TPTP Problem File: SCT190_5.p
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%------------------------------------------------------------------------------
% File : SCT190_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Social Choice Theory
% Problem : Arrow's Impossibility Theorem line 132
% 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_132 [Bla13]
% Status : Unknown
% Rating : 1.00 v6.4.0
% Syntax : Number of formulae : 168 ( 46 unt; 44 typ; 0 def)
% Number of atoms : 282 ( 103 equ)
% Maximal formula atoms : 12 ( 1 avg)
% Number of connectives : 220 ( 62 ~; 15 |; 15 &)
% ( 23 <=>; 105 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 7 avg)
% Maximal term depth : 10 ( 2 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 45 ( 25 >; 20 *; 0 +; 0 <<)
% Number of predicates : 8 ( 7 usr; 0 prp; 1-6 aty)
% Number of functors : 34 ( 34 usr; 7 con; 0-6 aty)
% Number of variables : 607 ( 557 !; 7 ?; 607 :)
% ( 43 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_UNK_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:16:24
%------------------------------------------------------------------------------
%----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_Otop,type,
top:
!>[A1: $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:
!>[B1: $tType,C: $tType,A1: $tType] : fun(fun(B1,C),fun(fun(A1,B1),fun(A1,C))) ).
tff(sy_c_COMBC,type,
combc:
!>[A1: $tType,B1: $tType,C: $tType] : ( ( fun(A1,fun(B1,C)) * B1 ) > fun(A1,C) ) ).
tff(sy_c_COMBI,type,
combi:
!>[A1: $tType] : fun(A1,A1) ).
tff(sy_c_COMBK,type,
combk:
!>[A1: $tType,B1: $tType] : ( A1 > fun(B1,A1) ) ).
tff(sy_c_COMBS,type,
combs:
!>[A1: $tType,B1: $tType,C: $tType] : ( ( fun(A1,fun(B1,C)) * fun(A1,B1) ) > fun(A1,C) ) ).
tff(sy_c_FuncSet_OPi,type,
pi:
!>[A1: $tType,B1: $tType] : ( ( fun(A1,bool) * fun(A1,fun(B1,bool)) ) > fun(fun(A1,B1),bool) ) ).
tff(sy_c_FuncSet_Ocompose,type,
compose:
!>[A1: $tType,B1: $tType,C: $tType] : ( ( fun(A1,bool) * fun(B1,C) * fun(A1,B1) ) > fun(A1,C) ) ).
tff(sy_c_HOL_OThe,type,
the:
!>[A1: $tType] : ( fun(A1,bool) > A1 ) ).
tff(sy_c_Order__Relation_Ostrict__linear__order__on,type,
order_215145569der_on:
!>[A1: $tType] : ( fun(A1,bool) > fun(fun(product_prod(A1,A1),bool),bool) ) ).
tff(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A1: $tType] : A1 ).
tff(sy_c_Orderings_Otop__class_Otop,type,
top_top:
!>[A1: $tType] : A1 ).
tff(sy_c_Predicate_Oinv__imagep,type,
inv_imagep:
!>[B1: $tType,A1: $tType] : ( ( fun(B1,fun(B1,bool)) * fun(A1,B1) * A1 * A1 ) > $o ) ).
tff(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A1: $tType,B1: $tType] : fun(A1,fun(B1,product_prod(A1,B1))) ).
tff(sy_c_Product__Type_Ocurry,type,
product_curry:
!>[A1: $tType,B1: $tType,C: $tType] : ( fun(product_prod(A1,B1),C) > fun(A1,fun(B1,C)) ) ).
tff(sy_c_Product__Type_Ointernal__split,type,
produc1605651328_split:
!>[A1: $tType,B1: $tType,C: $tType] : fun(fun(A1,fun(B1,C)),fun(product_prod(A1,B1),C)) ).
tff(sy_c_Product__Type_Oprod_Oprod__case,type,
product_prod_case:
!>[A1: $tType,B1: $tType,T: $tType] : fun(fun(A1,fun(B1,T)),fun(product_prod(A1,B1),T)) ).
tff(sy_c_Product__Type_Oprod_Oprod__rec,type,
product_prod_rec:
!>[A1: $tType,B1: $tType,T: $tType] : ( ( fun(A1,fun(B1,T)) * product_prod(A1,B1) ) > T ) ).
tff(sy_c_Set_OCollect,type,
collect:
!>[A1: $tType] : ( fun(A1,bool) > fun(A1,bool) ) ).
tff(sy_c_aa,type,
aa:
!>[A1: $tType,B1: $tType] : ( ( fun(A1,B1) * A1 ) > B1 ) ).
tff(sy_c_fFalse,type,
fFalse: bool ).
tff(sy_c_fNot,type,
fNot: bool > bool ).
tff(sy_c_fTrue,type,
fTrue: bool ).
tff(sy_c_fconj,type,
fconj: fun(bool,fun(bool,bool)) ).
tff(sy_c_fequal,type,
fequal:
!>[A1: $tType] : fun(A1,fun(A1,bool)) ).
tff(sy_c_fimplies,type,
fimplies: bool > fun(bool,bool) ).
tff(sy_c_member,type,
member:
!>[A1: $tType] : ( ( A1 * fun(A1,bool) ) > $o ) ).
tff(sy_c_pp,type,
pp: bool > $o ).
tff(sy_v_F,type,
f: 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)) ).
tff(sy_v_i,type,
i: arrow_159774573e_indi ).
%----Relevant facts (100)
tff(fact_0_notin__Lin__iff,axiom,
! [Y2: arrow_411405190le_alt,X4: arrow_411405190le_alt,L: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)] :
( member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),L,arrow_1985332922le_Lin)
=> ( ( X4 != Y2 )
=> ( ~ 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),X4),Y2),L)
<=> 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),Y2),X4),L) ) ) ) ).
tff(fact_1_Lin__irrefl,axiom,
! [B4: arrow_411405190le_alt,A3: arrow_411405190le_alt,L: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)] :
( member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),L,arrow_1985332922le_Lin)
=> ( 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),B4),L)
=> ~ 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),B4),A3),L) ) ) ).
tff(fact_2_dictator__def,axiom,
! [Ia: arrow_159774573e_indi,Fa: 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(Fa,Ia)
<=> ! [X2: fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))] :
( member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),X2,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),Fa,X2) = aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),X2,Ia) ) ) ) ).
tff(fact_3_PiE,axiom,
! [B1: $tType,A1: $tType,X4: A1,B2: fun(A1,fun(B1,bool)),A2: fun(A1,bool),F: fun(A1,B1)] :
( member(fun(A1,B1),F,pi(A1,B1,A2,B2))
=> ( ~ member(B1,aa(A1,B1,F,X4),aa(A1,fun(B1,bool),B2,X4))
=> ~ member(A1,X4,A2) ) ) ).
tff(fact_4_split__paired__All,axiom,
! [A1: $tType,B1: $tType,P1: fun(product_prod(A1,B1),bool)] :
( ! [X11: product_prod(A1,B1)] : pp(aa(product_prod(A1,B1),bool,P1,X11))
<=> ! [A5: A1,B5: B1] : pp(aa(product_prod(A1,B1),bool,P1,aa(B1,product_prod(A1,B1),aa(A1,fun(B1,product_prod(A1,B1)),product_Pair(A1,B1),A5),B5))) ) ).
tff(fact_5_Pair__eq,axiom,
! [A1: $tType,B1: $tType,B8: B1,A8: A1,B4: B1,A3: A1] :
( ( aa(B1,product_prod(A1,B1),aa(A1,fun(B1,product_prod(A1,B1)),product_Pair(A1,B1),A3),B4) = aa(B1,product_prod(A1,B1),aa(A1,fun(B1,product_prod(A1,B1)),product_Pair(A1,B1),A8),B8) )
<=> ( ( A3 = A8 )
& ( B4 = B8 ) ) ) ).
tff(fact_6_in__above,axiom,
! [Y2: arrow_411405190le_alt,X4: arrow_411405190le_alt,L: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),B4: arrow_411405190le_alt,A3: arrow_411405190le_alt] :
( ( A3 != B4 )
=> ( member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),L,arrow_1985332922le_Lin)
=> ( 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),X4),Y2),arrow_1158827142_above(L,A3,B4))
<=> ( ( X4 != Y2 )
& ( ( X4 = B4 )
=> 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),Y2),L) )
& ( ( X4 != B4 )
=> ( ( ( Y2 = B4 )
=> ( ( X4 = A3 )
| 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),X4),A3),L) ) )
& ( ( Y2 != B4 )
=> 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),X4),Y2),L) ) ) ) ) ) ) ) ).
tff(fact_7_in__below,axiom,
! [Y2: arrow_411405190le_alt,X4: arrow_411405190le_alt,L: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),B4: arrow_411405190le_alt,A3: arrow_411405190le_alt] :
( ( A3 != B4 )
=> ( member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),L,arrow_1985332922le_Lin)
=> ( 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),X4),Y2),arrow_319942042_below(L,A3,B4))
<=> ( ( X4 != Y2 )
& ( ( Y2 = A3 )
=> 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),X4),B4),L) )
& ( ( Y2 != A3 )
=> ( ( ( X4 = A3 )
=> ( ( Y2 = B4 )
| 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),B4),Y2),L) ) )
& ( ( X4 != A3 )
=> 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),X4),Y2),L) ) ) ) ) ) ) ) ).
tff(fact_8_unanimity__def,axiom,
! [Fa: 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(Fa)
<=> ! [X2: fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))] :
( member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),X2,arrow_610318064e_Prof)
=> ! [A5: arrow_411405190le_alt,B5: arrow_411405190le_alt] :
( ! [I: arrow_159774573e_indi] : 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),A5),B5),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),X2,I))
=> 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),A5),B5),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),Fa,X2)) ) ) ) ).
tff(fact_9_IIA__def,axiom,
! [Fa: 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(Fa)
<=> ! [X2: fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))] :
( member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),X2,arrow_610318064e_Prof)
=> ! [Xa1: fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))] :
( member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),Xa1,arrow_610318064e_Prof)
=> ! [A5: arrow_411405190le_alt,B5: arrow_411405190le_alt] :
( ! [I: arrow_159774573e_indi] :
( 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),A5),B5),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),X2,I))
<=> 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),A5),B5),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Xa1,I)) )
=> ( 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),A5),B5),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),Fa,X2))
<=> 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),A5),B5),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),Fa,Xa1)) ) ) ) ) ) ).
tff(fact_10_linear__alt,axiom,
? [L1: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)] : member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),L1,arrow_1985332922le_Lin) ).
tff(fact_11_funcset__id,axiom,
! [A1: $tType,A2: fun(A1,bool)] : member(fun(A1,A1),combi(A1),pi(A1,A1,A2,combk(fun(A1,bool),A1,A2))) ).
tff(fact_12_funcset__mem,axiom,
! [A1: $tType,B1: $tType,X4: A1,B2: fun(B1,bool),A2: fun(A1,bool),F: fun(A1,B1)] :
( member(fun(A1,B1),F,pi(A1,B1,A2,combk(fun(B1,bool),A1,B2)))
=> ( member(A1,X4,A2)
=> member(B1,aa(A1,B1,F,X4),B2) ) ) ).
tff(fact_13_above__Lin,axiom,
! [L: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Y2: arrow_411405190le_alt,X4: arrow_411405190le_alt] :
( ( X4 != Y2 )
=> ( member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),L,arrow_1985332922le_Lin)
=> member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),arrow_1158827142_above(L,X4,Y2),arrow_1985332922le_Lin) ) ) ).
tff(fact_14_below__Lin,axiom,
! [L: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Y2: arrow_411405190le_alt,X4: arrow_411405190le_alt] :
( ( X4 != Y2 )
=> ( member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),L,arrow_1985332922le_Lin)
=> member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),arrow_319942042_below(L,X4,Y2),arrow_1985332922le_Lin) ) ) ).
tff(fact_15_Pair__inject,axiom,
! [A1: $tType,B1: $tType,B7: B1,A7: A1,B6: B1,A6: A1] :
( ( aa(B1,product_prod(A1,B1),aa(A1,fun(B1,product_prod(A1,B1)),product_Pair(A1,B1),A6),B6) = aa(B1,product_prod(A1,B1),aa(A1,fun(B1,product_prod(A1,B1)),product_Pair(A1,B1),A7),B7) )
=> ~ ( ( A6 = A7 )
=> ( B6 != B7 ) ) ) ).
tff(fact_16_Pi__mem,axiom,
! [B1: $tType,A1: $tType,X4: A1,B2: fun(A1,fun(B1,bool)),A2: fun(A1,bool),F: fun(A1,B1)] :
( member(fun(A1,B1),F,pi(A1,B1,A2,B2))
=> ( member(A1,X4,A2)
=> member(B1,aa(A1,B1,F,X4),aa(A1,fun(B1,bool),B2,X4)) ) ) ).
tff(fact_17_Pi__I,axiom,
! [B1: $tType,A1: $tType,B2: fun(A1,fun(B1,bool)),F: fun(A1,B1),A2: fun(A1,bool)] :
( ! [X3: A1] :
( member(A1,X3,A2)
=> member(B1,aa(A1,B1,F,X3),aa(A1,fun(B1,bool),B2,X3)) )
=> member(fun(A1,B1),F,pi(A1,B1,A2,B2)) ) ).
tff(fact_18_split__paired__Ex,axiom,
! [A1: $tType,B1: $tType,P1: fun(product_prod(A1,B1),bool)] :
( ? [X11: product_prod(A1,B1)] : pp(aa(product_prod(A1,B1),bool,P1,X11))
<=> ? [A5: A1,B5: B1] : pp(aa(product_prod(A1,B1),bool,P1,aa(B1,product_prod(A1,B1),aa(A1,fun(B1,product_prod(A1,B1)),product_Pair(A1,B1),A5),B5))) ) ).
tff(fact_19_prod_Orecs,axiom,
! [B1: $tType,A1: $tType,C: $tType,B4: C,A3: B1,F1: fun(B1,fun(C,A1))] : product_prod_rec(B1,C,A1,F1,aa(C,product_prod(B1,C),aa(B1,fun(C,product_prod(B1,C)),product_Pair(B1,C),A3),B4)) = aa(C,A1,aa(B1,fun(C,A1),F1,A3),B4) ).
tff(fact_20_funcsetI,axiom,
! [A1: $tType,B1: $tType,B2: fun(B1,bool),F: fun(A1,B1),A2: fun(A1,bool)] :
( ! [X3: A1] :
( member(A1,X3,A2)
=> member(B1,aa(A1,B1,F,X3),B2) )
=> member(fun(A1,B1),F,pi(A1,B1,A2,combk(fun(B1,bool),A1,B2))) ) ).
tff(fact_21_Pi__cong,axiom,
! [B1: $tType,A1: $tType,B2: fun(A1,fun(B1,bool)),G: fun(A1,B1),F: fun(A1,B1),A2: fun(A1,bool)] :
( ! [W: A1] :
( member(A1,W,A2)
=> ( aa(A1,B1,F,W) = aa(A1,B1,G,W) ) )
=> ( member(fun(A1,B1),F,pi(A1,B1,A2,B2))
<=> member(fun(A1,B1),G,pi(A1,B1,A2,B2)) ) ) ).
tff(fact_22_Pi__I_H,axiom,
! [B1: $tType,A1: $tType,B2: fun(A1,fun(B1,bool)),F: fun(A1,B1),A2: fun(A1,bool)] :
( ! [X3: A1] :
( member(A1,X3,A2)
=> member(B1,aa(A1,B1,F,X3),aa(A1,fun(B1,bool),B2,X3)) )
=> member(fun(A1,B1),F,pi(A1,B1,A2,B2)) ) ).
tff(fact_23_mktop__Lin,axiom,
! [X4: arrow_411405190le_alt,L: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)] :
( member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),L,arrow_1985332922le_Lin)
=> member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),arrow_424895264_mktop(L,X4),arrow_1985332922le_Lin) ) ).
tff(fact_24_mkbot__Lin,axiom,
! [X4: arrow_411405190le_alt,L: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)] :
( member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),L,arrow_1985332922le_Lin)
=> member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),arrow_276188178_mkbot(L,X4),arrow_1985332922le_Lin) ) ).
tff(fact_25_in__mkbot,axiom,
! [Z: arrow_411405190le_alt,L: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Y2: arrow_411405190le_alt,X4: arrow_411405190le_alt] :
( 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),X4),Y2),arrow_276188178_mkbot(L,Z))
<=> ( ( Y2 != Z )
& ( ( X4 = Z )
=> ( X4 != Y2 ) )
& ( ( X4 != Z )
=> 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),X4),Y2),L) ) ) ) ).
tff(fact_26_in__mktop,axiom,
! [Z: arrow_411405190le_alt,L: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Y2: arrow_411405190le_alt,X4: arrow_411405190le_alt] :
( 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),X4),Y2),arrow_424895264_mktop(L,Z))
<=> ( ( X4 != Z )
& ( ( Y2 = Z )
=> ( X4 != Y2 ) )
& ( ( Y2 != Z )
=> 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),X4),Y2),L) ) ) ) ).
tff(fact_27_pred__equals__eq2,axiom,
! [B1: $tType,A1: $tType,S: fun(product_prod(A1,B1),bool),R2: fun(product_prod(A1,B1),bool)] :
( ! [X2: A1,Xa1: B1] :
( member(product_prod(A1,B1),aa(B1,product_prod(A1,B1),aa(A1,fun(B1,product_prod(A1,B1)),product_Pair(A1,B1),X2),Xa1),R2)
<=> member(product_prod(A1,B1),aa(B1,product_prod(A1,B1),aa(A1,fun(B1,product_prod(A1,B1)),product_Pair(A1,B1),X2),Xa1),S) )
<=> ( R2 = S ) ) ).
tff(fact_28_prod__induct6,axiom,
! [F2: $tType,E: $tType,D: $tType,C: $tType,B1: $tType,A1: $tType,X4: product_prod(A1,product_prod(B1,product_prod(C,product_prod(D,product_prod(E,F2))))),P1: fun(product_prod(A1,product_prod(B1,product_prod(C,product_prod(D,product_prod(E,F2))))),bool)] :
( ! [A4: A1,B3: B1,C3: C,D2: D,E1: E,F3: F2] : pp(aa(product_prod(A1,product_prod(B1,product_prod(C,product_prod(D,product_prod(E,F2))))),bool,P1,aa(product_prod(B1,product_prod(C,product_prod(D,product_prod(E,F2)))),product_prod(A1,product_prod(B1,product_prod(C,product_prod(D,product_prod(E,F2))))),aa(A1,fun(product_prod(B1,product_prod(C,product_prod(D,product_prod(E,F2)))),product_prod(A1,product_prod(B1,product_prod(C,product_prod(D,product_prod(E,F2)))))),product_Pair(A1,product_prod(B1,product_prod(C,product_prod(D,product_prod(E,F2))))),A4),aa(product_prod(C,product_prod(D,product_prod(E,F2))),product_prod(B1,product_prod(C,product_prod(D,product_prod(E,F2)))),aa(B1,fun(product_prod(C,product_prod(D,product_prod(E,F2))),product_prod(B1,product_prod(C,product_prod(D,product_prod(E,F2))))),product_Pair(B1,product_prod(C,product_prod(D,product_prod(E,F2)))),B3),aa(product_prod(D,product_prod(E,F2)),product_prod(C,product_prod(D,product_prod(E,F2))),aa(C,fun(product_prod(D,product_prod(E,F2)),product_prod(C,product_prod(D,product_prod(E,F2)))),product_Pair(C,product_prod(D,product_prod(E,F2))),C3),aa(product_prod(E,F2),product_prod(D,product_prod(E,F2)),aa(D,fun(product_prod(E,F2),product_prod(D,product_prod(E,F2))),product_Pair(D,product_prod(E,F2)),D2),aa(F2,product_prod(E,F2),aa(E,fun(F2,product_prod(E,F2)),product_Pair(E,F2),E1),F3)))))))
=> pp(aa(product_prod(A1,product_prod(B1,product_prod(C,product_prod(D,product_prod(E,F2))))),bool,P1,X4)) ) ).
tff(fact_29_prod__cases6,axiom,
! [A1: $tType,B1: $tType,C: $tType,D: $tType,E: $tType,F2: $tType,Y: product_prod(A1,product_prod(B1,product_prod(C,product_prod(D,product_prod(E,F2)))))] :
~ ! [A4: A1,B3: B1,C3: C,D2: D,E1: E,F3: F2] : Y != aa(product_prod(B1,product_prod(C,product_prod(D,product_prod(E,F2)))),product_prod(A1,product_prod(B1,product_prod(C,product_prod(D,product_prod(E,F2))))),aa(A1,fun(product_prod(B1,product_prod(C,product_prod(D,product_prod(E,F2)))),product_prod(A1,product_prod(B1,product_prod(C,product_prod(D,product_prod(E,F2)))))),product_Pair(A1,product_prod(B1,product_prod(C,product_prod(D,product_prod(E,F2))))),A4),aa(product_prod(C,product_prod(D,product_prod(E,F2))),product_prod(B1,product_prod(C,product_prod(D,product_prod(E,F2)))),aa(B1,fun(product_prod(C,product_prod(D,product_prod(E,F2))),product_prod(B1,product_prod(C,product_prod(D,product_prod(E,F2))))),product_Pair(B1,product_prod(C,product_prod(D,product_prod(E,F2)))),B3),aa(product_prod(D,product_prod(E,F2)),product_prod(C,product_prod(D,product_prod(E,F2))),aa(C,fun(product_prod(D,product_prod(E,F2)),product_prod(C,product_prod(D,product_prod(E,F2)))),product_Pair(C,product_prod(D,product_prod(E,F2))),C3),aa(product_prod(E,F2),product_prod(D,product_prod(E,F2)),aa(D,fun(product_prod(E,F2),product_prod(D,product_prod(E,F2))),product_Pair(D,product_prod(E,F2)),D2),aa(F2,product_prod(E,F2),aa(E,fun(F2,product_prod(E,F2)),product_Pair(E,F2),E1),F3))))) ).
tff(fact_30_prod__induct5,axiom,
! [E: $tType,D: $tType,C: $tType,B1: $tType,A1: $tType,X4: product_prod(A1,product_prod(B1,product_prod(C,product_prod(D,E)))),P1: fun(product_prod(A1,product_prod(B1,product_prod(C,product_prod(D,E)))),bool)] :
( ! [A4: A1,B3: B1,C3: C,D2: D,E1: E] : pp(aa(product_prod(A1,product_prod(B1,product_prod(C,product_prod(D,E)))),bool,P1,aa(product_prod(B1,product_prod(C,product_prod(D,E))),product_prod(A1,product_prod(B1,product_prod(C,product_prod(D,E)))),aa(A1,fun(product_prod(B1,product_prod(C,product_prod(D,E))),product_prod(A1,product_prod(B1,product_prod(C,product_prod(D,E))))),product_Pair(A1,product_prod(B1,product_prod(C,product_prod(D,E)))),A4),aa(product_prod(C,product_prod(D,E)),product_prod(B1,product_prod(C,product_prod(D,E))),aa(B1,fun(product_prod(C,product_prod(D,E)),product_prod(B1,product_prod(C,product_prod(D,E)))),product_Pair(B1,product_prod(C,product_prod(D,E))),B3),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)),C3),aa(E,product_prod(D,E),aa(D,fun(E,product_prod(D,E)),product_Pair(D,E),D2),E1))))))
=> pp(aa(product_prod(A1,product_prod(B1,product_prod(C,product_prod(D,E)))),bool,P1,X4)) ) ).
tff(fact_31_prod__cases5,axiom,
! [A1: $tType,B1: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod(A1,product_prod(B1,product_prod(C,product_prod(D,E))))] :
~ ! [A4: A1,B3: B1,C3: C,D2: D,E1: E] : Y != aa(product_prod(B1,product_prod(C,product_prod(D,E))),product_prod(A1,product_prod(B1,product_prod(C,product_prod(D,E)))),aa(A1,fun(product_prod(B1,product_prod(C,product_prod(D,E))),product_prod(A1,product_prod(B1,product_prod(C,product_prod(D,E))))),product_Pair(A1,product_prod(B1,product_prod(C,product_prod(D,E)))),A4),aa(product_prod(C,product_prod(D,E)),product_prod(B1,product_prod(C,product_prod(D,E))),aa(B1,fun(product_prod(C,product_prod(D,E)),product_prod(B1,product_prod(C,product_prod(D,E)))),product_Pair(B1,product_prod(C,product_prod(D,E))),B3),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)),C3),aa(E,product_prod(D,E),aa(D,fun(E,product_prod(D,E)),product_Pair(D,E),D2),E1)))) ).
tff(fact_32_prod__induct4,axiom,
! [D: $tType,C: $tType,B1: $tType,A1: $tType,X4: product_prod(A1,product_prod(B1,product_prod(C,D))),P1: fun(product_prod(A1,product_prod(B1,product_prod(C,D))),bool)] :
( ! [A4: A1,B3: B1,C3: C,D2: D] : pp(aa(product_prod(A1,product_prod(B1,product_prod(C,D))),bool,P1,aa(product_prod(B1,product_prod(C,D)),product_prod(A1,product_prod(B1,product_prod(C,D))),aa(A1,fun(product_prod(B1,product_prod(C,D)),product_prod(A1,product_prod(B1,product_prod(C,D)))),product_Pair(A1,product_prod(B1,product_prod(C,D))),A4),aa(product_prod(C,D),product_prod(B1,product_prod(C,D)),aa(B1,fun(product_prod(C,D),product_prod(B1,product_prod(C,D))),product_Pair(B1,product_prod(C,D)),B3),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),C3),D2)))))
=> pp(aa(product_prod(A1,product_prod(B1,product_prod(C,D))),bool,P1,X4)) ) ).
tff(fact_33_prod__cases4,axiom,
! [A1: $tType,B1: $tType,C: $tType,D: $tType,Y: product_prod(A1,product_prod(B1,product_prod(C,D)))] :
~ ! [A4: A1,B3: B1,C3: C,D2: D] : Y != aa(product_prod(B1,product_prod(C,D)),product_prod(A1,product_prod(B1,product_prod(C,D))),aa(A1,fun(product_prod(B1,product_prod(C,D)),product_prod(A1,product_prod(B1,product_prod(C,D)))),product_Pair(A1,product_prod(B1,product_prod(C,D))),A4),aa(product_prod(C,D),product_prod(B1,product_prod(C,D)),aa(B1,fun(product_prod(C,D),product_prod(B1,product_prod(C,D))),product_Pair(B1,product_prod(C,D)),B3),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),C3),D2))) ).
tff(fact_34_prod__cases3,axiom,
! [A1: $tType,B1: $tType,C: $tType,Y: product_prod(A1,product_prod(B1,C))] :
~ ! [A4: A1,B3: B1,C3: C] : Y != aa(product_prod(B1,C),product_prod(A1,product_prod(B1,C)),aa(A1,fun(product_prod(B1,C),product_prod(A1,product_prod(B1,C))),product_Pair(A1,product_prod(B1,C)),A4),aa(C,product_prod(B1,C),aa(B1,fun(C,product_prod(B1,C)),product_Pair(B1,C),B3),C3)) ).
tff(fact_35_prod__induct3,axiom,
! [C: $tType,B1: $tType,A1: $tType,X4: product_prod(A1,product_prod(B1,C)),P1: fun(product_prod(A1,product_prod(B1,C)),bool)] :
( ! [A4: A1,B3: B1,C3: C] : pp(aa(product_prod(A1,product_prod(B1,C)),bool,P1,aa(product_prod(B1,C),product_prod(A1,product_prod(B1,C)),aa(A1,fun(product_prod(B1,C),product_prod(A1,product_prod(B1,C))),product_Pair(A1,product_prod(B1,C)),A4),aa(C,product_prod(B1,C),aa(B1,fun(C,product_prod(B1,C)),product_Pair(B1,C),B3),C3))))
=> pp(aa(product_prod(A1,product_prod(B1,C)),bool,P1,X4)) ) ).
tff(fact_36_prod_Oexhaust,axiom,
! [A1: $tType,B1: $tType,Y: product_prod(A1,B1)] :
~ ! [A4: A1,B3: B1] : Y != aa(B1,product_prod(A1,B1),aa(A1,fun(B1,product_prod(A1,B1)),product_Pair(A1,B1),A4),B3) ).
tff(fact_37_PairE,axiom,
! [A1: $tType,B1: $tType,P3: product_prod(A1,B1)] :
~ ! [X3: A1,Y1: B1] : P3 != aa(B1,product_prod(A1,B1),aa(A1,fun(B1,product_prod(A1,B1)),product_Pair(A1,B1),X3),Y1) ).
tff(fact_38_internal__split__conv,axiom,
! [B1: $tType,A1: $tType,C: $tType,B4: C,A3: B1,C1: fun(B1,fun(C,A1))] : aa(product_prod(B1,C),A1,aa(fun(B1,fun(C,A1)),fun(product_prod(B1,C),A1),produc1605651328_split(B1,C,A1),C1),aa(C,product_prod(B1,C),aa(B1,fun(C,product_prod(B1,C)),product_Pair(B1,C),A3),B4)) = aa(C,A1,aa(B1,fun(C,A1),C1,A3),B4) ).
tff(fact_39_curry__conv,axiom,
! [A1: $tType,B1: $tType,C: $tType,B4: C,A3: B1,F: fun(product_prod(B1,C),A1)] : aa(C,A1,aa(B1,fun(C,A1),product_curry(B1,C,A1,F),A3),B4) = aa(product_prod(B1,C),A1,F,aa(C,product_prod(B1,C),aa(B1,fun(C,product_prod(B1,C)),product_Pair(B1,C),A3),B4)) ).
tff(fact_40_curryI,axiom,
! [A1: $tType,B1: $tType,B4: B1,A3: A1,F: fun(product_prod(A1,B1),bool)] :
( pp(aa(product_prod(A1,B1),bool,F,aa(B1,product_prod(A1,B1),aa(A1,fun(B1,product_prod(A1,B1)),product_Pair(A1,B1),A3),B4)))
=> pp(aa(B1,bool,aa(A1,fun(B1,bool),product_curry(A1,B1,bool,F),A3),B4)) ) ).
tff(fact_41_curryD,axiom,
! [A1: $tType,B1: $tType,B4: B1,A3: A1,F: fun(product_prod(A1,B1),bool)] :
( pp(aa(B1,bool,aa(A1,fun(B1,bool),product_curry(A1,B1,bool,F),A3),B4))
=> pp(aa(product_prod(A1,B1),bool,F,aa(B1,product_prod(A1,B1),aa(A1,fun(B1,product_prod(A1,B1)),product_Pair(A1,B1),A3),B4))) ) ).
tff(fact_42_curryE,axiom,
! [A1: $tType,B1: $tType,B4: B1,A3: A1,F: fun(product_prod(A1,B1),bool)] :
( pp(aa(B1,bool,aa(A1,fun(B1,bool),product_curry(A1,B1,bool,F),A3),B4))
=> pp(aa(product_prod(A1,B1),bool,F,aa(B1,product_prod(A1,B1),aa(A1,fun(B1,product_prod(A1,B1)),product_Pair(A1,B1),A3),B4))) ) ).
tff(fact_43_curry__def,axiom,
! [C: $tType,A1: $tType,B1: $tType,X: fun(product_prod(A1,B1),C),Xa: A1,Xb: B1] : aa(B1,C,aa(A1,fun(B1,C),product_curry(A1,B1,C,X),Xa),Xb) = aa(product_prod(A1,B1),C,X,aa(B1,product_prod(A1,B1),aa(A1,fun(B1,product_prod(A1,B1)),product_Pair(A1,B1),Xa),Xb)) ).
tff(fact_44_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_45_in__inv__imagep,axiom,
! [A1: $tType,B1: $tType,Y2: B1,X4: B1,F: fun(B1,A1),R1: fun(A1,fun(A1,bool))] :
( inv_imagep(A1,B1,R1,F,X4,Y2)
<=> pp(aa(A1,bool,aa(A1,fun(A1,bool),R1,aa(B1,A1,F,X4)),aa(B1,A1,F,Y2))) ) ).
tff(fact_46_internal__split__def,axiom,
! [C: $tType,B1: $tType,A1: $tType] : produc1605651328_split(A1,B1,C) = product_prod_case(A1,B1,C) ).
tff(fact_47_funcset__compose,axiom,
! [B1: $tType,A1: $tType,C: $tType,C2: fun(C,bool),G: fun(B1,C),B2: fun(B1,bool),A2: fun(A1,bool),F: fun(A1,B1)] :
( member(fun(A1,B1),F,pi(A1,B1,A2,combk(fun(B1,bool),A1,B2)))
=> ( member(fun(B1,C),G,pi(B1,C,B2,combk(fun(C,bool),B1,C2)))
=> member(fun(A1,C),compose(A1,B1,C,A2,G,F),pi(A1,C,A2,combk(fun(C,bool),A1,C2))) ) ) ).
tff(fact_48_compose__assoc,axiom,
! [C: $tType,D: $tType,B1: $tType,A1: $tType,D1: fun(D,bool),H: fun(C,D),C2: fun(C,bool),G: fun(B1,C),B2: fun(B1,bool),A2: fun(A1,bool),F: fun(A1,B1)] :
( member(fun(A1,B1),F,pi(A1,B1,A2,combk(fun(B1,bool),A1,B2)))
=> ( member(fun(B1,C),G,pi(B1,C,B2,combk(fun(C,bool),B1,C2)))
=> ( member(fun(C,D),H,pi(C,D,C2,combk(fun(D,bool),C,D1)))
=> ( compose(A1,C,D,A2,H,compose(A1,B1,C,A2,G,F)) = compose(A1,B1,D,A2,compose(B1,C,D,B2,H,G),F) ) ) ) ) ).
tff(fact_49_prod__caseI,axiom,
! [A1: $tType,B1: $tType,B4: B1,A3: A1,F1: fun(A1,fun(B1,bool))] :
( pp(aa(B1,bool,aa(A1,fun(B1,bool),F1,A3),B4))
=> pp(aa(product_prod(A1,B1),bool,aa(fun(A1,fun(B1,bool)),fun(product_prod(A1,B1),bool),product_prod_case(A1,B1,bool),F1),aa(B1,product_prod(A1,B1),aa(A1,fun(B1,product_prod(A1,B1)),product_Pair(A1,B1),A3),B4))) ) ).
tff(fact_50_splitI,axiom,
! [A1: $tType,B1: $tType,B4: B1,A3: A1,F: fun(A1,fun(B1,bool))] :
( pp(aa(B1,bool,aa(A1,fun(B1,bool),F,A3),B4))
=> pp(aa(product_prod(A1,B1),bool,aa(fun(A1,fun(B1,bool)),fun(product_prod(A1,B1),bool),product_prod_case(A1,B1,bool),F),aa(B1,product_prod(A1,B1),aa(A1,fun(B1,product_prod(A1,B1)),product_Pair(A1,B1),A3),B4))) ) ).
tff(fact_51_mem__splitI,axiom,
! [A1: $tType,B1: $tType,C: $tType,B4: C,A3: B1,C1: fun(B1,fun(C,fun(A1,bool))),Z: A1] :
( member(A1,Z,aa(C,fun(A1,bool),aa(B1,fun(C,fun(A1,bool)),C1,A3),B4))
=> member(A1,Z,aa(product_prod(B1,C),fun(A1,bool),aa(fun(B1,fun(C,fun(A1,bool))),fun(product_prod(B1,C),fun(A1,bool)),product_prod_case(B1,C,fun(A1,bool)),C1),aa(C,product_prod(B1,C),aa(B1,fun(C,product_prod(B1,C)),product_Pair(B1,C),A3),B4))) ) ).
tff(fact_52_top1I,axiom,
! [A1: $tType,X4: A1] : pp(aa(A1,bool,top_top(fun(A1,bool)),X4)) ).
tff(fact_53_split__conv,axiom,
! [B1: $tType,A1: $tType,C: $tType,B4: C,A3: B1,F: fun(B1,fun(C,A1))] : aa(product_prod(B1,C),A1,aa(fun(B1,fun(C,A1)),fun(product_prod(B1,C),A1),product_prod_case(B1,C,A1),F),aa(C,product_prod(B1,C),aa(B1,fun(C,product_prod(B1,C)),product_Pair(B1,C),A3),B4)) = aa(C,A1,aa(B1,fun(C,A1),F,A3),B4) ).
tff(fact_54_Pi__UNIV,axiom,
! [B1: $tType,A1: $tType,A2: fun(A1,bool)] : pi(A1,B1,A2,combk(fun(B1,bool),A1,top_top(fun(B1,bool)))) = top_top(fun(fun(A1,B1),bool)) ).
tff(fact_55_compose__eq,axiom,
! [B1: $tType,C: $tType,A1: $tType,F: fun(A1,C),G: fun(C,B1),A2: fun(A1,bool),X4: A1] :
( member(A1,X4,A2)
=> ( aa(A1,B1,compose(A1,C,B1,A2,G,F),X4) = aa(C,B1,G,aa(A1,C,F,X4)) ) ) ).
tff(fact_56_split__weak__cong,axiom,
! [C: $tType,B1: $tType,A1: $tType,C1: fun(A1,fun(B1,C)),Q1: product_prod(A1,B1),P2: product_prod(A1,B1)] :
( ( P2 = Q1 )
=> ( aa(product_prod(A1,B1),C,aa(fun(A1,fun(B1,C)),fun(product_prod(A1,B1),C),product_prod_case(A1,B1,C),C1),P2) = aa(product_prod(A1,B1),C,aa(fun(A1,fun(B1,C)),fun(product_prod(A1,B1),C),product_prod_case(A1,B1,C),C1),Q1) ) ) ).
tff(fact_57_split__twice,axiom,
! [B1: $tType,C: $tType,A1: $tType,E: $tType,D: $tType,P2: product_prod(D,E),G: fun(D,fun(E,product_prod(B1,C))),F: fun(B1,fun(C,A1))] : aa(product_prod(B1,C),A1,aa(fun(B1,fun(C,A1)),fun(product_prod(B1,C),A1),product_prod_case(B1,C,A1),F),aa(product_prod(D,E),product_prod(B1,C),aa(fun(D,fun(E,product_prod(B1,C))),fun(product_prod(D,E),product_prod(B1,C)),product_prod_case(D,E,product_prod(B1,C)),G),P2)) = aa(product_prod(D,E),A1,aa(fun(D,fun(E,A1)),fun(product_prod(D,E),A1),product_prod_case(D,E,A1),aa(fun(D,fun(E,product_prod(B1,C))),fun(D,fun(E,A1)),aa(fun(fun(E,product_prod(B1,C)),fun(E,A1)),fun(fun(D,fun(E,product_prod(B1,C))),fun(D,fun(E,A1))),combb(fun(E,product_prod(B1,C)),fun(E,A1),D),aa(fun(product_prod(B1,C),A1),fun(fun(E,product_prod(B1,C)),fun(E,A1)),combb(product_prod(B1,C),A1,E),aa(fun(B1,fun(C,A1)),fun(product_prod(B1,C),A1),product_prod_case(B1,C,A1),F))),G)),P2) ).
tff(fact_58_splitD,axiom,
! [A1: $tType,B1: $tType,B4: B1,A3: A1,F: fun(A1,fun(B1,bool))] :
( pp(aa(product_prod(A1,B1),bool,aa(fun(A1,fun(B1,bool)),fun(product_prod(A1,B1),bool),product_prod_case(A1,B1,bool),F),aa(B1,product_prod(A1,B1),aa(A1,fun(B1,product_prod(A1,B1)),product_Pair(A1,B1),A3),B4)))
=> pp(aa(B1,bool,aa(A1,fun(B1,bool),F,A3),B4)) ) ).
tff(fact_59_splitD_H,axiom,
! [B1: $tType,A1: $tType,C: $tType,C1: C,B4: B1,A3: A1,R2: fun(A1,fun(B1,fun(C,bool)))] :
( pp(aa(C,bool,aa(product_prod(A1,B1),fun(C,bool),aa(fun(A1,fun(B1,fun(C,bool))),fun(product_prod(A1,B1),fun(C,bool)),product_prod_case(A1,B1,fun(C,bool)),R2),aa(B1,product_prod(A1,B1),aa(A1,fun(B1,product_prod(A1,B1)),product_Pair(A1,B1),A3),B4)),C1))
=> pp(aa(C,bool,aa(B1,fun(C,bool),aa(A1,fun(B1,fun(C,bool)),R2,A3),B4),C1)) ) ).
tff(fact_60_prod_Osimps_I2_J,axiom,
! [B1: $tType,A1: $tType,C: $tType,B4: C,A3: B1,F1: fun(B1,fun(C,A1))] : aa(product_prod(B1,C),A1,aa(fun(B1,fun(C,A1)),fun(product_prod(B1,C),A1),product_prod_case(B1,C,A1),F1),aa(C,product_prod(B1,C),aa(B1,fun(C,product_prod(B1,C)),product_Pair(B1,C),A3),B4)) = aa(C,A1,aa(B1,fun(C,A1),F1,A3),B4) ).
tff(fact_61_split__eta,axiom,
! [C: $tType,B1: $tType,A1: $tType,F: fun(product_prod(A1,B1),C)] : aa(fun(A1,fun(B1,C)),fun(product_prod(A1,B1),C),product_prod_case(A1,B1,C),aa(fun(A1,fun(B1,product_prod(A1,B1))),fun(A1,fun(B1,C)),aa(fun(fun(B1,product_prod(A1,B1)),fun(B1,C)),fun(fun(A1,fun(B1,product_prod(A1,B1))),fun(A1,fun(B1,C))),combb(fun(B1,product_prod(A1,B1)),fun(B1,C),A1),aa(fun(product_prod(A1,B1),C),fun(fun(B1,product_prod(A1,B1)),fun(B1,C)),combb(product_prod(A1,B1),C,B1),F)),product_Pair(A1,B1))) = F ).
tff(fact_62_split__curry,axiom,
! [C: $tType,B1: $tType,A1: $tType,F: fun(product_prod(A1,B1),C)] : aa(fun(A1,fun(B1,C)),fun(product_prod(A1,B1),C),product_prod_case(A1,B1,C),product_curry(A1,B1,C,F)) = F ).
tff(fact_63_curry__split,axiom,
! [C: $tType,B1: $tType,A1: $tType,F: fun(A1,fun(B1,C))] : product_curry(A1,B1,C,aa(fun(A1,fun(B1,C)),fun(product_prod(A1,B1),C),product_prod_case(A1,B1,C),F)) = F ).
tff(fact_64_inv__imagep__def,axiom,
! [B1: $tType,A1: $tType,F: fun(A1,B1),R1: fun(B1,fun(B1,bool)),X: A1,Xa: A1] :
( inv_imagep(B1,A1,R1,F,X,Xa)
<=> pp(aa(B1,bool,aa(B1,fun(B1,bool),R1,aa(A1,B1,F,X)),aa(A1,B1,F,Xa))) ) ).
tff(fact_65_mem__splitI2,axiom,
! [C: $tType,B1: $tType,A1: $tType,C1: fun(A1,fun(B1,fun(C,bool))),Z: C,P2: product_prod(A1,B1)] :
( ! [A4: A1,B3: B1] :
( ( P2 = aa(B1,product_prod(A1,B1),aa(A1,fun(B1,product_prod(A1,B1)),product_Pair(A1,B1),A4),B3) )
=> member(C,Z,aa(B1,fun(C,bool),aa(A1,fun(B1,fun(C,bool)),C1,A4),B3)) )
=> member(C,Z,aa(product_prod(A1,B1),fun(C,bool),aa(fun(A1,fun(B1,fun(C,bool))),fun(product_prod(A1,B1),fun(C,bool)),product_prod_case(A1,B1,fun(C,bool)),C1),P2)) ) ).
tff(fact_66_splitI2_H,axiom,
! [A1: $tType,B1: $tType,C: $tType,X4: C,C1: fun(A1,fun(B1,fun(C,bool))),P2: product_prod(A1,B1)] :
( ! [A4: A1,B3: B1] :
( ( aa(B1,product_prod(A1,B1),aa(A1,fun(B1,product_prod(A1,B1)),product_Pair(A1,B1),A4),B3) = P2 )
=> pp(aa(C,bool,aa(B1,fun(C,bool),aa(A1,fun(B1,fun(C,bool)),C1,A4),B3),X4)) )
=> pp(aa(C,bool,aa(product_prod(A1,B1),fun(C,bool),aa(fun(A1,fun(B1,fun(C,bool))),fun(product_prod(A1,B1),fun(C,bool)),product_prod_case(A1,B1,fun(C,bool)),C1),P2),X4)) ) ).
tff(fact_67_mem__splitE,axiom,
! [B1: $tType,A1: $tType,C: $tType,P2: product_prod(B1,C),C1: fun(B1,fun(C,fun(A1,bool))),Z: A1] :
( member(A1,Z,aa(product_prod(B1,C),fun(A1,bool),aa(fun(B1,fun(C,fun(A1,bool))),fun(product_prod(B1,C),fun(A1,bool)),product_prod_case(B1,C,fun(A1,bool)),C1),P2))
=> ~ ! [X3: B1,Y1: C] :
( ( P2 = aa(C,product_prod(B1,C),aa(B1,fun(C,product_prod(B1,C)),product_Pair(B1,C),X3),Y1) )
=> ~ member(A1,Z,aa(C,fun(A1,bool),aa(B1,fun(C,fun(A1,bool)),C1,X3),Y1)) ) ) ).
tff(fact_68_splitE_H,axiom,
! [B1: $tType,A1: $tType,C: $tType,Z: C,P2: product_prod(A1,B1),C1: fun(A1,fun(B1,fun(C,bool)))] :
( pp(aa(C,bool,aa(product_prod(A1,B1),fun(C,bool),aa(fun(A1,fun(B1,fun(C,bool))),fun(product_prod(A1,B1),fun(C,bool)),product_prod_case(A1,B1,fun(C,bool)),C1),P2),Z))
=> ~ ! [X3: A1,Y1: B1] :
( ( P2 = aa(B1,product_prod(A1,B1),aa(A1,fun(B1,product_prod(A1,B1)),product_Pair(A1,B1),X3),Y1) )
=> ~ pp(aa(C,bool,aa(B1,fun(C,bool),aa(A1,fun(B1,fun(C,bool)),C1,X3),Y1),Z)) ) ) ).
tff(fact_69_split__part,axiom,
! [B1: $tType,A1: $tType,Q2: fun(A1,fun(B1,bool)),P1: bool,X: product_prod(A1,B1)] :
( pp(aa(product_prod(A1,B1),bool,aa(fun(A1,fun(B1,bool)),fun(product_prod(A1,B1),bool),product_prod_case(A1,B1,bool),aa(fun(A1,fun(B1,bool)),fun(A1,fun(B1,bool)),aa(fun(fun(B1,bool),fun(B1,bool)),fun(fun(A1,fun(B1,bool)),fun(A1,fun(B1,bool))),combb(fun(B1,bool),fun(B1,bool),A1),aa(fun(bool,bool),fun(fun(B1,bool),fun(B1,bool)),combb(bool,bool,B1),aa(bool,fun(bool,bool),fconj,P1))),Q2)),X))
<=> ( pp(P1)
& pp(aa(product_prod(A1,B1),bool,aa(fun(A1,fun(B1,bool)),fun(product_prod(A1,B1),bool),product_prod_case(A1,B1,bool),Q2),X)) ) ) ).
tff(fact_70_splitE,axiom,
! [A1: $tType,B1: $tType,P2: product_prod(A1,B1),C1: fun(A1,fun(B1,bool))] :
( pp(aa(product_prod(A1,B1),bool,aa(fun(A1,fun(B1,bool)),fun(product_prod(A1,B1),bool),product_prod_case(A1,B1,bool),C1),P2))
=> ~ ! [X3: A1,Y1: B1] :
( ( P2 = aa(B1,product_prod(A1,B1),aa(A1,fun(B1,product_prod(A1,B1)),product_Pair(A1,B1),X3),Y1) )
=> ~ pp(aa(B1,bool,aa(A1,fun(B1,bool),C1,X3),Y1)) ) ) ).
tff(fact_71_splitI2,axiom,
! [B1: $tType,A1: $tType,C1: fun(A1,fun(B1,bool)),P2: product_prod(A1,B1)] :
( ! [A4: A1,B3: B1] :
( ( P2 = aa(B1,product_prod(A1,B1),aa(A1,fun(B1,product_prod(A1,B1)),product_Pair(A1,B1),A4),B3) )
=> pp(aa(B1,bool,aa(A1,fun(B1,bool),C1,A4),B3)) )
=> pp(aa(product_prod(A1,B1),bool,aa(fun(A1,fun(B1,bool)),fun(product_prod(A1,B1),bool),product_prod_case(A1,B1,bool),C1),P2)) ) ).
tff(fact_72_cond__split__eta,axiom,
! [C: $tType,B1: $tType,A1: $tType,G: fun(product_prod(A1,B1),C),F: fun(A1,fun(B1,C))] :
( ! [X3: A1,Y1: B1] : aa(B1,C,aa(A1,fun(B1,C),F,X3),Y1) = aa(product_prod(A1,B1),C,G,aa(B1,product_prod(A1,B1),aa(A1,fun(B1,product_prod(A1,B1)),product_Pair(A1,B1),X3),Y1))
=> ( aa(fun(A1,fun(B1,C)),fun(product_prod(A1,B1),C),product_prod_case(A1,B1,C),F) = G ) ) ).
tff(fact_73_UNIV__I,axiom,
! [A1: $tType,X4: A1] : member(A1,X4,top_top(fun(A1,bool))) ).
tff(fact_74_iso__tuple__UNIV__I,axiom,
! [A1: $tType,X4: A1] : member(A1,X4,top_top(fun(A1,bool))) ).
tff(fact_75_ext,axiom,
! [B1: $tType,A1: $tType,G: fun(A1,B1),F: fun(A1,B1)] :
( ! [X3: A1] : aa(A1,B1,F,X3) = aa(A1,B1,G,X3)
=> ( F = G ) ) ).
tff(fact_76_mem__def,axiom,
! [A1: $tType,A2: fun(A1,bool),X4: A1] :
( member(A1,X4,A2)
<=> pp(aa(A1,bool,A2,X4)) ) ).
tff(fact_77_Collect__def,axiom,
! [A1: $tType,P1: fun(A1,bool)] : collect(A1,P1) = P1 ).
tff(fact_78_UNIV__def,axiom,
! [A1: $tType] : top_top(fun(A1,bool)) = collect(A1,combk(bool,A1,fTrue)) ).
tff(fact_79_splitE2,axiom,
! [B1: $tType,A1: $tType,C: $tType,Z: product_prod(B1,C),P1: fun(B1,fun(C,A1)),Q2: fun(A1,bool)] :
( pp(aa(A1,bool,Q2,aa(product_prod(B1,C),A1,aa(fun(B1,fun(C,A1)),fun(product_prod(B1,C),A1),product_prod_case(B1,C,A1),P1),Z)))
=> ~ ! [X3: B1,Y1: C] :
( ( Z = aa(C,product_prod(B1,C),aa(B1,fun(C,product_prod(B1,C)),product_Pair(B1,C),X3),Y1) )
=> ~ pp(aa(A1,bool,Q2,aa(C,A1,aa(B1,fun(C,A1),P1,X3),Y1))) ) ) ).
tff(fact_80_top__apply,axiom,
! [B1: $tType,A1: $tType] :
( top(A1)
=> ! [X4: B1] : aa(B1,A1,top_top(fun(B1,A1)),X4) = top_top(A1) ) ).
tff(fact_81_UNIV__eq__I,axiom,
! [A1: $tType,A2: fun(A1,bool)] :
( ! [X3: A1] : member(A1,X3,A2)
=> ( top_top(fun(A1,bool)) = A2 ) ) ).
tff(fact_82_UNIV__witness,axiom,
! [A1: $tType] :
? [X3: A1] : member(A1,X3,top_top(fun(A1,bool))) ).
tff(fact_83_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_84_The__split__eq,axiom,
! [A1: $tType,B1: $tType,Y2: B1,X4: A1] : the(product_prod(A1,B1),aa(fun(A1,fun(B1,bool)),fun(product_prod(A1,B1),bool),product_prod_case(A1,B1,bool),combc(A1,fun(B1,bool),fun(B1,bool),aa(fun(A1,fun(bool,bool)),fun(A1,fun(fun(B1,bool),fun(B1,bool))),aa(fun(fun(bool,bool),fun(fun(B1,bool),fun(B1,bool))),fun(fun(A1,fun(bool,bool)),fun(A1,fun(fun(B1,bool),fun(B1,bool)))),combb(fun(bool,bool),fun(fun(B1,bool),fun(B1,bool)),A1),combb(bool,bool,B1)),aa(fun(A1,bool),fun(A1,fun(bool,bool)),aa(fun(bool,fun(bool,bool)),fun(fun(A1,bool),fun(A1,fun(bool,bool))),combb(bool,fun(bool,bool),A1),fconj),aa(A1,fun(A1,bool),fequal(A1),X4))),aa(B1,fun(B1,bool),fequal(B1),Y2)))) = aa(B1,product_prod(A1,B1),aa(A1,fun(B1,product_prod(A1,B1)),product_Pair(A1,B1),X4),Y2) ).
tff(fact_85_split__paired__The,axiom,
! [B1: $tType,A1: $tType,P1: fun(product_prod(A1,B1),bool)] : the(product_prod(A1,B1),P1) = the(product_prod(A1,B1),aa(fun(A1,fun(B1,bool)),fun(product_prod(A1,B1),bool),product_prod_case(A1,B1,bool),aa(fun(A1,fun(B1,product_prod(A1,B1))),fun(A1,fun(B1,bool)),aa(fun(fun(B1,product_prod(A1,B1)),fun(B1,bool)),fun(fun(A1,fun(B1,product_prod(A1,B1))),fun(A1,fun(B1,bool))),combb(fun(B1,product_prod(A1,B1)),fun(B1,bool),A1),aa(fun(product_prod(A1,B1),bool),fun(fun(B1,product_prod(A1,B1)),fun(B1,bool)),combb(product_prod(A1,B1),bool,B1),P1)),product_Pair(A1,B1)))) ).
tff(fact_86_If__def,axiom,
! [A1: $tType,Y2: A1,X4: A1,P1: bool] :
( ( pp(P1)
=> ( X4 = the(A1,combs(A1,bool,bool,aa(fun(A1,bool),fun(A1,fun(bool,bool)),aa(fun(bool,fun(bool,bool)),fun(fun(A1,bool),fun(A1,fun(bool,bool))),combb(bool,fun(bool,bool),A1),fconj),aa(fun(A1,bool),fun(A1,bool),aa(fun(bool,bool),fun(fun(A1,bool),fun(A1,bool)),combb(bool,bool,A1),fimplies(P1)),combc(A1,A1,bool,fequal(A1),X4))),aa(fun(A1,bool),fun(A1,bool),aa(fun(bool,bool),fun(fun(A1,bool),fun(A1,bool)),combb(bool,bool,A1),fimplies(fNot(P1))),combc(A1,A1,bool,fequal(A1),Y2)))) ) )
& ( ~ pp(P1)
=> ( Y2 = the(A1,combs(A1,bool,bool,aa(fun(A1,bool),fun(A1,fun(bool,bool)),aa(fun(bool,fun(bool,bool)),fun(fun(A1,bool),fun(A1,fun(bool,bool))),combb(bool,fun(bool,bool),A1),fconj),aa(fun(A1,bool),fun(A1,bool),aa(fun(bool,bool),fun(fun(A1,bool),fun(A1,bool)),combb(bool,bool,A1),fimplies(P1)),combc(A1,A1,bool,fequal(A1),X4))),aa(fun(A1,bool),fun(A1,bool),aa(fun(bool,bool),fun(fun(A1,bool),fun(A1,bool)),combb(bool,bool,A1),fimplies(fNot(P1))),combc(A1,A1,bool,fequal(A1),Y2)))) ) ) ) ).
tff(fact_87_the__eq__trivial,axiom,
! [A1: $tType,A3: A1] : the(A1,combc(A1,A1,bool,fequal(A1),A3)) = A3 ).
tff(fact_88_the__sym__eq__trivial,axiom,
! [A1: $tType,X4: A1] : the(A1,aa(A1,fun(A1,bool),fequal(A1),X4)) = X4 ).
tff(fact_89_the__equality,axiom,
! [A1: $tType,A3: A1,P1: fun(A1,bool)] :
( pp(aa(A1,bool,P1,A3))
=> ( ! [X3: A1] :
( pp(aa(A1,bool,P1,X3))
=> ( X3 = A3 ) )
=> ( the(A1,P1) = A3 ) ) ) ).
tff(fact_90_theI,axiom,
! [A1: $tType,A3: A1,P1: fun(A1,bool)] :
( pp(aa(A1,bool,P1,A3))
=> ( ! [X3: A1] :
( pp(aa(A1,bool,P1,X3))
=> ( X3 = A3 ) )
=> pp(aa(A1,bool,P1,the(A1,P1))) ) ) ).
tff(fact_91_theI_H,axiom,
! [A1: $tType,P1: fun(A1,bool)] :
( ? [X: A1] :
( pp(aa(A1,bool,P1,X))
& ! [Y1: A1] :
( pp(aa(A1,bool,P1,Y1))
=> ( Y1 = X ) ) )
=> pp(aa(A1,bool,P1,the(A1,P1))) ) ).
tff(fact_92_the1__equality,axiom,
! [A1: $tType,A3: A1,P1: fun(A1,bool)] :
( ? [X: A1] :
( pp(aa(A1,bool,P1,X))
& ! [Y1: A1] :
( pp(aa(A1,bool,P1,Y1))
=> ( Y1 = X ) ) )
=> ( pp(aa(A1,bool,P1,A3))
=> ( the(A1,P1) = A3 ) ) ) ).
tff(fact_93_Pi__empty,axiom,
! [B1: $tType,A1: $tType,B2: fun(A1,fun(B1,bool))] : pi(A1,B1,bot_bot(fun(A1,bool)),B2) = top_top(fun(fun(A1,B1),bool)) ).
tff(fact_94_split__cong,axiom,
! [C: $tType,B1: $tType,A1: $tType,P2: product_prod(A1,B1),G: fun(A1,fun(B1,C)),F: fun(A1,fun(B1,C)),Q1: product_prod(A1,B1)] :
( ! [X3: A1,Y1: B1] :
( ( aa(B1,product_prod(A1,B1),aa(A1,fun(B1,product_prod(A1,B1)),product_Pair(A1,B1),X3),Y1) = Q1 )
=> ( aa(B1,C,aa(A1,fun(B1,C),F,X3),Y1) = aa(B1,C,aa(A1,fun(B1,C),G,X3),Y1) ) )
=> ( ( P2 = Q1 )
=> ( aa(product_prod(A1,B1),C,aa(fun(A1,fun(B1,C)),fun(product_prod(A1,B1),C),product_prod_case(A1,B1,C),F),P2) = aa(product_prod(A1,B1),C,aa(fun(A1,fun(B1,C)),fun(product_prod(A1,B1),C),product_prod_case(A1,B1,C),G),Q1) ) ) ) ).
tff(fact_95_emptyE,axiom,
! [A1: $tType,A3: A1] : ~ member(A1,A3,bot_bot(fun(A1,bool))) ).
tff(fact_96_Collect__empty__eq,axiom,
! [A1: $tType,P1: fun(A1,bool)] :
( ( collect(A1,P1) = bot_bot(fun(A1,bool)) )
<=> ! [X2: A1] : ~ pp(aa(A1,bool,P1,X2)) ) ).
tff(fact_97_empty__iff,axiom,
! [A1: $tType,C1: A1] : ~ member(A1,C1,bot_bot(fun(A1,bool))) ).
tff(fact_98_empty__Collect__eq,axiom,
! [A1: $tType,P1: fun(A1,bool)] :
( ( bot_bot(fun(A1,bool)) = collect(A1,P1) )
<=> ! [X2: A1] : ~ pp(aa(A1,bool,P1,X2)) ) ).
tff(fact_99_all__not__in__conv,axiom,
! [A1: $tType,A2: fun(A1,bool)] :
( ! [X2: A1] : ~ member(A1,X2,A2)
<=> ( A2 = bot_bot(fun(A1,bool)) ) ) ).
%----Arities (2)
tff(arity_fun___Orderings_Otop,axiom,
! [T_1: $tType,T_2: $tType] :
( top(T_2)
=> top(fun(T_1,T_2)) ) ).
tff(arity_HOL_Obool___Orderings_Otop,axiom,
top(bool) ).
%----Helper facts (19)
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(fNot(P))
| ~ pp(P) ) ).
tff(help_fNot_2_1_U,axiom,
! [P: bool] :
( pp(P)
| pp(fNot(P)) ) ).
tff(help_COMBB_1_1_U,axiom,
! [C: $tType,B1: $tType,A1: $tType,R: A1,Q: fun(A1,B1),P: fun(B1,C)] : aa(A1,C,aa(fun(A1,B1),fun(A1,C),aa(fun(B1,C),fun(fun(A1,B1),fun(A1,C)),combb(B1,C,A1),P),Q),R) = aa(B1,C,P,aa(A1,B1,Q,R)) ).
tff(help_COMBC_1_1_U,axiom,
! [A1: $tType,C: $tType,B1: $tType,R: A1,Q: B1,P: fun(A1,fun(B1,C))] : aa(A1,C,combc(A1,B1,C,P,Q),R) = aa(B1,C,aa(A1,fun(B1,C),P,R),Q) ).
tff(help_COMBI_1_1_U,axiom,
! [A1: $tType,P: A1] : aa(A1,A1,combi(A1),P) = P ).
tff(help_COMBK_1_1_U,axiom,
! [B1: $tType,A1: $tType,Q: B1,P: A1] : aa(B1,A1,combk(A1,B1,P),Q) = P ).
tff(help_COMBS_1_1_U,axiom,
! [C: $tType,B1: $tType,A1: $tType,R: A1,Q: fun(A1,B1),P: fun(A1,fun(B1,C))] : aa(A1,C,combs(A1,B1,C,P,Q),R) = aa(B1,C,aa(A1,fun(B1,C),P,R),aa(A1,B1,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_fequal_1_1_T,axiom,
! [A1: $tType,Y: A1,X1: A1] :
( ~ pp(aa(A1,bool,aa(A1,fun(A1,bool),fequal(A1),X1),Y))
| ( X1 = Y ) ) ).
tff(help_fequal_2_1_T,axiom,
! [A1: $tType,Y: A1,X1: A1] :
( ( X1 != Y )
| pp(aa(A1,bool,aa(A1,fun(A1,bool),fequal(A1),X1),Y)) ) ).
tff(help_fimplies_1_1_U,axiom,
! [Q: bool,P: bool] :
( pp(P)
| pp(aa(bool,bool,fimplies(P),Q)) ) ).
tff(help_fimplies_2_1_U,axiom,
! [P: bool,Q: bool] :
( ~ pp(Q)
| pp(aa(bool,bool,fimplies(P),Q)) ) ).
tff(help_fimplies_3_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(aa(bool,bool,fimplies(P),Q))
| ~ pp(P)
| pp(Q) ) ).
%----Conjectures (3)
tff(conj_0,hypothesis,
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)),f,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))) ).
tff(conj_1,hypothesis,
! [X: fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))] :
( member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),X,arrow_610318064e_Prof)
=> ! [A: arrow_411405190le_alt,B: arrow_411405190le_alt] :
( ( A != B )
=> ( 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),A),B),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),X,i))
=> 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),A),B),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),f,X)) ) ) ) ).
tff(conj_2,conjecture,
arrow_987702531ctator(f,i) ).
%------------------------------------------------------------------------------