TPTP Problem File: SCT188_5.p
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%------------------------------------------------------------------------------
% File : SCT188_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Social Choice Theory
% Problem : Arrow's Impossibility Theorem line 110
% Version : Especial.
% English : Formalization of two proofs of Arrow's impossibility theorem. One
% formalization is based on utility functions, the other one on
% strict partial orders.
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : arrow_110 [Bla13]
% Status : Unknown
% Rating : 1.00 v6.4.0
% Syntax : Number of formulae : 178 ( 57 unt; 40 typ; 0 def)
% Number of atoms : 277 ( 119 equ)
% Maximal formula atoms : 10 ( 1 avg)
% Number of connectives : 196 ( 57 ~; 25 |; 21 &)
% ( 22 <=>; 71 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 5 avg)
% Maximal term depth : 26 ( 2 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 24 ( 13 >; 11 *; 0 +; 0 <<)
% Number of predicates : 7 ( 6 usr; 0 prp; 1-2 aty)
% Number of functors : 32 ( 32 usr; 9 con; 0-4 aty)
% Number of variables : 474 ( 436 !; 5 ?; 474 :)
% ( 33 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_UNK_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:15:48
%------------------------------------------------------------------------------
%----Should-be-implicit typings (4)
tff(ty_tc_Arrow__Order__Mirabelle__qkbtqzkjxu_Oalt,type,
arrow_411405190le_alt: $tType ).
tff(ty_tc_HOL_Obool,type,
bool: $tType ).
tff(ty_tc_fun,type,
fun: ( $tType * $tType ) > $tType ).
tff(ty_tc_prod,type,
product_prod: ( $tType * $tType ) > $tType ).
%----Explicit typings (36)
tff(sy_cl_Lattices_Obounded__lattice,type,
bounded_lattice:
!>[A: $tType] : $o ).
tff(sy_cl_Lattices_Olattice,type,
lattice:
!>[A: $tType] : $o ).
tff(sy_cl_Lattices_Osemilattice__sup,type,
semilattice_sup:
!>[A: $tType] : $o ).
tff(sy_cl_Lattices_Obounded__lattice__bot,type,
bounded_lattice_bot:
!>[A: $tType] : $o ).
tff(sy_cl_Lattices_Obounded__lattice__top,type,
bounded_lattice_top:
!>[A: $tType] : $o ).
tff(sy_c_Arrow__Order__Mirabelle__qkbtqzkjxu_Oabove,type,
arrow_1158827142_above: ( fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool) * arrow_411405190le_alt * arrow_411405190le_alt ) > fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool) ).
tff(sy_c_Arrow__Order__Mirabelle__qkbtqzkjxu_Obelow,type,
arrow_319942042_below: ( fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool) * arrow_411405190le_alt * arrow_411405190le_alt ) > fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool) ).
tff(sy_c_Arrow__Order__Mirabelle__qkbtqzkjxu_Omkbot,type,
arrow_276188178_mkbot: ( fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool) * arrow_411405190le_alt ) > fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool) ).
tff(sy_c_Arrow__Order__Mirabelle__qkbtqzkjxu_Omktop,type,
arrow_424895264_mktop: ( fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool) * arrow_411405190le_alt ) > fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool) ).
tff(sy_c_COMBB,type,
combb:
!>[B: $tType,C: $tType,A: $tType] : fun(fun(B,C),fun(fun(A,B),fun(A,C))) ).
tff(sy_c_COMBC,type,
combc:
!>[A: $tType,B: $tType,C: $tType] : fun(fun(A,fun(B,C)),fun(B,fun(A,C))) ).
tff(sy_c_COMBI,type,
combi:
!>[A: $tType] : fun(A,A) ).
tff(sy_c_COMBK,type,
combk:
!>[A: $tType,B: $tType] : ( A > fun(B,A) ) ).
tff(sy_c_COMBS,type,
combs:
!>[A: $tType,B: $tType,C: $tType] : fun(fun(A,fun(B,C)),fun(fun(A,B),fun(A,C))) ).
tff(sy_c_Lattices_Osup__class_Osup,type,
sup_sup:
!>[A: $tType] : ( ( A * A ) > A ) ).
tff(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
tff(sy_c_Orderings_Otop__class_Otop,type,
top_top:
!>[A: $tType] : 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_Oprod_Oprod__case,type,
product_prod_case:
!>[A: $tType,B: $tType,T: $tType] : ( fun(A,fun(B,T)) > fun(product_prod(A,B),T) ) ).
tff(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( fun(A,bool) > fun(A,bool) ) ).
tff(sy_c_Set_Oinsert,type,
insert:
!>[A: $tType] : ( ( A * fun(A,bool) ) > fun(A,bool) ) ).
tff(sy_c_aa,type,
aa:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * A ) > B ) ).
tff(sy_c_fAll,type,
fAll:
!>[A: $tType] : fun(fun(A,bool),bool) ).
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: fun(bool,fun(bool,bool)) ).
tff(sy_c_fdisj,type,
fdisj: fun(bool,fun(bool,bool)) ).
tff(sy_c_fequal,type,
fequal:
!>[A: $tType] : fun(A,fun(A,bool)) ).
tff(sy_c_fimplies,type,
fimplies: fun(bool,fun(bool,bool)) ).
tff(sy_c_member,type,
member:
!>[A: $tType] : fun(A,fun(fun(A,bool),bool)) ).
tff(sy_c_pp,type,
pp: bool > $o ).
tff(sy_v_L,type,
l: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool) ).
tff(sy_v_x,type,
x: arrow_411405190le_alt ).
tff(sy_v_y,type,
y: arrow_411405190le_alt ).
%----Relevant facts (99)
tff(fact_0_above__def,axiom,
! [B4: arrow_411405190le_alt,A3: arrow_411405190le_alt,La: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)] : arrow_1158827142_above(La,A3,B4) = sup_sup(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),sup_sup(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),sup_sup(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),collect(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_prod_case(arrow_411405190le_alt,arrow_411405190le_alt,bool,aa(fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool)),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool)),aa(fun(arrow_411405190le_alt,fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool))),fun(fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool)),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool))),combs(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool)),aa(fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(bool,bool))),fun(arrow_411405190le_alt,fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool))),aa(fun(fun(arrow_411405190le_alt,fun(bool,bool)),fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool))),fun(fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(bool,bool))),fun(arrow_411405190le_alt,fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool)))),combb(fun(arrow_411405190le_alt,fun(bool,bool)),fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool)),arrow_411405190le_alt),combs(arrow_411405190le_alt,bool,bool)),aa(fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool)),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(bool,bool))),aa(fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,fun(bool,bool))),fun(fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool)),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(bool,bool)))),combb(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,fun(bool,bool)),arrow_411405190le_alt),aa(fun(bool,fun(bool,bool)),fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,fun(bool,bool))),combb(bool,fun(bool,bool),arrow_411405190le_alt),fconj)),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool)),aa(fun(arrow_411405190le_alt,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,bool))),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool))),combc(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,bool)),aa(fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool))),fun(arrow_411405190le_alt,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,bool))),aa(fun(fun(arrow_411405190le_alt,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool)),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,bool))),fun(fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool))),fun(arrow_411405190le_alt,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,bool)))),combb(fun(arrow_411405190le_alt,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool)),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,bool)),arrow_411405190le_alt),combc(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool)),aa(fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt))),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool))),aa(fun(fun(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),fun(arrow_411405190le_alt,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool))),fun(fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt))),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool)))),combb(fun(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),fun(arrow_411405190le_alt,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool)),arrow_411405190le_alt),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool)),fun(fun(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),fun(arrow_411405190le_alt,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool))),combb(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool),arrow_411405190le_alt),member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt)))),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt)))),La)))),aa(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool)),aa(fun(arrow_411405190le_alt,fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool))),fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool))),combc(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool)),aa(fun(arrow_411405190le_alt,fun(bool,bool)),fun(arrow_411405190le_alt,fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool))),aa(fun(fun(bool,bool),fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool))),fun(fun(arrow_411405190le_alt,fun(bool,bool)),fun(arrow_411405190le_alt,fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool)))),combb(fun(bool,bool),fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool)),arrow_411405190le_alt),combb(bool,bool,arrow_411405190le_alt)),aa(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,fun(bool,bool)),aa(fun(bool,fun(bool,bool)),fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,fun(bool,bool))),combb(bool,fun(bool,bool),arrow_411405190le_alt),fconj),aa(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool),aa(fun(bool,bool),fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool)),combb(bool,bool,arrow_411405190le_alt),fNot),aa(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool),aa(fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool)),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool)),combc(arrow_411405190le_alt,arrow_411405190le_alt,bool),fequal(arrow_411405190le_alt)),B4))))),aa(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool),aa(fun(bool,bool),fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool)),combb(bool,bool,arrow_411405190le_alt),fNot),aa(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool),aa(fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool)),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool)),combc(arrow_411405190le_alt,arrow_411405190le_alt,bool),fequal(arrow_411405190le_alt)),B4)))))),insert(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),aa(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt),aa(arrow_411405190le_alt,fun(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt),A3),B4),bot_bot(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)))),collect(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(arrow_411405190le_alt,bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),aa(fun(fun(arrow_411405190le_alt,bool),bool),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(arrow_411405190le_alt,bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combb(fun(arrow_411405190le_alt,bool),bool,product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),fEx(arrow_411405190le_alt)),aa(fun(arrow_411405190le_alt,bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(arrow_411405190le_alt,bool)),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool))),fun(fun(arrow_411405190le_alt,bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(arrow_411405190le_alt,bool))),combc(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool)),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(arrow_411405190le_alt,fun(bool,bool))),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool))),aa(fun(fun(arrow_411405190le_alt,fun(bool,bool)),fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool))),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(arrow_411405190le_alt,fun(bool,bool))),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool)))),combb(fun(arrow_411405190le_alt,fun(bool,bool)),fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool)),product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),combs(arrow_411405190le_alt,bool,bool)),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(arrow_411405190le_alt,bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(arrow_411405190le_alt,fun(bool,bool))),aa(fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,fun(bool,bool))),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(arrow_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).
tff(fact_1_below__def,axiom,
! [B4: arrow_411405190le_alt,A3: arrow_411405190le_alt,La: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)] : arrow_319942042_below(La,A3,B4) = 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).
tff(fact_2_ball__UNIV,axiom,
! [A: $tType,P2: fun(A,bool)] :
( ! [X3: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X3),top_top(fun(A,bool))))
=> pp(aa(A,bool,P2,X3)) )
<=> ! [X11: A] : pp(aa(A,bool,P2,X11)) ) ).
tff(fact_3_Collect__const,axiom,
! [A: $tType,P2: bool] :
( ( pp(P2)
=> ( collect(A,combk(bool,A,P2)) = top_top(fun(A,bool)) ) )
& ( ~ pp(P2)
=> ( collect(A,combk(bool,A,P2)) = bot_bot(fun(A,bool)) ) ) ) ).
tff(fact_4_singleton__conv2,axiom,
! [A: $tType,A3: A] : collect(A,aa(A,fun(A,bool),fequal(A),A3)) = insert(A,A3,bot_bot(fun(A,bool))) ).
tff(fact_5_singleton__conv,axiom,
! [A: $tType,A3: A] : collect(A,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),combc(A,A,bool),fequal(A)),A3)) = insert(A,A3,bot_bot(fun(A,bool))) ).
tff(fact_6_mkbot__def,axiom,
! [B4: arrow_411405190le_alt,La: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)] : arrow_276188178_mkbot(La,B4) = sup_sup(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),collect(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_prod_case(arrow_411405190le_alt,arrow_411405190le_alt,bool,aa(fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool)),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool)),aa(fun(arrow_411405190le_alt,fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool))),fun(fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool)),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool))),combs(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool)),aa(fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(bool,bool))),fun(arrow_411405190le_alt,fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool))),aa(fun(fun(arrow_411405190le_alt,fun(bool,bool)),fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool))),fun(fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(bool,bool))),fun(arrow_411405190le_alt,fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool)))),combb(fun(arrow_411405190le_alt,fun(bool,bool)),fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool)),arrow_411405190le_alt),combs(arrow_411405190le_alt,bool,bool)),aa(fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool)),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(bool,bool))),aa(fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,fun(bool,bool))),fun(fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool)),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(bool,bool)))),combb(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,fun(bool,bool)),arrow_411405190le_alt),aa(fun(bool,fun(bool,bool)),fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,fun(bool,bool))),combb(bool,fun(bool,bool),arrow_411405190le_alt),fconj)),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool)),aa(fun(arrow_411405190le_alt,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,bool))),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool))),combc(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,bool)),aa(fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool))),fun(arrow_411405190le_alt,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,bool))),aa(fun(fun(arrow_411405190le_alt,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool)),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,bool))),fun(fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool))),fun(arrow_411405190le_alt,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,bool)))),combb(fun(arrow_411405190le_alt,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool)),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,bool)),arrow_411405190le_alt),combc(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool)),aa(fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt))),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool))),aa(fun(fun(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),fun(arrow_411405190le_alt,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool))),fun(fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt))),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool)))),combb(fun(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),fun(arrow_411405190le_alt,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool)),arrow_411405190le_alt),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool)),fun(fun(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),fun(arrow_411405190le_alt,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool))),combb(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool),arrow_411405190le_alt),member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt)))),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt)))),La)))),aa(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool)),aa(fun(arrow_411405190le_alt,fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool))),fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool))),combc(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool)),aa(fun(arrow_411405190le_alt,fun(bool,bool)),fun(arrow_411405190le_alt,fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool))),aa(fun(fun(bool,bool),fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool))),fun(fun(arrow_411405190le_alt,fun(bool,bool)),fun(arrow_411405190le_alt,fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool)))),combb(fun(bool,bool),fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool)),arrow_411405190le_alt),combb(bool,bool,arrow_411405190le_alt)),aa(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,fun(bool,bool)),aa(fun(bool,fun(bool,bool)),fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,fun(bool,bool))),combb(bool,fun(bool,bool),arrow_411405190le_alt),fconj),aa(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool),aa(fun(bool,bool),fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool)),combb(bool,bool,arrow_411405190le_alt),fNot),aa(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool),aa(fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool)),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool)),combc(arrow_411405190le_alt,arrow_411405190le_alt,bool),fequal(arrow_411405190le_alt)),B4))))),aa(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool),aa(fun(bool,bool),fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool)),combb(bool,bool,arrow_411405190le_alt),fNot),aa(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool),aa(fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool)),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool)),combc(arrow_411405190le_alt,arrow_411405190le_alt,bool),fequal(arrow_411405190le_alt)),B4)))))),collect(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(arrow_411405190le_alt,bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),aa(fun(fun(arrow_411405190le_alt,bool),bool),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(arrow_411405190le_alt,bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combb(fun(arrow_411405190le_alt,bool),bool,product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),fEx(arrow_411405190le_alt)),aa(fun(arrow_411405190le_alt,bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(arrow_411405190le_alt,bool)),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool))),fun(fun(arrow_411405190le_alt,bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(arrow_411405190le_alt,bool))),combc(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool)),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(arrow_411405190le_alt,fun(bool,bool))),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool))),aa(fun(fun(arrow_411405190le_alt,fun(bool,bool)),fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool))),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(arrow_411405190le_alt,fun(bool,bool))),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool)))),combb(fun(arrow_411405190le_alt,fun(bool,bool)),fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool)),product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),combs(arrow_411405190le_alt,bool,bool)),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(arrow_411405190le_alt,bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(arrow_411405190le_alt,fun(bool,bool))),aa(fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,fun(bool,bool))),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(arrow_411405190le_alt,bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(arrow_411405190le_alt,fun(bool,bool)))),combb(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,fun(bool,bool)),product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),aa(fun(bool,fun(bool,bool)),fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,fun(bool,bool))),combb(bool,fun(bool,bool),arrow_411405190le_alt),fconj)),aa(fun(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(arrow_411405190le_alt,bool)),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(fun(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),fun(arrow_411405190le_alt,bool))),fun(fun(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(arrow_411405190le_alt,bool))),combc(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),fun(arrow_411405190le_alt,bool)),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(fun(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),fun(arrow_411405190le_alt,bool))),aa(fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),fun(arrow_411405190le_alt,bool))),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(fun(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),fun(arrow_411405190le_alt,bool)))),combb(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),fun(arrow_411405190le_alt,bool)),product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),combb(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool,arrow_411405190le_alt)),fequal(product_prod(arrow_411405190le_alt,arrow_411405190le_alt)))),aa(arrow_411405190le_alt,fun(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt),B4))))),aa(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool),aa(fun(bool,bool),fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool)),combb(bool,bool,arrow_411405190le_alt),fNot),aa(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool),aa(fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool)),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool)),combc(arrow_411405190le_alt,arrow_411405190le_alt,bool),fequal(arrow_411405190le_alt)),B4)))))) ).
tff(fact_7_mktop__def,axiom,
! [B4: arrow_411405190le_alt,La: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)] : arrow_424895264_mktop(La,B4) = sup_sup(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),collect(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_prod_case(arrow_411405190le_alt,arrow_411405190le_alt,bool,aa(fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool)),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool)),aa(fun(arrow_411405190le_alt,fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool))),fun(fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool)),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool))),combs(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool)),aa(fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(bool,bool))),fun(arrow_411405190le_alt,fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool))),aa(fun(fun(arrow_411405190le_alt,fun(bool,bool)),fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool))),fun(fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(bool,bool))),fun(arrow_411405190le_alt,fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool)))),combb(fun(arrow_411405190le_alt,fun(bool,bool)),fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool)),arrow_411405190le_alt),combs(arrow_411405190le_alt,bool,bool)),aa(fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool)),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(bool,bool))),aa(fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,fun(bool,bool))),fun(fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool)),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(bool,bool)))),combb(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,fun(bool,bool)),arrow_411405190le_alt),aa(fun(bool,fun(bool,bool)),fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,fun(bool,bool))),combb(bool,fun(bool,bool),arrow_411405190le_alt),fconj)),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool)),aa(fun(arrow_411405190le_alt,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,bool))),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool))),combc(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,bool)),aa(fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool))),fun(arrow_411405190le_alt,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,bool))),aa(fun(fun(arrow_411405190le_alt,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool)),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,bool))),fun(fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool))),fun(arrow_411405190le_alt,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,bool)))),combb(fun(arrow_411405190le_alt,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool)),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,bool)),arrow_411405190le_alt),combc(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool)),aa(fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt))),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool))),aa(fun(fun(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),fun(arrow_411405190le_alt,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool))),fun(fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt))),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool)))),combb(fun(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),fun(arrow_411405190le_alt,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool)),arrow_411405190le_alt),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool)),fun(fun(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),fun(arrow_411405190le_alt,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool))),combb(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool),arrow_411405190le_alt),member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt)))),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt)))),La)))),aa(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool)),aa(fun(arrow_411405190le_alt,fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool))),fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool))),combc(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool)),aa(fun(arrow_411405190le_alt,fun(bool,bool)),fun(arrow_411405190le_alt,fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool))),aa(fun(fun(bool,bool),fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool))),fun(fun(arrow_411405190le_alt,fun(bool,bool)),fun(arrow_411405190le_alt,fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool)))),combb(fun(bool,bool),fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool)),arrow_411405190le_alt),combb(bool,bool,arrow_411405190le_alt)),aa(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,fun(bool,bool)),aa(fun(bool,fun(bool,bool)),fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,fun(bool,bool))),combb(bool,fun(bool,bool),arrow_411405190le_alt),fconj),aa(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool),aa(fun(bool,bool),fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool)),combb(bool,bool,arrow_411405190le_alt),fNot),aa(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool),aa(fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool)),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool)),combc(arrow_411405190le_alt,arrow_411405190le_alt,bool),fequal(arrow_411405190le_alt)),B4))))),aa(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool),aa(fun(bool,bool),fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool)),combb(bool,bool,arrow_411405190le_alt),fNot),aa(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool),aa(fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool)),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool)),combc(arrow_411405190le_alt,arrow_411405190le_alt,bool),fequal(arrow_411405190le_alt)),B4)))))),collect(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(arrow_411405190le_alt,bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),aa(fun(fun(arrow_411405190le_alt,bool),bool),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(arrow_411405190le_alt,bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combb(fun(arrow_411405190le_alt,bool),bool,product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),fEx(arrow_411405190le_alt)),aa(fun(arrow_411405190le_alt,bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(arrow_411405190le_alt,bool)),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool))),fun(fun(arrow_411405190le_alt,bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(arrow_411405190le_alt,bool))),combc(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool)),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(arrow_411405190le_alt,fun(bool,bool))),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool))),aa(fun(fun(arrow_411405190le_alt,fun(bool,bool)),fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool))),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(arrow_411405190le_alt,fun(bool,bool))),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool)))),combb(fun(arrow_411405190le_alt,fun(bool,bool)),fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool)),product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),combs(arrow_411405190le_alt,bool,bool)),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(arrow_411405190le_alt,bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(arrow_411405190le_alt,fun(bool,bool))),aa(fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,fun(bool,bool))),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(arrow_411405190le_alt,bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(arrow_411405190le_alt,fun(bool,bool)))),combb(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,fun(bool,bool)),product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),aa(fun(bool,fun(bool,bool)),fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,fun(bool,bool))),combb(bool,fun(bool,bool),arrow_411405190le_alt),fconj)),aa(fun(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(arrow_411405190le_alt,bool)),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(fun(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),fun(arrow_411405190le_alt,bool))),fun(fun(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(arrow_411405190le_alt,bool))),combc(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),fun(arrow_411405190le_alt,bool)),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(fun(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),fun(arrow_411405190le_alt,bool))),aa(fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),fun(arrow_411405190le_alt,bool))),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(fun(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),fun(arrow_411405190le_alt,bool)))),combb(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),fun(arrow_411405190le_alt,bool)),product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),combb(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool,arrow_411405190le_alt)),fequal(product_prod(arrow_411405190le_alt,arrow_411405190le_alt)))),aa(arrow_411405190le_alt,fun(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),aa(fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt))),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt))),combc(arrow_411405190le_alt,arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt)),B4))))),aa(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool),aa(fun(bool,bool),fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool)),combb(bool,bool,arrow_411405190le_alt),fNot),aa(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool),aa(fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool)),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool)),combc(arrow_411405190le_alt,arrow_411405190le_alt,bool),fequal(arrow_411405190le_alt)),B4)))))) ).
tff(fact_8_splitI,axiom,
! [A: $tType,B: $tType,B4: B,A3: A,F: fun(A,fun(B,bool))] :
( pp(aa(B,bool,aa(A,fun(B,bool),F,A3),B4))
=> pp(aa(product_prod(A,B),bool,product_prod_case(A,B,bool,F),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B4))) ) ).
tff(fact_9_prod__caseI,axiom,
! [A: $tType,B: $tType,B4: B,A3: A,F1: fun(A,fun(B,bool))] :
( pp(aa(B,bool,aa(A,fun(B,bool),F1,A3),B4))
=> pp(aa(product_prod(A,B),bool,product_prod_case(A,B,bool,F1),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B4))) ) ).
tff(fact_10_split__conv,axiom,
! [B: $tType,A: $tType,C: $tType,B4: C,A3: B,F: fun(B,fun(C,A))] : aa(product_prod(B,C),A,product_prod_case(B,C,A,F),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A3),B4)) = aa(C,A,aa(B,fun(C,A),F,A3),B4) ).
tff(fact_11_Un__insert__right,axiom,
! [A: $tType,B3: fun(A,bool),A3: A,A4: fun(A,bool)] : sup_sup(fun(A,bool),A4,insert(A,A3,B3)) = insert(A,A3,sup_sup(fun(A,bool),A4,B3)) ).
tff(fact_12_mem__splitI,axiom,
! [A: $tType,B: $tType,C: $tType,B4: C,A3: B,C1: fun(B,fun(C,fun(A,bool))),Z: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Z),aa(C,fun(A,bool),aa(B,fun(C,fun(A,bool)),C1,A3),B4)))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Z),aa(product_prod(B,C),fun(A,bool),product_prod_case(B,C,fun(A,bool),C1),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A3),B4)))) ) ).
tff(fact_13_Pair__eq,axiom,
! [A: $tType,B: $tType,B7: B,A7: A,B4: B,A3: A] :
( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B4) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A7),B7) )
<=> ( ( A3 = A7 )
& ( B4 = B7 ) ) ) ).
tff(fact_14_split__paired__All,axiom,
! [A: $tType,B: $tType,P2: fun(product_prod(A,B),bool)] :
( ! [X11: product_prod(A,B)] : pp(aa(product_prod(A,B),bool,P2,X11))
<=> ! [A6: A,B6: 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),A6),B6))) ) ).
tff(fact_15_emptyE,axiom,
! [A: $tType,A3: A] : ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A3),bot_bot(fun(A,bool)))) ).
tff(fact_16_Collect__empty__eq,axiom,
! [A: $tType,P2: fun(A,bool)] :
( ( collect(A,P2) = bot_bot(fun(A,bool)) )
<=> ! [X3: A] : ~ pp(aa(A,bool,P2,X3)) ) ).
tff(fact_17_empty__iff,axiom,
! [A: $tType,C1: A] : ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),bot_bot(fun(A,bool)))) ).
tff(fact_18_empty__Collect__eq,axiom,
! [A: $tType,P2: fun(A,bool)] :
( ( bot_bot(fun(A,bool)) = collect(A,P2) )
<=> ! [X3: A] : ~ pp(aa(A,bool,P2,X3)) ) ).
tff(fact_19_all__not__in__conv,axiom,
! [A: $tType,A4: fun(A,bool)] :
( ! [X3: A] : ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X3),A4))
<=> ( A4 = bot_bot(fun(A,bool)) ) ) ).
tff(fact_20_insertCI,axiom,
! [A: $tType,B4: A,B3: fun(A,bool),A3: A] :
( ( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A3),B3))
=> ( A3 = B4 ) )
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A3),insert(A,B4,B3))) ) ).
tff(fact_21_insertE,axiom,
! [A: $tType,A4: fun(A,bool),B4: A,A3: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A3),insert(A,B4,A4)))
=> ( ( A3 != B4 )
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A3),A4)) ) ) ).
tff(fact_22_insert__iff,axiom,
! [A: $tType,A4: fun(A,bool),B4: A,A3: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A3),insert(A,B4,A4)))
<=> ( ( A3 = B4 )
| pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A3),A4)) ) ) ).
tff(fact_23_insert__absorb2,axiom,
! [A: $tType,A4: fun(A,bool),Xa: A] : insert(A,Xa,insert(A,Xa,A4)) = insert(A,Xa,A4) ).
tff(fact_24_UNIV__I,axiom,
! [A: $tType,Xa: A] : pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Xa),top_top(fun(A,bool)))) ).
tff(fact_25_UnCI,axiom,
! [A: $tType,A4: fun(A,bool),B3: fun(A,bool),C1: A] :
( ( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),B3))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),A4)) )
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),sup_sup(fun(A,bool),A4,B3))) ) ).
tff(fact_26_UnE,axiom,
! [A: $tType,B3: fun(A,bool),A4: fun(A,bool),C1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),sup_sup(fun(A,bool),A4,B3)))
=> ( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),A4))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),B3)) ) ) ).
tff(fact_27_Un__iff,axiom,
! [A: $tType,B3: fun(A,bool),A4: fun(A,bool),C1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),sup_sup(fun(A,bool),A4,B3)))
<=> ( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),A4))
| pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),B3)) ) ) ).
tff(fact_28_split__part,axiom,
! [B: $tType,A: $tType,Q1: fun(A,fun(B,bool)),P2: bool,X2: product_prod(A,B)] :
( pp(aa(product_prod(A,B),bool,product_prod_case(A,B,bool,aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(fun(B,bool),fun(B,bool)),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),combb(fun(B,bool),fun(B,bool),A),aa(fun(bool,bool),fun(fun(B,bool),fun(B,bool)),combb(bool,bool,B),aa(bool,fun(bool,bool),fconj,P2))),Q1)),X2))
<=> ( pp(P2)
& pp(aa(product_prod(A,B),bool,product_prod_case(A,B,bool,Q1),X2)) ) ) ).
tff(fact_29_ball__empty,axiom,
! [A: $tType,P2: fun(A,bool),X2: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X2),bot_bot(fun(A,bool))))
=> pp(aa(A,bool,P2,X2)) ) ).
tff(fact_30_Un__empty,axiom,
! [A: $tType,B3: fun(A,bool),A4: fun(A,bool)] :
( ( sup_sup(fun(A,bool),A4,B3) = bot_bot(fun(A,bool)) )
<=> ( ( A4 = bot_bot(fun(A,bool)) )
& ( B3 = bot_bot(fun(A,bool)) ) ) ) ).
tff(fact_31_Un__insert__left,axiom,
! [A: $tType,C2: fun(A,bool),B3: fun(A,bool),A3: A] : sup_sup(fun(A,bool),insert(A,A3,B3),C2) = insert(A,A3,sup_sup(fun(A,bool),B3,C2)) ).
tff(fact_32_splitD_H,axiom,
! [B: $tType,A: $tType,C: $tType,C1: C,B4: B,A3: A,R1: fun(A,fun(B,fun(C,bool)))] :
( pp(aa(C,bool,aa(product_prod(A,B),fun(C,bool),product_prod_case(A,B,fun(C,bool),R1),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B4)),C1))
=> pp(aa(C,bool,aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),R1,A3),B4),C1)) ) ).
tff(fact_33_Pair__inject,axiom,
! [A: $tType,B: $tType,B5: B,A5: A,B1: B,A1: A] :
( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A1),B1) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B5) )
=> ~ ( ( A1 = A5 )
=> ( B1 != B5 ) ) ) ).
tff(fact_34_equals0D,axiom,
! [A: $tType,A3: A,A4: fun(A,bool)] :
( ( A4 = bot_bot(fun(A,bool)) )
=> ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A3),A4)) ) ).
tff(fact_35_ex__in__conv,axiom,
! [A: $tType,A4: fun(A,bool)] :
( ? [X3: A] : pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X3),A4))
<=> ( A4 != bot_bot(fun(A,bool)) ) ) ).
tff(fact_36_empty__def,axiom,
! [A: $tType] : bot_bot(fun(A,bool)) = collect(A,combk(bool,A,fFalse)) ).
tff(fact_37_insert__absorb,axiom,
! [A: $tType,A4: fun(A,bool),A3: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A3),A4))
=> ( insert(A,A3,A4) = A4 ) ) ).
tff(fact_38_insertI2,axiom,
! [A: $tType,B4: A,B3: fun(A,bool),A3: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A3),B3))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A3),insert(A,B4,B3))) ) ).
tff(fact_39_insert__eq__iff,axiom,
! [A: $tType,B3: fun(A,bool),B4: A,A4: fun(A,bool),A3: A] :
( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A3),A4))
=> ( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),B4),B3))
=> ( ( insert(A,A3,A4) = insert(A,B4,B3) )
<=> ( ( ( A3 = B4 )
=> ( A4 = B3 ) )
& ( ( A3 != B4 )
=> ? [C3: fun(A,bool)] :
( ( A4 = insert(A,B4,C3) )
& ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),B4),C3))
& ( B3 = insert(A,A3,C3) )
& ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A3),C3)) ) ) ) ) ) ) ).
tff(fact_40_insert__ident,axiom,
! [A: $tType,B3: fun(A,bool),A4: fun(A,bool),Xa: A] :
( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Xa),A4))
=> ( ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Xa),B3))
=> ( ( insert(A,Xa,A4) = insert(A,Xa,B3) )
<=> ( A4 = B3 ) ) ) ) ).
tff(fact_41_insert__code,axiom,
! [A: $tType,Xa: A,A4: fun(A,bool),Ya: A] :
( pp(aa(A,bool,insert(A,Ya,A4),Xa))
<=> ( ( Ya = Xa )
| pp(aa(A,bool,A4,Xa)) ) ) ).
tff(fact_42_insert__commute,axiom,
! [A: $tType,A4: fun(A,bool),Ya: A,Xa: A] : insert(A,Xa,insert(A,Ya,A4)) = insert(A,Ya,insert(A,Xa,A4)) ).
tff(fact_43_insert__Collect,axiom,
! [A: $tType,P2: fun(A,bool),A3: A] : insert(A,A3,collect(A,P2)) = collect(A,aa(fun(A,bool),fun(A,bool),aa(fun(A,fun(bool,bool)),fun(fun(A,bool),fun(A,bool)),combs(A,bool,bool),aa(fun(A,bool),fun(A,fun(bool,bool)),aa(fun(bool,fun(bool,bool)),fun(fun(A,bool),fun(A,fun(bool,bool))),combb(bool,fun(bool,bool),A),fimplies),aa(fun(A,bool),fun(A,bool),aa(fun(bool,bool),fun(fun(A,bool),fun(A,bool)),combb(bool,bool,A),fNot),aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),combc(A,A,bool),fequal(A)),A3)))),P2)) ).
tff(fact_44_insert__compr,axiom,
! [A: $tType,B3: fun(A,bool),A3: A] : insert(A,A3,B3) = collect(A,aa(fun(A,bool),fun(A,bool),aa(fun(A,fun(bool,bool)),fun(fun(A,bool),fun(A,bool)),combs(A,bool,bool),aa(fun(A,bool),fun(A,fun(bool,bool)),aa(fun(bool,fun(bool,bool)),fun(fun(A,bool),fun(A,fun(bool,bool))),combb(bool,fun(bool,bool),A),fdisj),aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),combc(A,A,bool),fequal(A)),A3))),aa(fun(A,bool),fun(A,bool),aa(fun(A,fun(fun(A,bool),bool)),fun(fun(A,bool),fun(A,bool)),combc(A,fun(A,bool),bool),member(A)),B3))) ).
tff(fact_45_insertI1,axiom,
! [A: $tType,B3: fun(A,bool),A3: A] : pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A3),insert(A,A3,B3))) ).
tff(fact_46_UNIV__def,axiom,
! [A: $tType] : top_top(fun(A,bool)) = collect(A,combk(bool,A,fTrue)) ).
tff(fact_47_UnI2,axiom,
! [A: $tType,A4: fun(A,bool),B3: fun(A,bool),C1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),B3))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),sup_sup(fun(A,bool),A4,B3))) ) ).
tff(fact_48_UnI1,axiom,
! [A: $tType,B3: fun(A,bool),A4: fun(A,bool),C1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),A4))
=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),C1),sup_sup(fun(A,bool),A4,B3))) ) ).
tff(fact_49_ball__Un,axiom,
! [A: $tType,P2: fun(A,bool),B3: fun(A,bool),A4: fun(A,bool)] :
( ! [X3: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X3),sup_sup(fun(A,bool),A4,B3)))
=> pp(aa(A,bool,P2,X3)) )
<=> ( ! [X3: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X3),A4))
=> pp(aa(A,bool,P2,X3)) )
& ! [X3: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X3),B3))
=> pp(aa(A,bool,P2,X3)) ) ) ) ).
tff(fact_50_bex__Un,axiom,
! [A: $tType,P2: fun(A,bool),B3: fun(A,bool),A4: fun(A,bool)] :
( ? [X3: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X3),sup_sup(fun(A,bool),A4,B3)))
& pp(aa(A,bool,P2,X3)) )
<=> ( ? [X3: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X3),A4))
& pp(aa(A,bool,P2,X3)) )
| ? [X3: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X3),B3))
& pp(aa(A,bool,P2,X3)) ) ) ) ).
tff(fact_51_Un__assoc,axiom,
! [A: $tType,C2: fun(A,bool),B3: fun(A,bool),A4: fun(A,bool)] : sup_sup(fun(A,bool),sup_sup(fun(A,bool),A4,B3),C2) = sup_sup(fun(A,bool),A4,sup_sup(fun(A,bool),B3,C2)) ).
tff(fact_52_Un__left__commute,axiom,
! [A: $tType,C2: fun(A,bool),B3: fun(A,bool),A4: fun(A,bool)] : sup_sup(fun(A,bool),A4,sup_sup(fun(A,bool),B3,C2)) = sup_sup(fun(A,bool),B3,sup_sup(fun(A,bool),A4,C2)) ).
tff(fact_53_Un__left__absorb,axiom,
! [A: $tType,B3: fun(A,bool),A4: fun(A,bool)] : sup_sup(fun(A,bool),A4,sup_sup(fun(A,bool),A4,B3)) = sup_sup(fun(A,bool),A4,B3) ).
tff(fact_54_Un__commute,axiom,
! [A: $tType,B3: fun(A,bool),A4: fun(A,bool)] : sup_sup(fun(A,bool),A4,B3) = sup_sup(fun(A,bool),B3,A4) ).
tff(fact_55_Un__def,axiom,
! [A: $tType,B3: fun(A,bool),A4: fun(A,bool)] : sup_sup(fun(A,bool),A4,B3) = collect(A,aa(fun(A,bool),fun(A,bool),aa(fun(A,fun(bool,bool)),fun(fun(A,bool),fun(A,bool)),combs(A,bool,bool),aa(fun(A,bool),fun(A,fun(bool,bool)),aa(fun(bool,fun(bool,bool)),fun(fun(A,bool),fun(A,fun(bool,bool))),combb(bool,fun(bool,bool),A),fdisj),aa(fun(A,bool),fun(A,bool),aa(fun(A,fun(fun(A,bool),bool)),fun(fun(A,bool),fun(A,bool)),combc(A,fun(A,bool),bool),member(A)),A4))),aa(fun(A,bool),fun(A,bool),aa(fun(A,fun(fun(A,bool),bool)),fun(fun(A,bool),fun(A,bool)),combc(A,fun(A,bool),bool),member(A)),B3))) ).
tff(fact_56_Un__absorb,axiom,
! [A: $tType,A4: fun(A,bool)] : sup_sup(fun(A,bool),A4,A4) = A4 ).
tff(fact_57_split__weak__cong,axiom,
! [C: $tType,B: $tType,A: $tType,C1: fun(A,fun(B,C)),Q2: product_prod(A,B),P1: product_prod(A,B)] :
( ( P1 = Q2 )
=> ( aa(product_prod(A,B),C,product_prod_case(A,B,C,C1),P1) = aa(product_prod(A,B),C,product_prod_case(A,B,C,C1),Q2) ) ) ).
tff(fact_58_in__mktop,axiom,
! [Z: arrow_411405190le_alt,La: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Ya: arrow_411405190le_alt,Xa: arrow_411405190le_alt] :
( pp(aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool,aa(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool),member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),aa(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt),aa(arrow_411405190le_alt,fun(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt),Xa),Ya)),arrow_424895264_mktop(La,Z)))
<=> ( ( Xa != Z )
& ( ( Ya = Z )
=> ( Xa != Ya ) )
& ( ( Ya != Z )
=> pp(aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool,aa(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool),member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),aa(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt),aa(arrow_411405190le_alt,fun(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt),Xa),Ya)),La)) ) ) ) ).
tff(fact_59_in__mkbot,axiom,
! [Z: arrow_411405190le_alt,La: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Ya: arrow_411405190le_alt,Xa: arrow_411405190le_alt] :
( pp(aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool,aa(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool),member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),aa(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt),aa(arrow_411405190le_alt,fun(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt),Xa),Ya)),arrow_276188178_mkbot(La,Z)))
<=> ( ( Ya != Z )
& ( ( Xa = Z )
=> ( Xa != Ya ) )
& ( ( Xa != Z )
=> pp(aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool,aa(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool),member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),aa(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt),aa(arrow_411405190le_alt,fun(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt),Xa),Ya)),La)) ) ) ) ).
tff(fact_60_insert__compr__raw,axiom,
! [A: $tType,X2: A,Xa1: fun(A,bool)] : insert(A,X2,Xa1) = collect(A,aa(fun(A,bool),fun(A,bool),aa(fun(A,fun(bool,bool)),fun(fun(A,bool),fun(A,bool)),combs(A,bool,bool),aa(fun(A,bool),fun(A,fun(bool,bool)),aa(fun(bool,fun(bool,bool)),fun(fun(A,bool),fun(A,fun(bool,bool))),combb(bool,fun(bool,bool),A),fdisj),aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),combc(A,A,bool),fequal(A)),X2))),aa(fun(A,bool),fun(A,bool),aa(fun(A,fun(fun(A,bool),bool)),fun(fun(A,bool),fun(A,bool)),combc(A,fun(A,bool),bool),member(A)),Xa1))) ).
tff(fact_61_Collect__disj__eq,axiom,
! [A: $tType,Q1: fun(A,bool),P2: fun(A,bool)] : collect(A,aa(fun(A,bool),fun(A,bool),aa(fun(A,fun(bool,bool)),fun(fun(A,bool),fun(A,bool)),combs(A,bool,bool),aa(fun(A,bool),fun(A,fun(bool,bool)),aa(fun(bool,fun(bool,bool)),fun(fun(A,bool),fun(A,fun(bool,bool))),combb(bool,fun(bool,bool),A),fdisj),P2)),Q1)) = sup_sup(fun(A,bool),collect(A,P2),collect(A,Q1)) ).
tff(fact_62_split__twice,axiom,
! [B: $tType,C: $tType,A: $tType,E: $tType,D1: $tType,P1: product_prod(D1,E),G: fun(D1,fun(E,product_prod(B,C))),F: fun(B,fun(C,A))] : aa(product_prod(B,C),A,product_prod_case(B,C,A,F),aa(product_prod(D1,E),product_prod(B,C),product_prod_case(D1,E,product_prod(B,C),G),P1)) = aa(product_prod(D1,E),A,product_prod_case(D1,E,A,aa(fun(D1,fun(E,product_prod(B,C))),fun(D1,fun(E,A)),aa(fun(fun(E,product_prod(B,C)),fun(E,A)),fun(fun(D1,fun(E,product_prod(B,C))),fun(D1,fun(E,A))),combb(fun(E,product_prod(B,C)),fun(E,A),D1),aa(fun(product_prod(B,C),A),fun(fun(E,product_prod(B,C)),fun(E,A)),combb(product_prod(B,C),A,E),product_prod_case(B,C,A,F))),G)),P1) ).
tff(fact_63_singleton__inject,axiom,
! [A: $tType,B4: A,A3: A] :
( ( insert(A,A3,bot_bot(fun(A,bool))) = insert(A,B4,bot_bot(fun(A,bool))) )
=> ( A3 = B4 ) ) ).
tff(fact_64_singletonE,axiom,
! [A: $tType,A3: A,B4: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),B4),insert(A,A3,bot_bot(fun(A,bool)))))
=> ( B4 = A3 ) ) ).
tff(fact_65_doubleton__eq__iff,axiom,
! [A: $tType,D: A,C1: A,B4: A,A3: A] :
( ( insert(A,A3,insert(A,B4,bot_bot(fun(A,bool)))) = insert(A,C1,insert(A,D,bot_bot(fun(A,bool)))) )
<=> ( ( ( A3 = C1 )
& ( B4 = D ) )
| ( ( A3 = D )
& ( B4 = C1 ) ) ) ) ).
tff(fact_66_singleton__iff,axiom,
! [A: $tType,A3: A,B4: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),B4),insert(A,A3,bot_bot(fun(A,bool)))))
<=> ( B4 = A3 ) ) ).
tff(fact_67_insert__not__empty,axiom,
! [A: $tType,A4: fun(A,bool),A3: A] : insert(A,A3,A4) != bot_bot(fun(A,bool)) ).
tff(fact_68_empty__not__insert,axiom,
! [A: $tType,A4: fun(A,bool),A3: A] : bot_bot(fun(A,bool)) != insert(A,A3,A4) ).
tff(fact_69_UNIV__not__empty,axiom,
! [A: $tType] : top_top(fun(A,bool)) != bot_bot(fun(A,bool)) ).
tff(fact_70_Un__empty__right,axiom,
! [A: $tType,A4: fun(A,bool)] : sup_sup(fun(A,bool),A4,bot_bot(fun(A,bool))) = A4 ).
tff(fact_71_Un__empty__left,axiom,
! [A: $tType,B3: fun(A,bool)] : sup_sup(fun(A,bool),bot_bot(fun(A,bool)),B3) = B3 ).
tff(fact_72_prod_Osimps_I2_J,axiom,
! [B: $tType,A: $tType,C: $tType,B4: C,A3: B,F1: fun(B,fun(C,A))] : aa(product_prod(B,C),A,product_prod_case(B,C,A,F1),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A3),B4)) = aa(C,A,aa(B,fun(C,A),F1,A3),B4) ).
tff(fact_73_Un__UNIV__right,axiom,
! [A: $tType,A4: fun(A,bool)] : sup_sup(fun(A,bool),A4,top_top(fun(A,bool))) = top_top(fun(A,bool)) ).
tff(fact_74_ext,axiom,
! [B: $tType,A: $tType,G: fun(A,B),F: fun(A,B)] :
( ! [X1: A] : aa(A,B,F,X1) = aa(A,B,G,X1)
=> ( F = G ) ) ).
tff(fact_75_mem__def,axiom,
! [A: $tType,A4: fun(A,bool),Xa: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Xa),A4))
<=> pp(aa(A,bool,A4,Xa)) ) ).
tff(fact_76_Collect__def,axiom,
! [A: $tType,P2: fun(A,bool)] : collect(A,P2) = P2 ).
tff(fact_77_Un__UNIV__left,axiom,
! [A: $tType,B3: fun(A,bool)] : sup_sup(fun(A,bool),top_top(fun(A,bool)),B3) = top_top(fun(A,bool)) ).
tff(fact_78_splitD,axiom,
! [A: $tType,B: $tType,B4: B,A3: A,F: fun(A,fun(B,bool))] :
( pp(aa(product_prod(A,B),bool,product_prod_case(A,B,bool,F),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B4)))
=> pp(aa(B,bool,aa(A,fun(B,bool),F,A3),B4)) ) ).
tff(fact_79_Collect__conv__if,axiom,
! [A: $tType,A3: A,P2: fun(A,bool)] :
( ( pp(aa(A,bool,P2,A3))
=> ( collect(A,aa(fun(A,bool),fun(A,bool),aa(fun(A,fun(bool,bool)),fun(fun(A,bool),fun(A,bool)),combs(A,bool,bool),aa(fun(A,bool),fun(A,fun(bool,bool)),aa(fun(bool,fun(bool,bool)),fun(fun(A,bool),fun(A,fun(bool,bool))),combb(bool,fun(bool,bool),A),fconj),aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),combc(A,A,bool),fequal(A)),A3))),P2)) = insert(A,A3,bot_bot(fun(A,bool))) ) )
& ( ~ pp(aa(A,bool,P2,A3))
=> ( collect(A,aa(fun(A,bool),fun(A,bool),aa(fun(A,fun(bool,bool)),fun(fun(A,bool),fun(A,bool)),combs(A,bool,bool),aa(fun(A,bool),fun(A,fun(bool,bool)),aa(fun(bool,fun(bool,bool)),fun(fun(A,bool),fun(A,fun(bool,bool))),combb(bool,fun(bool,bool),A),fconj),aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),combc(A,A,bool),fequal(A)),A3))),P2)) = bot_bot(fun(A,bool)) ) ) ) ).
tff(fact_80_Collect__conv__if2,axiom,
! [A: $tType,A3: A,P2: fun(A,bool)] :
( ( pp(aa(A,bool,P2,A3))
=> ( collect(A,aa(fun(A,bool),fun(A,bool),aa(fun(A,fun(bool,bool)),fun(fun(A,bool),fun(A,bool)),combs(A,bool,bool),aa(fun(A,bool),fun(A,fun(bool,bool)),aa(fun(bool,fun(bool,bool)),fun(fun(A,bool),fun(A,fun(bool,bool))),combb(bool,fun(bool,bool),A),fconj),aa(A,fun(A,bool),fequal(A),A3))),P2)) = insert(A,A3,bot_bot(fun(A,bool))) ) )
& ( ~ pp(aa(A,bool,P2,A3))
=> ( collect(A,aa(fun(A,bool),fun(A,bool),aa(fun(A,fun(bool,bool)),fun(fun(A,bool),fun(A,bool)),combs(A,bool,bool),aa(fun(A,bool),fun(A,fun(bool,bool)),aa(fun(bool,fun(bool,bool)),fun(fun(A,bool),fun(A,fun(bool,bool))),combb(bool,fun(bool,bool),A),fconj),aa(A,fun(A,bool),fequal(A),A3))),P2)) = bot_bot(fun(A,bool)) ) ) ) ).
tff(fact_81_insert__def,axiom,
! [A: $tType,B3: fun(A,bool),A3: A] : insert(A,A3,B3) = sup_sup(fun(A,bool),collect(A,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),combc(A,A,bool),fequal(A)),A3)),B3) ).
tff(fact_82_split__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(product_prod(A,B),C)] : product_prod_case(A,B,C,aa(fun(A,fun(B,product_prod(A,B))),fun(A,fun(B,C)),aa(fun(fun(B,product_prod(A,B)),fun(B,C)),fun(fun(A,fun(B,product_prod(A,B))),fun(A,fun(B,C))),combb(fun(B,product_prod(A,B)),fun(B,C),A),aa(fun(product_prod(A,B),C),fun(fun(B,product_prod(A,B)),fun(B,C)),combb(product_prod(A,B),C,B),F)),product_Pair(A,B))) = F ).
tff(fact_83_insert__is__Un,axiom,
! [A: $tType,A4: fun(A,bool),A3: A] : insert(A,A3,A4) = sup_sup(fun(A,bool),insert(A,A3,bot_bot(fun(A,bool))),A4) ).
tff(fact_84_splitE,axiom,
! [A: $tType,B: $tType,P1: product_prod(A,B),C1: fun(A,fun(B,bool))] :
( pp(aa(product_prod(A,B),bool,product_prod_case(A,B,bool,C1),P1))
=> ~ ! [X1: A,Y1: B] :
( ( P1 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X1),Y1) )
=> ~ pp(aa(B,bool,aa(A,fun(B,bool),C1,X1),Y1)) ) ) ).
tff(fact_85_splitI2,axiom,
! [B: $tType,A: $tType,C1: fun(A,fun(B,bool)),P1: product_prod(A,B)] :
( ! [A2: A,B2: B] :
( ( P1 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2) )
=> pp(aa(B,bool,aa(A,fun(B,bool),C1,A2),B2)) )
=> pp(aa(product_prod(A,B),bool,product_prod_case(A,B,bool,C1),P1)) ) ).
tff(fact_86_sup__top__right,axiom,
! [A: $tType] :
( bounded_lattice_top(A)
=> ! [X: A] : sup_sup(A,X,top_top(A)) = top_top(A) ) ).
tff(fact_87_sup__top__left,axiom,
! [A: $tType] :
( bounded_lattice_top(A)
=> ! [X: A] : sup_sup(A,top_top(A),X) = top_top(A) ) ).
tff(fact_88_sup__eq__bot__iff,axiom,
! [A: $tType] :
( bounded_lattice_bot(A)
=> ! [Ya: A,Xa: A] :
( ( sup_sup(A,Xa,Ya) = bot_bot(A) )
<=> ( ( Xa = bot_bot(A) )
& ( Ya = bot_bot(A) ) ) ) ) ).
tff(fact_89_splitE_H,axiom,
! [B: $tType,A: $tType,C: $tType,Z: C,P1: product_prod(A,B),C1: fun(A,fun(B,fun(C,bool)))] :
( pp(aa(C,bool,aa(product_prod(A,B),fun(C,bool),product_prod_case(A,B,fun(C,bool),C1),P1),Z))
=> ~ ! [X1: A,Y1: B] :
( ( P1 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X1),Y1) )
=> ~ pp(aa(C,bool,aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),C1,X1),Y1),Z)) ) ) ).
tff(fact_90_mem__splitE,axiom,
! [B: $tType,A: $tType,C: $tType,P1: product_prod(B,C),C1: fun(B,fun(C,fun(A,bool))),Z: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Z),aa(product_prod(B,C),fun(A,bool),product_prod_case(B,C,fun(A,bool),C1),P1)))
=> ~ ! [X1: B,Y1: C] :
( ( P1 = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X1),Y1) )
=> ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Z),aa(C,fun(A,bool),aa(B,fun(C,fun(A,bool)),C1,X1),Y1))) ) ) ).
tff(fact_91_splitI2_H,axiom,
! [A: $tType,B: $tType,C: $tType,Xa: C,C1: fun(A,fun(B,fun(C,bool))),P1: product_prod(A,B)] :
( ! [A2: A,B2: B] :
( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2) = P1 )
=> pp(aa(C,bool,aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),C1,A2),B2),Xa)) )
=> pp(aa(C,bool,aa(product_prod(A,B),fun(C,bool),product_prod_case(A,B,fun(C,bool),C1),P1),Xa)) ) ).
tff(fact_92_mem__splitI2,axiom,
! [C: $tType,B: $tType,A: $tType,C1: fun(A,fun(B,fun(C,bool))),Z: C,P1: product_prod(A,B)] :
( ! [A2: A,B2: B] :
( ( P1 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2) )
=> pp(aa(fun(C,bool),bool,aa(C,fun(fun(C,bool),bool),member(C),Z),aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),C1,A2),B2))) )
=> pp(aa(fun(C,bool),bool,aa(C,fun(fun(C,bool),bool),member(C),Z),aa(product_prod(A,B),fun(C,bool),product_prod_case(A,B,fun(C,bool),C1),P1))) ) ).
tff(fact_93_sup_Oleft__idem,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [B1: A,A1: A] : sup_sup(A,A1,sup_sup(A,A1,B1)) = sup_sup(A,A1,B1) ) ).
tff(fact_94_sup__left__idem,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [Y: A,X: A] : sup_sup(A,X,sup_sup(A,X,Y)) = sup_sup(A,X,Y) ) ).
tff(fact_95_sup_Oidem,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [A1: A] : sup_sup(A,A1,A1) = A1 ) ).
tff(fact_96_sup__idem,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [X: A] : sup_sup(A,X,X) = X ) ).
tff(fact_97_sup_Ocommute,axiom,
! [A: $tType] :
( semilattice_sup(A)
=> ! [B1: A,A1: A] : sup_sup(A,A1,B1) = sup_sup(A,B1,A1) ) ).
tff(fact_98_inf__sup__aci_I5_J,axiom,
! [A: $tType] :
( lattice(A)
=> ! [Y: A,X: A] : sup_sup(A,X,Y) = sup_sup(A,Y,X) ) ).
%----Arities (10)
tff(arity_HOL_Obool___Lattices_Obounded__lattice,axiom,
bounded_lattice(bool) ).
tff(arity_fun___Lattices_Obounded__lattice,axiom,
! [T_1: $tType,T_2: $tType] :
( bounded_lattice(T_2)
=> bounded_lattice(fun(T_1,T_2)) ) ).
tff(arity_fun___Lattices_Obounded__lattice__top,axiom,
! [T_1: $tType,T_2: $tType] :
( bounded_lattice(T_2)
=> bounded_lattice_top(fun(T_1,T_2)) ) ).
tff(arity_fun___Lattices_Obounded__lattice__bot,axiom,
! [T_1: $tType,T_2: $tType] :
( bounded_lattice(T_2)
=> bounded_lattice_bot(fun(T_1,T_2)) ) ).
tff(arity_fun___Lattices_Osemilattice__sup,axiom,
! [T_1: $tType,T_2: $tType] :
( lattice(T_2)
=> semilattice_sup(fun(T_1,T_2)) ) ).
tff(arity_fun___Lattices_Olattice,axiom,
! [T_1: $tType,T_2: $tType] :
( lattice(T_2)
=> lattice(fun(T_1,T_2)) ) ).
tff(arity_HOL_Obool___Lattices_Obounded__lattice__top,axiom,
bounded_lattice_top(bool) ).
tff(arity_HOL_Obool___Lattices_Obounded__lattice__bot,axiom,
bounded_lattice_bot(bool) ).
tff(arity_HOL_Obool___Lattices_Osemilattice__sup,axiom,
semilattice_sup(bool) ).
tff(arity_HOL_Obool___Lattices_Olattice,axiom,
lattice(bool) ).
%----Helper facts (26)
tff(help_pp_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_pp_2_1_U,axiom,
pp(fTrue) ).
tff(help_fEx_1_1_U,axiom,
! [A: $tType,X: A,P: fun(A,bool)] :
( ~ pp(aa(A,bool,P,X))
| pp(aa(fun(A,bool),bool,fEx(A),P)) ) ).
tff(help_fAll_1_1_U,axiom,
! [A: $tType,X: A,P: fun(A,bool)] :
( ~ pp(aa(fun(A,bool),bool,fAll(A),P))
| pp(aa(A,bool,P,X)) ) ).
tff(help_fNot_1_1_U,axiom,
! [P: bool] :
( ~ pp(aa(bool,bool,fNot,P))
| ~ pp(P) ) ).
tff(help_fNot_2_1_U,axiom,
! [P: bool] :
( pp(P)
| pp(aa(bool,bool,fNot,P)) ) ).
tff(help_COMBB_1_1_U,axiom,
! [C: $tType,B: $tType,A: $tType,R: A,Q: fun(A,B),P: fun(B,C)] : aa(A,C,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),combb(B,C,A),P),Q),R) = aa(B,C,P,aa(A,B,Q,R)) ).
tff(help_COMBC_1_1_U,axiom,
! [A: $tType,C: $tType,B: $tType,R: A,Q: B,P: fun(A,fun(B,C))] : aa(A,C,aa(B,fun(A,C),aa(fun(A,fun(B,C)),fun(B,fun(A,C)),combc(A,B,C),P),Q),R) = aa(B,C,aa(A,fun(B,C),P,R),Q) ).
tff(help_COMBI_1_1_U,axiom,
! [A: $tType,P: A] : aa(A,A,combi(A),P) = P ).
tff(help_COMBK_1_1_U,axiom,
! [B: $tType,A: $tType,Q: B,P: A] : aa(B,A,combk(A,B,P),Q) = P ).
tff(help_COMBS_1_1_U,axiom,
! [C: $tType,B: $tType,A: $tType,R: A,Q: fun(A,B),P: fun(A,fun(B,C))] : aa(A,C,aa(fun(A,B),fun(A,C),aa(fun(A,fun(B,C)),fun(fun(A,B),fun(A,C)),combs(A,B,C),P),Q),R) = aa(B,C,aa(A,fun(B,C),P,R),aa(A,B,Q,R)) ).
tff(help_fTrue_1_1_U,axiom,
pp(fTrue) ).
tff(help_fTrue_1_1_T,axiom,
! [P: bool] :
( ( P = fTrue )
| ( P = fFalse ) ) ).
tff(help_fconj_1_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(P)
| ~ pp(Q)
| pp(aa(bool,bool,aa(bool,fun(bool,bool),fconj,P),Q)) ) ).
tff(help_fconj_2_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(aa(bool,bool,aa(bool,fun(bool,bool),fconj,P),Q))
| pp(P) ) ).
tff(help_fconj_3_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(aa(bool,bool,aa(bool,fun(bool,bool),fconj,P),Q))
| pp(Q) ) ).
tff(help_fdisj_1_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(P)
| pp(aa(bool,bool,aa(bool,fun(bool,bool),fdisj,P),Q)) ) ).
tff(help_fdisj_2_1_U,axiom,
! [P: bool,Q: bool] :
( ~ pp(Q)
| pp(aa(bool,bool,aa(bool,fun(bool,bool),fdisj,P),Q)) ) ).
tff(help_fdisj_3_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(aa(bool,bool,aa(bool,fun(bool,bool),fdisj,P),Q))
| pp(P)
| pp(Q) ) ).
tff(help_fFalse_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_fFalse_1_1_T,axiom,
! [P: bool] :
( ( P = fTrue )
| ( P = fFalse ) ) ).
tff(help_fequal_1_1_T,axiom,
! [A: $tType,Y: A,X: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),fequal(A),X),Y))
| ( X = Y ) ) ).
tff(help_fequal_2_1_T,axiom,
! [A: $tType,Y: A,X: A] :
( ( X != Y )
| pp(aa(A,bool,aa(A,fun(A,bool),fequal(A),X),Y)) ) ).
tff(help_fimplies_1_1_U,axiom,
! [Q: bool,P: bool] :
( pp(P)
| pp(aa(bool,bool,aa(bool,fun(bool,bool),fimplies,P),Q)) ) ).
tff(help_fimplies_2_1_U,axiom,
! [P: bool,Q: bool] :
( ~ pp(Q)
| pp(aa(bool,bool,aa(bool,fun(bool,bool),fimplies,P),Q)) ) ).
tff(help_fimplies_3_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(aa(bool,bool,aa(bool,fun(bool,bool),fimplies,P),Q))
| ~ pp(P)
| pp(Q) ) ).
%----Conjectures (3)
tff(conj_0,hypothesis,
x != y ).
tff(conj_1,hypothesis,
pp(aa(fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool),bool,aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool),bool),member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),l),collect(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),aa(fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool),aa(fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(bool,bool)),fun(fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool)),combs(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool,bool),aa(fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(bool,bool)),aa(fun(bool,fun(bool,bool)),fun(fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(bool,bool))),combb(bool,fun(bool,bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fconj),aa(fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,bool)),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool),aa(fun(fun(arrow_411405190le_alt,bool),bool),fun(fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,bool)),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool)),combb(fun(arrow_411405190le_alt,bool),bool,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fAll(arrow_411405190le_alt)),aa(fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool))),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,bool)),aa(fun(fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool)),fun(arrow_411405190le_alt,bool)),fun(fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool))),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,bool))),combb(fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool)),fun(arrow_411405190le_alt,bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),aa(fun(fun(arrow_411405190le_alt,bool),bool),fun(fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool)),fun(arrow_411405190le_alt,bool)),combb(fun(arrow_411405190le_alt,bool),bool,arrow_411405190le_alt),fAll(arrow_411405190le_alt))),aa(fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool))),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool))),aa(fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool)),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool)))),fun(fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool))),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool)))),combs(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool)),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool))),aa(fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool)))),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool)),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool)))),aa(fun(fun(arrow_411405190le_alt,fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool))),fun(fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool)),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool)))),fun(fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool)))),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool)),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool))))),combb(fun(arrow_411405190le_alt,fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool))),fun(fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool)),fun(arrow_411405190le_alt,fun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).
tff(conj_2,conjecture,
pp(aa(fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool),bool,aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool),bool),member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),sup_sup(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),sup_sup(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),sup_sup(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),collect(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_prod_case(arrow_411405190le_alt,arrow_411405190le_alt,bool,aa(fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool)),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool)),aa(fun(arrow_411405190le_alt,fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool))),fun(fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool)),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool))),combs(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool)),aa(fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(bool,bool))),fun(arrow_411405190le_alt,fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool))),aa(fun(fun(arrow_411405190le_alt,fun(bool,bool)),fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool))),fun(fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(bool,bool))),fun(arrow_411405190le_alt,fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool)))),combb(fun(arrow_411405190le_alt,fun(bool,bool)),fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,bool)),arrow_411405190le_alt),combs(arrow_411405190le_alt,bool,bool)),aa(fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool)),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(bool,bool))),aa(fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,fun(bool,bool))),fun(fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool)),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(bool,bool)))),combb(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,fun(bool,bool)),arrow_411405190le_alt),aa(fun(bool,fun(bool,bool)),fun(fun(arrow_411405190le_alt,bool),fun(arrow_411405190le_alt,fun(bool,bool))),combb(bool,fun(bool,bool),arrow_411405190le_alt),fconj)),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool)),aa(fun(arrow_411405190le_alt,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,bool))),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool))),combc(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,bool)),aa(fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool))),fun(arrow_411405190le_alt,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,bool))),aa(fun(fun(arrow_411405190le_alt,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool)),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,bool))),fun(fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool))),fun(arrow_411405190le_alt,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,bool)))),combb(fun(arrow_411405190le_alt,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool)),fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,bool)),arrow_411405190le_alt),combc(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool)),aa(fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt))),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool))),aa(fun(fun(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),fun(arrow_411405190le_alt,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool))),fun(fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt))),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool)))),combb(fun(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),fun(arrow_411405190le_alt,fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool)),arrow_411405190le_alt),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),fun(fun(product_prod(arrow_41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).
%------------------------------------------------------------------------------