TPTP Problem File: SCT197_5.p
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%------------------------------------------------------------------------------
% File : SCT197_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Social Choice Theory
% Problem : Arrow's Impossibility Theorem line 142
% 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_142 [Bla13]
% Status : Theorem
% Rating : 0.00 v7.5.0, 0.33 v7.4.0, 0.50 v7.1.0, 0.67 v6.4.0
% Syntax : Number of formulae : 164 ( 45 unt; 44 typ; 0 def)
% Number of atoms : 262 ( 91 equ)
% Maximal formula atoms : 12 ( 1 avg)
% Number of connectives : 200 ( 58 ~; 9 |; 15 &)
% ( 35 <=>; 83 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 6 avg)
% Maximal term depth : 13 ( 2 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 47 ( 26 >; 21 *; 0 +; 0 <<)
% Number of predicates : 9 ( 8 usr; 0 prp; 1-6 aty)
% Number of functors : 33 ( 33 usr; 6 con; 0-6 aty)
% Number of variables : 517 ( 464 !; 9 ?; 517 :)
% ( 44 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:17:58
%------------------------------------------------------------------------------
%----Should-be-implicit typings (5)
tff(ty_tc_Arrow__Order__Mirabelle__qkbtqzkjxu_Oalt,type,
arrow_411405190le_alt: $tType ).
tff(ty_tc_Arrow__Order__Mirabelle__qkbtqzkjxu_Oindi,type,
arrow_159774573e_indi: $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 (39)
tff(sy_cl_Orderings_Obot,type,
bot:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Otop,type,
top:
!>[A: $tType] : $o ).
tff(sy_c_Arrow__Order__Mirabelle__qkbtqzkjxu_OIIA,type,
arrow_1958449194le_IIA: fun(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)) > $o ).
tff(sy_c_Arrow__Order__Mirabelle__qkbtqzkjxu_OLin,type,
arrow_1985332922le_Lin: fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool) ).
tff(sy_c_Arrow__Order__Mirabelle__qkbtqzkjxu_OProf,type,
arrow_610318064e_Prof: fun(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),bool) ).
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_Odictator,type,
arrow_987702531ctator: ( fun(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)) * arrow_159774573e_indi ) > $o ).
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_Arrow__Order__Mirabelle__qkbtqzkjxu_Ounanimity,type,
arrow_2069624013nimity: fun(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)) > $o ).
tff(sy_c_COMBB,type,
combb:
!>[B: $tType,C: $tType,A: $tType] : ( 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_FuncSet_OPi,type,
pi:
!>[A: $tType,B: $tType] : ( ( fun(A,bool) * fun(A,fun(B,bool)) ) > fun(fun(A,B),bool) ) ).
tff(sy_c_FuncSet_Oextensional__funcset,type,
extensional_funcset:
!>[A: $tType,B: $tType] : ( ( fun(A,bool) * fun(B,bool) ) > fun(fun(A,B),bool) ) ).
tff(sy_c_Order__Relation_Ostrict__linear__order__on,type,
order_215145569der_on:
!>[A: $tType] : ( fun(A,bool) > fun(fun(product_prod(A,A),bool),bool) ) ).
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_Predicate_Oconversep,type,
conversep:
!>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > fun(B,fun(A,bool)) ) ).
tff(sy_c_Predicate_Oinv__imagep,type,
inv_imagep:
!>[B: $tType,A: $tType] : ( ( fun(B,fun(B,bool)) * fun(A,B) * A * A ) > $o ) ).
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_Ocurry,type,
product_curry:
!>[A: $tType,B: $tType,C: $tType] : ( ( fun(product_prod(A,B),C) * A * B ) > C ) ).
tff(sy_c_Product__Type_Ointernal__split,type,
produc1605651328_split:
!>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,fun(B,C)) * product_prod(A,B) ) > C ) ).
tff(sy_c_Product__Type_Oprod_Oprod__rec,type,
product_prod_rec:
!>[A: $tType,B: $tType,T1: $tType] : ( ( fun(A,fun(B,T1)) * product_prod(A,B) ) > T1 ) ).
tff(sy_c_Relation_OId__on,type,
id_on:
!>[A: $tType] : ( fun(A,bool) > fun(product_prod(A,A),bool) ) ).
tff(sy_c_Relation_Oconverse,type,
converse:
!>[A: $tType,B: $tType] : ( fun(product_prod(A,B),bool) > fun(product_prod(B,A),bool) ) ).
tff(sy_c_Relation_Oinv__image,type,
inv_image:
!>[B: $tType,A: $tType] : ( ( fun(product_prod(B,B),bool) * fun(A,B) ) > fun(product_prod(A,A),bool) ) ).
tff(sy_c_Relation_Ototal__on,type,
total_on:
!>[A: $tType] : ( ( fun(A,bool) * fun(product_prod(A,A),bool) ) > $o ) ).
tff(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( fun(A,bool) > fun(A,bool) ) ).
tff(sy_c_aa,type,
aa:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * A ) > B ) ).
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_fequal,type,
fequal:
!>[A: $tType] : fun(A,fun(A,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) ).
%----Relevant facts (100)
tff(fact_0_dictator__def,axiom,
! [I1: arrow_159774573e_indi,F3: fun(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))] :
( arrow_987702531ctator(F3,I1)
<=> ! [X3: fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))] :
( pp(aa(fun(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),bool),bool,aa(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(fun(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),bool),bool),member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),X3),arrow_610318064e_Prof))
=> ( aa(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),F3,X3) = aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),X3,I1) ) ) ) ).
tff(fact_1_linear__alt,axiom,
? [L: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)] : 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),arrow_1985332922le_Lin)) ).
tff(fact_2_above__Lin,axiom,
! [La: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Y1: arrow_411405190le_alt,X1: arrow_411405190le_alt] :
( ( X1 != Y1 )
=> ( 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)),La),arrow_1985332922le_Lin))
=> 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)),arrow_1158827142_above(La,X1,Y1)),arrow_1985332922le_Lin)) ) ) ).
tff(fact_3_below__Lin,axiom,
! [La: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Y1: arrow_411405190le_alt,X1: arrow_411405190le_alt] :
( ( X1 != Y1 )
=> ( 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)),La),arrow_1985332922le_Lin))
=> 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)),arrow_319942042_below(La,X1,Y1)),arrow_1985332922le_Lin)) ) ) ).
tff(fact_4_mkbot__Lin,axiom,
! [X1: arrow_411405190le_alt,La: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)] :
( 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)),La),arrow_1985332922le_Lin))
=> 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)),arrow_276188178_mkbot(La,X1)),arrow_1985332922le_Lin)) ) ).
tff(fact_5_mktop__Lin,axiom,
! [X1: arrow_411405190le_alt,La: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)] :
( 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)),La),arrow_1985332922le_Lin))
=> 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)),arrow_424895264_mktop(La,X1)),arrow_1985332922le_Lin)) ) ).
tff(fact_6_converse__in__Lin,axiom,
! [La: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)] :
( 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)),converse(arrow_411405190le_alt,arrow_411405190le_alt,La)),arrow_1985332922le_Lin))
<=> 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)),La),arrow_1985332922le_Lin)) ) ).
tff(fact_7_Prof__def,axiom,
arrow_610318064e_Prof = pi(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),top_top(fun(arrow_159774573e_indi,bool)),combk(fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool),arrow_159774573e_indi,arrow_1985332922le_Lin)) ).
tff(fact_8_notin__Lin__iff,axiom,
! [Y1: arrow_411405190le_alt,X1: arrow_411405190le_alt,La: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)] :
( 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)),La),arrow_1985332922le_Lin))
=> ( ( X1 != Y1 )
=> ( ~ 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),X1),Y1)),La))
<=> 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),Y1),X1)),La)) ) ) ) ).
tff(fact_9_Lin__irrefl,axiom,
! [B2: arrow_411405190le_alt,A4: arrow_411405190le_alt,La: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)] :
( 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)),La),arrow_1985332922le_Lin))
=> ( 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),A4),B2)),La))
=> ~ 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),B2),A4)),La)) ) ) ).
tff(fact_10_dictatorI,axiom,
! [I1: arrow_159774573e_indi,F3: fun(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))] :
( pp(aa(fun(fun(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),bool),bool,aa(fun(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(fun(fun(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),bool),bool),member(fun(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),F3),pi(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),arrow_610318064e_Prof,combk(fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool),fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),arrow_1985332922le_Lin))))
=> ( ! [X4: fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))] :
( pp(aa(fun(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),bool),bool,aa(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(fun(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),bool),bool),member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),X4),arrow_610318064e_Prof))
=> ! [A6: arrow_411405190le_alt,B4: arrow_411405190le_alt] :
( ( A6 != B4 )
=> ( 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),A6),B4)),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),X4,I1)))
=> 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),A6),B4)),aa(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),F3,X4))) ) ) )
=> arrow_987702531ctator(F3,I1) ) ) ).
tff(fact_11_unanimity__def,axiom,
! [F3: fun(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))] :
( arrow_2069624013nimity(F3)
<=> ! [X3: fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))] :
( pp(aa(fun(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),bool),bool,aa(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(fun(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),bool),bool),member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),X3),arrow_610318064e_Prof))
=> ! [A3: arrow_411405190le_alt,B1: arrow_411405190le_alt] :
( ! [I: arrow_159774573e_indi] : 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),A3),B1)),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),X3,I)))
=> 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),A3),B1)),aa(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),F3,X3))) ) ) ) ).
tff(fact_12_in__mkbot,axiom,
! [Z: arrow_411405190le_alt,La: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Y1: arrow_411405190le_alt,X1: 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),X1),Y1)),arrow_276188178_mkbot(La,Z)))
<=> ( ( Y1 != Z )
& ( ( X1 = Z )
=> ( X1 != Y1 ) )
& ( ( X1 != 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),X1),Y1)),La)) ) ) ) ).
tff(fact_13_in__mktop,axiom,
! [Z: arrow_411405190le_alt,La: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Y1: arrow_411405190le_alt,X1: 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),X1),Y1)),arrow_424895264_mktop(La,Z)))
<=> ( ( X1 != Z )
& ( ( Y1 = Z )
=> ( X1 != Y1 ) )
& ( ( Y1 != 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),X1),Y1)),La)) ) ) ) ).
tff(fact_14_in__below,axiom,
! [Y1: arrow_411405190le_alt,X1: arrow_411405190le_alt,La: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),B2: arrow_411405190le_alt,A4: arrow_411405190le_alt] :
( ( A4 != B2 )
=> ( 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)),La),arrow_1985332922le_Lin))
=> ( 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),X1),Y1)),arrow_319942042_below(La,A4,B2)))
<=> ( ( X1 != Y1 )
& ( ( Y1 = A4 )
=> 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),X1),B2)),La)) )
& ( ( Y1 != A4 )
=> ( ( ( X1 = A4 )
=> ( ( Y1 = B2 )
| 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),B2),Y1)),La)) ) )
& ( ( X1 != A4 )
=> 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),X1),Y1)),La)) ) ) ) ) ) ) ) ).
tff(fact_15_in__above,axiom,
! [Y1: arrow_411405190le_alt,X1: arrow_411405190le_alt,La: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),B2: arrow_411405190le_alt,A4: arrow_411405190le_alt] :
( ( A4 != B2 )
=> ( 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)),La),arrow_1985332922le_Lin))
=> ( 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),X1),Y1)),arrow_1158827142_above(La,A4,B2)))
<=> ( ( X1 != Y1 )
& ( ( X1 = B2 )
=> 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),A4),Y1)),La)) )
& ( ( X1 != B2 )
=> ( ( ( Y1 = B2 )
=> ( ( X1 = A4 )
| 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),X1),A4)),La)) ) )
& ( ( Y1 != B2 )
=> 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),X1),Y1)),La)) ) ) ) ) ) ) ) ).
tff(fact_16_Pi__UNIV,axiom,
! [B: $tType,A: $tType,A5: fun(A,bool)] : pi(A,B,A5,combk(fun(B,bool),A,top_top(fun(B,bool)))) = top_top(fun(fun(A,B),bool)) ).
tff(fact_17_converse__iff,axiom,
! [A: $tType,B: $tType,R1: fun(product_prod(B,A),bool),B2: B,A4: A] :
( pp(aa(fun(product_prod(A,B),bool),bool,aa(product_prod(A,B),fun(fun(product_prod(A,B),bool),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B2)),converse(B,A,R1)))
<=> pp(aa(fun(product_prod(B,A),bool),bool,aa(product_prod(B,A),fun(fun(product_prod(B,A),bool),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),A4)),R1)) ) ).
tff(fact_18_PiE,axiom,
! [B: $tType,A: $tType,X1: A,B3: fun(A,fun(B,bool)),A5: fun(A,bool),F: fun(A,B)] :
( pp(aa(fun(fun(A,B),bool),bool,aa(fun(A,B),fun(fun(fun(A,B),bool),bool),member(fun(A,B)),F),pi(A,B,A5,B3)))
=> ( ~ pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),aa(A,B,F,X1)),aa(A,fun(B,bool),B3,X1)))
=> ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X1),A5)) ) ) ).
tff(fact_19_top1I,axiom,
! [A: $tType,X1: A] : pp(aa(A,bool,top_top(fun(A,bool)),X1)) ).
tff(fact_20_UNIV__I,axiom,
! [A: $tType,X1: A] : pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X1),top_top(fun(A,bool)))) ).
tff(fact_21_iso__tuple__UNIV__I,axiom,
! [A: $tType,X1: A] : pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X1),top_top(fun(A,bool)))) ).
tff(fact_22_split__paired__All,axiom,
! [A: $tType,B: $tType,P1: fun(product_prod(A,B),bool)] :
( ! [X11: product_prod(A,B)] : pp(aa(product_prod(A,B),bool,P1,X11))
<=> ! [A3: A,B1: B] : pp(aa(product_prod(A,B),bool,P1,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B1))) ) ).
tff(fact_23_Pair__eq,axiom,
! [A: $tType,B: $tType,B7: B,A9: A,B2: B,A4: A] :
( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B2) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A9),B7) )
<=> ( ( A4 = A9 )
& ( B2 = B7 ) ) ) ).
tff(fact_24_converseI,axiom,
! [B: $tType,A: $tType,R1: fun(product_prod(A,B),bool),B2: B,A4: A] :
( pp(aa(fun(product_prod(A,B),bool),bool,aa(product_prod(A,B),fun(fun(product_prod(A,B),bool),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B2)),R1))
=> pp(aa(fun(product_prod(B,A),bool),bool,aa(product_prod(B,A),fun(fun(product_prod(B,A),bool),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),A4)),converse(A,B,R1))) ) ).
tff(fact_25_Pair__inject,axiom,
! [A: $tType,B: $tType,B6: B,A8: A,B5: B,A7: A] :
( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A7),B5) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A8),B6) )
=> ~ ( ( A7 = A8 )
=> ( B5 != B6 ) ) ) ).
tff(fact_26_UNIV__def,axiom,
! [A: $tType] : top_top(fun(A,bool)) = collect(A,combk(bool,A,fTrue)) ).
tff(fact_27_converse__converse,axiom,
! [B: $tType,A: $tType,R1: fun(product_prod(A,B),bool)] : converse(B,A,converse(A,B,R1)) = R1 ).
tff(fact_28_Pi__mem,axiom,
! [B: $tType,A: $tType,X1: A,B3: fun(A,fun(B,bool)),A5: fun(A,bool),F: fun(A,B)] :
( pp(aa(fun(fun(A,B),bool),bool,aa(fun(A,B),fun(fun(fun(A,B),bool),bool),member(fun(A,B)),F),pi(A,B,A5,B3)))
=> ( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X1),A5))
=> pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),aa(A,B,F,X1)),aa(A,fun(B,bool),B3,X1))) ) ) ).
tff(fact_29_pred__equals__eq2,axiom,
! [B: $tType,A: $tType,S: fun(product_prod(A,B),bool),R2: fun(product_prod(A,B),bool)] :
( ! [X3: A,Xa1: B] :
( pp(aa(fun(product_prod(A,B),bool),bool,aa(product_prod(A,B),fun(fun(product_prod(A,B),bool),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Xa1)),R2))
<=> pp(aa(fun(product_prod(A,B),bool),bool,aa(product_prod(A,B),fun(fun(product_prod(A,B),bool),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Xa1)),S)) )
<=> ( R2 = S ) ) ).
tff(fact_30_funcset__mem,axiom,
! [A: $tType,B: $tType,X1: A,B3: fun(B,bool),A5: fun(A,bool),F: fun(A,B)] :
( pp(aa(fun(fun(A,B),bool),bool,aa(fun(A,B),fun(fun(fun(A,B),bool),bool),member(fun(A,B)),F),pi(A,B,A5,combk(fun(B,bool),A,B3))))
=> ( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X1),A5))
=> pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),aa(A,B,F,X1)),B3)) ) ) ).
tff(fact_31_funcset__id,axiom,
! [A: $tType,A5: fun(A,bool)] : pp(aa(fun(fun(A,A),bool),bool,aa(fun(A,A),fun(fun(fun(A,A),bool),bool),member(fun(A,A)),combi(A)),pi(A,A,A5,combk(fun(A,bool),A,A5)))) ).
tff(fact_32_converseD,axiom,
! [A: $tType,B: $tType,R1: fun(product_prod(B,A),bool),B2: B,A4: A] :
( pp(aa(fun(product_prod(A,B),bool),bool,aa(product_prod(A,B),fun(fun(product_prod(A,B),bool),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B2)),converse(B,A,R1)))
=> pp(aa(fun(product_prod(B,A),bool),bool,aa(product_prod(B,A),fun(fun(product_prod(B,A),bool),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),A4)),R1)) ) ).
tff(fact_33_converseE,axiom,
! [A: $tType,B: $tType,R1: fun(product_prod(B,A),bool),Yx: product_prod(A,B)] :
( pp(aa(fun(product_prod(A,B),bool),bool,aa(product_prod(A,B),fun(fun(product_prod(A,B),bool),bool),member(product_prod(A,B)),Yx),converse(B,A,R1)))
=> ~ ! [X4: B,Y2: A] :
( ( Yx = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y2),X4) )
=> ~ pp(aa(fun(product_prod(B,A),bool),bool,aa(product_prod(B,A),fun(fun(product_prod(B,A),bool),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),Y2)),R1)) ) ) ).
tff(fact_34_Pi__I,axiom,
! [B: $tType,A: $tType,B3: fun(A,fun(B,bool)),F: fun(A,B),A5: fun(A,bool)] :
( ! [X4: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X4),A5))
=> pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),aa(A,B,F,X4)),aa(A,fun(B,bool),B3,X4))) )
=> pp(aa(fun(fun(A,B),bool),bool,aa(fun(A,B),fun(fun(fun(A,B),bool),bool),member(fun(A,B)),F),pi(A,B,A5,B3))) ) ).
tff(fact_35_split__paired__Ex,axiom,
! [A: $tType,B: $tType,P1: fun(product_prod(A,B),bool)] :
( ? [X11: product_prod(A,B)] : pp(aa(product_prod(A,B),bool,P1,X11))
<=> ? [A3: A,B1: B] : pp(aa(product_prod(A,B),bool,P1,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B1))) ) ).
tff(fact_36_prod_Orecs,axiom,
! [B: $tType,A: $tType,C: $tType,B2: C,A4: B,F11: fun(B,fun(C,A))] : product_prod_rec(B,C,A,F11,aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A4),B2)) = aa(C,A,aa(B,fun(C,A),F11,A4),B2) ).
tff(fact_37_top__apply,axiom,
! [B: $tType,A: $tType] :
( top(A)
=> ! [X1: B] : aa(B,A,top_top(fun(B,A)),X1) = top_top(A) ) ).
tff(fact_38_funcsetI,axiom,
! [A: $tType,B: $tType,B3: fun(B,bool),F: fun(A,B),A5: fun(A,bool)] :
( ! [X4: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X4),A5))
=> pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),aa(A,B,F,X4)),B3)) )
=> pp(aa(fun(fun(A,B),bool),bool,aa(fun(A,B),fun(fun(fun(A,B),bool),bool),member(fun(A,B)),F),pi(A,B,A5,combk(fun(B,bool),A,B3)))) ) ).
tff(fact_39_IIA__def,axiom,
! [F3: fun(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))] :
( arrow_1958449194le_IIA(F3)
<=> ! [X3: fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))] :
( pp(aa(fun(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),bool),bool,aa(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(fun(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),bool),bool),member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),X3),arrow_610318064e_Prof))
=> ! [Xa1: fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))] :
( pp(aa(fun(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),bool),bool,aa(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(fun(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),bool),bool),member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),Xa1),arrow_610318064e_Prof))
=> ! [A3: arrow_411405190le_alt,B1: arrow_411405190le_alt] :
( ! [I: arrow_159774573e_indi] :
( 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),A3),B1)),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),X3,I)))
<=> 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),A3),B1)),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Xa1,I))) )
=> ( 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),A3),B1)),aa(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),F3,X3)))
<=> 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),A3),B1)),aa(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),F3,Xa1))) ) ) ) ) ) ).
tff(fact_40_Pi__I_H,axiom,
! [B: $tType,A: $tType,B3: fun(A,fun(B,bool)),F: fun(A,B),A5: fun(A,bool)] :
( ! [X4: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X4),A5))
=> pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),aa(A,B,F,X4)),aa(A,fun(B,bool),B3,X4))) )
=> pp(aa(fun(fun(A,B),bool),bool,aa(fun(A,B),fun(fun(fun(A,B),bool),bool),member(fun(A,B)),F),pi(A,B,A5,B3))) ) ).
tff(fact_41_Pi__cong,axiom,
! [B: $tType,A: $tType,B3: fun(A,fun(B,bool)),G: fun(A,B),F: fun(A,B),A5: fun(A,bool)] :
( ! [W: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),W),A5))
=> ( aa(A,B,F,W) = aa(A,B,G,W) ) )
=> ( pp(aa(fun(fun(A,B),bool),bool,aa(fun(A,B),fun(fun(fun(A,B),bool),bool),member(fun(A,B)),F),pi(A,B,A5,B3)))
<=> pp(aa(fun(fun(A,B),bool),bool,aa(fun(A,B),fun(fun(fun(A,B),bool),bool),member(fun(A,B)),G),pi(A,B,A5,B3))) ) ) ).
tff(fact_42_UNIV__witness,axiom,
! [A: $tType] :
? [X4: A] : pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X4),top_top(fun(A,bool)))) ).
tff(fact_43_UNIV__eq__I,axiom,
! [A: $tType,A5: fun(A,bool)] :
( ! [X4: A] : pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X4),A5))
=> ( top_top(fun(A,bool)) = A5 ) ) ).
tff(fact_44_Lin__def,axiom,
arrow_1985332922le_Lin = collect(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),order_215145569der_on(arrow_411405190le_alt,top_top(fun(arrow_411405190le_alt,bool)))) ).
tff(fact_45_prod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod(A,B)] :
~ ! [A6: A,B4: B] : Y != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B4) ).
tff(fact_46_PairE,axiom,
! [A: $tType,B: $tType,P2: product_prod(A,B)] :
~ ! [X4: A,Y2: B] : P2 != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y2) ).
tff(fact_47_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod(A,product_prod(B,C))] :
~ ! [A6: A,B4: B,C2: C] : Y != 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)),A6),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B4),C2)) ).
tff(fact_48_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,X1: product_prod(A,product_prod(B,C)),P1: fun(product_prod(A,product_prod(B,C)),bool)] :
( ! [A6: A,B4: B,C2: C] : pp(aa(product_prod(A,product_prod(B,C)),bool,P1,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)),A6),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,P1,X1)) ) ).
tff(fact_49_prod__induct6,axiom,
! [F1: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,X1: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F1))))),P1: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F1))))),bool)] :
( ! [A6: A,B4: B,C2: C,D1: D,E1: E,F2: F1] : pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F1))))),bool,P1,aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F1)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F1))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F1)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F1)))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F1))))),A6),aa(product_prod(C,product_prod(D,product_prod(E,F1))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F1)))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,F1))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F1))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,F1)))),B4),aa(product_prod(D,product_prod(E,F1)),product_prod(C,product_prod(D,product_prod(E,F1))),aa(C,fun(product_prod(D,product_prod(E,F1)),product_prod(C,product_prod(D,product_prod(E,F1)))),product_Pair(C,product_prod(D,product_prod(E,F1))),C2),aa(product_prod(E,F1),product_prod(D,product_prod(E,F1)),aa(D,fun(product_prod(E,F1),product_prod(D,product_prod(E,F1))),product_Pair(D,product_prod(E,F1)),D1),aa(F1,product_prod(E,F1),aa(E,fun(F1,product_prod(E,F1)),product_Pair(E,F1),E1),F2)))))))
=> pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F1))))),bool,P1,X1)) ) ).
tff(fact_50_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F1: $tType,Y: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F1)))))] :
~ ! [A6: A,B4: B,C2: C,D1: D,E1: E,F2: F1] : Y != aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F1)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F1))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F1)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F1)))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F1))))),A6),aa(product_prod(C,product_prod(D,product_prod(E,F1))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F1)))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,F1))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F1))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,F1)))),B4),aa(product_prod(D,product_prod(E,F1)),product_prod(C,product_prod(D,product_prod(E,F1))),aa(C,fun(product_prod(D,product_prod(E,F1)),product_prod(C,product_prod(D,product_prod(E,F1)))),product_Pair(C,product_prod(D,product_prod(E,F1))),C2),aa(product_prod(E,F1),product_prod(D,product_prod(E,F1)),aa(D,fun(product_prod(E,F1),product_prod(D,product_prod(E,F1))),product_Pair(D,product_prod(E,F1)),D1),aa(F1,product_prod(E,F1),aa(E,fun(F1,product_prod(E,F1)),product_Pair(E,F1),E1),F2))))) ).
tff(fact_51_prod__induct5,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,X1: product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),P1: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),bool)] :
( ! [A6: A,B4: B,C2: C,D1: D,E1: E] : pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),bool,P1,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)))),A6),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),D1),E1))))))
=> pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),bool,P1,X1)) ) ).
tff(fact_52_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))] :
~ ! [A6: A,B4: B,C2: C,D1: D,E1: E] : Y != 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)))),A6),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),D1),E1)))) ).
tff(fact_53_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod(A,product_prod(B,product_prod(C,D)))] :
~ ! [A6: A,B4: B,C2: C,D1: D] : Y != 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))),A6),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),D1))) ).
tff(fact_54_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,X1: product_prod(A,product_prod(B,product_prod(C,D))),P1: fun(product_prod(A,product_prod(B,product_prod(C,D))),bool)] :
( ! [A6: A,B4: B,C2: C,D1: D] : pp(aa(product_prod(A,product_prod(B,product_prod(C,D))),bool,P1,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))),A6),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),D1)))))
=> pp(aa(product_prod(A,product_prod(B,product_prod(C,D))),bool,P1,X1)) ) ).
tff(fact_55_internal__split__conv,axiom,
! [B: $tType,A: $tType,C: $tType,B2: C,A4: B,C1: fun(B,fun(C,A))] : produc1605651328_split(B,C,A,C1,aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A4),B2)) = aa(C,A,aa(B,fun(C,A),C1,A4),B2) ).
tff(fact_56_curry__conv,axiom,
! [A: $tType,B: $tType,C: $tType,B2: C,A4: B,F: fun(product_prod(B,C),A)] : product_curry(B,C,A,F,A4,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),A4),B2)) ).
tff(fact_57_curryI,axiom,
! [A: $tType,B: $tType,B2: B,A4: A,F: fun(product_prod(A,B),bool)] :
( 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),A4),B2)))
=> pp(product_curry(A,B,bool,F,A4,B2)) ) ).
tff(fact_58_curryD,axiom,
! [A: $tType,B: $tType,B2: B,A4: A,F: fun(product_prod(A,B),bool)] :
( pp(product_curry(A,B,bool,F,A4,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),A4),B2))) ) ).
tff(fact_59_curryE,axiom,
! [A: $tType,B: $tType,B2: B,A4: A,F: fun(product_prod(A,B),bool)] :
( pp(product_curry(A,B,bool,F,A4,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),A4),B2))) ) ).
tff(fact_60_curry__def,axiom,
! [C: $tType,A: $tType,B: $tType,X2: fun(product_prod(A,B),C),Xa: A,Xb: B] : product_curry(A,B,C,X2,Xa,Xb) = aa(product_prod(A,B),C,X2,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Xb)) ).
tff(fact_61_Pi__empty,axiom,
! [B: $tType,A: $tType,B3: fun(A,fun(B,bool))] : pi(A,B,bot_bot(fun(A,bool)),B3) = top_top(fun(fun(A,B),bool)) ).
tff(fact_62_total__on__converse,axiom,
! [A: $tType,R1: fun(product_prod(A,A),bool),A5: fun(A,bool)] :
( total_on(A,A5,converse(A,A,R1))
<=> total_on(A,A5,R1) ) ).
tff(fact_63_converse__inv__image,axiom,
! [B: $tType,A: $tType,F: fun(A,B),R2: fun(product_prod(B,B),bool)] : converse(A,A,inv_image(B,A,R2,F)) = inv_image(B,A,converse(B,B,R2),F) ).
tff(fact_64_all__not__in__conv,axiom,
! [A: $tType,A5: fun(A,bool)] :
( ! [X3: A] : ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X3),A5))
<=> ( A5 = bot_bot(fun(A,bool)) ) ) ).
tff(fact_65_empty__Collect__eq,axiom,
! [A: $tType,P1: fun(A,bool)] :
( ( bot_bot(fun(A,bool)) = collect(A,P1) )
<=> ! [X3: A] : ~ pp(aa(A,bool,P1,X3)) ) ).
tff(fact_66_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_67_Collect__empty__eq,axiom,
! [A: $tType,P1: fun(A,bool)] :
( ( collect(A,P1) = bot_bot(fun(A,bool)) )
<=> ! [X3: A] : ~ pp(aa(A,bool,P1,X3)) ) ).
tff(fact_68_emptyE,axiom,
! [A: $tType,A4: A] : ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A4),bot_bot(fun(A,bool)))) ).
tff(fact_69_in__inv__image,axiom,
! [A: $tType,B: $tType,F: fun(A,B),R1: fun(product_prod(B,B),bool),Y1: A,X1: A] :
( pp(aa(fun(product_prod(A,A),bool),bool,aa(product_prod(A,A),fun(fun(product_prod(A,A),bool),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),Y1)),inv_image(B,A,R1,F)))
<=> pp(aa(fun(product_prod(B,B),bool),bool,aa(product_prod(B,B),fun(fun(product_prod(B,B),bool),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,X1)),aa(A,B,F,Y1))),R1)) ) ).
tff(fact_70_Collect__const,axiom,
! [A: $tType,P1: bool] :
( ( pp(P1)
=> ( collect(A,combk(bool,A,P1)) = top_top(fun(A,bool)) ) )
& ( ~ pp(P1)
=> ( collect(A,combk(bool,A,P1)) = bot_bot(fun(A,bool)) ) ) ) ).
tff(fact_71_Pi__eq__empty,axiom,
! [A: $tType,B: $tType,B3: fun(A,fun(B,bool)),A5: fun(A,bool)] :
( ( pi(A,B,A5,B3) = bot_bot(fun(fun(A,B),bool)) )
<=> ? [X3: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X3),A5))
& ( aa(A,fun(B,bool),B3,X3) = bot_bot(fun(B,bool)) ) ) ) ).
tff(fact_72_bot__fun__def,axiom,
! [A: $tType,B: $tType] :
( bot(B)
=> ! [X2: A] : aa(A,B,bot_bot(fun(A,B)),X2) = bot_bot(B) ) ).
tff(fact_73_bot__apply,axiom,
! [B: $tType,A: $tType] :
( bot(A)
=> ! [X1: B] : aa(B,A,bot_bot(fun(B,A)),X1) = bot_bot(A) ) ).
tff(fact_74_total__on__empty,axiom,
! [A: $tType,R1: fun(product_prod(A,A),bool)] : total_on(A,bot_bot(fun(A,bool)),R1) ).
tff(fact_75_ext,axiom,
! [B: $tType,A: $tType,G: fun(A,B),F: fun(A,B)] :
( ! [X4: A] : aa(A,B,F,X4) = aa(A,B,G,X4)
=> ( F = G ) ) ).
tff(fact_76_mem__def,axiom,
! [A: $tType,A5: fun(A,bool),X1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X1),A5))
<=> pp(aa(A,bool,A5,X1)) ) ).
tff(fact_77_Collect__def,axiom,
! [A: $tType,P1: fun(A,bool)] : collect(A,P1) = P1 ).
tff(fact_78_bot__empty__eq,axiom,
! [A: $tType,X2: A] :
( pp(aa(A,bool,bot_bot(fun(A,bool)),X2))
<=> pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X2),bot_bot(fun(A,bool)))) ) ).
tff(fact_79_empty__def,axiom,
! [A: $tType] : bot_bot(fun(A,bool)) = collect(A,combk(bool,A,fFalse)) ).
tff(fact_80_ex__in__conv,axiom,
! [A: $tType,A5: fun(A,bool)] :
( ? [X3: A] : pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X3),A5))
<=> ( A5 != bot_bot(fun(A,bool)) ) ) ).
tff(fact_81_equals0D,axiom,
! [A: $tType,A4: A,A5: fun(A,bool)] :
( ( A5 = bot_bot(fun(A,bool)) )
=> ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),A4),A5)) ) ).
tff(fact_82_UNIV__not__empty,axiom,
! [A: $tType] : top_top(fun(A,bool)) != bot_bot(fun(A,bool)) ).
tff(fact_83_total__on__def,axiom,
! [A: $tType,R1: fun(product_prod(A,A),bool),A5: fun(A,bool)] :
( total_on(A,A5,R1)
<=> ! [X3: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X3),A5))
=> ! [Xa1: A] :
( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Xa1),A5))
=> ( ( X3 != Xa1 )
=> ( pp(aa(fun(product_prod(A,A),bool),bool,aa(product_prod(A,A),fun(fun(product_prod(A,A),bool),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X3),Xa1)),R1))
| pp(aa(fun(product_prod(A,A),bool),bool,aa(product_prod(A,A),fun(fun(product_prod(A,A),bool),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa1),X3)),R1)) ) ) ) ) ) ).
tff(fact_84_equals0I,axiom,
! [A: $tType,A5: fun(A,bool)] :
( ! [Y2: A] : ~ pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),Y2),A5))
=> ( A5 = bot_bot(fun(A,bool)) ) ) ).
tff(fact_85_in__inv__imagep,axiom,
! [A: $tType,B: $tType,Y1: B,X1: B,F: fun(B,A),R1: fun(A,fun(A,bool))] :
( inv_imagep(A,B,R1,F,X1,Y1)
<=> pp(aa(A,bool,aa(A,fun(A,bool),R1,aa(B,A,F,X1)),aa(B,A,F,Y1))) ) ).
tff(fact_86_conversep__converse__eq,axiom,
! [A: $tType,B: $tType,R1: fun(product_prod(B,A),bool),X2: A,Xa: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),conversep(B,A,aa(fun(product_prod(B,A),bool),fun(B,fun(A,bool)),aa(fun(B,fun(fun(product_prod(B,A),bool),fun(A,bool))),fun(fun(product_prod(B,A),bool),fun(B,fun(A,bool))),combc(B,fun(product_prod(B,A),bool),fun(A,bool)),aa(fun(B,fun(A,fun(fun(product_prod(B,A),bool),bool))),fun(B,fun(fun(product_prod(B,A),bool),fun(A,bool))),combb(fun(A,fun(fun(product_prod(B,A),bool),bool)),fun(fun(product_prod(B,A),bool),fun(A,bool)),B,combc(A,fun(product_prod(B,A),bool),bool)),aa(fun(B,fun(A,product_prod(B,A))),fun(B,fun(A,fun(fun(product_prod(B,A),bool),bool))),combb(fun(A,product_prod(B,A)),fun(A,fun(fun(product_prod(B,A),bool),bool)),B,combb(product_prod(B,A),fun(fun(product_prod(B,A),bool),bool),A,member(product_prod(B,A)))),product_Pair(B,A)))),R1)),X2),Xa))
<=> pp(aa(fun(product_prod(A,B),bool),bool,aa(product_prod(A,B),fun(fun(product_prod(A,B),bool),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),Xa)),converse(B,A,R1))) ) ).
tff(fact_87_conversep__iff,axiom,
! [A: $tType,B: $tType,B2: A,A4: B,R1: fun(A,fun(B,bool))] :
( pp(aa(A,bool,aa(B,fun(A,bool),conversep(A,B,R1),A4),B2))
<=> pp(aa(B,bool,aa(A,fun(B,bool),R1,B2),A4)) ) ).
tff(fact_88_conversep__eq,axiom,
! [A: $tType] : conversep(A,A,fequal(A)) = fequal(A) ).
tff(fact_89_conversep__noteq,axiom,
! [A: $tType,X2: A,Xa: A] :
( pp(aa(A,bool,aa(A,fun(A,bool),conversep(A,A,aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),combb(fun(A,bool),fun(A,bool),A,combb(bool,bool,A,fNot)),fequal(A))),X2),Xa))
<=> ( X2 != Xa ) ) ).
tff(fact_90_bot__empty__eq2,axiom,
! [B: $tType,A: $tType,X2: A,Xa: B] :
( pp(aa(B,bool,aa(A,fun(B,bool),bot_bot(fun(A,fun(B,bool))),X2),Xa))
<=> pp(aa(fun(product_prod(A,B),bool),bool,aa(product_prod(A,B),fun(fun(product_prod(A,B),bool),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),Xa)),bot_bot(fun(product_prod(A,B),bool)))) ) ).
tff(fact_91_conversepD,axiom,
! [A: $tType,B: $tType,B2: A,A4: B,R1: fun(A,fun(B,bool))] :
( pp(aa(A,bool,aa(B,fun(A,bool),conversep(A,B,R1),A4),B2))
=> pp(aa(B,bool,aa(A,fun(B,bool),R1,B2),A4)) ) ).
tff(fact_92_conversep_Ointros,axiom,
! [B: $tType,A: $tType,B2: B,A4: A,R1: fun(A,fun(B,bool))] :
( pp(aa(B,bool,aa(A,fun(B,bool),R1,A4),B2))
=> pp(aa(A,bool,aa(B,fun(A,bool),conversep(A,B,R1),B2),A4)) ) ).
tff(fact_93_conversep__conversep,axiom,
! [B: $tType,A: $tType,R1: fun(A,fun(B,bool))] : conversep(B,A,conversep(A,B,R1)) = R1 ).
tff(fact_94_inv__imagep__def,axiom,
! [B: $tType,A: $tType,F: fun(A,B),R1: fun(B,fun(B,bool)),X2: A,Xa: A] :
( inv_imagep(B,A,R1,F,X2,Xa)
<=> pp(aa(B,bool,aa(B,fun(B,bool),R1,aa(A,B,F,X2)),aa(A,B,F,Xa))) ) ).
tff(fact_95_conversep_Osimps,axiom,
! [A: $tType,B: $tType,A2: A,A1: B,R1: fun(A,fun(B,bool))] :
( pp(aa(A,bool,aa(B,fun(A,bool),conversep(A,B,R1),A1),A2))
<=> ? [A3: A,B1: B] :
( ( A1 = B1 )
& ( A2 = A3 )
& pp(aa(B,bool,aa(A,fun(B,bool),R1,A3),B1)) ) ) ).
tff(fact_96_extensional__funcset__empty__range,axiom,
! [B: $tType,A: $tType,S: fun(A,bool)] :
( ( S != bot_bot(fun(A,bool)) )
=> ( extensional_funcset(A,B,S,bot_bot(fun(B,bool))) = bot_bot(fun(fun(A,B),bool)) ) ) ).
tff(fact_97_bot2E,axiom,
! [A: $tType,B: $tType,Y1: B,X1: A] : ~ pp(aa(B,bool,aa(A,fun(B,bool),bot_bot(fun(A,fun(B,bool))),X1),Y1)) ).
tff(fact_98_extensional__funcset__mem,axiom,
! [A: $tType,B: $tType,X1: A,T: fun(B,bool),S: fun(A,bool),F: fun(A,B)] :
( pp(aa(fun(fun(A,B),bool),bool,aa(fun(A,B),fun(fun(fun(A,B),bool),bool),member(fun(A,B)),F),extensional_funcset(A,B,S,T)))
=> ( pp(aa(fun(A,bool),bool,aa(A,fun(fun(A,bool),bool),member(A),X1),S))
=> pp(aa(fun(B,bool),bool,aa(B,fun(fun(B,bool),bool),member(B),aa(A,B,F,X1)),T)) ) ) ).
tff(fact_99_Id__on__empty,axiom,
! [A: $tType] : id_on(A,bot_bot(fun(A,bool))) = bot_bot(fun(product_prod(A,A),bool)) ).
%----Arities (4)
tff(arity_fun___Orderings_Otop,axiom,
! [T_1: $tType,T_2: $tType] :
( top(T_2)
=> top(fun(T_1,T_2)) ) ).
tff(arity_fun___Orderings_Obot,axiom,
! [T_1: $tType,T_2: $tType] :
( bot(T_2)
=> bot(fun(T_1,T_2)) ) ).
tff(arity_HOL_Obool___Orderings_Otop,axiom,
top(bool) ).
tff(arity_HOL_Obool___Orderings_Obot,axiom,
bot(bool) ).
%----Helper facts (14)
tff(help_pp_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_pp_2_1_U,axiom,
pp(fTrue) ).
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),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_fTrue_1_1_U,axiom,
pp(fTrue) ).
tff(help_fTrue_1_1_T,axiom,
! [P: bool] :
( ( P = fTrue )
| ( P = fFalse ) ) ).
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)) ) ).
%----Conjectures (2)
tff(conj_0,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),arrow_1985332922le_Lin)) ).
tff(conj_1,conjecture,
pp(aa(fun(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),bool),bool,aa(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(fun(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),bool),bool),member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),combk(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),arrow_159774573e_indi,l)),arrow_610318064e_Prof)) ).
%------------------------------------------------------------------------------