% option.rec_o_map
tff(fact_2141_extract__Cons__code,axiom,
! [A: $tType,P2: fun(A,bool),X2: A,Xs: list(A)] :
( ( pp(aa(A,bool,P2,X2))
=> ( extract(A,P2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs)) = aa(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A)))),some(product_prod(list(A),product_prod(A,list(A)))),aa(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A))),aa(list(A),fun(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A)))),product_Pair(list(A),product_prod(A,list(A))),nil(A)),aa(list(A),product_prod(A,list(A)),aa(A,fun(list(A),product_prod(A,list(A))),product_Pair(A,list(A)),X2),Xs))) ) )
& ( ~ pp(aa(A,bool,P2,X2))
=> ( extract(A,P2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),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)))),aa(fun(list(A),fun(product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A)))))),fun(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A))))),product_case_prod(list(A),product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A))))),aTP_Lamp_sr(A,fun(list(A),fun(product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A)))))),X2)),extract(A,P2,Xs)) ) ) ) ).
% extract_Cons_code
tff(fact_2142_insort__insert__insort__key,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F: fun(B,A),X2: B,Xs: list(B)] :
( ~ pp(aa(set(A),bool,member(A,aa(B,A,F,X2)),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),aa(list(B),set(B),set3(B),Xs))))
=> ( linord329482645794927042rt_key(B,A,F,X2,Xs) = aa(list(B),list(B),linorder_insort_key(B,A,F,X2),Xs) ) ) ) ).
% insort_insert_insort_key
tff(fact_2143_insort__insert__key__def,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F: fun(B,A),X2: B,Xs: list(B)] :
( ( pp(aa(set(A),bool,member(A,aa(B,A,F,X2)),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),aa(list(B),set(B),set3(B),Xs))))
=> ( linord329482645794927042rt_key(B,A,F,X2,Xs) = Xs ) )
& ( ~ pp(aa(set(A),bool,member(A,aa(B,A,F,X2)),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),aa(list(B),set(B),set3(B),Xs))))
=> ( linord329482645794927042rt_key(B,A,F,X2,Xs) = aa(list(B),list(B),linorder_insort_key(B,A,F,X2),Xs) ) ) ) ) ).
% insort_insert_key_def
tff(fact_2144_set__bind__option,axiom,
! [A: $tType,B: $tType,X2: option(B),F: fun(B,option(A))] : aa(option(A),set(A),set_option(A),bind2(B,A,X2,F)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),aa(fun(B,option(A)),fun(B,set(A)),aa(fun(option(A),set(A)),fun(fun(B,option(A)),fun(B,set(A))),comp(option(A),set(A),B),set_option(A)),F)),aa(option(B),set(B),set_option(B),X2))) ).
% set_bind_option
tff(fact_2145_image__uminus__greaterThanAtMost,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [X2: A,Y2: A] : aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),uminus_uminus(A)),set_or3652927894154168847AtMost(A,X2,Y2)) = set_or7035219750837199246ssThan(A,aa(A,A,uminus_uminus(A),Y2),aa(A,A,uminus_uminus(A),X2)) ) ).
% image_uminus_greaterThanAtMost
tff(fact_2146_image__uminus__atLeastLessThan,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [X2: A,Y2: A] : aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),uminus_uminus(A)),set_or7035219750837199246ssThan(A,X2,Y2)) = set_or3652927894154168847AtMost(A,aa(A,A,uminus_uminus(A),Y2),aa(A,A,uminus_uminus(A),X2)) ) ).
% image_uminus_atLeastLessThan
tff(fact_2147_image__add__greaterThanAtMost,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [C3: A,A3: A,B2: A] : aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aa(A,fun(A,A),plus_plus(A),C3)),set_or3652927894154168847AtMost(A,A3,B2)) = set_or3652927894154168847AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B2)) ) ).
% image_add_greaterThanAtMost
tff(fact_2148_image__diff__atLeastLessThan,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [C3: A,A3: A,B2: A] : aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aa(A,fun(A,A),minus_minus(A),C3)),set_or7035219750837199246ssThan(A,A3,B2)) = set_or3652927894154168847AtMost(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),B2),aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),A3)) ) ).
% image_diff_atLeastLessThan
tff(fact_2149_image__minus__const__greaterThanAtMost,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [C3: A,A3: A,B2: A] : aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aa(A,fun(A,A),minus_minus(A),C3)),set_or3652927894154168847AtMost(A,A3,B2)) = set_or7035219750837199246ssThan(A,aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),B2),aa(A,A,aa(A,fun(A,A),minus_minus(A),C3),A3)) ) ).
% image_minus_const_greaterThanAtMost
tff(fact_2150_option_Oset__map,axiom,
! [B: $tType,A: $tType,F: fun(A,B),V: option(A)] : aa(option(B),set(B),set_option(B),aa(option(A),option(B),map_option(A,B,F),V)) = aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),aa(option(A),set(A),set_option(A),V)) ).
% option.set_map
tff(fact_2151_insort__insert__key__triv,axiom,
! [A: $tType,B: $tType] :
( linorder(A)
=> ! [F: fun(B,A),X2: B,Xs: list(B)] :
( pp(aa(set(A),bool,member(A,aa(B,A,F,X2)),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),aa(list(B),set(B),set3(B),Xs))))
=> ( linord329482645794927042rt_key(B,A,F,X2,Xs) = Xs ) ) ) ).
% insort_insert_key_triv
tff(fact_2152_option_Oin__rel,axiom,
! [A: $tType,B: $tType,R3: fun(A,fun(B,bool)),A3: option(A),B2: option(B)] :
( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),rel_option(A,B,R3),A3),B2))
<=> ? [Z4: option(product_prod(A,B))] :
( pp(aa(set(option(product_prod(A,B))),bool,member(option(product_prod(A,B)),Z4),aa(fun(option(product_prod(A,B)),bool),set(option(product_prod(A,B))),collect(option(product_prod(A,B))),aTP_Lamp_ss(fun(A,fun(B,bool)),fun(option(product_prod(A,B)),bool),R3))))
& ( aa(option(product_prod(A,B)),option(A),map_option(product_prod(A,B),A,product_fst(A,B)),Z4) = A3 )
& ( aa(option(product_prod(A,B)),option(B),map_option(product_prod(A,B),B,product_snd(A,B)),Z4) = B2 ) ) ) ).
% option.in_rel
tff(fact_2153_option_Orel__compp__Grp,axiom,
! [B: $tType,A: $tType,R3: fun(A,fun(B,bool))] : rel_option(A,B,R3) = aa(fun(option(product_prod(A,B)),fun(option(B),bool)),fun(option(A),fun(option(B),bool)),aa(fun(option(A),fun(option(product_prod(A,B)),bool)),fun(fun(option(product_prod(A,B)),fun(option(B),bool)),fun(option(A),fun(option(B),bool))),relcompp(option(A),option(product_prod(A,B)),option(B)),conversep(option(product_prod(A,B)),option(A),bNF_Grp(option(product_prod(A,B)),option(A),aa(fun(option(product_prod(A,B)),bool),set(option(product_prod(A,B))),collect(option(product_prod(A,B))),aTP_Lamp_ss(fun(A,fun(B,bool)),fun(option(product_prod(A,B)),bool),R3)),map_option(product_prod(A,B),A,product_fst(A,B))))),bNF_Grp(option(product_prod(A,B)),option(B),aa(fun(option(product_prod(A,B)),bool),set(option(product_prod(A,B))),collect(option(product_prod(A,B))),aTP_Lamp_ss(fun(A,fun(B,bool)),fun(option(product_prod(A,B)),bool),R3)),map_option(product_prod(A,B),B,product_snd(A,B)))) ).
% option.rel_compp_Grp
tff(fact_2154_inv__fn__o__fn__is__id,axiom,
! [A: $tType,F: fun(A,A),N4: nat] :
( bij_betw(A,A,F,top_top(set(A)),top_top(set(A)))
=> ! [X4: A] : aa(A,A,aa(fun(A,A),fun(A,A),aa(fun(A,A),fun(fun(A,A),fun(A,A)),comp(A,A,A),compow(fun(A,A),N4,hilbert_inv_into(A,A,top_top(set(A)),F))),compow(fun(A,A),N4,F)),X4) = X4 ) ).
% inv_fn_o_fn_is_id
tff(fact_2155_fn__o__inv__fn__is__id,axiom,
! [A: $tType,F: fun(A,A),N4: nat] :
( bij_betw(A,A,F,top_top(set(A)),top_top(set(A)))
=> ! [X4: A] : aa(A,A,aa(fun(A,A),fun(A,A),aa(fun(A,A),fun(fun(A,A),fun(A,A)),comp(A,A,A),compow(fun(A,A),N4,F)),compow(fun(A,A),N4,hilbert_inv_into(A,A,top_top(set(A)),F))),X4) = X4 ) ).
% fn_o_inv_fn_is_id
tff(fact_2156_lfp__induct2,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,F: fun(set(product_prod(A,B)),set(product_prod(A,B))),P2: fun(A,fun(B,bool))] :
( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),complete_lattice_lfp(set(product_prod(A,B)),F)))
=> ( order_mono(set(product_prod(A,B)),set(product_prod(A,B)),F)
=> ( ! [A5: A,B4: B] :
( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4)),aa(set(product_prod(A,B)),set(product_prod(A,B)),F,aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),inf_inf(set(product_prod(A,B))),complete_lattice_lfp(set(product_prod(A,B)),F)),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),P2))))))
=> pp(aa(B,bool,aa(A,fun(B,bool),P2,A5),B4)) )
=> pp(aa(B,bool,aa(A,fun(B,bool),P2,A3),B2)) ) ) ) ).
% lfp_induct2
tff(fact_2157_Heap_Osize__gen__o__map,axiom,
! [B: $tType,A: $tType,F: fun(B,nat),G: fun(A,B)] : aa(fun(heap_Heap(A),heap_Heap(B)),fun(heap_Heap(A),nat),aa(fun(heap_Heap(B),nat),fun(fun(heap_Heap(A),heap_Heap(B)),fun(heap_Heap(A),nat)),comp(heap_Heap(B),nat,heap_Heap(A)),heap_size_Heap(B,F)),aa(fun(A,B),fun(heap_Heap(A),heap_Heap(B)),heap_map_Heap(A,B),G)) = heap_size_Heap(A,aa(fun(A,B),fun(A,nat),aa(fun(B,nat),fun(fun(A,B),fun(A,nat)),comp(B,nat,A),F),G)) ).
% Heap.size_gen_o_map
tff(fact_2158_lists__image,axiom,
! [A: $tType,B: $tType,F: fun(B,A),A6: set(B)] : lists(A,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6)) = aa(set(list(B)),set(list(A)),aa(fun(list(B),list(A)),fun(set(list(B)),set(list(A))),image2(list(B),list(A)),map(B,A,F)),lists(B,A6)) ).
% lists_image
tff(fact_2159_surj__fn,axiom,
! [A: $tType,F: fun(A,A),N4: nat] :
( ( aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),F),top_top(set(A))) = top_top(set(A)) )
=> ( aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),compow(fun(A,A),N4,F)),top_top(set(A))) = top_top(set(A)) ) ) ).
% surj_fn
tff(fact_2160_comp__funpow,axiom,
! [B: $tType,A: $tType,N4: nat,F: fun(A,A)] : compow(fun(fun(B,A),fun(B,A)),N4,aa(fun(A,A),fun(fun(B,A),fun(B,A)),comp(A,A,B),F)) = aa(fun(A,A),fun(fun(B,A),fun(B,A)),comp(A,A,B),compow(fun(A,A),N4,F)) ).
% comp_funpow
tff(fact_2161_Heap_Omap__ident,axiom,
! [A: $tType,T2: heap_Heap(A)] : aa(heap_Heap(A),heap_Heap(A),aa(fun(A,A),fun(heap_Heap(A),heap_Heap(A)),heap_map_Heap(A,A),aTP_Lamp_ab(A,A)),T2) = T2 ).
% Heap.map_ident
tff(fact_2162_relpowp__commute,axiom,
! [A: $tType,P2: fun(A,fun(A,bool)),N4: nat] : aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),relcompp(A,A,A),P2),compow(fun(A,fun(A,bool)),N4,P2)) = aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),relcompp(A,A,A),compow(fun(A,fun(A,bool)),N4,P2)),P2) ).
% relpowp_commute
tff(fact_2163_option_Orel__compp,axiom,
! [A: $tType,C: $tType,B: $tType,R3: fun(A,fun(B,bool)),S3: fun(B,fun(C,bool))] : rel_option(A,C,aa(fun(B,fun(C,bool)),fun(A,fun(C,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(C,bool)),fun(A,fun(C,bool))),relcompp(A,B,C),R3),S3)) = aa(fun(option(B),fun(option(C),bool)),fun(option(A),fun(option(C),bool)),aa(fun(option(A),fun(option(B),bool)),fun(fun(option(B),fun(option(C),bool)),fun(option(A),fun(option(C),bool))),relcompp(option(A),option(B),option(C)),rel_option(A,B,R3)),rel_option(B,C,S3)) ).
% option.rel_compp
tff(fact_2164_Heap_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,G: fun(B,C),F: fun(A,B),V: heap_Heap(A)] : aa(heap_Heap(B),heap_Heap(C),aa(fun(B,C),fun(heap_Heap(B),heap_Heap(C)),heap_map_Heap(B,C),G),aa(heap_Heap(A),heap_Heap(B),aa(fun(A,B),fun(heap_Heap(A),heap_Heap(B)),heap_map_Heap(A,B),F),V)) = aa(heap_Heap(A),heap_Heap(C),aa(fun(A,C),fun(heap_Heap(A),heap_Heap(C)),heap_map_Heap(A,C),aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),G),F)),V) ).
% Heap.map_comp
tff(fact_2165_Heap_Omap__id,axiom,
! [A: $tType,T2: heap_Heap(A)] : aa(heap_Heap(A),heap_Heap(A),aa(fun(A,A),fun(heap_Heap(A),heap_Heap(A)),heap_map_Heap(A,A),id(A)),T2) = T2 ).
% Heap.map_id
tff(fact_2166_Heap_Omap__id0,axiom,
! [A: $tType] : aa(fun(A,A),fun(heap_Heap(A),heap_Heap(A)),heap_map_Heap(A,A),id(A)) = id(heap_Heap(A)) ).
% Heap.map_id0
tff(fact_2167_Heap_Omap,axiom,
! [B: $tType,A: $tType,F: fun(A,B),X2: fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))))] : aa(heap_Heap(A),heap_Heap(B),aa(fun(A,B),fun(heap_Heap(A),heap_Heap(B)),heap_map_Heap(A,B),F),aa(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_Heap(A),heap_Heap2(A),X2)) = aa(fun(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit)))),heap_Heap(B),heap_Heap2(B),aa(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),fun(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit)))),aa(fun(option(product_prod(A,heap_ext(product_unit))),option(product_prod(B,heap_ext(product_unit)))),fun(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),fun(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit))))),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
tff(fact_2168_rel__option__iff,axiom,
! [A: $tType,B: $tType,R3: fun(A,fun(B,bool)),X2: option(A),Y2: option(B)] :
( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),rel_option(A,B,R3),X2),Y2))
<=> pp(aa(product_prod(option(A),option(B)),bool,aa(fun(option(A),fun(option(B),bool)),fun(product_prod(option(A),option(B)),bool),product_case_prod(option(A),option(B),bool),aTP_Lamp_sv(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),R3)),aa(option(B),product_prod(option(A),option(B)),aa(option(A),fun(option(B),product_prod(option(A),option(B))),product_Pair(option(A),option(B)),X2),Y2))) ) ).
% rel_option_iff
tff(fact_2169_relpowp_Osimps_I2_J,axiom,
! [A: $tType,N4: nat,R3: fun(A,fun(A,bool))] : compow(fun(A,fun(A,bool)),aa(nat,nat,suc,N4),R3) = aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),relcompp(A,A,A),compow(fun(A,fun(A,bool)),N4,R3)),R3) ).
% relpowp.simps(2)
tff(fact_2170_relpowp__add,axiom,
! [A: $tType,M4: nat,N4: nat,P2: fun(A,fun(A,bool))] : compow(fun(A,fun(A,bool)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M4),N4),P2) = aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),relcompp(A,A,A),compow(fun(A,fun(A,bool)),M4,P2)),compow(fun(A,fun(A,bool)),N4,P2)) ).
% relpowp_add
tff(fact_2171_funpow__Suc__right,axiom,
! [A: $tType,N4: nat,F: fun(A,A)] : compow(fun(A,A),aa(nat,nat,suc,N4),F) = aa(fun(A,A),fun(A,A),aa(fun(A,A),fun(fun(A,A),fun(A,A)),comp(A,A,A),compow(fun(A,A),N4,F)),F) ).
% funpow_Suc_right
tff(fact_2172_funpow_Osimps_I2_J,axiom,
! [A: $tType,N4: nat,F: fun(A,A)] : compow(fun(A,A),aa(nat,nat,suc,N4),F) = aa(fun(A,A),fun(A,A),aa(fun(A,A),fun(fun(A,A),fun(A,A)),comp(A,A,A),F),compow(fun(A,A),N4,F)) ).
% funpow.simps(2)
tff(fact_2173_funpow__add,axiom,
! [A: $tType,M4: nat,N4: nat,F: fun(A,A)] : compow(fun(A,A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M4),N4),F) = aa(fun(A,A),fun(A,A),aa(fun(A,A),fun(fun(A,A),fun(A,A)),comp(A,A,A),compow(fun(A,A),M4,F)),compow(fun(A,A),N4,F)) ).
% funpow_add
tff(fact_2174_relpowp__relpow__eq,axiom,
! [A: $tType,N4: nat,R3: set(product_prod(A,A)),X4: A,Xa4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),compow(fun(A,fun(A,bool)),N4,aTP_Lamp_hl(set(product_prod(A,A)),fun(A,fun(A,bool)),R3)),X4),Xa4))
<=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa4)),compow(set(product_prod(A,A)),N4,R3))) ) ).
% relpowp_relpow_eq
tff(fact_2175_Heap_Oinj__map,axiom,
! [B: $tType,A: $tType,F: fun(A,B)] :
( inj_on(A,B,F,top_top(set(A)))
=> inj_on(heap_Heap(A),heap_Heap(B),aa(fun(A,B),fun(heap_Heap(A),heap_Heap(B)),heap_map_Heap(A,B),F),top_top(set(heap_Heap(A)))) ) ).
% Heap.inj_map
tff(fact_2176_Heap_Orec__o__map,axiom,
! [C: $tType,B: $tType,A: $tType,G: fun(fun(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit)))),C),F: fun(A,B)] : aa(fun(heap_Heap(A),heap_Heap(B)),fun(heap_Heap(A),C),aa(fun(heap_Heap(B),C),fun(fun(heap_Heap(A),heap_Heap(B)),fun(heap_Heap(A),C)),comp(heap_Heap(B),C,heap_Heap(A)),aa(fun(fun(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit)))),C),fun(heap_Heap(B),C),heap_rec_Heap(B,C),G)),aa(fun(A,B),fun(heap_Heap(A),heap_Heap(B)),heap_map_Heap(A,B),F)) = aa(fun(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),C),fun(heap_Heap(A),C),heap_rec_Heap(A,C),aa(fun(A,B),fun(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),C),aTP_Lamp_sw(fun(fun(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit)))),C),fun(fun(A,B),fun(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),C)),G),F)) ).
% Heap.rec_o_map
tff(fact_2177_pcr__int__def,axiom,
pcr_int = aa(fun(product_prod(nat,nat),fun(int,bool)),fun(product_prod(nat,nat),fun(int,bool)),aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)),fun(fun(product_prod(nat,nat),fun(int,bool)),fun(product_prod(nat,nat),fun(int,bool))),relcompp(product_prod(nat,nat),product_prod(nat,nat),int),basic_rel_prod(nat,nat,nat,nat,fequal(nat),fequal(nat))),cr_int) ).
% pcr_int_def
tff(fact_2178_fold__graph__image,axiom,
! [C: $tType,B: $tType,A: $tType,G: fun(A,B),A6: set(A),F: fun(B,fun(C,C)),Z2: C] :
( inj_on(A,B,G,A6)
=> ( finite_fold_graph(B,C,F,Z2,aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),G),A6)) = finite_fold_graph(A,C,aa(fun(A,B),fun(A,fun(C,C)),aa(fun(B,fun(C,C)),fun(fun(A,B),fun(A,fun(C,C))),comp(B,fun(C,C),A),F),G),Z2,A6) ) ) ).
% fold_graph_image
tff(fact_2179_hom__Max__commute,axiom,
! [A: $tType] :
( linorder(A)
=> ! [H: fun(A,A),N: set(A)] :
( ! [X: A,Y: A] : aa(A,A,H,ord_max(A,X,Y)) = ord_max(A,aa(A,A,H,X),aa(A,A,H,Y))
=> ( finite_finite2(A,N)
=> ( ( N != bot_bot(set(A)) )
=> ( aa(A,A,H,lattic643756798349783984er_Max(A,N)) = lattic643756798349783984er_Max(A,aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),H),N)) ) ) ) ) ) ).
% hom_Max_commute
tff(fact_2180_fold__remove1__split,axiom,
! [B: $tType,A: $tType,Xs: list(A),F: fun(A,fun(B,B)),X2: A] :
( ! [X: A,Y: A] :
( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set3(A),Xs)))
=> ( pp(aa(set(A),bool,member(A,Y),aa(list(A),set(A),set3(A),Xs)))
=> ( aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X)),aa(A,fun(B,B),F,Y)) = aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,Y)),aa(A,fun(B,B),F,X)) ) ) )
=> ( pp(aa(set(A),bool,member(A,X2),aa(list(A),set(A),set3(A),Xs)))
=> ( fold(A,B,F,Xs) = aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),fold(A,B,F,remove1(A,X2,Xs))),aa(A,fun(B,B),F,X2)) ) ) ) ).
% fold_remove1_split
tff(fact_2181_Heap_Oset,axiom,
! [A: $tType,X2: fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))))] : aa(heap_Heap(A),set(A),heap_set_Heap(A),aa(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_Heap(A),heap_Heap2(A),X2)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(option(product_prod(A,heap_ext(product_unit)))),set(set(A)),aa(fun(option(product_prod(A,heap_ext(product_unit))),set(A)),fun(set(option(product_prod(A,heap_ext(product_unit)))),set(set(A))),image2(option(product_prod(A,heap_ext(product_unit))),set(A)),aTP_Lamp_sx(option(product_prod(A,heap_ext(product_unit))),set(A))),aa(set(heap_ext(product_unit)),set(option(product_prod(A,heap_ext(product_unit)))),aa(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),fun(set(heap_ext(product_unit)),set(option(product_prod(A,heap_ext(product_unit))))),image2(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),X2),top_top(set(heap_ext(product_unit)))))) ).
% Heap.set
tff(fact_2182_trancl__def,axiom,
! [A: $tType,X4: set(product_prod(A,A))] : transitive_trancl(A,X4) = aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),transitive_tranclp(A,aTP_Lamp_hl(set(product_prod(A,A)),fun(A,fun(A,bool)),X4)))) ).
% trancl_def
tff(fact_2183_tranclp__induct2,axiom,
! [A: $tType,B: $tType,R: fun(product_prod(A,B),fun(product_prod(A,B),bool)),Ax: A,Ay: B,Bx: A,By: B,P2: fun(A,fun(B,bool))] :
( pp(aa(product_prod(A,B),bool,aa(product_prod(A,B),fun(product_prod(A,B),bool),transitive_tranclp(product_prod(A,B),R),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By)))
=> ( ! [A5: A,B4: B] :
( pp(aa(product_prod(A,B),bool,aa(product_prod(A,B),fun(product_prod(A,B),bool),R,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4)))
=> pp(aa(B,bool,aa(A,fun(B,bool),P2,A5),B4)) )
=> ( ! [A5: A,B4: B,Aa2: A,Ba: B] :
( pp(aa(product_prod(A,B),bool,aa(product_prod(A,B),fun(product_prod(A,B),bool),transitive_tranclp(product_prod(A,B),R),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4)))
=> ( pp(aa(product_prod(A,B),bool,aa(product_prod(A,B),fun(product_prod(A,B),bool),R,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba)))
=> ( pp(aa(B,bool,aa(A,fun(B,bool),P2,A5),B4))
=> pp(aa(B,bool,aa(A,fun(B,bool),P2,Aa2),Ba)) ) ) )
=> pp(aa(B,bool,aa(A,fun(B,bool),P2,Bx),By)) ) ) ) ).
% tranclp_induct2
tff(fact_2184_Heap_Oset__intros,axiom,
! [A: $tType,Y2: option(product_prod(A,heap_ext(product_unit))),X2: fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),Ya: product_prod(A,heap_ext(product_unit)),Yb2: A] :
( pp(aa(set(option(product_prod(A,heap_ext(product_unit)))),bool,member(option(product_prod(A,heap_ext(product_unit))),Y2),aa(set(heap_ext(product_unit)),set(option(product_prod(A,heap_ext(product_unit)))),aa(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),fun(set(heap_ext(product_unit)),set(option(product_prod(A,heap_ext(product_unit))))),image2(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),X2),top_top(set(heap_ext(product_unit))))))
=> ( pp(aa(set(product_prod(A,heap_ext(product_unit))),bool,member(product_prod(A,heap_ext(product_unit)),Ya),aa(option(product_prod(A,heap_ext(product_unit))),set(product_prod(A,heap_ext(product_unit))),set_option(product_prod(A,heap_ext(product_unit))),Y2)))
=> ( pp(aa(set(A),bool,member(A,Yb2),aa(product_prod(A,heap_ext(product_unit)),set(A),basic_fsts(A,heap_ext(product_unit)),Ya)))
=> pp(aa(set(A),bool,member(A,Yb2),aa(heap_Heap(A),set(A),heap_set_Heap(A),aa(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_Heap(A),heap_Heap2(A),X2)))) ) ) ) ).
% Heap.set_intros
tff(fact_2185_Heap_Oset__cases,axiom,
! [A: $tType,E3: A,A3: heap_Heap(A)] :
( pp(aa(set(A),bool,member(A,E3),aa(heap_Heap(A),set(A),heap_set_Heap(A),A3)))
=> ~ ! [Z3: fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))))] :
( ( A3 = aa(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_Heap(A),heap_Heap2(A),Z3) )
=> ! [Xa: option(product_prod(A,heap_ext(product_unit)))] :
( pp(aa(set(option(product_prod(A,heap_ext(product_unit)))),bool,member(option(product_prod(A,heap_ext(product_unit))),Xa),aa(set(heap_ext(product_unit)),set(option(product_prod(A,heap_ext(product_unit)))),aa(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),fun(set(heap_ext(product_unit)),set(option(product_prod(A,heap_ext(product_unit))))),image2(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),Z3),top_top(set(heap_ext(product_unit))))))
=> ! [Xb4: product_prod(A,heap_ext(product_unit))] :
( pp(aa(set(product_prod(A,heap_ext(product_unit))),bool,member(product_prod(A,heap_ext(product_unit)),Xb4),aa(option(product_prod(A,heap_ext(product_unit))),set(product_prod(A,heap_ext(product_unit))),set_option(product_prod(A,heap_ext(product_unit))),Xa)))
=> ~ pp(aa(set(A),bool,member(A,E3),aa(product_prod(A,heap_ext(product_unit)),set(A),basic_fsts(A,heap_ext(product_unit)),Xb4))) ) ) ) ) ).
% Heap.set_cases
tff(fact_2186_Heap_Oinj__map__strong,axiom,
! [B: $tType,A: $tType,X2: heap_Heap(A),Xa2: heap_Heap(A),F: fun(A,B),Fa: fun(A,B)] :
( ! [Z3: A,Za: A] :
( pp(aa(set(A),bool,member(A,Z3),aa(heap_Heap(A),set(A),heap_set_Heap(A),X2)))
=> ( pp(aa(set(A),bool,member(A,Za),aa(heap_Heap(A),set(A),heap_set_Heap(A),Xa2)))
=> ( ( aa(A,B,F,Z3) = aa(A,B,Fa,Za) )
=> ( Z3 = Za ) ) ) )
=> ( ( aa(heap_Heap(A),heap_Heap(B),aa(fun(A,B),fun(heap_Heap(A),heap_Heap(B)),heap_map_Heap(A,B),F),X2) = aa(heap_Heap(A),heap_Heap(B),aa(fun(A,B),fun(heap_Heap(A),heap_Heap(B)),heap_map_Heap(A,B),Fa),Xa2) )
=> ( X2 = Xa2 ) ) ) ).
% Heap.inj_map_strong
tff(fact_2187_Heap_Omap__cong0,axiom,
! [B: $tType,A: $tType,X2: heap_Heap(A),F: fun(A,B),G: fun(A,B)] :
( ! [Z3: A] :
( pp(aa(set(A),bool,member(A,Z3),aa(heap_Heap(A),set(A),heap_set_Heap(A),X2)))
=> ( aa(A,B,F,Z3) = aa(A,B,G,Z3) ) )
=> ( aa(heap_Heap(A),heap_Heap(B),aa(fun(A,B),fun(heap_Heap(A),heap_Heap(B)),heap_map_Heap(A,B),F),X2) = aa(heap_Heap(A),heap_Heap(B),aa(fun(A,B),fun(heap_Heap(A),heap_Heap(B)),heap_map_Heap(A,B),G),X2) ) ) ).
% Heap.map_cong0
tff(fact_2188_Heap_Omap__cong,axiom,
! [B: $tType,A: $tType,X2: heap_Heap(A),Ya: heap_Heap(A),F: fun(A,B),G: fun(A,B)] :
( ( X2 = Ya )
=> ( ! [Z3: A] :
( pp(aa(set(A),bool,member(A,Z3),aa(heap_Heap(A),set(A),heap_set_Heap(A),Ya)))
=> ( aa(A,B,F,Z3) = aa(A,B,G,Z3) ) )
=> ( aa(heap_Heap(A),heap_Heap(B),aa(fun(A,B),fun(heap_Heap(A),heap_Heap(B)),heap_map_Heap(A,B),F),X2) = aa(heap_Heap(A),heap_Heap(B),aa(fun(A,B),fun(heap_Heap(A),heap_Heap(B)),heap_map_Heap(A,B),G),Ya) ) ) ) ).
% Heap.map_cong
tff(fact_2189_tranclp__trancl__eq,axiom,
! [A: $tType,R: set(product_prod(A,A)),X4: A,Xa4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),transitive_tranclp(A,aTP_Lamp_hl(set(product_prod(A,A)),fun(A,fun(A,bool)),R)),X4),Xa4))
<=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa4)),transitive_trancl(A,R))) ) ).
% tranclp_trancl_eq
tff(fact_2190_Heap_Orec,axiom,
! [C: $tType,A: $tType,F: fun(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),C),X2: fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))))] : aa(heap_Heap(A),C,aa(fun(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),C),fun(heap_Heap(A),C),heap_rec_Heap(A,C),F),aa(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_Heap(A),heap_Heap2(A),X2)) = aa(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),C,F,X2) ).
% Heap.rec
tff(fact_2191_Heap_Oset__map,axiom,
! [B: $tType,A: $tType,F: fun(A,B),V: heap_Heap(A)] : aa(heap_Heap(B),set(B),heap_set_Heap(B),aa(heap_Heap(A),heap_Heap(B),aa(fun(A,B),fun(heap_Heap(A),heap_Heap(B)),heap_map_Heap(A,B),F),V)) = aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),aa(heap_Heap(A),set(A),heap_set_Heap(A),V)) ).
% Heap.set_map
tff(fact_2192_Nitpick_Otranclp__unfold,axiom,
! [A: $tType,R: fun(A,fun(A,bool)),A3: A,B2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),transitive_tranclp(A,R),A3),B2))
<=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),transitive_trancl(A,aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),R))))) ) ).
% Nitpick.tranclp_unfold
tff(fact_2193_Heap_Orel__Grp,axiom,
! [B: $tType,A: $tType,A6: set(A),F: fun(A,B)] : aa(fun(A,fun(B,bool)),fun(heap_Heap(A),fun(heap_Heap(B),bool)),heap_rel_Heap(A,B),bNF_Grp(A,B,A6,F)) = bNF_Grp(heap_Heap(A),heap_Heap(B),aa(fun(heap_Heap(A),bool),set(heap_Heap(A)),collect(heap_Heap(A)),aTP_Lamp_sy(set(A),fun(heap_Heap(A),bool),A6)),aa(fun(A,B),fun(heap_Heap(A),heap_Heap(B)),heap_map_Heap(A,B),F)) ).
% Heap.rel_Grp
tff(fact_2194_Heap_Oin__rel,axiom,
! [A: $tType,B: $tType,R3: fun(A,fun(B,bool)),A3: heap_Heap(A),B2: heap_Heap(B)] :
( pp(aa(heap_Heap(B),bool,aa(heap_Heap(A),fun(heap_Heap(B),bool),aa(fun(A,fun(B,bool)),fun(heap_Heap(A),fun(heap_Heap(B),bool)),heap_rel_Heap(A,B),R3),A3),B2))
<=> ? [Z4: heap_Heap(product_prod(A,B))] :
( pp(aa(set(heap_Heap(product_prod(A,B))),bool,member(heap_Heap(product_prod(A,B)),Z4),aa(fun(heap_Heap(product_prod(A,B)),bool),set(heap_Heap(product_prod(A,B))),collect(heap_Heap(product_prod(A,B))),aTP_Lamp_sz(fun(A,fun(B,bool)),fun(heap_Heap(product_prod(A,B)),bool),R3))))
& ( aa(heap_Heap(product_prod(A,B)),heap_Heap(A),aa(fun(product_prod(A,B),A),fun(heap_Heap(product_prod(A,B)),heap_Heap(A)),heap_map_Heap(product_prod(A,B),A),product_fst(A,B)),Z4) = A3 )
& ( aa(heap_Heap(product_prod(A,B)),heap_Heap(B),aa(fun(product_prod(A,B),B),fun(heap_Heap(product_prod(A,B)),heap_Heap(B)),heap_map_Heap(product_prod(A,B),B),product_snd(A,B)),Z4) = B2 ) ) ) ).
% Heap.in_rel
tff(fact_2195_Heap_Orel__compp__Grp,axiom,
! [B: $tType,A: $tType,R3: fun(A,fun(B,bool))] : aa(fun(A,fun(B,bool)),fun(heap_Heap(A),fun(heap_Heap(B),bool)),heap_rel_Heap(A,B),R3) = aa(fun(heap_Heap(product_prod(A,B)),fun(heap_Heap(B),bool)),fun(heap_Heap(A),fun(heap_Heap(B),bool)),aa(fun(heap_Heap(A),fun(heap_Heap(product_prod(A,B)),bool)),fun(fun(heap_Heap(product_prod(A,B)),fun(heap_Heap(B),bool)),fun(heap_Heap(A),fun(heap_Heap(B),bool))),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),aa(fun(heap_Heap(product_prod(A,B)),bool),set(heap_Heap(product_prod(A,B))),collect(heap_Heap(product_prod(A,B))),aTP_Lamp_sz(fun(A,fun(B,bool)),fun(heap_Heap(product_prod(A,B)),bool),R3)),aa(fun(product_prod(A,B),A),fun(heap_Heap(product_prod(A,B)),heap_Heap(A)),heap_map_Heap(product_prod(A,B),A),product_fst(A,B))))),bNF_Grp(heap_Heap(product_prod(A,B)),heap_Heap(B),aa(fun(heap_Heap(product_prod(A,B)),bool),set(heap_Heap(product_prod(A,B))),collect(heap_Heap(product_prod(A,B))),aTP_Lamp_sz(fun(A,fun(B,bool)),fun(heap_Heap(product_prod(A,B)),bool),R3)),aa(fun(product_prod(A,B),B),fun(heap_Heap(product_prod(A,B)),heap_Heap(B)),heap_map_Heap(product_prod(A,B),B),product_snd(A,B)))) ).
% Heap.rel_compp_Grp
tff(fact_2196_reflp__refl__eq,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( reflp(A,aTP_Lamp_hl(set(product_prod(A,A)),fun(A,fun(A,bool)),R))
<=> refl_on(A,top_top(set(A)),R) ) ).
% reflp_refl_eq
tff(fact_2197_update__change,axiom,
! [A: $tType] :
( heap(A)
=> ! [R: ref(A),E3: A] : update2(A,R,E3) = heap_bind(A,product_unit,change(A,aTP_Lamp_ta(A,fun(A,A),E3),R),aTP_Lamp_tb(A,heap_Heap(product_unit))) ) ).
% update_change
tff(fact_2198_old_Orec__bool__def,axiom,
! [T: $tType,X4: T,Xa4: T,Xb: bool] : product_rec_bool(T,X4,Xa4,Xb) = the(T,product_rec_set_bool(T,X4,Xa4,Xb)) ).
% old.rec_bool_def
tff(fact_2199_unit__abs__eta__conv,axiom,
! [A: $tType,F: fun(product_unit,A)] : aTP_Lamp_tc(fun(product_unit,A),fun(product_unit,A),F) = F ).
% unit_abs_eta_conv
tff(fact_2200_old_Obool_Osimps_I5_J,axiom,
! [T: $tType,F1: T,F22: T] : product_rec_bool(T,F1,F22,fTrue) = F1 ).
% old.bool.simps(5)
tff(fact_2201_old_Obool_Osimps_I6_J,axiom,
! [T: $tType,F1: T,F22: T] : product_rec_bool(T,F1,F22,fFalse) = F22 ).
% old.bool.simps(6)
tff(fact_2202_old_Ounit_Orec,axiom,
! [T: $tType,F1: T] : product_rec_unit(T,F1,product_Unity) = F1 ).
% old.unit.rec
tff(fact_2203_UNIV__unit,axiom,
top_top(set(product_unit)) = aa(set(product_unit),set(product_unit),insert(product_unit,product_Unity),bot_bot(set(product_unit))) ).
% UNIV_unit
tff(fact_2204_top__unit__def,axiom,
top_top(product_unit) = product_Unity ).
% top_unit_def
tff(fact_2205_uminus__unit__def,axiom,
! [Uu: product_unit] : aa(product_unit,product_unit,uminus_uminus(product_unit),Uu) = product_Unity ).
% uminus_unit_def
tff(fact_2206_inf__unit__def,axiom,
! [Uu: product_unit,Uv: product_unit] : aa(product_unit,product_unit,aa(product_unit,fun(product_unit,product_unit),inf_inf(product_unit),Uu),Uv) = product_Unity ).
% inf_unit_def
tff(fact_2207_Heap_Orel__eq,axiom,
! [A: $tType] : aa(fun(A,fun(A,bool)),fun(heap_Heap(A),fun(heap_Heap(A),bool)),heap_rel_Heap(A,A),fequal(A)) = fequal(heap_Heap(A)) ).
% Heap.rel_eq
tff(fact_2208_Heap_Orel__refl,axiom,
! [B: $tType,Ra: fun(B,fun(B,bool)),X2: heap_Heap(B)] :
( ! [X: B] : pp(aa(B,bool,aa(B,fun(B,bool),Ra,X),X))
=> pp(aa(heap_Heap(B),bool,aa(heap_Heap(B),fun(heap_Heap(B),bool),aa(fun(B,fun(B,bool)),fun(heap_Heap(B),fun(heap_Heap(B),bool)),heap_rel_Heap(B,B),Ra),X2),X2)) ) ).
% Heap.rel_refl
tff(fact_2209_Heap_Orel__reflp,axiom,
! [A: $tType,R3: fun(A,fun(A,bool))] :
( reflp(A,R3)
=> reflp(heap_Heap(A),aa(fun(A,fun(A,bool)),fun(heap_Heap(A),fun(heap_Heap(A),bool)),heap_rel_Heap(A,A),R3)) ) ).
% Heap.rel_reflp
tff(fact_2210_old_Ounit_Oexhaust,axiom,
! [Y2: product_unit] : Y2 = product_Unity ).
% old.unit.exhaust
tff(fact_2211_Inf__unit__def,axiom,
! [Uu: set(product_unit)] : aa(set(product_unit),product_unit,complete_Inf_Inf(product_unit),Uu) = product_Unity ).
% Inf_unit_def
tff(fact_2212_Sup__unit__def,axiom,
! [Uu: set(product_unit)] : aa(set(product_unit),product_unit,complete_Sup_Sup(product_unit),Uu) = product_Unity ).
% Sup_unit_def
tff(fact_2213_sup__unit__def,axiom,
! [Uu: product_unit,Uv: product_unit] : aa(product_unit,product_unit,aa(product_unit,fun(product_unit,product_unit),sup_sup(product_unit),Uu),Uv) = product_Unity ).
% sup_unit_def
tff(fact_2214_bot__unit__def,axiom,
bot_bot(product_unit) = product_Unity ).
% bot_unit_def
tff(fact_2215_Heap_Orel__compp,axiom,
! [A: $tType,C: $tType,B: $tType,R3: fun(A,fun(B,bool)),S3: fun(B,fun(C,bool))] : aa(fun(A,fun(C,bool)),fun(heap_Heap(A),fun(heap_Heap(C),bool)),heap_rel_Heap(A,C),aa(fun(B,fun(C,bool)),fun(A,fun(C,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(C,bool)),fun(A,fun(C,bool))),relcompp(A,B,C),R3),S3)) = aa(fun(heap_Heap(B),fun(heap_Heap(C),bool)),fun(heap_Heap(A),fun(heap_Heap(C),bool)),aa(fun(heap_Heap(A),fun(heap_Heap(B),bool)),fun(fun(heap_Heap(B),fun(heap_Heap(C),bool)),fun(heap_Heap(A),fun(heap_Heap(C),bool))),relcompp(heap_Heap(A),heap_Heap(B),heap_Heap(C)),aa(fun(A,fun(B,bool)),fun(heap_Heap(A),fun(heap_Heap(B),bool)),heap_rel_Heap(A,B),R3)),aa(fun(B,fun(C,bool)),fun(heap_Heap(B),fun(heap_Heap(C),bool)),heap_rel_Heap(B,C),S3)) ).
% Heap.rel_compp
tff(fact_2216_Heap_Orel__flip,axiom,
! [A: $tType,B: $tType,R3: fun(A,fun(B,bool)),A3: heap_Heap(B),B2: heap_Heap(A)] :
( pp(aa(heap_Heap(A),bool,aa(heap_Heap(B),fun(heap_Heap(A),bool),aa(fun(B,fun(A,bool)),fun(heap_Heap(B),fun(heap_Heap(A),bool)),heap_rel_Heap(B,A),conversep(A,B,R3)),A3),B2))
<=> pp(aa(heap_Heap(B),bool,aa(heap_Heap(A),fun(heap_Heap(B),bool),aa(fun(A,fun(B,bool)),fun(heap_Heap(A),fun(heap_Heap(B),bool)),heap_rel_Heap(A,B),R3),B2),A3)) ) ).
% Heap.rel_flip
tff(fact_2217_Heap_Orel__conversep,axiom,
! [B: $tType,A: $tType,R3: fun(A,fun(B,bool))] : aa(fun(B,fun(A,bool)),fun(heap_Heap(B),fun(heap_Heap(A),bool)),heap_rel_Heap(B,A),conversep(A,B,R3)) = conversep(heap_Heap(A),heap_Heap(B),aa(fun(A,fun(B,bool)),fun(heap_Heap(A),fun(heap_Heap(B),bool)),heap_rel_Heap(A,B),R3)) ).
% Heap.rel_conversep
tff(fact_2218_Heap_Orel__mono,axiom,
! [B: $tType,A: $tType,R3: fun(A,fun(B,bool)),Ra: fun(A,fun(B,bool))] :
( pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),R3),Ra))
=> pp(aa(fun(heap_Heap(A),fun(heap_Heap(B),bool)),bool,aa(fun(heap_Heap(A),fun(heap_Heap(B),bool)),fun(fun(heap_Heap(A),fun(heap_Heap(B),bool)),bool),ord_less_eq(fun(heap_Heap(A),fun(heap_Heap(B),bool))),aa(fun(A,fun(B,bool)),fun(heap_Heap(A),fun(heap_Heap(B),bool)),heap_rel_Heap(A,B),R3)),aa(fun(A,fun(B,bool)),fun(heap_Heap(A),fun(heap_Heap(B),bool)),heap_rel_Heap(A,B),Ra))) ) ).
% Heap.rel_mono
tff(fact_2219_Heap_Obi__total__rel,axiom,
! [B: $tType,A: $tType,R3: fun(A,fun(B,bool))] :
( bi_total(A,B,R3)
=> bi_total(heap_Heap(A),heap_Heap(B),aa(fun(A,fun(B,bool)),fun(heap_Heap(A),fun(heap_Heap(B),bool)),heap_rel_Heap(A,B),R3)) ) ).
% Heap.bi_total_rel
tff(fact_2220_Heap_Oright__total__rel,axiom,
! [B: $tType,A: $tType,R3: fun(A,fun(B,bool))] :
( right_total(A,B,R3)
=> right_total(heap_Heap(A),heap_Heap(B),aa(fun(A,fun(B,bool)),fun(heap_Heap(A),fun(heap_Heap(B),bool)),heap_rel_Heap(A,B),R3)) ) ).
% Heap.right_total_rel
tff(fact_2221_reflp__eq,axiom,
! [A: $tType,R3: fun(A,fun(A,bool))] :
( reflp(A,R3)
<=> pp(aa(fun(A,fun(A,bool)),bool,aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),bool),ord_less_eq(fun(A,fun(A,bool))),fequal(A)),R3)) ) ).
% reflp_eq
tff(fact_2222_Heap_Oright__unique__rel,axiom,
! [B: $tType,A: $tType,R3: fun(A,fun(B,bool))] :
( right_unique(A,B,R3)
=> right_unique(heap_Heap(A),heap_Heap(B),aa(fun(A,fun(B,bool)),fun(heap_Heap(A),fun(heap_Heap(B),bool)),heap_rel_Heap(A,B),R3)) ) ).
% Heap.right_unique_rel
tff(fact_2223_Heap_Oleft__unique__rel,axiom,
! [B: $tType,A: $tType,R3: fun(A,fun(B,bool))] :
( left_unique(A,B,R3)
=> left_unique(heap_Heap(A),heap_Heap(B),aa(fun(A,fun(B,bool)),fun(heap_Heap(A),fun(heap_Heap(B),bool)),heap_rel_Heap(A,B),R3)) ) ).
% Heap.left_unique_rel
tff(fact_2224_Heap_Oleft__total__rel,axiom,
! [B: $tType,A: $tType,R3: fun(A,fun(B,bool))] :
( left_total(A,B,R3)
=> left_total(heap_Heap(A),heap_Heap(B),aa(fun(A,fun(B,bool)),fun(heap_Heap(A),fun(heap_Heap(B),bool)),heap_rel_Heap(A,B),R3)) ) ).
% Heap.left_total_rel
tff(fact_2225_Heap_Orel__cases,axiom,
! [A: $tType,B: $tType,R3: fun(A,fun(B,bool)),A3: heap_Heap(A),B2: heap_Heap(B)] :
( pp(aa(heap_Heap(B),bool,aa(heap_Heap(A),fun(heap_Heap(B),bool),aa(fun(A,fun(B,bool)),fun(heap_Heap(A),fun(heap_Heap(B),bool)),heap_rel_Heap(A,B),R3),A3),B2))
=> ~ ! [Xa: fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))))] :
( ( A3 = aa(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_Heap(A),heap_Heap2(A),Xa) )
=> ! [Ya2: fun(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit))))] :
( ( B2 = aa(fun(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit)))),heap_Heap(B),heap_Heap2(B),Ya2) )
=> ~ pp(aa(fun(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit)))),bool,aa(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),fun(fun(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit)))),bool),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))),fequal(heap_ext(product_unit)),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,fequal(heap_ext(product_unit))))),Xa),Ya2)) ) ) ) ).
% Heap.rel_cases
tff(fact_2226_Heap_Orel__induct,axiom,
! [A: $tType,B: $tType,R3: fun(A,fun(B,bool)),X2: heap_Heap(A),Y2: heap_Heap(B),Q2: fun(heap_Heap(A),fun(heap_Heap(B),bool))] :
( pp(aa(heap_Heap(B),bool,aa(heap_Heap(A),fun(heap_Heap(B),bool),aa(fun(A,fun(B,bool)),fun(heap_Heap(A),fun(heap_Heap(B),bool)),heap_rel_Heap(A,B),R3),X2),Y2))
=> ( ! [A5: fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),B4: fun(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit))))] :
( pp(aa(fun(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit)))),bool,aa(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),fun(fun(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit)))),bool),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))),fequal(heap_ext(product_unit)),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,fequal(heap_ext(product_unit))))),A5),B4))
=> pp(aa(heap_Heap(B),bool,aa(heap_Heap(A),fun(heap_Heap(B),bool),Q2,aa(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_Heap(A),heap_Heap2(A),A5)),aa(fun(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit)))),heap_Heap(B),heap_Heap2(B),B4))) )
=> pp(aa(heap_Heap(B),bool,aa(heap_Heap(A),fun(heap_Heap(B),bool),Q2,X2),Y2)) ) ) ).
% Heap.rel_induct
tff(fact_2227_Heap_Orel__inject,axiom,
! [A: $tType,B: $tType,R3: fun(A,fun(B,bool)),X2: fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),Y2: fun(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit))))] :
( pp(aa(heap_Heap(B),bool,aa(heap_Heap(A),fun(heap_Heap(B),bool),aa(fun(A,fun(B,bool)),fun(heap_Heap(A),fun(heap_Heap(B),bool)),heap_rel_Heap(A,B),R3),aa(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_Heap(A),heap_Heap2(A),X2)),aa(fun(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit)))),heap_Heap(B),heap_Heap2(B),Y2)))
<=> pp(aa(fun(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit)))),bool,aa(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),fun(fun(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit)))),bool),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))),fequal(heap_ext(product_unit)),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,fequal(heap_ext(product_unit))))),X2),Y2)) ) ).
% Heap.rel_inject
tff(fact_2228_Heap_Orel__intros,axiom,
! [A: $tType,B: $tType,R3: fun(A,fun(B,bool)),X2: fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),Y2: fun(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit))))] :
( pp(aa(fun(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit)))),bool,aa(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),fun(fun(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit)))),bool),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))),fequal(heap_ext(product_unit)),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,fequal(heap_ext(product_unit))))),X2),Y2))
=> pp(aa(heap_Heap(B),bool,aa(heap_Heap(A),fun(heap_Heap(B),bool),aa(fun(A,fun(B,bool)),fun(heap_Heap(A),fun(heap_Heap(B),bool)),heap_rel_Heap(A,B),R3),aa(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_Heap(A),heap_Heap2(A),X2)),aa(fun(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit)))),heap_Heap(B),heap_Heap2(B),Y2))) ) ).
% Heap.rel_intros
tff(fact_2229_Heap_Octr__transfer,axiom,
! [A: $tType,B: $tType,R3: fun(A,fun(B,bool))] : pp(aa(fun(fun(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit)))),heap_Heap(B)),bool,aa(fun(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_Heap(A)),fun(fun(fun(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit)))),heap_Heap(B)),bool),bNF_rel_fun(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),fun(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))),fequal(heap_ext(product_unit)),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,fequal(heap_ext(product_unit))))),aa(fun(A,fun(B,bool)),fun(heap_Heap(A),fun(heap_Heap(B),bool)),heap_rel_Heap(A,B),R3)),heap_Heap2(A)),heap_Heap2(B))) ).
% Heap.ctr_transfer
tff(fact_2230_Heap_Orel__cong,axiom,
! [A: $tType,B: $tType,X2: heap_Heap(A),Ya: heap_Heap(A),Y2: heap_Heap(B),Xa2: heap_Heap(B),R3: fun(A,fun(B,bool)),Ra: fun(A,fun(B,bool))] :
( ( X2 = Ya )
=> ( ( Y2 = Xa2 )
=> ( ! [Z3: A,Yb: B] :
( pp(aa(set(A),bool,member(A,Z3),aa(heap_Heap(A),set(A),heap_set_Heap(A),Ya)))
=> ( pp(aa(set(B),bool,member(B,Yb),aa(heap_Heap(B),set(B),heap_set_Heap(B),Xa2)))
=> ( pp(aa(B,bool,aa(A,fun(B,bool),R3,Z3),Yb))
<=> pp(aa(B,bool,aa(A,fun(B,bool),Ra,Z3),Yb)) ) ) )
=> ( pp(aa(heap_Heap(B),bool,aa(heap_Heap(A),fun(heap_Heap(B),bool),aa(fun(A,fun(B,bool)),fun(heap_Heap(A),fun(heap_Heap(B),bool)),heap_rel_Heap(A,B),R3),X2),Y2))
<=> pp(aa(heap_Heap(B),bool,aa(heap_Heap(A),fun(heap_Heap(B),bool),aa(fun(A,fun(B,bool)),fun(heap_Heap(A),fun(heap_Heap(B),bool)),heap_rel_Heap(A,B),Ra),Ya),Xa2)) ) ) ) ) ).
% Heap.rel_cong
tff(fact_2231_Heap_Orel__mono__strong,axiom,
! [A: $tType,B: $tType,R3: fun(A,fun(B,bool)),X2: heap_Heap(A),Y2: heap_Heap(B),Ra: fun(A,fun(B,bool))] :
( pp(aa(heap_Heap(B),bool,aa(heap_Heap(A),fun(heap_Heap(B),bool),aa(fun(A,fun(B,bool)),fun(heap_Heap(A),fun(heap_Heap(B),bool)),heap_rel_Heap(A,B),R3),X2),Y2))
=> ( ! [Z3: A,Yb: B] :
( pp(aa(set(A),bool,member(A,Z3),aa(heap_Heap(A),set(A),heap_set_Heap(A),X2)))
=> ( pp(aa(set(B),bool,member(B,Yb),aa(heap_Heap(B),set(B),heap_set_Heap(B),Y2)))
=> ( pp(aa(B,bool,aa(A,fun(B,bool),R3,Z3),Yb))
=> pp(aa(B,bool,aa(A,fun(B,bool),Ra,Z3),Yb)) ) ) )
=> pp(aa(heap_Heap(B),bool,aa(heap_Heap(A),fun(heap_Heap(B),bool),aa(fun(A,fun(B,bool)),fun(heap_Heap(A),fun(heap_Heap(B),bool)),heap_rel_Heap(A,B),Ra),X2),Y2)) ) ) ).
% Heap.rel_mono_strong
tff(fact_2232_Heap_Orel__refl__strong,axiom,
! [A: $tType,X2: heap_Heap(A),Ra: fun(A,fun(A,bool))] :
( ! [Z3: A] :
( pp(aa(set(A),bool,member(A,Z3),aa(heap_Heap(A),set(A),heap_set_Heap(A),X2)))
=> pp(aa(A,bool,aa(A,fun(A,bool),Ra,Z3),Z3)) )
=> pp(aa(heap_Heap(A),bool,aa(heap_Heap(A),fun(heap_Heap(A),bool),aa(fun(A,fun(A,bool)),fun(heap_Heap(A),fun(heap_Heap(A),bool)),heap_rel_Heap(A,A),Ra),X2),X2)) ) ).
% Heap.rel_refl_strong
tff(fact_2233_Heap_Orel__map_I2_J,axiom,
! [A: $tType,C: $tType,B: $tType,Sa: fun(A,fun(C,bool)),X2: heap_Heap(A),G: fun(B,C),Y2: heap_Heap(B)] :
( pp(aa(heap_Heap(C),bool,aa(heap_Heap(A),fun(heap_Heap(C),bool),aa(fun(A,fun(C,bool)),fun(heap_Heap(A),fun(heap_Heap(C),bool)),heap_rel_Heap(A,C),Sa),X2),aa(heap_Heap(B),heap_Heap(C),aa(fun(B,C),fun(heap_Heap(B),heap_Heap(C)),heap_map_Heap(B,C),G),Y2)))
<=> pp(aa(heap_Heap(B),bool,aa(heap_Heap(A),fun(heap_Heap(B),bool),aa(fun(A,fun(B,bool)),fun(heap_Heap(A),fun(heap_Heap(B),bool)),heap_rel_Heap(A,B),aa(fun(B,C),fun(A,fun(B,bool)),aTP_Lamp_ar(fun(A,fun(C,bool)),fun(fun(B,C),fun(A,fun(B,bool))),Sa),G)),X2),Y2)) ) ).
% Heap.rel_map(2)
tff(fact_2234_Heap_Orel__map_I1_J,axiom,
! [A: $tType,C: $tType,B: $tType,Sb: fun(C,fun(B,bool)),I: fun(A,C),X2: heap_Heap(A),Y2: heap_Heap(B)] :
( pp(aa(heap_Heap(B),bool,aa(heap_Heap(C),fun(heap_Heap(B),bool),aa(fun(C,fun(B,bool)),fun(heap_Heap(C),fun(heap_Heap(B),bool)),heap_rel_Heap(C,B),Sb),aa(heap_Heap(A),heap_Heap(C),aa(fun(A,C),fun(heap_Heap(A),heap_Heap(C)),heap_map_Heap(A,C),I),X2)),Y2))
<=> pp(aa(heap_Heap(B),bool,aa(heap_Heap(A),fun(heap_Heap(B),bool),aa(fun(A,fun(B,bool)),fun(heap_Heap(A),fun(heap_Heap(B),bool)),heap_rel_Heap(A,B),aa(fun(A,C),fun(A,fun(B,bool)),aTP_Lamp_aq(fun(C,fun(B,bool)),fun(fun(A,C),fun(A,fun(B,bool))),Sb),I)),X2),Y2)) ) ).
% Heap.rel_map(1)
tff(fact_2235_Heap_Orec__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,R3: fun(A,fun(B,bool)),S3: fun(C,fun(D,bool))] : pp(aa(fun(fun(fun(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit)))),D),fun(heap_Heap(B),D)),bool,aa(fun(fun(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),C),fun(heap_Heap(A),C)),fun(fun(fun(fun(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit)))),D),fun(heap_Heap(B),D)),bool),bNF_rel_fun(fun(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),C),fun(fun(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit)))),D),fun(heap_Heap(A),C),fun(heap_Heap(B),D),bNF_rel_fun(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),fun(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))),fequal(heap_ext(product_unit)),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,fequal(heap_ext(product_unit))))),S3),bNF_rel_fun(heap_Heap(A),heap_Heap(B),C,D,aa(fun(A,fun(B,bool)),fun(heap_Heap(A),fun(heap_Heap(B),bool)),heap_rel_Heap(A,B),R3),S3)),heap_rec_Heap(A,C)),heap_rec_Heap(B,D))) ).
% Heap.rec_transfer
tff(fact_2236_Heap_Orel__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,Sa: fun(A,fun(C,bool)),Sc: fun(B,fun(D,bool))] : pp(aa(fun(fun(C,fun(D,bool)),fun(heap_Heap(C),fun(heap_Heap(D),bool))),bool,aa(fun(fun(A,fun(B,bool)),fun(heap_Heap(A),fun(heap_Heap(B),bool))),fun(fun(fun(C,fun(D,bool)),fun(heap_Heap(C),fun(heap_Heap(D),bool))),bool),bNF_rel_fun(fun(A,fun(B,bool)),fun(C,fun(D,bool)),fun(heap_Heap(A),fun(heap_Heap(B),bool)),fun(heap_Heap(C),fun(heap_Heap(D),bool)),bNF_rel_fun(A,C,fun(B,bool),fun(D,bool),Sa,bNF_rel_fun(B,D,bool,bool,Sc,fequal(bool))),bNF_rel_fun(heap_Heap(A),heap_Heap(C),fun(heap_Heap(B),bool),fun(heap_Heap(D),bool),aa(fun(A,fun(C,bool)),fun(heap_Heap(A),fun(heap_Heap(C),bool)),heap_rel_Heap(A,C),Sa),bNF_rel_fun(heap_Heap(B),heap_Heap(D),bool,bool,aa(fun(B,fun(D,bool)),fun(heap_Heap(B),fun(heap_Heap(D),bool)),heap_rel_Heap(B,D),Sc),fequal(bool)))),heap_rel_Heap(A,B)),heap_rel_Heap(C,D))) ).
% Heap.rel_transfer
tff(fact_2237_Heap_Omap__transfer,axiom,
! [A: $tType,B: $tType,F2: $tType,E: $tType,Rb: fun(A,fun(E,bool)),Sd: fun(B,fun(F2,bool))] : pp(aa(fun(fun(E,F2),fun(heap_Heap(E),heap_Heap(F2))),bool,aa(fun(fun(A,B),fun(heap_Heap(A),heap_Heap(B))),fun(fun(fun(E,F2),fun(heap_Heap(E),heap_Heap(F2))),bool),bNF_rel_fun(fun(A,B),fun(E,F2),fun(heap_Heap(A),heap_Heap(B)),fun(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),aa(fun(A,fun(E,bool)),fun(heap_Heap(A),fun(heap_Heap(E),bool)),heap_rel_Heap(A,E),Rb),aa(fun(B,fun(F2,bool)),fun(heap_Heap(B),fun(heap_Heap(F2),bool)),heap_rel_Heap(B,F2),Sd))),heap_map_Heap(A,B)),heap_map_Heap(E,F2))) ).
% Heap.map_transfer
tff(fact_2238_Heap_OQuotient,axiom,
! [B: $tType,A: $tType,R3: fun(A,fun(A,bool)),Abs: fun(A,B),Rep: fun(B,A),T3: fun(A,fun(B,bool))] :
( quotient(A,B,R3,Abs,Rep,T3)
=> quotient(heap_Heap(A),heap_Heap(B),aa(fun(A,fun(A,bool)),fun(heap_Heap(A),fun(heap_Heap(A),bool)),heap_rel_Heap(A,A),R3),aa(fun(A,B),fun(heap_Heap(A),heap_Heap(B)),heap_map_Heap(A,B),Abs),aa(fun(B,A),fun(heap_Heap(B),heap_Heap(A)),heap_map_Heap(B,A),Rep),aa(fun(A,fun(B,bool)),fun(heap_Heap(A),fun(heap_Heap(B),bool)),heap_rel_Heap(A,B),T3)) ) ).
% Heap.Quotient
tff(fact_2239_execute__update,axiom,
! [A: $tType] :
( heap(A)
=> ! [R: ref(A),V: A,H: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(product_unit,heap_ext(product_unit))),aa(heap_Heap(product_unit),fun(heap_ext(product_unit),option(product_prod(product_unit,heap_ext(product_unit)))),heap_execute(product_unit),update2(A,R,V)),H) = aa(product_prod(product_unit,heap_ext(product_unit)),option(product_prod(product_unit,heap_ext(product_unit))),some(product_prod(product_unit,heap_ext(product_unit))),aa(heap_ext(product_unit),product_prod(product_unit,heap_ext(product_unit)),aa(product_unit,fun(heap_ext(product_unit),product_prod(product_unit,heap_ext(product_unit))),product_Pair(product_unit,heap_ext(product_unit)),product_Unity),aa(heap_ext(product_unit),heap_ext(product_unit),set4(A,R,V),H))) ) ).
% execute_update
tff(fact_2240_Ref_Oupdate__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [R: ref(A),V: A] : update2(A,R,V) = heap_heap(product_unit,aa(A,fun(heap_ext(product_unit),product_prod(product_unit,heap_ext(product_unit))),aTP_Lamp_td(ref(A),fun(A,fun(heap_ext(product_unit),product_prod(product_unit,heap_ext(product_unit)))),R),V)) ) ).
% Ref.update_def
tff(fact_2241_upd_H__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [A3: array(A),I: code_integer,X2: A] : upd2(A,A3,I,X2) = heap_bind(array(A),product_unit,upd(A,aa(code_integer,nat,code_nat_of_integer,I),X2,A3),aTP_Lamp_te(array(A),heap_Heap(product_unit))) ) ).
% upd'_def
tff(fact_2242_default__unit__def,axiom,
default_default(product_unit) = product_Unity ).
% default_unit_def
tff(fact_2243_rtranclp__is__Sup__relpowp,axiom,
! [A: $tType,P2: fun(A,fun(A,bool))] : transitive_rtranclp(A,P2) = aa(set(fun(A,fun(A,bool))),fun(A,fun(A,bool)),complete_Sup_Sup(fun(A,fun(A,bool))),aa(set(nat),set(fun(A,fun(A,bool))),aa(fun(nat,fun(A,fun(A,bool))),fun(set(nat),set(fun(A,fun(A,bool)))),image2(nat,fun(A,fun(A,bool))),aTP_Lamp_tf(fun(A,fun(A,bool)),fun(nat,fun(A,fun(A,bool))),P2)),top_top(set(nat)))) ).
% rtranclp_is_Sup_relpowp
tff(fact_2244_rtranclp__imp__Sup__relpowp,axiom,
! [A: $tType,P2: fun(A,fun(A,bool)),X2: A,Y2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),transitive_rtranclp(A,P2),X2),Y2))
=> pp(aa(A,bool,aa(A,fun(A,bool),aa(set(fun(A,fun(A,bool))),fun(A,fun(A,bool)),complete_Sup_Sup(fun(A,fun(A,bool))),aa(set(nat),set(fun(A,fun(A,bool))),aa(fun(nat,fun(A,fun(A,bool))),fun(set(nat),set(fun(A,fun(A,bool)))),image2(nat,fun(A,fun(A,bool))),aTP_Lamp_tf(fun(A,fun(A,bool)),fun(nat,fun(A,fun(A,bool))),P2)),top_top(set(nat)))),X2),Y2)) ) ).
% rtranclp_imp_Sup_relpowp
tff(fact_2245_map__option__o__case__sum,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F: fun(D,C),G: fun(A,option(D)),H: fun(B,option(D))] : aa(fun(sum_sum(A,B),option(D)),fun(sum_sum(A,B),option(C)),aa(fun(option(D),option(C)),fun(fun(sum_sum(A,B),option(D)),fun(sum_sum(A,B),option(C))),comp(option(D),option(C),sum_sum(A,B)),map_option(D,C,F)),sum_case_sum(A,option(D),B,G,H)) = sum_case_sum(A,option(C),B,aa(fun(A,option(D)),fun(A,option(C)),aa(fun(option(D),option(C)),fun(fun(A,option(D)),fun(A,option(C))),comp(option(D),option(C),A),map_option(D,C,F)),G),aa(fun(B,option(D)),fun(B,option(C)),aa(fun(option(D),option(C)),fun(fun(B,option(D)),fun(B,option(C))),comp(option(D),option(C),B),map_option(D,C,F)),H)) ).
% map_option_o_case_sum
tff(fact_2246_Heap_Opred__map,axiom,
! [B: $tType,A: $tType,Q2: fun(B,bool),F: fun(A,B),X2: heap_Heap(A)] :
( pp(aa(heap_Heap(B),bool,aa(fun(B,bool),fun(heap_Heap(B),bool),heap_pred_Heap(B),Q2),aa(heap_Heap(A),heap_Heap(B),aa(fun(A,B),fun(heap_Heap(A),heap_Heap(B)),heap_map_Heap(A,B),F),X2)))
<=> pp(aa(heap_Heap(A),bool,aa(fun(A,bool),fun(heap_Heap(A),bool),heap_pred_Heap(A),aa(fun(A,B),fun(A,bool),aa(fun(B,bool),fun(fun(A,B),fun(A,bool)),comp(B,bool,A),Q2),F)),X2)) ) ).
% Heap.pred_map
tff(fact_2247_Heap_Opred__True,axiom,
! [A: $tType,X4: heap_Heap(A)] : pp(aa(heap_Heap(A),bool,aa(fun(A,bool),fun(heap_Heap(A),bool),heap_pred_Heap(A),aTP_Lamp_ov(A,bool)),X4)) ).
% Heap.pred_True
tff(fact_2248_disjE__realizer,axiom,
! [C: $tType,B: $tType,A: $tType,P2: fun(A,bool),Q2: fun(B,bool),X2: sum_sum(A,B),R3: fun(C,bool),F: fun(A,C),G: fun(B,C)] :
( pp(aa(sum_sum(A,B),bool,sum_case_sum(A,bool,B,P2,Q2),X2))
=> ( ! [P8: A] :
( pp(aa(A,bool,P2,P8))
=> pp(aa(C,bool,R3,aa(A,C,F,P8))) )
=> ( ! [Q6: B] :
( pp(aa(B,bool,Q2,Q6))
=> pp(aa(C,bool,R3,aa(B,C,G,Q6))) )
=> pp(aa(C,bool,R3,aa(sum_sum(A,B),C,sum_case_sum(A,C,B,F,G),X2))) ) ) ) ).
% disjE_realizer
tff(fact_2249_converse__rtranclp__induct2,axiom,
! [A: $tType,B: $tType,R: fun(product_prod(A,B),fun(product_prod(A,B),bool)),Ax: A,Ay: B,Bx: A,By: B,P2: fun(A,fun(B,bool))] :
( pp(aa(product_prod(A,B),bool,aa(product_prod(A,B),fun(product_prod(A,B),bool),transitive_rtranclp(product_prod(A,B),R),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By)))
=> ( pp(aa(B,bool,aa(A,fun(B,bool),P2,Bx),By))
=> ( ! [A5: A,B4: B,Aa2: A,Ba: B] :
( pp(aa(product_prod(A,B),bool,aa(product_prod(A,B),fun(product_prod(A,B),bool),R,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba)))
=> ( pp(aa(product_prod(A,B),bool,aa(product_prod(A,B),fun(product_prod(A,B),bool),transitive_rtranclp(product_prod(A,B),R),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By)))
=> ( pp(aa(B,bool,aa(A,fun(B,bool),P2,Aa2),Ba))
=> pp(aa(B,bool,aa(A,fun(B,bool),P2,A5),B4)) ) ) )
=> pp(aa(B,bool,aa(A,fun(B,bool),P2,Ax),Ay)) ) ) ) ).
% converse_rtranclp_induct2
tff(fact_2250_converse__rtranclpE2,axiom,
! [A: $tType,B: $tType,R: fun(product_prod(A,B),fun(product_prod(A,B),bool)),Xa2: A,Xb3: B,Za2: A,Zb: B] :
( pp(aa(product_prod(A,B),bool,aa(product_prod(A,B),fun(product_prod(A,B),bool),transitive_rtranclp(product_prod(A,B),R),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa2),Xb3)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Za2),Zb)))
=> ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa2),Xb3) != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Za2),Zb) )
=> ~ ! [A5: A,B4: B] :
( pp(aa(product_prod(A,B),bool,aa(product_prod(A,B),fun(product_prod(A,B),bool),R,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa2),Xb3)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4)))
=> ~ pp(aa(product_prod(A,B),bool,aa(product_prod(A,B),fun(product_prod(A,B),bool),transitive_rtranclp(product_prod(A,B),R),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Za2),Zb))) ) ) ) ).
% converse_rtranclpE2
tff(fact_2251_rtranclp__induct2,axiom,
! [A: $tType,B: $tType,R: fun(product_prod(A,B),fun(product_prod(A,B),bool)),Ax: A,Ay: B,Bx: A,By: B,P2: fun(A,fun(B,bool))] :
( pp(aa(product_prod(A,B),bool,aa(product_prod(A,B),fun(product_prod(A,B),bool),transitive_rtranclp(product_prod(A,B),R),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Bx),By)))
=> ( pp(aa(B,bool,aa(A,fun(B,bool),P2,Ax),Ay))
=> ( ! [A5: A,B4: B,Aa2: A,Ba: B] :
( pp(aa(product_prod(A,B),bool,aa(product_prod(A,B),fun(product_prod(A,B),bool),transitive_rtranclp(product_prod(A,B),R),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Ax),Ay)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4)))
=> ( pp(aa(product_prod(A,B),bool,aa(product_prod(A,B),fun(product_prod(A,B),bool),R,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa2),Ba)))
=> ( pp(aa(B,bool,aa(A,fun(B,bool),P2,A5),B4))
=> pp(aa(B,bool,aa(A,fun(B,bool),P2,Aa2),Ba)) ) ) )
=> pp(aa(B,bool,aa(A,fun(B,bool),P2,Bx),By)) ) ) ) ).
% rtranclp_induct2
tff(fact_2252_o__case__sum,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,H: fun(D,C),F: fun(A,D),G: fun(B,D)] : aa(fun(sum_sum(A,B),D),fun(sum_sum(A,B),C),aa(fun(D,C),fun(fun(sum_sum(A,B),D),fun(sum_sum(A,B),C)),comp(D,C,sum_sum(A,B)),H),sum_case_sum(A,D,B,F,G)) = sum_case_sum(A,C,B,aa(fun(A,D),fun(A,C),aa(fun(D,C),fun(fun(A,D),fun(A,C)),comp(D,C,A),H),F),aa(fun(B,D),fun(B,C),aa(fun(D,C),fun(fun(B,D),fun(B,C)),comp(D,C,B),H),G)) ).
% o_case_sum
tff(fact_2253_Heap_Opred__mono,axiom,
! [A: $tType,P2: fun(A,bool),Pa: fun(A,bool)] :
( pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),P2),Pa))
=> pp(aa(fun(heap_Heap(A),bool),bool,aa(fun(heap_Heap(A),bool),fun(fun(heap_Heap(A),bool),bool),ord_less_eq(fun(heap_Heap(A),bool)),aa(fun(A,bool),fun(heap_Heap(A),bool),heap_pred_Heap(A),P2)),aa(fun(A,bool),fun(heap_Heap(A),bool),heap_pred_Heap(A),Pa))) ) ).
% Heap.pred_mono
tff(fact_2254_Heap_Opred__mono__strong,axiom,
! [A: $tType,P2: fun(A,bool),X2: heap_Heap(A),Pa: fun(A,bool)] :
( pp(aa(heap_Heap(A),bool,aa(fun(A,bool),fun(heap_Heap(A),bool),heap_pred_Heap(A),P2),X2))
=> ( ! [Z3: A] :
( pp(aa(set(A),bool,member(A,Z3),aa(heap_Heap(A),set(A),heap_set_Heap(A),X2)))
=> ( pp(aa(A,bool,P2,Z3))
=> pp(aa(A,bool,Pa,Z3)) ) )
=> pp(aa(heap_Heap(A),bool,aa(fun(A,bool),fun(heap_Heap(A),bool),heap_pred_Heap(A),Pa),X2)) ) ) ).
% Heap.pred_mono_strong
tff(fact_2255_Heap_Opred__cong,axiom,
! [A: $tType,X2: heap_Heap(A),Ya: heap_Heap(A),P2: fun(A,bool),Pa: fun(A,bool)] :
( ( X2 = Ya )
=> ( ! [Z3: A] :
( pp(aa(set(A),bool,member(A,Z3),aa(heap_Heap(A),set(A),heap_set_Heap(A),Ya)))
=> ( pp(aa(A,bool,P2,Z3))
<=> pp(aa(A,bool,Pa,Z3)) ) )
=> ( pp(aa(heap_Heap(A),bool,aa(fun(A,bool),fun(heap_Heap(A),bool),heap_pred_Heap(A),P2),X2))
<=> pp(aa(heap_Heap(A),bool,aa(fun(A,bool),fun(heap_Heap(A),bool),heap_pred_Heap(A),Pa),Ya)) ) ) ) ).
% Heap.pred_cong
tff(fact_2256_Transitive__Closure_Ortranclp__rtrancl__eq,axiom,
! [A: $tType,R: set(product_prod(A,A)),X4: A,Xa4: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),transitive_rtranclp(A,aTP_Lamp_hl(set(product_prod(A,A)),fun(A,fun(A,bool)),R)),X4),Xa4))
<=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),Xa4)),transitive_rtrancl(A,R))) ) ).
% Transitive_Closure.rtranclp_rtrancl_eq
tff(fact_2257_Heap_Omap__cong__pred,axiom,
! [B: $tType,A: $tType,X2: heap_Heap(A),Ya: heap_Heap(A),F: fun(A,B),G: fun(A,B)] :
( ( X2 = Ya )
=> ( pp(aa(heap_Heap(A),bool,aa(fun(A,bool),fun(heap_Heap(A),bool),heap_pred_Heap(A),aa(fun(A,B),fun(A,bool),aTP_Lamp_qh(fun(A,B),fun(fun(A,B),fun(A,bool)),F),G)),Ya))
=> ( aa(heap_Heap(A),heap_Heap(B),aa(fun(A,B),fun(heap_Heap(A),heap_Heap(B)),heap_map_Heap(A,B),F),X2) = aa(heap_Heap(A),heap_Heap(B),aa(fun(A,B),fun(heap_Heap(A),heap_Heap(B)),heap_map_Heap(A,B),G),Ya) ) ) ) ).
% Heap.map_cong_pred
tff(fact_2258_rtrancl__def,axiom,
! [A: $tType,X4: set(product_prod(A,A))] : transitive_rtrancl(A,X4) = aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),transitive_rtranclp(A,aTP_Lamp_hl(set(product_prod(A,A)),fun(A,fun(A,bool)),X4)))) ).
% rtrancl_def
tff(fact_2259_Enum_Ortranclp__rtrancl__eq,axiom,
! [A: $tType,R: fun(A,fun(A,bool)),X2: A,Y2: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),transitive_rtranclp(A,R),X2),Y2))
<=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),transitive_rtrancl(A,aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),R))))) ) ).
% Enum.rtranclp_rtrancl_eq
tff(fact_2260_Heap_Opred__transfer,axiom,
! [A: $tType,B: $tType,R3: fun(A,fun(B,bool))] : pp(aa(fun(fun(B,bool),fun(heap_Heap(B),bool)),bool,aa(fun(fun(A,bool),fun(heap_Heap(A),bool)),fun(fun(fun(B,bool),fun(heap_Heap(B),bool)),bool),bNF_rel_fun(fun(A,bool),fun(B,bool),fun(heap_Heap(A),bool),fun(heap_Heap(B),bool),bNF_rel_fun(A,B,bool,bool,R3,fequal(bool)),bNF_rel_fun(heap_Heap(A),heap_Heap(B),bool,bool,aa(fun(A,fun(B,bool)),fun(heap_Heap(A),fun(heap_Heap(B),bool)),heap_rel_Heap(A,B),R3),fequal(bool))),heap_pred_Heap(A)),heap_pred_Heap(B))) ).
% Heap.pred_transfer
tff(fact_2261_Heap_Opred__set,axiom,
! [A: $tType,P2: fun(A,bool),X4: heap_Heap(A)] :
( pp(aa(heap_Heap(A),bool,aa(fun(A,bool),fun(heap_Heap(A),bool),heap_pred_Heap(A),P2),X4))
<=> ! [Xa3: A] :
( pp(aa(set(A),bool,member(A,Xa3),aa(heap_Heap(A),set(A),heap_set_Heap(A),X4)))
=> pp(aa(A,bool,P2,Xa3)) ) ) ).
% Heap.pred_set
tff(fact_2262_Heap_Opred__rel,axiom,
! [A: $tType,P2: fun(A,bool),X2: heap_Heap(A)] :
( pp(aa(heap_Heap(A),bool,aa(fun(A,bool),fun(heap_Heap(A),bool),heap_pred_Heap(A),P2),X2))
<=> pp(aa(heap_Heap(A),bool,aa(heap_Heap(A),fun(heap_Heap(A),bool),aa(fun(A,fun(A,bool)),fun(heap_Heap(A),fun(heap_Heap(A),bool)),heap_rel_Heap(A,A),bNF_eq_onp(A,P2)),X2),X2)) ) ).
% Heap.pred_rel
tff(fact_2263_Heap_Orel__eq__onp,axiom,
! [A: $tType,P2: fun(A,bool)] : aa(fun(A,fun(A,bool)),fun(heap_Heap(A),fun(heap_Heap(A),bool)),heap_rel_Heap(A,A),bNF_eq_onp(A,P2)) = bNF_eq_onp(heap_Heap(A),aa(fun(A,bool),fun(heap_Heap(A),bool),heap_pred_Heap(A),P2)) ).
% Heap.rel_eq_onp
tff(fact_2264_Heap_ODomainp__rel,axiom,
! [B: $tType,A: $tType,R3: fun(A,fun(B,bool))] : domainp(heap_Heap(A),heap_Heap(B),aa(fun(A,fun(B,bool)),fun(heap_Heap(A),fun(heap_Heap(B),bool)),heap_rel_Heap(A,B),R3)) = aa(fun(A,bool),fun(heap_Heap(A),bool),heap_pred_Heap(A),domainp(A,B,R3)) ).
% Heap.Domainp_rel
tff(fact_2265_Heap_Ocase__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,R3: fun(A,fun(B,bool)),S3: fun(C,fun(D,bool))] : pp(aa(fun(fun(fun(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit)))),D),fun(heap_Heap(B),D)),bool,aa(fun(fun(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),C),fun(heap_Heap(A),C)),fun(fun(fun(fun(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit)))),D),fun(heap_Heap(B),D)),bool),bNF_rel_fun(fun(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),C),fun(fun(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit)))),D),fun(heap_Heap(A),C),fun(heap_Heap(B),D),bNF_rel_fun(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),fun(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))),fequal(heap_ext(product_unit)),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,fequal(heap_ext(product_unit))))),S3),bNF_rel_fun(heap_Heap(A),heap_Heap(B),C,D,aa(fun(A,fun(B,bool)),fun(heap_Heap(A),fun(heap_Heap(B),bool)),heap_rel_Heap(A,B),R3),S3)),heap_case_Heap(A,C)),heap_case_Heap(B,D))) ).
% Heap.case_transfer
tff(fact_2266_iterate_Oelims,axiom,
! [A: $tType,B: $tType,X2: code_natural,Xa2: fun(B,fun(A,product_prod(B,A))),Xb3: B,Y2: fun(A,product_prod(B,A))] :
( ( aa(B,fun(A,product_prod(B,A)),iterate(B,A,X2,Xa2),Xb3) = Y2 )
=> ( ( ( X2 = zero_zero(code_natural) )
=> ( Y2 = aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Xb3) ) )
& ( ( X2 != zero_zero(code_natural) )
=> ( Y2 = product_scomp(A,B,A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),Xa2,Xb3),iterate(B,A,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(code_natural),X2),one_one(code_natural)),Xa2)) ) ) ) ) ).
% iterate.elims
tff(fact_2267_iterate_Osimps,axiom,
! [A: $tType,B: $tType,K: code_natural,F: fun(B,fun(A,product_prod(B,A))),X2: B] :
( ( ( K = zero_zero(code_natural) )
=> ( aa(B,fun(A,product_prod(B,A)),iterate(B,A,K,F),X2) = aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X2) ) )
& ( ( K != zero_zero(code_natural) )
=> ( aa(B,fun(A,product_prod(B,A)),iterate(B,A,K,F),X2) = product_scomp(A,B,A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),F,X2),iterate(B,A,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(code_natural),K),one_one(code_natural)),F)) ) ) ) ).
% iterate.simps
tff(fact_2268_r__OO__conversep__into__equivclp,axiom,
! [A: $tType,R: fun(A,fun(A,bool))] : pp(aa(fun(A,fun(A,bool)),bool,aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),bool),ord_less_eq(fun(A,fun(A,bool))),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),relcompp(A,A,A),transitive_rtranclp(A,R)),transitive_rtranclp(A,conversep(A,A,R)))),equiv_equivclp(A,R))) ).
% r_OO_conversep_into_equivclp
tff(fact_2269_mult__inj__if__coprime__nat,axiom,
! [A: $tType,B: $tType,F: fun(A,nat),A6: set(A),G: fun(B,nat),B6: set(B)] :
( inj_on(A,nat,F,A6)
=> ( inj_on(B,nat,G,B6)
=> ( ! [A5: A,B4: B] :
( pp(aa(set(A),bool,member(A,A5),A6))
=> ( pp(aa(set(B),bool,member(B,B4),B6))
=> algebr8660921524188924756oprime(nat,aa(A,nat,F,A5),aa(B,nat,G,B4)) ) )
=> inj_on(product_prod(A,B),nat,aa(fun(A,fun(B,nat)),fun(product_prod(A,B),nat),product_case_prod(A,B,nat),aa(fun(B,nat),fun(A,fun(B,nat)),aTP_Lamp_tg(fun(A,nat),fun(fun(B,nat),fun(A,fun(B,nat))),F),G)),product_Sigma(A,B,A6,aTP_Lamp_bu(set(B),fun(A,set(B)),B6))) ) ) ) ).
% mult_inj_if_coprime_nat
tff(fact_2270_length__filter__map,axiom,
! [A: $tType,B: $tType,P2: fun(A,bool),F: fun(B,A),Xs: list(B)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,P2),aa(list(B),list(A),map(B,A,F),Xs))) = aa(list(B),nat,size_size(list(B)),aa(list(B),list(B),filter2(B,aa(fun(B,A),fun(B,bool),aa(fun(A,bool),fun(fun(B,A),fun(B,bool)),comp(A,bool,B),P2),F)),Xs)) ).
% length_filter_map
tff(fact_2271_partition__filter__conv,axiom,
! [A: $tType,F: fun(A,bool),Xs: list(A)] : partition(A,F,Xs) = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),filter2(A,F),Xs)),aa(list(A),list(A),filter2(A,aa(fun(A,bool),fun(A,bool),aa(fun(bool,bool),fun(fun(A,bool),fun(A,bool)),comp(bool,bool,A),fNot),F)),Xs)) ).
% partition_filter_conv
tff(fact_2272_partition__filter2,axiom,
! [A: $tType,P2: fun(A,bool),Xs: list(A)] : aa(product_prod(list(A),list(A)),list(A),product_snd(list(A),list(A)),partition(A,P2,Xs)) = aa(list(A),list(A),filter2(A,aa(fun(A,bool),fun(A,bool),aa(fun(bool,bool),fun(fun(A,bool),fun(A,bool)),comp(bool,bool,A),fNot),P2)),Xs) ).
% partition_filter2
tff(fact_2273_Heap_Ocase__distrib,axiom,
! [B: $tType,C: $tType,A: $tType,H: fun(B,C),F: fun(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),B),Heap: heap_Heap(A)] : aa(B,C,H,aa(heap_Heap(A),B,aa(fun(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),B),fun(heap_Heap(A),B),heap_case_Heap(A,B),F),Heap)) = aa(heap_Heap(A),C,aa(fun(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),C),fun(heap_Heap(A),C),heap_case_Heap(A,C),aa(fun(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),B),fun(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),C),aTP_Lamp_th(fun(B,C),fun(fun(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),B),fun(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),C)),H),F)),Heap) ).
% Heap.case_distrib
tff(fact_2274_filter__map,axiom,
! [A: $tType,B: $tType,P2: fun(A,bool),F: fun(B,A),Xs: list(B)] : aa(list(A),list(A),filter2(A,P2),aa(list(B),list(A),map(B,A,F),Xs)) = aa(list(B),list(A),map(B,A,F),aa(list(B),list(B),filter2(B,aa(fun(B,A),fun(B,bool),aa(fun(A,bool),fun(fun(B,A),fun(B,bool)),comp(A,bool,B),P2),F)),Xs)) ).
% filter_map
tff(fact_2275_filter__shuffles,axiom,
! [A: $tType,P2: fun(A,bool),Xs: list(A),Ys: list(A)] : aa(set(list(A)),set(list(A)),aa(fun(list(A),list(A)),fun(set(list(A)),set(list(A))),image2(list(A),list(A)),filter2(A,P2)),shuffles(A,Xs,Ys)) = shuffles(A,aa(list(A),list(A),filter2(A,P2),Xs),aa(list(A),list(A),filter2(A,P2),Ys)) ).
% filter_shuffles
tff(fact_2276_equivclp__OO__equivclp__le__equivclp,axiom,
! [A: $tType,R: fun(A,fun(A,bool))] : pp(aa(fun(A,fun(A,bool)),bool,aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),bool),ord_less_eq(fun(A,fun(A,bool))),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),relcompp(A,A,A),equiv_equivclp(A,R)),equiv_equivclp(A,R))),equiv_equivclp(A,R))) ).
% equivclp_OO_equivclp_le_equivclp
tff(fact_2277_map__of__filter__in,axiom,
! [B: $tType,A: $tType,Xs: list(product_prod(B,A)),K: B,Z2: A,P2: fun(B,fun(A,bool))] :
( ( aa(B,option(A),map_of(B,A,Xs),K) = aa(A,option(A),some(A),Z2) )
=> ( pp(aa(A,bool,aa(B,fun(A,bool),P2,K),Z2))
=> ( aa(B,option(A),map_of(B,A,aa(list(product_prod(B,A)),list(product_prod(B,A)),filter2(product_prod(B,A),aa(fun(B,fun(A,bool)),fun(product_prod(B,A),bool),product_case_prod(B,A,bool),P2)),Xs)),K) = aa(A,option(A),some(A),Z2) ) ) ) ).
% map_of_filter_in
tff(fact_2278_Heap_Ocase,axiom,
! [B: $tType,A: $tType,F: fun(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),B),X2: fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))))] : aa(heap_Heap(A),B,aa(fun(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),B),fun(heap_Heap(A),B),heap_case_Heap(A,B),F),aa(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_Heap(A),heap_Heap2(A),X2)) = aa(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),B,F,X2) ).
% Heap.case
tff(fact_2279_map__filter__def,axiom,
! [B: $tType,A: $tType,F: fun(A,option(B)),Xs: list(A)] : map_filter(A,B,F,Xs) = aa(list(A),list(B),map(A,B,aa(fun(A,option(B)),fun(A,B),aa(fun(option(B),B),fun(fun(A,option(B)),fun(A,B)),comp(option(B),B,A),the2(B)),F)),aa(list(A),list(A),filter2(A,aTP_Lamp_ti(fun(A,option(B)),fun(A,bool),F)),Xs)) ).
% map_filter_def
tff(fact_2280_iterate_Opelims,axiom,
! [A: $tType,B: $tType,X2: code_natural,Xa2: fun(B,fun(A,product_prod(B,A))),Xb3: B,Y2: fun(A,product_prod(B,A))] :
( ( aa(B,fun(A,product_prod(B,A)),iterate(B,A,X2,Xa2),Xb3) = Y2 )
=> ( pp(aa(product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),bool,accp(product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),iterate_rel(B,A)),aa(product_prod(fun(B,fun(A,product_prod(B,A))),B),product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),aa(code_natural,fun(product_prod(fun(B,fun(A,product_prod(B,A))),B),product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B))),product_Pair(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),X2),aa(B,product_prod(fun(B,fun(A,product_prod(B,A))),B),aa(fun(B,fun(A,product_prod(B,A))),fun(B,product_prod(fun(B,fun(A,product_prod(B,A))),B)),product_Pair(fun(B,fun(A,product_prod(B,A))),B),Xa2),Xb3))))
=> ~ ( ( ( ( X2 = zero_zero(code_natural) )
=> ( Y2 = aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Xb3) ) )
& ( ( X2 != zero_zero(code_natural) )
=> ( Y2 = product_scomp(A,B,A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),Xa2,Xb3),iterate(B,A,aa(code_natural,code_natural,aa(code_natural,fun(code_natural,code_natural),minus_minus(code_natural),X2),one_one(code_natural)),Xa2)) ) ) )
=> ~ pp(aa(product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),bool,accp(product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),iterate_rel(B,A)),aa(product_prod(fun(B,fun(A,product_prod(B,A))),B),product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),aa(code_natural,fun(product_prod(fun(B,fun(A,product_prod(B,A))),B),product_prod(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B))),product_Pair(code_natural,product_prod(fun(B,fun(A,product_prod(B,A))),B)),X2),aa(B,product_prod(fun(B,fun(A,product_prod(B,A))),B),aa(fun(B,fun(A,product_prod(B,A))),fun(B,product_prod(fun(B,fun(A,product_prod(B,A))),B)),product_Pair(fun(B,fun(A,product_prod(B,A))),B),Xa2),Xb3)))) ) ) ) ).
% iterate.pelims
tff(fact_2281_Gcd__eq__Max,axiom,
! [M: set(nat)] :
( finite_finite2(nat,M)
=> ( ( M != bot_bot(set(nat)) )
=> ( ~ pp(aa(set(nat),bool,member(nat,zero_zero(nat)),M))
=> ( gcd_Gcd(nat,M) = lattic643756798349783984er_Max(nat,aa(set(set(nat)),set(nat),complete_Inf_Inf(set(nat)),aa(set(nat),set(set(nat)),aa(fun(nat,set(nat)),fun(set(nat),set(set(nat))),image2(nat,set(nat)),aTP_Lamp_tk(nat,set(nat))),M))) ) ) ) ) ).
% Gcd_eq_Max
tff(fact_2282_select__weight__drop__zero,axiom,
! [A: $tType,Xs: list(product_prod(code_natural,A))] : select_weight(A,aa(list(product_prod(code_natural,A)),list(product_prod(code_natural,A)),filter2(product_prod(code_natural,A),aa(fun(code_natural,fun(A,bool)),fun(product_prod(code_natural,A),bool),product_case_prod(code_natural,A,bool),aTP_Lamp_tl(code_natural,fun(A,bool)))),Xs)) = select_weight(A,Xs) ).
% select_weight_drop_zero
tff(fact_2283_pick__drop__zero,axiom,
! [A: $tType,Xs: list(product_prod(code_natural,A))] : pick(A,aa(list(product_prod(code_natural,A)),list(product_prod(code_natural,A)),filter2(product_prod(code_natural,A),aa(fun(code_natural,fun(A,bool)),fun(product_prod(code_natural,A),bool),product_case_prod(code_natural,A,bool),aTP_Lamp_tl(code_natural,fun(A,bool)))),Xs)) = pick(A,Xs) ).
% pick_drop_zero
tff(fact_2284_case__sum__o__map__sum__id,axiom,
! [A: $tType,B: $tType,C: $tType,G: fun(B,A),F: fun(C,A),X2: sum_sum(C,B)] : aa(sum_sum(C,B),A,aa(fun(sum_sum(C,B),sum_sum(A,B)),fun(sum_sum(C,B),A),aa(fun(sum_sum(A,B),A),fun(fun(sum_sum(C,B),sum_sum(A,B)),fun(sum_sum(C,B),A)),comp(sum_sum(A,B),A,sum_sum(C,B)),sum_case_sum(A,A,B,id(A),G)),sum_map_sum(C,A,B,B,F,id(B))),X2) = aa(sum_sum(C,B),A,sum_case_sum(C,A,B,aa(fun(C,C),fun(C,A),aa(fun(C,A),fun(fun(C,C),fun(C,A)),comp(C,A,C),F),id(C)),G),X2) ).
% case_sum_o_map_sum_id
tff(fact_2285_Gcd__mono,axiom,
! [A: $tType,B: $tType] :
( semiring_Gcd(A)
=> ! [A6: set(B),F: fun(B,A),G: fun(B,A)] :
( ! [X: B] :
( pp(aa(set(B),bool,member(B,X),A6))
=> dvd_dvd(A,aa(B,A,F,X),aa(B,A,G,X)) )
=> dvd_dvd(A,gcd_Gcd(A,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6)),gcd_Gcd(A,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),G),A6))) ) ) ).
% Gcd_mono
tff(fact_2286_sum_Omap__comp,axiom,
! [D: $tType,F2: $tType,E: $tType,C: $tType,B: $tType,A: $tType,G1: fun(C,E),G22: fun(D,F2),F1: fun(A,C),F22: fun(B,D),V: sum_sum(A,B)] : aa(sum_sum(C,D),sum_sum(E,F2),sum_map_sum(C,E,D,F2,G1,G22),aa(sum_sum(A,B),sum_sum(C,D),sum_map_sum(A,C,B,D,F1,F22),V)) = aa(sum_sum(A,B),sum_sum(E,F2),sum_map_sum(A,E,B,F2,aa(fun(A,C),fun(A,E),aa(fun(C,E),fun(fun(A,C),fun(A,E)),comp(C,E,A),G1),F1),aa(fun(B,D),fun(B,F2),aa(fun(D,F2),fun(fun(B,D),fun(B,F2)),comp(D,F2,B),G22),F22)),V) ).
% sum.map_comp
tff(fact_2287_case__sum__o__map__sum,axiom,
! [A: $tType,D: $tType,C: $tType,E: $tType,B: $tType,F: fun(D,C),G: fun(E,C),H1: fun(A,D),H22: fun(B,E)] : aa(fun(sum_sum(A,B),sum_sum(D,E)),fun(sum_sum(A,B),C),aa(fun(sum_sum(D,E),C),fun(fun(sum_sum(A,B),sum_sum(D,E)),fun(sum_sum(A,B),C)),comp(sum_sum(D,E),C,sum_sum(A,B)),sum_case_sum(D,C,E,F,G)),sum_map_sum(A,D,B,E,H1,H22)) = sum_case_sum(A,C,B,aa(fun(A,D),fun(A,C),aa(fun(D,C),fun(fun(A,D),fun(A,C)),comp(D,C,A),F),H1),aa(fun(B,E),fun(B,C),aa(fun(E,C),fun(fun(B,E),fun(B,C)),comp(E,C,B),G),H22)) ).
% case_sum_o_map_sum
tff(fact_2288_case__sum__map__sum,axiom,
! [C: $tType,A: $tType,B: $tType,E: $tType,D: $tType,L: fun(B,A),R: fun(C,A),F: fun(D,B),G: fun(E,C),X2: sum_sum(D,E)] : aa(sum_sum(B,C),A,sum_case_sum(B,A,C,L,R),aa(sum_sum(D,E),sum_sum(B,C),sum_map_sum(D,B,E,C,F,G),X2)) = aa(sum_sum(D,E),A,sum_case_sum(D,A,E,aa(fun(D,B),fun(D,A),aa(fun(B,A),fun(fun(D,B),fun(D,A)),comp(B,A,D),L),F),aa(fun(E,C),fun(E,A),aa(fun(C,A),fun(fun(E,C),fun(E,A)),comp(C,A,E),R),G)),X2) ).
% case_sum_map_sum
tff(fact_2289_map__sum_Ocompositionality,axiom,
! [D: $tType,F2: $tType,E: $tType,C: $tType,B: $tType,A: $tType,F: fun(C,E),G: fun(D,F2),H: fun(A,C),I: fun(B,D),Sum: sum_sum(A,B)] : aa(sum_sum(C,D),sum_sum(E,F2),sum_map_sum(C,E,D,F2,F,G),aa(sum_sum(A,B),sum_sum(C,D),sum_map_sum(A,C,B,D,H,I),Sum)) = aa(sum_sum(A,B),sum_sum(E,F2),sum_map_sum(A,E,B,F2,aa(fun(A,C),fun(A,E),aa(fun(C,E),fun(fun(A,C),fun(A,E)),comp(C,E,A),F),H),aa(fun(B,D),fun(B,F2),aa(fun(D,F2),fun(fun(B,D),fun(B,F2)),comp(D,F2,B),G),I)),Sum) ).
% map_sum.compositionality
tff(fact_2290_map__sum_Ocomp,axiom,
! [A: $tType,C: $tType,E: $tType,F2: $tType,D: $tType,B: $tType,F: fun(C,E),G: fun(D,F2),H: fun(A,C),I: fun(B,D)] : aa(fun(sum_sum(A,B),sum_sum(C,D)),fun(sum_sum(A,B),sum_sum(E,F2)),aa(fun(sum_sum(C,D),sum_sum(E,F2)),fun(fun(sum_sum(A,B),sum_sum(C,D)),fun(sum_sum(A,B),sum_sum(E,F2))),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,aa(fun(A,C),fun(A,E),aa(fun(C,E),fun(fun(A,C),fun(A,E)),comp(C,E,A),F),H),aa(fun(B,D),fun(B,F2),aa(fun(D,F2),fun(fun(B,D),fun(B,F2)),comp(D,F2,B),G),I)) ).
% map_sum.comp
tff(fact_2291_sum_Osize__gen__o__map,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F: fun(C,nat),Fa: fun(D,nat),G: fun(A,C),Ga: fun(B,D)] : aa(fun(sum_sum(A,B),sum_sum(C,D)),fun(sum_sum(A,B),nat),aa(fun(sum_sum(C,D),nat),fun(fun(sum_sum(A,B),sum_sum(C,D)),fun(sum_sum(A,B),nat)),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,aa(fun(A,C),fun(A,nat),aa(fun(C,nat),fun(fun(A,C),fun(A,nat)),comp(C,nat,A),F),G),aa(fun(B,D),fun(B,nat),aa(fun(D,nat),fun(fun(B,D),fun(B,nat)),comp(D,nat,B),Fa),Ga)) ).
% sum.size_gen_o_map
tff(fact_2292_Gcd__int__eq,axiom,
! [N: set(nat)] : gcd_Gcd(int,aa(set(nat),set(int),aa(fun(nat,int),fun(set(nat),set(int)),image2(nat,int),semiring_1_of_nat(int)),N)) = aa(nat,int,semiring_1_of_nat(int),gcd_Gcd(nat,N)) ).
% Gcd_int_eq
tff(fact_2293_extract__def,axiom,
! [A: $tType,P2: fun(A,bool),Xs: list(A)] : extract(A,P2,Xs) = case_list(option(product_prod(list(A),product_prod(A,list(A)))),A,none(product_prod(list(A),product_prod(A,list(A)))),aa(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A)))))),aTP_Lamp_tm(fun(A,bool),fun(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))))),P2),Xs),dropWhile(A,aa(fun(A,bool),fun(A,bool),aa(fun(bool,bool),fun(fun(A,bool),fun(A,bool)),comp(bool,bool,A),fNot),P2),Xs)) ).
% extract_def
tff(fact_2294_sym__INTER,axiom,
! [B: $tType,A: $tType,S3: set(A),R: fun(A,set(product_prod(B,B)))] :
( ! [X: A] :
( pp(aa(set(A),bool,member(A,X),S3))
=> sym(B,aa(A,set(product_prod(B,B)),R,X)) )
=> sym(B,aa(set(set(product_prod(B,B))),set(product_prod(B,B)),complete_Inf_Inf(set(product_prod(B,B))),aa(set(A),set(set(product_prod(B,B))),aa(fun(A,set(product_prod(B,B))),fun(set(A),set(set(product_prod(B,B)))),image2(A,set(product_prod(B,B))),R),S3))) ) ).
% sym_INTER
tff(fact_2295_zip__takeWhile__fst,axiom,
! [A: $tType,B: $tType,P2: fun(A,bool),Xs: list(A),Ys: list(B)] : zip(A,B,takeWhile(A,P2,Xs),Ys) = takeWhile(product_prod(A,B),aa(fun(product_prod(A,B),A),fun(product_prod(A,B),bool),aa(fun(A,bool),fun(fun(product_prod(A,B),A),fun(product_prod(A,B),bool)),comp(A,bool,product_prod(A,B)),P2),product_fst(A,B)),zip(A,B,Xs,Ys)) ).
% zip_takeWhile_fst
tff(fact_2296_zip__takeWhile__snd,axiom,
! [A: $tType,B: $tType,Xs: list(A),P2: fun(B,bool),Ys: list(B)] : zip(A,B,Xs,takeWhile(B,P2,Ys)) = takeWhile(product_prod(A,B),aa(fun(product_prod(A,B),B),fun(product_prod(A,B),bool),aa(fun(B,bool),fun(fun(product_prod(A,B),B),fun(product_prod(A,B),bool)),comp(B,bool,product_prod(A,B)),P2),product_snd(A,B)),zip(A,B,Xs,Ys)) ).
% zip_takeWhile_snd
tff(fact_2297_sym__def,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( sym(A,R)
<=> ! [X5: A,Y4: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),Y4)),R))
=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),X5)),R)) ) ) ).
% sym_def
tff(fact_2298_symI,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( ! [A5: A,B4: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A5),B4)),R))
=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B4),A5)),R)) )
=> sym(A,R) ) ).
% symI
tff(fact_2299_symE,axiom,
! [A: $tType,R: set(product_prod(A,A)),B2: A,A3: A] :
( sym(A,R)
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A3)),R))
=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R)) ) ) ).
% symE
tff(fact_2300_symD,axiom,
! [A: $tType,R: set(product_prod(A,A)),B2: A,A3: A] :
( sym(A,R)
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A3)),R))
=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R)) ) ) ).
% symD
tff(fact_2301_takeWhile__map,axiom,
! [A: $tType,B: $tType,P2: fun(A,bool),F: fun(B,A),Xs: list(B)] : takeWhile(A,P2,aa(list(B),list(A),map(B,A,F),Xs)) = aa(list(B),list(A),map(B,A,F),takeWhile(B,aa(fun(B,A),fun(B,bool),aa(fun(A,bool),fun(fun(B,A),fun(B,bool)),comp(A,bool,B),P2),F),Xs)) ).
% takeWhile_map
tff(fact_2302_dropWhile__map,axiom,
! [A: $tType,B: $tType,P2: fun(A,bool),F: fun(B,A),Xs: list(B)] : dropWhile(A,P2,aa(list(B),list(A),map(B,A,F),Xs)) = aa(list(B),list(A),map(B,A,F),dropWhile(B,aa(fun(B,A),fun(B,bool),aa(fun(A,bool),fun(fun(B,A),fun(B,bool)),comp(A,bool,B),P2),F),Xs)) ).
% dropWhile_map
tff(fact_2303_sym__UNION,axiom,
! [B: $tType,A: $tType,S3: set(A),R: fun(A,set(product_prod(B,B)))] :
( ! [X: A] :
( pp(aa(set(A),bool,member(A,X),S3))
=> sym(B,aa(A,set(product_prod(B,B)),R,X)) )
=> sym(B,aa(set(set(product_prod(B,B))),set(product_prod(B,B)),complete_Sup_Sup(set(product_prod(B,B))),aa(set(A),set(set(product_prod(B,B))),aa(fun(A,set(product_prod(B,B))),fun(set(A),set(set(product_prod(B,B)))),image2(A,set(product_prod(B,B))),R),S3))) ) ).
% sym_UNION
tff(fact_2304_find__dropWhile,axiom,
! [A: $tType,P2: fun(A,bool),Xs: list(A)] : find(A,P2,Xs) = case_list(option(A),A,none(A),aTP_Lamp_tn(A,fun(list(A),option(A))),dropWhile(A,aa(fun(A,bool),fun(A,bool),aa(fun(bool,bool),fun(fun(A,bool),fun(A,bool)),comp(bool,bool,A),fNot),P2),Xs)) ).
% find_dropWhile
tff(fact_2305_Gcd__int__def,axiom,
! [K4: set(int)] : gcd_Gcd(int,K4) = aa(nat,int,semiring_1_of_nat(int),gcd_Gcd(nat,aa(set(int),set(nat),aa(fun(int,nat),fun(set(int),set(nat)),image2(int,nat),aa(fun(int,int),fun(int,nat),aa(fun(int,nat),fun(fun(int,int),fun(int,nat)),comp(int,nat,int),nat2),abs_abs(int))),K4))) ).
% Gcd_int_def
tff(fact_2306_Gcd__nat__abs__eq,axiom,
! [K4: set(int)] : gcd_Gcd(nat,aa(set(int),set(nat),aa(fun(int,nat),fun(set(int),set(nat)),image2(int,nat),aTP_Lamp_to(int,nat)),K4)) = aa(int,nat,nat2,gcd_Gcd(int,K4)) ).
% Gcd_nat_abs_eq
tff(fact_2307_sum_Osize__gen_I1_J,axiom,
! [B: $tType,A: $tType,Xa2: fun(A,nat),X2: fun(B,nat),X1: A] : aa(sum_sum(A,B),nat,basic_BNF_size_sum(A,B,Xa2,X2),aa(A,sum_sum(A,B),sum_Inl(A,B),X1)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,Xa2,X1)),aa(nat,nat,suc,zero_zero(nat))) ).
% sum.size_gen(1)
tff(fact_2308_sum_Osize__gen_I2_J,axiom,
! [A: $tType,B: $tType,Xa2: fun(A,nat),X2: fun(B,nat),X22: B] : aa(sum_sum(A,B),nat,basic_BNF_size_sum(A,B,Xa2,X2),aa(B,sum_sum(A,B),sum_Inr(B,A),X22)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(B,nat,X2,X22)),aa(nat,nat,suc,zero_zero(nat))) ).
% sum.size_gen(2)
tff(fact_2309_map__sum__o__inj_I1_J,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,F: fun(A,B),G: fun(D,C)] : aa(fun(A,sum_sum(A,D)),fun(A,sum_sum(B,C)),aa(fun(sum_sum(A,D),sum_sum(B,C)),fun(fun(A,sum_sum(A,D)),fun(A,sum_sum(B,C))),comp(sum_sum(A,D),sum_sum(B,C),A),sum_map_sum(A,B,D,C,F,G)),sum_Inl(A,D)) = aa(fun(A,B),fun(A,sum_sum(B,C)),aa(fun(B,sum_sum(B,C)),fun(fun(A,B),fun(A,sum_sum(B,C))),comp(B,sum_sum(B,C),A),sum_Inl(B,C)),F) ).
% map_sum_o_inj(1)
tff(fact_2310_range__abs__Nats,axiom,
aa(set(int),set(int),aa(fun(int,int),fun(set(int),set(int)),image2(int,int),abs_abs(int)),top_top(set(int))) = semiring_1_Nats(int) ).
% range_abs_Nats
tff(fact_2311_Gcd__abs__eq,axiom,
! [K4: set(int)] : gcd_Gcd(int,aa(set(int),set(int),aa(fun(int,int),fun(set(int),set(int)),image2(int,int),abs_abs(int)),K4)) = gcd_Gcd(int,K4) ).
% Gcd_abs_eq
tff(fact_2312_UNIV__sum,axiom,
! [A: $tType,B: $tType] : top_top(set(sum_sum(A,B))) = aa(set(sum_sum(A,B)),set(sum_sum(A,B)),aa(set(sum_sum(A,B)),fun(set(sum_sum(A,B)),set(sum_sum(A,B))),sup_sup(set(sum_sum(A,B))),aa(set(A),set(sum_sum(A,B)),aa(fun(A,sum_sum(A,B)),fun(set(A),set(sum_sum(A,B))),image2(A,sum_sum(A,B)),sum_Inl(A,B)),top_top(set(A)))),aa(set(B),set(sum_sum(A,B)),aa(fun(B,sum_sum(A,B)),fun(set(B),set(sum_sum(A,B))),image2(B,sum_sum(A,B)),sum_Inr(B,A)),top_top(set(B)))) ).
% UNIV_sum
tff(fact_2313_Basic__BNF__LFPs_OInr__def__alt,axiom,
! [B: $tType,A: $tType,X4: A] : aa(A,sum_sum(B,A),sum_Inr(A,B),X4) = aa(sum_sum(B,A),sum_sum(B,A),basic_BNF_xtor(sum_sum(B,A)),aa(sum_sum(B,A),sum_sum(B,A),bNF_id_bnf(sum_sum(B,A)),aa(A,sum_sum(B,A),sum_Inr(A,B),X4))) ).
% Basic_BNF_LFPs.Inr_def_alt
tff(fact_2314_Basic__BNF__LFPs_OInl__def__alt,axiom,
! [B: $tType,A: $tType,X4: A] : aa(A,sum_sum(A,B),sum_Inl(A,B),X4) = aa(sum_sum(A,B),sum_sum(A,B),basic_BNF_xtor(sum_sum(A,B)),aa(sum_sum(A,B),sum_sum(A,B),bNF_id_bnf(sum_sum(A,B)),aa(A,sum_sum(A,B),sum_Inl(A,B),X4))) ).
% Basic_BNF_LFPs.Inl_def_alt
tff(fact_2315_not__arg__cong__Inr,axiom,
! [B: $tType,A: $tType,X2: A,Y2: A] :
( ( X2 != Y2 )
=> ( aa(A,sum_sum(B,A),sum_Inr(A,B),X2) != aa(A,sum_sum(B,A),sum_Inr(A,B),Y2) ) ) ).
% not_arg_cong_Inr
tff(fact_2316_obj__sumE,axiom,
! [A: $tType,B: $tType,S: sum_sum(A,B)] :
( ! [X: A] : S != aa(A,sum_sum(A,B),sum_Inl(A,B),X)
=> ~ ! [X: B] : S != aa(B,sum_sum(A,B),sum_Inr(B,A),X) ) ).
% obj_sumE
tff(fact_2317_case__sum__expand__Inr,axiom,
! [B: $tType,C: $tType,A: $tType,F: fun(sum_sum(A,C),B),G: fun(A,B),X2: sum_sum(A,C)] :
( ( aa(fun(A,sum_sum(A,C)),fun(A,B),aa(fun(sum_sum(A,C),B),fun(fun(A,sum_sum(A,C)),fun(A,B)),comp(sum_sum(A,C),B,A),F),sum_Inl(A,C)) = G )
=> ( aa(sum_sum(A,C),B,F,X2) = aa(sum_sum(A,C),B,sum_case_sum(A,B,C,G,aa(fun(C,sum_sum(A,C)),fun(C,B),aa(fun(sum_sum(A,C),B),fun(fun(C,sum_sum(A,C)),fun(C,B)),comp(sum_sum(A,C),B,C),F),sum_Inr(C,A))),X2) ) ) ).
% case_sum_expand_Inr
tff(fact_2318_case__sum__expand__Inr_H,axiom,
! [B: $tType,C: $tType,A: $tType,F: fun(sum_sum(A,C),B),G: fun(A,B),H: fun(C,B)] :
( ( aa(fun(A,sum_sum(A,C)),fun(A,B),aa(fun(sum_sum(A,C),B),fun(fun(A,sum_sum(A,C)),fun(A,B)),comp(sum_sum(A,C),B,A),F),sum_Inl(A,C)) = G )
=> ( ( H = aa(fun(C,sum_sum(A,C)),fun(C,B),aa(fun(sum_sum(A,C),B),fun(fun(C,sum_sum(A,C)),fun(C,B)),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'
tff(fact_2319_case__sum__expand__Inr__pointfree,axiom,
! [B: $tType,C: $tType,A: $tType,F: fun(sum_sum(A,C),B),G: fun(A,B)] :
( ( aa(fun(A,sum_sum(A,C)),fun(A,B),aa(fun(sum_sum(A,C),B),fun(fun(A,sum_sum(A,C)),fun(A,B)),comp(sum_sum(A,C),B,A),F),sum_Inl(A,C)) = G )
=> ( sum_case_sum(A,B,C,G,aa(fun(C,sum_sum(A,C)),fun(C,B),aa(fun(sum_sum(A,C),B),fun(fun(C,sum_sum(A,C)),fun(C,B)),comp(sum_sum(A,C),B,C),F),sum_Inr(C,A))) = F ) ) ).
% case_sum_expand_Inr_pointfree
tff(fact_2320_finite__image__absD,axiom,
! [A: $tType] :
( linordered_ring(A)
=> ! [S3: set(A)] :
( finite_finite2(A,aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),abs_abs(A)),S3))
=> finite_finite2(A,S3) ) ) ).
% finite_image_absD
tff(fact_2321_case__sum__o__inj_I2_J,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(A,B),G: fun(C,B)] : aa(fun(C,sum_sum(A,C)),fun(C,B),aa(fun(sum_sum(A,C),B),fun(fun(C,sum_sum(A,C)),fun(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)
tff(fact_2322_case__sum__o__inj_I1_J,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(A,B),G: fun(C,B)] : aa(fun(A,sum_sum(A,C)),fun(A,B),aa(fun(sum_sum(A,C),B),fun(fun(A,sum_sum(A,C)),fun(A,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)
tff(fact_2323_int__in__range__abs,axiom,
! [N4: nat] : pp(aa(set(int),bool,member(int,aa(nat,int,semiring_1_of_nat(int),N4)),aa(set(int),set(int),aa(fun(int,int),fun(set(int),set(int)),image2(int,int),abs_abs(int)),top_top(set(int))))) ).
% int_in_range_abs
tff(fact_2324_map__sum__o__inj_I2_J,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,F: fun(A,B),G: fun(D,C)] : aa(fun(D,sum_sum(A,D)),fun(D,sum_sum(B,C)),aa(fun(sum_sum(A,D),sum_sum(B,C)),fun(fun(D,sum_sum(A,D)),fun(D,sum_sum(B,C))),comp(sum_sum(A,D),sum_sum(B,C),D),sum_map_sum(A,B,D,C,F,G)),sum_Inr(D,A)) = aa(fun(D,C),fun(D,sum_sum(B,C)),aa(fun(C,sum_sum(B,C)),fun(fun(D,C),fun(D,sum_sum(B,C))),comp(C,sum_sum(B,C),D),sum_Inr(C,B)),G) ).
% map_sum_o_inj(2)
tff(fact_2325_measure__function__int,axiom,
fun_is_measure(int,aa(fun(int,int),fun(int,nat),aa(fun(int,nat),fun(fun(int,int),fun(int,nat)),comp(int,nat,int),nat2),abs_abs(int))) ).
% measure_function_int
tff(fact_2326_sum_Osize_I3_J,axiom,
! [A: $tType,B: $tType,X1: A] : aa(sum_sum(A,B),nat,size_size(sum_sum(A,B)),aa(A,sum_sum(A,B),sum_Inl(A,B),X1)) = aa(nat,nat,suc,zero_zero(nat)) ).
% sum.size(3)
tff(fact_2327_sum_Osize_I4_J,axiom,
! [B: $tType,A: $tType,X22: B] : aa(sum_sum(A,B),nat,size_size(sum_sum(A,B)),aa(B,sum_sum(A,B),sum_Inr(B,A),X22)) = aa(nat,nat,suc,zero_zero(nat)) ).
% sum.size(4)
tff(fact_2328_prod_OPlus,axiom,
! [B: $tType,A: $tType,C: $tType] :
( comm_monoid_mult(A)
=> ! [A6: set(B),B6: set(C),G: fun(sum_sum(B,C),A)] :
( finite_finite2(B,A6)
=> ( finite_finite2(C,B6)
=> ( aa(set(sum_sum(B,C)),A,aa(fun(sum_sum(B,C),A),fun(set(sum_sum(B,C)),A),groups7121269368397514597t_prod(sum_sum(B,C),A),G),sum_Plus(B,C,A6,B6)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,sum_sum(B,C)),fun(B,A),aa(fun(sum_sum(B,C),A),fun(fun(B,sum_sum(B,C)),fun(B,A)),comp(sum_sum(B,C),A,B),G),sum_Inl(B,C))),A6)),aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),aa(fun(C,sum_sum(B,C)),fun(C,A),aa(fun(sum_sum(B,C),A),fun(fun(C,sum_sum(B,C)),fun(C,A)),comp(sum_sum(B,C),A,C),G),sum_Inr(C,B))),B6)) ) ) ) ) ).
% prod.Plus
tff(fact_2329_sum_OPlus,axiom,
! [B: $tType,A: $tType,C: $tType] :
( comm_monoid_add(A)
=> ! [A6: set(B),B6: set(C),G: fun(sum_sum(B,C),A)] :
( finite_finite2(B,A6)
=> ( finite_finite2(C,B6)
=> ( aa(set(sum_sum(B,C)),A,aa(fun(sum_sum(B,C),A),fun(set(sum_sum(B,C)),A),groups7311177749621191930dd_sum(sum_sum(B,C),A),G),sum_Plus(B,C,A6,B6)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(fun(B,sum_sum(B,C)),fun(B,A),aa(fun(sum_sum(B,C),A),fun(fun(B,sum_sum(B,C)),fun(B,A)),comp(sum_sum(B,C),A,B),G),sum_Inl(B,C))),A6)),aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),aa(fun(C,sum_sum(B,C)),fun(C,A),aa(fun(sum_sum(B,C),A),fun(fun(C,sum_sum(B,C)),fun(C,A)),comp(sum_sum(B,C),A,C),G),sum_Inr(C,B))),B6)) ) ) ) ) ).
% sum.Plus
tff(fact_2330_rat__abs__code,axiom,
! [P: rat] : quotient_of(aa(rat,rat,abs_abs(rat),P)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_tp(int,fun(int,product_prod(int,int)))),quotient_of(P)) ).
% rat_abs_code
tff(fact_2331_is__measure_Osimps,axiom,
! [A: $tType,A3: fun(A,nat)] :
( fun_is_measure(A,A3)
<=> ? [X_1: fun(A,nat)] : A3 = X_1 ) ).
% is_measure.simps
tff(fact_2332_is__measure__trivial,axiom,
! [A: $tType,F: fun(A,nat)] : fun_is_measure(A,F) ).
% is_measure_trivial
tff(fact_2333_measure__fst,axiom,
! [B: $tType,A: $tType,F: fun(A,nat)] :
( fun_is_measure(A,F)
=> fun_is_measure(product_prod(A,B),aTP_Lamp_tq(fun(A,nat),fun(product_prod(A,B),nat),F)) ) ).
% measure_fst
tff(fact_2334_measure__snd,axiom,
! [B: $tType,A: $tType,F: fun(A,nat)] :
( fun_is_measure(A,F)
=> fun_is_measure(product_prod(B,A),aTP_Lamp_tr(fun(A,nat),fun(product_prod(B,A),nat),F)) ) ).
% measure_snd
tff(fact_2335_Plus__def,axiom,
! [A: $tType,B: $tType,A6: set(A),B6: set(B)] : sum_Plus(A,B,A6,B6) = aa(set(sum_sum(A,B)),set(sum_sum(A,B)),aa(set(sum_sum(A,B)),fun(set(sum_sum(A,B)),set(sum_sum(A,B))),sup_sup(set(sum_sum(A,B))),aa(set(A),set(sum_sum(A,B)),aa(fun(A,sum_sum(A,B)),fun(set(A),set(sum_sum(A,B))),image2(A,sum_sum(A,B)),sum_Inl(A,B)),A6)),aa(set(B),set(sum_sum(A,B)),aa(fun(B,sum_sum(A,B)),fun(set(B),set(sum_sum(A,B))),image2(B,sum_sum(A,B)),sum_Inr(B,A)),B6)) ).
% Plus_def
tff(fact_2336_measure__size,axiom,
! [A: $tType] :
( size(A)
=> fun_is_measure(A,size_size(A)) ) ).
% measure_size
tff(fact_2337_sum_Osize__neq,axiom,
! [A: $tType,B: $tType,X2: sum_sum(A,B)] : aa(sum_sum(A,B),nat,size_size(sum_sum(A,B)),X2) != zero_zero(nat) ).
% sum.size_neq
tff(fact_2338_Field__csum,axiom,
! [A: $tType,B: $tType,R: set(product_prod(A,A)),S: set(product_prod(B,B))] : aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(sum_sum(A,B)),field2(sum_sum(A,B)),bNF_Cardinal_csum(A,B,R,S)) = aa(set(sum_sum(A,B)),set(sum_sum(A,B)),aa(set(sum_sum(A,B)),fun(set(sum_sum(A,B)),set(sum_sum(A,B))),sup_sup(set(sum_sum(A,B))),aa(set(A),set(sum_sum(A,B)),aa(fun(A,sum_sum(A,B)),fun(set(A),set(sum_sum(A,B))),image2(A,sum_sum(A,B)),sum_Inl(A,B)),aa(set(product_prod(A,A)),set(A),field2(A),R))),aa(set(B),set(sum_sum(A,B)),aa(fun(B,sum_sum(A,B)),fun(set(B),set(sum_sum(A,B))),image2(B,sum_sum(A,B)),sum_Inr(B,A)),aa(set(product_prod(B,B)),set(B),field2(B),S))) ).
% Field_csum
tff(fact_2339_surj__int__encode,axiom,
aa(set(int),set(nat),aa(fun(int,nat),fun(set(int),set(nat)),image2(int,nat),nat_int_encode),top_top(set(int))) = top_top(set(nat)) ).
% surj_int_encode
tff(fact_2340_surj__int__decode,axiom,
aa(set(nat),set(int),aa(fun(nat,int),fun(set(nat),set(int)),image2(nat,int),nat_int_decode),top_top(set(nat))) = top_top(set(int)) ).
% surj_int_decode
tff(fact_2341_Ints__def,axiom,
! [A: $tType] :
( ring_1(A)
=> ( ring_1_Ints(A) = aa(set(int),set(A),aa(fun(int,A),fun(set(int),set(A)),image2(int,A),ring_1_of_int(A)),top_top(set(int))) ) ) ).
% Ints_def
tff(fact_2342_Lcm__abs__eq,axiom,
! [K4: set(int)] : gcd_Lcm(int,aa(set(int),set(int),aa(fun(int,int),fun(set(int),set(int)),image2(int,int),abs_abs(int)),K4)) = gcd_Lcm(int,K4) ).
% Lcm_abs_eq
tff(fact_2343_Lcm__int__def,axiom,
! [K4: set(int)] : gcd_Lcm(int,K4) = aa(nat,int,semiring_1_of_nat(int),gcd_Lcm(nat,aa(set(int),set(nat),aa(fun(int,nat),fun(set(int),set(nat)),image2(int,nat),aa(fun(int,int),fun(int,nat),aa(fun(int,nat),fun(fun(int,int),fun(int,nat)),comp(int,nat,int),nat2),abs_abs(int))),K4))) ).
% Lcm_int_def
tff(fact_2344_Lcm__int__eq,axiom,
! [N: set(nat)] : gcd_Lcm(int,aa(set(nat),set(int),aa(fun(nat,int),fun(set(nat),set(int)),image2(nat,int),semiring_1_of_nat(int)),N)) = aa(nat,int,semiring_1_of_nat(int),gcd_Lcm(nat,N)) ).
% Lcm_int_eq
tff(fact_2345_Lcm__nat__abs__eq,axiom,
! [K4: set(int)] : gcd_Lcm(nat,aa(set(int),set(nat),aa(fun(int,nat),fun(set(int),set(nat)),image2(int,nat),aTP_Lamp_to(int,nat)),K4)) = aa(int,nat,nat2,gcd_Lcm(int,K4)) ).
% Lcm_nat_abs_eq
tff(fact_2346_Inr__Field__csum,axiom,
! [B: $tType,A: $tType,A3: A,S: set(product_prod(A,A)),R: set(product_prod(B,B))] :
( pp(aa(set(A),bool,member(A,A3),aa(set(product_prod(A,A)),set(A),field2(A),S)))
=> pp(aa(set(sum_sum(B,A)),bool,member(sum_sum(B,A),aa(A,sum_sum(B,A),sum_Inr(A,B),A3)),aa(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(sum_sum(B,A)),field2(sum_sum(B,A)),bNF_Cardinal_csum(B,A,R,S)))) ) ).
% Inr_Field_csum
tff(fact_2347_Inl__Field__csum,axiom,
! [A: $tType,B: $tType,A3: A,R: set(product_prod(A,A)),S: set(product_prod(B,B))] :
( pp(aa(set(A),bool,member(A,A3),aa(set(product_prod(A,A)),set(A),field2(A),R)))
=> pp(aa(set(sum_sum(A,B)),bool,member(sum_sum(A,B),aa(A,sum_sum(A,B),sum_Inl(A,B),A3)),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(sum_sum(A,B)),field2(sum_sum(A,B)),bNF_Cardinal_csum(A,B,R,S)))) ) ).
% Inl_Field_csum
tff(fact_2348_Lcm__mono,axiom,
! [A: $tType,B: $tType] :
( semiring_Gcd(A)
=> ! [A6: set(B),F: fun(B,A),G: fun(B,A)] :
( ! [X: B] :
( pp(aa(set(B),bool,member(B,X),A6))
=> dvd_dvd(A,aa(B,A,F,X),aa(B,A,G,X)) )
=> dvd_dvd(A,gcd_Lcm(A,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),A6)),gcd_Lcm(A,aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),G),A6))) ) ) ).
% Lcm_mono
tff(fact_2349_prod_Osize__neq,axiom,
! [A: $tType,B: $tType,X2: product_prod(A,B)] : aa(product_prod(A,B),nat,size_size(product_prod(A,B)),X2) != zero_zero(nat) ).
% prod.size_neq
tff(fact_2350_Lcm__mult,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [A6: set(A),C3: A] :
( ( A6 != bot_bot(set(A)) )
=> ( gcd_Lcm(A,aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aa(A,fun(A,A),times_times(A),C3)),A6)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),gcd_Lcm(A,A6))) ) ) ) ).
% Lcm_mult
tff(fact_2351_option_Opred__map,axiom,
! [B: $tType,A: $tType,Q2: fun(B,bool),F: fun(A,B),X2: option(A)] :
( pp(aa(option(B),bool,pred_option(B,Q2),aa(option(A),option(B),map_option(A,B,F),X2)))
<=> pp(aa(option(A),bool,pred_option(A,aa(fun(A,B),fun(A,bool),aa(fun(B,bool),fun(fun(A,B),fun(A,bool)),comp(B,bool,A),Q2),F)),X2)) ) ).
% option.pred_map
tff(fact_2352_rotate__add,axiom,
! [A: $tType,M4: nat,N4: nat] : rotate(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M4),N4)) = aa(fun(list(A),list(A)),fun(list(A),list(A)),aa(fun(list(A),list(A)),fun(fun(list(A),list(A)),fun(list(A),list(A))),comp(list(A),list(A),list(A)),rotate(A,M4)),rotate(A,N4)) ).
% rotate_add
tff(fact_2353_old_Ounit_Ocase,axiom,
! [A: $tType,F: A] : product_case_unit(A,F,product_Unity) = F ).
% old.unit.case
tff(fact_2354_accp__eq__acc,axiom,
! [A: $tType,R: fun(A,fun(A,bool)),X4: A] :
( pp(aa(A,bool,accp(A,R),X4))
<=> pp(aa(set(A),bool,member(A,X4),acc(A,aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),R))))) ) ).
% accp_eq_acc
tff(fact_2355_rat__inverse__code,axiom,
! [P: rat] : quotient_of(aa(rat,rat,inverse_inverse(rat),P)) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_ts(int,fun(int,product_prod(int,int)))),quotient_of(P)) ).
% rat_inverse_code
tff(fact_2356_Gcd__image__normalize,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [A6: set(A)] : gcd_Gcd(A,aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),normal6383669964737779283malize(A)),A6)) = gcd_Gcd(A,A6) ) ).
% Gcd_image_normalize
tff(fact_2357_unit_Ocase__distrib,axiom,
! [A: $tType,B: $tType,H: fun(A,B),F: A,Unit: product_unit] : aa(A,B,H,product_case_unit(A,F,Unit)) = product_case_unit(B,aa(A,B,H,F),Unit) ).
% unit.case_distrib
tff(fact_2358_acc__induct__rule,axiom,
! [A: $tType,A3: A,R: set(product_prod(A,A)),P2: fun(A,bool)] :
( pp(aa(set(A),bool,member(A,A3),acc(A,R)))
=> ( ! [X: A] :
( pp(aa(set(A),bool,member(A,X),acc(A,R)))
=> ( ! [Y5: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),X)),R))
=> pp(aa(A,bool,P2,Y5)) )
=> pp(aa(A,bool,P2,X)) ) )
=> pp(aa(A,bool,P2,A3)) ) ) ).
% acc_induct_rule
tff(fact_2359_not__acc__down,axiom,
! [A: $tType,X2: A,R3: set(product_prod(A,A))] :
( ~ pp(aa(set(A),bool,member(A,X2),acc(A,R3)))
=> ~ ! [Z3: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z3),X2)),R3))
=> pp(aa(set(A),bool,member(A,Z3),acc(A,R3))) ) ) ).
% not_acc_down
tff(fact_2360_acc__downward,axiom,
! [A: $tType,B2: A,R: set(product_prod(A,A)),A3: A] :
( pp(aa(set(A),bool,member(A,B2),acc(A,R)))
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A3),B2)),R))
=> pp(aa(set(A),bool,member(A,A3),acc(A,R))) ) ) ).
% acc_downward
tff(fact_2361_acc__induct,axiom,
! [A: $tType,A3: A,R: set(product_prod(A,A)),P2: fun(A,bool)] :
( pp(aa(set(A),bool,member(A,A3),acc(A,R)))
=> ( ! [X: A] :
( pp(aa(set(A),bool,member(A,X),acc(A,R)))
=> ( ! [Y5: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),X)),R))
=> pp(aa(A,bool,P2,Y5)) )
=> pp(aa(A,bool,P2,X)) ) )
=> pp(aa(A,bool,P2,A3)) ) ) ).
% acc_induct
tff(fact_2362_acc_Ointros,axiom,
! [A: $tType,X2: A,R: set(product_prod(A,A))] :
( ! [Y: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),X2)),R))
=> pp(aa(set(A),bool,member(A,Y),acc(A,R))) )
=> pp(aa(set(A),bool,member(A,X2),acc(A,R))) ) ).
% acc.intros
tff(fact_2363_acc_Osimps,axiom,
! [A: $tType,A3: A,R: set(product_prod(A,A))] :
( pp(aa(set(A),bool,member(A,A3),acc(A,R)))
<=> ? [X5: A] :
( ( A3 = X5 )
& ! [Xa3: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa3),X5)),R))
=> pp(aa(set(A),bool,member(A,Xa3),acc(A,R))) ) ) ) ).
% acc.simps
tff(fact_2364_acc_Ocases,axiom,
! [A: $tType,A3: A,R: set(product_prod(A,A))] :
( pp(aa(set(A),bool,member(A,A3),acc(A,R)))
=> ! [Y5: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y5),A3)),R))
=> pp(aa(set(A),bool,member(A,Y5),acc(A,R))) ) ) ).
% acc.cases
tff(fact_2365_prod__inversef,axiom,
! [A: $tType,B: $tType] :
( field(A)
=> ! [F: fun(B,A),A6: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,A),fun(B,A),aa(fun(A,A),fun(fun(B,A),fun(B,A)),comp(A,A,B),inverse_inverse(A)),F)),A6) = aa(A,A,inverse_inverse(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F),A6)) ) ).
% prod_inversef
tff(fact_2366_acc__subset__induct,axiom,
! [A: $tType,D3: set(A),R3: set(product_prod(A,A)),X2: A,P2: fun(A,bool)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),D3),acc(A,R3)))
=> ( ! [X: A,Z3: A] :
( pp(aa(set(A),bool,member(A,X),D3))
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z3),X)),R3))
=> pp(aa(set(A),bool,member(A,Z3),D3)) ) )
=> ( pp(aa(set(A),bool,member(A,X2),D3))
=> ( ! [X: A] :
( pp(aa(set(A),bool,member(A,X),D3))
=> ( ! [Z6: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z6),X)),R3))
=> pp(aa(A,bool,P2,Z6)) )
=> pp(aa(A,bool,P2,X)) ) )
=> pp(aa(A,bool,P2,X2)) ) ) ) ) ).
% acc_subset_induct
tff(fact_2367_acc__downwards__aux,axiom,
! [A: $tType,B2: A,A3: A,R: set(product_prod(A,A))] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A3)),transitive_rtrancl(A,R)))
=> ( pp(aa(set(A),bool,member(A,A3),acc(A,R)))
=> pp(aa(set(A),bool,member(A,B2),acc(A,R))) ) ) ).
% acc_downwards_aux
tff(fact_2368_acc__downwards,axiom,
! [A: $tType,A3: A,R: set(product_prod(A,A)),B2: A] :
( pp(aa(set(A),bool,member(A,A3),acc(A,R)))
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B2),A3)),transitive_rtrancl(A,R)))
=> pp(aa(set(A),bool,member(A,B2),acc(A,R))) ) ) ).
% acc_downwards
tff(fact_2369_Gcd__mult,axiom,
! [A: $tType] :
( semiring_Gcd(A)
=> ! [C3: A,A6: set(A)] : gcd_Gcd(A,aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aa(A,fun(A,A),times_times(A),C3)),A6)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),gcd_Gcd(A,A6))) ) ).
% Gcd_mult
tff(fact_2370_accp__acc__eq,axiom,
! [A: $tType,R: set(product_prod(A,A)),X4: A] :
( pp(aa(A,bool,accp(A,aTP_Lamp_hl(set(product_prod(A,A)),fun(A,fun(A,bool)),R)),X4))
<=> pp(aa(set(A),bool,member(A,X4),acc(A,R))) ) ).
% accp_acc_eq
tff(fact_2371_acc__def,axiom,
! [A: $tType,X4: set(product_prod(A,A))] : acc(A,X4) = aa(fun(A,bool),set(A),collect(A),accp(A,aTP_Lamp_hl(set(product_prod(A,A)),fun(A,fun(A,bool)),X4))) ).
% acc_def
tff(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,aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aa(A,fun(A,A),times_times(A),B2)),A6)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),semiring_gcd_Gcd_fin(A,A6))) ) ) ) ).
% Gcd_fin_mult
tff(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,aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),aa(A,fun(A,A),times_times(A),B2)),A6)) = aa(A,A,normal6383669964737779283malize(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),semiring_gcd_Lcm_fin(A,A6))) ) ) ) ).
% Lcm_fin_mult
tff(fact_2374_Heap_Opred__inject,axiom,
! [A: $tType,P2: fun(A,bool),A3: fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))))] :
( pp(aa(heap_Heap(A),bool,aa(fun(A,bool),fun(heap_Heap(A),bool),heap_pred_Heap(A),P2),aa(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_Heap(A),heap_Heap2(A),A3)))
<=> basic_pred_fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))),aTP_Lamp_tt(heap_ext(product_unit),bool),pred_option(product_prod(A,heap_ext(product_unit)),basic_pred_prod(A,heap_ext(product_unit),P2,top_top(fun(heap_ext(product_unit),bool)))),A3) ) ).
% Heap.pred_inject
tff(fact_2375_sum_Opred__map,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,Q1: fun(C,bool),Q22: fun(D,bool),F1: fun(A,C),F22: fun(B,D),X2: sum_sum(A,B)] :
( basic_pred_sum(C,D,Q1,Q22,aa(sum_sum(A,B),sum_sum(C,D),sum_map_sum(A,C,B,D,F1,F22),X2))
<=> basic_pred_sum(A,B,aa(fun(A,C),fun(A,bool),aa(fun(C,bool),fun(fun(A,C),fun(A,bool)),comp(C,bool,A),Q1),F1),aa(fun(B,D),fun(B,bool),aa(fun(D,bool),fun(fun(B,D),fun(B,bool)),comp(D,bool,B),Q22),F22),X2) ) ).
% sum.pred_map
tff(fact_2376_INF__parametric,axiom,
! [A: $tType,C: $tType,B: $tType] :
( complete_Inf(C)
=> ! [A6: fun(A,fun(B,bool))] : pp(aa(fun(set(B),fun(fun(B,C),C)),bool,aa(fun(set(A),fun(fun(A,C),C)),fun(fun(set(B),fun(fun(B,C),C)),bool),bNF_rel_fun(set(A),set(B),fun(fun(A,C),C),fun(fun(B,C),C),bNF_rel_set(A,B,A6),bNF_rel_fun(fun(A,C),fun(B,C),C,C,bNF_rel_fun(A,B,C,C,A6,fequal(C)),fequal(C))),aTP_Lamp_tu(set(A),fun(fun(A,C),C))),aTP_Lamp_tv(set(B),fun(fun(B,C),C)))) ) ).
% INF_parametric
tff(fact_2377_SUP__parametric,axiom,
! [A: $tType,C: $tType,B: $tType] :
( complete_Sup(C)
=> ! [R3: fun(A,fun(B,bool))] : pp(aa(fun(set(B),fun(fun(B,C),C)),bool,aa(fun(set(A),fun(fun(A,C),C)),fun(fun(set(B),fun(fun(B,C),C)),bool),bNF_rel_fun(set(A),set(B),fun(fun(A,C),C),fun(fun(B,C),C),bNF_rel_set(A,B,R3),bNF_rel_fun(fun(A,C),fun(B,C),C,C,bNF_rel_fun(A,B,C,C,R3,fequal(C)),fequal(C))),aTP_Lamp_tw(set(A),fun(fun(A,C),C))),aTP_Lamp_tx(set(B),fun(fun(B,C),C)))) ) ).
% SUP_parametric
tff(fact_2378_pred__prod__inject,axiom,
! [A: $tType,B: $tType,P12: fun(A,bool),P23: fun(B,bool),A3: A,B2: B] :
( pp(aa(product_prod(A,B),bool,basic_pred_prod(A,B,P12,P23),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)))
<=> ( pp(aa(A,bool,P12,A3))
& pp(aa(B,bool,P23,B2)) ) ) ).
% pred_prod_inject
tff(fact_2379_rel__setI,axiom,
! [A: $tType,B: $tType,A6: set(A),B6: set(B),R3: fun(A,fun(B,bool))] :
( ! [X: A] :
( pp(aa(set(A),bool,member(A,X),A6))
=> ? [Xa4: B] :
( pp(aa(set(B),bool,member(B,Xa4),B6))
& pp(aa(B,bool,aa(A,fun(B,bool),R3,X),Xa4)) ) )
=> ( ! [Y: B] :
( pp(aa(set(B),bool,member(B,Y),B6))
=> ? [X4: A] :
( pp(aa(set(A),bool,member(A,X4),A6))
& pp(aa(B,bool,aa(A,fun(B,bool),R3,X4),Y)) ) )
=> pp(aa(set(B),bool,aa(set(A),fun(set(B),bool),bNF_rel_set(A,B,R3),A6),B6)) ) ) ).
% rel_setI
tff(fact_2380_rel__set__OO,axiom,
! [A: $tType,B: $tType,C: $tType,R3: fun(A,fun(C,bool)),S3: fun(C,fun(B,bool))] : aa(fun(set(C),fun(set(B),bool)),fun(set(A),fun(set(B),bool)),aa(fun(set(A),fun(set(C),bool)),fun(fun(set(C),fun(set(B),bool)),fun(set(A),fun(set(B),bool))),relcompp(set(A),set(C),set(B)),bNF_rel_set(A,C,R3)),bNF_rel_set(C,B,S3)) = bNF_rel_set(A,B,aa(fun(C,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(C,bool)),fun(fun(C,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,C,B),R3),S3)) ).
% rel_set_OO
tff(fact_2381_image__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,A6: fun(A,fun(C,bool)),B6: fun(B,fun(D,bool))] : pp(aa(fun(fun(C,D),fun(set(C),set(D))),bool,aa(fun(fun(A,B),fun(set(A),set(B))),fun(fun(fun(C,D),fun(set(C),set(D))),bool),bNF_rel_fun(fun(A,B),fun(C,D),fun(set(A),set(B)),fun(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
tff(fact_2382_rel__set__UNION,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,Q2: fun(A,fun(B,bool)),A6: set(A),B6: set(B),R3: fun(C,fun(D,bool)),F: fun(A,set(C)),G: fun(B,set(D))] :
( pp(aa(set(B),bool,aa(set(A),fun(set(B),bool),bNF_rel_set(A,B,Q2),A6),B6))
=> ( pp(aa(fun(B,set(D)),bool,aa(fun(A,set(C)),fun(fun(B,set(D)),bool),bNF_rel_fun(A,B,set(C),set(D),Q2,bNF_rel_set(C,D,R3)),F),G))
=> pp(aa(set(D),bool,aa(set(C),fun(set(D),bool),bNF_rel_set(C,D,R3),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(A),set(set(C)),aa(fun(A,set(C)),fun(set(A),set(set(C))),image2(A,set(C)),F),A6))),aa(set(set(D)),set(D),complete_Sup_Sup(set(D)),aa(set(B),set(set(D)),aa(fun(B,set(D)),fun(set(B),set(set(D))),image2(B,set(D)),G),B6)))) ) ) ).
% rel_set_UNION
tff(fact_2383_Quotient__set,axiom,
! [B: $tType,A: $tType,R3: fun(A,fun(A,bool)),Abs: fun(A,B),Rep: fun(B,A),T3: fun(A,fun(B,bool))] :
( quotient(A,B,R3,Abs,Rep,T3)
=> quotient(set(A),set(B),bNF_rel_set(A,A,R3),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),Abs),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),Rep),bNF_rel_set(A,B,T3)) ) ).
% Quotient_set
tff(fact_2384_pred__prod_Ocases,axiom,
! [A: $tType,B: $tType,P12: fun(A,bool),P23: fun(B,bool),A3: product_prod(A,B)] :
( pp(aa(product_prod(A,B),bool,basic_pred_prod(A,B,P12,P23),A3))
=> ~ ! [A5: A,B4: B] :
( ( A3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4) )
=> ( pp(aa(A,bool,P12,A5))
=> ~ pp(aa(B,bool,P23,B4)) ) ) ) ).
% pred_prod.cases
tff(fact_2385_pred__prod_Osimps,axiom,
! [A: $tType,B: $tType,P12: fun(A,bool),P23: fun(B,bool),A3: product_prod(A,B)] :
( pp(aa(product_prod(A,B),bool,basic_pred_prod(A,B,P12,P23),A3))
<=> ? [A7: A,B5: B] :
( ( A3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A7),B5) )
& pp(aa(A,bool,P12,A7))
& pp(aa(B,bool,P23,B5)) ) ) ).
% pred_prod.simps
tff(fact_2386_pred__prod_Ointros,axiom,
! [A: $tType,B: $tType,P12: fun(A,bool),A3: A,P23: fun(B,bool),B2: B] :
( pp(aa(A,bool,P12,A3))
=> ( pp(aa(B,bool,P23,B2))
=> pp(aa(product_prod(A,B),bool,basic_pred_prod(A,B,P12,P23),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2))) ) ) ).
% pred_prod.intros
tff(fact_2387_pred__prod__split,axiom,
! [A: $tType,B: $tType,P2: fun(bool,bool),Q2: fun(A,bool),R3: fun(B,bool),Xy: product_prod(A,B)] :
( pp(aa(bool,bool,P2,aa(product_prod(A,B),bool,basic_pred_prod(A,B,Q2,R3),Xy)))
<=> ! [X5: A,Y4: B] :
( ( Xy = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X5),Y4) )
=> pp(aa(bool,bool,P2,fconj(aa(A,bool,Q2,X5),aa(B,bool,R3,Y4)))) ) ) ).
% pred_prod_split
tff(fact_2388_UNION__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A6: fun(A,fun(B,bool)),B6: fun(C,fun(D,bool))] : pp(aa(fun(set(B),fun(fun(B,set(D)),set(D))),bool,aa(fun(set(A),fun(fun(A,set(C)),set(C))),fun(fun(set(B),fun(fun(B,set(D)),set(D))),bool),bNF_rel_fun(set(A),set(B),fun(fun(A,set(C)),set(C)),fun(fun(B,set(D)),set(D)),bNF_rel_set(A,B,A6),bNF_rel_fun(fun(A,set(C)),fun(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))),aTP_Lamp_ty(set(A),fun(fun(A,set(C)),set(C)))),aTP_Lamp_tz(set(B),fun(fun(B,set(D)),set(D))))) ).
% UNION_transfer
tff(fact_2389_Heap_Oset__transfer,axiom,
! [A: $tType,B: $tType,R3: fun(A,fun(B,bool))] : pp(aa(fun(heap_Heap(B),set(B)),bool,aa(fun(heap_Heap(A),set(A)),fun(fun(heap_Heap(B),set(B)),bool),bNF_rel_fun(heap_Heap(A),heap_Heap(B),set(A),set(B),aa(fun(A,fun(B,bool)),fun(heap_Heap(A),fun(heap_Heap(B),bool)),heap_rel_Heap(A,B),R3),bNF_rel_set(A,B,R3)),heap_set_Heap(A)),heap_set_Heap(B))) ).
% Heap.set_transfer
tff(fact_2390_rel__set__def,axiom,
! [A: $tType,B: $tType,R3: fun(A,fun(B,bool)),X4: set(A),Xa4: set(B)] :
( pp(aa(set(B),bool,aa(set(A),fun(set(B),bool),bNF_rel_set(A,B,R3),X4),Xa4))
<=> ( ! [Xb2: A] :
( pp(aa(set(A),bool,member(A,Xb2),X4))
=> ? [Xc: B] :
( pp(aa(set(B),bool,member(B,Xc),Xa4))
& pp(aa(B,bool,aa(A,fun(B,bool),R3,Xb2),Xc)) ) )
& ! [Xb2: B] :
( pp(aa(set(B),bool,member(B,Xb2),Xa4))
=> ? [Xc: A] :
( pp(aa(set(A),bool,member(A,Xc),X4))
& pp(aa(B,bool,aa(A,fun(B,bool),R3,Xc),Xb2)) ) ) ) ) ).
% rel_set_def
tff(fact_2391_pred__prod__beta,axiom,
! [B: $tType,A: $tType,P2: fun(A,bool),Q2: fun(B,bool),Xy: product_prod(A,B)] :
( pp(aa(product_prod(A,B),bool,basic_pred_prod(A,B,P2,Q2),Xy))
<=> ( pp(aa(A,bool,P2,aa(product_prod(A,B),A,product_fst(A,B),Xy)))
& pp(aa(B,bool,Q2,aa(product_prod(A,B),B,product_snd(A,B),Xy))) ) ) ).
% pred_prod_beta
tff(fact_2392_fun_Oset__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,R3: fun(A,fun(B,bool))] : pp(aa(fun(fun(D,B),set(B)),bool,aa(fun(fun(D,A),set(A)),fun(fun(fun(D,B),set(B)),bool),bNF_rel_fun(fun(D,A),fun(D,B),set(A),set(B),bNF_rel_fun(D,D,A,B,fequal(D),R3),bNF_rel_set(A,B,R3)),aTP_Lamp_ua(fun(D,A),set(A))),aTP_Lamp_ub(fun(D,B),set(B)))) ).
% fun.set_transfer
tff(fact_2393_prod_Opred__map,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,Q1: fun(C,bool),Q22: fun(D,bool),F1: fun(A,C),F22: fun(B,D),X2: product_prod(A,B)] :
( pp(aa(product_prod(C,D),bool,basic_pred_prod(C,D,Q1,Q22),aa(product_prod(A,B),product_prod(C,D),product_map_prod(A,C,B,D,F1,F22),X2)))
<=> pp(aa(product_prod(A,B),bool,basic_pred_prod(A,B,aa(fun(A,C),fun(A,bool),aa(fun(C,bool),fun(fun(A,C),fun(A,bool)),comp(C,bool,A),Q1),F1),aa(fun(B,D),fun(B,bool),aa(fun(D,bool),fun(fun(B,D),fun(B,bool)),comp(D,bool,B),Q22),F22)),X2)) ) ).
% prod.pred_map
tff(fact_2394_fold__rev,axiom,
! [B: $tType,A: $tType,Xs: list(A),F: fun(A,fun(B,B))] :
( ! [X: A,Y: A] :
( pp(aa(set(A),bool,member(A,X),aa(list(A),set(A),set3(A),Xs)))
=> ( pp(aa(set(A),bool,member(A,Y),aa(list(A),set(A),set3(A),Xs)))
=> ( aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,Y)),aa(A,fun(B,B),F,X)) = aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X)),aa(A,fun(B,B),F,Y)) ) ) )
=> ( fold(A,B,F,rev(A,Xs)) = fold(A,B,F,Xs) ) ) ).
% fold_rev
tff(fact_2395_dir__image__def,axiom,
! [A2: $tType,A: $tType,R: set(product_prod(A,A)),F: fun(A,A2)] : bNF_We2720479622203943262_image(A,A2,R,F) = aa(fun(product_prod(A2,A2),bool),set(product_prod(A2,A2)),collect(product_prod(A2,A2)),aa(fun(A,A2),fun(product_prod(A2,A2),bool),aTP_Lamp_uc(set(product_prod(A,A)),fun(fun(A,A2),fun(product_prod(A2,A2),bool)),R),F)) ).
% dir_image_def
tff(fact_2396_card__order__csum__cone__cexp__def,axiom,
! [A: $tType,B: $tType,R: set(product_prod(A,A)),A12: 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,A12),bNF_Cardinal_cone),R) = bNF_Ca6860139660246222851ard_of(fun(A,sum_sum(B,product_unit)),bNF_Wellorder_Func(A,sum_sum(B,product_unit),top_top(set(A)),aa(set(sum_sum(B,product_unit)),set(sum_sum(B,product_unit)),aa(set(sum_sum(B,product_unit)),fun(set(sum_sum(B,product_unit)),set(sum_sum(B,product_unit))),sup_sup(set(sum_sum(B,product_unit))),aa(set(B),set(sum_sum(B,product_unit)),aa(fun(B,sum_sum(B,product_unit)),fun(set(B),set(sum_sum(B,product_unit))),image2(B,sum_sum(B,product_unit)),sum_Inl(B,product_unit)),A12)),aa(set(sum_sum(B,product_unit)),set(sum_sum(B,product_unit)),insert(sum_sum(B,product_unit),aa(product_unit,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
tff(fact_2397_bi__unique__rel__set__lemma,axiom,
! [B: $tType,A: $tType,R3: fun(A,fun(B,bool)),X6: set(A),Y3: set(B)] :
( bi_unique(A,B,R3)
=> ( pp(aa(set(B),bool,aa(set(A),fun(set(B),bool),bNF_rel_set(A,B,R3),X6),Y3))
=> ~ ! [F3: fun(A,B)] :
( ( Y3 = aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F3),X6) )
=> ( inj_on(A,B,F3,X6)
=> ~ ! [X4: A] :
( pp(aa(set(A),bool,member(A,X4),X6))
=> pp(aa(B,bool,aa(A,fun(B,bool),R3,X4),aa(A,B,F3,X4))) ) ) ) ) ) ).
% bi_unique_rel_set_lemma
tff(fact_2398_Heap__ordI,axiom,
! [A: $tType,X2: heap_Heap(A),Y2: heap_Heap(A)] :
( ! [H7: heap_ext(product_unit)] :
( ( aa(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))),aa(heap_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_execute(A),X2),H7) = none(product_prod(A,heap_ext(product_unit))) )
| ( aa(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))),aa(heap_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_execute(A),X2),H7) = aa(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))),aa(heap_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_execute(A),Y2),H7) ) )
=> pp(aa(heap_Heap(A),bool,aa(heap_Heap(A),fun(heap_Heap(A),bool),heap_Heap_ord(A),X2),Y2)) ) ).
% Heap_ordI
tff(fact_2399_Heap__ordE,axiom,
! [A: $tType,X2: heap_Heap(A),Y2: heap_Heap(A),H: heap_ext(product_unit)] :
( pp(aa(heap_Heap(A),bool,aa(heap_Heap(A),fun(heap_Heap(A),bool),heap_Heap_ord(A),X2),Y2))
=> ( ( aa(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))),aa(heap_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_execute(A),X2),H) != none(product_prod(A,heap_ext(product_unit))) )
=> ( aa(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))),aa(heap_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_execute(A),X2),H) = aa(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))),aa(heap_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_execute(A),Y2),H) ) ) ) ).
% Heap_ordE
tff(fact_2400_heap_Oleq__refl,axiom,
! [A: $tType,X2: heap_Heap(A)] : pp(aa(heap_Heap(A),bool,aa(heap_Heap(A),fun(heap_Heap(A),bool),heap_Heap_ord(A),X2),X2)) ).
% heap.leq_refl
tff(fact_2401_heap_Oleq__trans,axiom,
! [A: $tType,X2: heap_Heap(A),Y2: heap_Heap(A),Z2: heap_Heap(A)] :
( pp(aa(heap_Heap(A),bool,aa(heap_Heap(A),fun(heap_Heap(A),bool),heap_Heap_ord(A),X2),Y2))
=> ( pp(aa(heap_Heap(A),bool,aa(heap_Heap(A),fun(heap_Heap(A),bool),heap_Heap_ord(A),Y2),Z2))
=> pp(aa(heap_Heap(A),bool,aa(heap_Heap(A),fun(heap_Heap(A),bool),heap_Heap_ord(A),X2),Z2)) ) ) ).
% heap.leq_trans
tff(fact_2402_heap_Oleq__antisym,axiom,
! [A: $tType,X2: heap_Heap(A),Y2: heap_Heap(A)] :
( pp(aa(heap_Heap(A),bool,aa(heap_Heap(A),fun(heap_Heap(A),bool),heap_Heap_ord(A),X2),Y2))
=> ( pp(aa(heap_Heap(A),bool,aa(heap_Heap(A),fun(heap_Heap(A),bool),heap_Heap_ord(A),Y2),X2))
=> ( X2 = Y2 ) ) ) ).
% heap.leq_antisym
tff(fact_2403_Heap_Obi__unique__rel,axiom,
! [B: $tType,A: $tType,R3: fun(A,fun(B,bool))] :
( bi_unique(A,B,R3)
=> bi_unique(heap_Heap(A),heap_Heap(B),aa(fun(A,fun(B,bool)),fun(heap_Heap(A),fun(heap_Heap(B),bool)),heap_rel_Heap(A,B),R3)) ) ).
% Heap.bi_unique_rel
tff(fact_2404_bi__unique__OO,axiom,
! [A: $tType,C: $tType,B: $tType,A6: fun(A,fun(B,bool)),B6: fun(B,fun(C,bool))] :
( bi_unique(A,B,A6)
=> ( bi_unique(B,C,B6)
=> bi_unique(A,C,aa(fun(B,fun(C,bool)),fun(A,fun(C,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(C,bool)),fun(A,fun(C,bool))),relcompp(A,B,C),A6),B6)) ) ) ).
% bi_unique_OO
tff(fact_2405_dir__image__Field,axiom,
! [A: $tType,B: $tType,R: set(product_prod(B,B)),F: fun(B,A)] : aa(set(product_prod(A,A)),set(A),field2(A),bNF_We2720479622203943262_image(B,A,R,F)) = aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),F),aa(set(product_prod(B,B)),set(B),field2(B),R)) ).
% dir_image_Field
tff(fact_2406_heap__interpretation,axiom,
! [A: $tType] : partia7178651479351089652itions(heap_Heap(A),heap_Heap_ord(A),heap_Heap_lub(A)) ).
% heap_interpretation
tff(fact_2407_scomp__fcomp,axiom,
! [A: $tType,D: $tType,E: $tType,B: $tType,C: $tType,F: fun(A,product_prod(D,E)),G: fun(D,fun(E,C)),H: fun(C,B)] : fcomp(A,C,B,product_scomp(A,D,E,C,F,G),H) = product_scomp(A,D,E,B,F,aa(fun(C,B),fun(D,fun(E,B)),aTP_Lamp_ud(fun(D,fun(E,C)),fun(fun(C,B),fun(D,fun(E,B))),G),H)) ).
% scomp_fcomp
tff(fact_2408_image__uminus__greaterThanLessThan,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [X2: A,Y2: A] : aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),uminus_uminus(A)),set_or5935395276787703475ssThan(A,X2,Y2)) = set_or5935395276787703475ssThan(A,aa(A,A,uminus_uminus(A),Y2),aa(A,A,uminus_uminus(A),X2)) ) ).
% image_uminus_greaterThanLessThan
tff(fact_2409_euclidean__size__int__def,axiom,
euclid6346220572633701492n_size(int) = aa(fun(int,int),fun(int,nat),aa(fun(int,nat),fun(fun(int,int),fun(int,nat)),comp(int,nat,int),nat2),abs_abs(int)) ).
% euclidean_size_int_def
tff(fact_2410_symp__INF,axiom,
! [B: $tType,A: $tType,S3: set(A),R: fun(A,fun(B,fun(B,bool)))] :
( ! [X: A] :
( pp(aa(set(A),bool,member(A,X),S3))
=> symp(B,aa(A,fun(B,fun(B,bool)),R,X)) )
=> symp(B,aa(set(fun(B,fun(B,bool))),fun(B,fun(B,bool)),complete_Inf_Inf(fun(B,fun(B,bool))),aa(set(A),set(fun(B,fun(B,bool))),aa(fun(A,fun(B,fun(B,bool))),fun(set(A),set(fun(B,fun(B,bool)))),image2(A,fun(B,fun(B,bool))),R),S3))) ) ).
% symp_INF
tff(fact_2411_transp__INF,axiom,
! [B: $tType,A: $tType,S3: set(A),R: fun(A,fun(B,fun(B,bool)))] :
( ! [X: A] :
( pp(aa(set(A),bool,member(A,X),S3))
=> transp(B,aa(A,fun(B,fun(B,bool)),R,X)) )
=> transp(B,aa(set(fun(B,fun(B,bool))),fun(B,fun(B,bool)),complete_Inf_Inf(fun(B,fun(B,bool))),aa(set(A),set(fun(B,fun(B,bool))),aa(fun(A,fun(B,fun(B,bool))),fun(set(A),set(fun(B,fun(B,bool)))),image2(A,fun(B,fun(B,bool))),R),S3))) ) ).
% transp_INF
tff(fact_2412_Heap_Orel__symp,axiom,
! [A: $tType,R3: fun(A,fun(A,bool))] :
( symp(A,R3)
=> symp(heap_Heap(A),aa(fun(A,fun(A,bool)),fun(heap_Heap(A),fun(heap_Heap(A),bool)),heap_rel_Heap(A,A),R3)) ) ).
% Heap.rel_symp
tff(fact_2413_Heap_Orel__transp,axiom,
! [A: $tType,R3: fun(A,fun(A,bool))] :
( transp(A,R3)
=> transp(heap_Heap(A),aa(fun(A,fun(A,bool)),fun(heap_Heap(A),fun(heap_Heap(A),bool)),heap_rel_Heap(A,A),R3)) ) ).
% Heap.rel_transp
tff(fact_2414_fcomp__scomp,axiom,
! [A: $tType,E: $tType,B: $tType,D: $tType,C: $tType,F: fun(A,E),G: fun(E,product_prod(C,D)),H: fun(C,fun(D,B))] : product_scomp(A,C,D,B,fcomp(A,E,product_prod(C,D),F,G),H) = fcomp(A,E,B,F,product_scomp(E,C,D,B,G,H)) ).
% fcomp_scomp
tff(fact_2415_fcomp__comp,axiom,
! [B: $tType,C: $tType,A: $tType,F: fun(A,C),G: fun(C,B)] : fcomp(A,C,B,F,G) = aa(fun(A,C),fun(A,B),aa(fun(C,B),fun(fun(A,C),fun(A,B)),comp(C,B,A),G),F) ).
% fcomp_comp
tff(fact_2416_transp__relcompp__less__eq,axiom,
! [A: $tType,R: fun(A,fun(A,bool))] :
( transp(A,R)
=> pp(aa(fun(A,fun(A,bool)),bool,aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),bool),ord_less_eq(fun(A,fun(A,bool))),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),relcompp(A,A,A),R),R)),R)) ) ).
% transp_relcompp_less_eq
tff(fact_2417_transp__relcompp,axiom,
! [A: $tType,R: fun(A,fun(A,bool))] :
( transp(A,R)
<=> pp(aa(fun(A,fun(A,bool)),bool,aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),bool),ord_less_eq(fun(A,fun(A,bool))),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),relcompp(A,A,A),R),R)),R)) ) ).
% transp_relcompp
tff(fact_2418_symp__conversep,axiom,
! [A: $tType,R3: fun(A,fun(A,bool))] :
( symp(A,R3)
<=> pp(aa(fun(A,fun(A,bool)),bool,aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),bool),ord_less_eq(fun(A,fun(A,bool))),conversep(A,A,R3)),R3)) ) ).
% symp_conversep
tff(fact_2419_transp__trans__eq,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( transp(A,aTP_Lamp_hl(set(product_prod(A,A)),fun(A,fun(A,bool)),R))
<=> trans(A,R) ) ).
% transp_trans_eq
tff(fact_2420_symp__sym__eq,axiom,
! [A: $tType,R: set(product_prod(A,A))] :
( symp(A,aTP_Lamp_hl(set(product_prod(A,A)),fun(A,fun(A,bool)),R))
<=> sym(A,R) ) ).
% symp_sym_eq
tff(fact_2421_transp__trans,axiom,
! [A: $tType,R: fun(A,fun(A,bool))] :
( transp(A,R)
<=> trans(A,aa(fun(product_prod(A,A),bool),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),R))) ) ).
% transp_trans
tff(fact_2422_symp__SUP,axiom,
! [B: $tType,A: $tType,S3: set(A),R: fun(A,fun(B,fun(B,bool)))] :
( ! [X: A] :
( pp(aa(set(A),bool,member(A,X),S3))
=> symp(B,aa(A,fun(B,fun(B,bool)),R,X)) )
=> symp(B,aa(set(fun(B,fun(B,bool))),fun(B,fun(B,bool)),complete_Sup_Sup(fun(B,fun(B,bool))),aa(set(A),set(fun(B,fun(B,bool))),aa(fun(A,fun(B,fun(B,bool))),fun(set(A),set(fun(B,fun(B,bool)))),image2(A,fun(B,fun(B,bool))),R),S3))) ) ).
% symp_SUP
tff(fact_2423_sum_Oset__map_I1_J,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,F1: fun(A,C),F22: fun(B,D),V: sum_sum(A,B)] : basic_setl(C,D,aa(sum_sum(A,B),sum_sum(C,D),sum_map_sum(A,C,B,D,F1,F22),V)) = aa(set(A),set(C),aa(fun(A,C),fun(set(A),set(C)),image2(A,C),F1),basic_setl(A,B,V)) ).
% sum.set_map(1)
tff(fact_2424_sum_Oset__map_I2_J,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,F1: fun(A,C),F22: fun(B,D),V: sum_sum(A,B)] : basic_setr(C,D,aa(sum_sum(A,B),sum_sum(C,D),sum_map_sum(A,C,B,D,F1,F22),V)) = aa(set(B),set(D),aa(fun(B,D),fun(set(B),set(D)),image2(B,D),F22),basic_setr(A,B,V)) ).
% sum.set_map(2)
tff(fact_2425_heap_Olub__least,axiom,
! [A: $tType,A6: set(heap_Heap(A)),Z2: heap_Heap(A)] :
( comple1602240252501008431_chain(heap_Heap(A),heap_Heap_ord(A),A6)
=> ( ! [X: heap_Heap(A)] :
( pp(aa(set(heap_Heap(A)),bool,member(heap_Heap(A),X),A6))
=> pp(aa(heap_Heap(A),bool,aa(heap_Heap(A),fun(heap_Heap(A),bool),heap_Heap_ord(A),X),Z2)) )
=> pp(aa(heap_Heap(A),bool,aa(heap_Heap(A),fun(heap_Heap(A),bool),heap_Heap_ord(A),aa(set(heap_Heap(A)),heap_Heap(A),heap_Heap_lub(A),A6)),Z2)) ) ) ).
% heap.lub_least
tff(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)
=> ( pp(aa(set(heap_Heap(A)),bool,member(heap_Heap(A),X2),A6))
=> pp(aa(heap_Heap(A),bool,aa(heap_Heap(A),fun(heap_Heap(A),bool),heap_Heap_ord(A),X2),aa(set(heap_Heap(A)),heap_Heap(A),heap_Heap_lub(A),A6))) ) ) ).
% heap.lub_upper
tff(fact_2427_folding__idem__axioms_Ointro,axiom,
! [B: $tType,A: $tType,F: fun(A,fun(B,B))] :
( ! [X: A] : aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X)),aa(A,fun(B,B),F,X)) = aa(A,fun(B,B),F,X)
=> finite7837460588564673216axioms(A,B,F) ) ).
% folding_idem_axioms.intro
tff(fact_2428_comp__fun__idem__axioms_Ointro,axiom,
! [B: $tType,A: $tType,F: fun(A,fun(B,B))] :
( ! [X: A] : aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X)),aa(A,fun(B,B),F,X)) = aa(A,fun(B,B),F,X)
=> finite1981172056906289455axioms(A,B,F) ) ).
% comp_fun_idem_axioms.intro
tff(fact_2429_comp__fun__idem__axioms__def,axiom,
! [B: $tType,A: $tType,F: fun(A,fun(B,B))] :
( finite1981172056906289455axioms(A,B,F)
<=> ! [X5: A] : aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X5)),aa(A,fun(B,B),F,X5)) = aa(A,fun(B,B),F,X5) ) ).
% comp_fun_idem_axioms_def
tff(fact_2430_folding__idem__axioms__def,axiom,
! [B: $tType,A: $tType,F: fun(A,fun(B,B))] :
( finite7837460588564673216axioms(A,B,F)
<=> ! [X5: A] : aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X5)),aa(A,fun(B,B),F,X5)) = aa(A,fun(B,B),F,X5) ) ).
% folding_idem_axioms_def
tff(fact_2431_chain__imageI,axiom,
! [B: $tType,A: $tType,Le_a: fun(A,fun(A,bool)),Y3: set(A),Le_b: fun(B,fun(B,bool)),F: fun(A,B)] :
( comple1602240252501008431_chain(A,Le_a,Y3)
=> ( ! [X: A,Y: A] :
( pp(aa(set(A),bool,member(A,X),Y3))
=> ( pp(aa(set(A),bool,member(A,Y),Y3))
=> ( pp(aa(A,bool,aa(A,fun(A,bool),Le_a,X),Y))
=> pp(aa(B,bool,aa(B,fun(B,bool),Le_b,aa(A,B,F,X)),aa(A,B,F,Y))) ) ) )
=> comple1602240252501008431_chain(B,Le_b,aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),Y3)) ) ) ).
% chain_imageI
tff(fact_2432_sum_Orel__compp__Grp,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,R1: fun(A,fun(C,bool)),R22: fun(B,fun(D,bool))] : bNF_rel_sum(A,C,B,D,R1,R22) = aa(fun(sum_sum(product_prod(A,C),product_prod(B,D)),fun(sum_sum(C,D),bool)),fun(sum_sum(A,B),fun(sum_sum(C,D),bool)),aa(fun(sum_sum(A,B),fun(sum_sum(product_prod(A,C),product_prod(B,D)),bool)),fun(fun(sum_sum(product_prod(A,C),product_prod(B,D)),fun(sum_sum(C,D),bool)),fun(sum_sum(A,B),fun(sum_sum(C,D),bool))),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),aa(fun(sum_sum(product_prod(A,C),product_prod(B,D)),bool),set(sum_sum(product_prod(A,C),product_prod(B,D))),collect(sum_sum(product_prod(A,C),product_prod(B,D))),aa(fun(B,fun(D,bool)),fun(sum_sum(product_prod(A,C),product_prod(B,D)),bool),aTP_Lamp_ue(fun(A,fun(C,bool)),fun(fun(B,fun(D,bool)),fun(sum_sum(product_prod(A,C),product_prod(B,D)),bool)),R1),R22)),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),aa(fun(sum_sum(product_prod(A,C),product_prod(B,D)),bool),set(sum_sum(product_prod(A,C),product_prod(B,D))),collect(sum_sum(product_prod(A,C),product_prod(B,D))),aa(fun(B,fun(D,bool)),fun(sum_sum(product_prod(A,C),product_prod(B,D)),bool),aTP_Lamp_ue(fun(A,fun(C,bool)),fun(fun(B,fun(D,bool)),fun(sum_sum(product_prod(A,C),product_prod(B,D)),bool)),R1),R22)),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
tff(fact_2433_sum_Oin__rel,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,R1: fun(A,fun(C,bool)),R22: fun(B,fun(D,bool)),A3: sum_sum(A,B),B2: sum_sum(C,D)] :
( pp(aa(sum_sum(C,D),bool,aa(sum_sum(A,B),fun(sum_sum(C,D),bool),bNF_rel_sum(A,C,B,D,R1,R22),A3),B2))
<=> ? [Z4: sum_sum(product_prod(A,C),product_prod(B,D))] :
( pp(aa(set(sum_sum(product_prod(A,C),product_prod(B,D))),bool,member(sum_sum(product_prod(A,C),product_prod(B,D)),Z4),aa(fun(sum_sum(product_prod(A,C),product_prod(B,D)),bool),set(sum_sum(product_prod(A,C),product_prod(B,D))),collect(sum_sum(product_prod(A,C),product_prod(B,D))),aa(fun(B,fun(D,bool)),fun(sum_sum(product_prod(A,C),product_prod(B,D)),bool),aTP_Lamp_ue(fun(A,fun(C,bool)),fun(fun(B,fun(D,bool)),fun(sum_sum(product_prod(A,C),product_prod(B,D)),bool)),R1),R22))))
& ( aa(sum_sum(product_prod(A,C),product_prod(B,D)),sum_sum(A,B),sum_map_sum(product_prod(A,C),A,product_prod(B,D),B,product_fst(A,C),product_fst(B,D)),Z4) = A3 )
& ( aa(sum_sum(product_prod(A,C),product_prod(B,D)),sum_sum(C,D),sum_map_sum(product_prod(A,C),C,product_prod(B,D),D,product_snd(A,C),product_snd(B,D)),Z4) = B2 ) ) ) ).
% sum.in_rel
tff(fact_2434_comp__fun__idem__on__axioms__def,axiom,
! [B: $tType,A: $tType,S3: set(A),F: fun(A,fun(B,B))] :
( finite4980608107308702382axioms(A,B,S3,F)
<=> ! [X5: A] :
( pp(aa(set(A),bool,member(A,X5),S3))
=> ( aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X5)),aa(A,fun(B,B),F,X5)) = aa(A,fun(B,B),F,X5) ) ) ) ).
% comp_fun_idem_on_axioms_def
tff(fact_2435_folding__idem__on__axioms__def,axiom,
! [B: $tType,A: $tType,S3: set(A),F: fun(A,fun(B,B))] :
( finite6916993218817215295axioms(A,B,S3,F)
<=> ! [X5: A,Y4: A] :
( pp(aa(set(A),bool,member(A,X5),S3))
=> ( pp(aa(set(A),bool,member(A,Y4),S3))
=> ( aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X5)),aa(A,fun(B,B),F,X5)) = aa(A,fun(B,B),F,X5) ) ) ) ) ).
% folding_idem_on_axioms_def
tff(fact_2436_comp__fun__idem__on__axioms_Ointro,axiom,
! [B: $tType,A: $tType,S3: set(A),F: fun(A,fun(B,B))] :
( ! [X: A] :
( pp(aa(set(A),bool,member(A,X),S3))
=> ( aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X)),aa(A,fun(B,B),F,X)) = aa(A,fun(B,B),F,X) ) )
=> finite4980608107308702382axioms(A,B,S3,F) ) ).
% comp_fun_idem_on_axioms.intro
tff(fact_2437_sum_Orel__compp,axiom,
! [B: $tType,A: $tType,C: $tType,E: $tType,F2: $tType,D: $tType,R1: fun(A,fun(C,bool)),S1: fun(C,fun(E,bool)),R22: fun(B,fun(D,bool)),S22: fun(D,fun(F2,bool))] : bNF_rel_sum(A,E,B,F2,aa(fun(C,fun(E,bool)),fun(A,fun(E,bool)),aa(fun(A,fun(C,bool)),fun(fun(C,fun(E,bool)),fun(A,fun(E,bool))),relcompp(A,C,E),R1),S1),aa(fun(D,fun(F2,bool)),fun(B,fun(F2,bool)),aa(fun(B,fun(D,bool)),fun(fun(D,fun(F2,bool)),fun(B,fun(F2,bool))),relcompp(B,D,F2),R22),S22)) = aa(fun(sum_sum(C,D),fun(sum_sum(E,F2),bool)),fun(sum_sum(A,B),fun(sum_sum(E,F2),bool)),aa(fun(sum_sum(A,B),fun(sum_sum(C,D),bool)),fun(fun(sum_sum(C,D),fun(sum_sum(E,F2),bool)),fun(sum_sum(A,B),fun(sum_sum(E,F2),bool))),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
tff(fact_2438_rel__sum_Ointros_I2_J,axiom,
! [B: $tType,A: $tType,C: $tType,D: $tType,R22: fun(B,fun(D,bool)),B2: B,D4: D,R1: fun(A,fun(C,bool))] :
( pp(aa(D,bool,aa(B,fun(D,bool),R22,B2),D4))
=> pp(aa(sum_sum(C,D),bool,aa(sum_sum(A,B),fun(sum_sum(C,D),bool),bNF_rel_sum(A,C,B,D,R1,R22),aa(B,sum_sum(A,B),sum_Inr(B,A),B2)),aa(D,sum_sum(C,D),sum_Inr(D,C),D4))) ) ).
% rel_sum.intros(2)
tff(fact_2439_rel__sum_Ointros_I1_J,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,R1: fun(A,fun(C,bool)),A3: A,C3: C,R22: fun(B,fun(D,bool))] :
( pp(aa(C,bool,aa(A,fun(C,bool),R1,A3),C3))
=> pp(aa(sum_sum(C,D),bool,aa(sum_sum(A,B),fun(sum_sum(C,D),bool),bNF_rel_sum(A,C,B,D,R1,R22),aa(A,sum_sum(A,B),sum_Inl(A,B),A3)),aa(C,sum_sum(C,D),sum_Inl(C,D),C3))) ) ).
% rel_sum.intros(1)
tff(fact_2440_rel__sum_Osimps,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,R1: fun(A,fun(C,bool)),R22: fun(B,fun(D,bool)),A1: sum_sum(A,B),A22: sum_sum(C,D)] :
( pp(aa(sum_sum(C,D),bool,aa(sum_sum(A,B),fun(sum_sum(C,D),bool),bNF_rel_sum(A,C,B,D,R1,R22),A1),A22))
<=> ( ? [A7: A,C6: C] :
( ( A1 = aa(A,sum_sum(A,B),sum_Inl(A,B),A7) )
& ( A22 = aa(C,sum_sum(C,D),sum_Inl(C,D),C6) )
& pp(aa(C,bool,aa(A,fun(C,bool),R1,A7),C6)) )
| ? [B5: B,D5: D] :
( ( A1 = aa(B,sum_sum(A,B),sum_Inr(B,A),B5) )
& ( A22 = aa(D,sum_sum(C,D),sum_Inr(D,C),D5) )
& pp(aa(D,bool,aa(B,fun(D,bool),R22,B5),D5)) ) ) ) ).
% rel_sum.simps
tff(fact_2441_rel__sum_Ocases,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,R1: fun(A,fun(C,bool)),R22: fun(B,fun(D,bool)),A1: sum_sum(A,B),A22: sum_sum(C,D)] :
( pp(aa(sum_sum(C,D),bool,aa(sum_sum(A,B),fun(sum_sum(C,D),bool),bNF_rel_sum(A,C,B,D,R1,R22),A1),A22))
=> ( ! [A5: A] :
( ( A1 = aa(A,sum_sum(A,B),sum_Inl(A,B),A5) )
=> ! [C2: C] :
( ( A22 = aa(C,sum_sum(C,D),sum_Inl(C,D),C2) )
=> ~ pp(aa(C,bool,aa(A,fun(C,bool),R1,A5),C2)) ) )
=> ~ ! [B4: B] :
( ( A1 = aa(B,sum_sum(A,B),sum_Inr(B,A),B4) )
=> ! [D2: D] :
( ( A22 = aa(D,sum_sum(C,D),sum_Inr(D,C),D2) )
=> ~ pp(aa(D,bool,aa(B,fun(D,bool),R22,B4),D2)) ) ) ) ) ).
% rel_sum.cases
tff(fact_2442_folding__idem__on__axioms_Ointro,axiom,
! [B: $tType,A: $tType,S3: set(A),F: fun(A,fun(B,B))] :
( ! [X: A,Y: A] :
( pp(aa(set(A),bool,member(A,X),S3))
=> ( pp(aa(set(A),bool,member(A,Y),S3))
=> ( aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X)),aa(A,fun(B,B),F,X)) = aa(A,fun(B,B),F,X) ) ) )
=> finite6916993218817215295axioms(A,B,S3,F) ) ).
% folding_idem_on_axioms.intro
tff(fact_2443_sndsp_Ointros,axiom,
! [B: $tType,A: $tType,P: product_prod(A,B)] : basic_sndsp(A,B,P,aa(product_prod(A,B),B,product_snd(A,B),P)) ).
% sndsp.intros
tff(fact_2444_sndsp_Osimps,axiom,
! [B: $tType,A: $tType,P: product_prod(A,B),A3: B] :
( basic_sndsp(A,B,P,A3)
<=> ( A3 = aa(product_prod(A,B),B,product_snd(A,B),P) ) ) ).
% sndsp.simps
tff(fact_2445_sndsp_Ocases,axiom,
! [B: $tType,A: $tType,P: product_prod(A,B),A3: B] :
( basic_sndsp(A,B,P,A3)
=> ( A3 = aa(product_prod(A,B),B,product_snd(A,B),P) ) ) ).
% sndsp.cases
tff(fact_2446_fstsp_Ointros,axiom,
! [B: $tType,A: $tType,P: product_prod(A,B)] : basic_fstsp(A,B,P,aa(product_prod(A,B),A,product_fst(A,B),P)) ).
% fstsp.intros
tff(fact_2447_fstsp_Ocases,axiom,
! [B: $tType,A: $tType,P: product_prod(A,B),A3: A] :
( basic_fstsp(A,B,P,A3)
=> ( A3 = aa(product_prod(A,B),A,product_fst(A,B),P) ) ) ).
% fstsp.cases
tff(fact_2448_fstsp_Osimps,axiom,
! [B: $tType,A: $tType,P: product_prod(A,B),A3: A] :
( basic_fstsp(A,B,P,A3)
<=> ( A3 = aa(product_prod(A,B),A,product_fst(A,B),P) ) ) ).
% fstsp.simps
tff(fact_2449_comm__monoid__set_OatLeastAtMost__shift__0,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,M4: nat,N4: nat,G: fun(nat,A)] :
( groups778175481326437816id_set(A,F,Z2)
=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M4),N4))
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),set_or1337092689740270186AtMost(nat,M4,N4)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),aa(nat,fun(nat,nat),plus_plus(nat),M4))),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N4),M4))) ) ) ) ).
% comm_monoid_set.atLeastAtMost_shift_0
tff(fact_2450_list_Opred__map,axiom,
! [B: $tType,A: $tType,Q2: fun(B,bool),F: fun(A,B),X2: list(A)] :
( list_all(B,Q2,aa(list(A),list(B),map(A,B,F),X2))
<=> list_all(A,aa(fun(A,B),fun(A,bool),aa(fun(B,bool),fun(fun(A,B),fun(A,bool)),comp(B,bool,A),Q2),F),X2) ) ).
% list.pred_map
tff(fact_2451_Unity__def,axiom,
product_Unity = aa(bool,product_unit,product_Abs_unit,fTrue) ).
% Unity_def
tff(fact_2452_prod__encode__def,axiom,
nat_prod_encode = aa(fun(nat,fun(nat,nat)),fun(product_prod(nat,nat),nat),product_case_prod(nat,nat,nat),aTP_Lamp_uf(nat,fun(nat,nat))) ).
% prod_encode_def
tff(fact_2453_select__weight__def,axiom,
! [A: $tType,Xs: list(product_prod(code_natural,A))] : select_weight(A,Xs) = product_scomp(product_prod(code_natural,code_natural),code_natural,product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural)),range(groups8242544230860333062m_list(code_natural,aa(list(product_prod(code_natural,A)),list(code_natural),map(product_prod(code_natural,A),code_natural,product_fst(code_natural,A)),Xs))),aTP_Lamp_ug(list(product_prod(code_natural,A)),fun(code_natural,fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural)))),Xs)) ).
% select_weight_def
tff(fact_2454_adm__wf__def,axiom,
! [B: $tType,A: $tType,R3: set(product_prod(A,A)),F6: fun(fun(A,B),fun(A,B))] :
( adm_wf(A,B,R3,F6)
<=> ! [F7: fun(A,B),G5: fun(A,B),X5: A] :
( ! [Z4: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z4),X5)),R3))
=> ( aa(A,B,F7,Z4) = aa(A,B,G5,Z4) ) )
=> ( aa(A,B,aa(fun(A,B),fun(A,B),F6,F7),X5) = aa(A,B,aa(fun(A,B),fun(A,B),F6,G5),X5) ) ) ) ).
% adm_wf_def
tff(fact_2455_comm__monoid__set_OUnion__comp,axiom,
! [A: $tType,B: $tType,F: fun(A,fun(A,A)),Z2: A,B6: set(set(B)),G: fun(B,A)] :
( groups778175481326437816id_set(A,F,Z2)
=> ( ! [X: set(B)] :
( pp(aa(set(set(B)),bool,member(set(B),X),B6))
=> finite_finite2(B,X) )
=> ( ! [A15: set(B)] :
( pp(aa(set(set(B)),bool,member(set(B),A15),B6))
=> ! [A25: set(B)] :
( pp(aa(set(set(B)),bool,member(set(B),A25),B6))
=> ( ( A15 != A25 )
=> ! [X: B] :
( pp(aa(set(B),bool,member(B,X),A15))
=> ( pp(aa(set(B),bool,member(B,X),A25))
=> ( aa(B,A,G,X) = Z2 ) ) ) ) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),G),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),B6)) = aa(set(set(B)),A,aa(fun(B,A),fun(set(set(B)),A),aa(fun(fun(B,A),fun(set(B),A)),fun(fun(B,A),fun(set(set(B)),A)),aa(fun(fun(set(B),A),fun(set(set(B)),A)),fun(fun(fun(B,A),fun(set(B),A)),fun(fun(B,A),fun(set(set(B)),A))),comp(fun(set(B),A),fun(set(set(B)),A),fun(B,A)),groups_comm_monoid_F(A,set(B),F,Z2)),groups_comm_monoid_F(A,B,F,Z2)),G),B6) ) ) ) ) ).
% comm_monoid_set.Union_comp
tff(fact_2456_comm__monoid__set_Oreindex__cong,axiom,
! [C: $tType,A: $tType,B: $tType,F: fun(A,fun(A,A)),Z2: A,L: fun(B,C),B6: set(B),A6: set(C),G: fun(C,A),H: fun(B,A)] :
( groups778175481326437816id_set(A,F,Z2)
=> ( inj_on(B,C,L,B6)
=> ( ( A6 = aa(set(B),set(C),aa(fun(B,C),fun(set(B),set(C)),image2(B,C),L),B6) )
=> ( ! [X: B] :
( pp(aa(set(B),bool,member(B,X),B6))
=> ( aa(C,A,G,aa(B,C,L,X)) = aa(B,A,H,X) ) )
=> ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups_comm_monoid_F(A,C,F,Z2),G),A6) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),H),B6) ) ) ) ) ) ).
% comm_monoid_set.reindex_cong
tff(fact_2457_Abs__unit__cases,axiom,
! [X2: product_unit] :
~ ! [Y: bool] :
( ( X2 = aa(bool,product_unit,product_Abs_unit,Y) )
=> ~ pp(aa(set(bool),bool,member(bool,Y),aa(set(bool),set(bool),insert(bool,fTrue),bot_bot(set(bool))))) ) ).
% Abs_unit_cases
tff(fact_2458_Abs__unit__induct,axiom,
! [P2: fun(product_unit,bool),X2: product_unit] :
( ! [Y: bool] :
( pp(aa(set(bool),bool,member(bool,Y),aa(set(bool),set(bool),insert(bool,fTrue),bot_bot(set(bool)))))
=> pp(aa(product_unit,bool,P2,aa(bool,product_unit,product_Abs_unit,Y))) )
=> pp(aa(product_unit,bool,P2,X2)) ) ).
% Abs_unit_induct
tff(fact_2459_Abs__unit__inject,axiom,
! [X2: bool,Y2: bool] :
( pp(aa(set(bool),bool,member(bool,X2),aa(set(bool),set(bool),insert(bool,fTrue),bot_bot(set(bool)))))
=> ( pp(aa(set(bool),bool,member(bool,Y2),aa(set(bool),set(bool),insert(bool,fTrue),bot_bot(set(bool)))))
=> ( ( aa(bool,product_unit,product_Abs_unit,X2) = aa(bool,product_unit,product_Abs_unit,Y2) )
<=> ( pp(X2)
<=> pp(Y2) ) ) ) ) ).
% Abs_unit_inject
tff(fact_2460_comm__monoid__set_Ocartesian__product,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(A,fun(A,A)),Z2: A,G: fun(B,fun(C,A)),B6: set(C),A6: set(B)] :
( groups778175481326437816id_set(A,F,Z2)
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),aa(set(C),fun(B,A),aa(fun(B,fun(C,A)),fun(set(C),fun(B,A)),aa(A,fun(fun(B,fun(C,A)),fun(set(C),fun(B,A))),aTP_Lamp_uh(fun(A,fun(A,A)),fun(A,fun(fun(B,fun(C,A)),fun(set(C),fun(B,A)))),F),Z2),G),B6)),A6) = aa(set(product_prod(B,C)),A,aa(fun(product_prod(B,C),A),fun(set(product_prod(B,C)),A),groups_comm_monoid_F(A,product_prod(B,C),F,Z2),aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),G)),product_Sigma(B,C,A6,aTP_Lamp_hc(set(C),fun(B,set(C)),B6))) ) ) ).
% comm_monoid_set.cartesian_product
tff(fact_2461_comm__monoid__set_OSigma,axiom,
! [A: $tType,C: $tType,B: $tType,F: fun(A,fun(A,A)),Z2: A,A6: set(B),B6: fun(B,set(C)),G: fun(B,fun(C,A))] :
( groups778175481326437816id_set(A,F,Z2)
=> ( finite_finite2(B,A6)
=> ( ! [X: B] :
( pp(aa(set(B),bool,member(B,X),A6))
=> finite_finite2(C,aa(B,set(C),B6,X)) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),aa(fun(B,fun(C,A)),fun(B,A),aa(fun(B,set(C)),fun(fun(B,fun(C,A)),fun(B,A)),aa(A,fun(fun(B,set(C)),fun(fun(B,fun(C,A)),fun(B,A))),aTP_Lamp_ui(fun(A,fun(A,A)),fun(A,fun(fun(B,set(C)),fun(fun(B,fun(C,A)),fun(B,A)))),F),Z2),B6),G)),A6) = aa(set(product_prod(B,C)),A,aa(fun(product_prod(B,C),A),fun(set(product_prod(B,C)),A),groups_comm_monoid_F(A,product_prod(B,C),F,Z2),aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_case_prod(B,C,A),G)),product_Sigma(B,C,A6,B6)) ) ) ) ) ).
% comm_monoid_set.Sigma
tff(fact_2462_comm__monoid__set_Oeq__fold,axiom,
! [A: $tType,B: $tType,F: fun(A,fun(A,A)),Z2: A,G: fun(B,A),A6: set(B)] :
( groups778175481326437816id_set(A,F,Z2)
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),G),A6) = finite_fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),aa(fun(A,fun(A,A)),fun(fun(B,A),fun(B,fun(A,A))),comp(A,fun(A,A),B),F),G),Z2,A6) ) ) ).
% comm_monoid_set.eq_fold
tff(fact_2463_comm__monoid__set_Oimage__gen,axiom,
! [A: $tType,C: $tType,B: $tType,F: fun(A,fun(A,A)),Z2: A,S3: set(B),H: fun(B,A),G: fun(B,C)] :
( groups778175481326437816id_set(A,F,Z2)
=> ( finite_finite2(B,S3)
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),H),S3) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups_comm_monoid_F(A,C,F,Z2),aa(fun(B,C),fun(C,A),aa(fun(B,A),fun(fun(B,C),fun(C,A)),aa(set(B),fun(fun(B,A),fun(fun(B,C),fun(C,A))),aa(A,fun(set(B),fun(fun(B,A),fun(fun(B,C),fun(C,A)))),aTP_Lamp_uj(fun(A,fun(A,A)),fun(A,fun(set(B),fun(fun(B,A),fun(fun(B,C),fun(C,A))))),F),Z2),S3),H),G)),aa(set(B),set(C),aa(fun(B,C),fun(set(B),set(C)),image2(B,C),G),S3)) ) ) ) ).
% comm_monoid_set.image_gen
tff(fact_2464_comm__monoid__set_Oimage__eq,axiom,
! [A: $tType,B: $tType,F: fun(A,fun(A,A)),Z2: A,G: fun(B,A),A6: set(B)] :
( groups778175481326437816id_set(A,F,Z2)
=> ( inj_on(B,A,G,A6)
=> ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups_comm_monoid_F(A,A,F,Z2),aTP_Lamp_ab(A,A)),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),G),A6)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),G),A6) ) ) ) ).
% comm_monoid_set.image_eq
tff(fact_2465_comm__monoid__set_OatLeastAtMost__reindex,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [F: fun(A,fun(A,A)),Z2: A,H: fun(nat,B),M4: nat,N4: nat,G: fun(B,A)] :
( groups778175481326437816id_set(A,F,Z2)
=> ( bij_betw(nat,B,H,set_or1337092689740270186AtMost(nat,M4,N4),set_or1337092689740270186AtMost(B,aa(nat,B,H,M4),aa(nat,B,H,N4)))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),G),set_or1337092689740270186AtMost(B,aa(nat,B,H,M4),aa(nat,B,H,N4))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),aa(fun(nat,B),fun(nat,A),aa(fun(B,A),fun(fun(nat,B),fun(nat,A)),comp(B,A,nat),G),H)),set_or1337092689740270186AtMost(nat,M4,N4)) ) ) ) ) ).
% comm_monoid_set.atLeastAtMost_reindex
tff(fact_2466_comm__monoid__set_OatLeastLessThan__reindex,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [F: fun(A,fun(A,A)),Z2: A,H: fun(nat,B),M4: nat,N4: nat,G: fun(B,A)] :
( groups778175481326437816id_set(A,F,Z2)
=> ( bij_betw(nat,B,H,set_or7035219750837199246ssThan(nat,M4,N4),set_or7035219750837199246ssThan(B,aa(nat,B,H,M4),aa(nat,B,H,N4)))
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),G),set_or7035219750837199246ssThan(B,aa(nat,B,H,M4),aa(nat,B,H,N4))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),aa(fun(nat,B),fun(nat,A),aa(fun(B,A),fun(fun(nat,B),fun(nat,A)),comp(B,A,nat),G),H)),set_or7035219750837199246ssThan(nat,M4,N4)) ) ) ) ) ).
% comm_monoid_set.atLeastLessThan_reindex
tff(fact_2467_comm__monoid__set_Oreindex__nontrivial,axiom,
! [C: $tType,A: $tType,B: $tType,F: fun(A,fun(A,A)),Z2: A,A6: set(B),H: fun(B,C),G: fun(C,A)] :
( groups778175481326437816id_set(A,F,Z2)
=> ( finite_finite2(B,A6)
=> ( ! [X: B,Y: B] :
( pp(aa(set(B),bool,member(B,X),A6))
=> ( pp(aa(set(B),bool,member(B,Y),A6))
=> ( ( X != Y )
=> ( ( aa(B,C,H,X) = aa(B,C,H,Y) )
=> ( aa(C,A,G,aa(B,C,H,X)) = Z2 ) ) ) ) )
=> ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups_comm_monoid_F(A,C,F,Z2),G),aa(set(B),set(C),aa(fun(B,C),fun(set(B),set(C)),image2(B,C),H),A6)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),aa(fun(B,C),fun(B,A),aa(fun(C,A),fun(fun(B,C),fun(B,A)),comp(C,A,B),G),H)),A6) ) ) ) ) ).
% comm_monoid_set.reindex_nontrivial
tff(fact_2468_comm__monoid__set_Oreindex,axiom,
! [C: $tType,A: $tType,B: $tType,F: fun(A,fun(A,A)),Z2: A,H: fun(B,C),A6: set(B),G: fun(C,A)] :
( groups778175481326437816id_set(A,F,Z2)
=> ( inj_on(B,C,H,A6)
=> ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups_comm_monoid_F(A,C,F,Z2),G),aa(set(B),set(C),aa(fun(B,C),fun(set(B),set(C)),image2(B,C),H),A6)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),aa(fun(B,C),fun(B,A),aa(fun(C,A),fun(fun(B,C),fun(B,A)),comp(C,A,B),G),H)),A6) ) ) ) ).
% comm_monoid_set.reindex
tff(fact_2469_comm__monoid__set_OatLeast__Suc__atMost__Suc__shift,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,G: fun(nat,A),M4: nat,N4: nat] :
( groups778175481326437816id_set(A,F,Z2)
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M4),aa(nat,nat,suc,N4))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),suc)),set_or1337092689740270186AtMost(nat,M4,N4)) ) ) ).
% comm_monoid_set.atLeast_Suc_atMost_Suc_shift
tff(fact_2470_comm__monoid__set_OatLeast__Suc__lessThan__Suc__shift,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,G: fun(nat,A),M4: nat,N4: nat] :
( groups778175481326437816id_set(A,F,Z2)
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M4),aa(nat,nat,suc,N4))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),suc)),set_or7035219750837199246ssThan(nat,M4,N4)) ) ) ).
% comm_monoid_set.atLeast_Suc_lessThan_Suc_shift
tff(fact_2471_comm__monoid__set_OatLeastAtMost__shift__bounds,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,G: fun(nat,A),M4: nat,K: nat,N4: nat] :
( groups778175481326437816id_set(A,F,Z2)
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M4),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N4),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),aa(nat,fun(nat,nat),plus_plus(nat),K))),set_or1337092689740270186AtMost(nat,M4,N4)) ) ) ).
% comm_monoid_set.atLeastAtMost_shift_bounds
tff(fact_2472_comm__monoid__set_OatLeastLessThan__shift__bounds,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,G: fun(nat,A),M4: nat,K: nat,N4: nat] :
( groups778175481326437816id_set(A,F,Z2)
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M4),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N4),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),aa(nat,fun(nat,nat),plus_plus(nat),K))),set_or7035219750837199246ssThan(nat,M4,N4)) ) ) ).
% comm_monoid_set.atLeastLessThan_shift_bounds
tff(fact_2473_comm__monoid__set_Ogroup,axiom,
! [C: $tType,A: $tType,B: $tType,F: fun(A,fun(A,A)),Z2: A,S3: set(B),T3: set(C),G: fun(B,C),H: fun(B,A)] :
( groups778175481326437816id_set(A,F,Z2)
=> ( finite_finite2(B,S3)
=> ( finite_finite2(C,T3)
=> ( pp(aa(set(C),bool,aa(set(C),fun(set(C),bool),ord_less_eq(set(C)),aa(set(B),set(C),aa(fun(B,C),fun(set(B),set(C)),image2(B,C),G),S3)),T3))
=> ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups_comm_monoid_F(A,C,F,Z2),aa(fun(B,A),fun(C,A),aa(fun(B,C),fun(fun(B,A),fun(C,A)),aa(set(B),fun(fun(B,C),fun(fun(B,A),fun(C,A))),aa(A,fun(set(B),fun(fun(B,C),fun(fun(B,A),fun(C,A)))),aTP_Lamp_uk(fun(A,fun(A,A)),fun(A,fun(set(B),fun(fun(B,C),fun(fun(B,A),fun(C,A))))),F),Z2),S3),G),H)),T3) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),H),S3) ) ) ) ) ) ).
% comm_monoid_set.group
tff(fact_2474_comm__monoid__set_OPlus,axiom,
! [B: $tType,A: $tType,C: $tType,F: fun(A,fun(A,A)),Z2: A,A6: set(B),B6: set(C),G: fun(sum_sum(B,C),A)] :
( groups778175481326437816id_set(A,F,Z2)
=> ( finite_finite2(B,A6)
=> ( finite_finite2(C,B6)
=> ( aa(set(sum_sum(B,C)),A,aa(fun(sum_sum(B,C),A),fun(set(sum_sum(B,C)),A),groups_comm_monoid_F(A,sum_sum(B,C),F,Z2),G),sum_Plus(B,C,A6,B6)) = aa(A,A,aa(A,fun(A,A),F,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),aa(fun(B,sum_sum(B,C)),fun(B,A),aa(fun(sum_sum(B,C),A),fun(fun(B,sum_sum(B,C)),fun(B,A)),comp(sum_sum(B,C),A,B),G),sum_Inl(B,C))),A6)),aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups_comm_monoid_F(A,C,F,Z2),aa(fun(C,sum_sum(B,C)),fun(C,A),aa(fun(sum_sum(B,C),A),fun(fun(C,sum_sum(B,C)),fun(C,A)),comp(sum_sum(B,C),A,C),G),sum_Inr(C,B))),B6)) ) ) ) ) ).
% comm_monoid_set.Plus
tff(fact_2475_comm__monoid__set_OUnion__disjoint,axiom,
! [A: $tType,B: $tType,F: fun(A,fun(A,A)),Z2: A,C5: set(set(B)),G: fun(B,A)] :
( groups778175481326437816id_set(A,F,Z2)
=> ( ! [X: set(B)] :
( pp(aa(set(set(B)),bool,member(set(B),X),C5))
=> finite_finite2(B,X) )
=> ( ! [X: set(B)] :
( pp(aa(set(set(B)),bool,member(set(B),X),C5))
=> ! [Xa: set(B)] :
( pp(aa(set(set(B)),bool,member(set(B),Xa),C5))
=> ( ( X != Xa )
=> ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),X),Xa) = bot_bot(set(B)) ) ) ) )
=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),G),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),C5)) = aa(set(set(B)),A,aa(fun(B,A),fun(set(set(B)),A),aa(fun(fun(B,A),fun(set(B),A)),fun(fun(B,A),fun(set(set(B)),A)),aa(fun(fun(set(B),A),fun(set(set(B)),A)),fun(fun(fun(B,A),fun(set(B),A)),fun(fun(B,A),fun(set(set(B)),A))),comp(fun(set(B),A),fun(set(set(B)),A),fun(B,A)),groups_comm_monoid_F(A,set(B),F,Z2)),groups_comm_monoid_F(A,B,F,Z2)),G),C5) ) ) ) ) ).
% comm_monoid_set.Union_disjoint
tff(fact_2476_comm__monoid__set_OatLeast0__atMost__Suc__shift,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,G: fun(nat,A),N4: nat] :
( groups778175481326437816id_set(A,F,Z2)
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N4))) = aa(A,A,aa(A,fun(A,A),F,aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),suc)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N4))) ) ) ).
% comm_monoid_set.atLeast0_atMost_Suc_shift
tff(fact_2477_comm__monoid__set_OatLeast0__lessThan__Suc__shift,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,G: fun(nat,A),N4: nat] :
( groups778175481326437816id_set(A,F,Z2)
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N4))) = aa(A,A,aa(A,fun(A,A),F,aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),suc)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N4))) ) ) ).
% comm_monoid_set.atLeast0_lessThan_Suc_shift
tff(fact_2478_surj__prod__encode,axiom,
aa(set(product_prod(nat,nat)),set(nat),aa(fun(product_prod(nat,nat),nat),fun(set(product_prod(nat,nat)),set(nat)),image2(product_prod(nat,nat),nat),nat_prod_encode),top_top(set(product_prod(nat,nat)))) = top_top(set(nat)) ).
% surj_prod_encode
tff(fact_2479_comm__monoid__set_OatLeastLessThan__shift__0,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,G: fun(nat,A),M4: nat,N4: nat] :
( groups778175481326437816id_set(A,F,Z2)
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),set_or7035219750837199246ssThan(nat,M4,N4)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),aa(nat,fun(nat,nat),plus_plus(nat),M4))),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),N4),M4))) ) ) ).
% comm_monoid_set.atLeastLessThan_shift_0
tff(fact_2480_comm__monoid__set_OUNION__disjoint,axiom,
! [C: $tType,A: $tType,B: $tType,F: fun(A,fun(A,A)),Z2: A,I4: set(B),A6: fun(B,set(C)),G: fun(C,A)] :
( groups778175481326437816id_set(A,F,Z2)
=> ( finite_finite2(B,I4)
=> ( ! [X: B] :
( pp(aa(set(B),bool,member(B,X),I4))
=> finite_finite2(C,aa(B,set(C),A6,X)) )
=> ( ! [X: B] :
( pp(aa(set(B),bool,member(B,X),I4))
=> ! [Xa: B] :
( pp(aa(set(B),bool,member(B,Xa),I4))
=> ( ( X != Xa )
=> ( aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),aa(B,set(C),A6,X)),aa(B,set(C),A6,Xa)) = bot_bot(set(C)) ) ) ) )
=> ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups_comm_monoid_F(A,C,F,Z2),G),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),aa(fun(B,set(C)),fun(set(B),set(set(C))),image2(B,set(C)),A6),I4))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,F,Z2),aa(fun(C,A),fun(B,A),aa(fun(B,set(C)),fun(fun(C,A),fun(B,A)),aa(A,fun(fun(B,set(C)),fun(fun(C,A),fun(B,A))),aTP_Lamp_ul(fun(A,fun(A,A)),fun(A,fun(fun(B,set(C)),fun(fun(C,A),fun(B,A)))),F),Z2),A6),G)),I4) ) ) ) ) ) ).
% comm_monoid_set.UNION_disjoint
tff(fact_2481_comm__monoid__set_OatLeast__atMost__pred__shift,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,G: fun(nat,A),M4: nat,N4: nat] :
( groups778175481326437816id_set(A,F,Z2)
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),aTP_Lamp_lm(nat,nat))),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M4),aa(nat,nat,suc,N4))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),set_or1337092689740270186AtMost(nat,M4,N4)) ) ) ).
% comm_monoid_set.atLeast_atMost_pred_shift
tff(fact_2482_comm__monoid__set_OatLeast__lessThan__pred__shift,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,G: fun(nat,A),M4: nat,N4: nat] :
( groups778175481326437816id_set(A,F,Z2)
=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),G),aTP_Lamp_lm(nat,nat))),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M4),aa(nat,nat,suc,N4))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),G),set_or7035219750837199246ssThan(nat,M4,N4)) ) ) ).
% comm_monoid_set.atLeast_lessThan_pred_shift
tff(fact_2483_uminus__sum__list__map,axiom,
! [A: $tType,B: $tType] :
( ab_group_add(A)
=> ! [F: fun(B,A),Xs: list(B)] : aa(A,A,uminus_uminus(A),groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,F),Xs))) = groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,aa(fun(B,A),fun(B,A),aa(fun(A,A),fun(fun(B,A),fun(B,A)),comp(A,A,B),uminus_uminus(A)),F)),Xs)) ) ).
% uminus_sum_list_map
tff(fact_2484_comm__monoid__set_Otriangle__reindex,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,G: fun(nat,fun(nat,A)),N4: nat] :
( groups778175481326437816id_set(A,F,Z2)
=> ( aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups_comm_monoid_F(A,product_prod(nat,nat),F,Z2),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G)),aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_me(nat,fun(nat,fun(nat,bool)),N4)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),aa(fun(nat,fun(nat,A)),fun(nat,A),aa(A,fun(fun(nat,fun(nat,A)),fun(nat,A)),aTP_Lamp_un(fun(A,fun(A,A)),fun(A,fun(fun(nat,fun(nat,A)),fun(nat,A))),F),Z2),G)),aa(nat,set(nat),set_ord_lessThan(nat),N4)) ) ) ).
% comm_monoid_set.triangle_reindex
tff(fact_2485_comm__monoid__set_Otriangle__reindex__eq,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,G: fun(nat,fun(nat,A)),N4: nat] :
( groups778175481326437816id_set(A,F,Z2)
=> ( aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups_comm_monoid_F(A,product_prod(nat,nat),F,Z2),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G)),aa(fun(product_prod(nat,nat),bool),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aTP_Lamp_lv(nat,fun(nat,fun(nat,bool)),N4)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),aa(fun(nat,fun(nat,A)),fun(nat,A),aa(A,fun(fun(nat,fun(nat,A)),fun(nat,A)),aTP_Lamp_un(fun(A,fun(A,A)),fun(A,fun(fun(nat,fun(nat,A)),fun(nat,A))),F),Z2),G)),aa(nat,set(nat),set_ord_atMost(nat),N4)) ) ) ).
% comm_monoid_set.triangle_reindex_eq
tff(fact_2486_comm__monoid__set_OatLeast__int__lessThan__int__shift,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,G: fun(int,A),M4: nat,N4: nat] :
( groups778175481326437816id_set(A,F,Z2)
=> ( aa(set(int),A,aa(fun(int,A),fun(set(int),A),groups_comm_monoid_F(A,int,F,Z2),G),set_or7035219750837199246ssThan(int,aa(nat,int,semiring_1_of_nat(int),M4),aa(nat,int,semiring_1_of_nat(int),N4))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),aa(fun(nat,int),fun(nat,A),aa(fun(int,A),fun(fun(nat,int),fun(nat,A)),comp(int,A,nat),G),semiring_1_of_nat(int))),set_or7035219750837199246ssThan(nat,M4,N4)) ) ) ).
% comm_monoid_set.atLeast_int_lessThan_int_shift
tff(fact_2487_comm__monoid__set_OatLeast__int__atMost__int__shift,axiom,
! [A: $tType,F: fun(A,fun(A,A)),Z2: A,G: fun(int,A),M4: nat,N4: nat] :
( groups778175481326437816id_set(A,F,Z2)
=> ( aa(set(int),A,aa(fun(int,A),fun(set(int),A),groups_comm_monoid_F(A,int,F,Z2),G),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),M4),aa(nat,int,semiring_1_of_nat(int),N4))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,F,Z2),aa(fun(nat,int),fun(nat,A),aa(fun(int,A),fun(fun(nat,int),fun(nat,A)),comp(int,A,nat),G),semiring_1_of_nat(int))),set_or1337092689740270186AtMost(nat,M4,N4)) ) ) ).
% comm_monoid_set.atLeast_int_atMost_int_shift
tff(fact_2488_type__definition__unit,axiom,
type_definition(product_unit,bool,product_Rep_unit,product_Abs_unit,aa(set(bool),set(bool),insert(bool,fTrue),bot_bot(set(bool)))) ).
% type_definition_unit
tff(fact_2489_Rep__unit__inject,axiom,
! [X2: product_unit,Y2: product_unit] :
( ( pp(aa(product_unit,bool,product_Rep_unit,X2))
<=> pp(aa(product_unit,bool,product_Rep_unit,Y2)) )
<=> ( X2 = Y2 ) ) ).
% Rep_unit_inject
tff(fact_2490_Rep__unit__induct,axiom,
! [Y2: bool,P2: fun(bool,bool)] :
( pp(aa(set(bool),bool,member(bool,Y2),aa(set(bool),set(bool),insert(bool,fTrue),bot_bot(set(bool)))))
=> ( ! [X: product_unit] : pp(aa(bool,bool,P2,aa(product_unit,bool,product_Rep_unit,X)))
=> pp(aa(bool,bool,P2,Y2)) ) ) ).
% Rep_unit_induct
tff(fact_2491_Rep__unit__cases,axiom,
! [Y2: bool] :
( pp(aa(set(bool),bool,member(bool,Y2),aa(set(bool),set(bool),insert(bool,fTrue),bot_bot(set(bool)))))
=> ~ ! [X: product_unit] :
( pp(Y2)
<=> ~ pp(aa(product_unit,bool,product_Rep_unit,X)) ) ) ).
% Rep_unit_cases
tff(fact_2492_Rep__unit,axiom,
! [X2: product_unit] : pp(aa(set(bool),bool,member(bool,aa(product_unit,bool,product_Rep_unit,X2)),aa(set(bool),set(bool),insert(bool,fTrue),bot_bot(set(bool))))) ).
% Rep_unit
tff(fact_2493_Rep__unit__inverse,axiom,
! [X2: product_unit] : aa(bool,product_unit,product_Abs_unit,aa(product_unit,bool,product_Rep_unit,X2)) = X2 ).
% Rep_unit_inverse
tff(fact_2494_Abs__unit__inverse,axiom,
! [Y2: bool] :
( pp(aa(set(bool),bool,member(bool,Y2),aa(set(bool),set(bool),insert(bool,fTrue),bot_bot(set(bool)))))
=> ( pp(aa(product_unit,bool,product_Rep_unit,aa(bool,product_unit,product_Abs_unit,Y2)))
<=> pp(Y2) ) ) ).
% Abs_unit_inverse
tff(fact_2495_surj__prod__decode,axiom,
aa(set(nat),set(product_prod(nat,nat)),aa(fun(nat,product_prod(nat,nat)),fun(set(nat),set(product_prod(nat,nat))),image2(nat,product_prod(nat,nat)),nat_prod_decode),top_top(set(nat))) = top_top(set(product_prod(nat,nat))) ).
% surj_prod_decode
tff(fact_2496_select__def,axiom,
! [A: $tType,Xs: list(A)] : select(A,Xs) = product_scomp(product_prod(code_natural,code_natural),code_natural,product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural)),range(code_natural_of_nat(aa(list(A),nat,size_size(list(A)),Xs))),aTP_Lamp_uo(list(A),fun(code_natural,fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural)))),Xs)) ).
% select_def
tff(fact_2497_folding__idem_Ocomp__fun__idem,axiom,
! [B: $tType,A: $tType,F: fun(A,fun(B,B)),X2: A] :
( finite_folding_idem(A,B,F)
=> ( aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X2)),aa(A,fun(B,B),F,X2)) = aa(A,fun(B,B),F,X2) ) ) ).
% folding_idem.comp_fun_idem
tff(fact_2498_pair__less__def,axiom,
fun_pair_less = lex_prod(nat,nat,less_than,less_than) ).
% pair_less_def
tff(fact_2499_map__upds__fold__map__upd,axiom,
! [A: $tType,B: $tType,M4: fun(A,option(B)),Ks: list(A),Vs2: list(B)] : map_upds(A,B,M4,Ks,Vs2) = foldl(fun(A,option(B)),product_prod(A,B),aTP_Lamp_uq(fun(A,option(B)),fun(product_prod(A,B),fun(A,option(B)))),M4,zip(A,B,Ks,Vs2)) ).
% map_upds_fold_map_upd
tff(fact_2500_rec__nat__0__imp,axiom,
! [A: $tType,F: fun(nat,A),F1: A,F22: fun(nat,fun(A,A))] :
( ( F = rec_nat(A,F1,F22) )
=> ( aa(nat,A,F,zero_zero(nat)) = F1 ) ) ).
% rec_nat_0_imp
tff(fact_2501_rec__nat__Suc__imp,axiom,
! [A: $tType,F: fun(nat,A),F1: A,F22: fun(nat,fun(A,A)),N4: nat] :
( ( F = rec_nat(A,F1,F22) )
=> ( aa(nat,A,F,aa(nat,nat,suc,N4)) = aa(A,A,aa(nat,fun(A,A),F22,N4),aa(nat,A,F,N4)) ) ) ).
% rec_nat_Suc_imp
tff(fact_2502_list__decode_Oelims,axiom,
! [X2: nat,Y2: list(nat)] :
( ( aa(nat,list(nat),nat_list_decode,X2) = Y2 )
=> ( ( ( X2 = zero_zero(nat) )
=> ( Y2 != nil(nat) ) )
=> ~ ! [N2: nat] :
( ( X2 = aa(nat,nat,suc,N2) )
=> ( Y2 != aa(product_prod(nat,nat),list(nat),aa(fun(nat,fun(nat,list(nat))),fun(product_prod(nat,nat),list(nat)),product_case_prod(nat,nat,list(nat)),aTP_Lamp_ur(nat,fun(nat,list(nat)))),aa(nat,product_prod(nat,nat),nat_prod_decode,N2)) ) ) ) ) ).
% list_decode.elims
tff(fact_2503_nat__to__rat__surj__def,axiom,
! [N4: nat] : aa(nat,rat,nat_to_rat_surj,N4) = aa(product_prod(nat,nat),rat,aa(fun(nat,fun(nat,rat)),fun(product_prod(nat,nat),rat),product_case_prod(nat,nat,rat),aTP_Lamp_us(nat,fun(nat,rat))),aa(nat,product_prod(nat,nat),nat_prod_decode,N4)) ).
% nat_to_rat_surj_def
tff(fact_2504_list__decode_Osimps_I2_J,axiom,
! [N4: nat] : aa(nat,list(nat),nat_list_decode,aa(nat,nat,suc,N4)) = aa(product_prod(nat,nat),list(nat),aa(fun(nat,fun(nat,list(nat))),fun(product_prod(nat,nat),list(nat)),product_case_prod(nat,nat,list(nat)),aTP_Lamp_ur(nat,fun(nat,list(nat)))),aa(nat,product_prod(nat,nat),nat_prod_decode,N4)) ).
% list_decode.simps(2)
tff(fact_2505_list__decode_Opelims,axiom,
! [X2: nat,Y2: list(nat)] :
( ( aa(nat,list(nat),nat_list_decode,X2) = Y2 )
=> ( pp(aa(nat,bool,accp(nat,nat_list_decode_rel),X2))
=> ( ( ( X2 = zero_zero(nat) )
=> ( ( Y2 = nil(nat) )
=> ~ pp(aa(nat,bool,accp(nat,nat_list_decode_rel),zero_zero(nat))) ) )
=> ~ ! [N2: nat] :
( ( X2 = aa(nat,nat,suc,N2) )
=> ( ( Y2 = aa(product_prod(nat,nat),list(nat),aa(fun(nat,fun(nat,list(nat))),fun(product_prod(nat,nat),list(nat)),product_case_prod(nat,nat,list(nat)),aTP_Lamp_ur(nat,fun(nat,list(nat)))),aa(nat,product_prod(nat,nat),nat_prod_decode,N2)) )
=> ~ pp(aa(nat,bool,accp(nat,nat_list_decode_rel),aa(nat,nat,suc,N2))) ) ) ) ) ) ).
% list_decode.pelims
tff(fact_2506_folding_Ointro,axiom,
! [B: $tType,A: $tType,F: fun(A,fun(B,B))] :
( ! [Y: A,X: A] : aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,Y)),aa(A,fun(B,B),F,X)) = aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X)),aa(A,fun(B,B),F,Y))
=> finite_folding(A,B,F) ) ).
% folding.intro
tff(fact_2507_folding_Ocomp__fun__commute,axiom,
! [B: $tType,A: $tType,F: fun(A,fun(B,B)),Y2: A,X2: A] :
( finite_folding(A,B,F)
=> ( aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,Y2)),aa(A,fun(B,B),F,X2)) = aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X2)),aa(A,fun(B,B),F,Y2)) ) ) ).
% folding.comp_fun_commute
tff(fact_2508_surj__list__decode,axiom,
aa(set(nat),set(list(nat)),aa(fun(nat,list(nat)),fun(set(nat),set(list(nat))),image2(nat,list(nat)),nat_list_decode),top_top(set(nat))) = top_top(set(list(nat))) ).
% surj_list_decode
tff(fact_2509_list__decode_Opsimps_I2_J,axiom,
! [N4: nat] :
( pp(aa(nat,bool,accp(nat,nat_list_decode_rel),aa(nat,nat,suc,N4)))
=> ( aa(nat,list(nat),nat_list_decode,aa(nat,nat,suc,N4)) = aa(product_prod(nat,nat),list(nat),aa(fun(nat,fun(nat,list(nat))),fun(product_prod(nat,nat),list(nat)),product_case_prod(nat,nat,list(nat)),aTP_Lamp_ur(nat,fun(nat,list(nat)))),aa(nat,product_prod(nat,nat),nat_prod_decode,N4)) ) ) ).
% list_decode.psimps(2)
tff(fact_2510_folding__def,axiom,
! [B: $tType,A: $tType,F: fun(A,fun(B,B))] :
( finite_folding(A,B,F)
<=> ! [Y4: A,X5: A] : aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,Y4)),aa(A,fun(B,B),F,X5)) = aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X5)),aa(A,fun(B,B),F,Y4)) ) ).
% folding_def
tff(fact_2511_folding__idem__on_Ocomp__fun__idem__on,axiom,
! [B: $tType,A: $tType,S3: set(A),F: fun(A,fun(B,B)),X2: A,Y2: A] :
( finite1890593828518410140dem_on(A,B,S3,F)
=> ( pp(aa(set(A),bool,member(A,X2),S3))
=> ( pp(aa(set(A),bool,member(A,Y2),S3))
=> ( aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X2)),aa(A,fun(B,B),F,X2)) = aa(A,fun(B,B),F,X2) ) ) ) ) ).
% folding_idem_on.comp_fun_idem_on
tff(fact_2512_Heap__lub__def,axiom,
! [A: $tType] : heap_Heap_lub(A) = partial_img_lub(heap_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),fun(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
tff(fact_2513_heap_Oconst__mono,axiom,
! [B: $tType,A: $tType,Ord: fun(B,fun(B,bool)),C3: heap_Heap(A)] : comple7038119648293358887notone(B,heap_Heap(A),Ord,heap_Heap_ord(A),aTP_Lamp_ut(heap_Heap(A),fun(B,heap_Heap(A)),C3)) ).
% heap.const_mono
tff(fact_2514_Ref_Oget__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [H: heap_ext(product_unit)] : get2(A,H) = aa(fun(ref(A),nat),fun(ref(A),A),aa(fun(nat,A),fun(fun(ref(A),nat),fun(ref(A),A)),comp(nat,A,ref(A)),aa(fun(nat,nat),fun(nat,A),aa(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),comp(nat,A,nat),from_nat(A)),aa(typerep,fun(nat,nat),refs(product_unit,H),typerep_typerep(A,type2(A))))),addr_of_ref(A)) ) ).
% Ref.get_def
tff(fact_2515_Array_Oget__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [H: heap_ext(product_unit),A3: array(A)] : aa(array(A),list(A),get(A,H),A3) = aa(list(nat),list(A),map(nat,A,from_nat(A)),aa(nat,list(nat),aa(typerep,fun(nat,list(nat)),arrays(product_unit,H),typerep_typerep(A,type2(A))),addr_of_array(A,A3))) ) ).
% Array.get_def
tff(fact_2516_surj__sum__encode,axiom,
aa(set(sum_sum(nat,nat)),set(nat),aa(fun(sum_sum(nat,nat),nat),fun(set(sum_sum(nat,nat)),set(nat)),image2(sum_sum(nat,nat),nat),nat_sum_encode),top_top(set(sum_sum(nat,nat)))) = top_top(set(nat)) ).
% surj_sum_encode
tff(fact_2517_Array_Onoteq__def,axiom,
! [A: $tType,B: $tType] :
( ( heap(B)
& heap(A) )
=> ! [R: array(A),S: array(B)] :
( noteq(A,B,R,S)
<=> ( ( typerep_typerep(A,type2(A)) != typerep_typerep(B,type2(B)) )
| ( addr_of_array(A,R) != addr_of_array(B,S) ) ) ) ) ).
% Array.noteq_def
tff(fact_2518_Ref_Onoteq__def,axiom,
! [A: $tType,B: $tType] :
( ( heap(B)
& heap(A) )
=> ! [R: ref(A),S: ref(B)] :
( noteq2(A,B,R,S)
<=> ( ( typerep_typerep(A,type2(A)) != typerep_typerep(B,type2(B)) )
| ( aa(ref(A),nat,addr_of_ref(A),R) != aa(ref(B),nat,addr_of_ref(B),S) ) ) ) ) ).
% Ref.noteq_def
tff(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
tff(fact_2520_img__lub__def,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType,F: fun(A,C),G: fun(D,B),Lub: fun(set(C),D),X4: set(A)] : aa(set(A),B,partial_img_lub(A,C,D,B,F,G,Lub),X4) = aa(D,B,G,aa(set(C),D,Lub,aa(set(A),set(C),aa(fun(A,C),fun(set(A),set(C)),image2(A,C),F),X4))) ).
% img_lub_def
tff(fact_2521_typerep__Heap__def,axiom,
! [A: $tType] :
( typerep2(A)
=> ! [T3: itself(heap_Heap(A))] : typerep_typerep(heap_Heap(A),T3) = typerep3(literal2(fFalse,fFalse,fFalse,fTrue,fFalse,fFalse,fTrue,literal2(fTrue,fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,literal2(fTrue,fFalse,fFalse,fFalse,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fFalse,fFalse,fTrue,fTrue,fTrue,literal2(fTrue,fTrue,fTrue,fTrue,fTrue,fFalse,fTrue,literal2(fTrue,fFalse,fTrue,fTrue,fFalse,fFalse,fTrue,literal2(fTrue,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fFalse,fTrue,fTrue,fTrue,fFalse,fTrue,fTrue,literal2(fTrue,fFalse,fFalse,fFalse,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,literal2(fFalse,fTrue,fTrue,fTrue,fFalse,fTrue,fFalse,literal2(fFalse,fFalse,fFalse,fTrue,fFalse,fFalse,fTrue,literal2(fTrue,fFalse,fTrue,fFalse,fFalse,fTrue,fTrue,literal2(fTrue,fFalse,fFalse,fFalse,fFalse,fTrue,fTrue,literal2(fFalse,fFalse,fFalse,fFalse,fTrue,fTrue,fTrue,zero_zero(literal)))))))))))))))),aa(list(typerep),list(typerep),aa(typerep,fun(list(typerep),list(typerep)),cons(typerep),typerep_typerep(A,type2(A))),nil(typerep))) ) ).
% typerep_Heap_def
tff(fact_2522_Array_Oset__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [A3: array(A),X2: list(A)] : set2(A,A3,X2) = arrays_update(product_unit,aa(list(A),fun(fun(typerep,fun(nat,list(nat))),fun(typerep,fun(nat,list(nat)))),aTP_Lamp_uu(array(A),fun(list(A),fun(fun(typerep,fun(nat,list(nat))),fun(typerep,fun(nat,list(nat))))),A3),X2)) ) ).
% Array.set_def
tff(fact_2523_Ref_Oset__def,axiom,
! [A: $tType] :
( heap(A)
=> ! [R: ref(A),X2: A] : set4(A,R,X2) = refs_update(product_unit,aa(A,fun(fun(typerep,fun(nat,nat)),fun(typerep,fun(nat,nat))),aTP_Lamp_uv(ref(A),fun(A,fun(fun(typerep,fun(nat,nat)),fun(typerep,fun(nat,nat)))),R),X2)) ) ).
% Ref.set_def
tff(fact_2524_surj__list__encode,axiom,
aa(set(list(nat)),set(nat),aa(fun(list(nat),nat),fun(set(list(nat)),set(nat)),image2(list(nat),nat),nat_list_encode),top_top(set(list(nat)))) = top_top(set(nat)) ).
% surj_list_encode
tff(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
tff(fact_2526_admissible__image,axiom,
! [A: $tType,B: $tType,Le: fun(A,fun(A,bool)),Lub2: fun(set(A),A),P2: fun(B,bool),G: fun(A,B),F: fun(B,A)] :
( partia7178651479351089652itions(A,Le,Lub2)
=> ( comple1908693960933563346ssible(A,Lub2,Le,aa(fun(A,B),fun(A,bool),aa(fun(B,bool),fun(fun(A,B),fun(A,bool)),comp(B,bool,A),P2),G))
=> ( ! [X: B,Y: B] :
( ( aa(B,A,F,X) = aa(B,A,F,Y) )
=> ( X = Y ) )
=> ( ! [X: A] : aa(B,A,F,aa(A,B,G,X)) = X
=> comple1908693960933563346ssible(B,partial_img_lub(B,A,A,B,F,G,Lub2),partial_img_ord(B,A,bool,F,Le),P2) ) ) ) ) ).
% admissible_image
tff(fact_2527_surj__sum__decode,axiom,
aa(set(nat),set(sum_sum(nat,nat)),aa(fun(nat,sum_sum(nat,nat)),fun(set(nat),set(sum_sum(nat,nat))),image2(nat,sum_sum(nat,nat)),nat_sum_decode),top_top(set(nat))) = top_top(set(sum_sum(nat,nat))) ).
% surj_sum_decode
tff(fact_2528_Heap__Monad_Obind__mono,axiom,
! [D: $tType,B: $tType,A: $tType,C: $tType,B6: fun(fun(A,heap_Heap(B)),heap_Heap(C)),C5: fun(C,fun(fun(A,heap_Heap(B)),heap_Heap(D)))] :
( comple7038119648293358887notone(fun(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(fun(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),aa(C,fun(fun(A,heap_Heap(B)),heap_Heap(D)),C5,Y))
=> comple7038119648293358887notone(fun(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),aa(fun(C,fun(fun(A,heap_Heap(B)),heap_Heap(D))),fun(fun(A,heap_Heap(B)),heap_Heap(D)),aTP_Lamp_ux(fun(fun(A,heap_Heap(B)),heap_Heap(C)),fun(fun(C,fun(fun(A,heap_Heap(B)),heap_Heap(D))),fun(fun(A,heap_Heap(B)),heap_Heap(D))),B6),C5)) ) ) ).
% Heap_Monad.bind_mono
tff(fact_2529_heap__step__admissible,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,P2: fun(D,fun(A,fun(C,fun(B,bool)))),X2: D] : comple1908693960933563346ssible(fun(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)))),aa(D,fun(fun(A,option(product_prod(B,C))),bool),aTP_Lamp_uy(fun(D,fun(A,fun(C,fun(B,bool)))),fun(D,fun(fun(A,option(product_prod(B,C))),bool)),P2),X2)) ).
% heap_step_admissible
tff(fact_2530_semilattice__map2,axiom,
! [A: $tType,F: fun(A,fun(A,A))] :
( semilattice(A,F)
=> semilattice(list(A),aTP_Lamp_uz(fun(A,fun(A,A)),fun(list(A),fun(list(A),list(A))),F)) ) ).
% semilattice_map2
tff(fact_2531_Heap__ord__def,axiom,
! [A: $tType] : heap_Heap_ord(A) = partial_img_ord(heap_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),bool,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
tff(fact_2532_select__convs_I3_J,axiom,
! [Z: $tType,Arrays: fun(typerep,fun(nat,list(nat))),Refs: fun(typerep,fun(nat,nat)),Lim: nat,More: Z] : lim(Z,heap_ext2(Z,Arrays,Refs,Lim,More)) = Lim ).
% select_convs(3)
tff(fact_2533_admissible__heap,axiom,
! [B: $tType,A: $tType,P2: fun(A,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(B,bool))))] : comple1908693960933563346ssible(fun(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)),aTP_Lamp_va(fun(A,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(B,bool)))),fun(fun(A,heap_Heap(B)),bool),P2)) ).
% admissible_heap
tff(fact_2534_heap_Ofixp__induct__uc,axiom,
! [B: $tType,A: $tType,C: $tType,U4: fun(C,fun(B,heap_Heap(A))),F6: fun(C,C),C5: fun(fun(B,heap_Heap(A)),C),F: C,P2: fun(fun(B,heap_Heap(A)),bool)] :
( ! [X: B] : comple7038119648293358887notone(fun(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),aa(B,fun(fun(B,heap_Heap(A)),heap_Heap(A)),aa(fun(fun(B,heap_Heap(A)),C),fun(B,fun(fun(B,heap_Heap(A)),heap_Heap(A))),aa(fun(C,C),fun(fun(fun(B,heap_Heap(A)),C),fun(B,fun(fun(B,heap_Heap(A)),heap_Heap(A)))),aTP_Lamp_vb(fun(C,fun(B,heap_Heap(A))),fun(fun(C,C),fun(fun(fun(B,heap_Heap(A)),C),fun(B,fun(fun(B,heap_Heap(A)),heap_Heap(A))))),U4),F6),C5),X))
=> ( ( F = aa(fun(B,heap_Heap(A)),C,C5,comple187402453842119260l_fixp(fun(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)),aa(fun(fun(B,heap_Heap(A)),C),fun(fun(B,heap_Heap(A)),fun(B,heap_Heap(A))),aa(fun(C,C),fun(fun(fun(B,heap_Heap(A)),C),fun(fun(B,heap_Heap(A)),fun(B,heap_Heap(A)))),aTP_Lamp_vc(fun(C,fun(B,heap_Heap(A))),fun(fun(C,C),fun(fun(fun(B,heap_Heap(A)),C),fun(fun(B,heap_Heap(A)),fun(B,heap_Heap(A))))),U4),F6),C5))) )
=> ( ! [F3: fun(B,heap_Heap(A))] : aa(C,fun(B,heap_Heap(A)),U4,aa(fun(B,heap_Heap(A)),C,C5,F3)) = F3
=> ( comple1908693960933563346ssible(fun(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)
=> ( pp(aa(fun(B,heap_Heap(A)),bool,P2,aTP_Lamp_vd(B,heap_Heap(A))))
=> ( ! [F3: C] :
( pp(aa(fun(B,heap_Heap(A)),bool,P2,aa(C,fun(B,heap_Heap(A)),U4,F3)))
=> pp(aa(fun(B,heap_Heap(A)),bool,P2,aa(C,fun(B,heap_Heap(A)),U4,aa(C,C,F6,F3)))) )
=> pp(aa(fun(B,heap_Heap(A)),bool,P2,aa(C,fun(B,heap_Heap(A)),U4,F))) ) ) ) ) ) ) ).
% heap.fixp_induct_uc
tff(fact_2535_fixp__induct__heap,axiom,
! [C: $tType,B: $tType,A: $tType,U4: fun(C,fun(B,heap_Heap(A))),F6: fun(C,C),C5: fun(fun(B,heap_Heap(A)),C),F: C,P2: fun(B,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(A,bool)))),X2: B,H: heap_ext(product_unit),H2: heap_ext(product_unit),R: A] :
( ! [X: B] : comple7038119648293358887notone(fun(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),aa(B,fun(fun(B,heap_Heap(A)),heap_Heap(A)),aa(fun(fun(B,heap_Heap(A)),C),fun(B,fun(fun(B,heap_Heap(A)),heap_Heap(A))),aa(fun(C,C),fun(fun(fun(B,heap_Heap(A)),C),fun(B,fun(fun(B,heap_Heap(A)),heap_Heap(A)))),aTP_Lamp_vb(fun(C,fun(B,heap_Heap(A))),fun(fun(C,C),fun(fun(fun(B,heap_Heap(A)),C),fun(B,fun(fun(B,heap_Heap(A)),heap_Heap(A))))),U4),F6),C5),X))
=> ( ( F = aa(fun(B,heap_Heap(A)),C,C5,comple187402453842119260l_fixp(fun(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)),aa(fun(fun(B,heap_Heap(A)),C),fun(fun(B,heap_Heap(A)),fun(B,heap_Heap(A))),aa(fun(C,C),fun(fun(fun(B,heap_Heap(A)),C),fun(fun(B,heap_Heap(A)),fun(B,heap_Heap(A)))),aTP_Lamp_vc(fun(C,fun(B,heap_Heap(A))),fun(fun(C,C),fun(fun(fun(B,heap_Heap(A)),C),fun(fun(B,heap_Heap(A)),fun(B,heap_Heap(A))))),U4),F6),C5))) )
=> ( ! [F3: fun(B,heap_Heap(A))] : aa(C,fun(B,heap_Heap(A)),U4,aa(fun(B,heap_Heap(A)),C,C5,F3)) = F3
=> ( ! [F3: C,X: B,H7: heap_ext(product_unit),H4: heap_ext(product_unit),R4: A] :
( ! [Xa4: B,Ha: heap_ext(product_unit),H_a: heap_ext(product_unit),Ra3: A] :
( heap_effect(A,aa(B,heap_Heap(A),aa(C,fun(B,heap_Heap(A)),U4,F3),Xa4),Ha,H_a,Ra3)
=> pp(aa(A,bool,aa(heap_ext(product_unit),fun(A,bool),aa(heap_ext(product_unit),fun(heap_ext(product_unit),fun(A,bool)),aa(B,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(A,bool))),P2,Xa4),Ha),H_a),Ra3)) )
=> ( heap_effect(A,aa(B,heap_Heap(A),aa(C,fun(B,heap_Heap(A)),U4,aa(C,C,F6,F3)),X),H7,H4,R4)
=> pp(aa(A,bool,aa(heap_ext(product_unit),fun(A,bool),aa(heap_ext(product_unit),fun(heap_ext(product_unit),fun(A,bool)),aa(B,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(A,bool))),P2,X),H7),H4),R4)) ) )
=> ( heap_effect(A,aa(B,heap_Heap(A),aa(C,fun(B,heap_Heap(A)),U4,F),X2),H,H2,R)
=> pp(aa(A,bool,aa(heap_ext(product_unit),fun(A,bool),aa(heap_ext(product_unit),fun(heap_ext(product_unit),fun(A,bool)),aa(B,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(A,bool))),P2,X2),H),H2),R)) ) ) ) ) ) ).
% fixp_induct_heap
tff(fact_2536_heap_Omono__body__fixp,axiom,
! [A: $tType,B: $tType,F6: fun(fun(B,heap_Heap(A)),fun(B,heap_Heap(A)))] :
( ! [X: B] : comple7038119648293358887notone(fun(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),aa(B,fun(fun(B,heap_Heap(A)),heap_Heap(A)),aTP_Lamp_ve(fun(fun(B,heap_Heap(A)),fun(B,heap_Heap(A))),fun(B,fun(fun(B,heap_Heap(A)),heap_Heap(A))),F6),X))
=> ( comple187402453842119260l_fixp(fun(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)),F6) = aa(fun(B,heap_Heap(A)),fun(B,heap_Heap(A)),F6,comple187402453842119260l_fixp(fun(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)),F6)) ) ) ).
% heap.mono_body_fixp
tff(fact_2537_heap_Ofixp__rule__uc,axiom,
! [B: $tType,A: $tType,C: $tType,U4: fun(C,fun(B,heap_Heap(A))),F6: fun(C,C),C5: fun(fun(B,heap_Heap(A)),C),F: C] :
( ! [X: B] : comple7038119648293358887notone(fun(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),aa(B,fun(fun(B,heap_Heap(A)),heap_Heap(A)),aa(fun(fun(B,heap_Heap(A)),C),fun(B,fun(fun(B,heap_Heap(A)),heap_Heap(A))),aa(fun(C,C),fun(fun(fun(B,heap_Heap(A)),C),fun(B,fun(fun(B,heap_Heap(A)),heap_Heap(A)))),aTP_Lamp_vb(fun(C,fun(B,heap_Heap(A))),fun(fun(C,C),fun(fun(fun(B,heap_Heap(A)),C),fun(B,fun(fun(B,heap_Heap(A)),heap_Heap(A))))),U4),F6),C5),X))
=> ( ( F = aa(fun(B,heap_Heap(A)),C,C5,comple187402453842119260l_fixp(fun(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)),aa(fun(fun(B,heap_Heap(A)),C),fun(fun(B,heap_Heap(A)),fun(B,heap_Heap(A))),aa(fun(C,C),fun(fun(fun(B,heap_Heap(A)),C),fun(fun(B,heap_Heap(A)),fun(B,heap_Heap(A)))),aTP_Lamp_vc(fun(C,fun(B,heap_Heap(A))),fun(fun(C,C),fun(fun(fun(B,heap_Heap(A)),C),fun(fun(B,heap_Heap(A)),fun(B,heap_Heap(A))))),U4),F6),C5))) )
=> ( ! [F3: C] : aa(fun(B,heap_Heap(A)),C,C5,aa(C,fun(B,heap_Heap(A)),U4,F3)) = F3
=> ( F = aa(C,C,F6,F) ) ) ) ) ).
% heap.fixp_rule_uc
tff(fact_2538_divmod__step__int__def,axiom,
! [L: num,Qr: product_prod(int,int)] : unique1321980374590559556d_step(int,L,Qr) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_vf(num,fun(int,fun(int,product_prod(int,int))),L)),Qr) ).
% divmod_step_int_def
tff(fact_2539_divmod__step__eq,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [L: num,R: A,Q: A] :
( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),L)),R))
=> ( unique1321980374590559556d_step(A,L,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q),R)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one))),Q)),one_one(A))),aa(A,A,aa(A,fun(A,A),minus_minus(A),R),aa(num,A,numeral_numeral(A),L))) ) )
& ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),L)),R))
=> ( unique1321980374590559556d_step(A,L,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q),R)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one))),Q)),R) ) ) ) ) ).
% divmod_step_eq
tff(fact_2540_divmod__step__def,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [L: num,Qr: product_prod(A,A)] : unique1321980374590559556d_step(A,L,Qr) = aa(product_prod(A,A),product_prod(A,A),aa(fun(A,fun(A,product_prod(A,A))),fun(product_prod(A,A),product_prod(A,A)),product_case_prod(A,A,product_prod(A,A)),aTP_Lamp_vg(num,fun(A,fun(A,product_prod(A,A))),L)),Qr) ) ).
% divmod_step_def
tff(fact_2541_divmod__algorithm__code_I5_J,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [M4: num,N4: num] : unique8689654367752047608divmod(A,bit0(M4),bit0(N4)) = aa(product_prod(A,A),product_prod(A,A),aa(fun(A,fun(A,product_prod(A,A))),fun(product_prod(A,A),product_prod(A,A)),product_case_prod(A,A,product_prod(A,A)),aTP_Lamp_vh(A,fun(A,product_prod(A,A)))),unique8689654367752047608divmod(A,M4,N4)) ) ).
% divmod_algorithm_code(5)
tff(fact_2542_one__mod__numeral,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [N4: num] : modulo_modulo(A,one_one(A),aa(num,A,numeral_numeral(A),N4)) = aa(product_prod(A,A),A,product_snd(A,A),unique8689654367752047608divmod(A,one,N4)) ) ).
% one_mod_numeral
tff(fact_2543_one__div__numeral,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [N4: num] : aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),N4)) = aa(product_prod(A,A),A,product_fst(A,A),unique8689654367752047608divmod(A,one,N4)) ) ).
% one_div_numeral
tff(fact_2544_divmod__algorithm__code_I3_J,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [N4: num] : unique8689654367752047608divmod(A,one,bit0(N4)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),one)) ) ).
% divmod_algorithm_code(3)
tff(fact_2545_divmod__divmod__step,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [M4: num,N4: num] :
( ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M4),N4))
=> ( unique8689654367752047608divmod(A,M4,N4) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),M4)) ) )
& ( ~ pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M4),N4))
=> ( unique8689654367752047608divmod(A,M4,N4) = unique1321980374590559556d_step(A,N4,unique8689654367752047608divmod(A,M4,bit0(N4))) ) ) ) ) ).
% divmod_divmod_step
tff(fact_2546_divmod__algorithm__code_I2_J,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [M4: num] : unique8689654367752047608divmod(A,M4,one) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(num,A,numeral_numeral(A),M4)),zero_zero(A)) ) ).
% divmod_algorithm_code(2)
tff(fact_2547_numeral__div__numeral,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [K: num,L: num] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(num,A,numeral_numeral(A),K)),aa(num,A,numeral_numeral(A),L)) = aa(product_prod(A,A),A,product_fst(A,A),unique8689654367752047608divmod(A,K,L)) ) ).
% numeral_div_numeral
tff(fact_2548_numeral__mod__numeral,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [K: num,L: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),K),aa(num,A,numeral_numeral(A),L)) = aa(product_prod(A,A),A,product_snd(A,A),unique8689654367752047608divmod(A,K,L)) ) ).
% numeral_mod_numeral
tff(fact_2549_fst__divmod,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [M4: num,N4: num] : aa(product_prod(A,A),A,product_fst(A,A),unique8689654367752047608divmod(A,M4,N4)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(num,A,numeral_numeral(A),M4)),aa(num,A,numeral_numeral(A),N4)) ) ).
% fst_divmod
tff(fact_2550_snd__divmod,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [M4: num,N4: num] : aa(product_prod(A,A),A,product_snd(A,A),unique8689654367752047608divmod(A,M4,N4)) = modulo_modulo(A,aa(num,A,numeral_numeral(A),M4),aa(num,A,numeral_numeral(A),N4)) ) ).
% snd_divmod
tff(fact_2551_divmod__def,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [M4: num,N4: num] : unique8689654367752047608divmod(A,M4,N4) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(num,A,numeral_numeral(A),M4)),aa(num,A,numeral_numeral(A),N4))),modulo_modulo(A,aa(num,A,numeral_numeral(A),M4),aa(num,A,numeral_numeral(A),N4))) ) ).
% divmod_def
tff(fact_2552_divmod__step__nat__def,axiom,
! [L: num,Qr: product_prod(nat,nat)] : unique1321980374590559556d_step(nat,L,Qr) = aa(product_prod(nat,nat),product_prod(nat,nat),aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aTP_Lamp_vi(num,fun(nat,fun(nat,product_prod(nat,nat))),L)),Qr) ).
% divmod_step_nat_def
tff(fact_2553_divmod__algorithm__code_I6_J,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [M4: num,N4: num] : unique8689654367752047608divmod(A,aa(num,num,bit1,M4),bit0(N4)) = aa(product_prod(A,A),product_prod(A,A),aa(fun(A,fun(A,product_prod(A,A))),fun(product_prod(A,A),product_prod(A,A)),product_case_prod(A,A,product_prod(A,A)),aTP_Lamp_vj(A,fun(A,product_prod(A,A)))),unique8689654367752047608divmod(A,M4,N4)) ) ).
% divmod_algorithm_code(6)
tff(fact_2554_divmod__algorithm__code_I8_J,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [M4: num,N4: num] :
( ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M4),N4))
=> ( unique8689654367752047608divmod(A,aa(num,num,bit1,M4),aa(num,num,bit1,N4)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,M4))) ) )
& ( ~ pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M4),N4))
=> ( unique8689654367752047608divmod(A,aa(num,num,bit1,M4),aa(num,num,bit1,N4)) = unique1321980374590559556d_step(A,aa(num,num,bit1,N4),unique8689654367752047608divmod(A,aa(num,num,bit1,M4),bit0(aa(num,num,bit1,N4)))) ) ) ) ) ).
% divmod_algorithm_code(8)
tff(fact_2555_divmod__step__integer__def,axiom,
! [L: num,Qr: product_prod(code_integer,code_integer)] : unique1321980374590559556d_step(code_integer,L,Qr) = aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),product_case_prod(code_integer,code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_vk(num,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),Qr) ).
% divmod_step_integer_def
tff(fact_2556_nat__of__integer__code,axiom,
! [K: code_integer] :
( ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),K),zero_zero(code_integer)))
=> ( aa(code_integer,nat,code_nat_of_integer,K) = zero_zero(nat) ) )
& ( ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),K),zero_zero(code_integer)))
=> ( aa(code_integer,nat,code_nat_of_integer,K) = aa(product_prod(code_integer,code_integer),nat,aa(fun(code_integer,fun(code_integer,nat)),fun(product_prod(code_integer,code_integer),nat),product_case_prod(code_integer,code_integer,nat),aTP_Lamp_vl(code_integer,fun(code_integer,nat))),code_divmod_integer(K,aa(num,code_integer,numeral_numeral(code_integer),bit0(one)))) ) ) ) ).
% nat_of_integer_code
tff(fact_2557_divmod__algorithm__code_I7_J,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [M4: num,N4: num] :
( ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),M4),N4))
=> ( unique8689654367752047608divmod(A,bit0(M4),aa(num,num,bit1,N4)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),bit0(M4))) ) )
& ( ~ pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),M4),N4))
=> ( unique8689654367752047608divmod(A,bit0(M4),aa(num,num,bit1,N4)) = unique1321980374590559556d_step(A,aa(num,num,bit1,N4),unique8689654367752047608divmod(A,bit0(M4),bit0(aa(num,num,bit1,N4)))) ) ) ) ) ).
% divmod_algorithm_code(7)
tff(fact_2558_divmod__algorithm__code_I4_J,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [N4: num] : unique8689654367752047608divmod(A,one,aa(num,num,bit1,N4)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),one)) ) ).
% divmod_algorithm_code(4)
tff(fact_2559_num__of__integer__code,axiom,
! [K: code_integer] :
( ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),K),one_one(code_integer)))
=> ( aa(code_integer,num,code_num_of_integer,K) = one ) )
& ( ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),K),one_one(code_integer)))
=> ( aa(code_integer,num,code_num_of_integer,K) = aa(product_prod(code_integer,code_integer),num,aa(fun(code_integer,fun(code_integer,num)),fun(product_prod(code_integer,code_integer),num),product_case_prod(code_integer,code_integer,num),aTP_Lamp_vm(code_integer,fun(code_integer,num))),code_divmod_integer(K,aa(num,code_integer,numeral_numeral(code_integer),bit0(one)))) ) ) ) ).
% num_of_integer_code
tff(fact_2560_divmod__integer__code,axiom,
! [K: code_integer,L: code_integer] :
( ( ( K = zero_zero(code_integer) )
=> ( code_divmod_integer(K,L) = aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),zero_zero(code_integer)) ) )
& ( ( K != zero_zero(code_integer) )
=> ( ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),zero_zero(code_integer)),L))
=> ( ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),zero_zero(code_integer)),K))
=> ( code_divmod_integer(K,L) = code_divmod_abs(K,L) ) )
& ( ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),zero_zero(code_integer)),K))
=> ( code_divmod_integer(K,L) = aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),product_case_prod(code_integer,code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_vn(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),code_divmod_abs(K,L)) ) ) ) )
& ( ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),zero_zero(code_integer)),L))
=> ( ( ( L = zero_zero(code_integer) )
=> ( code_divmod_integer(K,L) = aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),K) ) )
& ( ( L != zero_zero(code_integer) )
=> ( code_divmod_integer(K,L) = aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(fun(code_integer,code_integer),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),product_apsnd(code_integer,code_integer,code_integer),uminus_uminus(code_integer)),if(product_prod(code_integer,code_integer),aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),K),zero_zero(code_integer)),code_divmod_abs(K,L),aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),product_case_prod(code_integer,code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_vo(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),code_divmod_abs(K,L)))) ) ) ) ) ) ) ) ).
% divmod_integer_code
tff(fact_2561_int__of__integer__code,axiom,
! [K: code_integer] :
( ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),K),zero_zero(code_integer)))
=> ( aa(code_integer,int,code_int_of_integer,K) = aa(int,int,uminus_uminus(int),aa(code_integer,int,code_int_of_integer,aa(code_integer,code_integer,uminus_uminus(code_integer),K))) ) )
& ( ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),K),zero_zero(code_integer)))
=> ( ( ( K = zero_zero(code_integer) )
=> ( aa(code_integer,int,code_int_of_integer,K) = zero_zero(int) ) )
& ( ( K != zero_zero(code_integer) )
=> ( aa(code_integer,int,code_int_of_integer,K) = aa(product_prod(code_integer,code_integer),int,aa(fun(code_integer,fun(code_integer,int)),fun(product_prod(code_integer,code_integer),int),product_case_prod(code_integer,code_integer,int),aTP_Lamp_vp(code_integer,fun(code_integer,int))),code_divmod_integer(K,aa(num,code_integer,numeral_numeral(code_integer),bit0(one)))) ) ) ) ) ) ).
% int_of_integer_code
tff(fact_2562_Code__Numeral_Onegative__def,axiom,
code_negative = aa(fun(num,code_integer),fun(num,code_integer),aa(fun(code_integer,code_integer),fun(fun(num,code_integer),fun(num,code_integer)),comp(code_integer,code_integer,num),uminus_uminus(code_integer)),numeral_numeral(code_integer)) ).
% Code_Numeral.negative_def
tff(fact_2563_bit__cut__integer__code,axiom,
! [K: code_integer] :
( ( ( K = zero_zero(code_integer) )
=> ( code_bit_cut_integer(K) = aa(bool,product_prod(code_integer,bool),aa(code_integer,fun(bool,product_prod(code_integer,bool)),product_Pair(code_integer,bool),zero_zero(code_integer)),fFalse) ) )
& ( ( K != zero_zero(code_integer) )
=> ( code_bit_cut_integer(K) = aa(product_prod(code_integer,code_integer),product_prod(code_integer,bool),aa(fun(code_integer,fun(code_integer,product_prod(code_integer,bool))),fun(product_prod(code_integer,code_integer),product_prod(code_integer,bool)),product_case_prod(code_integer,code_integer,product_prod(code_integer,bool)),aTP_Lamp_vq(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,bool))),K)),code_divmod_abs(K,aa(num,code_integer,numeral_numeral(code_integer),bit0(one)))) ) ) ) ).
% bit_cut_integer_code
tff(fact_2564_divmod__integer__eq__cases,axiom,
! [K: code_integer,L: code_integer] :
( ( ( K = zero_zero(code_integer) )
=> ( code_divmod_integer(K,L) = aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),zero_zero(code_integer)) ) )
& ( ( K != zero_zero(code_integer) )
=> ( ( ( L = zero_zero(code_integer) )
=> ( code_divmod_integer(K,L) = aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),K) ) )
& ( ( L != zero_zero(code_integer) )
=> ( code_divmod_integer(K,L) = aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),aa(fun(code_integer,code_integer),fun(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer))),aa(fun(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer))),fun(fun(code_integer,code_integer),fun(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)))),comp(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),code_integer),aa(fun(code_integer,fun(code_integer,code_integer)),fun(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer))),aa(fun(fun(code_integer,code_integer),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer))),fun(fun(code_integer,fun(code_integer,code_integer)),fun(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)))),comp(fun(code_integer,code_integer),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),code_integer),product_apsnd(code_integer,code_integer,code_integer)),times_times(code_integer))),sgn_sgn(code_integer)),L),if(product_prod(code_integer,code_integer),aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),aa(code_integer,code_integer,sgn_sgn(code_integer),K)),aa(code_integer,code_integer,sgn_sgn(code_integer),L)),code_divmod_abs(K,L),aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),product_case_prod(code_integer,code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_vr(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),code_divmod_abs(K,L)))) ) ) ) ) ) ).
% divmod_integer_eq_cases
tff(fact_2565_UNIV__char__of__nat,axiom,
top_top(set(char)) = aa(set(nat),set(char),aa(fun(nat,char),fun(set(nat),set(char)),image2(nat,char),unique5772411509450598832har_of(nat)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one))))))))))) ).
% UNIV_char_of_nat
tff(fact_2566_num__of__integer__def,axiom,
code_num_of_integer = aa(fun(int,num),fun(code_integer,num),map_fun(code_integer,int,num,num,code_int_of_integer,id(num)),aa(fun(int,nat),fun(int,num),aa(fun(nat,num),fun(fun(int,nat),fun(int,num)),comp(nat,num,int),num_of_nat),nat2)) ).
% num_of_integer_def
tff(fact_2567_range__nat__of__char,axiom,
aa(set(char),set(nat),aa(fun(char,nat),fun(set(char),set(nat)),image2(char,nat),comm_s6883823935334413003f_char(nat)),top_top(set(char))) = set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one)))))))))) ).
% range_nat_of_char
tff(fact_2568_weak__coinduct__image,axiom,
! [A: $tType,B: $tType,A3: A,X6: set(A),G: fun(A,B),F: fun(set(B),set(B))] :
( pp(aa(set(A),bool,member(A,A3),X6))
=> ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),G),X6)),aa(set(B),set(B),F,aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),G),X6))))
=> pp(aa(set(B),bool,member(B,aa(A,B,G,A3)),complete_lattice_gfp(set(B),F))) ) ) ).
% weak_coinduct_image
tff(fact_2569_last__zip,axiom,
! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] :
( ( Xs != nil(A) )
=> ( ( Ys != nil(B) )
=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
=> ( last(product_prod(A,B),zip(A,B,Xs,Ys)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),last(A,Xs)),last(B,Ys)) ) ) ) ) ).
% last_zip
tff(fact_2570_cuts__eq,axiom,
! [B: $tType,A: $tType,F: fun(A,B),R3: set(product_prod(A,A)),X2: A,G: fun(A,B)] :
( ( cut(A,B,F,R3,X2) = cut(A,B,G,R3,X2) )
<=> ! [Y4: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),X2)),R3))
=> ( aa(A,B,F,Y4) = aa(A,B,G,Y4) ) ) ) ).
% cuts_eq
tff(fact_2571_char__of__comp__of__char,axiom,
! [A: $tType] :
( bit_un5681908812861735899ations(A)
=> ( aa(fun(char,A),fun(char,char),aa(fun(A,char),fun(fun(char,A),fun(char,char)),comp(A,char,char),unique5772411509450598832har_of(A)),comm_s6883823935334413003f_char(A)) = id(char) ) ) ).
% char_of_comp_of_char
tff(fact_2572_num__of__integer_Orsp,axiom,
pp(aa(fun(int,num),bool,aa(fun(int,num),fun(fun(int,num),bool),bNF_rel_fun(int,int,num,num,fequal(int),fequal(num)),aa(fun(int,nat),fun(int,num),aa(fun(nat,num),fun(fun(int,nat),fun(int,num)),comp(nat,num,int),num_of_nat),nat2)),aa(fun(int,nat),fun(int,num),aa(fun(nat,num),fun(fun(int,nat),fun(int,num)),comp(nat,num,int),num_of_nat),nat2))) ).
% num_of_integer.rsp
tff(fact_2573_cut__apply,axiom,
! [B: $tType,A: $tType,X2: A,A3: A,R3: set(product_prod(A,A)),F: fun(A,B)] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),A3)),R3))
=> ( aa(A,B,cut(A,B,F,R3,A3),X2) = aa(A,B,F,X2) ) ) ).
% cut_apply
tff(fact_2574_num__of__integer_Otransfer,axiom,
pp(aa(fun(code_integer,num),bool,aa(fun(int,num),fun(fun(code_integer,num),bool),bNF_rel_fun(int,code_integer,num,num,code_pcr_integer,fequal(num)),aa(fun(int,nat),fun(int,num),aa(fun(nat,num),fun(fun(int,nat),fun(int,num)),comp(nat,num,int),num_of_nat),nat2)),code_num_of_integer)) ).
% num_of_integer.transfer
tff(fact_2575_cut__def,axiom,
! [A: $tType,B: $tType,X2: A,R3: set(product_prod(A,A)),F: fun(A,B),X4: A] :
( ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),X2)),R3))
=> ( aa(A,B,cut(A,B,F,R3,X2),X4) = aa(A,B,F,X4) ) )
& ( ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X4),X2)),R3))
=> ( aa(A,B,cut(A,B,F,R3,X2),X4) = undefined(B) ) ) ) ).
% cut_def
tff(fact_2576_NEG__pcr__rule,axiom,
! [C: $tType,A: $tType,D: $tType,B: $tType,A6: fun(A,fun(C,bool)),B6: fun(C,fun(B,bool)),C5: fun(A,fun(B,bool)),X6: fun(B,fun(D,bool))] :
( neg(A,B,aa(fun(C,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(C,bool)),fun(fun(C,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,C,B),A6),B6),C5)
=> neg(A,D,aa(fun(C,fun(D,bool)),fun(A,fun(D,bool)),aa(fun(A,fun(C,bool)),fun(fun(C,fun(D,bool)),fun(A,fun(D,bool))),relcompp(A,C,D),A6),aa(fun(B,fun(D,bool)),fun(C,fun(D,bool)),aa(fun(C,fun(B,bool)),fun(fun(B,fun(D,bool)),fun(C,fun(D,bool))),relcompp(C,B,D),B6),X6)),aa(fun(B,fun(D,bool)),fun(A,fun(D,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(D,bool)),fun(A,fun(D,bool))),relcompp(A,B,D),C5),X6)) ) ).
% NEG_pcr_rule
tff(fact_2577_neg__eq__OO,axiom,
! [B: $tType,A: $tType,A6: fun(A,fun(B,bool))] : neg(A,B,aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(A,bool)),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,A,B),fequal(A)),A6),A6) ).
% neg_eq_OO
tff(fact_2578_neg__OO__eq,axiom,
! [B: $tType,A: $tType,A6: fun(A,fun(B,bool))] : neg(A,B,aa(fun(B,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,B,B),A6),fequal(B)),A6) ).
% neg_OO_eq
tff(fact_2579_Product__Type_Oprod__def,axiom,
! [B: $tType,A: $tType] : product_prod2(A,B) = aa(fun(fun(A,fun(B,bool)),bool),set(fun(A,fun(B,bool))),collect(fun(A,fun(B,bool))),aTP_Lamp_vs(fun(A,fun(B,bool)),bool)) ).
% Product_Type.prod_def
tff(fact_2580_Pair__Rep__def,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,X4: A,Xa4: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),product_Pair_Rep(A,B,A3,B2),X4),Xa4))
<=> ( ( X4 = A3 )
& ( Xa4 = B2 ) ) ) ).
% Pair_Rep_def
tff(fact_2581_Pair__def,axiom,
! [A: $tType,B: $tType,A3: A,B2: B] : aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2) = aa(fun(A,fun(B,bool)),product_prod(A,B),product_Abs_prod(A,B),product_Pair_Rep(A,B,A3,B2)) ).
% Pair_def
tff(fact_2582_pcr__integer__def,axiom,
code_pcr_integer = aa(fun(int,fun(code_integer,bool)),fun(int,fun(code_integer,bool)),aa(fun(int,fun(int,bool)),fun(fun(int,fun(code_integer,bool)),fun(int,fun(code_integer,bool))),relcompp(int,int,code_integer),fequal(int)),code_cr_integer) ).
% pcr_integer_def
tff(fact_2583_curr__def,axiom,
! [A: $tType,C: $tType,B: $tType,A6: set(A),F: fun(product_prod(A,B),C),X4: A] :
( ( pp(aa(set(A),bool,member(A,X4),A6))
=> ( bNF_Wellorder_curr(A,B,C,A6,F,X4) = aa(A,fun(B,C),aTP_Lamp_as(fun(product_prod(A,B),C),fun(A,fun(B,C)),F),X4) ) )
& ( ~ pp(aa(set(A),bool,member(A,X4),A6))
=> ( bNF_Wellorder_curr(A,B,C,A6,F,X4) = undefined(fun(B,C)) ) ) ) ).
% curr_def
tff(fact_2584_Rep__prod__induct,axiom,
! [B: $tType,A: $tType,Y2: fun(A,fun(B,bool)),P2: fun(fun(A,fun(B,bool)),bool)] :
( pp(aa(set(fun(A,fun(B,bool))),bool,member(fun(A,fun(B,bool)),Y2),product_prod2(A,B)))
=> ( ! [X: product_prod(A,B)] : pp(aa(fun(A,fun(B,bool)),bool,P2,aa(product_prod(A,B),fun(A,fun(B,bool)),product_Rep_prod(A,B),X)))
=> pp(aa(fun(A,fun(B,bool)),bool,P2,Y2)) ) ) ).
% Rep_prod_induct
tff(fact_2585_Rep__prod__cases,axiom,
! [B: $tType,A: $tType,Y2: fun(A,fun(B,bool))] :
( pp(aa(set(fun(A,fun(B,bool))),bool,member(fun(A,fun(B,bool)),Y2),product_prod2(A,B)))
=> ~ ! [X: product_prod(A,B)] : Y2 != aa(product_prod(A,B),fun(A,fun(B,bool)),product_Rep_prod(A,B),X) ) ).
% Rep_prod_cases
tff(fact_2586_Rep__prod,axiom,
! [B: $tType,A: $tType,X2: product_prod(A,B)] : pp(aa(set(fun(A,fun(B,bool))),bool,member(fun(A,fun(B,bool)),aa(product_prod(A,B),fun(A,fun(B,bool)),product_Rep_prod(A,B),X2)),product_prod2(A,B))) ).
% Rep_prod
tff(fact_2587_Rep__prod__inverse,axiom,
! [B: $tType,A: $tType,X2: product_prod(A,B)] : aa(fun(A,fun(B,bool)),product_prod(A,B),product_Abs_prod(A,B),aa(product_prod(A,B),fun(A,fun(B,bool)),product_Rep_prod(A,B),X2)) = X2 ).
% Rep_prod_inverse
tff(fact_2588_Rep__prod__inject,axiom,
! [B: $tType,A: $tType,X2: product_prod(A,B),Y2: product_prod(A,B)] :
( ( aa(product_prod(A,B),fun(A,fun(B,bool)),product_Rep_prod(A,B),X2) = aa(product_prod(A,B),fun(A,fun(B,bool)),product_Rep_prod(A,B),Y2) )
<=> ( X2 = Y2 ) ) ).
% Rep_prod_inject
tff(fact_2589_Abs__prod__inverse,axiom,
! [B: $tType,A: $tType,Y2: fun(A,fun(B,bool))] :
( pp(aa(set(fun(A,fun(B,bool))),bool,member(fun(A,fun(B,bool)),Y2),product_prod2(A,B)))
=> ( aa(product_prod(A,B),fun(A,fun(B,bool)),product_Rep_prod(A,B),aa(fun(A,fun(B,bool)),product_prod(A,B),product_Abs_prod(A,B),Y2)) = Y2 ) ) ).
% Abs_prod_inverse
tff(fact_2590_type__definition__prod,axiom,
! [B: $tType,A: $tType] : type_definition(product_prod(A,B),fun(A,fun(B,bool)),product_Rep_prod(A,B),product_Abs_prod(A,B),product_prod2(A,B)) ).
% type_definition_prod
tff(fact_2591_Abs__prod__cases,axiom,
! [B: $tType,A: $tType,X2: product_prod(A,B)] :
~ ! [Y: fun(A,fun(B,bool))] :
( ( X2 = aa(fun(A,fun(B,bool)),product_prod(A,B),product_Abs_prod(A,B),Y) )
=> ~ pp(aa(set(fun(A,fun(B,bool))),bool,member(fun(A,fun(B,bool)),Y),product_prod2(A,B))) ) ).
% Abs_prod_cases
tff(fact_2592_Abs__prod__induct,axiom,
! [B: $tType,A: $tType,P2: fun(product_prod(A,B),bool),X2: product_prod(A,B)] :
( ! [Y: fun(A,fun(B,bool))] :
( pp(aa(set(fun(A,fun(B,bool))),bool,member(fun(A,fun(B,bool)),Y),product_prod2(A,B)))
=> pp(aa(product_prod(A,B),bool,P2,aa(fun(A,fun(B,bool)),product_prod(A,B),product_Abs_prod(A,B),Y))) )
=> pp(aa(product_prod(A,B),bool,P2,X2)) ) ).
% Abs_prod_induct
tff(fact_2593_Abs__prod__inject,axiom,
! [B: $tType,A: $tType,X2: fun(A,fun(B,bool)),Y2: fun(A,fun(B,bool))] :
( pp(aa(set(fun(A,fun(B,bool))),bool,member(fun(A,fun(B,bool)),X2),product_prod2(A,B)))
=> ( pp(aa(set(fun(A,fun(B,bool))),bool,member(fun(A,fun(B,bool)),Y2),product_prod2(A,B)))
=> ( ( aa(fun(A,fun(B,bool)),product_prod(A,B),product_Abs_prod(A,B),X2) = aa(fun(A,fun(B,bool)),product_prod(A,B),product_Abs_prod(A,B),Y2) )
<=> ( X2 = Y2 ) ) ) ) ).
% Abs_prod_inject
tff(fact_2594_ntrunc__UN1,axiom,
! [A: $tType,B: $tType,C: $tType,K: nat,F: fun(C,set(old_node(A,B)))] : aa(set(old_node(A,B)),set(old_node(A,B)),old_ntrunc(A,B,K),aa(set(set(old_node(A,B))),set(old_node(A,B)),complete_Sup_Sup(set(old_node(A,B))),aa(set(C),set(set(old_node(A,B))),aa(fun(C,set(old_node(A,B))),fun(set(C),set(set(old_node(A,B)))),image2(C,set(old_node(A,B))),F),top_top(set(C))))) = aa(set(set(old_node(A,B))),set(old_node(A,B)),complete_Sup_Sup(set(old_node(A,B))),aa(set(C),set(set(old_node(A,B))),aa(fun(C,set(old_node(A,B))),fun(set(C),set(set(old_node(A,B)))),image2(C,set(old_node(A,B))),aa(fun(C,set(old_node(A,B))),fun(C,set(old_node(A,B))),aTP_Lamp_vt(nat,fun(fun(C,set(old_node(A,B))),fun(C,set(old_node(A,B)))),K),F)),top_top(set(C)))) ).
% ntrunc_UN1
tff(fact_2595_POS__pcr__rule,axiom,
! [C: $tType,A: $tType,D: $tType,B: $tType,A6: fun(A,fun(C,bool)),B6: fun(C,fun(B,bool)),C5: fun(A,fun(B,bool)),X6: fun(B,fun(D,bool))] :
( pos(A,B,aa(fun(C,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(C,bool)),fun(fun(C,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,C,B),A6),B6),C5)
=> pos(A,D,aa(fun(C,fun(D,bool)),fun(A,fun(D,bool)),aa(fun(A,fun(C,bool)),fun(fun(C,fun(D,bool)),fun(A,fun(D,bool))),relcompp(A,C,D),A6),aa(fun(B,fun(D,bool)),fun(C,fun(D,bool)),aa(fun(C,fun(B,bool)),fun(fun(B,fun(D,bool)),fun(C,fun(D,bool))),relcompp(C,B,D),B6),X6)),aa(fun(B,fun(D,bool)),fun(A,fun(D,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(D,bool)),fun(A,fun(D,bool))),relcompp(A,B,D),C5),X6)) ) ).
% POS_pcr_rule
tff(fact_2596_pos__eq__OO,axiom,
! [B: $tType,A: $tType,A6: fun(A,fun(B,bool))] : pos(A,B,aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(A,bool)),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,A,B),fequal(A)),A6),A6) ).
% pos_eq_OO
tff(fact_2597_pos__OO__eq,axiom,
! [B: $tType,A: $tType,A6: fun(A,fun(B,bool))] : pos(A,B,aa(fun(B,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,B,B),A6),fequal(B)),A6) ).
% pos_OO_eq
tff(fact_2598_comp__fun__idem_Ocomp__comp__fun__idem,axiom,
! [B: $tType,A: $tType,C: $tType,F: fun(A,fun(B,B)),G: fun(C,A)] :
( finite_comp_fun_idem(A,B,F)
=> finite_comp_fun_idem(C,B,aa(fun(C,A),fun(C,fun(B,B)),aa(fun(A,fun(B,B)),fun(fun(C,A),fun(C,fun(B,B))),comp(A,fun(B,B),C),F),G)) ) ).
% comp_fun_idem.comp_comp_fun_idem
tff(fact_2599_In1__UN1,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(C,set(old_node(A,B)))] : aa(set(old_node(A,B)),set(old_node(A,B)),old_In1(A,B),aa(set(set(old_node(A,B))),set(old_node(A,B)),complete_Sup_Sup(set(old_node(A,B))),aa(set(C),set(set(old_node(A,B))),aa(fun(C,set(old_node(A,B))),fun(set(C),set(set(old_node(A,B)))),image2(C,set(old_node(A,B))),F),top_top(set(C))))) = aa(set(set(old_node(A,B))),set(old_node(A,B)),complete_Sup_Sup(set(old_node(A,B))),aa(set(C),set(set(old_node(A,B))),aa(fun(C,set(old_node(A,B))),fun(set(C),set(set(old_node(A,B)))),image2(C,set(old_node(A,B))),aTP_Lamp_vu(fun(C,set(old_node(A,B))),fun(C,set(old_node(A,B))),F)),top_top(set(C)))) ).
% In1_UN1
tff(fact_2600_ntrunc__o__equality,axiom,
! [C: $tType,B: $tType,A: $tType,H1: fun(A,set(old_node(B,C))),H22: fun(A,set(old_node(B,C)))] :
( ! [K3: nat] : aa(fun(A,set(old_node(B,C))),fun(A,set(old_node(B,C))),aa(fun(set(old_node(B,C)),set(old_node(B,C))),fun(fun(A,set(old_node(B,C))),fun(A,set(old_node(B,C)))),comp(set(old_node(B,C)),set(old_node(B,C)),A),old_ntrunc(B,C,K3)),H1) = aa(fun(A,set(old_node(B,C))),fun(A,set(old_node(B,C))),aa(fun(set(old_node(B,C)),set(old_node(B,C))),fun(fun(A,set(old_node(B,C))),fun(A,set(old_node(B,C)))),comp(set(old_node(B,C)),set(old_node(B,C)),A),old_ntrunc(B,C,K3)),H22)
=> ( H1 = H22 ) ) ).
% ntrunc_o_equality
tff(fact_2601_comp__fun__idem_Ocomp__fun__idem,axiom,
! [B: $tType,A: $tType,F: fun(A,fun(B,B)),X2: A] :
( finite_comp_fun_idem(A,B,F)
=> ( aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X2)),aa(A,fun(B,B),F,X2)) = aa(A,fun(B,B),F,X2) ) ) ).
% comp_fun_idem.comp_fun_idem
tff(fact_2602_In0__UN1,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(C,set(old_node(A,B)))] : aa(set(old_node(A,B)),set(old_node(A,B)),old_In0(A,B),aa(set(set(old_node(A,B))),set(old_node(A,B)),complete_Sup_Sup(set(old_node(A,B))),aa(set(C),set(set(old_node(A,B))),aa(fun(C,set(old_node(A,B))),fun(set(C),set(set(old_node(A,B)))),image2(C,set(old_node(A,B))),F),top_top(set(C))))) = aa(set(set(old_node(A,B))),set(old_node(A,B)),complete_Sup_Sup(set(old_node(A,B))),aa(set(C),set(set(old_node(A,B))),aa(fun(C,set(old_node(A,B))),fun(set(C),set(set(old_node(A,B)))),image2(C,set(old_node(A,B))),aTP_Lamp_vv(fun(C,set(old_node(A,B))),fun(C,set(old_node(A,B))),F)),top_top(set(C)))) ).
% In0_UN1
tff(fact_2603_Scons__UN1__x,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(C,set(old_node(A,B))),M: set(old_node(A,B))] : old_Scons(A,B,aa(set(set(old_node(A,B))),set(old_node(A,B)),complete_Sup_Sup(set(old_node(A,B))),aa(set(C),set(set(old_node(A,B))),aa(fun(C,set(old_node(A,B))),fun(set(C),set(set(old_node(A,B)))),image2(C,set(old_node(A,B))),F),top_top(set(C)))),M) = aa(set(set(old_node(A,B))),set(old_node(A,B)),complete_Sup_Sup(set(old_node(A,B))),aa(set(C),set(set(old_node(A,B))),aa(fun(C,set(old_node(A,B))),fun(set(C),set(set(old_node(A,B)))),image2(C,set(old_node(A,B))),aa(set(old_node(A,B)),fun(C,set(old_node(A,B))),aTP_Lamp_vw(fun(C,set(old_node(A,B))),fun(set(old_node(A,B)),fun(C,set(old_node(A,B)))),F),M)),top_top(set(C)))) ).
% Scons_UN1_x
tff(fact_2604_Scons__UN1__y,axiom,
! [A: $tType,B: $tType,C: $tType,M: set(old_node(A,B)),F: fun(C,set(old_node(A,B)))] : old_Scons(A,B,M,aa(set(set(old_node(A,B))),set(old_node(A,B)),complete_Sup_Sup(set(old_node(A,B))),aa(set(C),set(set(old_node(A,B))),aa(fun(C,set(old_node(A,B))),fun(set(C),set(set(old_node(A,B)))),image2(C,set(old_node(A,B))),F),top_top(set(C))))) = aa(set(set(old_node(A,B))),set(old_node(A,B)),complete_Sup_Sup(set(old_node(A,B))),aa(set(C),set(set(old_node(A,B))),aa(fun(C,set(old_node(A,B))),fun(set(C),set(set(old_node(A,B)))),image2(C,set(old_node(A,B))),aa(fun(C,set(old_node(A,B))),fun(C,set(old_node(A,B))),aTP_Lamp_vx(set(old_node(A,B)),fun(fun(C,set(old_node(A,B))),fun(C,set(old_node(A,B)))),M),F)),top_top(set(C)))) ).
% Scons_UN1_y
tff(fact_2605_Lim__def,axiom,
! [A: $tType,B: $tType,F: fun(B,set(old_node(A,B)))] : old_Lim(B,A,F) = aa(set(set(old_node(A,B))),set(old_node(A,B)),complete_Sup_Sup(set(old_node(A,B))),aa(fun(set(old_node(A,B)),bool),set(set(old_node(A,B))),collect(set(old_node(A,B))),aTP_Lamp_vy(fun(B,set(old_node(A,B))),fun(set(old_node(A,B)),bool),F))) ).
% Lim_def
tff(fact_2606_iso__tuple__snd__def,axiom,
! [C: $tType,B: $tType,A: $tType,Isom: tuple_isomorphism(A,B,C)] : iso_tuple_snd(A,B,C,Isom) = aa(fun(A,product_prod(B,C)),fun(A,C),aa(fun(product_prod(B,C),C),fun(fun(A,product_prod(B,C)),fun(A,C)),comp(product_prod(B,C),C,A),product_snd(B,C)),repr(A,B,C,Isom)) ).
% iso_tuple_snd_def
tff(fact_2607_iso__tuple__fst__def,axiom,
! [C: $tType,B: $tType,A: $tType,Isom: tuple_isomorphism(A,B,C)] : iso_tuple_fst(A,B,C,Isom) = aa(fun(A,product_prod(B,C)),fun(A,B),aa(fun(product_prod(B,C),B),fun(fun(A,product_prod(B,C)),fun(A,B)),comp(product_prod(B,C),B,A),product_fst(B,C)),repr(A,B,C,Isom)) ).
% iso_tuple_fst_def
tff(fact_2608_Scons__def,axiom,
! [B: $tType,A: $tType,M: set(old_node(A,B)),N: set(old_node(A,B))] : old_Scons(A,B,M,N) = aa(set(old_node(A,B)),set(old_node(A,B)),aa(set(old_node(A,B)),fun(set(old_node(A,B)),set(old_node(A,B))),sup_sup(set(old_node(A,B))),aa(set(old_node(A,B)),set(old_node(A,B)),aa(fun(old_node(A,B),old_node(A,B)),fun(set(old_node(A,B)),set(old_node(A,B))),image2(old_node(A,B),old_node(A,B)),old_Push_Node(B,A,aa(nat,sum_sum(B,nat),sum_Inr(nat,B),one_one(nat)))),M)),aa(set(old_node(A,B)),set(old_node(A,B)),aa(fun(old_node(A,B),old_node(A,B)),fun(set(old_node(A,B)),set(old_node(A,B))),image2(old_node(A,B),old_node(A,B)),old_Push_Node(B,A,aa(nat,sum_sum(B,nat),sum_Inr(nat,B),aa(nat,nat,suc,one_one(nat))))),N)) ).
% Scons_def
tff(fact_2609_dsum__def,axiom,
! [B: $tType,A: $tType,R: set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),S: set(product_prod(set(old_node(A,B)),set(old_node(A,B))))] : old_dsum(A,B,R,S) = aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),fun(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),sup_sup(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aa(set(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),complete_Sup_Sup(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),set(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aa(fun(product_prod(set(old_node(A,B)),set(old_node(A,B))),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),fun(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),set(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))),image2(product_prod(set(old_node(A,B)),set(old_node(A,B))),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aa(fun(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))),fun(product_prod(set(old_node(A,B)),set(old_node(A,B))),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),product_case_prod(set(old_node(A,B)),set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aTP_Lamp_vz(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))))),R))),aa(set(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),complete_Sup_Sup(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),set(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aa(fun(product_prod(set(old_node(A,B)),set(old_node(A,B))),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),fun(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),set(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))),image2(product_prod(set(old_node(A,B)),set(old_node(A,B))),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aa(fun(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))),fun(product_prod(set(old_node(A,B)),set(old_node(A,B))),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),product_case_prod(set(old_node(A,B)),set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aTP_Lamp_wa(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))))),S))) ).
% dsum_def
tff(fact_2610_usum__def,axiom,
! [B: $tType,A: $tType,A6: set(set(old_node(A,B))),B6: set(set(old_node(A,B)))] : old_usum(A,B,A6,B6) = aa(set(set(old_node(A,B))),set(set(old_node(A,B))),aa(set(set(old_node(A,B))),fun(set(set(old_node(A,B))),set(set(old_node(A,B)))),sup_sup(set(set(old_node(A,B)))),aa(set(set(old_node(A,B))),set(set(old_node(A,B))),aa(fun(set(old_node(A,B)),set(old_node(A,B))),fun(set(set(old_node(A,B))),set(set(old_node(A,B)))),image2(set(old_node(A,B)),set(old_node(A,B))),old_In0(A,B)),A6)),aa(set(set(old_node(A,B))),set(set(old_node(A,B))),aa(fun(set(old_node(A,B)),set(old_node(A,B))),fun(set(set(old_node(A,B))),set(set(old_node(A,B)))),image2(set(old_node(A,B)),set(old_node(A,B))),old_In1(A,B)),B6)) ).
% usum_def
tff(fact_2611_nth__item_Oelims,axiom,
! [A: $tType] :
( countable(A)
=> ! [X2: nat,Y2: set(old_node(A,product_unit))] :
( ( nth_item(A,X2) = Y2 )
=> ( ( ( X2 = zero_zero(nat) )
=> ( Y2 != undefined(set(old_node(A,product_unit))) ) )
=> ~ ! [N2: nat] :
( ( X2 = aa(nat,nat,suc,N2) )
=> ( Y2 != aa(sum_sum(nat,nat),set(old_node(A,product_unit)),sum_case_sum(nat,set(old_node(A,product_unit)),nat,aTP_Lamp_wd(nat,set(old_node(A,product_unit))),aTP_Lamp_wh(nat,set(old_node(A,product_unit)))),aa(nat,sum_sum(nat,nat),nat_sum_decode,N2)) ) ) ) ) ) ).
% nth_item.elims
tff(fact_2612_nth__item_Opelims,axiom,
! [A: $tType] :
( countable(A)
=> ! [X2: nat,Y2: set(old_node(A,product_unit))] :
( ( nth_item(A,X2) = Y2 )
=> ( pp(aa(nat,bool,accp(nat,nth_item_rel),X2))
=> ( ( ( X2 = zero_zero(nat) )
=> ( ( Y2 = undefined(set(old_node(A,product_unit))) )
=> ~ pp(aa(nat,bool,accp(nat,nth_item_rel),zero_zero(nat))) ) )
=> ~ ! [N2: nat] :
( ( X2 = aa(nat,nat,suc,N2) )
=> ( ( Y2 = aa(sum_sum(nat,nat),set(old_node(A,product_unit)),sum_case_sum(nat,set(old_node(A,product_unit)),nat,aTP_Lamp_wd(nat,set(old_node(A,product_unit))),aTP_Lamp_wh(nat,set(old_node(A,product_unit)))),aa(nat,sum_sum(nat,nat),nat_sum_decode,N2)) )
=> ~ pp(aa(nat,bool,accp(nat,nth_item_rel),aa(nat,nat,suc,N2))) ) ) ) ) ) ) ).
% nth_item.pelims
tff(fact_2613_iso__tuple__snd__update__def,axiom,
! [C: $tType,B: $tType,A: $tType,Isom: tuple_isomorphism(A,B,C),F: fun(C,C)] : aa(fun(C,C),fun(A,A),iso_tuple_snd_update(A,B,C,Isom),F) = aa(fun(A,product_prod(B,C)),fun(A,A),aa(fun(product_prod(B,C),A),fun(fun(A,product_prod(B,C)),fun(A,A)),comp(product_prod(B,C),A,A),aa(fun(product_prod(B,C),product_prod(B,C)),fun(product_prod(B,C),A),aa(fun(product_prod(B,C),A),fun(fun(product_prod(B,C),product_prod(B,C)),fun(product_prod(B,C),A)),comp(product_prod(B,C),A,product_prod(B,C)),abst(A,B,C,Isom)),aa(fun(C,C),fun(product_prod(B,C),product_prod(B,C)),product_apsnd(C,C,B),F))),repr(A,B,C,Isom)) ).
% iso_tuple_snd_update_def
tff(fact_2614_iso__tuple__fst__update__def,axiom,
! [C: $tType,B: $tType,A: $tType,Isom: tuple_isomorphism(A,B,C),F: fun(B,B)] : aa(fun(B,B),fun(A,A),iso_tuple_fst_update(A,B,C,Isom),F) = aa(fun(A,product_prod(B,C)),fun(A,A),aa(fun(product_prod(B,C),A),fun(fun(A,product_prod(B,C)),fun(A,A)),comp(product_prod(B,C),A,A),aa(fun(product_prod(B,C),product_prod(B,C)),fun(product_prod(B,C),A),aa(fun(product_prod(B,C),A),fun(fun(product_prod(B,C),product_prod(B,C)),fun(product_prod(B,C),A)),comp(product_prod(B,C),A,product_prod(B,C)),abst(A,B,C,Isom)),product_apfst(B,B,C,F))),repr(A,B,C,Isom)) ).
% iso_tuple_fst_update_def
tff(fact_2615_nth__item_Opsimps_I2_J,axiom,
! [A: $tType] :
( countable(A)
=> ! [N4: nat] :
( pp(aa(nat,bool,accp(nat,nth_item_rel),aa(nat,nat,suc,N4)))
=> ( nth_item(A,aa(nat,nat,suc,N4)) = aa(sum_sum(nat,nat),set(old_node(A,product_unit)),sum_case_sum(nat,set(old_node(A,product_unit)),nat,aTP_Lamp_wd(nat,set(old_node(A,product_unit))),aTP_Lamp_wh(nat,set(old_node(A,product_unit)))),aa(nat,sum_sum(nat,nat),nat_sum_decode,N4)) ) ) ) ).
% nth_item.psimps(2)
tff(fact_2616_nth__item_Osimps_I2_J,axiom,
! [A: $tType] :
( countable(A)
=> ! [N4: nat] : nth_item(A,aa(nat,nat,suc,N4)) = aa(sum_sum(nat,nat),set(old_node(A,product_unit)),sum_case_sum(nat,set(old_node(A,product_unit)),nat,aTP_Lamp_wd(nat,set(old_node(A,product_unit))),aTP_Lamp_wh(nat,set(old_node(A,product_unit)))),aa(nat,sum_sum(nat,nat),nat_sum_decode,N4)) ) ).
% nth_item.simps(2)
tff(fact_2617_dprod__def,axiom,
! [B: $tType,A: $tType,R: set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),S: set(product_prod(set(old_node(A,B)),set(old_node(A,B))))] : old_dprod(A,B,R,S) = aa(set(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),complete_Sup_Sup(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),set(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aa(fun(product_prod(set(old_node(A,B)),set(old_node(A,B))),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),fun(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),set(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))),image2(product_prod(set(old_node(A,B)),set(old_node(A,B))),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aa(fun(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))),fun(product_prod(set(old_node(A,B)),set(old_node(A,B))),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),product_case_prod(set(old_node(A,B)),set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aTP_Lamp_wj(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),fun(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))),S))),R)) ).
% dprod_def
tff(fact_2618_isomorphic__tuple_Oiso__tuple__access__update__snd__fst,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,Isom: tuple_isomorphism(A,B,C),F: fun(C,D),H: fun(E,fun(B,B)),G: E] :
( isomorphic_tuple(A,B,C,Isom)
=> ( aa(fun(A,A),fun(A,D),aa(fun(A,D),fun(fun(A,A),fun(A,D)),comp(A,D,A),aa(fun(A,C),fun(A,D),aa(fun(C,D),fun(fun(A,C),fun(A,D)),comp(C,D,A),F),iso_tuple_snd(A,B,C,Isom))),aa(E,fun(A,A),aa(fun(E,fun(B,B)),fun(E,fun(A,A)),aa(fun(fun(B,B),fun(A,A)),fun(fun(E,fun(B,B)),fun(E,fun(A,A))),comp(fun(B,B),fun(A,A),E),iso_tuple_fst_update(A,B,C,Isom)),H),G)) = aa(fun(A,D),fun(A,D),aa(fun(D,D),fun(fun(A,D),fun(A,D)),comp(D,D,A),id(D)),aa(fun(A,C),fun(A,D),aa(fun(C,D),fun(fun(A,C),fun(A,D)),comp(C,D,A),F),iso_tuple_snd(A,B,C,Isom))) ) ) ).
% isomorphic_tuple.iso_tuple_access_update_snd_fst
tff(fact_2619_isomorphic__tuple_Oiso__tuple__access__update__fst__snd,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,Isom: tuple_isomorphism(A,B,C),F: fun(B,D),H: fun(E,fun(C,C)),G: E] :
( isomorphic_tuple(A,B,C,Isom)
=> ( aa(fun(A,A),fun(A,D),aa(fun(A,D),fun(fun(A,A),fun(A,D)),comp(A,D,A),aa(fun(A,B),fun(A,D),aa(fun(B,D),fun(fun(A,B),fun(A,D)),comp(B,D,A),F),iso_tuple_fst(A,B,C,Isom))),aa(E,fun(A,A),aa(fun(E,fun(C,C)),fun(E,fun(A,A)),aa(fun(fun(C,C),fun(A,A)),fun(fun(E,fun(C,C)),fun(E,fun(A,A))),comp(fun(C,C),fun(A,A),E),iso_tuple_snd_update(A,B,C,Isom)),H),G)) = aa(fun(A,D),fun(A,D),aa(fun(D,D),fun(fun(A,D),fun(A,D)),comp(D,D,A),id(D)),aa(fun(A,B),fun(A,D),aa(fun(B,D),fun(fun(A,B),fun(A,D)),comp(B,D,A),F),iso_tuple_fst(A,B,C,Isom))) ) ) ).
% isomorphic_tuple.iso_tuple_access_update_fst_snd
tff(fact_2620_isomorphic__tuple_Oiso__tuple__access__update__snd__snd,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,Isom: tuple_isomorphism(A,B,C),F: fun(C,D),H: fun(E,fun(C,C)),G: E,J5: fun(D,D)] :
( isomorphic_tuple(A,B,C,Isom)
=> ( ( aa(fun(C,C),fun(C,D),aa(fun(C,D),fun(fun(C,C),fun(C,D)),comp(C,D,C),F),aa(E,fun(C,C),H,G)) = aa(fun(C,D),fun(C,D),aa(fun(D,D),fun(fun(C,D),fun(C,D)),comp(D,D,C),J5),F) )
=> ( aa(fun(A,A),fun(A,D),aa(fun(A,D),fun(fun(A,A),fun(A,D)),comp(A,D,A),aa(fun(A,C),fun(A,D),aa(fun(C,D),fun(fun(A,C),fun(A,D)),comp(C,D,A),F),iso_tuple_snd(A,B,C,Isom))),aa(E,fun(A,A),aa(fun(E,fun(C,C)),fun(E,fun(A,A)),aa(fun(fun(C,C),fun(A,A)),fun(fun(E,fun(C,C)),fun(E,fun(A,A))),comp(fun(C,C),fun(A,A),E),iso_tuple_snd_update(A,B,C,Isom)),H),G)) = aa(fun(A,D),fun(A,D),aa(fun(D,D),fun(fun(A,D),fun(A,D)),comp(D,D,A),J5),aa(fun(A,C),fun(A,D),aa(fun(C,D),fun(fun(A,C),fun(A,D)),comp(C,D,A),F),iso_tuple_snd(A,B,C,Isom))) ) ) ) ).
% isomorphic_tuple.iso_tuple_access_update_snd_snd
tff(fact_2621_isomorphic__tuple_Oiso__tuple__access__update__fst__fst,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,Isom: tuple_isomorphism(A,B,C),F: fun(B,D),H: fun(E,fun(B,B)),G: E,J5: fun(D,D)] :
( isomorphic_tuple(A,B,C,Isom)
=> ( ( aa(fun(B,B),fun(B,D),aa(fun(B,D),fun(fun(B,B),fun(B,D)),comp(B,D,B),F),aa(E,fun(B,B),H,G)) = aa(fun(B,D),fun(B,D),aa(fun(D,D),fun(fun(B,D),fun(B,D)),comp(D,D,B),J5),F) )
=> ( aa(fun(A,A),fun(A,D),aa(fun(A,D),fun(fun(A,A),fun(A,D)),comp(A,D,A),aa(fun(A,B),fun(A,D),aa(fun(B,D),fun(fun(A,B),fun(A,D)),comp(B,D,A),F),iso_tuple_fst(A,B,C,Isom))),aa(E,fun(A,A),aa(fun(E,fun(B,B)),fun(E,fun(A,A)),aa(fun(fun(B,B),fun(A,A)),fun(fun(E,fun(B,B)),fun(E,fun(A,A))),comp(fun(B,B),fun(A,A),E),iso_tuple_fst_update(A,B,C,Isom)),H),G)) = aa(fun(A,D),fun(A,D),aa(fun(D,D),fun(fun(A,D),fun(A,D)),comp(D,D,A),J5),aa(fun(A,B),fun(A,D),aa(fun(B,D),fun(fun(A,B),fun(A,D)),comp(B,D,A),F),iso_tuple_fst(A,B,C,Isom))) ) ) ) ).
% isomorphic_tuple.iso_tuple_access_update_fst_fst
tff(fact_2622_isomorphic__tuple_Oiso__tuple__update__swap__snd__fst,axiom,
! [E: $tType,B: $tType,C: $tType,A: $tType,D: $tType,Isom: tuple_isomorphism(A,B,C),H: fun(D,fun(B,B)),F: D,J5: fun(E,fun(C,C)),G: E] :
( isomorphic_tuple(A,B,C,Isom)
=> ( aa(fun(A,A),fun(A,A),aa(fun(A,A),fun(fun(A,A),fun(A,A)),comp(A,A,A),aa(D,fun(A,A),aa(fun(D,fun(B,B)),fun(D,fun(A,A)),aa(fun(fun(B,B),fun(A,A)),fun(fun(D,fun(B,B)),fun(D,fun(A,A))),comp(fun(B,B),fun(A,A),D),iso_tuple_fst_update(A,B,C,Isom)),H),F)),aa(E,fun(A,A),aa(fun(E,fun(C,C)),fun(E,fun(A,A)),aa(fun(fun(C,C),fun(A,A)),fun(fun(E,fun(C,C)),fun(E,fun(A,A))),comp(fun(C,C),fun(A,A),E),iso_tuple_snd_update(A,B,C,Isom)),J5),G)) = aa(fun(A,A),fun(A,A),aa(fun(A,A),fun(fun(A,A),fun(A,A)),comp(A,A,A),aa(E,fun(A,A),aa(fun(E,fun(C,C)),fun(E,fun(A,A)),aa(fun(fun(C,C),fun(A,A)),fun(fun(E,fun(C,C)),fun(E,fun(A,A))),comp(fun(C,C),fun(A,A),E),iso_tuple_snd_update(A,B,C,Isom)),J5),G)),aa(D,fun(A,A),aa(fun(D,fun(B,B)),fun(D,fun(A,A)),aa(fun(fun(B,B),fun(A,A)),fun(fun(D,fun(B,B)),fun(D,fun(A,A))),comp(fun(B,B),fun(A,A),D),iso_tuple_fst_update(A,B,C,Isom)),H),F)) ) ) ).
% isomorphic_tuple.iso_tuple_update_swap_snd_fst
tff(fact_2623_isomorphic__tuple_Oiso__tuple__update__swap__fst__fst,axiom,
! [E: $tType,B: $tType,C: $tType,A: $tType,D: $tType,Isom: tuple_isomorphism(A,B,C),H: fun(D,fun(B,B)),F: D,J5: fun(E,fun(B,B)),G: E] :
( isomorphic_tuple(A,B,C,Isom)
=> ( ( aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(D,fun(B,B),H,F)),aa(E,fun(B,B),J5,G)) = aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(E,fun(B,B),J5,G)),aa(D,fun(B,B),H,F)) )
=> ( aa(fun(A,A),fun(A,A),aa(fun(A,A),fun(fun(A,A),fun(A,A)),comp(A,A,A),aa(D,fun(A,A),aa(fun(D,fun(B,B)),fun(D,fun(A,A)),aa(fun(fun(B,B),fun(A,A)),fun(fun(D,fun(B,B)),fun(D,fun(A,A))),comp(fun(B,B),fun(A,A),D),iso_tuple_fst_update(A,B,C,Isom)),H),F)),aa(E,fun(A,A),aa(fun(E,fun(B,B)),fun(E,fun(A,A)),aa(fun(fun(B,B),fun(A,A)),fun(fun(E,fun(B,B)),fun(E,fun(A,A))),comp(fun(B,B),fun(A,A),E),iso_tuple_fst_update(A,B,C,Isom)),J5),G)) = aa(fun(A,A),fun(A,A),aa(fun(A,A),fun(fun(A,A),fun(A,A)),comp(A,A,A),aa(E,fun(A,A),aa(fun(E,fun(B,B)),fun(E,fun(A,A)),aa(fun(fun(B,B),fun(A,A)),fun(fun(E,fun(B,B)),fun(E,fun(A,A))),comp(fun(B,B),fun(A,A),E),iso_tuple_fst_update(A,B,C,Isom)),J5),G)),aa(D,fun(A,A),aa(fun(D,fun(B,B)),fun(D,fun(A,A)),aa(fun(fun(B,B),fun(A,A)),fun(fun(D,fun(B,B)),fun(D,fun(A,A))),comp(fun(B,B),fun(A,A),D),iso_tuple_fst_update(A,B,C,Isom)),H),F)) ) ) ) ).
% isomorphic_tuple.iso_tuple_update_swap_fst_fst
tff(fact_2624_isomorphic__tuple_Oiso__tuple__update__swap__snd__snd,axiom,
! [E: $tType,C: $tType,B: $tType,A: $tType,D: $tType,Isom: tuple_isomorphism(A,B,C),H: fun(D,fun(C,C)),F: D,J5: fun(E,fun(C,C)),G: E] :
( isomorphic_tuple(A,B,C,Isom)
=> ( ( aa(fun(C,C),fun(C,C),aa(fun(C,C),fun(fun(C,C),fun(C,C)),comp(C,C,C),aa(D,fun(C,C),H,F)),aa(E,fun(C,C),J5,G)) = aa(fun(C,C),fun(C,C),aa(fun(C,C),fun(fun(C,C),fun(C,C)),comp(C,C,C),aa(E,fun(C,C),J5,G)),aa(D,fun(C,C),H,F)) )
=> ( aa(fun(A,A),fun(A,A),aa(fun(A,A),fun(fun(A,A),fun(A,A)),comp(A,A,A),aa(D,fun(A,A),aa(fun(D,fun(C,C)),fun(D,fun(A,A)),aa(fun(fun(C,C),fun(A,A)),fun(fun(D,fun(C,C)),fun(D,fun(A,A))),comp(fun(C,C),fun(A,A),D),iso_tuple_snd_update(A,B,C,Isom)),H),F)),aa(E,fun(A,A),aa(fun(E,fun(C,C)),fun(E,fun(A,A)),aa(fun(fun(C,C),fun(A,A)),fun(fun(E,fun(C,C)),fun(E,fun(A,A))),comp(fun(C,C),fun(A,A),E),iso_tuple_snd_update(A,B,C,Isom)),J5),G)) = aa(fun(A,A),fun(A,A),aa(fun(A,A),fun(fun(A,A),fun(A,A)),comp(A,A,A),aa(E,fun(A,A),aa(fun(E,fun(C,C)),fun(E,fun(A,A)),aa(fun(fun(C,C),fun(A,A)),fun(fun(E,fun(C,C)),fun(E,fun(A,A))),comp(fun(C,C),fun(A,A),E),iso_tuple_snd_update(A,B,C,Isom)),J5),G)),aa(D,fun(A,A),aa(fun(D,fun(C,C)),fun(D,fun(A,A)),aa(fun(fun(C,C),fun(A,A)),fun(fun(D,fun(C,C)),fun(D,fun(A,A))),comp(fun(C,C),fun(A,A),D),iso_tuple_snd_update(A,B,C,Isom)),H),F)) ) ) ) ).
% isomorphic_tuple.iso_tuple_update_swap_snd_snd
tff(fact_2625_isomorphic__tuple_Oiso__tuple__update__compose__fst__fst,axiom,
! [B: $tType,C: $tType,A: $tType,E: $tType,D: $tType,F2: $tType,Isom: tuple_isomorphism(A,B,C),H: fun(fun(D,E),fun(B,B)),F: fun(D,E),J5: fun(fun(F2,D),fun(B,B)),G: fun(F2,D),K: fun(fun(F2,E),fun(B,B))] :
( isomorphic_tuple(A,B,C,Isom)
=> ( ( aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(fun(D,E),fun(B,B),H,F)),aa(fun(F2,D),fun(B,B),J5,G)) = aa(fun(F2,E),fun(B,B),K,aa(fun(F2,D),fun(F2,E),aa(fun(D,E),fun(fun(F2,D),fun(F2,E)),comp(D,E,F2),F),G)) )
=> ( aa(fun(A,A),fun(A,A),aa(fun(A,A),fun(fun(A,A),fun(A,A)),comp(A,A,A),aa(fun(D,E),fun(A,A),aa(fun(fun(D,E),fun(B,B)),fun(fun(D,E),fun(A,A)),aa(fun(fun(B,B),fun(A,A)),fun(fun(fun(D,E),fun(B,B)),fun(fun(D,E),fun(A,A))),comp(fun(B,B),fun(A,A),fun(D,E)),iso_tuple_fst_update(A,B,C,Isom)),H),F)),aa(fun(F2,D),fun(A,A),aa(fun(fun(F2,D),fun(B,B)),fun(fun(F2,D),fun(A,A)),aa(fun(fun(B,B),fun(A,A)),fun(fun(fun(F2,D),fun(B,B)),fun(fun(F2,D),fun(A,A))),comp(fun(B,B),fun(A,A),fun(F2,D)),iso_tuple_fst_update(A,B,C,Isom)),J5),G)) = aa(fun(F2,E),fun(A,A),aa(fun(fun(F2,E),fun(B,B)),fun(fun(F2,E),fun(A,A)),aa(fun(fun(B,B),fun(A,A)),fun(fun(fun(F2,E),fun(B,B)),fun(fun(F2,E),fun(A,A))),comp(fun(B,B),fun(A,A),fun(F2,E)),iso_tuple_fst_update(A,B,C,Isom)),K),aa(fun(F2,D),fun(F2,E),aa(fun(D,E),fun(fun(F2,D),fun(F2,E)),comp(D,E,F2),F),G)) ) ) ) ).
% isomorphic_tuple.iso_tuple_update_compose_fst_fst
tff(fact_2626_isomorphic__tuple_Oiso__tuple__update__compose__snd__snd,axiom,
! [C: $tType,B: $tType,A: $tType,E: $tType,D: $tType,F2: $tType,Isom: tuple_isomorphism(A,B,C),H: fun(fun(D,E),fun(C,C)),F: fun(D,E),J5: fun(fun(F2,D),fun(C,C)),G: fun(F2,D),K: fun(fun(F2,E),fun(C,C))] :
( isomorphic_tuple(A,B,C,Isom)
=> ( ( aa(fun(C,C),fun(C,C),aa(fun(C,C),fun(fun(C,C),fun(C,C)),comp(C,C,C),aa(fun(D,E),fun(C,C),H,F)),aa(fun(F2,D),fun(C,C),J5,G)) = aa(fun(F2,E),fun(C,C),K,aa(fun(F2,D),fun(F2,E),aa(fun(D,E),fun(fun(F2,D),fun(F2,E)),comp(D,E,F2),F),G)) )
=> ( aa(fun(A,A),fun(A,A),aa(fun(A,A),fun(fun(A,A),fun(A,A)),comp(A,A,A),aa(fun(D,E),fun(A,A),aa(fun(fun(D,E),fun(C,C)),fun(fun(D,E),fun(A,A)),aa(fun(fun(C,C),fun(A,A)),fun(fun(fun(D,E),fun(C,C)),fun(fun(D,E),fun(A,A))),comp(fun(C,C),fun(A,A),fun(D,E)),iso_tuple_snd_update(A,B,C,Isom)),H),F)),aa(fun(F2,D),fun(A,A),aa(fun(fun(F2,D),fun(C,C)),fun(fun(F2,D),fun(A,A)),aa(fun(fun(C,C),fun(A,A)),fun(fun(fun(F2,D),fun(C,C)),fun(fun(F2,D),fun(A,A))),comp(fun(C,C),fun(A,A),fun(F2,D)),iso_tuple_snd_update(A,B,C,Isom)),J5),G)) = aa(fun(F2,E),fun(A,A),aa(fun(fun(F2,E),fun(C,C)),fun(fun(F2,E),fun(A,A)),aa(fun(fun(C,C),fun(A,A)),fun(fun(fun(F2,E),fun(C,C)),fun(fun(F2,E),fun(A,A))),comp(fun(C,C),fun(A,A),fun(F2,E)),iso_tuple_snd_update(A,B,C,Isom)),K),aa(fun(F2,D),fun(F2,E),aa(fun(D,E),fun(fun(F2,D),fun(F2,E)),comp(D,E,F2),F),G)) ) ) ) ).
% isomorphic_tuple.iso_tuple_update_compose_snd_snd
tff(fact_2627_isomorphic__tuple_Oiso__tuple__update__swap__fst__snd,axiom,
! [E: $tType,C: $tType,B: $tType,A: $tType,D: $tType,Isom: tuple_isomorphism(A,B,C),H: fun(D,fun(C,C)),F: D,J5: fun(E,fun(B,B)),G: E] :
( isomorphic_tuple(A,B,C,Isom)
=> ( aa(fun(A,A),fun(A,A),aa(fun(A,A),fun(fun(A,A),fun(A,A)),comp(A,A,A),aa(D,fun(A,A),aa(fun(D,fun(C,C)),fun(D,fun(A,A)),aa(fun(fun(C,C),fun(A,A)),fun(fun(D,fun(C,C)),fun(D,fun(A,A))),comp(fun(C,C),fun(A,A),D),iso_tuple_snd_update(A,B,C,Isom)),H),F)),aa(E,fun(A,A),aa(fun(E,fun(B,B)),fun(E,fun(A,A)),aa(fun(fun(B,B),fun(A,A)),fun(fun(E,fun(B,B)),fun(E,fun(A,A))),comp(fun(B,B),fun(A,A),E),iso_tuple_fst_update(A,B,C,Isom)),J5),G)) = aa(fun(A,A),fun(A,A),aa(fun(A,A),fun(fun(A,A),fun(A,A)),comp(A,A,A),aa(E,fun(A,A),aa(fun(E,fun(B,B)),fun(E,fun(A,A)),aa(fun(fun(B,B),fun(A,A)),fun(fun(E,fun(B,B)),fun(E,fun(A,A))),comp(fun(B,B),fun(A,A),E),iso_tuple_fst_update(A,B,C,Isom)),J5),G)),aa(D,fun(A,A),aa(fun(D,fun(C,C)),fun(D,fun(A,A)),aa(fun(fun(C,C),fun(A,A)),fun(fun(D,fun(C,C)),fun(D,fun(A,A))),comp(fun(C,C),fun(A,A),D),iso_tuple_snd_update(A,B,C,Isom)),H),F)) ) ) ).
% isomorphic_tuple.iso_tuple_update_swap_fst_snd
tff(fact_2628_isomorphic__tuple_Oiso__tuple__snd__update__accessor__cong__assist,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,Isom: tuple_isomorphism(A,B,C),F: fun(fun(D,D),fun(C,C)),G: fun(C,D)] :
( isomorphic_tuple(A,B,C,Isom)
=> ( iso_tu6617205062836021118assist(D,C,F,G)
=> iso_tu6617205062836021118assist(D,A,aa(fun(fun(D,D),fun(C,C)),fun(fun(D,D),fun(A,A)),aa(fun(fun(C,C),fun(A,A)),fun(fun(fun(D,D),fun(C,C)),fun(fun(D,D),fun(A,A))),comp(fun(C,C),fun(A,A),fun(D,D)),iso_tuple_snd_update(A,B,C,Isom)),F),aa(fun(A,C),fun(A,D),aa(fun(C,D),fun(fun(A,C),fun(A,D)),comp(C,D,A),G),iso_tuple_snd(A,B,C,Isom))) ) ) ).
% isomorphic_tuple.iso_tuple_snd_update_accessor_cong_assist
tff(fact_2629_isomorphic__tuple_Oiso__tuple__fst__update__accessor__cong__assist,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,Isom: tuple_isomorphism(A,B,C),F: fun(fun(D,D),fun(B,B)),G: fun(B,D)] :
( isomorphic_tuple(A,B,C,Isom)
=> ( iso_tu6617205062836021118assist(D,B,F,G)
=> iso_tu6617205062836021118assist(D,A,aa(fun(fun(D,D),fun(B,B)),fun(fun(D,D),fun(A,A)),aa(fun(fun(B,B),fun(A,A)),fun(fun(fun(D,D),fun(B,B)),fun(fun(D,D),fun(A,A))),comp(fun(B,B),fun(A,A),fun(D,D)),iso_tuple_fst_update(A,B,C,Isom)),F),aa(fun(A,B),fun(A,D),aa(fun(B,D),fun(fun(A,B),fun(A,D)),comp(B,D,A),G),iso_tuple_fst(A,B,C,Isom))) ) ) ).
% isomorphic_tuple.iso_tuple_fst_update_accessor_cong_assist
tff(fact_2630_iso__tuple__cons__def,axiom,
! [C: $tType,B: $tType,A: $tType,Isom: tuple_isomorphism(A,B,C)] : iso_tuple_cons(A,B,C,Isom) = aa(fun(product_prod(B,C),A),fun(B,fun(C,A)),product_curry(B,C,A),abst(A,B,C,Isom)) ).
% iso_tuple_cons_def
tff(fact_2631_isomorphic__tuple_Oiso__tuple__snd__update__accessor__eq__assist,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,Isom: tuple_isomorphism(A,B,C),F: fun(fun(D,D),fun(C,C)),G: fun(C,D),B2: C,U: fun(D,D),B3: C,V: D,A3: B] :
( isomorphic_tuple(A,B,C,Isom)
=> ( iso_tu8710522896132340869assist(D,C,F,G,B2,U,B3,V)
=> iso_tu8710522896132340869assist(D,A,aa(fun(fun(D,D),fun(C,C)),fun(fun(D,D),fun(A,A)),aa(fun(fun(C,C),fun(A,A)),fun(fun(fun(D,D),fun(C,C)),fun(fun(D,D),fun(A,A))),comp(fun(C,C),fun(A,A),fun(D,D)),iso_tuple_snd_update(A,B,C,Isom)),F),aa(fun(A,C),fun(A,D),aa(fun(C,D),fun(fun(A,C),fun(A,D)),comp(C,D,A),G),iso_tuple_snd(A,B,C,Isom)),aa(C,A,aa(B,fun(C,A),iso_tuple_cons(A,B,C,Isom),A3),B2),U,aa(C,A,aa(B,fun(C,A),iso_tuple_cons(A,B,C,Isom),A3),B3),V) ) ) ).
% isomorphic_tuple.iso_tuple_snd_update_accessor_eq_assist
tff(fact_2632_isomorphic__tuple_Oiso__tuple__fst__update__accessor__eq__assist,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,Isom: tuple_isomorphism(A,B,C),F: fun(fun(D,D),fun(B,B)),G: fun(B,D),A3: B,U: fun(D,D),A4: B,V: D,B2: C] :
( isomorphic_tuple(A,B,C,Isom)
=> ( iso_tu8710522896132340869assist(D,B,F,G,A3,U,A4,V)
=> iso_tu8710522896132340869assist(D,A,aa(fun(fun(D,D),fun(B,B)),fun(fun(D,D),fun(A,A)),aa(fun(fun(B,B),fun(A,A)),fun(fun(fun(D,D),fun(B,B)),fun(fun(D,D),fun(A,A))),comp(fun(B,B),fun(A,A),fun(D,D)),iso_tuple_fst_update(A,B,C,Isom)),F),aa(fun(A,B),fun(A,D),aa(fun(B,D),fun(fun(A,B),fun(A,D)),comp(B,D,A),G),iso_tuple_fst(A,B,C,Isom)),aa(C,A,aa(B,fun(C,A),iso_tuple_cons(A,B,C,Isom),A3),B2),U,aa(C,A,aa(B,fun(C,A),iso_tuple_cons(A,B,C,Isom),A4),B2),V) ) ) ).
% isomorphic_tuple.iso_tuple_fst_update_accessor_eq_assist
tff(fact_2633_Leaf__def,axiom,
! [B: $tType,A: $tType] : old_Leaf(A,B) = aa(fun(A,sum_sum(A,nat)),fun(A,set(old_node(A,B))),aa(fun(sum_sum(A,nat),set(old_node(A,B))),fun(fun(A,sum_sum(A,nat)),fun(A,set(old_node(A,B)))),comp(sum_sum(A,nat),set(old_node(A,B)),A),old_Atom(A,B)),sum_Inl(A,nat)) ).
% Leaf_def
tff(fact_2634_update__accessor__noop__compE,axiom,
! [A: $tType,B: $tType,Upd: fun(fun(A,A),fun(B,B)),Ac: fun(B,A),F: fun(A,A),X2: B,G: fun(A,A)] :
( iso_tu6617205062836021118assist(A,B,Upd,Ac)
=> ( ( aa(A,A,F,aa(B,A,Ac,X2)) = aa(B,A,Ac,X2) )
=> ( aa(B,B,aa(fun(A,A),fun(B,B),Upd,aa(fun(A,A),fun(A,A),aa(fun(A,A),fun(fun(A,A),fun(A,A)),comp(A,A,A),G),F)),X2) = aa(B,B,aa(fun(A,A),fun(B,B),Upd,G),X2) ) ) ) ).
% update_accessor_noop_compE
tff(fact_2635_isomorphic__tuple_Oiso__tuple__surjective__proof__assist__step,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType,Isom: tuple_isomorphism(A,B,C),V: D,A3: B,F: fun(D,A),B2: C] :
( isomorphic_tuple(A,B,C,Isom)
=> ( iso_tu562120763810524177assist(D,B,V,A3,aa(fun(D,A),fun(D,B),aa(fun(A,B),fun(fun(D,A),fun(D,B)),comp(A,B,D),iso_tuple_fst(A,B,C,Isom)),F))
=> ( iso_tu562120763810524177assist(D,C,V,B2,aa(fun(D,A),fun(D,C),aa(fun(A,C),fun(fun(D,A),fun(D,C)),comp(A,C,D),iso_tuple_snd(A,B,C,Isom)),F))
=> iso_tu562120763810524177assist(D,A,V,aa(C,A,aa(B,fun(C,A),iso_tuple_cons(A,B,C,Isom),A3),B2),F) ) ) ) ).
% isomorphic_tuple.iso_tuple_surjective_proof_assist_step
tff(fact_2636_Numb__def,axiom,
! [B: $tType,A: $tType] : old_Numb(A,B) = aa(fun(nat,sum_sum(A,nat)),fun(nat,set(old_node(A,B))),aa(fun(sum_sum(A,nat),set(old_node(A,B))),fun(fun(nat,sum_sum(A,nat)),fun(nat,set(old_node(A,B)))),comp(sum_sum(A,nat),set(old_node(A,B)),nat),old_Atom(A,B)),sum_Inr(nat,A)) ).
% Numb_def
tff(fact_2637_ndepth__def,axiom,
! [B: $tType,A: $tType,N4: old_node(A,B)] : old_ndepth(A,B,N4) = aa(product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat)),nat,aa(fun(fun(nat,sum_sum(B,nat)),fun(sum_sum(A,nat),nat)),fun(product_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat)),nat),product_case_prod(fun(nat,sum_sum(B,nat)),sum_sum(A,nat),nat),aTP_Lamp_wl(fun(nat,sum_sum(B,nat)),fun(sum_sum(A,nat),nat))),old_Rep_Node(A,B,N4)) ).
% ndepth_def
tff(fact_2638_uprod__def,axiom,
! [B: $tType,A: $tType,A6: set(set(old_node(A,B))),B6: set(set(old_node(A,B)))] : old_uprod(A,B,A6,B6) = aa(set(set(set(old_node(A,B)))),set(set(old_node(A,B))),complete_Sup_Sup(set(set(old_node(A,B)))),aa(set(set(old_node(A,B))),set(set(set(old_node(A,B)))),aa(fun(set(old_node(A,B)),set(set(old_node(A,B)))),fun(set(set(old_node(A,B))),set(set(set(old_node(A,B))))),image2(set(old_node(A,B)),set(set(old_node(A,B)))),aTP_Lamp_wn(set(set(old_node(A,B))),fun(set(old_node(A,B)),set(set(old_node(A,B)))),B6)),A6)) ).
% uprod_def
tff(fact_2639_ext__surjective,axiom,
! [Z: $tType,X2: heap_ext(Z)] : X2 = heap_ext2(Z,aa(heap_ext(Z),fun(typerep,fun(nat,list(nat))),aa(fun(heap_ext(Z),product_prod(fun(typerep,fun(nat,list(nat))),product_prod(fun(typerep,fun(nat,nat)),nat))),fun(heap_ext(Z),fun(typerep,fun(nat,list(nat)))),aa(fun(product_prod(fun(typerep,fun(nat,list(nat))),product_prod(fun(typerep,fun(nat,nat)),nat)),fun(typerep,fun(nat,list(nat)))),fun(fun(heap_ext(Z),product_prod(fun(typerep,fun(nat,list(nat))),product_prod(fun(typerep,fun(nat,nat)),nat))),fun(heap_ext(Z),fun(typerep,fun(nat,list(nat))))),comp(product_prod(fun(typerep,fun(nat,list(nat))),product_prod(fun(typerep,fun(nat,nat)),nat)),fun(typerep,fun(nat,list(nat))),heap_ext(Z)),iso_tuple_fst(product_prod(fun(typerep,fun(nat,list(nat))),product_prod(fun(typerep,fun(nat,nat)),nat)),fun(typerep,fun(nat,list(nat))),product_prod(fun(typerep,fun(nat,nat)),nat),tuple_iso_tuple(fun(typerep,fun(nat,list(nat))),product_prod(fun(typerep,fun(nat,nat)),nat)))),aa(fun(heap_ext(Z),heap_ext(Z)),fun(heap_ext(Z),product_prod(fun(typerep,fun(nat,list(nat))),product_prod(fun(typerep,fun(nat,nat)),nat))),aa(fun(heap_ext(Z),product_prod(fun(typerep,fun(nat,list(nat))),product_prod(fun(typerep,fun(nat,nat)),nat))),fun(fun(heap_ext(Z),heap_ext(Z)),fun(heap_ext(Z),product_prod(fun(typerep,fun(nat,list(nat))),product_prod(fun(typerep,fun(nat,nat)),nat)))),comp(heap_ext(Z),product_prod(fun(typerep,fun(nat,list(nat))),product_prod(fun(typerep,fun(nat,nat)),nat)),heap_ext(Z)),iso_tuple_fst(heap_ext(Z),product_prod(fun(typerep,fun(nat,list(nat))),product_prod(fun(typerep,fun(nat,nat)),nat)),Z,heap_h6279449762549114721le_Iso(Z))),id(heap_ext(Z)))),X2),aa(heap_ext(Z),fun(typerep,fun(nat,nat)),aa(fun(heap_ext(Z),product_prod(fun(typerep,fun(nat,nat)),nat)),fun(heap_ext(Z),fun(typerep,fun(nat,nat))),aa(fun(product_prod(fun(typerep,fun(nat,nat)),nat),fun(typerep,fun(nat,nat))),fun(fun(heap_ext(Z),product_prod(fun(typerep,fun(nat,nat)),nat)),fun(heap_ext(Z),fun(typerep,fun(nat,nat)))),comp(product_prod(fun(typerep,fun(nat,nat)),nat),fun(typerep,fun(nat,nat)),heap_ext(Z)),iso_tuple_fst(product_prod(fun(typerep,fun(nat,nat)),nat),fun(typerep,fun(nat,nat)),nat,tuple_iso_tuple(fun(typerep,fun(nat,nat)),nat))),aa(fun(heap_ext(Z),product_prod(fun(typerep,fun(nat,list(nat))),product_prod(fun(typerep,fun(nat,nat)),nat))),fun(heap_ext(Z),product_prod(fun(typerep,fun(nat,nat)),nat)),aa(fun(product_prod(fun(typerep,fun(nat,list(nat))),product_prod(fun(typerep,fun(nat,nat)),nat)),product_prod(fun(typerep,fun(nat,nat)),nat)),fun(fun(heap_ext(Z),product_prod(fun(typerep,fun(nat,list(nat))),product_prod(fun(typerep,fun(nat,nat)),nat))),fun(heap_ext(Z),product_prod(fun(typerep,fun(nat,nat)),nat))),comp(product_prod(fun(typerep,fun(nat,list(nat))),product_prod(fun(typerep,fun(nat,nat)),nat)),product_prod(fun(typerep,fun(nat,nat)),nat),heap_ext(Z)),iso_tuple_snd(product_prod(fun(typerep,fun(nat,list(nat))),product_prod(fun(typerep,fun(nat,nat)),nat)),fun(typerep,fun(nat,list(nat))),product_prod(fun(typerep,fun(nat,nat)),nat),tuple_iso_tuple(fun(typerep,fun(nat,list(nat))),product_prod(fun(typerep,fun(nat,nat)),nat)))),aa(fun(heap_ext(Z),heap_ext(Z)),fun(heap_ext(Z),product_prod(fun(typerep,fun(nat,list(nat))),product_prod(fun(typerep,fun(nat,nat)),nat))),aa(fun(heap_ext(Z),product_prod(fun(typerep,fun(nat,list(nat))),product_prod(fun(typerep,fun(nat,nat)),nat))),fun(fun(heap_ext(Z),heap_ext(Z)),fun(heap_ext(Z),product_prod(fun(typerep,fun(nat,list(nat))),product_prod(fun(typerep,fun(nat,nat)),nat)))),comp(heap_ext(Z),product_prod(fun(typerep,fun(nat,list(nat))),product_prod(fun(typerep,fun(nat,nat)),nat)),heap_ext(Z)),iso_tuple_fst(heap_ext(Z),product_prod(fun(typerep,fun(nat,list(nat))),product_prod(fun(typerep,fun(nat,nat)),nat)),Z,heap_h6279449762549114721le_Iso(Z))),id(heap_ext(Z))))),X2),aa(heap_ext(Z),nat,aa(fun(heap_ext(Z),product_prod(fun(typerep,fun(nat,nat)),nat)),fun(heap_ext(Z),nat),aa(fun(product_prod(fun(typerep,fun(nat,nat)),nat),nat),fun(fun(heap_ext(Z),product_prod(fun(typerep,fun(nat,nat)),nat)),fun(heap_ext(Z),nat)),comp(product_prod(fun(typerep,fun(nat,nat)),nat),nat,heap_ext(Z)),iso_tuple_snd(product_prod(fun(typerep,fun(nat,nat)),nat),fun(typerep,fun(nat,nat)),nat,tuple_iso_tuple(fun(typerep,fun(nat,nat)),nat))),aa(fun(heap_ext(Z),product_prod(fun(typerep,fun(nat,list(nat))),product_prod(fun(typerep,fun(nat,nat)),nat))),fun(heap_ext(Z),product_prod(fun(typerep,fun(nat,nat)),nat)),aa(fun(product_prod(fun(typerep,fun(nat,list(nat))),product_prod(fun(typerep,fun(nat,nat)),nat)),product_prod(fun(typerep,fun(nat,nat)),nat)),fun(fun(heap_ext(Z),product_prod(fun(typerep,fun(nat,list(nat))),product_prod(fun(typerep,fun(nat,nat)),nat))),fun(heap_ext(Z),product_prod(fun(typerep,fun(nat,nat)),nat))),comp(product_prod(fun(typerep,fun(nat,list(nat))),product_prod(fun(typerep,fun(nat,nat)),nat)),product_prod(fun(typerep,fun(nat,nat)),nat),heap_ext(Z)),iso_tuple_snd(product_prod(fun(typerep,fun(nat,list(nat))),product_prod(fun(typerep,fun(nat,nat)),nat)),fun(typerep,fun(nat,list(nat))),product_prod(fun(typerep,fun(nat,nat)),nat),tuple_iso_tuple(fun(typerep,fun(nat,list(nat))),product_prod(fun(typerep,fun(nat,nat)),nat)))),aa(fun(heap_ext(Z),heap_ext(Z)),fun(heap_ext(Z),product_prod(fun(typerep,fun(nat,list(nat))),product_prod(fun(typerep,fun(nat,nat)),nat))),aa(fun(heap_ext(Z),product_prod(fun(typerep,fun(nat,list(nat))),product_prod(fun(typerep,fun(nat,nat)),nat))),fun(fun(heap_ext(Z),heap_ext(Z)),fun(heap_ext(Z),product_prod(fun(typerep,fun(nat,list(nat))),product_prod(fun(typerep,fun(nat,nat)),nat)))),comp(heap_ext(Z),product_prod(fun(typerep,fun(nat,list(nat))),product_prod(fun(typerep,fun(nat,nat)),nat)),heap_ext(Z)),iso_tuple_fst(heap_ext(Z),product_prod(fun(typerep,fun(nat,list(nat))),product_prod(fun(typerep,fun(nat,nat)),nat)),Z,heap_h6279449762549114721le_Iso(Z))),id(heap_ext(Z))))),X2),aa(heap_ext(Z),Z,aa(fun(heap_ext(Z),heap_ext(Z)),fun(heap_ext(Z),Z),aa(fun(heap_ext(Z),Z),fun(fun(heap_ext(Z),heap_ext(Z)),fun(heap_ext(Z),Z)),comp(heap_ext(Z),Z,heap_ext(Z)),iso_tuple_snd(heap_ext(Z),product_prod(fun(typerep,fun(nat,list(nat))),product_prod(fun(typerep,fun(nat,nat)),nat)),Z,heap_h6279449762549114721le_Iso(Z))),id(heap_ext(Z))),X2)) ).
% ext_surjective
tff(fact_2640_eval__SUP,axiom,
! [A: $tType,B: $tType,F: fun(B,pred(A)),A6: set(B)] : aa(pred(A),fun(A,bool),eval(A),aa(set(pred(A)),pred(A),complete_Sup_Sup(pred(A)),aa(set(B),set(pred(A)),aa(fun(B,pred(A)),fun(set(B),set(pred(A))),image2(B,pred(A)),F),A6))) = aa(set(fun(A,bool)),fun(A,bool),complete_Sup_Sup(fun(A,bool)),aa(set(B),set(fun(A,bool)),aa(fun(B,fun(A,bool)),fun(set(B),set(fun(A,bool))),image2(B,fun(A,bool)),aa(fun(B,pred(A)),fun(B,fun(A,bool)),aa(fun(pred(A),fun(A,bool)),fun(fun(B,pred(A)),fun(B,fun(A,bool))),comp(pred(A),fun(A,bool),B),eval(A)),F)),A6)) ).
% eval_SUP
tff(fact_2641_eval__Inf,axiom,
! [A: $tType,A6: set(pred(A))] : aa(pred(A),fun(A,bool),eval(A),aa(set(pred(A)),pred(A),complete_Inf_Inf(pred(A)),A6)) = aa(set(fun(A,bool)),fun(A,bool),complete_Inf_Inf(fun(A,bool)),aa(set(pred(A)),set(fun(A,bool)),aa(fun(pred(A),fun(A,bool)),fun(set(pred(A)),set(fun(A,bool))),image2(pred(A),fun(A,bool)),eval(A)),A6)) ).
% eval_Inf
tff(fact_2642_eval__Sup,axiom,
! [A: $tType,A6: set(pred(A))] : aa(pred(A),fun(A,bool),eval(A),aa(set(pred(A)),pred(A),complete_Sup_Sup(pred(A)),A6)) = aa(set(fun(A,bool)),fun(A,bool),complete_Sup_Sup(fun(A,bool)),aa(set(pred(A)),set(fun(A,bool)),aa(fun(pred(A),fun(A,bool)),fun(set(pred(A)),set(fun(A,bool))),image2(pred(A),fun(A,bool)),eval(A)),A6)) ).
% eval_Sup
tff(fact_2643_eval__INF,axiom,
! [A: $tType,B: $tType,F: fun(B,pred(A)),A6: set(B)] : aa(pred(A),fun(A,bool),eval(A),aa(set(pred(A)),pred(A),complete_Inf_Inf(pred(A)),aa(set(B),set(pred(A)),aa(fun(B,pred(A)),fun(set(B),set(pred(A))),image2(B,pred(A)),F),A6))) = aa(set(fun(A,bool)),fun(A,bool),complete_Inf_Inf(fun(A,bool)),aa(set(B),set(fun(A,bool)),aa(fun(B,fun(A,bool)),fun(set(B),set(fun(A,bool))),image2(B,fun(A,bool)),aa(fun(B,pred(A)),fun(B,fun(A,bool)),aa(fun(pred(A),fun(A,bool)),fun(fun(B,pred(A)),fun(B,fun(A,bool))),comp(pred(A),fun(A,bool),B),eval(A)),F)),A6)) ).
% eval_INF
tff(fact_2644_eval__map,axiom,
! [A: $tType,B: $tType,F: fun(B,A),P2: pred(B)] : aa(pred(A),fun(A,bool),eval(A),aa(pred(B),pred(A),map2(B,A,F),P2)) = aa(set(fun(A,bool)),fun(A,bool),complete_Sup_Sup(fun(A,bool)),aa(set(B),set(fun(A,bool)),aa(fun(B,fun(A,bool)),fun(set(B),set(fun(A,bool))),image2(B,fun(A,bool)),aTP_Lamp_wo(fun(B,A),fun(B,fun(A,bool)),F)),aa(fun(B,bool),set(B),collect(B),aa(pred(B),fun(B,bool),eval(B),P2)))) ).
% eval_map
tff(fact_2645_eval__bind,axiom,
! [A: $tType,B: $tType,P2: pred(B),F: fun(B,pred(A))] : aa(pred(A),fun(A,bool),eval(A),bind3(B,A,P2,F)) = aa(pred(A),fun(A,bool),eval(A),aa(set(pred(A)),pred(A),complete_Sup_Sup(pred(A)),aa(set(B),set(pred(A)),aa(fun(B,pred(A)),fun(set(B),set(pred(A))),image2(B,pred(A)),F),aa(fun(B,bool),set(B),collect(B),aa(pred(B),fun(B,bool),eval(B),P2))))) ).
% eval_bind
tff(fact_2646_Predicate_Obind__def,axiom,
! [B: $tType,A: $tType,P2: pred(A),F: fun(A,pred(B))] : bind3(A,B,P2,F) = aa(set(pred(B)),pred(B),complete_Sup_Sup(pred(B)),aa(set(A),set(pred(B)),aa(fun(A,pred(B)),fun(set(A),set(pred(B))),image2(A,pred(B)),F),aa(fun(A,bool),set(A),collect(A),aa(pred(A),fun(A,bool),eval(A),P2)))) ).
% Predicate.bind_def
tff(fact_2647_Sup__pred__def,axiom,
! [A: $tType,A6: set(pred(A))] : aa(set(pred(A)),pred(A),complete_Sup_Sup(pred(A)),A6) = aa(fun(A,bool),pred(A),pred2(A),aa(set(fun(A,bool)),fun(A,bool),complete_Sup_Sup(fun(A,bool)),aa(set(pred(A)),set(fun(A,bool)),aa(fun(pred(A),fun(A,bool)),fun(set(pred(A)),set(fun(A,bool))),image2(pred(A),fun(A,bool)),eval(A)),A6))) ).
% Sup_pred_def
tff(fact_2648_Inf__pred__def,axiom,
! [A: $tType,A6: set(pred(A))] : aa(set(pred(A)),pred(A),complete_Inf_Inf(pred(A)),A6) = aa(fun(A,bool),pred(A),pred2(A),aa(set(fun(A,bool)),fun(A,bool),complete_Inf_Inf(fun(A,bool)),aa(set(pred(A)),set(fun(A,bool)),aa(fun(pred(A),fun(A,bool)),fun(set(pred(A)),set(fun(A,bool))),image2(pred(A),fun(A,bool)),eval(A)),A6))) ).
% Inf_pred_def
tff(fact_2649_Sup__bind,axiom,
! [A: $tType,B: $tType,A6: set(pred(B)),F: fun(B,pred(A))] : bind3(B,A,aa(set(pred(B)),pred(B),complete_Sup_Sup(pred(B)),A6),F) = aa(set(pred(A)),pred(A),complete_Sup_Sup(pred(A)),aa(set(pred(B)),set(pred(A)),aa(fun(pred(B),pred(A)),fun(set(pred(B)),set(pred(A))),image2(pred(B),pred(A)),aTP_Lamp_wp(fun(B,pred(A)),fun(pred(B),pred(A)),F)),A6)) ).
% Sup_bind
tff(fact_2650_Predicate_Omap_Ocomp,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(B,C),G: fun(A,B)] : aa(fun(pred(A),pred(B)),fun(pred(A),pred(C)),aa(fun(pred(B),pred(C)),fun(fun(pred(A),pred(B)),fun(pred(A),pred(C))),comp(pred(B),pred(C),pred(A)),map2(B,C,F)),map2(A,B,G)) = map2(A,C,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),F),G)) ).
% Predicate.map.comp
tff(fact_2651_Predicate_Omap_Ocompositionality,axiom,
! [B: $tType,C: $tType,A: $tType,F: fun(B,C),G: fun(A,B),Pred: pred(A)] : aa(pred(B),pred(C),map2(B,C,F),aa(pred(A),pred(B),map2(A,B,G),Pred)) = aa(pred(A),pred(C),map2(A,C,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),F),G)),Pred) ).
% Predicate.map.compositionality
tff(fact_2652_Predicate_Omap__def,axiom,
! [B: $tType,A: $tType,F: fun(A,B),P2: pred(A)] : aa(pred(A),pred(B),map2(A,B,F),P2) = bind3(A,B,P2,aa(fun(A,B),fun(A,pred(B)),aa(fun(B,pred(B)),fun(fun(A,B),fun(A,pred(B))),comp(B,pred(B),A),single(B)),F)) ).
% Predicate.map_def
tff(fact_2653_pred__of__set__def,axiom,
! [A: $tType] : pred_of_set(A) = aa(fun(set(A),fun(A,bool)),fun(set(A),pred(A)),aa(fun(fun(A,bool),pred(A)),fun(fun(set(A),fun(A,bool)),fun(set(A),pred(A))),comp(fun(A,bool),pred(A),set(A)),pred2(A)),aTP_Lamp_a(set(A),fun(A,bool))) ).
% pred_of_set_def
tff(fact_2654_set__of__pred__def,axiom,
! [A: $tType] : set_of_pred(A) = aa(fun(pred(A),fun(A,bool)),fun(pred(A),set(A)),aa(fun(fun(A,bool),set(A)),fun(fun(pred(A),fun(A,bool)),fun(pred(A),set(A))),comp(fun(A,bool),set(A),pred(A)),collect(A)),eval(A)) ).
% set_of_pred_def
tff(fact_2655_Random__Pred_Obind__def,axiom,
! [B: $tType,A: $tType,R3: fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),F: fun(A,fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)))),X4: product_prod(code_natural,code_natural)] : aa(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)),random_bind(A,B,R3,F),X4) = aa(product_prod(pred(A),product_prod(code_natural,code_natural)),product_prod(pred(B),product_prod(code_natural,code_natural)),aa(fun(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)))),fun(product_prod(pred(A),product_prod(code_natural,code_natural)),product_prod(pred(B),product_prod(code_natural,code_natural))),product_case_prod(pred(A),product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural))),aTP_Lamp_ws(fun(A,fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)))),fun(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)))),F)),aa(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)),R3,X4)) ).
% Random_Pred.bind_def
tff(fact_2656_pred__of__set__fold__sup,axiom,
! [A: $tType,A6: set(A)] :
( finite_finite2(A,A6)
=> ( aa(set(A),pred(A),pred_of_set(A),A6) = finite_fold(pred(A),pred(A),sup_sup(pred(A)),bot_bot(pred(A)),aa(set(A),set(pred(A)),aa(fun(A,pred(A)),fun(set(A),set(pred(A))),image2(A,pred(A)),single(A)),A6)) ) ) ).
% pred_of_set_fold_sup
tff(fact_2657_wfrec__rel_Ocases,axiom,
! [B: $tType,A: $tType,R3: set(product_prod(A,A)),F6: fun(fun(A,B),fun(A,B)),A1: A,A22: B] :
( wfrec_rel(A,B,R3,F6,A1,A22)
=> ~ ! [G3: fun(A,B)] :
( ( A22 = aa(A,B,aa(fun(A,B),fun(A,B),F6,G3),A1) )
=> ~ ! [Z6: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z6),A1)),R3))
=> wfrec_rel(A,B,R3,F6,Z6,aa(A,B,G3,Z6)) ) ) ) ).
% wfrec_rel.cases
tff(fact_2658_wfrecI,axiom,
! [B: $tType,A: $tType,X2: A,R3: set(product_prod(A,A)),F6: fun(fun(A,B),fun(A,B)),G: fun(A,B)] :
( ! [Z3: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z3),X2)),R3))
=> wfrec_rel(A,B,R3,F6,Z3,aa(A,B,G,Z3)) )
=> wfrec_rel(A,B,R3,F6,X2,aa(A,B,aa(fun(A,B),fun(A,B),F6,G),X2)) ) ).
% wfrecI
tff(fact_2659_wfrec__rel_Osimps,axiom,
! [B: $tType,A: $tType,R3: set(product_prod(A,A)),F6: fun(fun(A,B),fun(A,B)),A1: A,A22: B] :
( wfrec_rel(A,B,R3,F6,A1,A22)
<=> ? [X5: A,G5: fun(A,B)] :
( ( A1 = X5 )
& ( A22 = aa(A,B,aa(fun(A,B),fun(A,B),F6,G5),X5) )
& ! [Xa3: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa3),X5)),R3))
=> wfrec_rel(A,B,R3,F6,Xa3,aa(A,B,G5,Xa3)) ) ) ) ).
% wfrec_rel.simps
tff(fact_2660_Random__Pred_Ounion__def,axiom,
! [A: $tType,R1: fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),R22: fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),X4: product_prod(code_natural,code_natural)] : random_union(A,R1,R22,X4) = aa(product_prod(pred(A),product_prod(code_natural,code_natural)),product_prod(pred(A),product_prod(code_natural,code_natural)),aa(fun(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),fun(product_prod(pred(A),product_prod(code_natural,code_natural)),product_prod(pred(A),product_prod(code_natural,code_natural))),product_case_prod(pred(A),product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),aTP_Lamp_wu(fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),fun(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),R22)),aa(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)),R1,X4)) ).
% Random_Pred.union_def
tff(fact_2661_Random__Pred_ORandom__def,axiom,
! [A: $tType,G: fun(product_prod(code_natural,code_natural),product_prod(product_prod(A,fun(product_unit,code_term)),product_prod(code_natural,code_natural)))] : random_Random(A,G) = product_scomp(product_prod(code_natural,code_natural),product_prod(A,fun(product_unit,code_term)),product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)),G,aa(fun(product_prod(A,fun(product_unit,code_term)),pred(A)),fun(product_prod(A,fun(product_unit,code_term)),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),aa(fun(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),fun(fun(product_prod(A,fun(product_unit,code_term)),pred(A)),fun(product_prod(A,fun(product_unit,code_term)),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))))),comp(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),product_prod(A,fun(product_unit,code_term))),product_Pair(pred(A),product_prod(code_natural,code_natural))),aa(fun(product_prod(A,fun(product_unit,code_term)),A),fun(product_prod(A,fun(product_unit,code_term)),pred(A)),aa(fun(A,pred(A)),fun(fun(product_prod(A,fun(product_unit,code_term)),A),fun(product_prod(A,fun(product_unit,code_term)),pred(A))),comp(A,pred(A),product_prod(A,fun(product_unit,code_term))),single(A)),product_fst(A,fun(product_unit,code_term))))) ).
% Random_Pred.Random_def
tff(fact_2662_Random__Pred_Onot__randompred__def,axiom,
! [P2: fun(product_prod(code_natural,code_natural),product_prod(pred(product_unit),product_prod(code_natural,code_natural))),X4: product_prod(code_natural,code_natural)] : random6974930770145893639ompred(P2,X4) = aa(product_prod(pred(product_unit),product_prod(code_natural,code_natural)),product_prod(pred(product_unit),product_prod(code_natural,code_natural)),aa(fun(pred(product_unit),fun(product_prod(code_natural,code_natural),product_prod(pred(product_unit),product_prod(code_natural,code_natural)))),fun(product_prod(pred(product_unit),product_prod(code_natural,code_natural)),product_prod(pred(product_unit),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))),aTP_Lamp_wv(pred(product_unit),fun(product_prod(code_natural,code_natural),product_prod(pred(product_unit),product_prod(code_natural,code_natural))))),aa(product_prod(code_natural,code_natural),product_prod(pred(product_unit),product_prod(code_natural,code_natural)),P2,X4)) ).
% Random_Pred.not_randompred_def
tff(fact_2663_Random__Pred_Omap__def,axiom,
! [B: $tType,A: $tType,F: fun(A,B),P2: fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))] : random_map(A,B,F,P2) = random_bind(A,B,P2,aa(fun(A,B),fun(A,fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)))),aa(fun(B,fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)))),fun(fun(A,B),fun(A,fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural))))),comp(B,fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural))),A),random_single(B)),F)) ).
% Random_Pred.map_def
tff(fact_2664_folding__on__def,axiom,
! [B: $tType,A: $tType,S3: set(A),F: fun(A,fun(B,B))] :
( finite_folding_on(A,B,S3,F)
<=> ! [X5: A,Y4: A] :
( pp(aa(set(A),bool,member(A,X5),S3))
=> ( pp(aa(set(A),bool,member(A,Y4),S3))
=> ( aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,Y4)),aa(A,fun(B,B),F,X5)) = aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X5)),aa(A,fun(B,B),F,Y4)) ) ) ) ) ).
% folding_on_def
tff(fact_2665_folding__on_Ocomp__fun__commute__on,axiom,
! [B: $tType,A: $tType,S3: set(A),F: fun(A,fun(B,B)),X2: A,Y2: A] :
( finite_folding_on(A,B,S3,F)
=> ( pp(aa(set(A),bool,member(A,X2),S3))
=> ( pp(aa(set(A),bool,member(A,Y2),S3))
=> ( aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,Y2)),aa(A,fun(B,B),F,X2)) = aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X2)),aa(A,fun(B,B),F,Y2)) ) ) ) ) ).
% folding_on.comp_fun_commute_on
tff(fact_2666_folding__on_Ointro,axiom,
! [B: $tType,A: $tType,S3: set(A),F: fun(A,fun(B,B))] :
( ! [X: A,Y: A] :
( pp(aa(set(A),bool,member(A,X),S3))
=> ( pp(aa(set(A),bool,member(A,Y),S3))
=> ( aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,Y)),aa(A,fun(B,B),F,X)) = aa(fun(B,B),fun(B,B),aa(fun(B,B),fun(fun(B,B),fun(B,B)),comp(B,B,B),aa(A,fun(B,B),F,X)),aa(A,fun(B,B),F,Y)) ) ) )
=> finite_folding_on(A,B,S3,F) ) ).
% folding_on.intro
tff(fact_2667_asym_Ocases,axiom,
! [A: $tType,A3: set(product_prod(A,A))] :
( asym(A,A3)
=> ! [A17: A,B11: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A17),B11)),A3))
=> ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B11),A17)),A3)) ) ) ).
% asym.cases
tff(fact_2668_asym_Osimps,axiom,
! [A: $tType,A3: set(product_prod(A,A))] :
( asym(A,A3)
<=> ? [R8: set(product_prod(A,A))] :
( ( A3 = R8 )
& ! [X5: A,Xa3: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),Xa3)),R8))
=> ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa3),X5)),R8)) ) ) ) ).
% asym.simps
tff(fact_2669_asym_Ointros,axiom,
! [A: $tType,R3: set(product_prod(A,A))] :
( ! [A5: A,B4: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A5),B4)),R3))
=> ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B4),A5)),R3)) )
=> asym(A,R3) ) ).
% asym.intros
tff(fact_2670_asymD,axiom,
! [A: $tType,R3: set(product_prod(A,A)),X2: A,Y2: A] :
( asym(A,R3)
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),Y2)),R3))
=> ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y2),X2)),R3)) ) ) ).
% asymD
tff(fact_2671_asym__iff,axiom,
! [A: $tType,R3: set(product_prod(A,A))] :
( asym(A,R3)
<=> ! [X5: A,Y4: A] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),Y4)),R3))
=> ~ pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y4),X5)),R3)) ) ) ).
% asym_iff
tff(fact_2672_asymp__asym__eq,axiom,
! [A: $tType,R3: set(product_prod(A,A))] :
( asymp(A,aTP_Lamp_hl(set(product_prod(A,A)),fun(A,fun(A,bool)),R3))
<=> asym(A,R3) ) ).
% asymp_asym_eq
tff(fact_2673_semilattice__set_Ohom__commute,axiom,
! [A: $tType,F: fun(A,fun(A,A)),H: fun(A,A),N: set(A)] :
( lattic149705377957585745ce_set(A,F)
=> ( ! [X: A,Y: A] : aa(A,A,H,aa(A,A,aa(A,fun(A,A),F,X),Y)) = aa(A,A,aa(A,fun(A,A),F,aa(A,A,H,X)),aa(A,A,H,Y))
=> ( finite_finite2(A,N)
=> ( ( N != bot_bot(set(A)) )
=> ( aa(A,A,H,lattic1715443433743089157tice_F(A,F,N)) = lattic1715443433743089157tice_F(A,F,aa(set(A),set(A),aa(fun(A,A),fun(set(A),set(A)),image2(A,A),H),N)) ) ) ) ) ) ).
% semilattice_set.hom_commute
tff(fact_2674_ref_Orel__compp__Grp,axiom,
! [B: $tType,A: $tType,R3: fun(A,fun(B,bool))] : rel_ref(A,B,R3) = aa(fun(ref(product_prod(A,B)),fun(ref(B),bool)),fun(ref(A),fun(ref(B),bool)),aa(fun(ref(A),fun(ref(product_prod(A,B)),bool)),fun(fun(ref(product_prod(A,B)),fun(ref(B),bool)),fun(ref(A),fun(ref(B),bool))),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),aa(fun(ref(product_prod(A,B)),bool),set(ref(product_prod(A,B))),collect(ref(product_prod(A,B))),aTP_Lamp_ww(fun(A,fun(B,bool)),fun(ref(product_prod(A,B)),bool),R3)),map_ref(product_prod(A,B),A,product_fst(A,B))))),bNF_Grp(ref(product_prod(A,B)),ref(B),aa(fun(ref(product_prod(A,B)),bool),set(ref(product_prod(A,B))),collect(ref(product_prod(A,B))),aTP_Lamp_ww(fun(A,fun(B,bool)),fun(ref(product_prod(A,B)),bool),R3)),map_ref(product_prod(A,B),B,product_snd(A,B)))) ).
% ref.rel_compp_Grp
tff(fact_2675_array_Orel__compp__Grp,axiom,
! [B: $tType,A: $tType,R3: fun(A,fun(B,bool))] : rel_array(A,B,R3) = aa(fun(array(product_prod(A,B)),fun(array(B),bool)),fun(array(A),fun(array(B),bool)),aa(fun(array(A),fun(array(product_prod(A,B)),bool)),fun(fun(array(product_prod(A,B)),fun(array(B),bool)),fun(array(A),fun(array(B),bool))),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),aa(fun(array(product_prod(A,B)),bool),set(array(product_prod(A,B))),collect(array(product_prod(A,B))),aTP_Lamp_wx(fun(A,fun(B,bool)),fun(array(product_prod(A,B)),bool),R3)),map_array(product_prod(A,B),A,product_fst(A,B))))),bNF_Grp(array(product_prod(A,B)),array(B),aa(fun(array(product_prod(A,B)),bool),set(array(product_prod(A,B))),collect(array(product_prod(A,B))),aTP_Lamp_wx(fun(A,fun(B,bool)),fun(array(product_prod(A,B)),bool),R3)),map_array(product_prod(A,B),B,product_snd(A,B)))) ).
% array.rel_compp_Grp
tff(fact_2676_array_Oin__rel,axiom,
! [A: $tType,B: $tType,R3: fun(A,fun(B,bool)),A3: array(A),B2: array(B)] :
( pp(aa(array(B),bool,aa(array(A),fun(array(B),bool),rel_array(A,B,R3),A3),B2))
<=> ? [Z4: array(product_prod(A,B))] :
( pp(aa(set(array(product_prod(A,B))),bool,member(array(product_prod(A,B)),Z4),aa(fun(array(product_prod(A,B)),bool),set(array(product_prod(A,B))),collect(array(product_prod(A,B))),aTP_Lamp_wx(fun(A,fun(B,bool)),fun(array(product_prod(A,B)),bool),R3))))
& ( aa(array(product_prod(A,B)),array(A),map_array(product_prod(A,B),A,product_fst(A,B)),Z4) = A3 )
& ( aa(array(product_prod(A,B)),array(B),map_array(product_prod(A,B),B,product_snd(A,B)),Z4) = B2 ) ) ) ).
% array.in_rel
tff(fact_2677_ref_Oin__rel,axiom,
! [A: $tType,B: $tType,R3: fun(A,fun(B,bool)),A3: ref(A),B2: ref(B)] :
( pp(aa(ref(B),bool,aa(ref(A),fun(ref(B),bool),rel_ref(A,B,R3),A3),B2))
<=> ? [Z4: ref(product_prod(A,B))] :
( pp(aa(set(ref(product_prod(A,B))),bool,member(ref(product_prod(A,B)),Z4),aa(fun(ref(product_prod(A,B)),bool),set(ref(product_prod(A,B))),collect(ref(product_prod(A,B))),aTP_Lamp_ww(fun(A,fun(B,bool)),fun(ref(product_prod(A,B)),bool),R3))))
& ( aa(ref(product_prod(A,B)),ref(A),map_ref(product_prod(A,B),A,product_fst(A,B)),Z4) = A3 )
& ( aa(ref(product_prod(A,B)),ref(B),map_ref(product_prod(A,B),B,product_snd(A,B)),Z4) = B2 ) ) ) ).
% ref.in_rel
tff(fact_2678_ref_Oset__map,axiom,
! [B: $tType,A: $tType,F: fun(A,B),V: ref(A)] : set_ref(B,aa(ref(A),ref(B),map_ref(A,B,F),V)) = aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),set_ref(A,V)) ).
% ref.set_map
tff(fact_2679_array_Oset__map,axiom,
! [B: $tType,A: $tType,F: fun(A,B),V: array(A)] : set_array(B,aa(array(A),array(B),map_array(A,B,F),V)) = aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),F),set_array(A,V)) ).
% array.set_map
tff(fact_2680_array_Orel__compp,axiom,
! [A: $tType,C: $tType,B: $tType,R3: fun(A,fun(B,bool)),S3: fun(B,fun(C,bool))] : rel_array(A,C,aa(fun(B,fun(C,bool)),fun(A,fun(C,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(C,bool)),fun(A,fun(C,bool))),relcompp(A,B,C),R3),S3)) = aa(fun(array(B),fun(array(C),bool)),fun(array(A),fun(array(C),bool)),aa(fun(array(A),fun(array(B),bool)),fun(fun(array(B),fun(array(C),bool)),fun(array(A),fun(array(C),bool))),relcompp(array(A),array(B),array(C)),rel_array(A,B,R3)),rel_array(B,C,S3)) ).
% array.rel_compp
tff(fact_2681_ref_Orel__compp,axiom,
! [A: $tType,C: $tType,B: $tType,R3: fun(A,fun(B,bool)),S3: fun(B,fun(C,bool))] : rel_ref(A,C,aa(fun(B,fun(C,bool)),fun(A,fun(C,bool)),aa(fun(A,fun(B,bool)),fun(fun(B,fun(C,bool)),fun(A,fun(C,bool))),relcompp(A,B,C),R3),S3)) = aa(fun(ref(B),fun(ref(C),bool)),fun(ref(A),fun(ref(C),bool)),aa(fun(ref(A),fun(ref(B),bool)),fun(fun(ref(B),fun(ref(C),bool)),fun(ref(A),fun(ref(C),bool))),relcompp(ref(A),ref(B),ref(C)),rel_ref(A,B,R3)),rel_ref(B,C,S3)) ).
% ref.rel_compp
tff(fact_2682_ref_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,G: fun(B,C),F: fun(A,B),V: ref(A)] : aa(ref(B),ref(C),map_ref(B,C,G),aa(ref(A),ref(B),map_ref(A,B,F),V)) = aa(ref(A),ref(C),map_ref(A,C,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),G),F)),V) ).
% ref.map_comp
tff(fact_2683_array_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,G: fun(B,C),F: fun(A,B),V: array(A)] : aa(array(B),array(C),map_array(B,C,G),aa(array(A),array(B),map_array(A,B,F),V)) = aa(array(A),array(C),map_array(A,C,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),comp(B,C,A),G),F)),V) ).
% array.map_comp
tff(fact_2684_ref_Osize__gen__o__map,axiom,
! [B: $tType,A: $tType,F: fun(B,nat),G: fun(A,B)] : aa(fun(ref(A),ref(B)),fun(ref(A),nat),aa(fun(ref(B),nat),fun(fun(ref(A),ref(B)),fun(ref(A),nat)),comp(ref(B),nat,ref(A)),size_ref(B,F)),map_ref(A,B,G)) = size_ref(A,aa(fun(A,B),fun(A,nat),aa(fun(B,nat),fun(fun(A,B),fun(A,nat)),comp(B,nat,A),F),G)) ).
% ref.size_gen_o_map
tff(fact_2685_array_Osize__gen__o__map,axiom,
! [B: $tType,A: $tType,F: fun(B,nat),G: fun(A,B)] : aa(fun(array(A),array(B)),fun(array(A),nat),aa(fun(array(B),nat),fun(fun(array(A),array(B)),fun(array(A),nat)),comp(array(B),nat,array(A)),size_array(B,F)),map_array(A,B,G)) = size_array(A,aa(fun(A,B),fun(A,nat),aa(fun(B,nat),fun(fun(A,B),fun(A,nat)),comp(B,nat,A),F),G)) ).
% array.size_gen_o_map
tff(fact_2686_ref_Orec__o__map,axiom,
! [B: $tType,A: $tType,C: $tType,G: fun(nat,C),F: fun(A,B)] : aa(fun(ref(A),ref(B)),fun(ref(A),C),aa(fun(ref(B),C),fun(fun(ref(A),ref(B)),fun(ref(A),C)),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
tff(fact_2687_array_Orec__o__map,axiom,
! [B: $tType,A: $tType,C: $tType,G: fun(nat,C),F: fun(A,B)] : aa(fun(array(A),array(B)),fun(array(A),C),aa(fun(array(B),C),fun(fun(array(A),array(B)),fun(array(A),C)),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
tff(fact_2688_ref_Opred__map,axiom,
! [B: $tType,A: $tType,Q2: fun(B,bool),F: fun(A,B),X2: ref(A)] :
( pred_ref(B,Q2,aa(ref(A),ref(B),map_ref(A,B,F),X2))
<=> pred_ref(A,aa(fun(A,B),fun(A,bool),aa(fun(B,bool),fun(fun(A,B),fun(A,bool)),comp(B,bool,A),Q2),F),X2) ) ).
% ref.pred_map
tff(fact_2689_array_Opred__map,axiom,
! [B: $tType,A: $tType,Q2: fun(B,bool),F: fun(A,B),X2: array(A)] :
( pred_array(B,Q2,aa(array(A),array(B),map_array(A,B,F),X2))
<=> pred_array(A,aa(fun(A,B),fun(A,bool),aa(fun(B,bool),fun(fun(A,B),fun(A,bool)),comp(B,bool,A),Q2),F),X2) ) ).
% array.pred_map
tff(fact_2690_isl__map__sum,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,F: fun(C,A),G: fun(D,B),S: sum_sum(C,D)] :
( pp(aa(sum_sum(A,B),bool,sum_isl(A,B),aa(sum_sum(C,D),sum_sum(A,B),sum_map_sum(C,A,D,B,F,G),S)))
<=> pp(aa(sum_sum(C,D),bool,sum_isl(C,D),S)) ) ).
% isl_map_sum
tff(fact_2691_isl__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,A6: fun(A,fun(C,bool)),B6: fun(B,fun(D,bool))] : pp(aa(fun(sum_sum(C,D),bool),bool,aa(fun(sum_sum(A,B),bool),fun(fun(sum_sum(C,D),bool),bool),bNF_rel_fun(sum_sum(A,B),sum_sum(C,D),bool,bool,bNF_rel_sum(A,C,B,D,A6,B6),fequal(bool)),sum_isl(A,B)),sum_isl(C,D))) ).
% isl_transfer
tff(fact_2692_rat__floor__code,axiom,
! [P: rat] : archim6421214686448440834_floor(rat,P) = aa(product_prod(int,int),int,aa(fun(int,fun(int,int)),fun(product_prod(int,int),int),product_case_prod(int,int,int),divide_divide(int)),quotient_of(P)) ).
% rat_floor_code
tff(fact_2693_Divides_Oadjust__div__def,axiom,
! [Qr: product_prod(int,int)] : adjust_div(Qr) = aa(product_prod(int,int),int,aa(fun(int,fun(int,int)),fun(product_prod(int,int),int),product_case_prod(int,int,int),aTP_Lamp_wy(int,fun(int,int))),Qr) ).
% Divides.adjust_div_def
tff(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)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),hd(A,Xs)),hd(B,Ys)) ) ) ) ).
% hd_zip
tff(fact_2695_set__of__seq__def,axiom,
! [A: $tType] : set_of_seq(A) = aa(fun(seq(A),pred(A)),fun(seq(A),set(A)),aa(fun(pred(A),set(A)),fun(fun(seq(A),pred(A)),fun(seq(A),set(A))),comp(pred(A),set(A),seq(A)),set_of_pred(A)),pred_of_seq(A)) ).
% set_of_seq_def
tff(fact_2696_set__sum__sel_I1_J,axiom,
! [B: $tType,A: $tType,S: sum_sum(A,B)] :
( pp(aa(sum_sum(A,B),bool,sum_isl(A,B),S))
=> pp(aa(set(A),bool,member(A,sum_projl(A,B,S)),basic_setl(A,B,S))) ) ).
% set_sum_sel(1)
tff(fact_2697_set__sum__sel_I2_J,axiom,
! [B: $tType,A: $tType,S: sum_sum(A,B)] :
( ~ pp(aa(sum_sum(A,B),bool,sum_isl(A,B),S))
=> pp(aa(set(B),bool,member(B,sum_projr(A,B,S)),basic_setr(A,B,S))) ) ).
% set_sum_sel(2)
tff(fact_2698_map__sum__sel_I1_J,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,S: sum_sum(A,B),F: fun(A,C),G: fun(B,D)] :
( pp(aa(sum_sum(A,B),bool,sum_isl(A,B),S))
=> ( sum_projl(C,D,aa(sum_sum(A,B),sum_sum(C,D),sum_map_sum(A,C,B,D,F,G),S)) = aa(A,C,F,sum_projl(A,B,S)) ) ) ).
% map_sum_sel(1)
tff(fact_2699_map__sum__sel_I2_J,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,S: sum_sum(A,B),F: fun(A,C),G: fun(B,D)] :
( ~ pp(aa(sum_sum(A,B),bool,sum_isl(A,B),S))
=> ( sum_projr(C,D,aa(sum_sum(A,B),sum_sum(C,D),sum_map_sum(A,C,B,D,F,G),S)) = aa(B,D,G,sum_projr(A,B,S)) ) ) ).
% map_sum_sel(2)
tff(fact_2700_rel__sum__sel,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,R1: fun(A,fun(B,bool)),R22: fun(C,fun(D,bool)),A3: sum_sum(A,C),B2: sum_sum(B,D)] :
( pp(aa(sum_sum(B,D),bool,aa(sum_sum(A,C),fun(sum_sum(B,D),bool),bNF_rel_sum(A,B,C,D,R1,R22),A3),B2))
<=> ( ( pp(aa(sum_sum(A,C),bool,sum_isl(A,C),A3))
<=> pp(aa(sum_sum(B,D),bool,sum_isl(B,D),B2)) )
& ( pp(aa(sum_sum(A,C),bool,sum_isl(A,C),A3))
=> ( pp(aa(sum_sum(B,D),bool,sum_isl(B,D),B2))
=> pp(aa(B,bool,aa(A,fun(B,bool),R1,sum_projl(A,C,A3)),sum_projl(B,D,B2))) ) )
& ( ~ pp(aa(sum_sum(A,C),bool,sum_isl(A,C),A3))
=> ( ~ pp(aa(sum_sum(B,D),bool,sum_isl(B,D),B2))
=> pp(aa(D,bool,aa(C,fun(D,bool),R22,sum_projr(A,C,A3)),sum_projr(B,D,B2))) ) ) ) ) ).
% rel_sum_sel
tff(fact_2701_take__bit__num__code,axiom,
! [N4: nat,M4: num] : bit_take_bit_num(N4,M4) = aa(product_prod(nat,num),option(num),aa(fun(nat,fun(num,option(num))),fun(product_prod(nat,num),option(num)),product_case_prod(nat,num,option(num)),aTP_Lamp_xd(nat,fun(num,option(num)))),aa(num,product_prod(nat,num),aa(nat,fun(num,product_prod(nat,num)),product_Pair(nat,num),N4),M4)) ).
% take_bit_num_code
tff(fact_2702_nths__drop,axiom,
! [A: $tType,N4: nat,Xs: list(A),I4: set(nat)] : nths(A,drop(A,N4,Xs),I4) = nths(A,Xs,aa(set(nat),set(nat),aa(fun(nat,nat),fun(set(nat),set(nat)),image2(nat,nat),aa(nat,fun(nat,nat),plus_plus(nat),N4)),I4)) ).
% nths_drop
tff(fact_2703_folding__insort__key_Oinsort__key__commute,axiom,
! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S3: set(B),F: fun(B,A),X2: B,Y2: B] :
( folding_insort_key(A,B,Less_eq,Less,S3,F)
=> ( pp(aa(set(B),bool,member(B,X2),S3))
=> ( pp(aa(set(B),bool,member(B,Y2),S3))
=> ( aa(fun(list(B),list(B)),fun(list(B),list(B)),aa(fun(list(B),list(B)),fun(fun(list(B),list(B)),fun(list(B),list(B))),comp(list(B),list(B),list(B)),insort_key(A,B,Less_eq,F,Y2)),insort_key(A,B,Less_eq,F,X2)) = aa(fun(list(B),list(B)),fun(list(B),list(B)),aa(fun(list(B),list(B)),fun(fun(list(B),list(B)),fun(list(B),list(B))),comp(list(B),list(B),list(B)),insort_key(A,B,Less_eq,F,X2)),insort_key(A,B,Less_eq,F,Y2)) ) ) ) ) ).
% folding_insort_key.insort_key_commute
tff(fact_2704_list__ex__iff__not__all__inverval__nat,axiom,
! [P2: fun(nat,bool),I: nat,J5: nat] :
( list_ex(nat,P2,upt(I,J5))
<=> ~ all_interval_nat(aa(fun(nat,bool),fun(nat,bool),aa(fun(bool,bool),fun(fun(nat,bool),fun(nat,bool)),comp(bool,bool,nat),fNot),P2),I,J5) ) ).
% list_ex_iff_not_all_inverval_nat
tff(fact_2705_Inl__def,axiom,
! [B: $tType,A: $tType] : sum_Inl(A,B) = aa(fun(A,fun(A,fun(B,fun(bool,bool)))),fun(A,sum_sum(A,B)),aa(fun(fun(A,fun(B,fun(bool,bool))),sum_sum(A,B)),fun(fun(A,fun(A,fun(B,fun(bool,bool)))),fun(A,sum_sum(A,B))),comp(fun(A,fun(B,fun(bool,bool))),sum_sum(A,B),A),sum_Abs_sum(A,B)),sum_Inl_Rep(A,B)) ).
% Inl_def
tff(fact_2706_fundef__ex1__iff,axiom,
! [A: $tType,B: $tType,F: fun(A,B),D4: fun(A,B),G6: fun(A,fun(B,bool)),X2: A,Y2: B] :
( ! [X: A] : aa(A,B,F,X) = fun_THE_default(B,aa(A,B,D4,X),aa(A,fun(B,bool),G6,X))
=> ( ? [X4: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),G6,X2),X4))
& ! [Y: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),G6,X2),Y))
=> ( Y = X4 ) ) )
=> ( pp(aa(B,bool,aa(A,fun(B,bool),G6,X2),Y2))
<=> ( aa(A,B,F,X2) = Y2 ) ) ) ) ).
% fundef_ex1_iff
tff(fact_2707_fundef__default__value,axiom,
! [B: $tType,A: $tType,F: fun(A,B),D4: fun(A,B),G6: fun(A,fun(B,bool)),D3: fun(A,bool),X2: A] :
( ! [X: A] : aa(A,B,F,X) = fun_THE_default(B,aa(A,B,D4,X),aa(A,fun(B,bool),G6,X))
=> ( ! [X: A,Y: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),G6,X),Y))
=> pp(aa(A,bool,D3,X)) )
=> ( ~ pp(aa(A,bool,D3,X2))
=> ( aa(A,B,F,X2) = aa(A,B,D4,X2) ) ) ) ) ).
% fundef_default_value
tff(fact_2708_THE__default1__equality,axiom,
! [A: $tType,P2: fun(A,bool),A3: A,D4: A] :
( ? [X4: A] :
( pp(aa(A,bool,P2,X4))
& ! [Y: A] :
( pp(aa(A,bool,P2,Y))
=> ( Y = X4 ) ) )
=> ( pp(aa(A,bool,P2,A3))
=> ( fun_THE_default(A,D4,P2) = A3 ) ) ) ).
% THE_default1_equality
tff(fact_2709_THE__default__none,axiom,
! [A: $tType,P2: fun(A,bool),D4: A] :
( ~ ? [X: A] :
( pp(aa(A,bool,P2,X))
& ! [Y5: A] :
( pp(aa(A,bool,P2,Y5))
=> ( Y5 = X ) ) )
=> ( fun_THE_default(A,D4,P2) = D4 ) ) ).
% THE_default_none
tff(fact_2710_THE__defaultI_H,axiom,
! [A: $tType,P2: fun(A,bool),D4: A] :
( ? [X4: A] :
( pp(aa(A,bool,P2,X4))
& ! [Y: A] :
( pp(aa(A,bool,P2,Y))
=> ( Y = X4 ) ) )
=> pp(aa(A,bool,P2,fun_THE_default(A,D4,P2))) ) ).
% THE_defaultI'
tff(fact_2711_fundef__ex1__uniqueness,axiom,
! [B: $tType,A: $tType,F: fun(A,B),D4: fun(A,B),G6: fun(A,fun(B,bool)),X2: A,H: fun(A,B)] :
( ! [X: A] : aa(A,B,F,X) = fun_THE_default(B,aa(A,B,D4,X),aa(A,fun(B,bool),G6,X))
=> ( ? [X4: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),G6,X2),X4))
& ! [Y: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),G6,X2),Y))
=> ( Y = X4 ) ) )
=> ( pp(aa(B,bool,aa(A,fun(B,bool),G6,X2),aa(A,B,H,X2)))
=> ( aa(A,B,H,X2) = aa(A,B,F,X2) ) ) ) ) ).
% fundef_ex1_uniqueness
tff(fact_2712_fundef__ex1__existence,axiom,
! [B: $tType,A: $tType,F: fun(A,B),D4: fun(A,B),G6: fun(A,fun(B,bool)),X2: A] :
( ! [X: A] : aa(A,B,F,X) = fun_THE_default(B,aa(A,B,D4,X),aa(A,fun(B,bool),G6,X))
=> ( ? [X4: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),G6,X2),X4))
& ! [Y: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),G6,X2),Y))
=> ( Y = X4 ) ) )
=> pp(aa(B,bool,aa(A,fun(B,bool),G6,X2),aa(A,B,F,X2))) ) ) ).
% fundef_ex1_existence
tff(fact_2713_Inr__def,axiom,
! [A: $tType,B: $tType] : sum_Inr(B,A) = aa(fun(B,fun(A,fun(B,fun(bool,bool)))),fun(B,sum_sum(A,B)),aa(fun(fun(A,fun(B,fun(bool,bool))),sum_sum(A,B)),fun(fun(B,fun(A,fun(B,fun(bool,bool)))),fun(B,sum_sum(A,B))),comp(fun(A,fun(B,fun(bool,bool))),sum_sum(A,B),B),sum_Abs_sum(A,B)),sum_Inr_Rep(B,A)) ).
% Inr_def
tff(fact_2714_THE__default__def,axiom,
! [A: $tType,P2: fun(A,bool),D4: A] :
( ( ? [X4: A] :
( pp(aa(A,bool,P2,X4))
& ! [Y: A] :
( pp(aa(A,bool,P2,Y))
=> ( Y = X4 ) ) )
=> ( fun_THE_default(A,D4,P2) = the(A,P2) ) )
& ( ~ ? [X: A] :
( pp(aa(A,bool,P2,X))
& ! [Y5: A] :
( pp(aa(A,bool,P2,Y5))
=> ( Y5 = X ) ) )
=> ( fun_THE_default(A,D4,P2) = D4 ) ) ) ).
% THE_default_def
tff(fact_2715_ATP_Olambda__1,axiom,
! [A: $tType] :
( countable(A)
=> ! [Uu: nat] : aa(nat,set(old_node(A,product_unit)),aTP_Lamp_wh(nat,set(old_node(A,product_unit))),Uu) = aa(sum_sum(nat,nat),set(old_node(A,product_unit)),sum_case_sum(nat,set(old_node(A,product_unit)),nat,aTP_Lamp_we(nat,set(old_node(A,product_unit))),aTP_Lamp_wg(nat,set(old_node(A,product_unit)))),aa(nat,sum_sum(nat,nat),nat_sum_decode,Uu)) ) ).
% ATP.lambda_1
tff(fact_2716_ATP_Olambda__2,axiom,
! [A: $tType] :
( countable(A)
=> ! [Uu: nat] : aa(nat,set(old_node(A,product_unit)),aTP_Lamp_wd(nat,set(old_node(A,product_unit))),Uu) = aa(sum_sum(nat,nat),set(old_node(A,product_unit)),sum_case_sum(nat,set(old_node(A,product_unit)),nat,aTP_Lamp_wb(nat,set(old_node(A,product_unit))),aTP_Lamp_wc(nat,set(old_node(A,product_unit)))),aa(nat,sum_sum(nat,nat),nat_sum_decode,Uu)) ) ).
% ATP.lambda_2
tff(fact_2717_ATP_Olambda__3,axiom,
! [A: $tType,Uu: A] : aa(A,set(product_prod(A,A)),aTP_Lamp_dm(A,set(product_prod(A,A))),Uu) = aa(set(product_prod(A,A)),set(product_prod(A,A)),insert(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),Uu)),bot_bot(set(product_prod(A,A)))) ).
% ATP.lambda_3
tff(fact_2718_ATP_Olambda__4,axiom,
! [A: $tType,Uu: product_prod(A,A)] :
( pp(aa(product_prod(A,A),bool,aTP_Lamp_hk(product_prod(A,A),bool),Uu))
<=> ( aa(product_prod(A,A),A,product_fst(A,A),Uu) = aa(product_prod(A,A),A,product_snd(A,A),Uu) ) ) ).
% ATP.lambda_4
tff(fact_2719_ATP_Olambda__5,axiom,
! [Uu: nat] : aa(nat,nat,aTP_Lamp_lm(nat,nat),Uu) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uu),aa(nat,nat,suc,zero_zero(nat))) ).
% ATP.lambda_5
tff(fact_2720_ATP_Olambda__6,axiom,
! [B: $tType,Uu: B] : aa(B,product_prod(B,B),aTP_Lamp_bm(B,product_prod(B,B)),Uu) = aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Uu),Uu) ).
% ATP.lambda_6
tff(fact_2721_ATP_Olambda__7,axiom,
! [A: $tType,Uu: A] : aa(A,product_prod(A,A),aTP_Lamp_bi(A,product_prod(A,A)),Uu) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),Uu) ).
% ATP.lambda_7
tff(fact_2722_ATP_Olambda__8,axiom,
! [A: $tType,Uu: A] : aa(A,list(A),aTP_Lamp_ps(A,list(A)),Uu) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uu),nil(A)) ).
% ATP.lambda_8
tff(fact_2723_ATP_Olambda__9,axiom,
! [A: $tType,Uu: A] : aa(A,set(A),aTP_Lamp_cz(A,set(A)),Uu) = aa(set(A),set(A),insert(A,Uu),bot_bot(set(A))) ).
% ATP.lambda_9
tff(fact_2724_ATP_Olambda__10,axiom,
! [B: $tType,D: $tType,Uu: fun(D,B)] : aa(fun(D,B),set(B),aTP_Lamp_ub(fun(D,B),set(B)),Uu) = aa(set(D),set(B),aa(fun(D,B),fun(set(D),set(B)),image2(D,B),Uu),top_top(set(D))) ).
% ATP.lambda_10
tff(fact_2725_ATP_Olambda__11,axiom,
! [A: $tType,D: $tType,Uu: fun(D,A)] : aa(fun(D,A),set(A),aTP_Lamp_ua(fun(D,A),set(A)),Uu) = aa(set(D),set(A),aa(fun(D,A),fun(set(D),set(A)),image2(D,A),Uu),top_top(set(D))) ).
% ATP.lambda_11
tff(fact_2726_ATP_Olambda__12,axiom,
! [A: $tType] :
( countable(A)
=> ! [Uu: nat] : aa(nat,set(old_node(A,product_unit)),aTP_Lamp_wg(nat,set(old_node(A,product_unit))),Uu) = aa(product_prod(nat,nat),set(old_node(A,product_unit)),aa(fun(nat,fun(nat,set(old_node(A,product_unit)))),fun(product_prod(nat,nat),set(old_node(A,product_unit))),product_case_prod(nat,nat,set(old_node(A,product_unit))),aTP_Lamp_wf(nat,fun(nat,set(old_node(A,product_unit))))),aa(nat,product_prod(nat,nat),nat_prod_decode,Uu)) ) ).
% ATP.lambda_12
tff(fact_2727_ATP_Olambda__13,axiom,
! [A: $tType,Uu: A] : aa(A,fun(set(product_prod(A,A)),set(product_prod(A,A))),aTP_Lamp_hu(A,fun(set(product_prod(A,A)),set(product_prod(A,A)))),Uu) = insert(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),Uu)) ).
% ATP.lambda_13
tff(fact_2728_ATP_Olambda__14,axiom,
! [A: $tType,Uu: option(product_prod(A,heap_ext(product_unit)))] : aa(option(product_prod(A,heap_ext(product_unit))),set(A),aTP_Lamp_sx(option(product_prod(A,heap_ext(product_unit))),set(A)),Uu) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(product_prod(A,heap_ext(product_unit))),set(set(A)),aa(fun(product_prod(A,heap_ext(product_unit)),set(A)),fun(set(product_prod(A,heap_ext(product_unit))),set(set(A))),image2(product_prod(A,heap_ext(product_unit)),set(A)),basic_fsts(A,heap_ext(product_unit))),aa(option(product_prod(A,heap_ext(product_unit))),set(product_prod(A,heap_ext(product_unit))),set_option(product_prod(A,heap_ext(product_unit))),Uu))) ).
% ATP.lambda_14
tff(fact_2729_ATP_Olambda__15,axiom,
! [B: $tType,A: $tType,Uu: fun(A,option(B))] : aa(fun(A,option(B)),fun(product_prod(A,B),fun(A,option(B))),aTP_Lamp_uq(fun(A,option(B)),fun(product_prod(A,B),fun(A,option(B)))),Uu) = aa(fun(A,fun(B,fun(A,option(B)))),fun(product_prod(A,B),fun(A,option(B))),product_case_prod(A,B,fun(A,option(B))),aTP_Lamp_up(fun(A,option(B)),fun(A,fun(B,fun(A,option(B)))),Uu)) ).
% ATP.lambda_15
tff(fact_2730_ATP_Olambda__16,axiom,
! [A: $tType,C: $tType,B: $tType,Uu: B] : aa(B,fun(product_prod(A,C),product_prod(A,product_prod(B,C))),aTP_Lamp_pf(B,fun(product_prod(A,C),product_prod(A,product_prod(B,C)))),Uu) = aa(fun(A,fun(C,product_prod(A,product_prod(B,C)))),fun(product_prod(A,C),product_prod(A,product_prod(B,C))),product_case_prod(A,C,product_prod(A,product_prod(B,C))),aTP_Lamp_pe(B,fun(A,fun(C,product_prod(A,product_prod(B,C)))),Uu)) ).
% ATP.lambda_16
tff(fact_2731_ATP_Olambda__17,axiom,
! [B: $tType,C: $tType,A: $tType,Uu: A] : aa(A,fun(product_prod(B,C),product_prod(product_prod(A,B),C)),aTP_Lamp_jb(A,fun(product_prod(B,C),product_prod(product_prod(A,B),C))),Uu) = aa(fun(B,fun(C,product_prod(product_prod(A,B),C))),fun(product_prod(B,C),product_prod(product_prod(A,B),C)),product_case_prod(B,C,product_prod(product_prod(A,B),C)),aTP_Lamp_ja(A,fun(B,fun(C,product_prod(product_prod(A,B),C))),Uu)) ).
% ATP.lambda_17
tff(fact_2732_ATP_Olambda__18,axiom,
! [Uu: nat] : aa(nat,set(nat),aTP_Lamp_tk(nat,set(nat)),Uu) = aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_tj(nat,fun(nat,bool),Uu)) ).
% ATP.lambda_18
tff(fact_2733_ATP_Olambda__19,axiom,
! [B: $tType,A: $tType,Uu: fun(A,B)] : aa(fun(A,B),set(product_prod(A,B)),aTP_Lamp_he(fun(A,B),set(product_prod(A,B))),Uu) = aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aTP_Lamp_hd(fun(A,B),fun(A,fun(B,bool)),Uu))) ).
% ATP.lambda_19
tff(fact_2734_ATP_Olambda__20,axiom,
! [Uu: num] : aa(num,option(num),aTP_Lamp_wz(num,option(num)),Uu) = aa(num,option(num),some(num),bit0(Uu)) ).
% ATP.lambda_20
tff(fact_2735_ATP_Olambda__21,axiom,
! [Uu: int] : aa(int,fun(int,product_prod(int,int)),aTP_Lamp_rk(int,fun(int,product_prod(int,int))),Uu) = aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),Uu)) ).
% ATP.lambda_21
tff(fact_2736_ATP_Olambda__22,axiom,
! [Uu: int] : aa(int,fun(int,product_prod(int,int)),aTP_Lamp_tp(int,fun(int,product_prod(int,int))),Uu) = aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,abs_abs(int),Uu)) ).
% ATP.lambda_22
tff(fact_2737_ATP_Olambda__23,axiom,
! [Uu: nat] : aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_lz(nat,fun(nat,product_prod(nat,nat))),Uu) = aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,suc,Uu)) ).
% ATP.lambda_23
tff(fact_2738_ATP_Olambda__24,axiom,
! [A: $tType] :
( countable(A)
=> ! [Uu: nat] : aa(nat,set(old_node(A,product_unit)),aTP_Lamp_we(nat,set(old_node(A,product_unit))),Uu) = aa(A,set(old_node(A,product_unit)),old_Leaf(A,product_unit),aa(nat,A,from_nat(A),Uu)) ) ).
% ATP.lambda_24
tff(fact_2739_ATP_Olambda__25,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [Uu: nat] : aa(nat,heap_Heap(A),aTP_Lamp_nr(nat,heap_Heap(A)),Uu) = aa(A,heap_Heap(A),heap_return(A),aa(nat,A,semiring_1_of_nat(A),Uu)) ) ).
% ATP.lambda_25
tff(fact_2740_ATP_Olambda__26,axiom,
! [A: $tType] :
( countable(A)
=> ! [Uu: nat] : aa(nat,set(old_node(A,product_unit)),aTP_Lamp_wc(nat,set(old_node(A,product_unit))),Uu) = aa(set(old_node(A,product_unit)),set(old_node(A,product_unit)),old_In1(A,product_unit),nth_item(A,Uu)) ) ).
% ATP.lambda_26
tff(fact_2741_ATP_Olambda__27,axiom,
! [A: $tType] :
( countable(A)
=> ! [Uu: nat] : aa(nat,set(old_node(A,product_unit)),aTP_Lamp_wb(nat,set(old_node(A,product_unit))),Uu) = aa(set(old_node(A,product_unit)),set(old_node(A,product_unit)),old_In0(A,product_unit),nth_item(A,Uu)) ) ).
% ATP.lambda_27
tff(fact_2742_ATP_Olambda__28,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Uu: A] : aa(A,filter(A),aTP_Lamp_oq(A,filter(A)),Uu) = principal(A,aa(A,set(A),set_ord_atLeast(A),Uu)) ) ).
% ATP.lambda_28
tff(fact_2743_ATP_Olambda__29,axiom,
! [A: $tType] :
( order(A)
=> ! [Uu: A] : aa(A,filter(A),aTP_Lamp_op(A,filter(A)),Uu) = principal(A,aa(A,set(A),set_ord_atLeast(A),Uu)) ) ).
% ATP.lambda_29
tff(fact_2744_ATP_Olambda__30,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Uu: A] : aa(A,filter(A),aTP_Lamp_mf(A,filter(A)),Uu) = principal(A,aa(A,set(A),set_ord_atMost(A),Uu)) ) ).
% ATP.lambda_30
tff(fact_2745_ATP_Olambda__31,axiom,
! [A: $tType] :
( order(A)
=> ! [Uu: A] : aa(A,filter(A),aTP_Lamp_md(A,filter(A)),Uu) = principal(A,aa(A,set(A),set_ord_atMost(A),Uu)) ) ).
% ATP.lambda_31
tff(fact_2746_ATP_Olambda__32,axiom,
! [Uu: int] : aa(int,nat,aTP_Lamp_to(int,nat),Uu) = aa(int,nat,nat2,aa(int,int,abs_abs(int),Uu)) ).
% ATP.lambda_32
tff(fact_2747_ATP_Olambda__33,axiom,
! [A: $tType,Uu: product_prod(A,A)] :
( pp(aa(product_prod(A,A),bool,aTP_Lamp_nc(product_prod(A,A),bool),Uu))
<=> ? [X5: A] : Uu = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),X5) ) ).
% ATP.lambda_33
tff(fact_2748_ATP_Olambda__34,axiom,
! [A: $tType,B: $tType,Uu: fun(A,fun(B,bool))] :
( pp(aa(fun(A,fun(B,bool)),bool,aTP_Lamp_vs(fun(A,fun(B,bool)),bool),Uu))
<=> ? [A7: A,B5: B] : Uu = product_Pair_Rep(A,B,A7,B5) ) ).
% ATP.lambda_34
tff(fact_2749_ATP_Olambda__35,axiom,
! [B: $tType,A: $tType,Uu: B] : aa(B,heap_Heap(A),aTP_Lamp_vd(B,heap_Heap(A)),Uu) = aa(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_Heap(A),heap_Heap2(A),aTP_Lamp_sm(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))))) ).
% ATP.lambda_35
tff(fact_2750_ATP_Olambda__36,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: array(A)] : aa(array(A),heap_Heap(product_unit),aTP_Lamp_te(array(A),heap_Heap(product_unit)),Uu) = aa(product_unit,heap_Heap(product_unit),heap_return(product_unit),product_Unity) ) ).
% ATP.lambda_36
tff(fact_2751_ATP_Olambda__37,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: A] : aa(A,heap_Heap(product_unit),aTP_Lamp_tb(A,heap_Heap(product_unit)),Uu) = aa(product_unit,heap_Heap(product_unit),heap_return(product_unit),product_Unity) ) ).
% ATP.lambda_37
tff(fact_2752_ATP_Olambda__38,axiom,
! [Uu: num,Uua: nat] : aa(nat,option(num),aTP_Lamp_xc(num,fun(nat,option(num)),Uu),Uua) = case_num(option(num),aa(num,option(num),some(num),one),aTP_Lamp_xa(nat,fun(num,option(num)),Uua),aTP_Lamp_xb(nat,fun(num,option(num)),Uua),Uu) ).
% ATP.lambda_38
tff(fact_2753_ATP_Olambda__39,axiom,
! [C: $tType,B: $tType,A: $tType,Uu: product_prod(A,C),Uua: product_prod(C,B)] : aa(product_prod(C,B),list(product_prod(A,B)),aTP_Lamp_po(product_prod(A,C),fun(product_prod(C,B),list(product_prod(A,B))),Uu),Uua) = if(list(product_prod(A,B)),aa(C,bool,aa(C,fun(C,bool),fequal(C),aa(product_prod(A,C),C,product_snd(A,C),Uu)),aa(product_prod(C,B),C,product_fst(C,B),Uua)),aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(A,C),A,product_fst(A,C),Uu)),aa(product_prod(C,B),B,product_snd(C,B),Uua))),nil(product_prod(A,B))),nil(product_prod(A,B))) ).
% ATP.lambda_39
tff(fact_2754_ATP_Olambda__40,axiom,
! [Uu: code_integer,Uua: code_integer] : aa(code_integer,int,aa(code_integer,fun(code_integer,int),aTP_Lamp_vp(code_integer,fun(code_integer,int)),Uu),Uua) = if(int,aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),Uua),zero_zero(code_integer)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one))),aa(code_integer,int,code_int_of_integer,Uu)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one))),aa(code_integer,int,code_int_of_integer,Uu))),one_one(int))) ).
% ATP.lambda_40
tff(fact_2755_ATP_Olambda__41,axiom,
! [Uu: code_integer,Uua: code_integer] : aa(code_integer,num,aa(code_integer,fun(code_integer,num),aTP_Lamp_vm(code_integer,fun(code_integer,num)),Uu),Uua) = if(num,aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),Uua),zero_zero(code_integer)),aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(code_integer,num,code_num_of_integer,Uu)),aa(code_integer,num,code_num_of_integer,Uu)),aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(code_integer,num,code_num_of_integer,Uu)),aa(code_integer,num,code_num_of_integer,Uu))),one)) ).
% ATP.lambda_41
tff(fact_2756_ATP_Olambda__42,axiom,
! [Uu: code_integer,Uua: code_integer] : aa(code_integer,nat,aa(code_integer,fun(code_integer,nat),aTP_Lamp_vl(code_integer,fun(code_integer,nat)),Uu),Uua) = if(nat,aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),Uua),zero_zero(code_integer)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(code_integer,nat,code_nat_of_integer,Uu)),aa(code_integer,nat,code_nat_of_integer,Uu)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(code_integer,nat,code_nat_of_integer,Uu)),aa(code_integer,nat,code_nat_of_integer,Uu))),one_one(nat))) ).
% ATP.lambda_42
tff(fact_2757_ATP_Olambda__43,axiom,
! [Uu: pred(product_unit),Uua: product_prod(code_natural,code_natural)] : aa(product_prod(code_natural,code_natural),product_prod(pred(product_unit),product_prod(code_natural,code_natural)),aa(pred(product_unit),fun(product_prod(code_natural,code_natural),product_prod(pred(product_unit),product_prod(code_natural,code_natural))),aTP_Lamp_wv(pred(product_unit),fun(product_prod(code_natural,code_natural),product_prod(pred(product_unit),product_prod(code_natural,code_natural)))),Uu),Uua) = if(product_prod(pred(product_unit),product_prod(code_natural,code_natural)),aa(product_unit,bool,aa(pred(product_unit),fun(product_unit,bool),eval(product_unit),Uu),product_Unity),aa(product_prod(code_natural,code_natural),product_prod(pred(product_unit),product_prod(code_natural,code_natural)),aa(pred(product_unit),fun(product_prod(code_natural,code_natural),product_prod(pred(product_unit),product_prod(code_natural,code_natural))),product_Pair(pred(product_unit),product_prod(code_natural,code_natural)),bot_bot(pred(product_unit))),Uua),aa(product_prod(code_natural,code_natural),product_prod(pred(product_unit),product_prod(code_natural,code_natural)),aa(pred(product_unit),fun(product_prod(code_natural,code_natural),product_prod(pred(product_unit),product_prod(code_natural,code_natural))),product_Pair(pred(product_unit),product_prod(code_natural,code_natural)),aa(product_unit,pred(product_unit),single(product_unit),product_Unity)),Uua)) ).
% ATP.lambda_43
tff(fact_2758_ATP_Olambda__44,axiom,
! [Uu: int,Uua: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_ts(int,fun(int,product_prod(int,int))),Uu),Uua) = if(product_prod(int,int),aa(int,bool,aa(int,fun(int,bool),fequal(int),Uu),zero_zero(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),Uu)),Uua)),aa(int,int,abs_abs(int),Uu))) ).
% ATP.lambda_44
tff(fact_2759_ATP_Olambda__45,axiom,
! [Uu: nat,Uua: num] : aa(num,option(num),aa(nat,fun(num,option(num)),aTP_Lamp_xd(nat,fun(num,option(num))),Uu),Uua) = case_nat(option(num),none(num),aTP_Lamp_xc(num,fun(nat,option(num)),Uua),Uu) ).
% ATP.lambda_45
tff(fact_2760_ATP_Olambda__46,axiom,
! [Uu: nat,Uua: num] : aa(num,option(num),aTP_Lamp_xa(nat,fun(num,option(num)),Uu),Uua) = case_option(option(num),num,none(num),aTP_Lamp_wz(num,option(num)),bit_take_bit_num(Uu,Uua)) ).
% ATP.lambda_46
tff(fact_2761_ATP_Olambda__47,axiom,
! [B: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: product_prod(A,B)] :
( pp(aa(product_prod(A,B),bool,aTP_Lamp_bx(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),Uu),Uua))
<=> pp(aa(B,bool,aa(A,fun(B,bool),Uu,aa(product_prod(A,B),A,product_fst(A,B),Uua)),aa(product_prod(A,B),B,product_snd(A,B),Uua))) ) ).
% ATP.lambda_47
tff(fact_2762_ATP_Olambda__48,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_nt(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_ns(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua)),aa(nat,set(nat),set_ord_atMost(nat),Uua)) ) ).
% ATP.lambda_48
tff(fact_2763_ATP_Olambda__49,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_lx(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_lw(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua)),aa(nat,set(nat),set_ord_atMost(nat),Uua)) ) ).
% ATP.lambda_49
tff(fact_2764_ATP_Olambda__50,axiom,
! [B: $tType,A: $tType,Uu: set(old_node(A,B)),Uua: set(old_node(A,B))] : aa(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aTP_Lamp_wa(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))),Uu),Uua) = aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),insert(product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),set(old_node(A,B)),old_In1(A,B),Uu)),aa(set(old_node(A,B)),set(old_node(A,B)),old_In1(A,B),Uua))),bot_bot(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))) ).
% ATP.lambda_50
tff(fact_2765_ATP_Olambda__51,axiom,
! [B: $tType,A: $tType,Uu: set(old_node(A,B)),Uua: set(old_node(A,B))] : aa(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aTP_Lamp_vz(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))),Uu),Uua) = aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),insert(product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),set(old_node(A,B)),old_In0(A,B),Uu)),aa(set(old_node(A,B)),set(old_node(A,B)),old_In0(A,B),Uua))),bot_bot(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))) ).
% ATP.lambda_51
tff(fact_2766_ATP_Olambda__52,axiom,
! [A2: $tType,A: $tType,Uu: set(product_prod(A,A)),Uua: set(product_prod(A2,A2))] :
( pp(aa(set(product_prod(A2,A2)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A2,A2)),bool),aTP_Lamp_nw(set(product_prod(A,A)),fun(set(product_prod(A2,A2)),bool)),Uu),Uua))
<=> ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),Uu),Uu)
& order_well_order_on(A2,aa(set(product_prod(A2,A2)),set(A2),field2(A2),Uua),Uua)
& ? [X_1: fun(A,A2)] : bNF_Wellorder_embedS(A,A2,Uu,Uua,X_1) ) ) ).
% ATP.lambda_52
tff(fact_2767_ATP_Olambda__53,axiom,
! [A2: $tType,A: $tType,Uu: set(product_prod(A,A)),Uua: set(product_prod(A2,A2))] :
( pp(aa(set(product_prod(A2,A2)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A2,A2)),bool),aTP_Lamp_oc(set(product_prod(A,A)),fun(set(product_prod(A2,A2)),bool)),Uu),Uua))
<=> ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),Uu),Uu)
& order_well_order_on(A2,aa(set(product_prod(A2,A2)),set(A2),field2(A2),Uua),Uua)
& ? [X_1: fun(A,A2)] : pp(aa(fun(A,A2),bool,bNF_Wellorder_embed(A,A2,Uu,Uua),X_1)) ) ) ).
% ATP.lambda_53
tff(fact_2768_ATP_Olambda__54,axiom,
! [A2: $tType,A: $tType,Uu: set(product_prod(A,A)),Uua: set(product_prod(A2,A2))] :
( pp(aa(set(product_prod(A2,A2)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A2,A2)),bool),aTP_Lamp_ob(set(product_prod(A,A)),fun(set(product_prod(A2,A2)),bool)),Uu),Uua))
<=> ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),Uu),Uu)
& order_well_order_on(A2,aa(set(product_prod(A2,A2)),set(A2),field2(A2),Uua),Uua)
& ? [X_1: fun(A,A2)] : bNF_Wellorder_iso(A,A2,Uu,Uua,X_1) ) ) ).
% ATP.lambda_54
tff(fact_2769_ATP_Olambda__55,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: A] : aa(A,set(set(A)),aTP_Lamp_iw(set(product_prod(A,A)),fun(A,set(set(A))),Uu),Uua) = aa(set(set(A)),set(set(A)),insert(set(A),aa(set(A),set(A),image(A,A,Uu),aa(set(A),set(A),insert(A,Uua),bot_bot(set(A))))),bot_bot(set(set(A)))) ).
% ATP.lambda_55
tff(fact_2770_ATP_Olambda__56,axiom,
! [A: $tType,Uu: set(option(A)),Uua: option(A)] :
( pp(aa(option(A),bool,aTP_Lamp_rw(set(option(A)),fun(option(A),bool),Uu),Uua))
<=> ( pp(aa(set(option(A)),bool,member(option(A),Uua),Uu))
& ( Uua != none(A) ) ) ) ).
% ATP.lambda_56
tff(fact_2771_ATP_Olambda__57,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(product_prod(A,A))] :
( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),aTP_Lamp_ou(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool)),Uu),Uua))
<=> ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),Uu),Uua))
& ! [A7: A,B5: A,C6: A] :
( ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A7),B5)),Uua))
& pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B5),C6)),Uu)) )
=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A7),B5)),Uu)) ) ) ) ).
% ATP.lambda_57
tff(fact_2772_ATP_Olambda__58,axiom,
! [A: $tType,Uu: set(A),Uua: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aTP_Lamp_kd(set(A),fun(set(A),bool)),Uu),Uua))
<=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),Uu),Uua))
& finite_finite2(A,Uua) ) ) ).
% ATP.lambda_58
tff(fact_2773_ATP_Olambda__59,axiom,
! [B: $tType,A: $tType,Uu: set(old_node(A,B)),Uua: set(old_node(A,B))] : aa(set(old_node(A,B)),set(set(old_node(A,B))),aTP_Lamp_wm(set(old_node(A,B)),fun(set(old_node(A,B)),set(set(old_node(A,B)))),Uu),Uua) = aa(set(set(old_node(A,B))),set(set(old_node(A,B))),insert(set(old_node(A,B)),old_Scons(A,B,Uu,Uua)),bot_bot(set(set(old_node(A,B))))) ).
% ATP.lambda_59
tff(fact_2774_ATP_Olambda__60,axiom,
! [A: $tType,B: $tType,Uu: B,Uua: A] : aa(A,set(product_prod(B,A)),aTP_Lamp_hn(B,fun(A,set(product_prod(B,A))),Uu),Uua) = aa(set(product_prod(B,A)),set(product_prod(B,A)),insert(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uu),Uua)),bot_bot(set(product_prod(B,A)))) ).
% ATP.lambda_60
tff(fact_2775_ATP_Olambda__61,axiom,
! [B: $tType,A: $tType,Uu: A,Uua: B] : aa(B,set(product_prod(A,B)),aTP_Lamp_df(A,fun(B,set(product_prod(A,B))),Uu),Uua) = aa(set(product_prod(A,B)),set(product_prod(A,B)),insert(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uu),Uua)),bot_bot(set(product_prod(A,B)))) ).
% ATP.lambda_61
tff(fact_2776_ATP_Olambda__62,axiom,
! [B: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: product_prod(A,B)] :
( pp(aa(product_prod(A,B),bool,aTP_Lamp_rt(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),Uu),Uua))
<=> pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),aa(set(product_prod(A,B)),set(product_prod(A,B)),insert(product_prod(A,B),Uua),bot_bot(set(product_prod(A,B))))),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),Uu)))) ) ).
% ATP.lambda_62
tff(fact_2777_ATP_Olambda__63,axiom,
! [D: $tType,B: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: fun(D,product_prod(A,B))] :
( pp(aa(fun(D,product_prod(A,B)),bool,aTP_Lamp_aa(fun(A,fun(B,bool)),fun(fun(D,product_prod(A,B)),bool),Uu),Uua))
<=> pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),aa(set(D),set(product_prod(A,B)),aa(fun(D,product_prod(A,B)),fun(set(D),set(product_prod(A,B))),image2(D,product_prod(A,B)),Uua),top_top(set(D)))),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),Uu)))) ) ).
% ATP.lambda_63
tff(fact_2778_ATP_Olambda__64,axiom,
! [A: $tType,B: $tType,Uu: set(B),Uua: fun(A,B)] :
( pp(aa(fun(A,B),bool,aTP_Lamp_oo(set(B),fun(fun(A,B),bool),Uu),Uua))
<=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),Uua),top_top(set(A)))),Uu)) ) ).
% ATP.lambda_64
tff(fact_2779_ATP_Olambda__65,axiom,
! [D: $tType,A: $tType,Uu: set(A),Uua: fun(D,A)] :
( pp(aa(fun(D,A),bool,aTP_Lamp_bn(set(A),fun(fun(D,A),bool),Uu),Uua))
<=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(D),set(A),aa(fun(D,A),fun(set(D),set(A)),image2(D,A),Uua),top_top(set(D)))),Uu)) ) ).
% ATP.lambda_65
tff(fact_2780_ATP_Olambda__66,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat] :
( pp(aa(nat,bool,aTP_Lamp_kc(set(product_prod(A,A)),fun(nat,bool),Uu),Uua))
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Uua))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Uua),aa(set(product_prod(A,A)),nat,finite_card(product_prod(A,A)),Uu))) ) ) ).
% ATP.lambda_66
tff(fact_2781_ATP_Olambda__67,axiom,
! [Uu: nat,Uua: nat] :
( pp(aa(nat,bool,aTP_Lamp_km(nat,fun(nat,bool),Uu),Uua))
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Uua))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Uua),aa(nat,nat,suc,Uu))) ) ) ).
% ATP.lambda_67
tff(fact_2782_ATP_Olambda__68,axiom,
! [B: $tType,Uu: fun(nat,sum_sum(B,nat)),Uua: nat] :
( pp(aa(nat,bool,aTP_Lamp_wk(fun(nat,sum_sum(B,nat)),fun(nat,bool),Uu),Uua))
<=> ( aa(nat,sum_sum(B,nat),Uu,Uua) = aa(nat,sum_sum(B,nat),sum_Inr(nat,B),zero_zero(nat)) ) ) ).
% ATP.lambda_68
tff(fact_2783_ATP_Olambda__69,axiom,
! [B: $tType,A: $tType,Uu: fun(B,A),Uua: B] : aa(B,set(A),aTP_Lamp_du(fun(B,A),fun(B,set(A)),Uu),Uua) = aa(set(A),set(A),insert(A,aa(B,A,Uu,Uua)),bot_bot(set(A))) ).
% ATP.lambda_69
tff(fact_2784_ATP_Olambda__70,axiom,
! [Uu: nat,Uua: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_uf(nat,fun(nat,nat)),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat_triangle(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uua))),Uu) ).
% ATP.lambda_70
tff(fact_2785_ATP_Olambda__71,axiom,
! [A: $tType,Uu: set(A),Uua: set(A)] :
( pp(aa(set(A),bool,aTP_Lamp_jv(set(A),fun(set(A),bool),Uu),Uua))
<=> ( finite_finite2(A,Uua)
& pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Uua),Uu)) ) ) ).
% ATP.lambda_71
tff(fact_2786_ATP_Olambda__72,axiom,
! [B: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: option(product_prod(A,B))] :
( pp(aa(option(product_prod(A,B)),bool,aTP_Lamp_ss(fun(A,fun(B,bool)),fun(option(product_prod(A,B)),bool),Uu),Uua))
<=> pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),aa(option(product_prod(A,B)),set(product_prod(A,B)),set_option(product_prod(A,B)),Uua)),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),Uu)))) ) ).
% ATP.lambda_72
tff(fact_2787_ATP_Olambda__73,axiom,
! [B: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: heap_Heap(product_prod(A,B))] :
( pp(aa(heap_Heap(product_prod(A,B)),bool,aTP_Lamp_sz(fun(A,fun(B,bool)),fun(heap_Heap(product_prod(A,B)),bool),Uu),Uua))
<=> pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),aa(heap_Heap(product_prod(A,B)),set(product_prod(A,B)),heap_set_Heap(product_prod(A,B)),Uua)),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),Uu)))) ) ).
% ATP.lambda_73
tff(fact_2788_ATP_Olambda__74,axiom,
! [B: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: array(product_prod(A,B))] :
( pp(aa(array(product_prod(A,B)),bool,aTP_Lamp_wx(fun(A,fun(B,bool)),fun(array(product_prod(A,B)),bool),Uu),Uua))
<=> pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),set_array(product_prod(A,B),Uua)),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),Uu)))) ) ).
% ATP.lambda_74
tff(fact_2789_ATP_Olambda__75,axiom,
! [B: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: ref(product_prod(A,B))] :
( pp(aa(ref(product_prod(A,B)),bool,aTP_Lamp_ww(fun(A,fun(B,bool)),fun(ref(product_prod(A,B)),bool),Uu),Uua))
<=> pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),set_ref(product_prod(A,B),Uua)),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),Uu)))) ) ).
% ATP.lambda_75
tff(fact_2790_ATP_Olambda__76,axiom,
! [B: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: list(product_prod(A,B))] :
( pp(aa(list(product_prod(A,B)),bool,aTP_Lamp_pi(fun(A,fun(B,bool)),fun(list(product_prod(A,B)),bool),Uu),Uua))
<=> pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),aa(list(product_prod(A,B)),set(product_prod(A,B)),set3(product_prod(A,B)),Uua)),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),Uu)))) ) ).
% ATP.lambda_76
tff(fact_2791_ATP_Olambda__77,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [Uu: nat,Uua: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_rd(nat,fun(nat,A)),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(nat,A,semiring_1_of_nat(A),Uu)),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).
% ATP.lambda_77
tff(fact_2792_ATP_Olambda__78,axiom,
! [A: $tType] :
( countable(A)
=> ! [Uu: nat,Uua: nat] : aa(nat,set(old_node(A,product_unit)),aa(nat,fun(nat,set(old_node(A,product_unit))),aTP_Lamp_wf(nat,fun(nat,set(old_node(A,product_unit)))),Uu),Uua) = old_Scons(A,product_unit,nth_item(A,Uu),nth_item(A,Uua)) ) ).
% ATP.lambda_78
tff(fact_2793_ATP_Olambda__79,axiom,
! [Uu: nat,Uua: nat] : aa(nat,rat,aa(nat,fun(nat,rat),aTP_Lamp_us(nat,fun(nat,rat)),Uu),Uua) = fract(aa(nat,int,nat_int_decode,Uu),aa(nat,int,nat_int_decode,Uua)) ).
% ATP.lambda_79
tff(fact_2794_ATP_Olambda__80,axiom,
! [A: $tType,Uu: list(A),Uua: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aTP_Lamp_pk(list(A),fun(list(A),bool)),Uu),Uua))
<=> ( aa(list(A),nat,size_size(list(A)),Uu) = aa(list(A),nat,size_size(list(A)),Uua) ) ) ).
% ATP.lambda_80
tff(fact_2795_ATP_Olambda__81,axiom,
! [A: $tType,Uu: set(A),Uua: heap_Heap(A)] :
( pp(aa(heap_Heap(A),bool,aTP_Lamp_sy(set(A),fun(heap_Heap(A),bool),Uu),Uua))
<=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(heap_Heap(A),set(A),heap_set_Heap(A),Uua)),Uu)) ) ).
% ATP.lambda_81
tff(fact_2796_ATP_Olambda__82,axiom,
! [A: $tType,B: $tType,Uu: list(B),Uua: A] : aa(A,list(product_prod(A,B)),aTP_Lamp_ph(list(B),fun(A,list(product_prod(A,B))),Uu),Uua) = aa(list(B),list(product_prod(A,B)),map(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua)),Uu) ).
% ATP.lambda_82
tff(fact_2797_ATP_Olambda__83,axiom,
! [Uu: heap_ext(product_unit),Uua: nat] : aa(nat,nat,aTP_Lamp_mm(heap_ext(product_unit),fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),lim(product_unit,Uu)),one_one(nat)) ).
% ATP.lambda_83
tff(fact_2798_ATP_Olambda__84,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [Uu: A,Uua: A] : aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),aTP_Lamp_vj(A,fun(A,product_prod(A,A))),Uu),Uua) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one))),Uua)),one_one(A))) ) ).
% ATP.lambda_84
tff(fact_2799_ATP_Olambda__85,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [Uu: A,Uua: A] : aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),aTP_Lamp_vh(A,fun(A,product_prod(A,A))),Uu),Uua) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one))),Uua)) ) ).
% ATP.lambda_85
tff(fact_2800_ATP_Olambda__86,axiom,
! [A: $tType,Uu: A,Uua: set(set(A))] : aa(set(set(A)),set(set(A)),aa(A,fun(set(set(A)),set(set(A))),aTP_Lamp_hm(A,fun(set(set(A)),set(set(A)))),Uu),Uua) = aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),sup_sup(set(set(A))),Uua),aa(set(set(A)),set(set(A)),aa(fun(set(A),set(A)),fun(set(set(A)),set(set(A))),image2(set(A),set(A)),insert(A,Uu)),Uua)) ).
% ATP.lambda_86
tff(fact_2801_ATP_Olambda__87,axiom,
! [A: $tType,B: $tType,Uu: set(product_prod(B,A)),Uua: B] : aa(B,set(A),aTP_Lamp_it(set(product_prod(B,A)),fun(B,set(A)),Uu),Uua) = aa(set(B),set(A),image(B,A,Uu),aa(set(B),set(B),insert(B,Uua),bot_bot(set(B)))) ).
% ATP.lambda_87
tff(fact_2802_ATP_Olambda__88,axiom,
! [A: $tType,B: $tType,Uu: fun(A,B),Uua: B] : aa(B,set(A),aTP_Lamp_hg(fun(A,B),fun(B,set(A)),Uu),Uua) = aa(set(B),set(A),vimage(A,B,Uu),aa(set(B),set(B),insert(B,Uua),bot_bot(set(B)))) ).
% ATP.lambda_88
tff(fact_2803_ATP_Olambda__89,axiom,
! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,product_prod(A,B),aTP_Lamp_fq(fun(A,B),fun(A,product_prod(A,B)),Uu),Uua) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),aa(A,B,Uu,Uua)) ).
% ATP.lambda_89
tff(fact_2804_ATP_Olambda__90,axiom,
! [B: $tType,A: $tType,Uu: fun(A,option(B)),Uua: A] : aa(A,product_prod(A,B),aTP_Lamp_sc(fun(A,option(B)),fun(A,product_prod(A,B)),Uu),Uua) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),aa(option(B),B,the2(B),aa(A,option(B),Uu,Uua))) ).
% ATP.lambda_90
tff(fact_2805_ATP_Olambda__91,axiom,
! [Uu: int,Uua: int] : aa(int,int,aa(int,fun(int,int),aTP_Lamp_wy(int,fun(int,int)),Uu),Uua) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Uu),zero_neq_one_of_bool(int,aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),fequal(int),Uua),zero_zero(int))))) ).
% ATP.lambda_91
tff(fact_2806_ATP_Olambda__92,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat] :
( pp(aa(nat,bool,aTP_Lamp_iz(set(product_prod(A,A)),fun(nat,bool),Uu),Uua))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Uua),aa(set(product_prod(A,A)),nat,finite_card(product_prod(A,A)),Uu))) ) ).
% ATP.lambda_92
tff(fact_2807_ATP_Olambda__93,axiom,
! [Uu: nat,Uua: nat] : aa(nat,list(nat),aa(nat,fun(nat,list(nat)),aTP_Lamp_ur(nat,fun(nat,list(nat))),Uu),Uua) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),Uu),aa(nat,list(nat),nat_list_decode,Uua)) ).
% ATP.lambda_93
tff(fact_2808_ATP_Olambda__94,axiom,
! [Uu: nat,Uua: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_kn(nat,fun(nat,bool)),Uu),Uua))
<=> ( Uua = aa(nat,nat,suc,Uu) ) ) ).
% ATP.lambda_94
tff(fact_2809_ATP_Olambda__95,axiom,
! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_li(nat,fun(nat,nat),Uu),Uua) = modulo_modulo(nat,Uua,Uu) ).
% ATP.lambda_95
tff(fact_2810_ATP_Olambda__96,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_mu(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uua),Uu) ) ).
% ATP.lambda_96
tff(fact_2811_ATP_Olambda__97,axiom,
! [Uu: nat,Uua: nat] :
( pp(aa(nat,bool,aTP_Lamp_jw(nat,fun(nat,bool),Uu),Uua))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uua),Uu)) ) ).
% ATP.lambda_97
tff(fact_2812_ATP_Olambda__98,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_mh(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uu) ) ).
% ATP.lambda_98
tff(fact_2813_ATP_Olambda__99,axiom,
! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_le(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uu) ).
% ATP.lambda_99
tff(fact_2814_ATP_Olambda__100,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_ma(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),Uua),Uu) ) ).
% ATP.lambda_100
tff(fact_2815_ATP_Olambda__101,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_eo(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),minus_minus(A),Uua),Uu) ) ).
% ATP.lambda_101
tff(fact_2816_ATP_Olambda__102,axiom,
! [A: $tType] :
( comple592849572758109894attice(A)
=> ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_gb(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Uua),Uu) ) ).
% ATP.lambda_102
tff(fact_2817_ATP_Olambda__103,axiom,
! [A: $tType,Uu: set(A),Uua: set(A)] : aa(set(A),set(A),aTP_Lamp_gs(set(A),fun(set(A),set(A)),Uu),Uua) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Uua),Uu) ).
% ATP.lambda_103
tff(fact_2818_ATP_Olambda__104,axiom,
! [A: $tType] :
( comple592849572758109894attice(A)
=> ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_fw(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Uua),Uu) ) ).
% ATP.lambda_104
tff(fact_2819_ATP_Olambda__105,axiom,
! [Uu: int,Uua: int] : aa(int,int,aTP_Lamp_nu(int,fun(int,int),Uu),Uua) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Uua),Uu) ).
% ATP.lambda_105
tff(fact_2820_ATP_Olambda__106,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_kt(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu) ) ).
% ATP.lambda_106
tff(fact_2821_ATP_Olambda__107,axiom,
! [Uu: nat,Uua: nat] :
( pp(aa(nat,bool,aTP_Lamp_tj(nat,fun(nat,bool),Uu),Uua))
<=> dvd_dvd(nat,Uua,Uu) ) ).
% ATP.lambda_107
tff(fact_2822_ATP_Olambda__108,axiom,
! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: set(A)] : aa(set(A),set(product_prod(A,B)),aTP_Lamp_dd(fun(A,set(B)),fun(set(A),set(product_prod(A,B))),Uu),Uua) = product_Sigma(A,B,Uua,Uu) ).
% ATP.lambda_108
tff(fact_2823_ATP_Olambda__109,axiom,
! [Uu: nat,Uua: nat] : aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_ra(nat,fun(nat,product_prod(nat,nat))),Uu),Uua) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Uua),Uu) ).
% ATP.lambda_109
tff(fact_2824_ATP_Olambda__110,axiom,
! [A: $tType,B: $tType,Uu: B,Uua: A] : aa(A,product_prod(A,B),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_bk(B,fun(A,product_prod(A,B))),Uu),Uua) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uu) ).
% ATP.lambda_110
tff(fact_2825_ATP_Olambda__111,axiom,
! [B: $tType,A: $tType,Uu: A,Uua: B] : aa(B,product_prod(B,A),aa(A,fun(B,product_prod(B,A)),aTP_Lamp_ba(A,fun(B,product_prod(B,A))),Uu),Uua) = aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uua),Uu) ).
% ATP.lambda_111
tff(fact_2826_ATP_Olambda__112,axiom,
! [A: $tType,B: $tType,Uu: fun(B,pred(A)),Uua: pred(B)] : aa(pred(B),pred(A),aTP_Lamp_wp(fun(B,pred(A)),fun(pred(B),pred(A)),Uu),Uua) = bind3(B,A,Uua,Uu) ).
% ATP.lambda_112
tff(fact_2827_ATP_Olambda__113,axiom,
! [A: $tType,Uu: list(A),Uua: A] : aa(A,list(A),aa(list(A),fun(A,list(A)),aTP_Lamp_pj(list(A),fun(A,list(A))),Uu),Uua) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uua),Uu) ).
% ATP.lambda_113
tff(fact_2828_ATP_Olambda__114,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: array(A),Uua: heap_ext(product_unit)] : aa(heap_ext(product_unit),nat,aTP_Lamp_az(array(A),fun(heap_ext(product_unit),nat),Uu),Uua) = aa(array(A),nat,length(A,Uua),Uu) ) ).
% ATP.lambda_114
tff(fact_2829_ATP_Olambda__115,axiom,
! [B: $tType,Uu: set(B),Uua: B] :
( pp(aa(B,bool,aTP_Lamp_iv(set(B),fun(B,bool),Uu),Uua))
<=> pp(aa(set(B),bool,member(B,Uua),Uu)) ) ).
% ATP.lambda_115
tff(fact_2830_ATP_Olambda__116,axiom,
! [A: $tType] :
( order(A)
=> ! [Uu: set(A),Uua: A] :
( pp(aa(A,bool,aTP_Lamp_jr(set(A),fun(A,bool),Uu),Uua))
<=> pp(aa(set(A),bool,member(A,Uua),Uu)) ) ) ).
% ATP.lambda_116
tff(fact_2831_ATP_Olambda__117,axiom,
! [A: $tType,Uu: set(A),Uua: A] :
( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_a(set(A),fun(A,bool)),Uu),Uua))
<=> pp(aa(set(A),bool,member(A,Uua),Uu)) ) ).
% ATP.lambda_117
tff(fact_2832_ATP_Olambda__118,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat] : aa(nat,set(product_prod(A,A)),aTP_Lamp_iy(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),Uu),Uua) = compow(set(product_prod(A,A)),Uua,Uu) ).
% ATP.lambda_118
tff(fact_2833_ATP_Olambda__119,axiom,
! [A: $tType,Uu: fun(A,fun(A,bool)),Uua: nat] : aa(nat,fun(A,fun(A,bool)),aTP_Lamp_tf(fun(A,fun(A,bool)),fun(nat,fun(A,fun(A,bool))),Uu),Uua) = compow(fun(A,fun(A,bool)),Uua,Uu) ).
% ATP.lambda_119
tff(fact_2834_ATP_Olambda__120,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: array(A),Uua: heap_ext(product_unit)] : aa(heap_ext(product_unit),list(A),aTP_Lamp_bb(array(A),fun(heap_ext(product_unit),list(A)),Uu),Uua) = aa(array(A),list(A),get(A,Uua),Uu) ) ).
% ATP.lambda_120
tff(fact_2835_ATP_Olambda__121,axiom,
! [B: $tType,C: $tType,A: $tType,Uu: fun(A,C),Uua: fun(C,set(B))] : aa(fun(C,set(B)),fun(A,set(B)),aTP_Lamp_cl(fun(A,C),fun(fun(C,set(B)),fun(A,set(B))),Uu),Uua) = aa(fun(A,C),fun(A,set(B)),aa(fun(C,set(B)),fun(fun(A,C),fun(A,set(B))),comp(C,set(B),A),Uua),Uu) ).
% ATP.lambda_121
tff(fact_2836_ATP_Olambda__122,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: ref(A),Uua: heap_ext(product_unit)] : aa(heap_ext(product_unit),A,aTP_Lamp_bj(ref(A),fun(heap_ext(product_unit),A),Uu),Uua) = aa(ref(A),A,get2(A,Uua),Uu) ) ).
% ATP.lambda_122
tff(fact_2837_ATP_Olambda__123,axiom,
! [A: $tType,Uu: code_natural,Uua: A] :
( pp(aa(A,bool,aa(code_natural,fun(A,bool),aTP_Lamp_tl(code_natural,fun(A,bool)),Uu),Uua))
<=> pp(aa(code_natural,bool,aa(code_natural,fun(code_natural,bool),ord_less(code_natural),zero_zero(code_natural)),Uu)) ) ).
% ATP.lambda_123
tff(fact_2838_ATP_Olambda__124,axiom,
! [Uu: nat,Uua: nat] : aa(nat,set(nat),aTP_Lamp_ko(nat,fun(nat,set(nat)),Uu),Uua) = order_underS(nat,bNF_Ca8665028551170535155natLeq,Uu) ).
% ATP.lambda_124
tff(fact_2839_ATP_Olambda__125,axiom,
! [B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: A] : aa(A,set(B),aTP_Lamp_bv(set(product_prod(A,B)),fun(A,set(B)),Uu),Uua) = aa(set(product_prod(A,B)),set(B),aa(fun(product_prod(A,B),B),fun(set(product_prod(A,B)),set(B)),image2(product_prod(A,B),B),product_snd(A,B)),Uu) ).
% ATP.lambda_125
tff(fact_2840_ATP_Olambda__126,axiom,
! [A: $tType,B: $tType,Uu: fun(A,nat),Uua: product_prod(B,A)] : aa(product_prod(B,A),nat,aTP_Lamp_tr(fun(A,nat),fun(product_prod(B,A),nat),Uu),Uua) = aa(A,nat,Uu,aa(product_prod(B,A),A,product_snd(B,A),Uua)) ).
% ATP.lambda_126
tff(fact_2841_ATP_Olambda__127,axiom,
! [B: $tType,A: $tType,Uu: fun(A,nat),Uua: product_prod(A,B)] : aa(product_prod(A,B),nat,aTP_Lamp_tq(fun(A,nat),fun(product_prod(A,B),nat),Uu),Uua) = aa(A,nat,Uu,aa(product_prod(A,B),A,product_fst(A,B),Uua)) ).
% ATP.lambda_127
tff(fact_2842_ATP_Olambda__128,axiom,
! [A: $tType,C: $tType,Uu: C,Uua: fun(C,set(set(A)))] : aa(fun(C,set(set(A))),set(set(A)),aTP_Lamp_oe(C,fun(fun(C,set(set(A))),set(set(A))),Uu),Uua) = aa(C,set(set(A)),Uua,Uu) ).
% ATP.lambda_128
tff(fact_2843_ATP_Olambda__129,axiom,
! [A: $tType,B: $tType,Uu: B,Uua: fun(B,set(A))] : aa(fun(B,set(A)),set(A),aTP_Lamp_dv(B,fun(fun(B,set(A)),set(A)),Uu),Uua) = aa(B,set(A),Uua,Uu) ).
% ATP.lambda_129
tff(fact_2844_ATP_Olambda__130,axiom,
! [B: $tType,A: $tType] :
( complete_Sup(B)
=> ! [Uu: A,Uua: fun(A,B)] : aa(fun(A,B),B,aTP_Lamp_cr(A,fun(fun(A,B),B),Uu),Uua) = aa(A,B,Uua,Uu) ) ).
% ATP.lambda_130
tff(fact_2845_ATP_Olambda__131,axiom,
! [B: $tType,A: $tType] :
( complete_Inf(B)
=> ! [Uu: A,Uua: fun(A,B)] : aa(fun(A,B),B,aTP_Lamp_cs(A,fun(fun(A,B),B),Uu),Uua) = aa(A,B,Uua,Uu) ) ).
% ATP.lambda_131
tff(fact_2846_ATP_Olambda__132,axiom,
! [A: $tType,Uu: fun(product_unit,A),Uua: product_unit] : aa(product_unit,A,aTP_Lamp_tc(fun(product_unit,A),fun(product_unit,A),Uu),Uua) = aa(product_unit,A,Uu,product_Unity) ).
% ATP.lambda_132
tff(fact_2847_ATP_Olambda__133,axiom,
! [Uu: nat,Uua: num] : aa(num,option(num),aTP_Lamp_xb(nat,fun(num,option(num)),Uu),Uua) = aa(num,option(num),some(num),case_option(num,num,one,bit1,bit_take_bit_num(Uu,Uua))) ).
% ATP.lambda_133
tff(fact_2848_ATP_Olambda__134,axiom,
! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),fun(product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A))))),aTP_Lamp_sr(A,fun(list(A),fun(product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A)))))),Uu),Uua) = aa(fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A)))))),fun(product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A))))),product_case_prod(A,list(A),option(product_prod(list(A),product_prod(A,list(A))))),aa(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A)))))),aTP_Lamp_sq(A,fun(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))))),Uu),Uua)) ).
% ATP.lambda_134
tff(fact_2849_ATP_Olambda__135,axiom,
! [Uu: nat,Uua: nat] : aa(nat,fun(product_prod(nat,nat),bool),aa(nat,fun(nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_rf(nat,fun(nat,fun(product_prod(nat,nat),bool))),Uu),Uua) = aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aa(nat,fun(nat,fun(nat,bool)),aTP_Lamp_re(nat,fun(nat,fun(nat,fun(nat,bool))),Uu),Uua)) ).
% ATP.lambda_135
tff(fact_2850_ATP_Olambda__136,axiom,
! [Uu: nat,Uua: nat] : aa(nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_rc(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),Uu),Uua) = aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aa(nat,fun(nat,fun(nat,product_prod(nat,nat))),aTP_Lamp_rb(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua)) ).
% ATP.lambda_136
tff(fact_2851_ATP_Olambda__137,axiom,
! [Uu: nat,Uua: nat] : aa(nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_qz(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),Uu),Uua) = aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aa(nat,fun(nat,fun(nat,product_prod(nat,nat))),aTP_Lamp_qy(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua)) ).
% ATP.lambda_137
tff(fact_2852_ATP_Olambda__138,axiom,
! [Uu: nat,Uua: nat] : aa(nat,fun(product_prod(nat,nat),product_prod(nat,nat)),aa(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))),aTP_Lamp_qx(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat)))),Uu),Uua) = aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aa(nat,fun(nat,fun(nat,product_prod(nat,nat))),aTP_Lamp_qw(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua)) ).
% ATP.lambda_138
tff(fact_2853_ATP_Olambda__139,axiom,
! [Uu: nat,Uua: nat] : aa(nat,fun(product_prod(nat,nat),bool),aa(nat,fun(nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_qt(nat,fun(nat,fun(product_prod(nat,nat),bool))),Uu),Uua) = aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aa(nat,fun(nat,fun(nat,bool)),aTP_Lamp_qs(nat,fun(nat,fun(nat,fun(nat,bool))),Uu),Uua)) ).
% ATP.lambda_139
tff(fact_2854_ATP_Olambda__140,axiom,
! [Uu: nat,Uua: nat] : aa(nat,fun(product_prod(nat,nat),bool),aa(nat,fun(nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_qr(nat,fun(nat,fun(product_prod(nat,nat),bool))),Uu),Uua) = aa(fun(nat,fun(nat,bool)),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,bool),aa(nat,fun(nat,fun(nat,bool)),aTP_Lamp_qq(nat,fun(nat,fun(nat,fun(nat,bool))),Uu),Uua)) ).
% ATP.lambda_140
tff(fact_2855_ATP_Olambda__141,axiom,
! [A: $tType,B: $tType,Uu: fun(A,fun(B,bool)),Uua: B] : aa(B,set(A),aTP_Lamp_kz(fun(A,fun(B,bool)),fun(B,set(A)),Uu),Uua) = aa(fun(A,bool),set(A),collect(A),aa(B,fun(A,bool),aTP_Lamp_ky(fun(A,fun(B,bool)),fun(B,fun(A,bool)),Uu),Uua)) ).
% ATP.lambda_141
tff(fact_2856_ATP_Olambda__142,axiom,
! [C: $tType,A: $tType,B: $tType] :
( comple592849572758109894attice(A)
=> ! [Uu: fun(C,fun(B,A)),Uua: fun(B,C)] : aa(fun(B,C),A,aTP_Lamp_eg(fun(C,fun(B,A)),fun(fun(B,C),A),Uu),Uua) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),aa(fun(B,C),fun(B,A),aTP_Lamp_ef(fun(C,fun(B,A)),fun(fun(B,C),fun(B,A)),Uu),Uua)),top_top(set(B)))) ) ).
% ATP.lambda_142
tff(fact_2857_ATP_Olambda__143,axiom,
! [B: $tType,A: $tType,C: $tType] :
( comple592849572758109894attice(A)
=> ! [Uu: fun(C,fun(B,A)),Uua: B] : aa(B,A,aTP_Lamp_eh(fun(C,fun(B,A)),fun(B,A),Uu),Uua) = aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),aa(fun(C,A),fun(set(C),set(A)),image2(C,A),aa(B,fun(C,A),aTP_Lamp_ed(fun(C,fun(B,A)),fun(B,fun(C,A)),Uu),Uua)),top_top(set(C)))) ) ).
% ATP.lambda_143
tff(fact_2858_ATP_Olambda__144,axiom,
! [C: $tType,A: $tType,B: $tType] :
( comple592849572758109894attice(A)
=> ! [Uu: fun(C,fun(B,A)),Uua: fun(B,C)] : aa(fun(B,C),A,aTP_Lamp_ei(fun(C,fun(B,A)),fun(fun(B,C),A),Uu),Uua) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),aa(fun(B,C),fun(B,A),aTP_Lamp_ef(fun(C,fun(B,A)),fun(fun(B,C),fun(B,A)),Uu),Uua)),top_top(set(B)))) ) ).
% ATP.lambda_144
tff(fact_2859_ATP_Olambda__145,axiom,
! [B: $tType,A: $tType,C: $tType] :
( comple592849572758109894attice(A)
=> ! [Uu: fun(C,fun(B,A)),Uua: B] : aa(B,A,aTP_Lamp_ee(fun(C,fun(B,A)),fun(B,A),Uu),Uua) = aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),aa(fun(C,A),fun(set(C),set(A)),image2(C,A),aa(B,fun(C,A),aTP_Lamp_ed(fun(C,fun(B,A)),fun(B,fun(C,A)),Uu),Uua)),top_top(set(C)))) ) ).
% ATP.lambda_145
tff(fact_2860_ATP_Olambda__146,axiom,
! [A: $tType,Uu: fun(heap_ext(product_unit),A),Uua: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))),aTP_Lamp_sh(fun(heap_ext(product_unit),A),fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),Uu),Uua) = aa(product_prod(A,heap_ext(product_unit)),option(product_prod(A,heap_ext(product_unit))),some(product_prod(A,heap_ext(product_unit))),aa(heap_ext(product_unit),product_prod(A,heap_ext(product_unit)),aa(A,fun(heap_ext(product_unit),product_prod(A,heap_ext(product_unit))),product_Pair(A,heap_ext(product_unit)),aa(heap_ext(product_unit),A,Uu,Uua)),Uua)) ).
% ATP.lambda_146
tff(fact_2861_ATP_Olambda__147,axiom,
! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: A] :
( pp(aa(A,bool,aTP_Lamp_ti(fun(A,option(B)),fun(A,bool),Uu),Uua))
<=> ( aa(A,option(B),Uu,Uua) != none(B) ) ) ).
% ATP.lambda_147
tff(fact_2862_ATP_Olambda__148,axiom,
! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A] : aa(A,set(product_prod(A,B)),aTP_Lamp_dg(fun(A,set(B)),fun(A,set(product_prod(A,B))),Uu),Uua) = aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(B),set(set(product_prod(A,B))),aa(fun(B,set(product_prod(A,B))),fun(set(B),set(set(product_prod(A,B)))),image2(B,set(product_prod(A,B))),aTP_Lamp_df(A,fun(B,set(product_prod(A,B))),Uua)),aa(A,set(B),Uu,Uua))) ).
% ATP.lambda_148
tff(fact_2863_ATP_Olambda__149,axiom,
! [B: $tType,A: $tType,Uu: set(set(old_node(A,B))),Uua: set(old_node(A,B))] : aa(set(old_node(A,B)),set(set(old_node(A,B))),aTP_Lamp_wn(set(set(old_node(A,B))),fun(set(old_node(A,B)),set(set(old_node(A,B)))),Uu),Uua) = aa(set(set(set(old_node(A,B)))),set(set(old_node(A,B))),complete_Sup_Sup(set(set(old_node(A,B)))),aa(set(set(old_node(A,B))),set(set(set(old_node(A,B)))),aa(fun(set(old_node(A,B)),set(set(old_node(A,B)))),fun(set(set(old_node(A,B))),set(set(set(old_node(A,B))))),image2(set(old_node(A,B)),set(set(old_node(A,B)))),aTP_Lamp_wm(set(old_node(A,B)),fun(set(old_node(A,B)),set(set(old_node(A,B)))),Uua)),Uu)) ).
% ATP.lambda_149
tff(fact_2864_ATP_Olambda__150,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: list(product_prod(C,B)),Uua: product_prod(A,C)] : aa(product_prod(A,C),list(product_prod(A,B)),aTP_Lamp_pp(list(product_prod(C,B)),fun(product_prod(A,C),list(product_prod(A,B))),Uu),Uua) = concat(product_prod(A,B),aa(list(product_prod(C,B)),list(list(product_prod(A,B))),map(product_prod(C,B),list(product_prod(A,B)),aTP_Lamp_po(product_prod(A,C),fun(product_prod(C,B),list(product_prod(A,B))),Uua)),Uu)) ).
% ATP.lambda_150
tff(fact_2865_ATP_Olambda__151,axiom,
! [A: $tType,Uu: set(set(A)),Uua: A] :
( pp(aa(A,bool,aTP_Lamp_eu(set(set(A)),fun(A,bool),Uu),Uua))
<=> pp(aa(set(bool),bool,complete_Sup_Sup(bool),aa(set(set(A)),set(bool),aa(fun(set(A),bool),fun(set(set(A)),set(bool)),image2(set(A),bool),member(A,Uua)),Uu))) ) ).
% ATP.lambda_151
tff(fact_2866_ATP_Olambda__152,axiom,
! [A: $tType,Uu: set(set(A)),Uua: A] :
( pp(aa(A,bool,aTP_Lamp_ev(set(set(A)),fun(A,bool),Uu),Uua))
<=> pp(aa(set(bool),bool,complete_Inf_Inf(bool),aa(set(set(A)),set(bool),aa(fun(set(A),bool),fun(set(set(A)),set(bool)),image2(set(A),bool),member(A,Uua)),Uu))) ) ).
% ATP.lambda_152
tff(fact_2867_ATP_Olambda__153,axiom,
! [A: $tType,Uu: fun(nat,set(A)),Uua: nat] : aa(nat,set(A),aTP_Lamp_kq(fun(nat,set(A)),fun(nat,set(A)),Uu),Uua) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),aa(fun(nat,set(A)),fun(set(nat),set(set(A))),image2(nat,set(A)),Uu),set_or7035219750837199246ssThan(nat,zero_zero(nat),Uua))) ).
% ATP.lambda_153
tff(fact_2868_ATP_Olambda__154,axiom,
! [A: $tType,Uu: list(A),Uua: code_natural] : aa(code_natural,fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural))),aTP_Lamp_uo(list(A),fun(code_natural,fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural)))),Uu),Uua) = aa(A,fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural))),product_Pair(A,product_prod(code_natural,code_natural)),aa(nat,A,nth3(A,Uu),code_nat_of_natural(Uua))) ).
% ATP.lambda_154
tff(fact_2869_ATP_Olambda__155,axiom,
! [P3: $tType,O: $tType,Uu: fun(O,set(P3)),Uua: set(O)] : aa(set(O),set(P3),aTP_Lamp_eb(fun(O,set(P3)),fun(set(O),set(P3)),Uu),Uua) = aa(set(set(P3)),set(P3),complete_Sup_Sup(set(P3)),aa(set(O),set(set(P3)),aa(fun(O,set(P3)),fun(set(O),set(set(P3))),image2(O,set(P3)),Uu),Uua)) ).
% ATP.lambda_155
tff(fact_2870_ATP_Olambda__156,axiom,
! [A: $tType,B: $tType] :
( comple592849572758109894attice(A)
=> ! [Uu: fun(B,A),Uua: set(B)] : aa(set(B),A,aTP_Lamp_nn(fun(B,A),fun(set(B),A),Uu),Uua) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),Uu),Uua)) ) ).
% ATP.lambda_156
tff(fact_2871_ATP_Olambda__157,axiom,
! [D: $tType,B: $tType,Uu: set(B),Uua: fun(B,set(D))] : aa(fun(B,set(D)),set(D),aa(set(B),fun(fun(B,set(D)),set(D)),aTP_Lamp_tz(set(B),fun(fun(B,set(D)),set(D))),Uu),Uua) = aa(set(set(D)),set(D),complete_Sup_Sup(set(D)),aa(set(B),set(set(D)),aa(fun(B,set(D)),fun(set(B),set(set(D))),image2(B,set(D)),Uua),Uu)) ).
% ATP.lambda_157
tff(fact_2872_ATP_Olambda__158,axiom,
! [C: $tType,B: $tType] :
( complete_Sup(C)
=> ! [Uu: set(B),Uua: fun(B,C)] : aa(fun(B,C),C,aa(set(B),fun(fun(B,C),C),aTP_Lamp_tx(set(B),fun(fun(B,C),C)),Uu),Uua) = aa(set(C),C,complete_Sup_Sup(C),aa(set(B),set(C),aa(fun(B,C),fun(set(B),set(C)),image2(B,C),Uua),Uu)) ) ).
% ATP.lambda_158
tff(fact_2873_ATP_Olambda__159,axiom,
! [A: $tType,Uu: set(A),Uua: fun(A,bool)] :
( pp(aa(fun(A,bool),bool,aa(set(A),fun(fun(A,bool),bool),aTP_Lamp_oh(set(A),fun(fun(A,bool),bool)),Uu),Uua))
<=> pp(aa(set(bool),bool,complete_Sup_Sup(bool),aa(set(A),set(bool),aa(fun(A,bool),fun(set(A),set(bool)),image2(A,bool),Uua),Uu))) ) ).
% ATP.lambda_159
tff(fact_2874_ATP_Olambda__160,axiom,
! [C: $tType,A: $tType,Uu: set(A),Uua: fun(A,set(C))] : aa(fun(A,set(C)),set(C),aa(set(A),fun(fun(A,set(C)),set(C)),aTP_Lamp_ty(set(A),fun(fun(A,set(C)),set(C))),Uu),Uua) = aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(A),set(set(C)),aa(fun(A,set(C)),fun(set(A),set(set(C))),image2(A,set(C)),Uua),Uu)) ).
% ATP.lambda_160
tff(fact_2875_ATP_Olambda__161,axiom,
! [C: $tType,A: $tType] :
( complete_Sup(C)
=> ! [Uu: set(A),Uua: fun(A,C)] : aa(fun(A,C),C,aa(set(A),fun(fun(A,C),C),aTP_Lamp_tw(set(A),fun(fun(A,C),C)),Uu),Uua) = aa(set(C),C,complete_Sup_Sup(C),aa(set(A),set(C),aa(fun(A,C),fun(set(A),set(C)),image2(A,C),Uua),Uu)) ) ).
% ATP.lambda_161
tff(fact_2876_ATP_Olambda__162,axiom,
! [P3: $tType,O: $tType,Uu: fun(O,set(P3)),Uua: set(O)] : aa(set(O),set(P3),aTP_Lamp_dl(fun(O,set(P3)),fun(set(O),set(P3)),Uu),Uua) = aa(set(set(P3)),set(P3),complete_Inf_Inf(set(P3)),aa(set(O),set(set(P3)),aa(fun(O,set(P3)),fun(set(O),set(set(P3))),image2(O,set(P3)),Uu),Uua)) ).
% ATP.lambda_162
tff(fact_2877_ATP_Olambda__163,axiom,
! [A: $tType,B: $tType] :
( comple592849572758109894attice(A)
=> ! [Uu: fun(B,A),Uua: set(B)] : aa(set(B),A,aTP_Lamp_no(fun(B,A),fun(set(B),A),Uu),Uua) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),Uu),Uua)) ) ).
% ATP.lambda_163
tff(fact_2878_ATP_Olambda__164,axiom,
! [C: $tType,B: $tType] :
( complete_Inf(C)
=> ! [Uu: set(B),Uua: fun(B,C)] : aa(fun(B,C),C,aa(set(B),fun(fun(B,C),C),aTP_Lamp_tv(set(B),fun(fun(B,C),C)),Uu),Uua) = aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),aa(fun(B,C),fun(set(B),set(C)),image2(B,C),Uua),Uu)) ) ).
% ATP.lambda_164
tff(fact_2879_ATP_Olambda__165,axiom,
! [A: $tType,Uu: set(A),Uua: fun(A,bool)] :
( pp(aa(fun(A,bool),bool,aa(set(A),fun(fun(A,bool),bool),aTP_Lamp_kr(set(A),fun(fun(A,bool),bool)),Uu),Uua))
<=> pp(aa(set(bool),bool,complete_Inf_Inf(bool),aa(set(A),set(bool),aa(fun(A,bool),fun(set(A),set(bool)),image2(A,bool),Uua),Uu))) ) ).
% ATP.lambda_165
tff(fact_2880_ATP_Olambda__166,axiom,
! [C: $tType,A: $tType] :
( complete_Inf(C)
=> ! [Uu: set(A),Uua: fun(A,C)] : aa(fun(A,C),C,aa(set(A),fun(fun(A,C),C),aTP_Lamp_tu(set(A),fun(fun(A,C),C)),Uu),Uua) = aa(set(C),C,complete_Inf_Inf(C),aa(set(A),set(C),aa(fun(A,C),fun(set(A),set(C)),image2(A,C),Uua),Uu)) ) ).
% ATP.lambda_166
tff(fact_2881_ATP_Olambda__167,axiom,
! [A: $tType,Uu: pred(A),Uua: pred(A)] : aa(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),aTP_Lamp_wt(pred(A),fun(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),Uu),Uua) = aa(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),product_Pair(pred(A),product_prod(code_natural,code_natural)),aa(pred(A),pred(A),aa(pred(A),fun(pred(A),pred(A)),sup_sup(pred(A)),Uu),Uua)) ).
% ATP.lambda_167
tff(fact_2882_ATP_Olambda__168,axiom,
! [C: $tType,A: $tType,B: $tType,Uu: A,Uua: B] : aa(B,fun(C,product_prod(product_prod(A,B),C)),aTP_Lamp_ja(A,fun(B,fun(C,product_prod(product_prod(A,B),C))),Uu),Uua) = aa(product_prod(A,B),fun(C,product_prod(product_prod(A,B),C)),product_Pair(product_prod(A,B),C),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uu),Uua)) ).
% ATP.lambda_168
tff(fact_2883_ATP_Olambda__169,axiom,
! [A: $tType,Uu: list(product_prod(code_natural,A)),Uua: code_natural] : aa(code_natural,fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural))),aTP_Lamp_ug(list(product_prod(code_natural,A)),fun(code_natural,fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural)))),Uu),Uua) = aa(A,fun(product_prod(code_natural,code_natural),product_prod(A,product_prod(code_natural,code_natural))),product_Pair(A,product_prod(code_natural,code_natural)),aa(code_natural,A,pick(A,Uu),Uua)) ).
% ATP.lambda_169
tff(fact_2884_ATP_Olambda__170,axiom,
! [B: $tType,Uu: B,Uua: list(B)] : aa(list(B),heap_Heap(list(B)),aTP_Lamp_rx(B,fun(list(B),heap_Heap(list(B))),Uu),Uua) = aa(list(B),heap_Heap(list(B)),heap_return(list(B)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Uu),Uua)) ).
% ATP.lambda_170
tff(fact_2885_ATP_Olambda__171,axiom,
! [A: $tType,Uu: list(A),Uua: list(A)] : aa(list(A),heap_Heap(list(A)),aTP_Lamp_rz(list(A),fun(list(A),heap_Heap(list(A))),Uu),Uua) = aa(list(A),heap_Heap(list(A)),heap_return(list(A)),append(A,Uu,Uua)) ).
% ATP.lambda_171
tff(fact_2886_ATP_Olambda__172,axiom,
! [B: $tType,A: $tType,Uu: B,Uua: A] : aa(A,fun(set(product_prod(B,A)),set(product_prod(B,A))),aTP_Lamp_ho(B,fun(A,fun(set(product_prod(B,A)),set(product_prod(B,A)))),Uu),Uua) = insert(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uu),Uua)) ).
% ATP.lambda_172
tff(fact_2887_ATP_Olambda__173,axiom,
! [A: $tType,B: $tType,Uu: A,Uua: B] : aa(B,fun(set(product_prod(A,B)),set(product_prod(A,B))),aTP_Lamp_hs(A,fun(B,fun(set(product_prod(A,B)),set(product_prod(A,B)))),Uu),Uua) = insert(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uu),Uua)) ).
% ATP.lambda_173
tff(fact_2888_ATP_Olambda__174,axiom,
! [A: $tType,Uu: A,Uua: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_qg(A,fun(A,bool)),Uu),Uua))
<=> ( Uu != Uua ) ) ).
% ATP.lambda_174
tff(fact_2889_ATP_Olambda__175,axiom,
! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: B] : aa(B,set(A),aTP_Lamp_ny(fun(B,set(A)),fun(B,set(A)),Uu),Uua) = aa(set(A),set(A),uminus_uminus(set(A)),aa(B,set(A),Uu,Uua)) ).
% ATP.lambda_175
tff(fact_2890_ATP_Olambda__176,axiom,
! [A: $tType,B: $tType] :
( comple489889107523837845lgebra(A)
=> ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_nz(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(B,A,Uu,Uua)) ) ).
% ATP.lambda_176
tff(fact_2891_ATP_Olambda__177,axiom,
! [A: $tType,B: $tType,Uu: fun(B,heap_Heap(A)),Uua: B] : aa(B,fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),aTP_Lamp_sf(fun(B,heap_Heap(A)),fun(B,fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))))),Uu),Uua) = aa(heap_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_execute(A),aa(B,heap_Heap(A),Uu,Uua)) ).
% ATP.lambda_177
tff(fact_2892_ATP_Olambda__178,axiom,
! [B: $tType,A: $tType,Uu: fun(A,heap_Heap(B)),Uua: A] : aa(A,fun(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit)))),aTP_Lamp_sk(fun(A,heap_Heap(B)),fun(A,fun(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit))))),Uu),Uua) = aa(heap_Heap(B),fun(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit)))),heap_execute(B),aa(A,heap_Heap(B),Uu,Uua)) ).
% ATP.lambda_178
tff(fact_2893_ATP_Olambda__179,axiom,
! [B: $tType,A: $tType,C: $tType,Uu: fun(C,set(product_prod(B,A))),Uua: C] : aa(C,set(product_prod(A,B)),aTP_Lamp_ip(fun(C,set(product_prod(B,A))),fun(C,set(product_prod(A,B))),Uu),Uua) = converse(B,A,aa(C,set(product_prod(B,A)),Uu,Uua)) ).
% ATP.lambda_179
tff(fact_2894_ATP_Olambda__180,axiom,
! [B: $tType,A: $tType,C: $tType,Uu: fun(C,A),Uua: C] : aa(C,fun(B,product_prod(A,B)),aTP_Lamp_pa(fun(C,A),fun(C,fun(B,product_prod(A,B))),Uu),Uua) = aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,Uu,Uua)) ).
% ATP.lambda_180
tff(fact_2895_ATP_Olambda__181,axiom,
! [C: $tType,B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,fun(C,product_prod(B,C)),aTP_Lamp_rv(fun(A,B),fun(A,fun(C,product_prod(B,C))),Uu),Uua) = aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(A,B,Uu,Uua)) ).
% ATP.lambda_181
tff(fact_2896_ATP_Olambda__182,axiom,
! [A: $tType,B: $tType,Uu: fun(B,A),Uua: B] : aa(B,heap_Heap(A),aTP_Lamp_bl(fun(B,A),fun(B,heap_Heap(A)),Uu),Uua) = aa(A,heap_Heap(A),heap_return(A),aa(B,A,Uu,Uua)) ).
% ATP.lambda_182
tff(fact_2897_ATP_Olambda__183,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: fun(C,set(old_node(A,B))),Uua: C] : aa(C,set(old_node(A,B)),aTP_Lamp_vu(fun(C,set(old_node(A,B))),fun(C,set(old_node(A,B))),Uu),Uua) = aa(set(old_node(A,B)),set(old_node(A,B)),old_In1(A,B),aa(C,set(old_node(A,B)),Uu,Uua)) ).
% ATP.lambda_183
tff(fact_2898_ATP_Olambda__184,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: fun(C,set(old_node(A,B))),Uua: C] : aa(C,set(old_node(A,B)),aTP_Lamp_vv(fun(C,set(old_node(A,B))),fun(C,set(old_node(A,B))),Uu),Uua) = aa(set(old_node(A,B)),set(old_node(A,B)),old_In0(A,B),aa(C,set(old_node(A,B)),Uu,Uua)) ).
% ATP.lambda_184
tff(fact_2899_ATP_Olambda__185,axiom,
! [C: $tType,D: $tType,Uu: fun(D,set(C)),Uua: D] : aa(D,filter(C),aTP_Lamp_im(fun(D,set(C)),fun(D,filter(C)),Uu),Uua) = principal(C,aa(D,set(C),Uu,Uua)) ).
% ATP.lambda_185
tff(fact_2900_ATP_Olambda__186,axiom,
! [D: $tType,C: $tType,Uu: fun(C,set(D)),Uua: C] : aa(C,filter(D),aTP_Lamp_is(fun(C,set(D)),fun(C,filter(D)),Uu),Uua) = principal(D,aa(C,set(D),Uu,Uua)) ).
% ATP.lambda_186
tff(fact_2901_ATP_Olambda__187,axiom,
! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: B] : aa(B,filter(A),aTP_Lamp_if(fun(B,set(A)),fun(B,filter(A)),Uu),Uua) = principal(A,aa(B,set(A),Uu,Uua)) ).
% ATP.lambda_187
tff(fact_2902_ATP_Olambda__188,axiom,
! [E: $tType,A: $tType,Uu: fun(A,set(E)),Uua: A] : aa(A,filter(E),aTP_Lamp_ir(fun(A,set(E)),fun(A,filter(E)),Uu),Uua) = principal(E,aa(A,set(E),Uu,Uua)) ).
% ATP.lambda_188
tff(fact_2903_ATP_Olambda__189,axiom,
! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A] : aa(A,filter(B),aTP_Lamp_id(fun(A,set(B)),fun(A,filter(B)),Uu),Uua) = principal(B,aa(A,set(B),Uu,Uua)) ).
% ATP.lambda_189
tff(fact_2904_ATP_Olambda__190,axiom,
! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A] : aa(A,nat,aTP_Lamp_lr(fun(A,set(B)),fun(A,nat),Uu),Uua) = aa(set(B),nat,finite_card(B),aa(A,set(B),Uu,Uua)) ).
% ATP.lambda_190
tff(fact_2905_ATP_Olambda__191,axiom,
! [A: $tType,B: $tType,Uu: fun(B,list(A)),Uua: B] : aa(B,set(A),aTP_Lamp_pc(fun(B,list(A)),fun(B,set(A)),Uu),Uua) = aa(list(A),set(A),set3(A),aa(B,list(A),Uu,Uua)) ).
% ATP.lambda_191
tff(fact_2906_ATP_Olambda__192,axiom,
! [B: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: A] : aa(A,set(B),aTP_Lamp_ce(fun(A,fun(B,bool)),fun(A,set(B)),Uu),Uua) = aa(fun(B,bool),set(B),collect(B),aa(A,fun(B,bool),Uu,Uua)) ).
% ATP.lambda_192
tff(fact_2907_ATP_Olambda__193,axiom,
! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,fun(set(B),set(B)),aTP_Lamp_hr(fun(A,B),fun(A,fun(set(B),set(B))),Uu),Uua) = insert(B,aa(A,B,Uu,Uua)) ).
% ATP.lambda_193
tff(fact_2908_ATP_Olambda__194,axiom,
! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: B] : aa(B,set(set(A)),aTP_Lamp_dk(fun(B,set(A)),fun(B,set(set(A))),Uu),Uua) = pow(A,aa(B,set(A),Uu,Uua)) ).
% ATP.lambda_194
tff(fact_2909_ATP_Olambda__195,axiom,
! [B: $tType,Uu: fun(B,bool),Uua: B] :
( pp(aa(B,bool,aTP_Lamp_fh(fun(B,bool),fun(B,bool),Uu),Uua))
<=> ~ pp(aa(B,bool,Uu,Uua)) ) ).
% ATP.lambda_195
tff(fact_2910_ATP_Olambda__196,axiom,
! [A: $tType,B: $tType,Uu: fun(A,fun(B,bool)),Uua: A] :
( pp(aa(A,bool,aTP_Lamp_ot(fun(A,fun(B,bool)),fun(A,bool),Uu),Uua))
<=> ! [X_1: B] : pp(aa(B,bool,aa(A,fun(B,bool),Uu,Uua),X_1)) ) ).
% ATP.lambda_196
tff(fact_2911_ATP_Olambda__197,axiom,
! [A: $tType,B: $tType,Uu: fun(B,A),Uua: B] : aa(B,fun(A,bool),aTP_Lamp_wo(fun(B,A),fun(B,fun(A,bool)),Uu),Uua) = aa(A,fun(A,bool),fequal(A),aa(B,A,Uu,Uua)) ).
% ATP.lambda_197
tff(fact_2912_ATP_Olambda__198,axiom,
! [A: $tType,B: $tType,Uu: fun(A,fun(B,bool)),Uua: A] :
( pp(aa(A,bool,aTP_Lamp_ms(fun(A,fun(B,bool)),fun(A,bool),Uu),Uua))
<=> ? [X_1: B] : pp(aa(B,bool,aa(A,fun(B,bool),Uu,Uua),X_1)) ) ).
% ATP.lambda_198
tff(fact_2913_ATP_Olambda__199,axiom,
! [A: $tType,B: $tType,Uu: fun(nat,sum_sum(B,nat)),Uua: sum_sum(A,nat)] : aa(sum_sum(A,nat),nat,aa(fun(nat,sum_sum(B,nat)),fun(sum_sum(A,nat),nat),aTP_Lamp_wl(fun(nat,sum_sum(B,nat)),fun(sum_sum(A,nat),nat)),Uu),Uua) = ord_Least(nat,aTP_Lamp_wk(fun(nat,sum_sum(B,nat)),fun(nat,bool),Uu)) ).
% ATP.lambda_199
tff(fact_2914_ATP_Olambda__200,axiom,
! [Uu: nat,Uua: nat] : aa(nat,set(nat),aTP_Lamp_kp(nat,fun(nat,set(nat)),Uu),Uua) = aa(fun(nat,bool),set(nat),collect(nat),aTP_Lamp_jw(nat,fun(nat,bool),Uu)) ).
% ATP.lambda_200
tff(fact_2915_ATP_Olambda__201,axiom,
! [A: $tType,Uu: set(A),Uua: set(A)] : aa(set(A),filter(set(A)),aTP_Lamp_ju(set(A),fun(set(A),filter(set(A))),Uu),Uua) = principal(set(A),aa(fun(set(A),bool),set(set(A)),collect(set(A)),aa(set(A),fun(set(A),bool),aTP_Lamp_jt(set(A),fun(set(A),fun(set(A),bool)),Uu),Uua))) ).
% ATP.lambda_201
tff(fact_2916_ATP_Olambda__202,axiom,
! [A: $tType,B: $tType,Uu: fun(A,bool),Uua: fun(B,bool)] : aa(fun(B,bool),filter(product_prod(A,B)),aa(fun(A,bool),fun(fun(B,bool),filter(product_prod(A,B))),aTP_Lamp_jd(fun(A,bool),fun(fun(B,bool),filter(product_prod(A,B)))),Uu),Uua) = principal(product_prod(A,B),aa(fun(product_prod(A,B),bool),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aa(fun(B,bool),fun(A,fun(B,bool)),aTP_Lamp_by(fun(A,bool),fun(fun(B,bool),fun(A,fun(B,bool))),Uu),Uua)))) ).
% ATP.lambda_202
tff(fact_2917_ATP_Olambda__203,axiom,
! [B: $tType,A: $tType,Uu: fun(B,fun(A,bool)),Uua: A] :
( pp(aa(A,bool,aTP_Lamp_mt(fun(B,fun(A,bool)),fun(A,bool),Uu),Uua))
<=> ? [X5: B] : pp(aa(A,bool,aa(B,fun(A,bool),Uu,X5),Uua)) ) ).
% ATP.lambda_203
tff(fact_2918_ATP_Olambda__204,axiom,
! [B: $tType,Uu: set(set(B)),Uua: set(B)] :
( pp(aa(set(B),bool,aTP_Lamp_np(set(set(B)),fun(set(B),bool),Uu),Uua))
<=> ? [F7: fun(set(B),B)] :
( ( Uua = aa(set(set(B)),set(B),aa(fun(set(B),B),fun(set(set(B)),set(B)),image2(set(B),B),F7),Uu) )
& ! [X5: set(B)] :
( pp(aa(set(set(B)),bool,member(set(B),X5),Uu))
=> pp(aa(set(B),bool,member(B,aa(set(B),B,F7,X5)),X5)) ) ) ) ).
% ATP.lambda_204
tff(fact_2919_ATP_Olambda__205,axiom,
! [A: $tType] :
( comple592849572758109894attice(A)
=> ! [Uu: set(set(A)),Uua: set(A)] :
( pp(aa(set(A),bool,aTP_Lamp_nk(set(set(A)),fun(set(A),bool),Uu),Uua))
<=> ? [F7: fun(set(A),A)] :
( ( Uua = aa(set(set(A)),set(A),aa(fun(set(A),A),fun(set(set(A)),set(A)),image2(set(A),A),F7),Uu) )
& ! [X5: set(A)] :
( pp(aa(set(set(A)),bool,member(set(A),X5),Uu))
=> pp(aa(set(A),bool,member(A,aa(set(A),A,F7,X5)),X5)) ) ) ) ) ).
% ATP.lambda_205
tff(fact_2920_ATP_Olambda__206,axiom,
! [A: $tType] :
( comple6319245703460814977attice(A)
=> ! [Uu: set(set(A)),Uua: set(A)] :
( pp(aa(set(A),bool,aTP_Lamp_nl(set(set(A)),fun(set(A),bool),Uu),Uua))
<=> ? [F7: fun(set(A),A)] :
( ( Uua = aa(set(set(A)),set(A),aa(fun(set(A),A),fun(set(set(A)),set(A)),image2(set(A),A),F7),Uu) )
& ! [X5: set(A)] :
( pp(aa(set(set(A)),bool,member(set(A),X5),Uu))
=> pp(aa(set(A),bool,member(A,aa(set(A),A,F7,X5)),X5)) ) ) ) ) ).
% ATP.lambda_206
tff(fact_2921_ATP_Olambda__207,axiom,
! [A: $tType] :
( finite8700451911770168679attice(A)
=> ! [Uu: set(set(A)),Uua: set(A)] :
( pp(aa(set(A),bool,aTP_Lamp_nm(set(set(A)),fun(set(A),bool),Uu),Uua))
<=> ? [F7: fun(set(A),A)] :
( ( Uua = aa(set(set(A)),set(A),aa(fun(set(A),A),fun(set(set(A)),set(A)),image2(set(A),A),F7),Uu) )
& ! [X5: set(A)] :
( pp(aa(set(set(A)),bool,member(set(A),X5),Uu))
=> pp(aa(set(A),bool,member(A,aa(set(A),A,F7,X5)),X5)) ) ) ) ) ).
% ATP.lambda_207
tff(fact_2922_ATP_Olambda__208,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A)] :
( pp(aa(set(A),bool,aTP_Lamp_lk(set(product_prod(A,A)),fun(set(A),bool),Uu),Uua))
<=> ! [X5: A] :
( pp(aa(set(A),bool,member(A,X5),Uua))
=> ! [Xa3: A] :
( pp(aa(set(A),bool,member(A,Xa3),Uua))
=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),Xa3)),Uu))
| pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa3),X5)),Uu)) ) ) ) ) ).
% ATP.lambda_208
tff(fact_2923_ATP_Olambda__209,axiom,
! [B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: A] :
( pp(aa(A,bool,aTP_Lamp_nd(set(product_prod(A,B)),fun(A,bool),Uu),Uua))
<=> ? [Y4: B] : pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Y4)),Uu)) ) ).
% ATP.lambda_209
tff(fact_2924_ATP_Olambda__210,axiom,
! [A: $tType,B: $tType,Uu: fun(B,set(old_node(A,B))),Uua: set(old_node(A,B))] :
( pp(aa(set(old_node(A,B)),bool,aTP_Lamp_vy(fun(B,set(old_node(A,B))),fun(set(old_node(A,B)),bool),Uu),Uua))
<=> ? [X5: B] : Uua = aa(set(old_node(A,B)),set(old_node(A,B)),aa(fun(old_node(A,B),old_node(A,B)),fun(set(old_node(A,B)),set(old_node(A,B))),image2(old_node(A,B),old_node(A,B)),old_Push_Node(B,A,aa(B,sum_sum(B,nat),sum_Inl(B,nat),X5))),aa(B,set(old_node(A,B)),Uu,X5)) ) ).
% ATP.lambda_210
tff(fact_2925_ATP_Olambda__211,axiom,
! [A: $tType,B: $tType,Uu: fun(B,A),Uua: A] :
( pp(aa(A,bool,aTP_Lamp_nb(fun(B,A),fun(A,bool),Uu),Uua))
<=> ? [X5: B] : Uua = aa(B,A,Uu,X5) ) ).
% ATP.lambda_211
tff(fact_2926_ATP_Olambda__212,axiom,
! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: product_prod(A,B)] :
( pp(aa(product_prod(A,B),bool,aTP_Lamp_sd(fun(A,option(B)),fun(product_prod(A,B),bool),Uu),Uua))
<=> ? [A7: A,B5: B] :
( ( Uua = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A7),B5) )
& ( aa(A,option(B),Uu,A7) = aa(B,option(B),some(B),B5) ) ) ) ).
% ATP.lambda_212
tff(fact_2927_ATP_Olambda__213,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: product_prod(set(A),set(A))] :
( pp(aa(product_prod(set(A),set(A)),bool,aTP_Lamp_of(set(product_prod(A,A)),fun(product_prod(set(A),set(A)),bool),Uu),Uua))
<=> ? [X9: set(A),Y8: set(A)] :
( ( Uua = aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),X9),Y8) )
& ( X9 != bot_bot(set(A)) )
& ! [X5: A] :
( pp(aa(set(A),bool,member(A,X5),Y8))
=> ? [Xa3: A] :
( pp(aa(set(A),bool,member(A,Xa3),X9))
& pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa3),X5)),Uu)) ) ) ) ) ).
% ATP.lambda_213
tff(fact_2928_ATP_Olambda__214,axiom,
! [A: $tType,B: $tType,Uu: fun(A,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(B,bool)))),Uua: fun(A,heap_Heap(B))] :
( pp(aa(fun(A,heap_Heap(B)),bool,aTP_Lamp_va(fun(A,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(B,bool)))),fun(fun(A,heap_Heap(B)),bool),Uu),Uua))
<=> ! [X5: A,H8: heap_ext(product_unit),H9: heap_ext(product_unit),R9: B] :
( heap_effect(B,aa(A,heap_Heap(B),Uua,X5),H8,H9,R9)
=> pp(aa(B,bool,aa(heap_ext(product_unit),fun(B,bool),aa(heap_ext(product_unit),fun(heap_ext(product_unit),fun(B,bool)),aa(A,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(B,bool))),Uu,X5),H8),H9),R9)) ) ) ).
% ATP.lambda_214
tff(fact_2929_ATP_Olambda__215,axiom,
! [A: $tType,B: $tType,Uu: set(B),Uua: A] : aa(A,set(B),aTP_Lamp_nx(set(B),fun(A,set(B)),Uu),Uua) = aa(set(B),set(B),uminus_uminus(set(B)),Uu) ).
% ATP.lambda_215
tff(fact_2930_ATP_Olambda__216,axiom,
! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),option(A),aa(A,fun(list(A),option(A)),aTP_Lamp_tn(A,fun(list(A),option(A))),Uu),Uua) = aa(A,option(A),some(A),Uu) ).
% ATP.lambda_216
tff(fact_2931_ATP_Olambda__217,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: array(A),Uua: array(A)] : aa(array(A),heap_Heap(array(A)),aTP_Lamp_kh(array(A),fun(array(A),heap_Heap(array(A))),Uu),Uua) = aa(array(A),heap_Heap(array(A)),heap_return(array(A)),Uu) ) ).
% ATP.lambda_217
tff(fact_2932_ATP_Olambda__218,axiom,
! [A: $tType,B: $tType,Uu: fun(B,bool),Uua: A] : aa(A,set(B),aTP_Lamp_bz(fun(B,bool),fun(A,set(B)),Uu),Uua) = aa(fun(B,bool),set(B),collect(B),Uu) ).
% ATP.lambda_218
tff(fact_2933_ATP_Olambda__219,axiom,
! [B: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: option(A),Uub: option(B)] :
( pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),aTP_Lamp_sv(fun(A,fun(B,bool)),fun(option(A),fun(option(B),bool)),Uu),Uua),Uub))
<=> pp(case_option(bool,A,case_option(bool,B,fTrue,aTP_Lamp_st(B,bool),Uub),aa(option(B),fun(A,bool),aTP_Lamp_su(fun(A,fun(B,bool)),fun(option(B),fun(A,bool)),Uu),Uub),Uua)) ) ).
% ATP.lambda_219
tff(fact_2934_ATP_Olambda__220,axiom,
! [Uu: num,Uua: code_integer,Uub: code_integer] : aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_vk(num,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = if(product_prod(code_integer,code_integer),aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),aa(num,code_integer,numeral_numeral(code_integer),Uu)),Uub),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(one))),Uua)),one_one(code_integer))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),Uub),aa(num,code_integer,numeral_numeral(code_integer),Uu))),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(one))),Uua)),Uub)) ).
% ATP.lambda_220
tff(fact_2935_ATP_Olambda__221,axiom,
! [Uu: num,Uua: nat,Uub: nat] : aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_vi(num,fun(nat,fun(nat,product_prod(nat,nat))),Uu),Uua),Uub) = if(product_prod(nat,nat),aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),Uu)),Uub),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one))),Uua)),one_one(nat))),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uub),aa(num,nat,numeral_numeral(nat),Uu))),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one))),Uua)),Uub)) ).
% ATP.lambda_221
tff(fact_2936_ATP_Olambda__222,axiom,
! [Uu: num,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_vf(num,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = if(product_prod(int,int),aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),Uu)),Uub),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one))),Uua)),one_one(int))),aa(int,int,aa(int,fun(int,int),minus_minus(int),Uub),aa(num,int,numeral_numeral(int),Uu))),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one))),Uua)),Uub)) ).
% ATP.lambda_222
tff(fact_2937_ATP_Olambda__223,axiom,
! [A: $tType] :
( unique1627219031080169319umeral(A)
=> ! [Uu: num,Uua: A,Uub: A] : aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),aTP_Lamp_vg(num,fun(A,fun(A,product_prod(A,A))),Uu),Uua),Uub) = if(product_prod(A,A),aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),Uu)),Uub),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one))),Uua)),one_one(A))),aa(A,A,aa(A,fun(A,A),minus_minus(A),Uub),aa(num,A,numeral_numeral(A),Uu))),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one))),Uua)),Uub)) ) ).
% ATP.lambda_223
tff(fact_2938_ATP_Olambda__224,axiom,
! [Uu: code_integer,Uua: code_integer,Uub: code_integer] : aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_vo(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = if(product_prod(code_integer,code_integer),aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),Uub),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),one_one(code_integer))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uu)),Uub))) ).
% ATP.lambda_224
tff(fact_2939_ATP_Olambda__225,axiom,
! [Uu: code_integer,Uua: code_integer,Uub: code_integer] : aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_vr(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = if(product_prod(code_integer,code_integer),aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),Uub),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),one_one(code_integer))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,abs_abs(code_integer),Uu)),Uub))) ).
% ATP.lambda_225
tff(fact_2940_ATP_Olambda__226,axiom,
! [Uu: code_integer,Uua: code_integer,Uub: code_integer] : aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_vn(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = if(product_prod(code_integer,code_integer),aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),Uub),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),one_one(code_integer))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),Uu),Uub))) ).
% ATP.lambda_226
tff(fact_2941_ATP_Olambda__227,axiom,
! [A: $tType,Uu: fun(heap_ext(product_unit),bool),Uua: fun(heap_ext(product_unit),product_prod(A,heap_ext(product_unit))),Uub: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))),aa(fun(heap_ext(product_unit),product_prod(A,heap_ext(product_unit))),fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),aTP_Lamp_sg(fun(heap_ext(product_unit),bool),fun(fun(heap_ext(product_unit),product_prod(A,heap_ext(product_unit))),fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))))),Uu),Uua),Uub) = if(option(product_prod(A,heap_ext(product_unit))),aa(heap_ext(product_unit),bool,Uu,Uub),aa(product_prod(A,heap_ext(product_unit)),option(product_prod(A,heap_ext(product_unit))),some(product_prod(A,heap_ext(product_unit))),aa(heap_ext(product_unit),product_prod(A,heap_ext(product_unit)),Uua,Uub)),none(product_prod(A,heap_ext(product_unit)))) ).
% ATP.lambda_227
tff(fact_2942_ATP_Olambda__228,axiom,
! [A: $tType,B: $tType,Uu: set(B),Uua: A,Uub: set(product_prod(A,B))] : aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(A,fun(set(product_prod(A,B)),set(product_prod(A,B))),aTP_Lamp_ht(set(B),fun(A,fun(set(product_prod(A,B)),set(product_prod(A,B)))),Uu),Uua),Uub) = finite_fold(B,set(product_prod(A,B)),aTP_Lamp_hs(A,fun(B,fun(set(product_prod(A,B)),set(product_prod(A,B)))),Uua),Uub,Uu) ).
% ATP.lambda_228
tff(fact_2943_ATP_Olambda__229,axiom,
! [B: $tType,A: $tType,Uu: set(A),Uua: B,Uub: set(product_prod(B,A))] : aa(set(product_prod(B,A)),set(product_prod(B,A)),aa(B,fun(set(product_prod(B,A)),set(product_prod(B,A))),aTP_Lamp_hy(set(A),fun(B,fun(set(product_prod(B,A)),set(product_prod(B,A)))),Uu),Uua),Uub) = finite_fold(A,set(product_prod(B,A)),aTP_Lamp_ho(B,fun(A,fun(set(product_prod(B,A)),set(product_prod(B,A)))),Uua),Uub,Uu) ).
% ATP.lambda_229
tff(fact_2944_ATP_Olambda__230,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: array(A),Uua: list(A),Uub: fun(typerep,fun(nat,list(nat)))] : aa(fun(typerep,fun(nat,list(nat))),fun(typerep,fun(nat,list(nat))),aa(list(A),fun(fun(typerep,fun(nat,list(nat))),fun(typerep,fun(nat,list(nat)))),aTP_Lamp_uu(array(A),fun(list(A),fun(fun(typerep,fun(nat,list(nat))),fun(typerep,fun(nat,list(nat))))),Uu),Uua),Uub) = fun_upd(typerep,fun(nat,list(nat)),Uub,typerep_typerep(A,type2(A)),fun_upd(nat,list(nat),aa(typerep,fun(nat,list(nat)),Uub,typerep_typerep(A,type2(A))),addr_of_array(A,Uu),aa(list(A),list(nat),map(A,nat,to_nat(A)),Uua))) ) ).
% ATP.lambda_230
tff(fact_2945_ATP_Olambda__231,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: ref(A),Uua: A,Uub: fun(typerep,fun(nat,nat))] : aa(fun(typerep,fun(nat,nat)),fun(typerep,fun(nat,nat)),aa(A,fun(fun(typerep,fun(nat,nat)),fun(typerep,fun(nat,nat))),aTP_Lamp_uv(ref(A),fun(A,fun(fun(typerep,fun(nat,nat)),fun(typerep,fun(nat,nat)))),Uu),Uua),Uub) = fun_upd(typerep,fun(nat,nat),Uub,typerep_typerep(A,type2(A)),fun_upd(nat,nat,aa(typerep,fun(nat,nat),Uub,typerep_typerep(A,type2(A))),aa(ref(A),nat,addr_of_ref(A),Uu),aa(A,nat,to_nat(A),Uua))) ) ).
% ATP.lambda_231
tff(fact_2946_ATP_Olambda__232,axiom,
! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: A,Uub: B] : aa(B,fun(A,option(B)),aa(A,fun(B,fun(A,option(B))),aTP_Lamp_up(fun(A,option(B)),fun(A,fun(B,fun(A,option(B)))),Uu),Uua),Uub) = fun_upd(A,option(B),Uu,Uua,aa(B,option(B),some(B),Uub)) ).
% ATP.lambda_232
tff(fact_2947_ATP_Olambda__233,axiom,
! [A: $tType,B: $tType,Uu: A,Uua: B,Uub: fun(A,option(B))] : aa(fun(A,option(B)),fun(A,option(B)),aa(B,fun(fun(A,option(B)),fun(A,option(B))),aa(A,fun(B,fun(fun(A,option(B)),fun(A,option(B)))),aTP_Lamp_sn(A,fun(B,fun(fun(A,option(B)),fun(A,option(B))))),Uu),Uua),Uub) = fun_upd(A,option(B),Uub,Uu,aa(B,option(B),some(B),Uua)) ).
% ATP.lambda_233
tff(fact_2948_ATP_Olambda__234,axiom,
! [A: $tType,Uu: set(A),Uua: set(A),Uub: bool] : aa(bool,set(A),aa(set(A),fun(bool,set(A)),aTP_Lamp_fr(set(A),fun(set(A),fun(bool,set(A))),Uu),Uua),Uub) = if(set(A),Uub,Uu,Uua) ).
% ATP.lambda_234
tff(fact_2949_ATP_Olambda__235,axiom,
! [A: $tType,B: $tType,Uu: fun(A,fun(B,bool)),Uua: option(B),Uub: A] :
( pp(aa(A,bool,aa(option(B),fun(A,bool),aTP_Lamp_su(fun(A,fun(B,bool)),fun(option(B),fun(A,bool)),Uu),Uua),Uub))
<=> pp(case_option(bool,B,fFalse,aa(A,fun(B,bool),Uu,Uub),Uua)) ) ).
% ATP.lambda_235
tff(fact_2950_ATP_Olambda__236,axiom,
! [B: $tType,A: $tType,Uu: heap_Heap(A),Uua: fun(A,heap_Heap(B)),Uub: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit))),aa(fun(A,heap_Heap(B)),fun(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit)))),aTP_Lamp_sl(heap_Heap(A),fun(fun(A,heap_Heap(B)),fun(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit))))),Uu),Uua),Uub) = 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))),aa(fun(A,fun(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit))))),fun(product_prod(A,heap_ext(product_unit)),option(product_prod(B,heap_ext(product_unit)))),product_case_prod(A,heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit)))),aTP_Lamp_sk(fun(A,heap_Heap(B)),fun(A,fun(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit))))),Uua)),aa(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))),aa(heap_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),heap_execute(A),Uu),Uub)) ).
% ATP.lambda_236
tff(fact_2951_ATP_Olambda__237,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,Uu: fun(A,fun(B,C)),Uua: fun(D,A),Uub: product_prod(D,B)] : aa(product_prod(D,B),C,aa(fun(D,A),fun(product_prod(D,B),C),aTP_Lamp_be(fun(A,fun(B,C)),fun(fun(D,A),fun(product_prod(D,B),C)),Uu),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu,aa(D,A,Uua,aa(product_prod(D,B),D,product_fst(D,B),Uub))),aa(product_prod(D,B),B,product_snd(D,B),Uub)) ).
% ATP.lambda_237
tff(fact_2952_ATP_Olambda__238,axiom,
! [C: $tType,A: $tType,B: $tType] :
( comple592849572758109894attice(A)
=> ! [Uu: fun(C,fun(B,A)),Uua: fun(B,C),Uub: B] : aa(B,A,aa(fun(B,C),fun(B,A),aTP_Lamp_ef(fun(C,fun(B,A)),fun(fun(B,C),fun(B,A)),Uu),Uua),Uub) = aa(B,A,aa(C,fun(B,A),Uu,aa(B,C,Uua,Uub)),Uub) ) ).
% ATP.lambda_238
tff(fact_2953_ATP_Olambda__239,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ns(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,aa(nat,fun(nat,A),Uu,Uub),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub)) ) ).
% ATP.lambda_239
tff(fact_2954_ATP_Olambda__240,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [Uu: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_lw(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,aa(nat,fun(nat,A),Uu,Uub),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub)) ) ).
% ATP.lambda_240
tff(fact_2955_ATP_Olambda__241,axiom,
! [A: $tType,Uu: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_um(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,aa(nat,fun(nat,A),Uu,Uub),aa(nat,nat,aa(nat,fun(nat,nat),minus_minus(nat),Uua),Uub)) ).
% ATP.lambda_241
tff(fact_2956_ATP_Olambda__242,axiom,
! [A: $tType,B: $tType,Uu: fun(fun(B,heap_Heap(A)),fun(B,heap_Heap(A))),Uua: B,Uub: fun(B,heap_Heap(A))] : aa(fun(B,heap_Heap(A)),heap_Heap(A),aa(B,fun(fun(B,heap_Heap(A)),heap_Heap(A)),aTP_Lamp_ve(fun(fun(B,heap_Heap(A)),fun(B,heap_Heap(A))),fun(B,fun(fun(B,heap_Heap(A)),heap_Heap(A))),Uu),Uua),Uub) = aa(B,heap_Heap(A),aa(fun(B,heap_Heap(A)),fun(B,heap_Heap(A)),Uu,Uub),Uua) ).
% ATP.lambda_242
tff(fact_2957_ATP_Olambda__243,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,Uu: fun(C,fun(fun(A,heap_Heap(B)),heap_Heap(D))),Uua: fun(A,heap_Heap(B)),Uub: C] : aa(C,heap_Heap(D),aa(fun(A,heap_Heap(B)),fun(C,heap_Heap(D)),aTP_Lamp_uw(fun(C,fun(fun(A,heap_Heap(B)),heap_Heap(D))),fun(fun(A,heap_Heap(B)),fun(C,heap_Heap(D))),Uu),Uua),Uub) = aa(fun(A,heap_Heap(B)),heap_Heap(D),aa(C,fun(fun(A,heap_Heap(B)),heap_Heap(D)),Uu,Uub),Uua) ).
% ATP.lambda_243
tff(fact_2958_ATP_Olambda__244,axiom,
! [C: $tType,A: $tType,B: $tType] :
( comple592849572758109894attice(A)
=> ! [Uu: fun(C,fun(B,A)),Uua: B,Uub: C] : aa(C,A,aa(B,fun(C,A),aTP_Lamp_ed(fun(C,fun(B,A)),fun(B,fun(C,A)),Uu),Uua),Uub) = aa(B,A,aa(C,fun(B,A),Uu,Uub),Uua) ) ).
% ATP.lambda_244
tff(fact_2959_ATP_Olambda__245,axiom,
! [C: $tType,A: $tType,B: $tType] :
( complete_Sup(A)
=> ! [Uu: fun(C,fun(B,A)),Uua: B,Uub: C] : aa(C,A,aa(B,fun(C,A),aTP_Lamp_cw(fun(C,fun(B,A)),fun(B,fun(C,A)),Uu),Uua),Uub) = aa(B,A,aa(C,fun(B,A),Uu,Uub),Uua) ) ).
% ATP.lambda_245
tff(fact_2960_ATP_Olambda__246,axiom,
! [C: $tType,A: $tType,B: $tType] :
( complete_Inf(A)
=> ! [Uu: fun(C,fun(B,A)),Uua: B,Uub: C] : aa(C,A,aa(B,fun(C,A),aTP_Lamp_cx(fun(C,fun(B,A)),fun(B,fun(C,A)),Uu),Uua),Uub) = aa(B,A,aa(C,fun(B,A),Uu,Uub),Uua) ) ).
% ATP.lambda_246
tff(fact_2961_ATP_Olambda__247,axiom,
! [C: $tType,A: $tType,B: $tType,Uu: fun(C,fun(B,A)),Uua: B,Uub: C] : aa(C,A,aa(B,fun(C,A),aTP_Lamp_af(fun(C,fun(B,A)),fun(B,fun(C,A)),Uu),Uua),Uub) = aa(B,A,aa(C,fun(B,A),Uu,Uub),Uua) ).
% ATP.lambda_247
tff(fact_2962_ATP_Olambda__248,axiom,
! [B: $tType,A: $tType,C: $tType] :
( comple6319245703460814977attice(A)
=> ! [Uu: fun(B,fun(C,A)),Uua: C,Uub: B] : aa(B,A,aa(C,fun(B,A),aTP_Lamp_do(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu),Uua),Uub) = aa(C,A,aa(B,fun(C,A),Uu,Uub),Uua) ) ).
% ATP.lambda_248
tff(fact_2963_ATP_Olambda__249,axiom,
! [A: $tType,B: $tType,Uu: fun(A,fun(B,bool)),Uua: B,Uub: A] :
( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_ky(fun(A,fun(B,bool)),fun(B,fun(A,bool)),Uu),Uua),Uub))
<=> pp(aa(B,bool,aa(A,fun(B,bool),Uu,Uub),Uua)) ) ).
% ATP.lambda_249
tff(fact_2964_ATP_Olambda__250,axiom,
! [Uu: code_integer,Uua: code_integer,Uub: code_integer] : aa(code_integer,product_prod(code_integer,bool),aa(code_integer,fun(code_integer,product_prod(code_integer,bool)),aTP_Lamp_vq(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,bool))),Uu),Uua),Uub) = aa(bool,product_prod(code_integer,bool),aa(code_integer,fun(bool,product_prod(code_integer,bool)),product_Pair(code_integer,bool),if(code_integer,aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),zero_zero(code_integer)),Uu),Uua,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),minus_minus(code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),Uub))),aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),Uub),one_one(code_integer))) ).
% ATP.lambda_250
tff(fact_2965_ATP_Olambda__251,axiom,
! [A: $tType,B: $tType,Uu: fun(A,fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)))),Uua: pred(A),Uub: product_prod(code_natural,code_natural)] : aa(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)),aa(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural))),aTP_Lamp_ws(fun(A,fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)))),fun(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)))),Uu),Uua),Uub) = aa(product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)),product_prod(pred(B),product_prod(code_natural,code_natural)),aa(fun(product_prod(code_natural,code_natural),fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)))),fun(product_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural)),product_prod(pred(B),product_prod(code_natural,code_natural))),product_case_prod(product_prod(code_natural,code_natural),product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural))),aa(pred(A),fun(product_prod(code_natural,code_natural),fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)))),aTP_Lamp_wr(fun(A,fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)))),fun(pred(A),fun(product_prod(code_natural,code_natural),fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural))))),Uu),Uua)),split_seed(Uub)) ).
% ATP.lambda_251
tff(fact_2966_ATP_Olambda__252,axiom,
! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_rs(rat,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_rr(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).
% ATP.lambda_252
tff(fact_2967_ATP_Olambda__253,axiom,
! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_rq(rat,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_rp(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).
% ATP.lambda_253
tff(fact_2968_ATP_Olambda__254,axiom,
! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_ro(rat,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_rn(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).
% ATP.lambda_254
tff(fact_2969_ATP_Olambda__255,axiom,
! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_rm(rat,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_rl(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).
% ATP.lambda_255
tff(fact_2970_ATP_Olambda__256,axiom,
! [Uu: rat,Uua: int,Uub: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_rj(rat,fun(int,fun(int,bool)),Uu),Uua),Uub))
<=> pp(aa(product_prod(int,int),bool,aa(fun(int,fun(int,bool)),fun(product_prod(int,int),bool),product_case_prod(int,int,bool),aa(int,fun(int,fun(int,bool)),aTP_Lamp_ri(int,fun(int,fun(int,fun(int,bool))),Uua),Uub)),quotient_of(Uu))) ) ).
% ATP.lambda_256
tff(fact_2971_ATP_Olambda__257,axiom,
! [Uu: rat,Uua: int,Uub: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_rh(rat,fun(int,fun(int,bool)),Uu),Uua),Uub))
<=> pp(aa(product_prod(int,int),bool,aa(fun(int,fun(int,bool)),fun(product_prod(int,int),bool),product_case_prod(int,int,bool),aa(int,fun(int,fun(int,bool)),aTP_Lamp_rg(int,fun(int,fun(int,fun(int,bool))),Uua),Uub)),quotient_of(Uu))) ) ).
% ATP.lambda_257
tff(fact_2972_ATP_Olambda__258,axiom,
! [D: $tType,E: $tType,A: $tType,C: $tType,B: $tType,Uu: fun(B,fun(C,fun(D,fun(E,set(A))))),Uua: product_prod(B,C),Uub: product_prod(D,E)] : aa(product_prod(D,E),set(A),aa(product_prod(B,C),fun(product_prod(D,E),set(A)),aTP_Lamp_el(fun(B,fun(C,fun(D,fun(E,set(A))))),fun(product_prod(B,C),fun(product_prod(D,E),set(A))),Uu),Uua),Uub) = aa(product_prod(B,C),set(A),aa(fun(B,fun(C,set(A))),fun(product_prod(B,C),set(A)),product_case_prod(B,C,set(A)),aa(product_prod(D,E),fun(B,fun(C,set(A))),aTP_Lamp_ek(fun(B,fun(C,fun(D,fun(E,set(A))))),fun(product_prod(D,E),fun(B,fun(C,set(A)))),Uu),Uub)),Uua) ).
% ATP.lambda_258
tff(fact_2973_ATP_Olambda__259,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: array(A),Uua: nat,Uub: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(A,heap_ext(product_unit)),aa(nat,fun(heap_ext(product_unit),product_prod(A,heap_ext(product_unit))),aTP_Lamp_kl(array(A),fun(nat,fun(heap_ext(product_unit),product_prod(A,heap_ext(product_unit)))),Uu),Uua),Uub) = aa(heap_ext(product_unit),product_prod(A,heap_ext(product_unit)),aa(A,fun(heap_ext(product_unit),product_prod(A,heap_ext(product_unit))),product_Pair(A,heap_ext(product_unit)),aa(nat,A,nth3(A,aa(array(A),list(A),get(A,Uub),Uu)),Uua)),Uub) ) ).
% ATP.lambda_259
tff(fact_2974_ATP_Olambda__260,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,Uu: fun(A,fun(C,bool)),Uua: fun(B,fun(D,bool)),Uub: sum_sum(product_prod(A,C),product_prod(B,D))] :
( pp(aa(sum_sum(product_prod(A,C),product_prod(B,D)),bool,aa(fun(B,fun(D,bool)),fun(sum_sum(product_prod(A,C),product_prod(B,D)),bool),aTP_Lamp_ue(fun(A,fun(C,bool)),fun(fun(B,fun(D,bool)),fun(sum_sum(product_prod(A,C),product_prod(B,D)),bool)),Uu),Uua),Uub))
<=> ( pp(aa(set(product_prod(A,C)),bool,aa(set(product_prod(A,C)),fun(set(product_prod(A,C)),bool),ord_less_eq(set(product_prod(A,C))),basic_setl(product_prod(A,C),product_prod(B,D),Uub)),aa(fun(product_prod(A,C),bool),set(product_prod(A,C)),collect(product_prod(A,C)),aa(fun(A,fun(C,bool)),fun(product_prod(A,C),bool),product_case_prod(A,C,bool),Uu))))
& pp(aa(set(product_prod(B,D)),bool,aa(set(product_prod(B,D)),fun(set(product_prod(B,D)),bool),ord_less_eq(set(product_prod(B,D))),basic_setr(product_prod(A,C),product_prod(B,D),Uub)),aa(fun(product_prod(B,D),bool),set(product_prod(B,D)),collect(product_prod(B,D)),aa(fun(B,fun(D,bool)),fun(product_prod(B,D),bool),product_case_prod(B,D,bool),Uua)))) ) ) ).
% ATP.lambda_260
tff(fact_2975_ATP_Olambda__261,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,Uu: fun(A,fun(C,bool)),Uua: fun(B,fun(D,bool)),Uub: product_prod(product_prod(A,C),product_prod(B,D))] :
( pp(aa(product_prod(product_prod(A,C),product_prod(B,D)),bool,aa(fun(B,fun(D,bool)),fun(product_prod(product_prod(A,C),product_prod(B,D)),bool),aTP_Lamp_qa(fun(A,fun(C,bool)),fun(fun(B,fun(D,bool)),fun(product_prod(product_prod(A,C),product_prod(B,D)),bool)),Uu),Uua),Uub))
<=> ( pp(aa(set(product_prod(A,C)),bool,aa(set(product_prod(A,C)),fun(set(product_prod(A,C)),bool),ord_less_eq(set(product_prod(A,C))),aa(product_prod(product_prod(A,C),product_prod(B,D)),set(product_prod(A,C)),basic_fsts(product_prod(A,C),product_prod(B,D)),Uub)),aa(fun(product_prod(A,C),bool),set(product_prod(A,C)),collect(product_prod(A,C)),aa(fun(A,fun(C,bool)),fun(product_prod(A,C),bool),product_case_prod(A,C,bool),Uu))))
& pp(aa(set(product_prod(B,D)),bool,aa(set(product_prod(B,D)),fun(set(product_prod(B,D)),bool),ord_less_eq(set(product_prod(B,D))),basic_snds(product_prod(A,C),product_prod(B,D),Uub)),aa(fun(product_prod(B,D),bool),set(product_prod(B,D)),collect(product_prod(B,D)),aa(fun(B,fun(D,bool)),fun(product_prod(B,D),bool),product_case_prod(B,D,bool),Uua)))) ) ) ).
% ATP.lambda_261
tff(fact_2976_ATP_Olambda__262,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aTP_Lamp_pr(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),Uu),Uua),Uub))
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Uua)),aa(list(A),nat,size_size(list(A)),Uub)))
| ( ( aa(list(A),nat,size_size(list(A)),Uua) = aa(list(A),nat,size_size(list(A)),Uub) )
& pp(aa(set(product_prod(list(A),list(A))),bool,member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Uua),Uub)),lex(A,Uu))) ) ) ) ).
% ATP.lambda_262
tff(fact_2977_ATP_Olambda__263,axiom,
! [A: $tType,Uu: nat,Uua: list(A),Uub: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aTP_Lamp_pq(nat,fun(list(A),fun(list(A),bool)),Uu),Uua),Uub))
<=> ( ( aa(list(A),nat,size_size(list(A)),Uua) = aa(nat,nat,suc,Uu) )
& ( aa(list(A),nat,size_size(list(A)),Uub) = aa(nat,nat,suc,Uu) ) ) ) ).
% ATP.lambda_263
tff(fact_2978_ATP_Olambda__264,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aTP_Lamp_pw(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),Uu),Uua),Uub))
<=> ( ( aa(list(A),nat,size_size(list(A)),Uua) = aa(list(A),nat,size_size(list(A)),Uub) )
& ? [Xys2: list(A),X5: A,Y4: A,Xs5: list(A),Ys5: list(A)] :
( ( Uua = append(A,Xys2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X5),Xs5)) )
& ( Uub = append(A,Xys2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys5)) )
& pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),Y4)),Uu)) ) ) ) ).
% ATP.lambda_264
tff(fact_2979_ATP_Olambda__265,axiom,
! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
( pp(aa(list(A),bool,aa(nat,fun(list(A),bool),aTP_Lamp_pl(set(A),fun(nat,fun(list(A),bool)),Uu),Uua),Uub))
<=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set3(A),Uub)),Uu))
& ( aa(list(A),nat,size_size(list(A)),Uub) = aa(nat,nat,suc,Uua) ) ) ) ).
% ATP.lambda_265
tff(fact_2980_ATP_Olambda__266,axiom,
! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
( pp(aa(list(A),bool,aa(nat,fun(list(A),bool),aTP_Lamp_pm(set(A),fun(nat,fun(list(A),bool)),Uu),Uua),Uub))
<=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set3(A),Uub)),Uu))
& ( aa(list(A),nat,size_size(list(A)),Uub) = Uua ) ) ) ).
% ATP.lambda_266
tff(fact_2981_ATP_Olambda__267,axiom,
! [A: $tType,Uu: set(list(A)),Uua: list(A),Uub: A] :
( pp(aa(A,bool,aa(list(A),fun(A,bool),aTP_Lamp_py(set(list(A)),fun(list(A),fun(A,bool)),Uu),Uua),Uub))
<=> pp(aa(set(list(A)),bool,member(list(A),append(A,Uua,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uub),nil(A)))),Uu)) ) ).
% ATP.lambda_267
tff(fact_2982_ATP_Olambda__268,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: fun(A,A),Uua: ref(A),Uub: A] : aa(A,heap_Heap(A),aa(ref(A),fun(A,heap_Heap(A)),aTP_Lamp_bp(fun(A,A),fun(ref(A),fun(A,heap_Heap(A))),Uu),Uua),Uub) = heap_bind(product_unit,A,update2(A,Uua,aa(A,A,Uu,Uub)),aa(A,fun(product_unit,heap_Heap(A)),aTP_Lamp_bo(fun(A,A),fun(A,fun(product_unit,heap_Heap(A))),Uu),Uub)) ) ).
% ATP.lambda_268
tff(fact_2983_ATP_Olambda__269,axiom,
! [B: $tType,A: $tType,Uu: fun(A,option(B)),Uua: A,Uub: product_prod(A,B)] :
( pp(aa(product_prod(A,B),bool,aa(A,fun(product_prod(A,B),bool),aTP_Lamp_se(fun(A,option(B)),fun(A,fun(product_prod(A,B),bool)),Uu),Uua),Uub))
<=> ( pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),Uub),graph(A,B,Uu)))
& ( aa(product_prod(A,B),A,product_fst(A,B),Uub) != Uua ) ) ) ).
% ATP.lambda_269
tff(fact_2984_ATP_Olambda__270,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: A] :
( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_lg(set(product_prod(A,A)),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
<=> ( pp(aa(set(A),bool,member(A,Uub),aa(set(product_prod(A,A)),set(A),field2(A),Uu)))
& ! [X5: A] :
( pp(aa(set(A),bool,member(A,X5),Uua))
=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),X5)),Uu)) ) ) ) ).
% ATP.lambda_270
tff(fact_2985_ATP_Olambda__271,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: A] :
( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_lh(set(product_prod(A,A)),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
<=> ( pp(aa(set(A),bool,member(A,Uub),aa(set(product_prod(A,A)),set(A),field2(A),Uu)))
& ! [X5: A] :
( pp(aa(set(A),bool,member(A,X5),Uua))
=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),Uub)),Uu)) ) ) ) ).
% ATP.lambda_271
tff(fact_2986_ATP_Olambda__272,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: A] :
( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_lf(set(product_prod(A,A)),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
<=> ( pp(aa(set(A),bool,member(A,Uub),aa(set(product_prod(A,A)),set(A),field2(A),Uu)))
& ! [X5: A] :
( pp(aa(set(A),bool,member(A,X5),Uua))
=> ( ( Uub != X5 )
& pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),X5)),Uu)) ) ) ) ) ).
% ATP.lambda_272
tff(fact_2987_ATP_Olambda__273,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: A] :
( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_lj(set(product_prod(A,A)),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
<=> ( pp(aa(set(A),bool,member(A,Uub),aa(set(product_prod(A,A)),set(A),field2(A),Uu)))
& ! [X5: A] :
( pp(aa(set(A),bool,member(A,X5),Uua))
=> ( ( Uub != X5 )
& pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),Uub)),Uu)) ) ) ) ) ).
% ATP.lambda_273
tff(fact_2988_ATP_Olambda__274,axiom,
! [Uu: nat,Uua: nat,Uub: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_jx(nat,fun(nat,fun(nat,bool)),Uu),Uua),Uub))
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uua),Uu))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uub),Uu))
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Uua),Uub)) ) ) ).
% ATP.lambda_274
tff(fact_2989_ATP_Olambda__275,axiom,
! [B: $tType,A: $tType] :
( linorder(B)
=> ! [Uu: set(A),Uua: fun(A,B),Uub: A] :
( pp(aa(A,bool,aa(fun(A,B),fun(A,bool),aTP_Lamp_qo(set(A),fun(fun(A,B),fun(A,bool)),Uu),Uua),Uub))
<=> ( pp(aa(set(A),bool,member(A,Uub),Uu))
& ( aa(A,B,Uua,Uub) = lattic643756798350308766er_Min(B,aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),Uua),Uu)) ) ) ) ) ).
% ATP.lambda_275
tff(fact_2990_ATP_Olambda__276,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aTP_Lamp_mj(set(product_prod(A,A)),fun(set(A),fun(set(A),bool)),Uu),Uua),Uub))
<=> ( order_ofilter(A,Uu,Uua)
& ( Uua != aa(set(product_prod(A,A)),set(A),field2(A),Uu) )
& order_ofilter(A,Uu,Uub)
& ( Uub != aa(set(product_prod(A,A)),set(A),field2(A),Uu) )
& pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),Uua),Uub)) ) ) ).
% ATP.lambda_276
tff(fact_2991_ATP_Olambda__277,axiom,
! [A: $tType,B: $tType,Uu: set(product_prod(A,A)),Uua: set(product_prod(A,A)),Uub: fun(B,A)] : aa(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(A,A)),fun(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),fun(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B))))),aTP_Lamp_ch(set(product_prod(A,A)),fun(set(product_prod(A,A)),fun(fun(B,A),product_prod(set(product_prod(B,B)),set(product_prod(B,B)))))),Uu),Uua),Uub) = aa(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B)))),product_Pair(set(product_prod(B,B)),set(product_prod(B,B))),inv_image(A,B,Uu,Uub)),inv_image(A,B,Uua,Uub)) ).
% ATP.lambda_277
tff(fact_2992_ATP_Olambda__278,axiom,
! [Uu: int,Uua: int,Uub: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_qv(int,fun(int,fun(int,bool)),Uu),Uua),Uub))
<=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Uu),Uua))
& pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Uua),Uub)) ) ) ).
% ATP.lambda_278
tff(fact_2993_ATP_Olambda__279,axiom,
! [Uu: int,Uua: int,Uub: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_qu(int,fun(int,fun(int,bool)),Uu),Uua),Uub))
<=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Uu),Uub))
& pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Uua),Uub)) ) ) ).
% ATP.lambda_279
tff(fact_2994_ATP_Olambda__280,axiom,
! [A: $tType,Uu: set(A),Uua: set(A),Uub: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aTP_Lamp_kb(set(A),fun(set(A),fun(set(A),bool)),Uu),Uua),Uub))
<=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),Uua),Uub))
& pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Uub),Uu)) ) ) ).
% ATP.lambda_280
tff(fact_2995_ATP_Olambda__281,axiom,
! [A: $tType,Uu: set(A),Uua: A,Uub: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_ld(set(A),fun(A,fun(A,bool)),Uu),Uua),Uub))
<=> ( pp(aa(set(A),bool,member(A,Uua),Uu))
& pp(aa(set(A),bool,member(A,Uub),Uu)) ) ) ).
% ATP.lambda_281
tff(fact_2996_ATP_Olambda__282,axiom,
! [A: $tType,Uu: set(A),Uua: fun(A,bool),Uub: A] :
( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ak(set(A),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
<=> ( pp(aa(set(A),bool,member(A,Uub),Uu))
& pp(aa(A,bool,Uua,Uub)) ) ) ).
% ATP.lambda_282
tff(fact_2997_ATP_Olambda__283,axiom,
! [A: $tType,Uu: fun(A,bool),Uua: A,Uub: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_aj(fun(A,bool),fun(A,fun(A,bool)),Uu),Uua),Uub))
<=> ( ( Uua = Uub )
& pp(aa(A,bool,Uu,Uua)) ) ) ).
% ATP.lambda_283
tff(fact_2998_ATP_Olambda__284,axiom,
! [B: $tType,A: $tType,Uu: fun(B,heap_Heap(A)),Uua: list(B),Uub: list(A)] : aa(list(A),heap_Heap(list(A)),aa(list(B),fun(list(A),heap_Heap(list(A))),aTP_Lamp_sa(fun(B,heap_Heap(A)),fun(list(B),fun(list(A),heap_Heap(list(A)))),Uu),Uua),Uub) = heap_bind(list(A),list(A),heap_fold_map(B,A,Uu,Uua),aTP_Lamp_rz(list(A),fun(list(A),heap_Heap(list(A))),Uub)) ).
% ATP.lambda_284
tff(fact_2999_ATP_Olambda__285,axiom,
! [A: $tType,B: $tType,Uu: fun(A,heap_Heap(B)),Uua: list(A),Uub: B] : aa(B,heap_Heap(list(B)),aa(list(A),fun(B,heap_Heap(list(B))),aTP_Lamp_ry(fun(A,heap_Heap(B)),fun(list(A),fun(B,heap_Heap(list(B)))),Uu),Uua),Uub) = heap_bind(list(B),list(B),heap_fold_map(A,B,Uu,Uua),aTP_Lamp_rx(B,fun(list(B),heap_Heap(list(B))),Uub)) ).
% ATP.lambda_285
tff(fact_3000_ATP_Olambda__286,axiom,
! [A: $tType,B: $tType,Uu: set(A),Uua: fun(A,set(B)),Uub: A] :
( pp(aa(A,bool,aa(fun(A,set(B)),fun(A,bool),aTP_Lamp_ck(set(A),fun(fun(A,set(B)),fun(A,bool)),Uu),Uua),Uub))
<=> ( pp(aa(set(A),bool,member(A,Uub),Uu))
& ( aa(A,set(B),Uua,Uub) != bot_bot(set(B)) ) ) ) ).
% ATP.lambda_286
tff(fact_3001_ATP_Olambda__287,axiom,
! [A: $tType,B: $tType,Uu: list(product_prod(A,B)),Uua: A,Uub: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_sb(list(product_prod(A,B)),fun(A,fun(B,bool)),Uu),Uua),Uub))
<=> ( aa(A,option(B),map_of(A,B,Uu),Uua) = aa(B,option(B),some(B),Uub) ) ) ).
% ATP.lambda_287
tff(fact_3002_ATP_Olambda__288,axiom,
! [Uu: nat,Uua: nat,Uub: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_lv(nat,fun(nat,fun(nat,bool)),Uu),Uua),Uub))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)),Uu)) ) ).
% ATP.lambda_288
tff(fact_3003_ATP_Olambda__289,axiom,
! [Uu: nat,Uua: nat,Uub: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_me(nat,fun(nat,fun(nat,bool)),Uu),Uua),Uub))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)),Uu)) ) ).
% ATP.lambda_289
tff(fact_3004_ATP_Olambda__290,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_mn(A,fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Uub),Uu)),Uua) ) ).
% ATP.lambda_290
tff(fact_3005_ATP_Olambda__291,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_mg(A,fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),minus_minus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Uu),Uub)),Uua) ) ).
% ATP.lambda_291
tff(fact_3006_ATP_Olambda__292,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_mo(A,fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Uub),Uu)),Uua) ) ).
% ATP.lambda_292
tff(fact_3007_ATP_Olambda__293,axiom,
! [A: $tType] :
( linordered_field(A)
=> ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_mi(A,fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Uu),Uub)),Uua) ) ).
% ATP.lambda_293
tff(fact_3008_ATP_Olambda__294,axiom,
! [C: $tType,B: $tType,Uu: set(product_prod(B,C)),Uua: B,Uub: C] :
( pp(aa(C,bool,aa(B,fun(C,bool),aTP_Lamp_ih(set(product_prod(B,C)),fun(B,fun(C,bool)),Uu),Uua),Uub))
<=> pp(aa(set(product_prod(B,C)),bool,member(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),Uua),Uub)),Uu)) ) ).
% ATP.lambda_294
tff(fact_3009_ATP_Olambda__295,axiom,
! [A: $tType,B: $tType,Uu: set(product_prod(B,A)),Uua: B,Uub: A] :
( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_iq(set(product_prod(B,A)),fun(B,fun(A,bool)),Uu),Uua),Uub))
<=> pp(aa(set(product_prod(B,A)),bool,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uua),Uub)),Uu)) ) ).
% ATP.lambda_295
tff(fact_3010_ATP_Olambda__296,axiom,
! [B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: A,Uub: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aa(set(product_prod(A,B)),fun(A,fun(B,bool)),aTP_Lamp_bg(set(product_prod(A,B)),fun(A,fun(B,bool))),Uu),Uua),Uub))
<=> pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uub)),Uu)) ) ).
% ATP.lambda_296
tff(fact_3011_ATP_Olambda__297,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_hl(set(product_prod(A,A)),fun(A,fun(A,bool)),Uu),Uua),Uub))
<=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uub)),Uu)) ) ).
% ATP.lambda_297
tff(fact_3012_ATP_Olambda__298,axiom,
! [A: $tType,B: $tType,Uu: set(product_prod(B,A)),Uua: A,Uub: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_io(set(product_prod(B,A)),fun(A,fun(B,bool)),Uu),Uua),Uub))
<=> pp(aa(set(product_prod(B,A)),bool,member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uub),Uua)),Uu)) ) ).
% ATP.lambda_298
tff(fact_3013_ATP_Olambda__299,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_nv(set(product_prod(A,A)),fun(A,fun(A,bool)),Uu),Uua),Uub))
<=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uua)),Uu)) ) ).
% ATP.lambda_299
tff(fact_3014_ATP_Olambda__300,axiom,
! [A: $tType,Uu: set(list(A)),Uua: A,Uub: list(A)] :
( pp(aa(list(A),bool,aa(A,fun(list(A),bool),aTP_Lamp_qe(set(list(A)),fun(A,fun(list(A),bool)),Uu),Uua),Uub))
<=> pp(aa(set(list(A)),bool,member(list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uua),Uub)),Uu)) ) ).
% ATP.lambda_300
tff(fact_3015_ATP_Olambda__301,axiom,
! [B: $tType,A: $tType,Uu: fun(A,bool),Uua: fun(B,bool),Uub: product_prod(A,B)] :
( pp(aa(product_prod(A,B),bool,aa(fun(B,bool),fun(product_prod(A,B),bool),aTP_Lamp_jf(fun(A,bool),fun(fun(B,bool),fun(product_prod(A,B),bool)),Uu),Uua),Uub))
<=> ( pp(aa(A,bool,Uu,aa(product_prod(A,B),A,product_fst(A,B),Uub)))
& pp(aa(B,bool,Uua,aa(product_prod(A,B),B,product_snd(A,B),Uub))) ) ) ).
% ATP.lambda_301
tff(fact_3016_ATP_Olambda__302,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,Uu: fun(fun(A,heap_Heap(B)),heap_Heap(C)),Uua: fun(C,fun(fun(A,heap_Heap(B)),heap_Heap(D))),Uub: fun(A,heap_Heap(B))] : aa(fun(A,heap_Heap(B)),heap_Heap(D),aa(fun(C,fun(fun(A,heap_Heap(B)),heap_Heap(D))),fun(fun(A,heap_Heap(B)),heap_Heap(D)),aTP_Lamp_ux(fun(fun(A,heap_Heap(B)),heap_Heap(C)),fun(fun(C,fun(fun(A,heap_Heap(B)),heap_Heap(D))),fun(fun(A,heap_Heap(B)),heap_Heap(D))),Uu),Uua),Uub) = heap_bind(C,D,aa(fun(A,heap_Heap(B)),heap_Heap(C),Uu,Uub),aa(fun(A,heap_Heap(B)),fun(C,heap_Heap(D)),aTP_Lamp_uw(fun(C,fun(fun(A,heap_Heap(B)),heap_Heap(D))),fun(fun(A,heap_Heap(B)),fun(C,heap_Heap(D))),Uua),Uub)) ).
% ATP.lambda_302
tff(fact_3017_ATP_Olambda__303,axiom,
! [A: $tType,C: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: fun(B,set(C)),Uua: fun(B,fun(C,A)),Uub: B] : aa(B,A,aa(fun(B,fun(C,A)),fun(B,A),aTP_Lamp_my(fun(B,set(C)),fun(fun(B,fun(C,A)),fun(B,A)),Uu),Uua),Uub) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),aa(B,fun(C,A),Uua,Uub)),aa(B,set(C),Uu,Uub)) ) ).
% ATP.lambda_303
tff(fact_3018_ATP_Olambda__304,axiom,
! [A: $tType,C: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [Uu: fun(B,set(C)),Uua: fun(B,fun(C,A)),Uub: B] : aa(B,A,aa(fun(B,fun(C,A)),fun(B,A),aTP_Lamp_lo(fun(B,set(C)),fun(fun(B,fun(C,A)),fun(B,A)),Uu),Uua),Uub) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),aa(B,fun(C,A),Uua,Uub)),aa(B,set(C),Uu,Uub)) ) ).
% ATP.lambda_304
tff(fact_3019_ATP_Olambda__305,axiom,
! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: fun(A,set(B)),Uub: A] : aa(A,set(B),aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_eq(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),minus_minus(set(B)),aa(A,set(B),Uu,Uub)),aa(A,set(B),Uua,Uub)) ).
% ATP.lambda_305
tff(fact_3020_ATP_Olambda__306,axiom,
! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: fun(B,set(A)),Uub: B] : aa(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_gg(fun(B,set(A)),fun(fun(B,set(A)),fun(B,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(B,set(A),Uu,Uub)),aa(B,set(A),Uua,Uub)) ).
% ATP.lambda_306
tff(fact_3021_ATP_Olambda__307,axiom,
! [A: $tType,B: $tType] :
( condit1219197933456340205attice(A)
=> ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_fs(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).
% ATP.lambda_307
tff(fact_3022_ATP_Olambda__308,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_fu(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).
% ATP.lambda_308
tff(fact_3023_ATP_Olambda__309,axiom,
! [A: $tType,B: $tType] :
( lattice(A)
=> ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_fj(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).
% ATP.lambda_309
tff(fact_3024_ATP_Olambda__310,axiom,
! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: fun(A,set(B)),Uub: A] : aa(A,set(B),aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_fn(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(A,set(B),Uu,Uub)),aa(A,set(B),Uua,Uub)) ).
% ATP.lambda_310
tff(fact_3025_ATP_Olambda__311,axiom,
! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: fun(B,set(A)),Uub: B] : aa(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_gr(fun(B,set(A)),fun(fun(B,set(A)),fun(B,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(B,set(A),Uu,Uub)),aa(B,set(A),Uua,Uub)) ).
% ATP.lambda_311
tff(fact_3026_ATP_Olambda__312,axiom,
! [A: $tType,B: $tType] :
( condit1219197933456340205attice(A)
=> ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ft(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).
% ATP.lambda_312
tff(fact_3027_ATP_Olambda__313,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_gf(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).
% ATP.lambda_313
tff(fact_3028_ATP_Olambda__314,axiom,
! [A: $tType,B: $tType] :
( lattice(A)
=> ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_fi(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).
% ATP.lambda_314
tff(fact_3029_ATP_Olambda__315,axiom,
! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: fun(A,set(B)),Uub: A] : aa(A,set(B),aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_fp(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(A,set(B),Uu,Uub)),aa(A,set(B),Uua,Uub)) ).
% ATP.lambda_315
tff(fact_3030_ATP_Olambda__316,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: fun(C,A),Uua: fun(C,B),Uub: C] : aa(C,product_prod(A,B),aa(fun(C,B),fun(C,product_prod(A,B)),aTP_Lamp_gw(fun(C,A),fun(fun(C,B),fun(C,product_prod(A,B))),Uu),Uua),Uub) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,Uu,Uub)),aa(C,B,Uua,Uub)) ).
% ATP.lambda_316
tff(fact_3031_ATP_Olambda__317,axiom,
! [B: $tType,C: $tType,A: $tType,Uu: fun(A,B),Uua: fun(A,C),Uub: A] : aa(A,product_prod(B,C),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_jc(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),Uu),Uua),Uub) = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(A,B,Uu,Uub)),aa(A,C,Uua,Uub)) ).
% ATP.lambda_317
tff(fact_3032_ATP_Olambda__318,axiom,
! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A,Uub: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_ly(fun(A,set(B)),fun(A,fun(A,bool)),Uu),Uua),Uub))
<=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),disjnt(B),aa(A,set(B),Uu,Uua)),aa(A,set(B),Uu,Uub))) ) ).
% ATP.lambda_318
tff(fact_3033_ATP_Olambda__319,axiom,
! [B: $tType,Uu: fun(B,bool),Uua: fun(B,bool),Uub: B] :
( pp(aa(B,bool,aa(fun(B,bool),fun(B,bool),aTP_Lamp_ll(fun(B,bool),fun(fun(B,bool),fun(B,bool)),Uu),Uua),Uub))
<=> ( pp(aa(B,bool,Uu,Uub))
& pp(aa(B,bool,Uua,Uub)) ) ) ).
% ATP.lambda_319
tff(fact_3034_ATP_Olambda__320,axiom,
! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A,Uub: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_la(fun(A,B),fun(A,fun(A,bool)),Uu),Uua),Uub))
<=> ( aa(A,B,Uu,Uua) = aa(A,B,Uu,Uub) ) ) ).
% ATP.lambda_320
tff(fact_3035_ATP_Olambda__321,axiom,
! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
( pp(aa(A,bool,aa(fun(A,B),fun(A,bool),aTP_Lamp_qh(fun(A,B),fun(fun(A,B),fun(A,bool)),Uu),Uua),Uub))
<=> ( aa(A,B,Uu,Uub) = aa(A,B,Uua,Uub) ) ) ).
% ATP.lambda_321
tff(fact_3036_ATP_Olambda__322,axiom,
! [B: $tType,A: $tType,C: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: fun(B,fun(C,A)),Uua: set(C),Uub: B] : aa(B,A,aa(set(C),fun(B,A),aTP_Lamp_mv(fun(B,fun(C,A)),fun(set(C),fun(B,A)),Uu),Uua),Uub) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),aa(B,fun(C,A),Uu,Uub)),Uua) ) ).
% ATP.lambda_322
tff(fact_3037_ATP_Olambda__323,axiom,
! [B: $tType,A: $tType,C: $tType] :
( comm_monoid_add(A)
=> ! [Uu: fun(B,fun(C,A)),Uua: set(C),Uub: B] : aa(B,A,aa(set(C),fun(B,A),aTP_Lamp_lu(fun(B,fun(C,A)),fun(set(C),fun(B,A)),Uu),Uua),Uub) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),aa(B,fun(C,A),Uu,Uub)),Uua) ) ).
% ATP.lambda_323
tff(fact_3038_ATP_Olambda__324,axiom,
! [B: $tType,A: $tType,Uu: fun(B,set(A)),Uua: set(A),Uub: B] :
( pp(aa(B,bool,aa(set(A),fun(B,bool),aTP_Lamp_ea(fun(B,set(A)),fun(set(A),fun(B,bool)),Uu),Uua),Uub))
<=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(B,set(A),Uu,Uub)),Uua)) ) ).
% ATP.lambda_324
tff(fact_3039_ATP_Olambda__325,axiom,
! [K2: $tType,L3: $tType,Uu: fun(K2,set(L3)),Uua: set(L3),Uub: K2] : aa(K2,set(L3),aa(set(L3),fun(K2,set(L3)),aTP_Lamp_ex(fun(K2,set(L3)),fun(set(L3),fun(K2,set(L3))),Uu),Uua),Uub) = aa(set(L3),set(L3),aa(set(L3),fun(set(L3),set(L3)),minus_minus(set(L3)),aa(K2,set(L3),Uu,Uub)),Uua) ).
% ATP.lambda_325
tff(fact_3040_ATP_Olambda__326,axiom,
! [E: $tType,F2: $tType,Uu: fun(E,set(F2)),Uua: set(F2),Uub: E] : aa(E,set(F2),aa(set(F2),fun(E,set(F2)),aTP_Lamp_ep(fun(E,set(F2)),fun(set(F2),fun(E,set(F2))),Uu),Uua),Uub) = aa(set(F2),set(F2),aa(set(F2),fun(set(F2),set(F2)),minus_minus(set(F2)),aa(E,set(F2),Uu,Uub)),Uua) ).
% ATP.lambda_326
tff(fact_3041_ATP_Olambda__327,axiom,
! [K2: $tType,L3: $tType,Uu: fun(K2,set(L3)),Uua: set(L3),Uub: K2] : aa(K2,set(L3),aa(set(L3),fun(K2,set(L3)),aTP_Lamp_gp(fun(K2,set(L3)),fun(set(L3),fun(K2,set(L3))),Uu),Uua),Uub) = aa(set(L3),set(L3),aa(set(L3),fun(set(L3),set(L3)),sup_sup(set(L3)),aa(K2,set(L3),Uu,Uub)),Uua) ).
% ATP.lambda_327
tff(fact_3042_ATP_Olambda__328,axiom,
! [C: $tType,D: $tType,Uu: fun(C,set(D)),Uua: set(D),Uub: C] : aa(C,set(D),aa(set(D),fun(C,set(D)),aTP_Lamp_fl(fun(C,set(D)),fun(set(D),fun(C,set(D))),Uu),Uua),Uub) = aa(set(D),set(D),aa(set(D),fun(set(D),set(D)),sup_sup(set(D)),aa(C,set(D),Uu,Uub)),Uua) ).
% ATP.lambda_328
tff(fact_3043_ATP_Olambda__329,axiom,
! [B: $tType,A: $tType] :
( comple592849572758109894attice(A)
=> ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_ga(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,Uu,Uub)),Uua) ) ).
% ATP.lambda_329
tff(fact_3044_ATP_Olambda__330,axiom,
! [G2: $tType,H5: $tType,Uu: fun(G2,set(H5)),Uua: set(H5),Uub: G2] : aa(G2,set(H5),aa(set(H5),fun(G2,set(H5)),aTP_Lamp_gk(fun(G2,set(H5)),fun(set(H5),fun(G2,set(H5))),Uu),Uua),Uub) = aa(set(H5),set(H5),aa(set(H5),fun(set(H5),set(H5)),inf_inf(set(H5)),aa(G2,set(H5),Uu,Uub)),Uua) ).
% ATP.lambda_330
tff(fact_3045_ATP_Olambda__331,axiom,
! [B: $tType,A: $tType] :
( comple592849572758109894attice(A)
=> ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_fv(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,Uu,Uub)),Uua) ) ).
% ATP.lambda_331
tff(fact_3046_ATP_Olambda__332,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_gt(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,Uu,Uub)),Uua) ) ).
% ATP.lambda_332
tff(fact_3047_ATP_Olambda__333,axiom,
! [A: $tType,B: $tType,Uu: fun(A,set(B)),Uua: set(B),Uub: A] : aa(A,set(B),aa(set(B),fun(A,set(B)),aTP_Lamp_fc(fun(A,set(B)),fun(set(B),fun(A,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(A,set(B),Uu,Uub)),Uua) ).
% ATP.lambda_333
tff(fact_3048_ATP_Olambda__334,axiom,
! [B: $tType,A: $tType] :
( linord4140545234300271783up_add(A)
=> ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_ks(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,Uu,Uub)),Uua) ) ).
% ATP.lambda_334
tff(fact_3049_ATP_Olambda__335,axiom,
! [E: $tType,F2: $tType,B: $tType,D: $tType,C: $tType,Uu: fun(E,fun(F2,product_prod(C,D))),Uua: fun(C,fun(D,B)),Uub: E] : aa(E,fun(F2,B),aa(fun(C,fun(D,B)),fun(E,fun(F2,B)),aTP_Lamp_ay(fun(E,fun(F2,product_prod(C,D))),fun(fun(C,fun(D,B)),fun(E,fun(F2,B))),Uu),Uua),Uub) = product_scomp(F2,C,D,B,aa(E,fun(F2,product_prod(C,D)),Uu,Uub),Uua) ).
% ATP.lambda_335
tff(fact_3050_ATP_Olambda__336,axiom,
! [C: $tType,B: $tType,A: $tType,Uu: fun(C,set(old_node(A,B))),Uua: set(old_node(A,B)),Uub: C] : aa(C,set(old_node(A,B)),aa(set(old_node(A,B)),fun(C,set(old_node(A,B))),aTP_Lamp_vw(fun(C,set(old_node(A,B))),fun(set(old_node(A,B)),fun(C,set(old_node(A,B)))),Uu),Uua),Uub) = old_Scons(A,B,aa(C,set(old_node(A,B)),Uu,Uub),Uua) ).
% ATP.lambda_336
tff(fact_3051_ATP_Olambda__337,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: fun(A,filter(B)),Uua: filter(C),Uub: A] : aa(A,filter(product_prod(B,C)),aa(filter(C),fun(A,filter(product_prod(B,C))),aTP_Lamp_jj(fun(A,filter(B)),fun(filter(C),fun(A,filter(product_prod(B,C)))),Uu),Uua),Uub) = prod_filter(B,C,aa(A,filter(B),Uu,Uub),Uua) ).
% ATP.lambda_337
tff(fact_3052_ATP_Olambda__338,axiom,
! [D: $tType,A: $tType,B: $tType,C: $tType,Uu: fun(D,fun(A,fun(C,bool))),Uua: fun(C,fun(B,bool)),Uub: D] : aa(D,fun(A,fun(B,bool)),aa(fun(C,fun(B,bool)),fun(D,fun(A,fun(B,bool))),aTP_Lamp_et(fun(D,fun(A,fun(C,bool))),fun(fun(C,fun(B,bool)),fun(D,fun(A,fun(B,bool)))),Uu),Uua),Uub) = aa(fun(C,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(C,bool)),fun(fun(C,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,C,B),aa(D,fun(A,fun(C,bool)),Uu,Uub)),Uua) ).
% ATP.lambda_338
tff(fact_3053_ATP_Olambda__339,axiom,
! [D: $tType,A: $tType,B: $tType,C: $tType,Uu: fun(D,set(product_prod(A,C))),Uua: set(product_prod(C,B)),Uub: D] : aa(D,set(product_prod(A,B)),aa(set(product_prod(C,B)),fun(D,set(product_prod(A,B))),aTP_Lamp_ij(fun(D,set(product_prod(A,C))),fun(set(product_prod(C,B)),fun(D,set(product_prod(A,B)))),Uu),Uua),Uub) = relcomp(A,C,B,aa(D,set(product_prod(A,C)),Uu,Uub),Uua) ).
% ATP.lambda_339
tff(fact_3054_ATP_Olambda__340,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: fun(A,heap_Heap(C)),Uua: fun(C,heap_Heap(B)),Uub: A] : aa(A,heap_Heap(B),aa(fun(C,heap_Heap(B)),fun(A,heap_Heap(B)),aTP_Lamp_ae(fun(A,heap_Heap(C)),fun(fun(C,heap_Heap(B)),fun(A,heap_Heap(B))),Uu),Uua),Uub) = heap_bind(C,B,aa(A,heap_Heap(C),Uu,Uub),Uua) ).
% ATP.lambda_340
tff(fact_3055_ATP_Olambda__341,axiom,
! [C: $tType,A: $tType,B: $tType,Uu: fun(C,set(product_prod(B,A))),Uua: set(B),Uub: C] : aa(C,set(A),aa(set(B),fun(C,set(A)),aTP_Lamp_iu(fun(C,set(product_prod(B,A))),fun(set(B),fun(C,set(A))),Uu),Uua),Uub) = aa(set(B),set(A),image(B,A,aa(C,set(product_prod(B,A)),Uu,Uub)),Uua) ).
% ATP.lambda_341
tff(fact_3056_ATP_Olambda__342,axiom,
! [D: $tType,E: $tType,B: $tType,C: $tType,Uu: fun(D,fun(E,C)),Uua: fun(C,B),Uub: D] : aa(D,fun(E,B),aa(fun(C,B),fun(D,fun(E,B)),aTP_Lamp_ud(fun(D,fun(E,C)),fun(fun(C,B),fun(D,fun(E,B))),Uu),Uua),Uub) = fcomp(E,C,B,aa(D,fun(E,C),Uu,Uub),Uua) ).
% ATP.lambda_342
tff(fact_3057_ATP_Olambda__343,axiom,
! [A: $tType,Uu: fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),Uua: pred(A),Uub: product_prod(code_natural,code_natural)] : aa(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)),aa(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),aTP_Lamp_wu(fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),fun(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),Uu),Uua),Uub) = aa(product_prod(pred(A),product_prod(code_natural,code_natural)),product_prod(pred(A),product_prod(code_natural,code_natural)),aa(fun(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),fun(product_prod(pred(A),product_prod(code_natural,code_natural)),product_prod(pred(A),product_prod(code_natural,code_natural))),product_case_prod(pred(A),product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural))),aTP_Lamp_wt(pred(A),fun(pred(A),fun(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)))),Uua)),aa(product_prod(code_natural,code_natural),product_prod(pred(A),product_prod(code_natural,code_natural)),Uu,Uub)) ).
% ATP.lambda_343
tff(fact_3058_ATP_Olambda__344,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_hh(set(product_prod(A,A)),fun(A,fun(A,bool)),Uu),Uua),Uub))
<=> ( ( Uub != Uua )
& pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uub)),Uu)) ) ) ).
% ATP.lambda_344
tff(fact_3059_ATP_Olambda__345,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_jy(set(product_prod(A,A)),fun(A,fun(A,bool)),Uu),Uua),Uub))
<=> ( ( Uub != Uua )
& pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uua)),Uu)) ) ) ).
% ATP.lambda_345
tff(fact_3060_ATP_Olambda__346,axiom,
! [A: $tType,Uu: set(A),Uua: set(A),Uub: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aTP_Lamp_jt(set(A),fun(set(A),fun(set(A),bool)),Uu),Uua),Uub))
<=> ( finite_finite2(A,Uub)
& pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Uua),Uub))
& pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Uub),Uu)) ) ) ).
% ATP.lambda_346
tff(fact_3061_ATP_Olambda__347,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: set(A)] :
( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aTP_Lamp_or(set(product_prod(A,A)),fun(set(A),fun(set(A),bool)),Uu),Uua),Uub))
<=> ( finite_finite2(A,Uua)
& finite_finite2(A,Uub)
& ( Uub != bot_bot(set(A)) )
& ! [X5: A] :
( pp(aa(set(A),bool,member(A,X5),Uua))
=> ? [Xa3: A] :
( pp(aa(set(A),bool,member(A,Xa3),Uub))
& pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),Xa3)),Uu)) ) ) ) ) ).
% ATP.lambda_347
tff(fact_3062_ATP_Olambda__348,axiom,
! [A: $tType,Uu: fun(A,fun(A,A)),Uua: list(A),Uub: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),aTP_Lamp_uz(fun(A,fun(A,A)),fun(list(A),fun(list(A),list(A))),Uu),Uua),Uub) = aa(list(product_prod(A,A)),list(A),map(product_prod(A,A),A,aa(fun(A,fun(A,A)),fun(product_prod(A,A),A),product_case_prod(A,A,A),Uu)),zip(A,A,Uua,Uub)) ).
% ATP.lambda_348
tff(fact_3063_ATP_Olambda__349,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: nat,Uua: array(A),Uub: heap_ext(product_unit)] :
( pp(aa(heap_ext(product_unit),bool,aa(array(A),fun(heap_ext(product_unit),bool),aTP_Lamp_kf(nat,fun(array(A),fun(heap_ext(product_unit),bool)),Uu),Uua),Uub))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uu),aa(array(A),nat,length(A,Uub),Uua))) ) ) ).
% ATP.lambda_349
tff(fact_3064_ATP_Olambda__350,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: array(A),Uua: nat,Uub: heap_ext(product_unit)] :
( pp(aa(heap_ext(product_unit),bool,aa(nat,fun(heap_ext(product_unit),bool),aTP_Lamp_kk(array(A),fun(nat,fun(heap_ext(product_unit),bool)),Uu),Uua),Uub))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uua),aa(array(A),nat,length(A,Uub),Uu))) ) ) ).
% ATP.lambda_350
tff(fact_3065_ATP_Olambda__351,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: A,Uua: B,Uub: C] : aa(C,product_prod(A,product_prod(B,C)),aa(B,fun(C,product_prod(A,product_prod(B,C))),aa(A,fun(B,fun(C,product_prod(A,product_prod(B,C)))),aTP_Lamp_pg(A,fun(B,fun(C,product_prod(A,product_prod(B,C))))),Uu),Uua),Uub) = aa(product_prod(B,C),product_prod(A,product_prod(B,C)),aa(A,fun(product_prod(B,C),product_prod(A,product_prod(B,C))),product_Pair(A,product_prod(B,C)),Uu),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),Uua),Uub)) ).
% ATP.lambda_351
tff(fact_3066_ATP_Olambda__352,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: B,Uua: A,Uub: C] : aa(C,product_prod(A,product_prod(B,C)),aa(A,fun(C,product_prod(A,product_prod(B,C))),aTP_Lamp_pe(B,fun(A,fun(C,product_prod(A,product_prod(B,C)))),Uu),Uua),Uub) = aa(product_prod(B,C),product_prod(A,product_prod(B,C)),aa(A,fun(product_prod(B,C),product_prod(A,product_prod(B,C))),product_Pair(A,product_prod(B,C)),Uua),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),Uu),Uub)) ).
% ATP.lambda_352
tff(fact_3067_ATP_Olambda__353,axiom,
! [B: $tType,A: $tType] :
( ( order(A)
& order(B) )
=> ! [Uu: fun(A,B),Uua: set(A),Uub: B] :
( pp(aa(B,bool,aa(set(A),fun(B,bool),aTP_Lamp_jq(fun(A,B),fun(set(A),fun(B,bool)),Uu),Uua),Uub))
<=> pp(aa(set(B),bool,member(B,Uub),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),Uu),Uua))) ) ) ).
% ATP.lambda_353
tff(fact_3068_ATP_Olambda__354,axiom,
! [A: $tType,C: $tType,B: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: fun(B,set(C)),Uua: fun(C,A),Uub: B] : aa(B,A,aa(fun(C,A),fun(B,A),aTP_Lamp_nq(fun(B,set(C)),fun(fun(C,A),fun(B,A)),Uu),Uua),Uub) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),Uua),aa(B,set(C),Uu,Uub)) ) ).
% ATP.lambda_354
tff(fact_3069_ATP_Olambda__355,axiom,
! [A: $tType,C: $tType,B: $tType] :
( comm_monoid_add(A)
=> ! [Uu: fun(B,set(C)),Uua: fun(C,A),Uub: B] : aa(B,A,aa(fun(C,A),fun(B,A),aTP_Lamp_ln(fun(B,set(C)),fun(fun(C,A),fun(B,A)),Uu),Uua),Uub) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7311177749621191930dd_sum(C,A),Uua),aa(B,set(C),Uu,Uub)) ) ).
% ATP.lambda_355
tff(fact_3070_ATP_Olambda__356,axiom,
! [A: $tType,C: $tType,B: $tType,Uu: fun(C,A),Uua: fun(B,option(C)),Uub: B] : aa(B,option(A),aa(fun(B,option(C)),fun(B,option(A)),aTP_Lamp_sj(fun(C,A),fun(fun(B,option(C)),fun(B,option(A))),Uu),Uua),Uub) = aa(option(C),option(A),map_option(C,A,Uu),aa(B,option(C),Uua,Uub)) ).
% ATP.lambda_356
tff(fact_3071_ATP_Olambda__357,axiom,
! [M5: $tType,N3: $tType,Uu: set(M5),Uua: fun(N3,set(M5)),Uub: N3] : aa(N3,set(M5),aa(fun(N3,set(M5)),fun(N3,set(M5)),aTP_Lamp_ey(set(M5),fun(fun(N3,set(M5)),fun(N3,set(M5))),Uu),Uua),Uub) = aa(set(M5),set(M5),aa(set(M5),fun(set(M5),set(M5)),minus_minus(set(M5)),Uu),aa(N3,set(M5),Uua,Uub)) ).
% ATP.lambda_357
tff(fact_3072_ATP_Olambda__358,axiom,
! [G2: $tType,H5: $tType,Uu: set(G2),Uua: fun(H5,set(G2)),Uub: H5] : aa(H5,set(G2),aa(fun(H5,set(G2)),fun(H5,set(G2)),aTP_Lamp_en(set(G2),fun(fun(H5,set(G2)),fun(H5,set(G2))),Uu),Uua),Uub) = aa(set(G2),set(G2),aa(set(G2),fun(set(G2),set(G2)),minus_minus(set(G2)),Uu),aa(H5,set(G2),Uua,Uub)) ).
% ATP.lambda_358
tff(fact_3073_ATP_Olambda__359,axiom,
! [M5: $tType,N3: $tType,Uu: set(M5),Uua: fun(N3,set(M5)),Uub: N3] : aa(N3,set(M5),aa(fun(N3,set(M5)),fun(N3,set(M5)),aTP_Lamp_gq(set(M5),fun(fun(N3,set(M5)),fun(N3,set(M5))),Uu),Uua),Uub) = aa(set(M5),set(M5),aa(set(M5),fun(set(M5),set(M5)),sup_sup(set(M5)),Uu),aa(N3,set(M5),Uua,Uub)) ).
% ATP.lambda_359
tff(fact_3074_ATP_Olambda__360,axiom,
! [E: $tType,F2: $tType,Uu: set(E),Uua: fun(F2,set(E)),Uub: F2] : aa(F2,set(E),aa(fun(F2,set(E)),fun(F2,set(E)),aTP_Lamp_fk(set(E),fun(fun(F2,set(E)),fun(F2,set(E))),Uu),Uua),Uub) = aa(set(E),set(E),aa(set(E),fun(set(E),set(E)),sup_sup(set(E)),Uu),aa(F2,set(E),Uua,Uub)) ).
% ATP.lambda_360
tff(fact_3075_ATP_Olambda__361,axiom,
! [A: $tType,B: $tType,Uu: set(A),Uua: fun(B,set(A)),Uub: B] : aa(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_go(set(A),fun(fun(B,set(A)),fun(B,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Uu),aa(B,set(A),Uua,Uub)) ).
% ATP.lambda_361
tff(fact_3076_ATP_Olambda__362,axiom,
! [A: $tType,B: $tType] :
( comple592849572758109894attice(A)
=> ! [Uu: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_gc(A,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Uu),aa(B,A,Uua,Uub)) ) ).
% ATP.lambda_362
tff(fact_3077_ATP_Olambda__363,axiom,
! [I5: $tType,J2: $tType,Uu: set(I5),Uua: fun(J2,set(I5)),Uub: J2] : aa(J2,set(I5),aa(fun(J2,set(I5)),fun(J2,set(I5)),aTP_Lamp_gl(set(I5),fun(fun(J2,set(I5)),fun(J2,set(I5))),Uu),Uua),Uub) = aa(set(I5),set(I5),aa(set(I5),fun(set(I5),set(I5)),inf_inf(set(I5)),Uu),aa(J2,set(I5),Uua,Uub)) ).
% ATP.lambda_363
tff(fact_3078_ATP_Olambda__364,axiom,
! [C: $tType,D: $tType,Uu: set(C),Uua: fun(D,set(C)),Uub: D] : aa(D,set(C),aa(fun(D,set(C)),fun(D,set(C)),aTP_Lamp_fb(set(C),fun(fun(D,set(C)),fun(D,set(C))),Uu),Uua),Uub) = aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),Uu),aa(D,set(C),Uua,Uub)) ).
% ATP.lambda_364
tff(fact_3079_ATP_Olambda__365,axiom,
! [A: $tType,B: $tType,Uu: set(A),Uua: fun(B,set(A)),Uub: B] : aa(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_gj(set(A),fun(fun(B,set(A)),fun(B,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Uu),aa(B,set(A),Uua,Uub)) ).
% ATP.lambda_365
tff(fact_3080_ATP_Olambda__366,axiom,
! [A: $tType,B: $tType] :
( comple592849572758109894attice(A)
=> ! [Uu: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_fx(A,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Uu),aa(B,A,Uua,Uub)) ) ).
% ATP.lambda_366
tff(fact_3081_ATP_Olambda__367,axiom,
! [A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [Uu: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_gu(A,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Uu),aa(B,A,Uua,Uub)) ) ).
% ATP.lambda_367
tff(fact_3082_ATP_Olambda__368,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: nat,Uua: fun(C,set(old_node(A,B))),Uub: C] : aa(C,set(old_node(A,B)),aa(fun(C,set(old_node(A,B))),fun(C,set(old_node(A,B))),aTP_Lamp_vt(nat,fun(fun(C,set(old_node(A,B))),fun(C,set(old_node(A,B)))),Uu),Uua),Uub) = aa(set(old_node(A,B)),set(old_node(A,B)),old_ntrunc(A,B,Uu),aa(C,set(old_node(A,B)),Uua,Uub)) ).
% ATP.lambda_368
tff(fact_3083_ATP_Olambda__369,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: set(old_node(A,B)),Uua: fun(C,set(old_node(A,B))),Uub: C] : aa(C,set(old_node(A,B)),aa(fun(C,set(old_node(A,B))),fun(C,set(old_node(A,B))),aTP_Lamp_vx(set(old_node(A,B)),fun(fun(C,set(old_node(A,B))),fun(C,set(old_node(A,B)))),Uu),Uua),Uub) = old_Scons(A,B,Uu,aa(C,set(old_node(A,B)),Uua,Uub)) ).
% ATP.lambda_369
tff(fact_3084_ATP_Olambda__370,axiom,
! [B: $tType,C: $tType,A: $tType,Uu: filter(B),Uua: fun(A,filter(C)),Uub: A] : aa(A,filter(product_prod(B,C)),aa(fun(A,filter(C)),fun(A,filter(product_prod(B,C))),aTP_Lamp_ji(filter(B),fun(fun(A,filter(C)),fun(A,filter(product_prod(B,C)))),Uu),Uua),Uub) = prod_filter(B,C,Uu,aa(A,filter(C),Uua,Uub)) ).
% ATP.lambda_370
tff(fact_3085_ATP_Olambda__371,axiom,
! [A: $tType,C: $tType,B: $tType,Uu: fun(A,C),Uua: fun(B,filter(C)),Uub: B] : aa(B,filter(A),aa(fun(B,filter(C)),fun(B,filter(A)),aTP_Lamp_js(fun(A,C),fun(fun(B,filter(C)),fun(B,filter(A))),Uu),Uua),Uub) = filtercomap(A,C,Uu,aa(B,filter(C),Uua,Uub)) ).
% ATP.lambda_371
tff(fact_3086_ATP_Olambda__372,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: fun(A,B),Uua: fun(C,filter(B)),Uub: C] : aa(C,filter(A),aa(fun(C,filter(B)),fun(C,filter(A)),aTP_Lamp_ka(fun(A,B),fun(fun(C,filter(B)),fun(C,filter(A))),Uu),Uua),Uub) = filtercomap(A,B,Uu,aa(C,filter(B),Uua,Uub)) ).
% ATP.lambda_372
tff(fact_3087_ATP_Olambda__373,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,Uu: fun(A,fun(C,bool)),Uua: fun(D,fun(C,fun(B,bool))),Uub: D] : aa(D,fun(A,fun(B,bool)),aa(fun(D,fun(C,fun(B,bool))),fun(D,fun(A,fun(B,bool))),aTP_Lamp_es(fun(A,fun(C,bool)),fun(fun(D,fun(C,fun(B,bool))),fun(D,fun(A,fun(B,bool)))),Uu),Uua),Uub) = aa(fun(C,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(A,fun(C,bool)),fun(fun(C,fun(B,bool)),fun(A,fun(B,bool))),relcompp(A,C,B),Uu),aa(D,fun(C,fun(B,bool)),Uua,Uub)) ).
% ATP.lambda_373
tff(fact_3088_ATP_Olambda__374,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: fun(C,B),Uua: A,Uub: C] : aa(C,product_prod(A,B),aa(A,fun(C,product_prod(A,B)),aTP_Lamp_oz(fun(C,B),fun(A,fun(C,product_prod(A,B))),Uu),Uua),Uub) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),aa(C,B,Uu,Uub)) ).
% ATP.lambda_374
tff(fact_3089_ATP_Olambda__375,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,Uu: set(product_prod(A,C)),Uua: fun(D,set(product_prod(C,B))),Uub: D] : aa(D,set(product_prod(A,B)),aa(fun(D,set(product_prod(C,B))),fun(D,set(product_prod(A,B))),aTP_Lamp_ii(set(product_prod(A,C)),fun(fun(D,set(product_prod(C,B))),fun(D,set(product_prod(A,B)))),Uu),Uua),Uub) = relcomp(A,C,B,Uu,aa(D,set(product_prod(C,B)),Uua,Uub)) ).
% ATP.lambda_375
tff(fact_3090_ATP_Olambda__376,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: fun(B,A),Uua: fun(C,filter(B)),Uub: C] : aa(C,filter(A),aa(fun(C,filter(B)),fun(C,filter(A)),aTP_Lamp_ix(fun(B,A),fun(fun(C,filter(B)),fun(C,filter(A))),Uu),Uua),Uub) = filtermap(B,A,Uu,aa(C,filter(B),Uua,Uub)) ).
% ATP.lambda_376
tff(fact_3091_ATP_Olambda__377,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: set(product_prod(B,A)),Uua: fun(C,set(B)),Uub: C] : aa(C,set(A),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_in(set(product_prod(B,A)),fun(fun(C,set(B)),fun(C,set(A))),Uu),Uua),Uub) = aa(set(B),set(A),image(B,A,Uu),aa(C,set(B),Uua,Uub)) ).
% ATP.lambda_377
tff(fact_3092_ATP_Olambda__378,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: fun(A,B),Uua: fun(C,set(B)),Uub: C] : aa(C,set(A),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_hb(fun(A,B),fun(fun(C,set(B)),fun(C,set(A))),Uu),Uua),Uub) = aa(set(B),set(A),vimage(A,B,Uu),aa(C,set(B),Uua,Uub)) ).
% ATP.lambda_378
tff(fact_3093_ATP_Olambda__379,axiom,
! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: B,Uub: A] :
( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_ew(fun(B,set(A)),fun(B,fun(A,bool)),Uu),Uua),Uub))
<=> pp(aa(set(A),bool,member(A,Uub),aa(B,set(A),Uu,Uua))) ) ).
% ATP.lambda_379
tff(fact_3094_ATP_Olambda__380,axiom,
! [I5: $tType,J2: $tType,Uu: I5,Uua: fun(J2,set(I5)),Uub: J2] : aa(J2,set(I5),aa(fun(J2,set(I5)),fun(J2,set(I5)),aTP_Lamp_dc(I5,fun(fun(J2,set(I5)),fun(J2,set(I5))),Uu),Uua),Uub) = aa(set(I5),set(I5),insert(I5,Uu),aa(J2,set(I5),Uua,Uub)) ).
% ATP.lambda_380
tff(fact_3095_ATP_Olambda__381,axiom,
! [B: $tType,A: $tType,Uu: B,Uua: fun(A,set(B)),Uub: A] : aa(A,set(B),aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_db(B,fun(fun(A,set(B)),fun(A,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),insert(B,Uu),aa(A,set(B),Uua,Uub)) ).
% ATP.lambda_381
tff(fact_3096_ATP_Olambda__382,axiom,
! [A: $tType,B: $tType,Uu: A,Uua: fun(B,set(A)),Uub: B] : aa(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_cy(A,fun(fun(B,set(A)),fun(B,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),insert(A,Uu),aa(B,set(A),Uua,Uub)) ).
% ATP.lambda_382
tff(fact_3097_ATP_Olambda__383,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: fun(B,A),Uua: fun(C,set(B)),Uub: C] : aa(C,set(A),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_dy(fun(B,A),fun(fun(C,set(B)),fun(C,set(A))),Uu),Uua),Uub) = aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),Uu),aa(C,set(B),Uua,Uub)) ).
% ATP.lambda_383
tff(fact_3098_ATP_Olambda__384,axiom,
! [B: $tType,A: $tType,C: $tType,Uu: fun(A,B),Uua: fun(C,set(A)),Uub: C] : aa(C,set(B),aa(fun(C,set(A)),fun(C,set(B)),aTP_Lamp_gv(fun(A,B),fun(fun(C,set(A)),fun(C,set(B))),Uu),Uua),Uub) = aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),Uu),aa(C,set(A),Uua,Uub)) ).
% ATP.lambda_384
tff(fact_3099_ATP_Olambda__385,axiom,
! [B: $tType,D: $tType,C: $tType] :
( condit1219197933456340205attice(B)
=> ! [Uu: fun(C,set(D)),Uua: fun(D,B),Uub: C] : aa(C,set(B),aa(fun(D,B),fun(C,set(B)),aTP_Lamp_fd(fun(C,set(D)),fun(fun(D,B),fun(C,set(B))),Uu),Uua),Uub) = aa(set(D),set(B),aa(fun(D,B),fun(set(D),set(B)),image2(D,B),Uua),aa(C,set(D),Uu,Uub)) ) ).
% ATP.lambda_385
tff(fact_3100_ATP_Olambda__386,axiom,
! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A,Uub: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_hd(fun(A,B),fun(A,fun(B,bool)),Uu),Uua),Uub))
<=> ( Uub = aa(A,B,Uu,Uua) ) ) ).
% ATP.lambda_386
tff(fact_3101_ATP_Olambda__387,axiom,
! [A: $tType,B: $tType,Uu: set(product_prod(B,B)),Uua: fun(B,A),Uub: A] : aa(A,set(A),aa(fun(B,A),fun(A,set(A)),aTP_Lamp_oa(set(product_prod(B,B)),fun(fun(B,A),fun(A,set(A))),Uu),Uua),Uub) = aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),Uua),aa(set(product_prod(B,B)),set(B),field2(B),Uu)) ).
% ATP.lambda_387
tff(fact_3102_ATP_Olambda__388,axiom,
! [A: $tType,B: $tType,Uu: set(B),Uua: set(B),Uub: A] : aa(A,set(B),aa(set(B),fun(A,set(B)),aTP_Lamp_fo(set(B),fun(set(B),fun(A,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),Uu),Uua) ).
% ATP.lambda_388
tff(fact_3103_ATP_Olambda__389,axiom,
! [A: $tType,B: $tType,Uu: B,Uua: set(B),Uub: A] : aa(A,set(B),aa(set(B),fun(A,set(B)),aTP_Lamp_fa(B,fun(set(B),fun(A,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),insert(B,Uu),Uua) ).
% ATP.lambda_389
tff(fact_3104_ATP_Olambda__390,axiom,
! [A: $tType,B: $tType,D: $tType,Uu: fun(D,B),Uua: set(D),Uub: A] : aa(A,set(B),aa(set(D),fun(A,set(B)),aTP_Lamp_bt(fun(D,B),fun(set(D),fun(A,set(B))),Uu),Uua),Uub) = aa(set(D),set(B),aa(fun(D,B),fun(set(D),set(B)),image2(D,B),Uu),Uua) ).
% ATP.lambda_390
tff(fact_3105_ATP_Olambda__391,axiom,
! [A: $tType,B: $tType,Uu: fun(B,A),Uua: set(B),Uub: A] : aa(A,set(A),aa(set(B),fun(A,set(A)),aTP_Lamp_hv(fun(B,A),fun(set(B),fun(A,set(A))),Uu),Uua),Uub) = aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),Uu),Uua) ).
% ATP.lambda_391
tff(fact_3106_ATP_Olambda__392,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: ref(A),Uua: A,Uub: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(product_unit,heap_ext(product_unit)),aa(A,fun(heap_ext(product_unit),product_prod(product_unit,heap_ext(product_unit))),aTP_Lamp_td(ref(A),fun(A,fun(heap_ext(product_unit),product_prod(product_unit,heap_ext(product_unit)))),Uu),Uua),Uub) = aa(heap_ext(product_unit),product_prod(product_unit,heap_ext(product_unit)),aa(product_unit,fun(heap_ext(product_unit),product_prod(product_unit,heap_ext(product_unit))),product_Pair(product_unit,heap_ext(product_unit)),product_Unity),aa(heap_ext(product_unit),heap_ext(product_unit),set4(A,Uu,Uua),Uub)) ) ).
% ATP.lambda_392
tff(fact_3107_ATP_Olambda__393,axiom,
! [A: $tType,B: $tType,Uu: fun(A,B),Uua: fun(A,bool),Uub: B] :
( pp(aa(B,bool,aa(fun(A,bool),fun(B,bool),aTP_Lamp_od(fun(A,B),fun(fun(A,bool),fun(B,bool)),Uu),Uua),Uub))
<=> pp(aa(A,bool,Uua,aa(B,A,hilbert_inv_into(A,B,top_top(set(A)),Uu),Uub))) ) ).
% ATP.lambda_393
tff(fact_3108_ATP_Olambda__394,axiom,
! [C: $tType,B: $tType,A: $tType,Uu: fun(fun(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit)))),C),Uua: fun(A,B),Uub: fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))))] : aa(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),C,aa(fun(A,B),fun(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),C),aTP_Lamp_sw(fun(fun(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit)))),C),fun(fun(A,B),fun(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),C)),Uu),Uua),Uub) = aa(fun(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit)))),C,Uu,aa(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),fun(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit)))),aa(fun(option(product_prod(A,heap_ext(product_unit))),option(product_prod(B,heap_ext(product_unit)))),fun(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),fun(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit))))),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),Uua,id(heap_ext(product_unit))))),Uub)) ).
% ATP.lambda_394
tff(fact_3109_ATP_Olambda__395,axiom,
! [A: $tType,Uu: fun(nat,set(A)),Uua: nat,Uub: nat] : aa(nat,set(A),aa(nat,fun(nat,set(A)),aTP_Lamp_mb(fun(nat,set(A)),fun(nat,fun(nat,set(A))),Uu),Uua),Uub) = aa(nat,set(A),Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ).
% ATP.lambda_395
tff(fact_3110_ATP_Olambda__396,axiom,
! [A: $tType,B: $tType,Uu: fun(product_prod(A,B),bool),Uua: A,Uub: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_ql(fun(product_prod(A,B),bool),fun(A,fun(B,bool)),Uu),Uua),Uub))
<=> pp(aa(product_prod(A,B),bool,Uu,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uub))) ) ).
% ATP.lambda_396
tff(fact_3111_ATP_Olambda__397,axiom,
! [C: $tType,A: $tType,B: $tType,Uu: fun(product_prod(A,B),C),Uua: A,Uub: B] : aa(B,C,aa(A,fun(B,C),aTP_Lamp_as(fun(product_prod(A,B),C),fun(A,fun(B,C)),Uu),Uua),Uub) = aa(product_prod(A,B),C,Uu,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uub)) ).
% ATP.lambda_397
tff(fact_3112_ATP_Olambda__398,axiom,
! [V4: $tType,U2: $tType,T: $tType,Uu: fun(U2,set(V4)),Uua: fun(T,U2),Uub: T] : aa(T,set(V4),aa(fun(T,U2),fun(T,set(V4)),aTP_Lamp_dw(fun(U2,set(V4)),fun(fun(T,U2),fun(T,set(V4))),Uu),Uua),Uub) = aa(U2,set(V4),Uu,aa(T,U2,Uua,Uub)) ).
% ATP.lambda_398
tff(fact_3113_ATP_Olambda__399,axiom,
! [B: $tType,C: $tType,A: $tType,Uu: fun(C,fun(B,bool)),Uua: fun(A,C),Uub: A] : aa(A,fun(B,bool),aa(fun(A,C),fun(A,fun(B,bool)),aTP_Lamp_aq(fun(C,fun(B,bool)),fun(fun(A,C),fun(A,fun(B,bool))),Uu),Uua),Uub) = aa(C,fun(B,bool),Uu,aa(A,C,Uua,Uub)) ).
% ATP.lambda_399
tff(fact_3114_ATP_Olambda__400,axiom,
! [A: $tType,C: $tType,B: $tType,Uu: fun(C,A),Uua: fun(B,C),Uub: B] : aa(B,A,aa(fun(B,C),fun(B,A),aTP_Lamp_ap(fun(C,A),fun(fun(B,C),fun(B,A)),Uu),Uua),Uub) = aa(C,A,Uu,aa(B,C,Uua,Uub)) ).
% ATP.lambda_400
tff(fact_3115_ATP_Olambda__401,axiom,
! [C: $tType,B: $tType,A: $tType,Uu: fun(B,C),Uua: fun(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),B),Uub: fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))))] : aa(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),C,aa(fun(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),B),fun(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),C),aTP_Lamp_th(fun(B,C),fun(fun(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),B),fun(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),C)),Uu),Uua),Uub) = aa(B,C,Uu,aa(fun(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),B,Uua,Uub)) ).
% ATP.lambda_401
tff(fact_3116_ATP_Olambda__402,axiom,
! [C: $tType,B: $tType,A: $tType,Uu: fun(B,C),Uua: fun(A,B),Uub: A] : aa(A,C,aa(fun(A,B),fun(A,C),aTP_Lamp_sp(fun(B,C),fun(fun(A,B),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uu,aa(A,B,Uua,Uub)) ).
% ATP.lambda_402
tff(fact_3117_ATP_Olambda__403,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: fun(B,A),Uua: fun(C,B),Uub: C] : aa(C,A,aa(fun(C,B),fun(C,A),aTP_Lamp_ag(fun(B,A),fun(fun(C,B),fun(C,A)),Uu),Uua),Uub) = aa(B,A,Uu,aa(C,B,Uua,Uub)) ).
% ATP.lambda_403
tff(fact_3118_ATP_Olambda__404,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( condit1219197933456340205attice(A)
& condit1219197933456340205attice(B) )
=> ! [Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_ha(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).
% ATP.lambda_404
tff(fact_3119_ATP_Olambda__405,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comple6319245703460814977attice(A)
& comple6319245703460814977attice(B) )
=> ! [Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_hf(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).
% ATP.lambda_405
tff(fact_3120_ATP_Olambda__406,axiom,
! [B: $tType,C: $tType,A: $tType,D: $tType,Uu: fun(A,fun(B,C)),Uua: fun(D,A),Uub: D] : aa(D,fun(B,C),aa(fun(D,A),fun(D,fun(B,C)),aTP_Lamp_bf(fun(A,fun(B,C)),fun(fun(D,A),fun(D,fun(B,C))),Uu),Uua),Uub) = aa(A,fun(B,C),Uu,aa(D,A,Uua,Uub)) ).
% ATP.lambda_406
tff(fact_3121_ATP_Olambda__407,axiom,
! [C: $tType,D: $tType,Uu: fun(D,C),Uua: fun(C,bool),Uub: D] :
( pp(aa(D,bool,aa(fun(C,bool),fun(D,bool),aTP_Lamp_bd(fun(D,C),fun(fun(C,bool),fun(D,bool)),Uu),Uua),Uub))
<=> pp(aa(C,bool,Uua,aa(D,C,Uu,Uub))) ) ).
% ATP.lambda_407
tff(fact_3122_ATP_Olambda__408,axiom,
! [D: $tType,C: $tType,Uu: fun(C,D),Uua: fun(D,bool),Uub: C] :
( pp(aa(C,bool,aa(fun(D,bool),fun(C,bool),aTP_Lamp_bc(fun(C,D),fun(fun(D,bool),fun(C,bool)),Uu),Uua),Uub))
<=> pp(aa(D,bool,Uua,aa(C,D,Uu,Uub))) ) ).
% ATP.lambda_408
tff(fact_3123_ATP_Olambda__409,axiom,
! [C: $tType,A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(A,C),Uub: B] : aa(B,C,aa(fun(A,C),fun(B,C),aTP_Lamp_hj(fun(B,A),fun(fun(A,C),fun(B,C)),Uu),Uua),Uub) = aa(A,C,Uua,aa(B,A,Uu,Uub)) ).
% ATP.lambda_409
tff(fact_3124_ATP_Olambda__410,axiom,
! [C: $tType,B: $tType,A: $tType] :
( semiring_1(C)
=> ! [Uu: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_mp(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).
% ATP.lambda_410
tff(fact_3125_ATP_Olambda__411,axiom,
! [A: $tType,B: $tType,D: $tType,Uu: fun(D,set(B)),Uua: D,Uub: A] : aa(A,set(B),aa(D,fun(A,set(B)),aTP_Lamp_dh(fun(D,set(B)),fun(D,fun(A,set(B))),Uu),Uua),Uub) = aa(D,set(B),Uu,Uua) ).
% ATP.lambda_411
tff(fact_3126_ATP_Olambda__412,axiom,
! [A: $tType,B: $tType,Uu: fun(B,bool),Uua: B,Uub: A] :
( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_jh(fun(B,bool),fun(B,fun(A,bool)),Uu),Uua),Uub))
<=> pp(aa(B,bool,Uu,Uua)) ) ).
% ATP.lambda_412
tff(fact_3127_ATP_Olambda__413,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] : aa(A,fun(product_prod(A,A),bool),aa(A,fun(A,fun(product_prod(A,A),bool)),aTP_Lamp_hq(set(product_prod(A,A)),fun(A,fun(A,fun(product_prod(A,A),bool))),Uu),Uua),Uub) = aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(A,fun(A,fun(A,bool)),aa(A,fun(A,fun(A,fun(A,bool))),aTP_Lamp_hp(set(product_prod(A,A)),fun(A,fun(A,fun(A,fun(A,bool)))),Uu),Uua),Uub)) ).
% ATP.lambda_413
tff(fact_3128_ATP_Olambda__414,axiom,
! [A: $tType,B: $tType,Uu: fun(A,fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)))),Uua: product_prod(code_natural,code_natural),Uub: A] : aa(A,pred(B),aa(product_prod(code_natural,code_natural),fun(A,pred(B)),aTP_Lamp_wq(fun(A,fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)))),fun(product_prod(code_natural,code_natural),fun(A,pred(B))),Uu),Uua),Uub) = aa(product_prod(pred(B),product_prod(code_natural,code_natural)),pred(B),product_fst(pred(B),product_prod(code_natural,code_natural)),aa(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)),aa(A,fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural))),Uu,Uub),Uua)) ).
% ATP.lambda_414
tff(fact_3129_ATP_Olambda__415,axiom,
! [B: $tType,C: $tType,A: $tType,E: $tType,D: $tType,Uu: fun(B,fun(C,fun(D,fun(E,set(A))))),Uua: set(product_prod(D,E)),Uub: product_prod(B,C)] : aa(product_prod(B,C),set(A),aa(set(product_prod(D,E)),fun(product_prod(B,C),set(A)),aTP_Lamp_em(fun(B,fun(C,fun(D,fun(E,set(A))))),fun(set(product_prod(D,E)),fun(product_prod(B,C),set(A))),Uu),Uua),Uub) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(product_prod(D,E)),set(set(A)),aa(fun(product_prod(D,E),set(A)),fun(set(product_prod(D,E)),set(set(A))),image2(product_prod(D,E),set(A)),aa(product_prod(B,C),fun(product_prod(D,E),set(A)),aTP_Lamp_el(fun(B,fun(C,fun(D,fun(E,set(A))))),fun(product_prod(B,C),fun(product_prod(D,E),set(A))),Uu),Uub)),Uua)) ).
% ATP.lambda_415
tff(fact_3130_ATP_Olambda__416,axiom,
! [C: $tType,A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [Uu: fun(B,fun(C,A)),Uua: set(B),Uub: C] : aa(C,A,aa(set(B),fun(C,A),aTP_Lamp_dr(fun(B,fun(C,A)),fun(set(B),fun(C,A)),Uu),Uua),Uub) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),aa(C,fun(B,A),aTP_Lamp_do(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu),Uub)),Uua)) ) ).
% ATP.lambda_416
tff(fact_3131_ATP_Olambda__417,axiom,
! [C: $tType,A: $tType,B: $tType] :
( comple6319245703460814977attice(A)
=> ! [Uu: fun(B,fun(C,A)),Uua: set(B),Uub: C] : aa(C,A,aa(set(B),fun(C,A),aTP_Lamp_dp(fun(B,fun(C,A)),fun(set(B),fun(C,A)),Uu),Uua),Uub) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),aa(C,fun(B,A),aTP_Lamp_do(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu),Uub)),Uua)) ) ).
% ATP.lambda_417
tff(fact_3132_ATP_Olambda__418,axiom,
! [B: $tType,A: $tType,C: $tType] :
( comple6319245703460814977attice(A)
=> ! [Uu: fun(B,fun(C,A)),Uua: set(C),Uub: B] : aa(B,A,aa(set(C),fun(B,A),aTP_Lamp_dq(fun(B,fun(C,A)),fun(set(C),fun(B,A)),Uu),Uua),Uub) = aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),aa(fun(C,A),fun(set(C),set(A)),image2(C,A),aa(B,fun(C,A),Uu,Uub)),Uua)) ) ).
% ATP.lambda_418
tff(fact_3133_ATP_Olambda__419,axiom,
! [A: $tType,C: $tType,B: $tType,Uu: fun(A,fun(B,set(C))),Uua: set(B),Uub: A] : aa(A,set(C),aa(set(B),fun(A,set(C)),aTP_Lamp_jn(fun(A,fun(B,set(C))),fun(set(B),fun(A,set(C))),Uu),Uua),Uub) = aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),aa(fun(B,set(C)),fun(set(B),set(set(C))),image2(B,set(C)),aa(A,fun(B,set(C)),Uu,Uub)),Uua)) ).
% ATP.lambda_419
tff(fact_3134_ATP_Olambda__420,axiom,
! [B: $tType,A: $tType,C: $tType] :
( comple6319245703460814977attice(A)
=> ! [Uu: fun(B,fun(C,A)),Uua: set(C),Uub: B] : aa(B,A,aa(set(C),fun(B,A),aTP_Lamp_dn(fun(B,fun(C,A)),fun(set(C),fun(B,A)),Uu),Uua),Uub) = aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),aa(fun(C,A),fun(set(C),set(A)),image2(C,A),aa(B,fun(C,A),Uu,Uub)),Uua)) ) ).
% ATP.lambda_420
tff(fact_3135_ATP_Olambda__421,axiom,
! [B: $tType,A: $tType,Uu: set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),Uua: set(old_node(A,B)),Uub: set(old_node(A,B))] : aa(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aTP_Lamp_wj(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),fun(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))),Uu),Uua),Uub) = aa(set(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),complete_Sup_Sup(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),set(set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aa(fun(product_prod(set(old_node(A,B)),set(old_node(A,B))),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),fun(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),set(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))),image2(product_prod(set(old_node(A,B)),set(old_node(A,B))),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aa(fun(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))),fun(product_prod(set(old_node(A,B)),set(old_node(A,B))),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),product_case_prod(set(old_node(A,B)),set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))),aTP_Lamp_wi(set(old_node(A,B)),fun(set(old_node(A,B)),fun(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))))),Uua),Uub))),Uu)) ).
% ATP.lambda_421
tff(fact_3136_ATP_Olambda__422,axiom,
! [A: $tType,B: $tType,Uu: fun(A,fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)))),Uua: pred(A),Uub: product_prod(code_natural,code_natural)] : aa(product_prod(code_natural,code_natural),fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural))),aa(pred(A),fun(product_prod(code_natural,code_natural),fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)))),aTP_Lamp_wr(fun(A,fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)))),fun(pred(A),fun(product_prod(code_natural,code_natural),fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural))))),Uu),Uua),Uub) = aa(pred(B),fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural))),product_Pair(pred(B),product_prod(code_natural,code_natural)),bind3(A,B,Uua,aa(product_prod(code_natural,code_natural),fun(A,pred(B)),aTP_Lamp_wq(fun(A,fun(product_prod(code_natural,code_natural),product_prod(pred(B),product_prod(code_natural,code_natural)))),fun(product_prod(code_natural,code_natural),fun(A,pred(B))),Uu),Uub))) ).
% ATP.lambda_422
tff(fact_3137_ATP_Olambda__423,axiom,
! [S5: $tType,R6: $tType,Q3: $tType,Uu: fun(R6,set(S5)),Uua: fun(Q3,set(R6)),Uub: Q3] : aa(Q3,set(S5),aa(fun(Q3,set(R6)),fun(Q3,set(S5)),aTP_Lamp_dt(fun(R6,set(S5)),fun(fun(Q3,set(R6)),fun(Q3,set(S5))),Uu),Uua),Uub) = aa(set(set(S5)),set(S5),complete_Sup_Sup(set(S5)),aa(set(R6),set(set(S5)),aa(fun(R6,set(S5)),fun(set(R6),set(set(S5))),image2(R6,set(S5)),Uu),aa(Q3,set(R6),Uua,Uub))) ).
% ATP.lambda_423
tff(fact_3138_ATP_Olambda__424,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: fun(B,set(A)),Uua: fun(C,set(B)),Uub: C] : aa(C,set(A),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_ds(fun(B,set(A)),fun(fun(C,set(B)),fun(C,set(A))),Uu),Uua),Uub) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),aa(fun(B,set(A)),fun(set(B),set(set(A))),image2(B,set(A)),Uu),aa(C,set(B),Uua,Uub))) ).
% ATP.lambda_424
tff(fact_3139_ATP_Olambda__425,axiom,
! [A: $tType,B: $tType,C: $tType] :
( comple6319245703460814977attice(A)
=> ! [Uu: fun(B,A),Uua: fun(C,set(B)),Uub: C] : aa(C,A,aa(fun(C,set(B)),fun(C,A),aTP_Lamp_de(fun(B,A),fun(fun(C,set(B)),fun(C,A)),Uu),Uua),Uub) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),Uu),aa(C,set(B),Uua,Uub))) ) ).
% ATP.lambda_425
tff(fact_3140_ATP_Olambda__426,axiom,
! [B: $tType,D: $tType,C: $tType] :
( condit1219197933456340205attice(B)
=> ! [Uu: fun(C,set(D)),Uua: fun(D,B),Uub: C] : aa(C,B,aa(fun(D,B),fun(C,B),aTP_Lamp_fe(fun(C,set(D)),fun(fun(D,B),fun(C,B)),Uu),Uua),Uub) = aa(set(B),B,complete_Sup_Sup(B),aa(set(D),set(B),aa(fun(D,B),fun(set(D),set(B)),image2(D,B),Uua),aa(C,set(D),Uu,Uub))) ) ).
% ATP.lambda_426
tff(fact_3141_ATP_Olambda__427,axiom,
! [S5: $tType,R6: $tType,Q3: $tType,Uu: fun(R6,set(S5)),Uua: fun(Q3,set(R6)),Uub: Q3] : aa(Q3,set(S5),aa(fun(Q3,set(R6)),fun(Q3,set(S5)),aTP_Lamp_da(fun(R6,set(S5)),fun(fun(Q3,set(R6)),fun(Q3,set(S5))),Uu),Uua),Uub) = aa(set(set(S5)),set(S5),complete_Inf_Inf(set(S5)),aa(set(R6),set(set(S5)),aa(fun(R6,set(S5)),fun(set(R6),set(set(S5))),image2(R6,set(S5)),Uu),aa(Q3,set(R6),Uua,Uub))) ).
% ATP.lambda_427
tff(fact_3142_ATP_Olambda__428,axiom,
! [B: $tType,D: $tType,C: $tType] :
( condit1219197933456340205attice(B)
=> ! [Uu: fun(C,set(D)),Uua: fun(D,B),Uub: C] : aa(C,B,aa(fun(D,B),fun(C,B),aTP_Lamp_ff(fun(C,set(D)),fun(fun(D,B),fun(C,B)),Uu),Uua),Uub) = aa(set(B),B,complete_Inf_Inf(B),aa(set(D),set(B),aa(fun(D,B),fun(set(D),set(B)),image2(D,B),Uua),aa(C,set(D),Uu,Uub))) ) ).
% ATP.lambda_428
tff(fact_3143_ATP_Olambda__429,axiom,
! [A: $tType,B: $tType,D: $tType,Uu: fun(D,set(B)),Uua: set(D),Uub: A] : aa(A,set(B),aa(set(D),fun(A,set(B)),aTP_Lamp_dj(fun(D,set(B)),fun(set(D),fun(A,set(B))),Uu),Uua),Uub) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(D),set(set(B)),aa(fun(D,set(B)),fun(set(D),set(set(B))),image2(D,set(B)),Uu),Uua)) ).
% ATP.lambda_429
tff(fact_3144_ATP_Olambda__430,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: fun(A,A),Uua: A,Uub: product_unit] : aa(product_unit,heap_Heap(A),aa(A,fun(product_unit,heap_Heap(A)),aTP_Lamp_bo(fun(A,A),fun(A,fun(product_unit,heap_Heap(A))),Uu),Uua),Uub) = aa(A,heap_Heap(A),heap_return(A),aa(A,A,Uu,Uua)) ) ).
% ATP.lambda_430
tff(fact_3145_ATP_Olambda__431,axiom,
! [A: $tType,B: $tType,Uu: fun(A,heap_Heap(B)),Uua: heap_ext(product_unit),Uub: A] : aa(A,B,aa(heap_ext(product_unit),fun(A,B),aTP_Lamp_ru(fun(A,heap_Heap(B)),fun(heap_ext(product_unit),fun(A,B)),Uu),Uua),Uub) = aa(product_prod(B,heap_ext(product_unit)),B,product_fst(B,heap_ext(product_unit)),aa(option(product_prod(B,heap_ext(product_unit))),product_prod(B,heap_ext(product_unit)),the2(product_prod(B,heap_ext(product_unit))),aa(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit))),aa(heap_Heap(B),fun(heap_ext(product_unit),option(product_prod(B,heap_ext(product_unit)))),heap_execute(B),aa(A,heap_Heap(B),Uu,Uub)),Uua))) ).
% ATP.lambda_431
tff(fact_3146_ATP_Olambda__432,axiom,
! [A: $tType,B: $tType,Uu: list(A),Uua: list(B),Uub: product_prod(A,B)] :
( pp(aa(product_prod(A,B),bool,aa(list(B),fun(product_prod(A,B),bool),aTP_Lamp_pd(list(A),fun(list(B),fun(product_prod(A,B),bool)),Uu),Uua),Uub))
<=> ? [I3: nat] :
( ( Uub = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(nat,A,nth3(A,Uu),I3)),aa(nat,B,nth3(B,Uua),I3)) )
& pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),ord_min(nat,aa(list(A),nat,size_size(list(A)),Uu),aa(list(B),nat,size_size(list(B)),Uua)))) ) ) ).
% ATP.lambda_432
tff(fact_3147_ATP_Olambda__433,axiom,
! [B: $tType,A: $tType,Uu: set(A),Uua: fun(A,B),Uub: product_prod(A,B)] :
( pp(aa(product_prod(A,B),bool,aa(fun(A,B),fun(product_prod(A,B),bool),aTP_Lamp_ne(set(A),fun(fun(A,B),fun(product_prod(A,B),bool)),Uu),Uua),Uub))
<=> ? [A7: A] :
( ( Uub = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A7),aa(A,B,Uua,A7)) )
& pp(aa(set(A),bool,member(A,A7),Uu)) ) ) ).
% ATP.lambda_433
tff(fact_3148_ATP_Olambda__434,axiom,
! [A: $tType,B: $tType,Uu: set(A),Uua: fun(A,set(B)),Uub: B] :
( pp(aa(B,bool,aa(fun(A,set(B)),fun(B,bool),aTP_Lamp_on(set(A),fun(fun(A,set(B)),fun(B,bool)),Uu),Uua),Uub))
<=> ? [X5: set(B)] :
( pp(aa(set(set(B)),bool,member(set(B),X5),aa(set(A),set(set(B)),aa(fun(A,set(B)),fun(set(A),set(set(B))),image2(A,set(B)),Uua),Uu)))
& pp(aa(set(B),bool,member(B,Uub),X5)) ) ) ).
% ATP.lambda_434
tff(fact_3149_ATP_Olambda__435,axiom,
! [A: $tType,B: $tType,Uu: fun(B,A),Uua: set(B),Uub: A] :
( pp(aa(A,bool,aa(set(B),fun(A,bool),aTP_Lamp_mx(fun(B,A),fun(set(B),fun(A,bool)),Uu),Uua),Uub))
<=> ? [X5: B] :
( ( Uub = aa(B,A,Uu,X5) )
& pp(aa(set(B),bool,member(B,X5),Uua)) ) ) ).
% ATP.lambda_435
tff(fact_3150_ATP_Olambda__436,axiom,
! [B: $tType,C: $tType,Uu: fun(B,set(C)),Uua: fun(C,bool),Uub: B] :
( pp(aa(B,bool,aa(fun(C,bool),fun(B,bool),aTP_Lamp_kv(fun(B,set(C)),fun(fun(C,bool),fun(B,bool)),Uu),Uua),Uub))
<=> ! [X5: C] :
( pp(aa(set(C),bool,member(C,X5),aa(B,set(C),Uu,Uub)))
=> pp(aa(C,bool,Uua,X5)) ) ) ).
% ATP.lambda_436
tff(fact_3151_ATP_Olambda__437,axiom,
! [A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(B,bool),Uub: A] :
( pp(aa(A,bool,aa(fun(B,bool),fun(A,bool),aTP_Lamp_mw(fun(B,A),fun(fun(B,bool),fun(A,bool)),Uu),Uua),Uub))
<=> ? [X5: B] :
( ( Uub = aa(B,A,Uu,X5) )
& pp(aa(B,bool,Uua,X5)) ) ) ).
% ATP.lambda_437
tff(fact_3152_ATP_Olambda__438,axiom,
! [A: $tType,B: $tType,Uu: set(B),Uua: fun(A,fun(B,bool)),Uub: A] :
( pp(aa(A,bool,aa(fun(A,fun(B,bool)),fun(A,bool),aTP_Lamp_kx(set(B),fun(fun(A,fun(B,bool)),fun(A,bool)),Uu),Uua),Uub))
<=> ! [X5: B] :
( pp(aa(set(B),bool,member(B,X5),Uu))
=> pp(aa(B,bool,aa(A,fun(B,bool),Uua,Uub),X5)) ) ) ).
% ATP.lambda_438
tff(fact_3153_ATP_Olambda__439,axiom,
! [A: $tType,B: $tType,Uu: set(B),Uua: fun(A,fun(B,bool)),Uub: A] :
( pp(aa(A,bool,aa(fun(A,fun(B,bool)),fun(A,bool),aTP_Lamp_ok(set(B),fun(fun(A,fun(B,bool)),fun(A,bool)),Uu),Uua),Uub))
<=> ? [X5: B] :
( pp(aa(set(B),bool,member(B,X5),Uu))
& pp(aa(B,bool,aa(A,fun(B,bool),Uua,Uub),X5)) ) ) ).
% ATP.lambda_439
tff(fact_3154_ATP_Olambda__440,axiom,
! [B: $tType,A: $tType,Uu: fun(B,fun(A,bool)),Uua: set(B),Uub: A] :
( pp(aa(A,bool,aa(set(B),fun(A,bool),aTP_Lamp_og(fun(B,fun(A,bool)),fun(set(B),fun(A,bool)),Uu),Uua),Uub))
<=> ? [X5: B] :
( pp(aa(set(B),bool,member(B,X5),Uua))
& pp(aa(A,bool,aa(B,fun(A,bool),Uu,X5),Uub)) ) ) ).
% ATP.lambda_440
tff(fact_3155_ATP_Olambda__441,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_qn(set(product_prod(A,A)),fun(A,fun(A,bool)),Uu),Uua),Uub))
<=> ! [X5: product_prod(A,A)] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),X5),Uu))
=> pp(aa(product_prod(A,A),bool,aa(fun(A,fun(A,bool)),fun(product_prod(A,A),bool),product_case_prod(A,A,bool),aa(A,fun(A,fun(A,bool)),aa(A,fun(A,fun(A,fun(A,bool))),aTP_Lamp_qm(set(product_prod(A,A)),fun(A,fun(A,fun(A,fun(A,bool)))),Uu),Uua),Uub)),X5)) ) ) ).
% ATP.lambda_441
tff(fact_3156_ATP_Olambda__442,axiom,
! [B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: set(A),Uub: B] :
( pp(aa(B,bool,aa(set(A),fun(B,bool),aTP_Lamp_ol(set(product_prod(A,B)),fun(set(A),fun(B,bool)),Uu),Uua),Uub))
<=> ? [X5: A] :
( pp(aa(set(A),bool,member(A,X5),Uua))
& pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X5),Uub)),Uu)) ) ) ).
% ATP.lambda_442
tff(fact_3157_ATP_Olambda__443,axiom,
! [B: $tType,A: $tType,Uu: fun(A,B),Uua: set(A),Uub: A] :
( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_om(fun(A,B),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
<=> ? [X5: A] :
( pp(aa(set(A),bool,member(A,X5),Uua))
& ( aa(A,B,Uu,X5) = aa(A,B,Uu,Uub) ) ) ) ).
% ATP.lambda_443
tff(fact_3158_ATP_Olambda__444,axiom,
! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: set(B),Uub: A] :
( pp(aa(A,bool,aa(set(B),fun(A,bool),aTP_Lamp_kw(fun(B,set(A)),fun(set(B),fun(A,bool)),Uu),Uua),Uub))
<=> ! [X5: B] :
( pp(aa(set(B),bool,member(B,X5),Uua))
=> pp(aa(set(A),bool,member(A,Uub),aa(B,set(A),Uu,X5))) ) ) ).
% ATP.lambda_444
tff(fact_3159_ATP_Olambda__445,axiom,
! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: set(B),Uub: A] :
( pp(aa(A,bool,aa(set(B),fun(A,bool),aTP_Lamp_oj(fun(B,set(A)),fun(set(B),fun(A,bool)),Uu),Uua),Uub))
<=> ? [X5: B] :
( pp(aa(set(B),bool,member(B,X5),Uua))
& pp(aa(set(A),bool,member(A,Uub),aa(B,set(A),Uu,X5))) ) ) ).
% ATP.lambda_445
tff(fact_3160_ATP_Olambda__446,axiom,
! [B: $tType,A: $tType,Uu: fun(A,B),Uua: set(A),Uub: B] :
( pp(aa(B,bool,aa(set(A),fun(B,bool),aTP_Lamp_oi(fun(A,B),fun(set(A),fun(B,bool)),Uu),Uua),Uub))
<=> ? [X5: A] :
( pp(aa(set(A),bool,member(A,X5),Uua))
& ( Uub = aa(A,B,Uu,X5) ) ) ) ).
% ATP.lambda_446
tff(fact_3161_ATP_Olambda__447,axiom,
! [A: $tType,B: $tType,Uu: set(product_prod(B,B)),Uua: fun(B,A),Uub: product_prod(A,A)] :
( pp(aa(product_prod(A,A),bool,aa(fun(B,A),fun(product_prod(A,A),bool),aTP_Lamp_nf(set(product_prod(B,B)),fun(fun(B,A),fun(product_prod(A,A),bool)),Uu),Uua),Uub))
<=> ? [A18: B,A26: B] :
( ( Uub = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(B,A,Uua,A18)),aa(B,A,Uua,A26)) )
& pp(aa(set(product_prod(B,B)),bool,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),A18),A26)),Uu)) ) ) ).
% ATP.lambda_447
tff(fact_3162_ATP_Olambda__448,axiom,
! [A2: $tType,A: $tType,Uu: set(product_prod(A,A)),Uua: fun(A,A2),Uub: product_prod(A2,A2)] :
( pp(aa(product_prod(A2,A2),bool,aa(fun(A,A2),fun(product_prod(A2,A2),bool),aTP_Lamp_uc(set(product_prod(A,A)),fun(fun(A,A2),fun(product_prod(A2,A2),bool)),Uu),Uua),Uub))
<=> ? [A7: A,B5: A] :
( ( Uub = aa(A2,product_prod(A2,A2),aa(A2,fun(A2,product_prod(A2,A2)),product_Pair(A2,A2),aa(A,A2,Uua,A7)),aa(A,A2,Uua,B5)) )
& pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A7),B5)),Uu)) ) ) ).
% ATP.lambda_448
tff(fact_3163_ATP_Olambda__449,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aTP_Lamp_px(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),Uu),Uua),Uub))
<=> ? [A7: A,V6: list(A)] :
( ( Uub = append(A,Uua,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A7),V6)) )
| ? [U5: list(A),Aa3: A,B5: A,Va: list(A),W3: list(A)] :
( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Aa3),B5)),Uu))
& ( Uua = append(A,U5,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Aa3),Va)) )
& ( Uub = append(A,U5,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B5),W3)) ) ) ) ) ).
% ATP.lambda_449
tff(fact_3164_ATP_Olambda__450,axiom,
! [C: $tType,D: $tType,A: $tType,B: $tType,Uu: fun(D,fun(A,fun(C,fun(B,bool)))),Uua: D,Uub: fun(A,option(product_prod(B,C)))] :
( pp(aa(fun(A,option(product_prod(B,C))),bool,aa(D,fun(fun(A,option(product_prod(B,C))),bool),aTP_Lamp_uy(fun(D,fun(A,fun(C,fun(B,bool)))),fun(D,fun(fun(A,option(product_prod(B,C))),bool)),Uu),Uua),Uub))
<=> ! [H8: A,H9: C,R9: B] :
( ( aa(A,option(product_prod(B,C)),Uub,H8) = aa(product_prod(B,C),option(product_prod(B,C)),some(product_prod(B,C)),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),R9),H9)) )
=> pp(aa(B,bool,aa(C,fun(B,bool),aa(A,fun(C,fun(B,bool)),aa(D,fun(A,fun(C,fun(B,bool))),Uu,Uua),H8),H9),R9)) ) ) ).
% ATP.lambda_450
tff(fact_3165_ATP_Olambda__451,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aTP_Lamp_pz(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),Uu),Uua),Uub))
<=> ? [Us3: list(A),Z4: A,Z10: A,Vs3: list(A)] :
( ( Uua = append(A,Us3,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z4),Vs3)) )
& pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Z4),Z10)),Uu))
& ( Uub = append(A,Us3,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z10),Vs3)) ) ) ) ).
% ATP.lambda_451
tff(fact_3166_ATP_Olambda__452,axiom,
! [A: $tType,Uu: fun(A,fun(A,A)),Uua: A,Uub: fun(nat,fun(nat,A)),Uuc: nat] : aa(nat,A,aa(fun(nat,fun(nat,A)),fun(nat,A),aa(A,fun(fun(nat,fun(nat,A)),fun(nat,A)),aTP_Lamp_un(fun(A,fun(A,A)),fun(A,fun(fun(nat,fun(nat,A)),fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups_comm_monoid_F(A,nat,Uu,Uua),aa(nat,fun(nat,A),aTP_Lamp_um(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uub),Uuc)),aa(nat,set(nat),set_ord_atMost(nat),Uuc)) ).
% ATP.lambda_452
tff(fact_3167_ATP_Olambda__453,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,Uu: fun(B,fun(C,fun(D,A))),Uua: D,Uub: B,Uuc: C] : aa(C,A,aa(B,fun(C,A),aa(D,fun(B,fun(C,A)),aTP_Lamp_al(fun(B,fun(C,fun(D,A))),fun(D,fun(B,fun(C,A))),Uu),Uua),Uub),Uuc) = aa(D,A,aa(C,fun(D,A),aa(B,fun(C,fun(D,A)),Uu,Uub),Uuc),Uua) ).
% ATP.lambda_453
tff(fact_3168_ATP_Olambda__454,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: product_prod(C,A),Uua: A,Uub: B,Uuc: set(product_prod(C,B))] : aa(set(product_prod(C,B)),set(product_prod(C,B)),aa(B,fun(set(product_prod(C,B)),set(product_prod(C,B))),aa(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aTP_Lamp_ie(product_prod(C,A),fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),Uu),Uua),Uub),Uuc) = if(set(product_prod(C,B)),aa(A,bool,aa(A,fun(A,bool),fequal(A),aa(product_prod(C,A),A,product_snd(C,A),Uu)),Uua),aa(set(product_prod(C,B)),set(product_prod(C,B)),insert(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),aa(product_prod(C,A),C,product_fst(C,A),Uu)),Uub)),Uuc),Uuc) ).
% ATP.lambda_454
tff(fact_3169_ATP_Olambda__455,axiom,
! [B: $tType,A: $tType,Uu: fun(A,B),Uua: set(A),Uub: A,Uuc: B] : aa(B,A,aa(A,fun(B,A),aa(set(A),fun(A,fun(B,A)),aTP_Lamp_gz(fun(A,B),fun(set(A),fun(A,fun(B,A))),Uu),Uua),Uub),Uuc) = if(A,aa(set(B),bool,member(B,Uuc),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),Uu),Uua)),aa(B,A,the_inv_into(A,B,Uua,Uu),Uuc),Uub) ).
% ATP.lambda_455
tff(fact_3170_ATP_Olambda__456,axiom,
! [B: $tType,A: $tType,Uu: set(A),Uua: fun(A,B),Uub: A,Uuc: B] : aa(B,A,aa(A,fun(B,A),aa(fun(A,B),fun(A,fun(B,A)),aTP_Lamp_gy(set(A),fun(fun(A,B),fun(A,fun(B,A))),Uu),Uua),Uub),Uuc) = if(A,aa(set(B),bool,member(B,Uuc),aa(set(A),set(B),aa(fun(A,B),fun(set(A),set(B)),image2(A,B),Uua),Uu)),aa(B,A,the_inv_into(A,B,Uu,Uua),Uuc),Uub) ).
% ATP.lambda_456
tff(fact_3171_ATP_Olambda__457,axiom,
! [A: $tType,B: $tType,Uu: set(A),Uua: A,Uub: B,Uuc: set(B)] : aa(set(B),set(B),aa(B,fun(set(B),set(B)),aa(A,fun(B,fun(set(B),set(B))),aTP_Lamp_ig(set(A),fun(A,fun(B,fun(set(B),set(B)))),Uu),Uua),Uub),Uuc) = if(set(B),aa(set(A),bool,member(A,Uua),Uu),aa(set(B),set(B),insert(B,Uub),Uuc),Uuc) ).
% ATP.lambda_457
tff(fact_3172_ATP_Olambda__458,axiom,
! [B: $tType,A: $tType,Uu: fun(A,B),Uua: set(A),Uub: fun(A,B),Uuc: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(set(A),fun(fun(A,B),fun(A,B)),aTP_Lamp_fm(fun(A,B),fun(set(A),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = if(B,aa(set(A),bool,member(A,Uuc),Uua),aa(A,B,Uu,Uuc),aa(A,B,Uub,Uuc)) ).
% ATP.lambda_458
tff(fact_3173_ATP_Olambda__459,axiom,
! [A: $tType,Uu: fun(A,bool),Uua: A,Uub: list(A),Uuc: list(A)] : aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),aa(A,fun(list(A),fun(list(A),product_prod(list(A),list(A)))),aTP_Lamp_qp(fun(A,bool),fun(A,fun(list(A),fun(list(A),product_prod(list(A),list(A))))),Uu),Uua),Uub),Uuc) = if(product_prod(list(A),list(A)),aa(A,bool,Uu,Uua),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uua),Uub)),Uuc),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Uub),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uua),Uuc))) ).
% ATP.lambda_459
tff(fact_3174_ATP_Olambda__460,axiom,
! [A: $tType,B: $tType,Uu: fun(B,bool),Uua: fun(B,A),Uub: fun(B,A),Uuc: B] : aa(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_fg(fun(B,bool),fun(fun(B,A),fun(fun(B,A),fun(B,A))),Uu),Uua),Uub),Uuc) = if(A,aa(B,bool,Uu,Uuc),aa(B,A,Uua,Uuc),aa(B,A,Uub,Uuc)) ).
% ATP.lambda_460
tff(fact_3175_ATP_Olambda__461,axiom,
! [A: $tType,C: $tType,B: $tType,Uu: set(product_prod(B,C)),Uua: A,Uub: B,Uuc: set(product_prod(A,C))] : aa(set(product_prod(A,C)),set(product_prod(A,C)),aa(B,fun(set(product_prod(A,C)),set(product_prod(A,C))),aa(A,fun(B,fun(set(product_prod(A,C)),set(product_prod(A,C)))),aTP_Lamp_il(set(product_prod(B,C)),fun(A,fun(B,fun(set(product_prod(A,C)),set(product_prod(A,C))))),Uu),Uua),Uub),Uuc) = finite_fold(product_prod(B,C),set(product_prod(A,C)),aa(fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C))))),fun(product_prod(B,C),fun(set(product_prod(A,C)),set(product_prod(A,C)))),product_case_prod(B,C,fun(set(product_prod(A,C)),set(product_prod(A,C)))),aa(B,fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C))))),aTP_Lamp_ik(A,fun(B,fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C)))))),Uua),Uub)),Uuc,Uu) ).
% ATP.lambda_461
tff(fact_3176_ATP_Olambda__462,axiom,
! [C: $tType,B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: C,Uub: A,Uuc: set(product_prod(C,B))] : aa(set(product_prod(C,B)),set(product_prod(C,B)),aa(A,fun(set(product_prod(C,B)),set(product_prod(C,B))),aa(C,fun(A,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aTP_Lamp_ia(set(product_prod(A,B)),fun(C,fun(A,fun(set(product_prod(C,B)),set(product_prod(C,B))))),Uu),Uua),Uub),Uuc) = finite_fold(product_prod(A,B),set(product_prod(C,B)),aa(fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),fun(product_prod(A,B),fun(set(product_prod(C,B)),set(product_prod(C,B)))),product_case_prod(A,B,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aa(A,fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),aTP_Lamp_hz(C,fun(A,fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B)))))),Uua),Uub)),Uuc,Uu) ).
% ATP.lambda_462
tff(fact_3177_ATP_Olambda__463,axiom,
! [C: $tType,B: $tType,A: $tType,Uu: fun(B,fun(list(B),fun(C,C))),Uua: fun(A,B),Uub: A,Uuc: list(A)] : aa(list(A),fun(C,C),aa(A,fun(list(A),fun(C,C)),aa(fun(A,B),fun(A,fun(list(A),fun(C,C))),aTP_Lamp_qd(fun(B,fun(list(B),fun(C,C))),fun(fun(A,B),fun(A,fun(list(A),fun(C,C)))),Uu),Uua),Uub),Uuc) = aa(list(B),fun(C,C),aa(B,fun(list(B),fun(C,C)),Uu,aa(A,B,Uua,Uub)),aa(list(A),list(B),map(A,B,Uua),Uuc)) ).
% ATP.lambda_463
tff(fact_3178_ATP_Olambda__464,axiom,
! [B: $tType,A: $tType,C: $tType,D: $tType,Uu: fun(B,fun(C,A)),Uua: fun(D,B),Uub: fun(D,C),Uuc: D] : aa(D,A,aa(fun(D,C),fun(D,A),aa(fun(D,B),fun(fun(D,C),fun(D,A)),aTP_Lamp_ow(fun(B,fun(C,A)),fun(fun(D,B),fun(fun(D,C),fun(D,A))),Uu),Uua),Uub),Uuc) = aa(C,A,aa(B,fun(C,A),Uu,aa(D,B,Uua,Uuc)),aa(D,C,Uub,Uuc)) ).
% ATP.lambda_464
tff(fact_3179_ATP_Olambda__465,axiom,
! [A: $tType,C: $tType,B: $tType,Uu: fun(A,fun(C,bool)),Uua: fun(B,C),Uub: A,Uuc: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(B,C),fun(A,fun(B,bool)),aTP_Lamp_ar(fun(A,fun(C,bool)),fun(fun(B,C),fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
<=> pp(aa(C,bool,aa(A,fun(C,bool),Uu,Uub),aa(B,C,Uua,Uuc))) ) ).
% ATP.lambda_465
tff(fact_3180_ATP_Olambda__466,axiom,
! [C: $tType,A: $tType,B: $tType,E: $tType,D: $tType,Uu: fun(B,fun(C,fun(D,fun(E,set(A))))),Uua: product_prod(D,E),Uub: B,Uuc: C] : aa(C,set(A),aa(B,fun(C,set(A)),aa(product_prod(D,E),fun(B,fun(C,set(A))),aTP_Lamp_ek(fun(B,fun(C,fun(D,fun(E,set(A))))),fun(product_prod(D,E),fun(B,fun(C,set(A)))),Uu),Uua),Uub),Uuc) = aa(product_prod(D,E),set(A),aa(fun(D,fun(E,set(A))),fun(product_prod(D,E),set(A)),product_case_prod(D,E,set(A)),aa(C,fun(D,fun(E,set(A))),aa(B,fun(C,fun(D,fun(E,set(A)))),Uu,Uub),Uuc)),Uua) ).
% ATP.lambda_466
tff(fact_3181_ATP_Olambda__467,axiom,
! [A: $tType,Uu: fun(A,fun(A,bool)),Uua: set(A),Uub: A,Uuc: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aa(set(A),fun(A,fun(A,bool)),aTP_Lamp_ez(fun(A,fun(A,bool)),fun(set(A),fun(A,fun(A,bool))),Uu),Uua),Uub),Uuc))
<=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uuc)),product_Sigma(A,A,Uua,aTP_Lamp_co(set(A),fun(A,set(A)),Uua))))
& pp(aa(A,bool,aa(A,fun(A,bool),Uu,Uub),Uuc)) ) ) ).
% ATP.lambda_467
tff(fact_3182_ATP_Olambda__468,axiom,
! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aa(nat,fun(nat,fun(nat,product_prod(nat,nat))),aTP_Lamp_rb(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua),Uub),Uuc) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uu),Uub)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),Uuc))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uu),Uuc)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),Uub))) ).
% ATP.lambda_468
tff(fact_3183_ATP_Olambda__469,axiom,
! [B: $tType,A: $tType,Uu: set(old_node(A,B)),Uua: set(old_node(A,B)),Uub: set(old_node(A,B)),Uuc: set(old_node(A,B))] : aa(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))),aTP_Lamp_wi(set(old_node(A,B)),fun(set(old_node(A,B)),fun(set(old_node(A,B)),fun(set(old_node(A,B)),set(product_prod(set(old_node(A,B)),set(old_node(A,B))))))),Uu),Uua),Uub),Uuc) = aa(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),set(product_prod(set(old_node(A,B)),set(old_node(A,B)))),insert(product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B))),aa(set(old_node(A,B)),fun(set(old_node(A,B)),product_prod(set(old_node(A,B)),set(old_node(A,B)))),product_Pair(set(old_node(A,B)),set(old_node(A,B))),old_Scons(A,B,Uu,Uub)),old_Scons(A,B,Uua,Uuc))),bot_bot(set(product_prod(set(old_node(A,B)),set(old_node(A,B)))))) ).
% ATP.lambda_469
tff(fact_3184_ATP_Olambda__470,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: nat,Uua: A,Uub: array(A),Uuc: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(A,heap_ext(product_unit)),aa(array(A),fun(heap_ext(product_unit),product_prod(A,heap_ext(product_unit))),aa(A,fun(array(A),fun(heap_ext(product_unit),product_prod(A,heap_ext(product_unit)))),aTP_Lamp_ki(nat,fun(A,fun(array(A),fun(heap_ext(product_unit),product_prod(A,heap_ext(product_unit))))),Uu),Uua),Uub),Uuc) = aa(heap_ext(product_unit),product_prod(A,heap_ext(product_unit)),aa(A,fun(heap_ext(product_unit),product_prod(A,heap_ext(product_unit))),product_Pair(A,heap_ext(product_unit)),aa(nat,A,nth3(A,aa(array(A),list(A),get(A,Uuc),Uub)),Uu)),update(A,Uub,Uu,Uua,Uuc)) ) ).
% ATP.lambda_470
tff(fact_3185_ATP_Olambda__471,axiom,
! [A: $tType,Uu: fun(A,nat),Uua: set(product_prod(A,A)),Uub: A,Uuc: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aa(set(product_prod(A,A)),fun(A,fun(A,bool)),aTP_Lamp_ke(fun(A,nat),fun(set(product_prod(A,A)),fun(A,fun(A,bool))),Uu),Uua),Uub),Uuc))
<=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,Uu,Uub)),aa(A,nat,Uu,Uuc)))
| ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,Uu,Uub)),aa(A,nat,Uu,Uuc)))
& pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uuc)),Uua)) ) ) ) ).
% ATP.lambda_471
tff(fact_3186_ATP_Olambda__472,axiom,
! [A: $tType,B: $tType,Uu: set(product_prod(B,B)),Uua: fun(A,B),Uub: A,Uuc: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,B),fun(A,fun(A,bool)),aTP_Lamp_bw(set(product_prod(B,B)),fun(fun(A,B),fun(A,fun(A,bool))),Uu),Uua),Uub),Uuc))
<=> pp(aa(set(product_prod(B,B)),bool,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,Uua,Uub)),aa(A,B,Uua,Uuc))),Uu)) ) ).
% ATP.lambda_472
tff(fact_3187_ATP_Olambda__473,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat,Uub: list(A),Uuc: list(A)] :
( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(nat,fun(list(A),fun(list(A),bool)),aTP_Lamp_pv(set(product_prod(A,A)),fun(nat,fun(list(A),fun(list(A),bool))),Uu),Uua),Uub),Uuc))
<=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uua )
& ( aa(list(A),nat,size_size(list(A)),Uuc) = Uua )
& ? [Xys2: list(A),X5: A,Y4: A,Xs5: list(A),Ys5: list(A)] :
( ( Uub = append(A,Xys2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X5),Xs5)) )
& ( Uuc = append(A,Xys2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys5)) )
& pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),Y4)),Uu)) ) ) ) ).
% ATP.lambda_473
tff(fact_3188_ATP_Olambda__474,axiom,
! [B: $tType,A: $tType,Uu: fun(B,A),Uua: set(B),Uub: fun(A,bool),Uuc: A] :
( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aa(set(B),fun(fun(A,bool),fun(A,bool)),aTP_Lamp_ah(fun(B,A),fun(set(B),fun(fun(A,bool),fun(A,bool))),Uu),Uua),Uub),Uuc))
<=> ( pp(aa(set(A),bool,member(A,Uuc),aa(set(B),set(A),aa(fun(B,A),fun(set(B),set(A)),image2(B,A),Uu),Uua)))
& pp(aa(A,bool,Uub,Uuc)) ) ) ).
% ATP.lambda_474
tff(fact_3189_ATP_Olambda__475,axiom,
! [A: $tType,B: $tType,Uu: set(A),Uua: set(product_prod(B,B)),Uub: B,Uuc: A] :
( pp(aa(A,bool,aa(B,fun(A,bool),aa(set(product_prod(B,B)),fun(B,fun(A,bool)),aTP_Lamp_qj(set(A),fun(set(product_prod(B,B)),fun(B,fun(A,bool))),Uu),Uua),Uub),Uuc))
<=> ( pp(aa(set(A),bool,member(A,Uuc),Uu))
& ( aa(A,B,bNF_Greatest_toCard(A,B,Uu,Uua),Uuc) = Uub ) ) ) ).
% ATP.lambda_475
tff(fact_3190_ATP_Olambda__476,axiom,
! [B: $tType,A: $tType,Uu: set(A),Uua: fun(A,B),Uub: A,Uuc: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,B),fun(A,fun(A,bool)),aTP_Lamp_hi(set(A),fun(fun(A,B),fun(A,fun(A,bool))),Uu),Uua),Uub),Uuc))
<=> ( pp(aa(set(A),bool,member(A,Uuc),Uu))
& ( aa(A,B,Uua,Uuc) = aa(A,B,Uua,Uub) ) ) ) ).
% ATP.lambda_476
tff(fact_3191_ATP_Olambda__477,axiom,
! [B: $tType,C: $tType,Uu: set(B),Uua: fun(B,C),Uub: C,Uuc: B] :
( pp(aa(B,bool,aa(C,fun(B,bool),aa(fun(B,C),fun(C,fun(B,bool)),aTP_Lamp_lp(set(B),fun(fun(B,C),fun(C,fun(B,bool))),Uu),Uua),Uub),Uuc))
<=> ( pp(aa(set(B),bool,member(B,Uuc),Uu))
& ( aa(B,C,Uua,Uuc) = Uub ) ) ) ).
% ATP.lambda_477
tff(fact_3192_ATP_Olambda__478,axiom,
! [A: $tType,B: $tType,Uu: set(A),Uua: fun(A,B),Uub: B,Uuc: A] :
( pp(aa(A,bool,aa(B,fun(A,bool),aa(fun(A,B),fun(B,fun(A,bool)),aTP_Lamp_mq(set(A),fun(fun(A,B),fun(B,fun(A,bool))),Uu),Uua),Uub),Uuc))
<=> ( pp(aa(set(A),bool,member(A,Uuc),Uu))
& ( aa(A,B,Uua,Uuc) = Uub ) ) ) ).
% ATP.lambda_478
tff(fact_3193_ATP_Olambda__479,axiom,
! [Uu: int,Uua: int,Uub: int,Uuc: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),aa(int,fun(int,fun(int,bool)),aTP_Lamp_ri(int,fun(int,fun(int,fun(int,bool))),Uu),Uua),Uub),Uuc))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uub))) ) ).
% ATP.lambda_479
tff(fact_3194_ATP_Olambda__480,axiom,
! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),aa(nat,fun(nat,fun(nat,bool)),aTP_Lamp_qq(nat,fun(nat,fun(nat,fun(nat,bool))),Uu),Uua),Uub),Uuc))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uuc)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua))) ) ).
% ATP.lambda_480
tff(fact_3195_ATP_Olambda__481,axiom,
! [Uu: int,Uua: int,Uub: int,Uuc: int] :
( pp(aa(int,bool,aa(int,fun(int,bool),aa(int,fun(int,fun(int,bool)),aTP_Lamp_rg(int,fun(int,fun(int,fun(int,bool))),Uu),Uua),Uub),Uuc))
<=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uub))) ) ).
% ATP.lambda_481
tff(fact_3196_ATP_Olambda__482,axiom,
! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),aa(nat,fun(nat,fun(nat,bool)),aTP_Lamp_qs(nat,fun(nat,fun(nat,fun(nat,bool))),Uu),Uua),Uub),Uuc))
<=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uuc)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua))) ) ).
% ATP.lambda_482
tff(fact_3197_ATP_Olambda__483,axiom,
! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aa(nat,fun(nat,fun(nat,product_prod(nat,nat))),aTP_Lamp_qy(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua),Uub),Uuc) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uub)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uuc)) ).
% ATP.lambda_483
tff(fact_3198_ATP_Olambda__484,axiom,
! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aa(nat,fun(nat,fun(nat,product_prod(nat,nat))),aTP_Lamp_qw(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua),Uub),Uuc) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uuc)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)) ).
% ATP.lambda_484
tff(fact_3199_ATP_Olambda__485,axiom,
! [A: $tType,B: $tType,Uu: A,Uua: list(A),Uub: B,Uuc: list(B)] : aa(list(B),list(product_prod(A,B)),aa(B,fun(list(B),list(product_prod(A,B))),aa(list(A),fun(B,fun(list(B),list(product_prod(A,B)))),aTP_Lamp_pt(A,fun(list(A),fun(B,fun(list(B),list(product_prod(A,B))))),Uu),Uua),Uub),Uuc) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uu),Uub)),zip(A,B,Uua,Uuc)) ).
% ATP.lambda_485
tff(fact_3200_ATP_Olambda__486,axiom,
! [A: $tType,B: $tType,Uu: B,Uua: list(B),Uub: A,Uuc: list(A)] : aa(list(A),list(product_prod(A,B)),aa(A,fun(list(A),list(product_prod(A,B))),aa(list(B),fun(A,fun(list(A),list(product_prod(A,B)))),aTP_Lamp_pu(B,fun(list(B),fun(A,fun(list(A),list(product_prod(A,B))))),Uu),Uua),Uub),Uuc) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uub),Uu)),zip(A,B,Uuc,Uua)) ).
% ATP.lambda_486
tff(fact_3201_ATP_Olambda__487,axiom,
! [A: $tType,B: $tType,Uu: filter(A),Uua: filter(B),Uub: fun(A,bool),Uuc: fun(B,bool)] :
( pp(aa(fun(B,bool),bool,aa(fun(A,bool),fun(fun(B,bool),bool),aa(filter(B),fun(fun(A,bool),fun(fun(B,bool),bool)),aTP_Lamp_je(filter(A),fun(filter(B),fun(fun(A,bool),fun(fun(B,bool),bool))),Uu),Uua),Uub),Uuc))
<=> ( eventually(A,Uub,Uu)
& eventually(B,Uuc,Uua) ) ) ).
% ATP.lambda_487
tff(fact_3202_ATP_Olambda__488,axiom,
! [A: $tType,B: $tType,Uu: A,Uua: B,Uub: A,Uuc: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aa(B,fun(A,fun(B,bool)),aTP_Lamp_cb(A,fun(B,fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
<=> ( ( Uu = Uub )
& ( Uua = Uuc ) ) ) ).
% ATP.lambda_488
tff(fact_3203_ATP_Olambda__489,axiom,
! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
( pp(aa(nat,bool,aa(nat,fun(nat,bool),aa(nat,fun(nat,fun(nat,bool)),aTP_Lamp_re(nat,fun(nat,fun(nat,fun(nat,bool))),Uu),Uua),Uub),Uuc))
<=> ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua) ) ) ).
% ATP.lambda_489
tff(fact_3204_ATP_Olambda__490,axiom,
! [A: $tType,B: $tType,Uu: fun(B,A),Uua: set(B),Uub: fun(A,bool),Uuc: B] :
( pp(aa(B,bool,aa(fun(A,bool),fun(B,bool),aa(set(B),fun(fun(A,bool),fun(B,bool)),aTP_Lamp_ai(fun(B,A),fun(set(B),fun(fun(A,bool),fun(B,bool))),Uu),Uua),Uub),Uuc))
<=> ( pp(aa(set(B),bool,member(B,Uuc),Uua))
& pp(aa(A,bool,Uub,aa(B,A,Uu,Uuc))) ) ) ).
% ATP.lambda_490
tff(fact_3205_ATP_Olambda__491,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: fun(C,set(product_prod(A,B))),Uua: C,Uub: A,Uuc: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aa(C,fun(A,fun(B,bool)),aTP_Lamp_er(fun(C,set(product_prod(A,B))),fun(C,fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
<=> pp(aa(set(product_prod(A,B)),bool,member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uub),Uuc)),aa(C,set(product_prod(A,B)),Uu,Uua))) ) ).
% ATP.lambda_491
tff(fact_3206_ATP_Olambda__492,axiom,
! [A: $tType,B: $tType,Uu: fun(A,bool),Uua: fun(A,fun(B,bool)),Uub: A,Uuc: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aTP_Lamp_cd(fun(A,bool),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
<=> ( pp(aa(A,bool,Uu,Uub))
& pp(aa(B,bool,aa(A,fun(B,bool),Uua,Uub),Uuc)) ) ) ).
% ATP.lambda_492
tff(fact_3207_ATP_Olambda__493,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType,Uu: fun(C,set(A)),Uua: fun(D,set(B)),Uub: C,Uuc: D] : aa(D,set(product_prod(A,B)),aa(C,fun(D,set(product_prod(A,B))),aa(fun(D,set(B)),fun(C,fun(D,set(product_prod(A,B)))),aTP_Lamp_di(fun(C,set(A)),fun(fun(D,set(B)),fun(C,fun(D,set(product_prod(A,B))))),Uu),Uua),Uub),Uuc) = product_Sigma(A,B,aa(C,set(A),Uu,Uub),aa(D,fun(A,set(B)),aTP_Lamp_dh(fun(D,set(B)),fun(D,fun(A,set(B))),Uua),Uuc)) ).
% ATP.lambda_493
tff(fact_3208_ATP_Olambda__494,axiom,
! [A: $tType,Uu: fun(A,bool),Uua: fun(A,A),Uub: A,Uuc: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,A),fun(A,fun(A,bool)),aTP_Lamp_jz(fun(A,bool),fun(fun(A,A),fun(A,fun(A,bool))),Uu),Uua),Uub),Uuc))
<=> ( pp(aa(A,bool,Uu,Uuc))
& ( Uub = aa(A,A,Uua,Uuc) ) ) ) ).
% ATP.lambda_494
tff(fact_3209_ATP_Olambda__495,axiom,
! [A: $tType,B: $tType,Uu: fun(A,nat),Uua: fun(B,nat),Uub: A,Uuc: B] : aa(B,nat,aa(A,fun(B,nat),aa(fun(B,nat),fun(A,fun(B,nat)),aTP_Lamp_tg(fun(A,nat),fun(fun(B,nat),fun(A,fun(B,nat))),Uu),Uua),Uub),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(A,nat,Uu,Uub)),aa(B,nat,Uua,Uuc)) ).
% ATP.lambda_495
tff(fact_3210_ATP_Olambda__496,axiom,
! [A: $tType,B: $tType,C: $tType] :
( semiring_0(B)
=> ! [Uu: fun(A,B),Uua: fun(C,B),Uub: A,Uuc: C] : aa(C,B,aa(A,fun(C,B),aa(fun(C,B),fun(A,fun(C,B)),aTP_Lamp_mk(fun(A,B),fun(fun(C,B),fun(A,fun(C,B))),Uu),Uua),Uub),Uuc) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(C,B,Uua,Uuc)) ) ).
% ATP.lambda_496
tff(fact_3211_ATP_Olambda__497,axiom,
! [B: $tType,A: $tType,C: $tType,Uu: fun(B,set(A)),Uua: fun(C,set(A)),Uub: B,Uuc: C] : aa(C,set(A),aa(B,fun(C,set(A)),aa(fun(C,set(A)),fun(B,fun(C,set(A))),aTP_Lamp_gm(fun(B,set(A)),fun(fun(C,set(A)),fun(B,fun(C,set(A)))),Uu),Uua),Uub),Uuc) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(B,set(A),Uu,Uub)),aa(C,set(A),Uua,Uuc)) ).
% ATP.lambda_497
tff(fact_3212_ATP_Olambda__498,axiom,
! [B: $tType,A: $tType,C: $tType] :
( comple592849572758109894attice(A)
=> ! [Uu: fun(B,A),Uua: fun(C,A),Uub: B,Uuc: C] : aa(C,A,aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_gd(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,Uu,Uub)),aa(C,A,Uua,Uuc)) ) ).
% ATP.lambda_498
tff(fact_3213_ATP_Olambda__499,axiom,
! [B: $tType,A: $tType,C: $tType,Uu: fun(B,set(A)),Uua: fun(C,set(A)),Uub: B,Uuc: C] : aa(C,set(A),aa(B,fun(C,set(A)),aa(fun(C,set(A)),fun(B,fun(C,set(A))),aTP_Lamp_gh(fun(B,set(A)),fun(fun(C,set(A)),fun(B,fun(C,set(A)))),Uu),Uua),Uub),Uuc) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(B,set(A),Uu,Uub)),aa(C,set(A),Uua,Uuc)) ).
% ATP.lambda_499
tff(fact_3214_ATP_Olambda__500,axiom,
! [B: $tType,A: $tType,C: $tType] :
( comple592849572758109894attice(A)
=> ! [Uu: fun(B,A),Uua: fun(C,A),Uub: B,Uuc: C] : aa(C,A,aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_fy(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,Uu,Uub)),aa(C,A,Uua,Uuc)) ) ).
% ATP.lambda_500
tff(fact_3215_ATP_Olambda__501,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,Uu: fun(A,filter(C)),Uua: fun(B,filter(D)),Uub: A,Uuc: B] : aa(B,filter(product_prod(C,D)),aa(A,fun(B,filter(product_prod(C,D))),aa(fun(B,filter(D)),fun(A,fun(B,filter(product_prod(C,D)))),aTP_Lamp_jk(fun(A,filter(C)),fun(fun(B,filter(D)),fun(A,fun(B,filter(product_prod(C,D))))),Uu),Uua),Uub),Uuc) = prod_filter(C,D,aa(A,filter(C),Uu,Uub),aa(B,filter(D),Uua,Uuc)) ).
% ATP.lambda_501
tff(fact_3216_ATP_Olambda__502,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,Uu: fun(C,A),Uua: fun(D,B),Uub: C,Uuc: D] : aa(D,product_prod(A,B),aa(C,fun(D,product_prod(A,B)),aa(fun(D,B),fun(C,fun(D,product_prod(A,B))),aTP_Lamp_br(fun(C,A),fun(fun(D,B),fun(C,fun(D,product_prod(A,B)))),Uu),Uua),Uub),Uuc) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,Uu,Uub)),aa(D,B,Uua,Uuc)) ).
% ATP.lambda_502
tff(fact_3217_ATP_Olambda__503,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,Uu: fun(A,C),Uua: fun(B,D),Uub: A,Uuc: B] : aa(B,product_prod(C,D),aa(A,fun(B,product_prod(C,D)),aa(fun(B,D),fun(A,fun(B,product_prod(C,D))),aTP_Lamp_av(fun(A,C),fun(fun(B,D),fun(A,fun(B,product_prod(C,D)))),Uu),Uua),Uub),Uuc) = aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),aa(A,C,Uu,Uub)),aa(B,D,Uua,Uuc)) ).
% ATP.lambda_503
tff(fact_3218_ATP_Olambda__504,axiom,
! [A: $tType,B: $tType,Uu: fun(A,bool),Uua: fun(B,bool),Uub: A,Uuc: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(B,bool),fun(A,fun(B,bool)),aTP_Lamp_by(fun(A,bool),fun(fun(B,bool),fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
<=> ( pp(aa(A,bool,Uu,Uub))
& pp(aa(B,bool,Uua,Uuc)) ) ) ).
% ATP.lambda_504
tff(fact_3219_ATP_Olambda__505,axiom,
! [A: $tType,B: $tType,Uu: fun(A,fun(A,bool)),Uua: fun(B,A),Uub: B,Uuc: B] :
( pp(aa(B,bool,aa(B,fun(B,bool),aa(fun(B,A),fun(B,fun(B,bool)),aTP_Lamp_mc(fun(A,fun(A,bool)),fun(fun(B,A),fun(B,fun(B,bool))),Uu),Uua),Uub),Uuc))
<=> ( ( aa(B,A,Uua,Uub) != aa(B,A,Uua,Uuc) )
=> pp(aa(A,bool,aa(A,fun(A,bool),Uu,aa(B,A,Uua,Uub)),aa(B,A,Uua,Uuc))) ) ) ).
% ATP.lambda_505
tff(fact_3220_ATP_Olambda__506,axiom,
! [A: $tType,C: $tType,B: $tType] :
( semiring_1(C)
=> ! [Uu: set(A),Uua: fun(A,B),Uub: fun(B,C),Uuc: B] : aa(B,C,aa(fun(B,C),fun(B,C),aa(fun(A,B),fun(fun(B,C),fun(B,C)),aTP_Lamp_mr(set(A),fun(fun(A,B),fun(fun(B,C),fun(B,C))),Uu),Uua),Uub),Uuc) = aa(C,C,aa(C,fun(C,C),times_times(C),aa(nat,C,semiring_1_of_nat(C),aa(set(A),nat,finite_card(A),aa(fun(A,bool),set(A),collect(A),aa(B,fun(A,bool),aa(fun(A,B),fun(B,fun(A,bool)),aTP_Lamp_mq(set(A),fun(fun(A,B),fun(B,fun(A,bool))),Uu),Uua),Uuc))))),aa(B,C,Uub,Uuc)) ) ).
% ATP.lambda_506
tff(fact_3221_ATP_Olambda__507,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: nat,Uua: fun(A,A),Uub: array(A),Uuc: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit)),aa(array(A),fun(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit))),aa(fun(A,A),fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit)))),aTP_Lamp_kj(nat,fun(fun(A,A),fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit))))),Uu),Uua),Uub),Uuc) = aa(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit)),aa(array(A),fun(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit))),product_Pair(array(A),heap_ext(product_unit)),Uub),update(A,Uub,Uu,aa(A,A,Uua,aa(nat,A,nth3(A,aa(array(A),list(A),get(A,Uuc),Uub)),Uu)),Uuc)) ) ).
% ATP.lambda_507
tff(fact_3222_ATP_Olambda__508,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: nat,Uua: A,Uub: array(A),Uuc: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit)),aa(array(A),fun(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit))),aa(A,fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit)))),aTP_Lamp_kg(nat,fun(A,fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit))))),Uu),Uua),Uub),Uuc) = aa(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit)),aa(array(A),fun(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit))),product_Pair(array(A),heap_ext(product_unit)),Uub),update(A,Uub,Uu,Uua,Uuc)) ) ).
% ATP.lambda_508
tff(fact_3223_ATP_Olambda__509,axiom,
! [A: $tType,B: $tType,Uu: bool,Uua: fun(A,fun(B,bool)),Uub: A,Uuc: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aTP_Lamp_ac(bool,fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
<=> ( pp(Uu)
& pp(aa(B,bool,aa(A,fun(B,bool),Uua,Uub),Uuc)) ) ) ).
% ATP.lambda_509
tff(fact_3224_ATP_Olambda__510,axiom,
! [B: $tType,A: $tType,C: $tType,Uu: fun(C,A),Uua: fun(C,B),Uub: set(C),Uuc: A] : aa(A,set(B),aa(set(C),fun(A,set(B)),aa(fun(C,B),fun(set(C),fun(A,set(B))),aTP_Lamp_gx(fun(C,A),fun(fun(C,B),fun(set(C),fun(A,set(B)))),Uu),Uua),Uub),Uuc) = aa(set(C),set(B),aa(fun(C,B),fun(set(C),set(B)),image2(C,B),Uua),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),aa(set(A),set(C),vimage(C,A,Uu),aa(set(A),set(A),insert(A,Uuc),bot_bot(set(A))))),Uub)) ).
% ATP.lambda_510
tff(fact_3225_ATP_Olambda__511,axiom,
! [A: $tType,B: $tType,C: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: set(B),Uua: fun(B,A),Uub: fun(B,C),Uuc: C] : aa(C,A,aa(fun(B,C),fun(C,A),aa(fun(B,A),fun(fun(B,C),fun(C,A)),aTP_Lamp_mz(set(B),fun(fun(B,A),fun(fun(B,C),fun(C,A))),Uu),Uua),Uub),Uuc) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),Uua),aa(fun(B,bool),set(B),collect(B),aa(C,fun(B,bool),aa(fun(B,C),fun(C,fun(B,bool)),aTP_Lamp_lp(set(B),fun(fun(B,C),fun(C,fun(B,bool))),Uu),Uub),Uuc))) ) ).
% ATP.lambda_511
tff(fact_3226_ATP_Olambda__512,axiom,
! [A: $tType,B: $tType,C: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: set(B),Uua: fun(B,C),Uub: fun(B,A),Uuc: C] : aa(C,A,aa(fun(B,A),fun(C,A),aa(fun(B,C),fun(fun(B,A),fun(C,A)),aTP_Lamp_nh(set(B),fun(fun(B,C),fun(fun(B,A),fun(C,A))),Uu),Uua),Uub),Uuc) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),Uub),aa(fun(B,bool),set(B),collect(B),aa(C,fun(B,bool),aa(fun(B,C),fun(C,fun(B,bool)),aTP_Lamp_lp(set(B),fun(fun(B,C),fun(C,fun(B,bool))),Uu),Uua),Uuc))) ) ).
% ATP.lambda_512
tff(fact_3227_ATP_Olambda__513,axiom,
! [A: $tType,B: $tType,C: $tType] :
( comm_monoid_add(A)
=> ! [Uu: set(B),Uua: fun(B,A),Uub: fun(B,C),Uuc: C] : aa(C,A,aa(fun(B,C),fun(C,A),aa(fun(B,A),fun(fun(B,C),fun(C,A)),aTP_Lamp_ls(set(B),fun(fun(B,A),fun(fun(B,C),fun(C,A))),Uu),Uua),Uub),Uuc) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),Uua),aa(fun(B,bool),set(B),collect(B),aa(C,fun(B,bool),aa(fun(B,C),fun(C,fun(B,bool)),aTP_Lamp_lp(set(B),fun(fun(B,C),fun(C,fun(B,bool))),Uu),Uub),Uuc))) ) ).
% ATP.lambda_513
tff(fact_3228_ATP_Olambda__514,axiom,
! [A: $tType,B: $tType,C: $tType] :
( comm_monoid_add(A)
=> ! [Uu: set(B),Uua: fun(B,C),Uub: fun(B,A),Uuc: C] : aa(C,A,aa(fun(B,A),fun(C,A),aa(fun(B,C),fun(fun(B,A),fun(C,A)),aTP_Lamp_lq(set(B),fun(fun(B,C),fun(fun(B,A),fun(C,A))),Uu),Uua),Uub),Uuc) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),Uub),aa(fun(B,bool),set(B),collect(B),aa(C,fun(B,bool),aa(fun(B,C),fun(C,fun(B,bool)),aTP_Lamp_lp(set(B),fun(fun(B,C),fun(C,fun(B,bool))),Uu),Uua),Uuc))) ) ).
% ATP.lambda_514
tff(fact_3229_ATP_Olambda__515,axiom,
! [D: $tType,A: $tType,C: $tType,B: $tType,Uu: fun(C,D),Uua: fun(A,fun(B,C)),Uub: A,Uuc: B] : aa(B,D,aa(A,fun(B,D),aa(fun(A,fun(B,C)),fun(A,fun(B,D)),aTP_Lamp_am(fun(C,D),fun(fun(A,fun(B,C)),fun(A,fun(B,D))),Uu),Uua),Uub),Uuc) = aa(C,D,Uu,aa(B,C,aa(A,fun(B,C),Uua,Uub),Uuc)) ).
% ATP.lambda_515
tff(fact_3230_ATP_Olambda__516,axiom,
! [A: $tType,D: $tType,B: $tType,C: $tType,Uu: fun(product_prod(B,C),A),Uua: fun(D,B),Uub: D,Uuc: C] : aa(C,A,aa(D,fun(C,A),aa(fun(D,B),fun(D,fun(C,A)),aTP_Lamp_ox(fun(product_prod(B,C),A),fun(fun(D,B),fun(D,fun(C,A))),Uu),Uua),Uub),Uuc) = aa(product_prod(B,C),A,Uu,aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(D,B,Uua,Uub)),Uuc)) ).
% ATP.lambda_516
tff(fact_3231_ATP_Olambda__517,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Uu: fun(product_prod(B,C),A),Uua: fun(D,C),Uub: B,Uuc: D] : aa(D,A,aa(B,fun(D,A),aa(fun(D,C),fun(B,fun(D,A)),aTP_Lamp_oy(fun(product_prod(B,C),A),fun(fun(D,C),fun(B,fun(D,A))),Uu),Uua),Uub),Uuc) = aa(product_prod(B,C),A,Uu,aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),Uub),aa(D,C,Uua,Uuc))) ).
% ATP.lambda_517
tff(fact_3232_ATP_Olambda__518,axiom,
! [C: $tType,A: $tType,B: $tType,Uu: fun(C,fun(B,heap_Heap(A))),Uua: fun(C,C),Uub: fun(fun(B,heap_Heap(A)),C),Uuc: fun(B,heap_Heap(A))] : aa(fun(B,heap_Heap(A)),fun(B,heap_Heap(A)),aa(fun(fun(B,heap_Heap(A)),C),fun(fun(B,heap_Heap(A)),fun(B,heap_Heap(A))),aa(fun(C,C),fun(fun(fun(B,heap_Heap(A)),C),fun(fun(B,heap_Heap(A)),fun(B,heap_Heap(A)))),aTP_Lamp_vc(fun(C,fun(B,heap_Heap(A))),fun(fun(C,C),fun(fun(fun(B,heap_Heap(A)),C),fun(fun(B,heap_Heap(A)),fun(B,heap_Heap(A))))),Uu),Uua),Uub),Uuc) = aa(C,fun(B,heap_Heap(A)),Uu,aa(C,C,Uua,aa(fun(B,heap_Heap(A)),C,Uub,Uuc))) ).
% ATP.lambda_518
tff(fact_3233_ATP_Olambda__519,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,Uu: fun(A,fun(C,bool)),Uua: fun(B,fun(D,bool)),Uub: A,Uuc: B] : aa(B,fun(product_prod(C,D),bool),aa(A,fun(B,fun(product_prod(C,D),bool)),aa(fun(B,fun(D,bool)),fun(A,fun(B,fun(product_prod(C,D),bool))),aTP_Lamp_qc(fun(A,fun(C,bool)),fun(fun(B,fun(D,bool)),fun(A,fun(B,fun(product_prod(C,D),bool)))),Uu),Uua),Uub),Uuc) = aa(fun(C,fun(D,bool)),fun(product_prod(C,D),bool),product_case_prod(C,D,bool),aa(B,fun(C,fun(D,bool)),aa(A,fun(B,fun(C,fun(D,bool))),aa(fun(B,fun(D,bool)),fun(A,fun(B,fun(C,fun(D,bool)))),aTP_Lamp_qb(fun(A,fun(C,bool)),fun(fun(B,fun(D,bool)),fun(A,fun(B,fun(C,fun(D,bool))))),Uu),Uua),Uub),Uuc)) ).
% ATP.lambda_519
tff(fact_3234_ATP_Olambda__520,axiom,
! [A: $tType,B: $tType,Uu: set(product_prod(A,A)),Uua: set(product_prod(B,B)),Uub: A,Uuc: B] : aa(B,fun(product_prod(A,B),bool),aa(A,fun(B,fun(product_prod(A,B),bool)),aa(set(product_prod(B,B)),fun(A,fun(B,fun(product_prod(A,B),bool))),aTP_Lamp_ic(set(product_prod(A,A)),fun(set(product_prod(B,B)),fun(A,fun(B,fun(product_prod(A,B),bool)))),Uu),Uua),Uub),Uuc) = aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aa(B,fun(A,fun(B,bool)),aa(A,fun(B,fun(A,fun(B,bool))),aa(set(product_prod(B,B)),fun(A,fun(B,fun(A,fun(B,bool)))),aTP_Lamp_ib(set(product_prod(A,A)),fun(set(product_prod(B,B)),fun(A,fun(B,fun(A,fun(B,bool))))),Uu),Uua),Uub),Uuc)) ).
% ATP.lambda_520
tff(fact_3235_ATP_Olambda__521,axiom,
! [A: $tType,B: $tType,Uu: fun(A,bool),Uua: fun(A,set(product_prod(B,B))),Uub: A,Uuc: B] : aa(B,fun(product_prod(A,B),bool),aa(A,fun(B,fun(product_prod(A,B),bool)),aa(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(product_prod(A,B),bool))),aTP_Lamp_hx(fun(A,bool),fun(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(product_prod(A,B),bool)))),Uu),Uua),Uub),Uuc) = aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_case_prod(A,B,bool),aa(B,fun(A,fun(B,bool)),aa(A,fun(B,fun(A,fun(B,bool))),aa(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(A,fun(B,bool)))),aTP_Lamp_hw(fun(A,bool),fun(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(A,fun(B,bool))))),Uu),Uua),Uub),Uuc)) ).
% ATP.lambda_521
tff(fact_3236_ATP_Olambda__522,axiom,
! [B: $tType,A: $tType,C: $tType,Uu: fun(B,set(A)),Uua: fun(C,set(A)),Uub: set(C),Uuc: B] : aa(B,set(A),aa(set(C),fun(B,set(A)),aa(fun(C,set(A)),fun(set(C),fun(B,set(A))),aTP_Lamp_gi(fun(B,set(A)),fun(fun(C,set(A)),fun(set(C),fun(B,set(A)))),Uu),Uua),Uub),Uuc) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),aa(fun(C,set(A)),fun(set(C),set(set(A))),image2(C,set(A)),aa(B,fun(C,set(A)),aa(fun(C,set(A)),fun(B,fun(C,set(A))),aTP_Lamp_gh(fun(B,set(A)),fun(fun(C,set(A)),fun(B,fun(C,set(A)))),Uu),Uua),Uuc)),Uub)) ).
% ATP.lambda_522
tff(fact_3237_ATP_Olambda__523,axiom,
! [B: $tType,A: $tType,C: $tType] :
( comple592849572758109894attice(A)
=> ! [Uu: fun(B,A),Uua: fun(C,A),Uub: set(C),Uuc: B] : aa(B,A,aa(set(C),fun(B,A),aa(fun(C,A),fun(set(C),fun(B,A)),aTP_Lamp_fz(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),Uu),Uua),Uub),Uuc) = aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),aa(fun(C,A),fun(set(C),set(A)),image2(C,A),aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_fy(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uuc)),Uub)) ) ).
% ATP.lambda_523
tff(fact_3238_ATP_Olambda__524,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,Uu: set(B),Uua: fun(A,filter(C)),Uub: fun(B,filter(D)),Uuc: A] : aa(A,filter(product_prod(C,D)),aa(fun(B,filter(D)),fun(A,filter(product_prod(C,D))),aa(fun(A,filter(C)),fun(fun(B,filter(D)),fun(A,filter(product_prod(C,D)))),aTP_Lamp_jl(set(B),fun(fun(A,filter(C)),fun(fun(B,filter(D)),fun(A,filter(product_prod(C,D))))),Uu),Uua),Uub),Uuc) = aa(set(filter(product_prod(C,D))),filter(product_prod(C,D)),complete_Inf_Inf(filter(product_prod(C,D))),aa(set(B),set(filter(product_prod(C,D))),aa(fun(B,filter(product_prod(C,D))),fun(set(B),set(filter(product_prod(C,D)))),image2(B,filter(product_prod(C,D))),aa(A,fun(B,filter(product_prod(C,D))),aa(fun(B,filter(D)),fun(A,fun(B,filter(product_prod(C,D)))),aTP_Lamp_jk(fun(A,filter(C)),fun(fun(B,filter(D)),fun(A,fun(B,filter(product_prod(C,D))))),Uua),Uub),Uuc)),Uu)) ).
% ATP.lambda_524
tff(fact_3239_ATP_Olambda__525,axiom,
! [B: $tType,A: $tType,C: $tType,Uu: fun(B,set(A)),Uua: fun(C,set(A)),Uub: set(C),Uuc: B] : aa(B,set(A),aa(set(C),fun(B,set(A)),aa(fun(C,set(A)),fun(set(C),fun(B,set(A))),aTP_Lamp_gn(fun(B,set(A)),fun(fun(C,set(A)),fun(set(C),fun(B,set(A)))),Uu),Uua),Uub),Uuc) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(C),set(set(A)),aa(fun(C,set(A)),fun(set(C),set(set(A))),image2(C,set(A)),aa(B,fun(C,set(A)),aa(fun(C,set(A)),fun(B,fun(C,set(A))),aTP_Lamp_gm(fun(B,set(A)),fun(fun(C,set(A)),fun(B,fun(C,set(A)))),Uu),Uua),Uuc)),Uub)) ).
% ATP.lambda_525
tff(fact_3240_ATP_Olambda__526,axiom,
! [B: $tType,A: $tType,C: $tType] :
( comple592849572758109894attice(A)
=> ! [Uu: fun(B,A),Uua: fun(C,A),Uub: set(C),Uuc: B] : aa(B,A,aa(set(C),fun(B,A),aa(fun(C,A),fun(set(C),fun(B,A)),aTP_Lamp_ge(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),Uu),Uua),Uub),Uuc) = aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),aa(fun(C,A),fun(set(C),set(A)),image2(C,A),aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_gd(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uuc)),Uub)) ) ).
% ATP.lambda_526
tff(fact_3241_ATP_Olambda__527,axiom,
! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_rl(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uu),Uua),Uub),Uuc) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),minus_minus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uub),Uua))),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uuc))) ).
% ATP.lambda_527
tff(fact_3242_ATP_Olambda__528,axiom,
! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_rn(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uu),Uua),Uub),Uuc) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uub),Uua))),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uuc))) ).
% ATP.lambda_528
tff(fact_3243_ATP_Olambda__529,axiom,
! [A: $tType,Uu: fun(A,bool),Uua: list(A),Uub: A,Uuc: list(A)] : aa(list(A),option(product_prod(list(A),product_prod(A,list(A)))),aa(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))),aa(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A)))))),aTP_Lamp_tm(fun(A,bool),fun(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))))),Uu),Uua),Uub),Uuc) = aa(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A)))),some(product_prod(list(A),product_prod(A,list(A)))),aa(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A))),aa(list(A),fun(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A)))),product_Pair(list(A),product_prod(A,list(A))),takeWhile(A,aa(fun(A,bool),fun(A,bool),aa(fun(bool,bool),fun(fun(A,bool),fun(A,bool)),comp(bool,bool,A),fNot),Uu),Uua)),aa(list(A),product_prod(A,list(A)),aa(A,fun(list(A),product_prod(A,list(A))),product_Pair(A,list(A)),Uub),Uuc))) ).
% ATP.lambda_529
tff(fact_3244_ATP_Olambda__530,axiom,
! [A: $tType,Uu: A,Uua: list(A),Uub: A,Uuc: list(A)] : aa(list(A),option(product_prod(list(A),product_prod(A,list(A)))),aa(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))),aa(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A)))))),aTP_Lamp_sq(A,fun(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))))),Uu),Uua),Uub),Uuc) = aa(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A)))),some(product_prod(list(A),product_prod(A,list(A)))),aa(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A))),aa(list(A),fun(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A)))),product_Pair(list(A),product_prod(A,list(A))),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uu),Uua)),aa(list(A),product_prod(A,list(A)),aa(A,fun(list(A),product_prod(A,list(A))),product_Pair(A,list(A)),Uub),Uuc))) ).
% ATP.lambda_530
tff(fact_3245_ATP_Olambda__531,axiom,
! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_rr(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uu),Uua),Uub),Uuc) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uub)),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uuc))) ).
% ATP.lambda_531
tff(fact_3246_ATP_Olambda__532,axiom,
! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_rp(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uu),Uua),Uub),Uuc) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uub))) ).
% ATP.lambda_532
tff(fact_3247_ATP_Olambda__533,axiom,
! [A: $tType,C: $tType,B: $tType,Uu: set(product_prod(B,B)),Uua: fun(A,fun(B,set(C))),Uub: B,Uuc: A] : aa(A,set(C),aa(B,fun(A,set(C)),aa(fun(A,fun(B,set(C))),fun(B,fun(A,set(C))),aTP_Lamp_jm(set(product_prod(B,B)),fun(fun(A,fun(B,set(C))),fun(B,fun(A,set(C)))),Uu),Uua),Uub),Uuc) = aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),aa(fun(B,set(C)),fun(set(B),set(set(C))),image2(B,set(C)),aa(A,fun(B,set(C)),Uua,Uuc)),aa(set(B),set(B),image(B,B,Uu),aa(set(B),set(B),insert(B,Uub),bot_bot(set(B)))))) ).
% ATP.lambda_533
tff(fact_3248_ATP_Olambda__534,axiom,
! [C: $tType,A: $tType,B: $tType,E: $tType,D: $tType,Uu: fun(B,fun(C,fun(D,fun(E,set(A))))),Uua: set(product_prod(D,E)),Uub: B,Uuc: C] : aa(C,set(A),aa(B,fun(C,set(A)),aa(set(product_prod(D,E)),fun(B,fun(C,set(A))),aTP_Lamp_ej(fun(B,fun(C,fun(D,fun(E,set(A))))),fun(set(product_prod(D,E)),fun(B,fun(C,set(A)))),Uu),Uua),Uub),Uuc) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(product_prod(D,E)),set(set(A)),aa(fun(product_prod(D,E),set(A)),fun(set(product_prod(D,E)),set(set(A))),image2(product_prod(D,E),set(A)),aa(fun(D,fun(E,set(A))),fun(product_prod(D,E),set(A)),product_case_prod(D,E,set(A)),aa(C,fun(D,fun(E,set(A))),aa(B,fun(C,fun(D,fun(E,set(A)))),Uu,Uub),Uuc))),Uua)) ).
% ATP.lambda_534
tff(fact_3249_ATP_Olambda__535,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: set(product_prod(A,C)),Uua: set(product_prod(C,B)),Uub: A,Uuc: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aa(set(product_prod(C,B)),fun(A,fun(B,bool)),aTP_Lamp_nj(set(product_prod(A,C)),fun(set(product_prod(C,B)),fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
<=> ? [Y4: C] :
( pp(aa(set(product_prod(A,C)),bool,member(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),Uub),Y4)),Uu))
& pp(aa(set(product_prod(C,B)),bool,member(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),Y4),Uuc)),Uua)) ) ) ).
% ATP.lambda_535
tff(fact_3250_ATP_Olambda__536,axiom,
! [B: $tType,A: $tType,C: $tType,Uu: set(C),Uua: fun(C,A),Uub: fun(C,B),Uuc: product_prod(A,B)] :
( pp(aa(product_prod(A,B),bool,aa(fun(C,B),fun(product_prod(A,B),bool),aa(fun(C,A),fun(fun(C,B),fun(product_prod(A,B),bool)),aTP_Lamp_ni(set(C),fun(fun(C,A),fun(fun(C,B),fun(product_prod(A,B),bool))),Uu),Uua),Uub),Uuc))
<=> ? [A7: C] :
( ( Uuc = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,Uua,A7)),aa(C,B,Uub,A7)) )
& pp(aa(set(C),bool,member(C,A7),Uu)) ) ) ).
% ATP.lambda_536
tff(fact_3251_ATP_Olambda__537,axiom,
! [A: $tType,C: $tType,B: $tType,Uu: set(product_prod(B,B)),Uua: fun(A,fun(B,C)),Uub: A,Uuc: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(B,C)),fun(A,fun(A,bool)),aTP_Lamp_lc(set(product_prod(B,B)),fun(fun(A,fun(B,C)),fun(A,fun(A,bool))),Uu),Uua),Uub),Uuc))
<=> ! [X5: product_prod(B,B)] :
( pp(aa(set(product_prod(B,B)),bool,member(product_prod(B,B),X5),Uu))
=> pp(aa(product_prod(B,B),bool,aa(fun(B,fun(B,bool)),fun(product_prod(B,B),bool),product_case_prod(B,B,bool),aa(A,fun(B,fun(B,bool)),aa(A,fun(A,fun(B,fun(B,bool))),aTP_Lamp_lb(fun(A,fun(B,C)),fun(A,fun(A,fun(B,fun(B,bool)))),Uua),Uub),Uuc)),X5)) ) ) ).
% ATP.lambda_537
tff(fact_3252_ATP_Olambda__538,axiom,
! [A: $tType,B: $tType,Uu: set(A),Uua: set(product_prod(B,B)),Uub: fun(A,B),Uuc: product_prod(A,A)] :
( pp(aa(product_prod(A,A),bool,aa(fun(A,B),fun(product_prod(A,A),bool),aa(set(product_prod(B,B)),fun(fun(A,B),fun(product_prod(A,A),bool)),aTP_Lamp_ng(set(A),fun(set(product_prod(B,B)),fun(fun(A,B),fun(product_prod(A,A),bool))),Uu),Uua),Uub),Uuc))
<=> ? [A18: A,A26: A] :
( ( Uuc = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A18),A26) )
& pp(aa(set(A),bool,member(A,A18),Uu))
& pp(aa(set(A),bool,member(A,A26),Uu))
& pp(aa(set(product_prod(B,B)),bool,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,Uub,A18)),aa(A,B,Uub,A26))),Uua)) ) ) ).
% ATP.lambda_538
tff(fact_3253_ATP_Olambda__539,axiom,
! [A: $tType,C: $tType,B: $tType,Uu: fun(A,fun(A,A)),Uua: A,Uub: fun(B,set(C)),Uuc: fun(B,fun(C,A)),Uud: B] : aa(B,A,aa(fun(B,fun(C,A)),fun(B,A),aa(fun(B,set(C)),fun(fun(B,fun(C,A)),fun(B,A)),aa(A,fun(fun(B,set(C)),fun(fun(B,fun(C,A)),fun(B,A))),aTP_Lamp_ui(fun(A,fun(A,A)),fun(A,fun(fun(B,set(C)),fun(fun(B,fun(C,A)),fun(B,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups_comm_monoid_F(A,C,Uu,Uua),aa(B,fun(C,A),Uuc,Uud)),aa(B,set(C),Uub,Uud)) ).
% ATP.lambda_539
tff(fact_3254_ATP_Olambda__540,axiom,
! [B: $tType,A: $tType,C: $tType,Uu: fun(A,fun(A,A)),Uua: A,Uub: fun(B,fun(C,A)),Uuc: set(C),Uud: B] : aa(B,A,aa(set(C),fun(B,A),aa(fun(B,fun(C,A)),fun(set(C),fun(B,A)),aa(A,fun(fun(B,fun(C,A)),fun(set(C),fun(B,A))),aTP_Lamp_uh(fun(A,fun(A,A)),fun(A,fun(fun(B,fun(C,A)),fun(set(C),fun(B,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups_comm_monoid_F(A,C,Uu,Uua),aa(B,fun(C,A),Uub,Uud)),Uuc) ).
% ATP.lambda_540
tff(fact_3255_ATP_Olambda__541,axiom,
! [A: $tType,C: $tType,B: $tType,Uu: fun(A,fun(A,A)),Uua: A,Uub: fun(B,set(C)),Uuc: fun(C,A),Uud: B] : aa(B,A,aa(fun(C,A),fun(B,A),aa(fun(B,set(C)),fun(fun(C,A),fun(B,A)),aa(A,fun(fun(B,set(C)),fun(fun(C,A),fun(B,A))),aTP_Lamp_ul(fun(A,fun(A,A)),fun(A,fun(fun(B,set(C)),fun(fun(C,A),fun(B,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups_comm_monoid_F(A,C,Uu,Uua),Uuc),aa(B,set(C),Uub,Uud)) ).
% ATP.lambda_541
tff(fact_3256_ATP_Olambda__542,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: C,Uua: A,Uub: A,Uuc: B,Uud: set(product_prod(C,B))] : aa(set(product_prod(C,B)),set(product_prod(C,B)),aa(B,fun(set(product_prod(C,B)),set(product_prod(C,B))),aa(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aa(A,fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),aTP_Lamp_hz(C,fun(A,fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B)))))),Uu),Uua),Uub),Uuc),Uud) = if(set(product_prod(C,B)),aa(A,bool,aa(A,fun(A,bool),fequal(A),Uua),Uub),aa(set(product_prod(C,B)),set(product_prod(C,B)),insert(product_prod(C,B),aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),Uu),Uuc)),Uud),Uud) ).
% ATP.lambda_542
tff(fact_3257_ATP_Olambda__543,axiom,
! [B: $tType,C: $tType,A: $tType,Uu: A,Uua: B,Uub: B,Uuc: C,Uud: set(product_prod(A,C))] : aa(set(product_prod(A,C)),set(product_prod(A,C)),aa(C,fun(set(product_prod(A,C)),set(product_prod(A,C))),aa(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C)))),aa(B,fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C))))),aTP_Lamp_ik(A,fun(B,fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C)))))),Uu),Uua),Uub),Uuc),Uud) = if(set(product_prod(A,C)),aa(B,bool,aa(B,fun(B,bool),fequal(B),Uua),Uub),aa(set(product_prod(A,C)),set(product_prod(A,C)),insert(product_prod(A,C),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),Uu),Uuc)),Uud),Uud) ).
% ATP.lambda_543
tff(fact_3258_ATP_Olambda__544,axiom,
! [A: $tType,C: $tType,B: $tType,Uu: fun(C,fun(B,heap_Heap(A))),Uua: fun(C,C),Uub: fun(fun(B,heap_Heap(A)),C),Uuc: B,Uud: fun(B,heap_Heap(A))] : aa(fun(B,heap_Heap(A)),heap_Heap(A),aa(B,fun(fun(B,heap_Heap(A)),heap_Heap(A)),aa(fun(fun(B,heap_Heap(A)),C),fun(B,fun(fun(B,heap_Heap(A)),heap_Heap(A))),aa(fun(C,C),fun(fun(fun(B,heap_Heap(A)),C),fun(B,fun(fun(B,heap_Heap(A)),heap_Heap(A)))),aTP_Lamp_vb(fun(C,fun(B,heap_Heap(A))),fun(fun(C,C),fun(fun(fun(B,heap_Heap(A)),C),fun(B,fun(fun(B,heap_Heap(A)),heap_Heap(A))))),Uu),Uua),Uub),Uuc),Uud) = aa(B,heap_Heap(A),aa(C,fun(B,heap_Heap(A)),Uu,aa(C,C,Uua,aa(fun(B,heap_Heap(A)),C,Uub,Uud))),Uuc) ).
% ATP.lambda_544
tff(fact_3259_ATP_Olambda__545,axiom,
! [A: $tType,D: $tType,C: $tType,E: $tType,B: $tType,Uu: fun(D,fun(E,C)),Uua: fun(A,D),Uub: fun(B,E),Uuc: A,Uud: B] : aa(B,C,aa(A,fun(B,C),aa(fun(B,E),fun(A,fun(B,C)),aa(fun(A,D),fun(fun(B,E),fun(A,fun(B,C))),aTP_Lamp_bh(fun(D,fun(E,C)),fun(fun(A,D),fun(fun(B,E),fun(A,fun(B,C)))),Uu),Uua),Uub),Uuc),Uud) = aa(E,C,aa(D,fun(E,C),Uu,aa(A,D,Uua,Uuc)),aa(B,E,Uub,Uud)) ).
% ATP.lambda_545
tff(fact_3260_ATP_Olambda__546,axiom,
! [D: $tType,B: $tType,A: $tType,C: $tType,E: $tType,Uu: fun(B,fun(C,A)),Uua: fun(D,B),Uub: fun(E,C),Uuc: D,Uud: E] : aa(E,A,aa(D,fun(E,A),aa(fun(E,C),fun(D,fun(E,A)),aa(fun(D,B),fun(fun(E,C),fun(D,fun(E,A))),aTP_Lamp_aw(fun(B,fun(C,A)),fun(fun(D,B),fun(fun(E,C),fun(D,fun(E,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(C,A,aa(B,fun(C,A),Uu,aa(D,B,Uua,Uuc)),aa(E,C,Uub,Uud)) ).
% ATP.lambda_546
tff(fact_3261_ATP_Olambda__547,axiom,
! [B: $tType,A: $tType,C: $tType,Uu: fun(B,fun(A,bool)),Uua: fun(A,fun(C,bool)),Uub: B,Uuc: C,Uud: A] :
( pp(aa(A,bool,aa(C,fun(A,bool),aa(B,fun(C,fun(A,bool)),aa(fun(A,fun(C,bool)),fun(B,fun(C,fun(A,bool))),aTP_Lamp_qk(fun(B,fun(A,bool)),fun(fun(A,fun(C,bool)),fun(B,fun(C,fun(A,bool)))),Uu),Uua),Uub),Uuc),Uud))
<=> ( pp(aa(A,bool,aa(B,fun(A,bool),Uu,Uub),Uud))
& pp(aa(C,bool,aa(A,fun(C,bool),Uua,Uud),Uuc)) ) ) ).
% ATP.lambda_547
tff(fact_3262_ATP_Olambda__548,axiom,
! [A: $tType,C: $tType,B: $tType,Uu: fun(A,fun(B,C)),Uua: A,Uub: A,Uuc: B,Uud: B] :
( pp(aa(B,bool,aa(B,fun(B,bool),aa(A,fun(B,fun(B,bool)),aa(A,fun(A,fun(B,fun(B,bool))),aTP_Lamp_lb(fun(A,fun(B,C)),fun(A,fun(A,fun(B,fun(B,bool)))),Uu),Uua),Uub),Uuc),Uud))
<=> ( aa(B,C,aa(A,fun(B,C),Uu,Uua),Uuc) = aa(B,C,aa(A,fun(B,C),Uu,Uub),Uud) ) ) ).
% ATP.lambda_548
tff(fact_3263_ATP_Olambda__549,axiom,
! [A: $tType,Uu: A,Uua: A,Uub: set(product_prod(A,A)),Uuc: A,Uud: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aa(set(product_prod(A,A)),fun(A,fun(A,bool)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool))),aTP_Lamp_jp(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool)))),Uu),Uua),Uub),Uuc),Uud))
<=> ( ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uuc),Uu)),transitive_trancl(A,Uub)))
| ( Uuc = Uu ) )
& ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uud)),transitive_trancl(A,Uub)))
| ( Uud = Uua ) ) ) ) ).
% ATP.lambda_549
tff(fact_3264_ATP_Olambda__550,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A,Uuc: A,Uud: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aa(A,fun(A,fun(A,bool)),aa(A,fun(A,fun(A,fun(A,bool))),aTP_Lamp_hp(set(product_prod(A,A)),fun(A,fun(A,fun(A,fun(A,bool)))),Uu),Uua),Uub),Uuc),Uud))
<=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),insert(A,Uua),aa(set(A),set(A),insert(A,Uub),aa(set(A),set(A),insert(A,Uuc),aa(set(A),set(A),insert(A,Uud),bot_bot(set(A))))))),aa(set(product_prod(A,A)),set(A),field2(A),Uu)))
& ( ( ( Uua = Uuc )
& ( Uub = Uud ) )
| pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),bNF_We1388413361240627857o_max2(A,Uu,Uua,Uub)),bNF_We1388413361240627857o_max2(A,Uu,Uuc,Uud))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),Uu),id2(A))))
| ( ( bNF_We1388413361240627857o_max2(A,Uu,Uua,Uub) = bNF_We1388413361240627857o_max2(A,Uu,Uuc,Uud) )
& pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uuc)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),Uu),id2(A)))) )
| ( ( bNF_We1388413361240627857o_max2(A,Uu,Uua,Uub) = bNF_We1388413361240627857o_max2(A,Uu,Uuc,Uud) )
& ( Uua = Uuc )
& pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uud)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),minus_minus(set(product_prod(A,A))),Uu),id2(A)))) ) ) ) ) ).
% ATP.lambda_550
tff(fact_3265_ATP_Olambda__551,axiom,
! [A: $tType,Uu: A,Uua: A,Uub: set(product_prod(A,A)),Uuc: A,Uud: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aa(set(product_prod(A,A)),fun(A,fun(A,bool)),aa(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool))),aTP_Lamp_jo(A,fun(A,fun(set(product_prod(A,A)),fun(A,fun(A,bool)))),Uu),Uua),Uub),Uuc),Uud))
<=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uuc),Uu)),transitive_rtrancl(A,Uub)))
& pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uud)),transitive_rtrancl(A,Uub))) ) ) ).
% ATP.lambda_551
tff(fact_3266_ATP_Olambda__552,axiom,
! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A,Uuc: A,Uud: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),aa(A,fun(A,fun(A,bool)),aa(A,fun(A,fun(A,fun(A,bool))),aTP_Lamp_qm(set(product_prod(A,A)),fun(A,fun(A,fun(A,fun(A,bool)))),Uu),Uua),Uub),Uuc),Uud))
<=> ( ( Uub = Uuc )
=> pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uua),Uud)),Uu)) ) ) ).
% ATP.lambda_552
tff(fact_3267_ATP_Olambda__553,axiom,
! [C: $tType,B: $tType,A: $tType,E: $tType,Uu: fun(A,fun(B,bool)),Uua: fun(C,fun(A,bool)),Uub: fun(A,fun(E,bool)),Uuc: C,Uud: E] :
( pp(aa(E,bool,aa(C,fun(E,bool),aa(fun(A,fun(E,bool)),fun(C,fun(E,bool)),aa(fun(C,fun(A,bool)),fun(fun(A,fun(E,bool)),fun(C,fun(E,bool))),aTP_Lamp_os(fun(A,fun(B,bool)),fun(fun(C,fun(A,bool)),fun(fun(A,fun(E,bool)),fun(C,fun(E,bool)))),Uu),Uua),Uub),Uuc),Uud))
<=> ? [X5: A] :
( pp(aa(set(A),bool,member(A,X5),aa(fun(A,bool),set(A),collect(A),domainp(A,B,Uu))))
& pp(aa(A,bool,aa(C,fun(A,bool),Uua,Uuc),X5))
& pp(aa(E,bool,aa(A,fun(E,bool),Uub,X5),Uud)) ) ) ).
% ATP.lambda_553
tff(fact_3268_ATP_Olambda__554,axiom,
! [A: $tType,B: $tType,C: $tType] :
( semiring_0(B)
=> ! [Uu: fun(A,B),Uua: fun(C,B),Uub: set(A),Uuc: set(C),Uud: B] :
( pp(aa(B,bool,aa(set(C),fun(B,bool),aa(set(A),fun(set(C),fun(B,bool)),aa(fun(C,B),fun(set(A),fun(set(C),fun(B,bool))),aTP_Lamp_ml(fun(A,B),fun(fun(C,B),fun(set(A),fun(set(C),fun(B,bool)))),Uu),Uua),Uub),Uuc),Uud))
<=> ? [A7: A,B5: C] :
( ( Uud = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,A7)),aa(C,B,Uua,B5)) )
& pp(aa(set(A),bool,member(A,A7),Uub))
& pp(aa(set(C),bool,member(C,B5),Uuc)) ) ) ) ).
% ATP.lambda_554
tff(fact_3269_ATP_Olambda__555,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: fun(A,fun(A,A)),Uua: A,Uub: set(B),Uuc: fun(B,A),Uud: fun(B,C),Uue: C] : aa(C,A,aa(fun(B,C),fun(C,A),aa(fun(B,A),fun(fun(B,C),fun(C,A)),aa(set(B),fun(fun(B,A),fun(fun(B,C),fun(C,A))),aa(A,fun(set(B),fun(fun(B,A),fun(fun(B,C),fun(C,A)))),aTP_Lamp_uj(fun(A,fun(A,A)),fun(A,fun(set(B),fun(fun(B,A),fun(fun(B,C),fun(C,A))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,Uu,Uua),Uuc),aa(fun(B,bool),set(B),collect(B),aa(C,fun(B,bool),aa(fun(B,C),fun(C,fun(B,bool)),aTP_Lamp_lp(set(B),fun(fun(B,C),fun(C,fun(B,bool))),Uub),Uud),Uue))) ).
% ATP.lambda_555
tff(fact_3270_ATP_Olambda__556,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: fun(A,fun(A,A)),Uua: A,Uub: set(B),Uuc: fun(B,C),Uud: fun(B,A),Uue: C] : aa(C,A,aa(fun(B,A),fun(C,A),aa(fun(B,C),fun(fun(B,A),fun(C,A)),aa(set(B),fun(fun(B,C),fun(fun(B,A),fun(C,A))),aa(A,fun(set(B),fun(fun(B,C),fun(fun(B,A),fun(C,A)))),aTP_Lamp_uk(fun(A,fun(A,A)),fun(A,fun(set(B),fun(fun(B,C),fun(fun(B,A),fun(C,A))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups_comm_monoid_F(A,B,Uu,Uua),Uud),aa(fun(B,bool),set(B),collect(B),aa(C,fun(B,bool),aa(fun(B,C),fun(C,fun(B,bool)),aTP_Lamp_lp(set(B),fun(fun(B,C),fun(C,fun(B,bool))),Uub),Uuc),Uue))) ).
% ATP.lambda_556
tff(fact_3271_ATP_Olambda__557,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,Uu: fun(A,fun(C,bool)),Uua: fun(B,fun(D,bool)),Uub: A,Uuc: B,Uud: C,Uue: D] :
( pp(aa(D,bool,aa(C,fun(D,bool),aa(B,fun(C,fun(D,bool)),aa(A,fun(B,fun(C,fun(D,bool))),aa(fun(B,fun(D,bool)),fun(A,fun(B,fun(C,fun(D,bool)))),aTP_Lamp_qb(fun(A,fun(C,bool)),fun(fun(B,fun(D,bool)),fun(A,fun(B,fun(C,fun(D,bool))))),Uu),Uua),Uub),Uuc),Uud),Uue))
<=> ( pp(aa(C,bool,aa(A,fun(C,bool),Uu,Uub),Uud))
& pp(aa(D,bool,aa(B,fun(D,bool),Uua,Uuc),Uue)) ) ) ).
% ATP.lambda_557
tff(fact_3272_ATP_Olambda__558,axiom,
! [A: $tType,B: $tType,Uu: set(product_prod(A,A)),Uua: set(product_prod(B,B)),Uub: A,Uuc: B,Uud: A,Uue: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aa(B,fun(A,fun(B,bool)),aa(A,fun(B,fun(A,fun(B,bool))),aa(set(product_prod(B,B)),fun(A,fun(B,fun(A,fun(B,bool)))),aTP_Lamp_ib(set(product_prod(A,A)),fun(set(product_prod(B,B)),fun(A,fun(B,fun(A,fun(B,bool))))),Uu),Uua),Uub),Uuc),Uud),Uue))
<=> ( pp(aa(set(product_prod(A,A)),bool,member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uub),Uud)),Uu))
| ( ( Uub = Uud )
& pp(aa(set(product_prod(B,B)),bool,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Uuc),Uue)),Uua)) ) ) ) ).
% ATP.lambda_558
tff(fact_3273_ATP_Olambda__559,axiom,
! [B: $tType,A: $tType,Uu: fun(A,bool),Uua: fun(A,set(product_prod(B,B))),Uub: A,Uuc: B,Uud: A,Uue: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aa(B,fun(A,fun(B,bool)),aa(A,fun(B,fun(A,fun(B,bool))),aa(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(A,fun(B,bool)))),aTP_Lamp_hw(fun(A,bool),fun(fun(A,set(product_prod(B,B))),fun(A,fun(B,fun(A,fun(B,bool))))),Uu),Uua),Uub),Uuc),Uud),Uue))
<=> ( ( Uub = Uud )
& pp(aa(A,bool,Uu,Uud))
& pp(aa(set(product_prod(B,B)),bool,member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Uuc),Uue)),aa(A,set(product_prod(B,B)),Uua,Uud))) ) ) ).
% ATP.lambda_559
tff(fact_3274_ATP_Olambda__560,axiom,
! [B: $tType,A: $tType,Uu: bool,Uua: A,Uub: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_cj(bool,fun(A,fun(B,bool)),Uu),Uua),Uub))
<=> pp(Uu) ) ).
% ATP.lambda_560
tff(fact_3275_ATP_Olambda__561,axiom,
! [B: $tType,A: $tType] :
( heap(B)
=> ! [Uu: heap_Heap(A),Uua: B] : aa(B,heap_Heap(A),aTP_Lamp_ax(heap_Heap(A),fun(B,heap_Heap(A)),Uu),Uua) = Uu ) ).
% ATP.lambda_561
tff(fact_3276_ATP_Olambda__562,axiom,
! [B: $tType,A: $tType,Uu: heap_Heap(A),Uua: B] : aa(B,heap_Heap(A),aTP_Lamp_ut(heap_Heap(A),fun(B,heap_Heap(A)),Uu),Uua) = Uu ).
% ATP.lambda_562
tff(fact_3277_ATP_Olambda__563,axiom,
! [C: $tType,D: $tType,Uu: set(D),Uua: C] : aa(C,set(D),aTP_Lamp_bs(set(D),fun(C,set(D)),Uu),Uua) = Uu ).
% ATP.lambda_563
tff(fact_3278_ATP_Olambda__564,axiom,
! [B: $tType,D: $tType,Uu: set(D),Uua: B] : aa(B,set(D),aTP_Lamp_cf(set(D),fun(B,set(D)),Uu),Uua) = Uu ).
% ATP.lambda_564
tff(fact_3279_ATP_Olambda__565,axiom,
! [B: $tType,C: $tType,Uu: set(C),Uua: B] : aa(B,set(C),aTP_Lamp_hc(set(C),fun(B,set(C)),Uu),Uua) = Uu ).
% ATP.lambda_565
tff(fact_3280_ATP_Olambda__566,axiom,
! [A: $tType,C: $tType,Uu: set(C),Uua: A] : aa(A,set(C),aTP_Lamp_cg(set(C),fun(A,set(C)),Uu),Uua) = Uu ).
% ATP.lambda_566
tff(fact_3281_ATP_Olambda__567,axiom,
! [A: $tType,B: $tType,Uu: set(B),Uua: A] : aa(A,set(B),aTP_Lamp_bu(set(B),fun(A,set(B)),Uu),Uua) = Uu ).
% ATP.lambda_567
tff(fact_3282_ATP_Olambda__568,axiom,
! [A: $tType,Uu: set(A),Uua: list(A)] : aa(list(A),set(A),aTP_Lamp_pn(set(A),fun(list(A),set(A)),Uu),Uua) = Uu ).
% ATP.lambda_568
tff(fact_3283_ATP_Olambda__569,axiom,
! [B: $tType,A: $tType,Uu: set(A),Uua: B] : aa(B,set(A),aTP_Lamp_cc(set(A),fun(B,set(A)),Uu),Uua) = Uu ).
% ATP.lambda_569
tff(fact_3284_ATP_Olambda__570,axiom,
! [A: $tType,Uu: set(A),Uua: A] : aa(A,set(A),aTP_Lamp_co(set(A),fun(A,set(A)),Uu),Uua) = Uu ).
% ATP.lambda_570
tff(fact_3285_ATP_Olambda__571,axiom,
! [A: $tType,B: $tType,Uu: fun(B,bool),Uua: A] : aa(A,fun(B,bool),aTP_Lamp_jg(fun(B,bool),fun(A,fun(B,bool)),Uu),Uua) = Uu ).
% ATP.lambda_571
tff(fact_3286_ATP_Olambda__572,axiom,
! [A: $tType,B: $tType,C: $tType,Uu: C,Uua: product_prod(A,B)] : aa(product_prod(A,B),C,aTP_Lamp_ao(C,fun(product_prod(A,B),C),Uu),Uua) = Uu ).
% ATP.lambda_572
tff(fact_3287_ATP_Olambda__573,axiom,
! [C: $tType,B: $tType,Uu: B,Uua: C] : aa(C,B,aTP_Lamp_bq(B,fun(C,B),Uu),Uua) = Uu ).
% ATP.lambda_573
tff(fact_3288_ATP_Olambda__574,axiom,
! [A: $tType,B: $tType,Uu: B,Uua: A] : aa(A,B,aTP_Lamp_dx(B,fun(A,B),Uu),Uua) = Uu ).
% ATP.lambda_574
tff(fact_3289_ATP_Olambda__575,axiom,
! [B: $tType,A: $tType] :
( condit1219197933456340205attice(A)
=> ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_ec(A,fun(B,A),Uu),Uua) = Uu ) ).
% ATP.lambda_575
tff(fact_3290_ATP_Olambda__576,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_cu(A,fun(B,A),Uu),Uua) = Uu ) ).
% ATP.lambda_576
tff(fact_3291_ATP_Olambda__577,axiom,
! [B: $tType,A: $tType] :
( linorder(A)
=> ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_ku(A,fun(B,A),Uu),Uua) = Uu ) ).
% ATP.lambda_577
tff(fact_3292_ATP_Olambda__578,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_pb(A,fun(nat,A),Uu),Uua) = Uu ) ).
% ATP.lambda_578
tff(fact_3293_ATP_Olambda__579,axiom,
! [A: $tType] :
( heap(A)
=> ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_ta(A,fun(A,A),Uu),Uua) = Uu ) ).
% ATP.lambda_579
tff(fact_3294_ATP_Olambda__580,axiom,
! [B: $tType,A: $tType,Uu: A,Uua: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_at(A,fun(B,A)),Uu),Uua) = Uu ).
% ATP.lambda_580
tff(fact_3295_ATP_Olambda__581,axiom,
! [A: $tType,B: $tType,Uu: A,Uua: B] : aa(B,B,aa(A,fun(B,B),aTP_Lamp_au(A,fun(B,B)),Uu),Uua) = Uua ).
% ATP.lambda_581
tff(fact_3296_ATP_Olambda__582,axiom,
! [B: $tType,A: $tType,Uu: A,Uua: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_an(A,fun(B,bool)),Uu),Uua))
<=> $true ) ).
% ATP.lambda_582
tff(fact_3297_ATP_Olambda__583,axiom,
! [B: $tType,Uu: B] : aa(B,B,aTP_Lamp_ad(B,B),Uu) = Uu ).
% ATP.lambda_583
tff(fact_3298_ATP_Olambda__584,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [Uu: A] : aa(A,A,aTP_Lamp_na(A,A),Uu) = Uu ) ).
% ATP.lambda_584
tff(fact_3299_ATP_Olambda__585,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [Uu: A] : aa(A,A,aTP_Lamp_lt(A,A),Uu) = Uu ) ).
% ATP.lambda_585
tff(fact_3300_ATP_Olambda__586,axiom,
! [A: $tType] :
( complete_Sup(A)
=> ! [Uu: A] : aa(A,A,aTP_Lamp_cp(A,A),Uu) = Uu ) ).
% ATP.lambda_586
tff(fact_3301_ATP_Olambda__587,axiom,
! [A: $tType] :
( complete_Inf(A)
=> ! [Uu: A] : aa(A,A,aTP_Lamp_cq(A,A),Uu) = Uu ) ).
% ATP.lambda_587
tff(fact_3302_ATP_Olambda__588,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Uu: A] : aa(A,A,aTP_Lamp_so(A,A),Uu) = Uu ) ).
% ATP.lambda_588
tff(fact_3303_ATP_Olambda__589,axiom,
! [A: $tType,Uu: A] : aa(A,A,aTP_Lamp_ab(A,A),Uu) = Uu ).
% ATP.lambda_589
tff(fact_3304_ATP_Olambda__590,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [Uu: B] : aa(B,A,aTP_Lamp_cv(B,A),Uu) = top_top(A) ) ).
% ATP.lambda_590
tff(fact_3305_ATP_Olambda__591,axiom,
! [A: $tType,B: $tType,Uu: A] : aa(A,set(B),aTP_Lamp_ca(A,set(B)),Uu) = top_top(set(B)) ).
% ATP.lambda_591
tff(fact_3306_ATP_Olambda__592,axiom,
! [C: $tType,B: $tType,Uu: C] : aa(C,set(B),aTP_Lamp_cm(C,set(B)),Uu) = bot_bot(set(B)) ).
% ATP.lambda_592
tff(fact_3307_ATP_Olambda__593,axiom,
! [B: $tType,A: $tType,Uu: B] : aa(B,set(A),aTP_Lamp_dz(B,set(A)),Uu) = bot_bot(set(A)) ).
% ATP.lambda_593
tff(fact_3308_ATP_Olambda__594,axiom,
! [B: $tType,A: $tType] :
( comple6319245703460814977attice(A)
=> ! [Uu: B] : aa(B,A,aTP_Lamp_ct(B,A),Uu) = bot_bot(A) ) ).
% ATP.lambda_594
tff(fact_3309_ATP_Olambda__595,axiom,
! [A: $tType,D: $tType,Uu: A] : aa(A,set(D),aTP_Lamp_cn(A,set(D)),Uu) = bot_bot(set(D)) ).
% ATP.lambda_595
tff(fact_3310_ATP_Olambda__596,axiom,
! [A: $tType,B: $tType,Uu: A] : aa(A,set(B),aTP_Lamp_ci(A,set(B)),Uu) = bot_bot(set(B)) ).
% ATP.lambda_596
tff(fact_3311_ATP_Olambda__597,axiom,
! [A: $tType,Uu: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit))),aTP_Lamp_sm(heap_ext(product_unit),option(product_prod(A,heap_ext(product_unit)))),Uu) = none(product_prod(A,heap_ext(product_unit))) ).
% ATP.lambda_597
tff(fact_3312_ATP_Olambda__598,axiom,
! [A: $tType,C: $tType,Uu: A] : aa(A,option(C),aTP_Lamp_si(A,option(C)),Uu) = none(C) ).
% ATP.lambda_598
tff(fact_3313_ATP_Olambda__599,axiom,
! [B: $tType,Uu: B] :
( pp(aa(B,bool,aTP_Lamp_st(B,bool),Uu))
<=> $false ) ).
% ATP.lambda_599
tff(fact_3314_ATP_Olambda__600,axiom,
! [Uu: heap_ext(product_unit)] :
( pp(aa(heap_ext(product_unit),bool,aTP_Lamp_tt(heap_ext(product_unit),bool),Uu))
<=> $true ) ).
% ATP.lambda_600
tff(fact_3315_ATP_Olambda__601,axiom,
! [D: $tType,Uu: D] :
( pp(aa(D,bool,aTP_Lamp_qf(D,bool),Uu))
<=> $true ) ).
% ATP.lambda_601
tff(fact_3316_ATP_Olambda__602,axiom,
! [C: $tType,Uu: C] :
( pp(aa(C,bool,aTP_Lamp_qi(C,bool),Uu))
<=> $true ) ).
% ATP.lambda_602
tff(fact_3317_ATP_Olambda__603,axiom,
! [A: $tType,Uu: A] :
( pp(aa(A,bool,aTP_Lamp_ov(A,bool),Uu))
<=> $true ) ).
% ATP.lambda_603
% Subclasses (3)
tff(subcl_Heap_Oheap___HOL_Otype,axiom,
! [A: $tType] :
( heap(A)
=> type(A) ) ).
tff(subcl_Heap_Oheap___Typerep_Otyperep,axiom,
! [A: $tType] :
( heap(A)
=> typerep2(A) ) ).
tff(subcl_Heap_Oheap___Countable_Ocountable,axiom,
! [A: $tType] :
( heap(A)
=> countable(A) ) ).
% Type constructors (282)
tff(tcon_Product__Type_Ounit___Finite__Set_Ofinite,axiom,
finite_finite(product_unit) ).
tff(tcon_Product__Type_Oprod___Finite__Set_Ofinite_1,axiom,
! [A19: $tType,A20: $tType] :
( ( finite_finite(A19)
& finite_finite(A20) )
=> finite_finite(product_prod(A19,A20)) ) ).
tff(tcon_Option_Ooption___Finite__Set_Ofinite_2,axiom,
! [A19: $tType] :
( finite_finite(A19)
=> finite_finite(option(A19)) ) ).
tff(tcon_Sum__Type_Osum___Finite__Set_Ofinite_3,axiom,
! [A19: $tType,A20: $tType] :
( ( finite_finite(A19)
& finite_finite(A20) )
=> finite_finite(sum_sum(A19,A20)) ) ).
tff(tcon_String_Ochar___Finite__Set_Ofinite_4,axiom,
finite_finite(char) ).
tff(tcon_HOL_Obool___Finite__Set_Ofinite_5,axiom,
finite_finite(bool) ).
tff(tcon_Set_Oset___Finite__Set_Ofinite_6,axiom,
! [A19: $tType] :
( finite_finite(A19)
=> finite_finite(set(A19)) ) ).
tff(tcon_fun___Finite__Set_Ofinite_7,axiom,
! [A19: $tType,A20: $tType] :
( ( finite_finite(A19)
& finite_finite(A20) )
=> finite_finite(fun(A19,A20)) ) ).
tff(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
! [A19: $tType,A20: $tType] :
( comple6319245703460814977attice(A20)
=> condit1219197933456340205attice(fun(A19,A20)) ) ).
tff(tcon_fun___Complete__Lattices_Ocomplete__distrib__lattice,axiom,
! [A19: $tType,A20: $tType] :
( comple592849572758109894attice(A20)
=> comple592849572758109894attice(fun(A19,A20)) ) ).
tff(tcon_fun___Complete__Lattices_Ocomplete__boolean__algebra,axiom,
! [A19: $tType,A20: $tType] :
( comple489889107523837845lgebra(A20)
=> comple489889107523837845lgebra(fun(A19,A20)) ) ).
tff(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
! [A19: $tType,A20: $tType] :
( comple6319245703460814977attice(A20)
=> comple6319245703460814977attice(fun(A19,A20)) ) ).
tff(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
! [A19: $tType,A20: $tType] :
( boolea8198339166811842893lgebra(A20)
=> boolea8198339166811842893lgebra(fun(A19,A20)) ) ).
tff(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A19: $tType,A20: $tType] :
( semilattice_sup(A20)
=> semilattice_sup(fun(A19,A20)) ) ).
tff(tcon_fun___Lattices_Osemilattice__inf,axiom,
! [A19: $tType,A20: $tType] :
( semilattice_inf(A20)
=> semilattice_inf(fun(A19,A20)) ) ).
tff(tcon_fun___Complete__Lattices_OSup,axiom,
! [A19: $tType,A20: $tType] :
( complete_Sup(A20)
=> complete_Sup(fun(A19,A20)) ) ).
tff(tcon_fun___Complete__Lattices_OInf,axiom,
! [A19: $tType,A20: $tType] :
( complete_Inf(A20)
=> complete_Inf(fun(A19,A20)) ) ).
tff(tcon_fun___Countable_Ocountable,axiom,
! [A19: $tType,A20: $tType] :
( ( finite_finite(A19)
& countable(A20) )
=> countable(fun(A19,A20)) ) ).
tff(tcon_fun___Orderings_Opreorder,axiom,
! [A19: $tType,A20: $tType] :
( preorder(A20)
=> preorder(fun(A19,A20)) ) ).
tff(tcon_fun___Lattices_Olattice,axiom,
! [A19: $tType,A20: $tType] :
( lattice(A20)
=> lattice(fun(A19,A20)) ) ).
tff(tcon_fun___Typerep_Otyperep,axiom,
! [A19: $tType,A20: $tType] :
( ( typerep2(A19)
& typerep2(A20) )
=> typerep2(fun(A19,A20)) ) ).
tff(tcon_fun___Orderings_Oorder,axiom,
! [A19: $tType,A20: $tType] :
( order(A20)
=> order(fun(A19,A20)) ) ).
tff(tcon_fun___Orderings_Oord,axiom,
! [A19: $tType,A20: $tType] :
( ord(A20)
=> ord(fun(A19,A20)) ) ).
tff(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
condit6923001295902523014norder(int) ).
tff(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_8,axiom,
condit1219197933456340205attice(int) ).
tff(tcon_Int_Oint___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations,axiom,
bit_un5681908812861735899ations(int) ).
tff(tcon_Int_Oint___Divides_Ounique__euclidean__semiring__numeral,axiom,
unique1627219031080169319umeral(int) ).
tff(tcon_Int_Oint___Groups_Olinordered__ab__semigroup__add,axiom,
linord4140545234300271783up_add(int) ).
tff(tcon_Int_Oint___Groups_Oordered__comm__monoid__add,axiom,
ordere6911136660526730532id_add(int) ).
tff(tcon_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
linord5086331880401160121up_add(int) ).
tff(tcon_Int_Oint___Groups_Oordered__ab__group__add,axiom,
ordered_ab_group_add(int) ).
tff(tcon_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add(int) ).
tff(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
linordered_semidom(int) ).
tff(tcon_Int_Oint___Lattices_Osemilattice__sup_9,axiom,
semilattice_sup(int) ).
tff(tcon_Int_Oint___Lattices_Osemilattice__inf_10,axiom,
semilattice_inf(int) ).
tff(tcon_Int_Oint___Groups_Ocomm__monoid__mult,axiom,
comm_monoid_mult(int) ).
tff(tcon_Int_Oint___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add(int) ).
tff(tcon_Int_Oint___Rings_Olinordered__ring,axiom,
linordered_ring(int) ).
tff(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
linordered_idom(int) ).
tff(tcon_Int_Oint___Complete__Lattices_OSup_11,axiom,
complete_Sup(int) ).
tff(tcon_Int_Oint___Complete__Lattices_OInf_12,axiom,
complete_Inf(int) ).
tff(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
semiring_char_0(int) ).
tff(tcon_Int_Oint___Groups_Oab__group__add,axiom,
ab_group_add(int) ).
tff(tcon_Int_Oint___Countable_Ocountable_13,axiom,
countable(int) ).
tff(tcon_Int_Oint___Orderings_Opreorder_14,axiom,
preorder(int) ).
tff(tcon_Int_Oint___Orderings_Olinorder,axiom,
linorder(int) ).
tff(tcon_Int_Oint___Rings_Osemiring__1,axiom,
semiring_1(int) ).
tff(tcon_Int_Oint___Rings_Osemiring__0,axiom,
semiring_0(int) ).
tff(tcon_Int_Oint___Lattices_Olattice_15,axiom,
lattice(int) ).
tff(tcon_Int_Oint___Groups_Ogroup__add,axiom,
group_add(int) ).
tff(tcon_Int_Oint___GCD_Osemiring__gcd,axiom,
semiring_gcd(int) ).
tff(tcon_Int_Oint___GCD_Osemiring__Gcd,axiom,
semiring_Gcd(int) ).
tff(tcon_Int_Oint___Typerep_Otyperep_16,axiom,
typerep2(int) ).
tff(tcon_Int_Oint___Orderings_Oorder_17,axiom,
order(int) ).
tff(tcon_Int_Oint___Orderings_Oord_18,axiom,
ord(int) ).
tff(tcon_Int_Oint___Rings_Oring__1,axiom,
ring_1(int) ).
tff(tcon_Int_Oint___Heap_Oheap,axiom,
heap(int) ).
tff(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_19,axiom,
condit6923001295902523014norder(nat) ).
tff(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_20,axiom,
condit1219197933456340205attice(nat) ).
tff(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_21,axiom,
bit_un5681908812861735899ations(nat) ).
tff(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_22,axiom,
unique1627219031080169319umeral(nat) ).
tff(tcon_Nat_Onat___Groups_Olinordered__ab__semigroup__add_23,axiom,
linord4140545234300271783up_add(nat) ).
tff(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_24,axiom,
ordere6911136660526730532id_add(nat) ).
tff(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_25,axiom,
cancel_semigroup_add(nat) ).
tff(tcon_Nat_Onat___Rings_Olinordered__semidom_26,axiom,
linordered_semidom(nat) ).
tff(tcon_Nat_Onat___Lattices_Osemilattice__sup_27,axiom,
semilattice_sup(nat) ).
tff(tcon_Nat_Onat___Lattices_Osemilattice__inf_28,axiom,
semilattice_inf(nat) ).
tff(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_29,axiom,
comm_monoid_mult(nat) ).
tff(tcon_Nat_Onat___Groups_Ocomm__monoid__add_30,axiom,
comm_monoid_add(nat) ).
tff(tcon_Nat_Onat___Complete__Lattices_OSup_31,axiom,
complete_Sup(nat) ).
tff(tcon_Nat_Onat___Complete__Lattices_OInf_32,axiom,
complete_Inf(nat) ).
tff(tcon_Nat_Onat___Orderings_Owellorder,axiom,
wellorder(nat) ).
tff(tcon_Nat_Onat___Nat_Osemiring__char__0_33,axiom,
semiring_char_0(nat) ).
tff(tcon_Nat_Onat___Countable_Ocountable_34,axiom,
countable(nat) ).
tff(tcon_Nat_Onat___Orderings_Opreorder_35,axiom,
preorder(nat) ).
tff(tcon_Nat_Onat___Orderings_Olinorder_36,axiom,
linorder(nat) ).
tff(tcon_Nat_Onat___Rings_Osemiring__1_37,axiom,
semiring_1(nat) ).
tff(tcon_Nat_Onat___Rings_Osemiring__0_38,axiom,
semiring_0(nat) ).
tff(tcon_Nat_Onat___Lattices_Olattice_39,axiom,
lattice(nat) ).
tff(tcon_Nat_Onat___GCD_Osemiring__gcd_40,axiom,
semiring_gcd(nat) ).
tff(tcon_Nat_Onat___GCD_Osemiring__Gcd_41,axiom,
semiring_Gcd(nat) ).
tff(tcon_Nat_Onat___Typerep_Otyperep_42,axiom,
typerep2(nat) ).
tff(tcon_Nat_Onat___Orderings_Oorder_43,axiom,
order(nat) ).
tff(tcon_Nat_Onat___Orderings_Oord_44,axiom,
ord(nat) ).
tff(tcon_Nat_Onat___Heap_Oheap_45,axiom,
heap(nat) ).
tff(tcon_Nat_Onat___Nat_Osize,axiom,
size(nat) ).
tff(tcon_Num_Onum___Orderings_Opreorder_46,axiom,
preorder(num) ).
tff(tcon_Num_Onum___Orderings_Olinorder_47,axiom,
linorder(num) ).
tff(tcon_Num_Onum___Typerep_Otyperep_48,axiom,
typerep2(num) ).
tff(tcon_Num_Onum___Orderings_Oorder_49,axiom,
order(num) ).
tff(tcon_Num_Onum___Orderings_Oord_50,axiom,
ord(num) ).
tff(tcon_Num_Onum___Nat_Osize_51,axiom,
size(num) ).
tff(tcon_Rat_Orat___Groups_Olinordered__ab__semigroup__add_52,axiom,
linord4140545234300271783up_add(rat) ).
tff(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_53,axiom,
ordere6911136660526730532id_add(rat) ).
tff(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_54,axiom,
linord5086331880401160121up_add(rat) ).
tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add_55,axiom,
ordered_ab_group_add(rat) ).
tff(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_56,axiom,
cancel_semigroup_add(rat) ).
tff(tcon_Rat_Orat___Rings_Olinordered__semidom_57,axiom,
linordered_semidom(rat) ).
tff(tcon_Rat_Orat___Lattices_Osemilattice__sup_58,axiom,
semilattice_sup(rat) ).
tff(tcon_Rat_Orat___Lattices_Osemilattice__inf_59,axiom,
semilattice_inf(rat) ).
tff(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_60,axiom,
comm_monoid_mult(rat) ).
tff(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
linordered_field(rat) ).
tff(tcon_Rat_Orat___Groups_Ocomm__monoid__add_61,axiom,
comm_monoid_add(rat) ).
tff(tcon_Rat_Orat___Rings_Olinordered__ring_62,axiom,
linordered_ring(rat) ).
tff(tcon_Rat_Orat___Rings_Olinordered__idom_63,axiom,
linordered_idom(rat) ).
tff(tcon_Rat_Orat___Nat_Osemiring__char__0_64,axiom,
semiring_char_0(rat) ).
tff(tcon_Rat_Orat___Groups_Oab__group__add_65,axiom,
ab_group_add(rat) ).
tff(tcon_Rat_Orat___Fields_Ofield__char__0,axiom,
field_char_0(rat) ).
tff(tcon_Rat_Orat___Countable_Ocountable_66,axiom,
countable(rat) ).
tff(tcon_Rat_Orat___Orderings_Opreorder_67,axiom,
preorder(rat) ).
tff(tcon_Rat_Orat___Orderings_Olinorder_68,axiom,
linorder(rat) ).
tff(tcon_Rat_Orat___Rings_Osemiring__1_69,axiom,
semiring_1(rat) ).
tff(tcon_Rat_Orat___Rings_Osemiring__0_70,axiom,
semiring_0(rat) ).
tff(tcon_Rat_Orat___Lattices_Olattice_71,axiom,
lattice(rat) ).
tff(tcon_Rat_Orat___Groups_Ogroup__add_72,axiom,
group_add(rat) ).
tff(tcon_Rat_Orat___Typerep_Otyperep_73,axiom,
typerep2(rat) ).
tff(tcon_Rat_Orat___Orderings_Oorder_74,axiom,
order(rat) ).
tff(tcon_Rat_Orat___Orderings_Oord_75,axiom,
ord(rat) ).
tff(tcon_Rat_Orat___Rings_Oring__1_76,axiom,
ring_1(rat) ).
tff(tcon_Rat_Orat___Fields_Ofield,axiom,
field(rat) ).
tff(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_77,axiom,
! [A19: $tType] : condit1219197933456340205attice(set(A19)) ).
tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__distrib__lattice_78,axiom,
! [A19: $tType] : comple592849572758109894attice(set(A19)) ).
tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__boolean__algebra_79,axiom,
! [A19: $tType] : comple489889107523837845lgebra(set(A19)) ).
tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_80,axiom,
! [A19: $tType] : comple6319245703460814977attice(set(A19)) ).
tff(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_81,axiom,
! [A19: $tType] : boolea8198339166811842893lgebra(set(A19)) ).
tff(tcon_Set_Oset___Lattices_Osemilattice__sup_82,axiom,
! [A19: $tType] : semilattice_sup(set(A19)) ).
tff(tcon_Set_Oset___Lattices_Osemilattice__inf_83,axiom,
! [A19: $tType] : semilattice_inf(set(A19)) ).
tff(tcon_Set_Oset___Complete__Lattices_OSup_84,axiom,
! [A19: $tType] : complete_Sup(set(A19)) ).
tff(tcon_Set_Oset___Complete__Lattices_OInf_85,axiom,
! [A19: $tType] : complete_Inf(set(A19)) ).
tff(tcon_Set_Oset___Countable_Ocountable_86,axiom,
! [A19: $tType] :
( finite_finite(A19)
=> countable(set(A19)) ) ).
tff(tcon_Set_Oset___Orderings_Opreorder_87,axiom,
! [A19: $tType] : preorder(set(A19)) ).
tff(tcon_Set_Oset___Lattices_Olattice_88,axiom,
! [A19: $tType] : lattice(set(A19)) ).
tff(tcon_Set_Oset___Typerep_Otyperep_89,axiom,
! [A19: $tType] :
( typerep2(A19)
=> typerep2(set(A19)) ) ).
tff(tcon_Set_Oset___Orderings_Oorder_90,axiom,
! [A19: $tType] : order(set(A19)) ).
tff(tcon_Set_Oset___Orderings_Oord_91,axiom,
! [A19: $tType] : ord(set(A19)) ).
tff(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_92,axiom,
condit1219197933456340205attice(bool) ).
tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__distrib__lattice_93,axiom,
comple592849572758109894attice(bool) ).
tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__boolean__algebra_94,axiom,
comple489889107523837845lgebra(bool) ).
tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_95,axiom,
comple6319245703460814977attice(bool) ).
tff(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_96,axiom,
boolea8198339166811842893lgebra(bool) ).
tff(tcon_HOL_Obool___Lattices_Osemilattice__sup_97,axiom,
semilattice_sup(bool) ).
tff(tcon_HOL_Obool___Lattices_Osemilattice__inf_98,axiom,
semilattice_inf(bool) ).
tff(tcon_HOL_Obool___Complete__Lattices_OSup_99,axiom,
complete_Sup(bool) ).
tff(tcon_HOL_Obool___Complete__Lattices_OInf_100,axiom,
complete_Inf(bool) ).
tff(tcon_HOL_Obool___Countable_Ocountable_101,axiom,
countable(bool) ).
tff(tcon_HOL_Obool___Orderings_Opreorder_102,axiom,
preorder(bool) ).
tff(tcon_HOL_Obool___Orderings_Olinorder_103,axiom,
linorder(bool) ).
tff(tcon_HOL_Obool___Lattices_Olattice_104,axiom,
lattice(bool) ).
tff(tcon_HOL_Obool___Typerep_Otyperep_105,axiom,
typerep2(bool) ).
tff(tcon_HOL_Obool___Orderings_Oorder_106,axiom,
order(bool) ).
tff(tcon_HOL_Obool___Orderings_Oord_107,axiom,
ord(bool) ).
tff(tcon_HOL_Obool___Heap_Oheap_108,axiom,
heap(bool) ).
tff(tcon_Heap_Oref___Countable_Ocountable_109,axiom,
! [A19: $tType] : countable(ref(A19)) ).
tff(tcon_Heap_Oref___Typerep_Otyperep_110,axiom,
! [A19: $tType] : typerep2(ref(A19)) ).
tff(tcon_Heap_Oref___Heap_Oheap_111,axiom,
! [A19: $tType] : heap(ref(A19)) ).
tff(tcon_Heap_Oref___Nat_Osize_112,axiom,
! [A19: $tType] : size(ref(A19)) ).
tff(tcon_List_Olist___Countable_Ocountable_113,axiom,
! [A19: $tType] :
( countable(A19)
=> countable(list(A19)) ) ).
tff(tcon_List_Olist___Typerep_Otyperep_114,axiom,
! [A19: $tType] :
( typerep2(A19)
=> typerep2(list(A19)) ) ).
tff(tcon_List_Olist___Heap_Oheap_115,axiom,
! [A19: $tType] :
( heap(A19)
=> heap(list(A19)) ) ).
tff(tcon_List_Olist___Nat_Osize_116,axiom,
! [A19: $tType] : size(list(A19)) ).
tff(tcon_Heap_Oarray___Countable_Ocountable_117,axiom,
! [A19: $tType] : countable(array(A19)) ).
tff(tcon_Heap_Oarray___Typerep_Otyperep_118,axiom,
! [A19: $tType] : typerep2(array(A19)) ).
tff(tcon_Heap_Oarray___Heap_Oheap_119,axiom,
! [A19: $tType] : heap(array(A19)) ).
tff(tcon_Heap_Oarray___Nat_Osize_120,axiom,
! [A19: $tType] : size(array(A19)) ).
tff(tcon_String_Ochar___Countable_Ocountable_121,axiom,
countable(char) ).
tff(tcon_String_Ochar___Typerep_Otyperep_122,axiom,
typerep2(char) ).
tff(tcon_String_Ochar___Heap_Oheap_123,axiom,
heap(char) ).
tff(tcon_String_Ochar___Nat_Osize_124,axiom,
size(char) ).
tff(tcon_Sum__Type_Osum___Countable_Ocountable_125,axiom,
! [A19: $tType,A20: $tType] :
( ( countable(A19)
& countable(A20) )
=> countable(sum_sum(A19,A20)) ) ).
tff(tcon_Sum__Type_Osum___Typerep_Otyperep_126,axiom,
! [A19: $tType,A20: $tType] :
( ( typerep2(A19)
& typerep2(A20) )
=> typerep2(sum_sum(A19,A20)) ) ).
tff(tcon_Sum__Type_Osum___Heap_Oheap_127,axiom,
! [A19: $tType,A20: $tType] :
( ( heap(A19)
& heap(A20) )
=> heap(sum_sum(A19,A20)) ) ).
tff(tcon_Sum__Type_Osum___Nat_Osize_128,axiom,
! [A19: $tType,A20: $tType] : size(sum_sum(A19,A20)) ).
tff(tcon_Filter_Ofilter___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_129,axiom,
! [A19: $tType] : condit1219197933456340205attice(filter(A19)) ).
tff(tcon_Filter_Ofilter___Complete__Lattices_Ocomplete__lattice_130,axiom,
! [A19: $tType] : comple6319245703460814977attice(filter(A19)) ).
tff(tcon_Filter_Ofilter___Lattices_Osemilattice__sup_131,axiom,
! [A19: $tType] : semilattice_sup(filter(A19)) ).
tff(tcon_Filter_Ofilter___Lattices_Osemilattice__inf_132,axiom,
! [A19: $tType] : semilattice_inf(filter(A19)) ).
tff(tcon_Filter_Ofilter___Complete__Lattices_OSup_133,axiom,
! [A19: $tType] : complete_Sup(filter(A19)) ).
tff(tcon_Filter_Ofilter___Complete__Lattices_OInf_134,axiom,
! [A19: $tType] : complete_Inf(filter(A19)) ).
tff(tcon_Filter_Ofilter___Orderings_Opreorder_135,axiom,
! [A19: $tType] : preorder(filter(A19)) ).
tff(tcon_Filter_Ofilter___Lattices_Olattice_136,axiom,
! [A19: $tType] : lattice(filter(A19)) ).
tff(tcon_Filter_Ofilter___Typerep_Otyperep_137,axiom,
! [A19: $tType] :
( typerep2(A19)
=> typerep2(filter(A19)) ) ).
tff(tcon_Filter_Ofilter___Orderings_Oorder_138,axiom,
! [A19: $tType] : order(filter(A19)) ).
tff(tcon_Filter_Ofilter___Orderings_Oord_139,axiom,
! [A19: $tType] : ord(filter(A19)) ).
tff(tcon_Option_Ooption___Countable_Ocountable_140,axiom,
! [A19: $tType] :
( countable(A19)
=> countable(option(A19)) ) ).
tff(tcon_Option_Ooption___Typerep_Otyperep_141,axiom,
! [A19: $tType] :
( typerep2(A19)
=> typerep2(option(A19)) ) ).
tff(tcon_Option_Ooption___Heap_Oheap_142,axiom,
! [A19: $tType] :
( heap(A19)
=> heap(option(A19)) ) ).
tff(tcon_Option_Ooption___Nat_Osize_143,axiom,
! [A19: $tType] : size(option(A19)) ).
tff(tcon_Predicate_Oseq___Typerep_Otyperep_144,axiom,
! [A19: $tType] :
( typerep2(A19)
=> typerep2(seq(A19)) ) ).
tff(tcon_Predicate_Opred___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_145,axiom,
! [A19: $tType] : condit1219197933456340205attice(pred(A19)) ).
tff(tcon_Predicate_Opred___Complete__Lattices_Ocomplete__distrib__lattice_146,axiom,
! [A19: $tType] : comple592849572758109894attice(pred(A19)) ).
tff(tcon_Predicate_Opred___Complete__Lattices_Ocomplete__boolean__algebra_147,axiom,
! [A19: $tType] : comple489889107523837845lgebra(pred(A19)) ).
tff(tcon_Predicate_Opred___Complete__Lattices_Ocomplete__lattice_148,axiom,
! [A19: $tType] : comple6319245703460814977attice(pred(A19)) ).
tff(tcon_Predicate_Opred___Boolean__Algebras_Oboolean__algebra_149,axiom,
! [A19: $tType] : boolea8198339166811842893lgebra(pred(A19)) ).
tff(tcon_Predicate_Opred___Lattices_Osemilattice__sup_150,axiom,
! [A19: $tType] : semilattice_sup(pred(A19)) ).
tff(tcon_Predicate_Opred___Lattices_Osemilattice__inf_151,axiom,
! [A19: $tType] : semilattice_inf(pred(A19)) ).
tff(tcon_Predicate_Opred___Complete__Lattices_OSup_152,axiom,
! [A19: $tType] : complete_Sup(pred(A19)) ).
tff(tcon_Predicate_Opred___Complete__Lattices_OInf_153,axiom,
! [A19: $tType] : complete_Inf(pred(A19)) ).
tff(tcon_Predicate_Opred___Orderings_Opreorder_154,axiom,
! [A19: $tType] : preorder(pred(A19)) ).
tff(tcon_Predicate_Opred___Lattices_Olattice_155,axiom,
! [A19: $tType] : lattice(pred(A19)) ).
tff(tcon_Predicate_Opred___Typerep_Otyperep_156,axiom,
! [A19: $tType] :
( typerep2(A19)
=> typerep2(pred(A19)) ) ).
tff(tcon_Predicate_Opred___Orderings_Oorder_157,axiom,
! [A19: $tType] : order(pred(A19)) ).
tff(tcon_Predicate_Opred___Orderings_Oord_158,axiom,
! [A19: $tType] : ord(pred(A19)) ).
tff(tcon_String_Oliteral___Countable_Ocountable_159,axiom,
countable(literal) ).
tff(tcon_String_Oliteral___Orderings_Opreorder_160,axiom,
preorder(literal) ).
tff(tcon_String_Oliteral___Orderings_Olinorder_161,axiom,
linorder(literal) ).
tff(tcon_String_Oliteral___Typerep_Otyperep_162,axiom,
typerep2(literal) ).
tff(tcon_String_Oliteral___Orderings_Oorder_163,axiom,
order(literal) ).
tff(tcon_String_Oliteral___Orderings_Oord_164,axiom,
ord(literal) ).
tff(tcon_String_Oliteral___Heap_Oheap_165,axiom,
heap(literal) ).
tff(tcon_String_Oliteral___Nat_Osize_166,axiom,
size(literal) ).
tff(tcon_Heap__Monad_OHeap___Typerep_Otyperep_167,axiom,
! [A19: $tType] :
( typerep2(A19)
=> typerep2(heap_Heap(A19)) ) ).
tff(tcon_Heap__Monad_OHeap___Nat_Osize_168,axiom,
! [A19: $tType] : size(heap_Heap(A19)) ).
tff(tcon_Typerep_Otyperep___Countable_Ocountable_169,axiom,
countable(typerep) ).
tff(tcon_Typerep_Otyperep___Typerep_Otyperep_170,axiom,
typerep2(typerep) ).
tff(tcon_Typerep_Otyperep___Heap_Oheap_171,axiom,
heap(typerep) ).
tff(tcon_Typerep_Otyperep___Nat_Osize_172,axiom,
size(typerep) ).
tff(tcon_Old__Datatype_Onode___Typerep_Otyperep_173,axiom,
! [A19: $tType,A20: $tType] :
( ( typerep2(A19)
& typerep2(A20) )
=> typerep2(old_node(A19,A20)) ) ).
tff(tcon_Product__Type_Oprod___Countable_Ocountable_174,axiom,
! [A19: $tType,A20: $tType] :
( ( countable(A19)
& countable(A20) )
=> countable(product_prod(A19,A20)) ) ).
tff(tcon_Product__Type_Oprod___Typerep_Otyperep_175,axiom,
! [A19: $tType,A20: $tType] :
( ( typerep2(A19)
& typerep2(A20) )
=> typerep2(product_prod(A19,A20)) ) ).
tff(tcon_Product__Type_Oprod___Heap_Oheap_176,axiom,
! [A19: $tType,A20: $tType] :
( ( heap(A19)
& heap(A20) )
=> heap(product_prod(A19,A20)) ) ).
tff(tcon_Product__Type_Oprod___Nat_Osize_177,axiom,
! [A19: $tType,A20: $tType] : size(product_prod(A19,A20)) ).
tff(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_178,axiom,
condit6923001295902523014norder(product_unit) ).
tff(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_179,axiom,
condit1219197933456340205attice(product_unit) ).
tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__distrib__lattice_180,axiom,
comple592849572758109894attice(product_unit) ).
tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__boolean__algebra_181,axiom,
comple489889107523837845lgebra(product_unit) ).
tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__linorder,axiom,
comple5582772986160207858norder(product_unit) ).
tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_182,axiom,
comple6319245703460814977attice(product_unit) ).
tff(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_183,axiom,
boolea8198339166811842893lgebra(product_unit) ).
tff(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_184,axiom,
semilattice_sup(product_unit) ).
tff(tcon_Product__Type_Ounit___Lattices_Osemilattice__inf_185,axiom,
semilattice_inf(product_unit) ).
tff(tcon_Product__Type_Ounit___Complete__Lattices_OSup_186,axiom,
complete_Sup(product_unit) ).
tff(tcon_Product__Type_Ounit___Complete__Lattices_OInf_187,axiom,
complete_Inf(product_unit) ).
tff(tcon_Product__Type_Ounit___Orderings_Owellorder_188,axiom,
wellorder(product_unit) ).
tff(tcon_Product__Type_Ounit___Countable_Ocountable_189,axiom,
countable(product_unit) ).
tff(tcon_Product__Type_Ounit___Orderings_Opreorder_190,axiom,
preorder(product_unit) ).
tff(tcon_Product__Type_Ounit___Orderings_Olinorder_191,axiom,
linorder(product_unit) ).
tff(tcon_Product__Type_Ounit___Lattices_Olattice_192,axiom,
lattice(product_unit) ).
tff(tcon_Product__Type_Ounit___Typerep_Otyperep_193,axiom,
typerep2(product_unit) ).
tff(tcon_Product__Type_Ounit___Orderings_Oorder_194,axiom,
order(product_unit) ).
tff(tcon_Product__Type_Ounit___Orderings_Oord_195,axiom,
ord(product_unit) ).
tff(tcon_Product__Type_Ounit___Heap_Oheap_196,axiom,
heap(product_unit) ).
tff(tcon_Heap_Oheap_Oheap__ext___Typerep_Otyperep_197,axiom,
! [A19: $tType] :
( typerep2(A19)
=> typerep2(heap_ext(A19)) ) ).
tff(tcon_Code__Evaluation_Oterm___Typerep_Otyperep_198,axiom,
typerep2(code_term) ).
tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_199,axiom,
bit_un5681908812861735899ations(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_200,axiom,
unique1627219031080169319umeral(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__semigroup__add_201,axiom,
linord4140545234300271783up_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_202,axiom,
ordere6911136660526730532id_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_203,axiom,
linord5086331880401160121up_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_204,axiom,
ordered_ab_group_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_205,axiom,
cancel_semigroup_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_206,axiom,
linordered_semidom(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_207,axiom,
comm_monoid_mult(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_208,axiom,
comm_monoid_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_209,axiom,
linordered_ring(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_210,axiom,
linordered_idom(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_211,axiom,
semiring_char_0(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_212,axiom,
ab_group_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_213,axiom,
preorder(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_214,axiom,
linorder(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_215,axiom,
semiring_1(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_216,axiom,
semiring_0(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_217,axiom,
group_add(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Typerep_Otyperep_218,axiom,
typerep2(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Orderings_Oorder_219,axiom,
order(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Orderings_Oord_220,axiom,
ord(code_integer) ).
tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1_221,axiom,
ring_1(code_integer) ).
tff(tcon_Code__Numeral_Onatural___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_222,axiom,
bit_un5681908812861735899ations(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Olinordered__ab__semigroup__add_223,axiom,
linord4140545234300271783up_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Oordered__comm__monoid__add_224,axiom,
ordere6911136660526730532id_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Ocancel__semigroup__add_225,axiom,
cancel_semigroup_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Olinordered__semidom_226,axiom,
linordered_semidom(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__mult_227,axiom,
comm_monoid_mult(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__add_228,axiom,
comm_monoid_add(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Nat_Osemiring__char__0_229,axiom,
semiring_char_0(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Orderings_Opreorder_230,axiom,
preorder(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Orderings_Olinorder_231,axiom,
linorder(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__1_232,axiom,
semiring_1(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Rings_Osemiring__0_233,axiom,
semiring_0(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Typerep_Otyperep_234,axiom,
typerep2(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Orderings_Oorder_235,axiom,
order(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Orderings_Oord_236,axiom,
ord(code_natural) ).
tff(tcon_Code__Numeral_Onatural___Nat_Osize_237,axiom,
size(code_natural) ).
% Helper facts (13)
tff(help_If_2_1_T,axiom,
! [A: $tType,X2: A,Y2: A] : if(A,fFalse,X2,Y2) = Y2 ).
tff(help_If_1_1_T,axiom,
! [A: $tType,X2: A,Y2: A] : if(A,fTrue,X2,Y2) = X2 ).
tff(help_fNot_2_1_U,axiom,
! [P2: bool] :
( pp(P2)
| pp(aa(bool,bool,fNot,P2)) ) ).
tff(help_fNot_1_1_U,axiom,
! [P2: bool] :
( ~ pp(aa(bool,bool,fNot,P2))
| ~ pp(P2) ) ).
tff(help_fTrue_1_1_U,axiom,
pp(fTrue) ).
tff(help_fconj_3_1_U,axiom,
! [P2: bool,Q2: bool] :
( ~ pp(fconj(P2,Q2))
| pp(Q2) ) ).
tff(help_fconj_2_1_U,axiom,
! [P2: bool,Q2: bool] :
( ~ pp(fconj(P2,Q2))
| pp(P2) ) ).
tff(help_fconj_1_1_U,axiom,
! [P2: bool,Q2: bool] :
( ~ pp(P2)
| ~ pp(Q2)
| pp(fconj(P2,Q2)) ) ).
tff(help_fFalse_1_1_T,axiom,
! [P2: bool] :
( ( P2 = fTrue )
| ( P2 = fFalse ) ) ).
tff(help_fFalse_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_fequal_2_1_T,axiom,
! [A: $tType,X2: A,Y2: A] :
( ( X2 != Y2 )
| pp(aa(A,bool,aa(A,fun(A,bool),fequal(A),X2),Y2)) ) ).
tff(help_fequal_1_1_T,axiom,
! [A: $tType,X2: A,Y2: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),fequal(A),X2),Y2))
| ( X2 = Y2 ) ) ).
tff(help_fChoice_1_1_T,axiom,
! [A: $tType,P2: fun(A,bool)] : aa(A,bool,P2,fChoice(A,P2)) = aa(fun(A,bool),bool,fEx(A),P2) ).
% Free types (1)
tff(tfree_0,hypothesis,
heap(a) ).
% Conjectures (2)
tff(conj_0,hypothesis,
$true ).
tff(conj_1,conjecture,
thesis ).
%------------------------------------------------------------------------------