TPTP Problem File: SCT201_5.p
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%------------------------------------------------------------------------------
% File : SCT201_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Social Choice Theory
% Problem : Arrow's Impossibility Theorem line 154
% 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_154 [Bla13]
% Status : Unknown
% Rating : 1.00 v6.4.0
% Syntax : Number of formulae : 156 ( 45 unt; 40 typ; 0 def)
% Number of atoms : 259 ( 93 equ)
% Maximal formula atoms : 12 ( 1 avg)
% Number of connectives : 202 ( 59 ~; 9 |; 15 &)
% ( 24 <=>; 95 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 7 avg)
% Maximal term depth : 10 ( 2 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 42 ( 23 >; 19 *; 0 +; 0 <<)
% Number of predicates : 8 ( 7 usr; 0 prp; 1-6 aty)
% Number of functors : 30 ( 30 usr; 6 con; 0-6 aty)
% Number of variables : 568 ( 520 !; 8 ?; 568 :)
% ( 40 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_UNK_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:18:47
%------------------------------------------------------------------------------
%----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 (35)
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(fun(B,C),fun(fun(A,B),fun(A,C))) ).
tff(sy_c_COMBC,type,
combc:
!>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,fun(B,C)) * 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_Ocompose,type,
compose:
!>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,bool) * fun(B,C) * fun(A,B) ) > fun(A,C) ) ).
tff(sy_c_HOL_OThe,type,
the:
!>[A: $tType] : ( fun(A,bool) > A ) ).
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_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) > fun(A,fun(B,C)) ) ).
tff(sy_c_Product__Type_Ointernal__split,type,
produc1605651328_split:
!>[A: $tType,B: $tType,C: $tType] : fun(fun(A,fun(B,C)),fun(product_prod(A,B),C)) ).
tff(sy_c_Product__Type_Oprod_Oprod__case,type,
product_prod_case:
!>[A: $tType,B: $tType,T: $tType] : fun(fun(A,fun(B,T)),fun(product_prod(A,B),T)) ).
tff(sy_c_Product__Type_Oprod_Oprod__rec,type,
product_prod_rec:
!>[A: $tType,B: $tType,T: $tType] : ( ( fun(A,fun(B,T)) * product_prod(A,B) ) > T ) ).
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_fTrue,type,
fTrue: bool ).
tff(sy_c_fconj,type,
fconj: fun(bool,fun(bool,bool)) ).
tff(sy_c_fequal,type,
fequal:
!>[A: $tType] : ( A > fun(A,bool) ) ).
tff(sy_c_member,type,
member:
!>[A: $tType] : ( ( A * fun(A,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)) ).
%----Relevant facts (100)
tff(fact_0_assms_I3_J,axiom,
arrow_1958449194le_IIA(f) ).
tff(fact_1_u,axiom,
arrow_2069624013nimity(f) ).
tff(fact_2_dictator__def,axiom,
! [I2: 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,I2)
<=> ! [X1: 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)),X1,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,X1) = aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),X1,I2) ) ) ) ).
tff(fact_3_assms_I1_J,axiom,
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(fact_4_const__Lin__Prof,axiom,
! [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(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) ) ).
tff(fact_5_PiE,axiom,
! [B: $tType,A: $tType,X2: A,B1: fun(A,fun(B,bool)),A1: fun(A,bool),F: fun(A,B)] :
( member(fun(A,B),F,pi(A,B,A1,B1))
=> ( ~ member(B,aa(A,B,F,X2),aa(A,fun(B,bool),B1,X2))
=> ~ member(A,X2,A1) ) ) ).
tff(fact_6_dictatorI,axiom,
! [I2: 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))] :
( 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)),Fa,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)))
=> ( ! [X3: 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)),X3,arrow_610318064e_Prof)
=> ! [A3: arrow_411405190le_alt,B2: arrow_411405190le_alt] :
( ( A3 != B2 )
=> ( 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),B2),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),X3,I2))
=> 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),B2),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,X3)) ) ) )
=> arrow_987702531ctator(Fa,I2) ) ) ).
tff(fact_7_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_8_funcset__id,axiom,
! [A: $tType,A1: fun(A,bool)] : member(fun(A,A),combi(A),pi(A,A,A1,combk(fun(A,bool),A,A1))) ).
tff(fact_9_funcset__mem,axiom,
! [A: $tType,B: $tType,X2: A,B1: fun(B,bool),A1: fun(A,bool),F: fun(A,B)] :
( member(fun(A,B),F,pi(A,B,A1,combk(fun(B,bool),A,B1)))
=> ( member(A,X2,A1)
=> member(B,aa(A,B,F,X2),B1) ) ) ).
tff(fact_10_above__Lin,axiom,
! [L: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Y1: arrow_411405190le_alt,X2: arrow_411405190le_alt] :
( ( X2 != Y1 )
=> ( 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,X2,Y1),arrow_1985332922le_Lin) ) ) ).
tff(fact_11_below__Lin,axiom,
! [L: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Y1: arrow_411405190le_alt,X2: arrow_411405190le_alt] :
( ( X2 != Y1 )
=> ( 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,X2,Y1),arrow_1985332922le_Lin) ) ) ).
tff(fact_12_mkbot__Lin,axiom,
! [X2: 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,X2),arrow_1985332922le_Lin) ) ).
tff(fact_13_mktop__Lin,axiom,
! [X2: 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,X2),arrow_1985332922le_Lin) ) ).
tff(fact_14_Pi__mem,axiom,
! [B: $tType,A: $tType,X2: A,B1: fun(A,fun(B,bool)),A1: fun(A,bool),F: fun(A,B)] :
( member(fun(A,B),F,pi(A,B,A1,B1))
=> ( member(A,X2,A1)
=> member(B,aa(A,B,F,X2),aa(A,fun(B,bool),B1,X2)) ) ) ).
tff(fact_15_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)
<=> ! [X1: 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)),X1,arrow_610318064e_Prof)
=> ! [A4: arrow_411405190le_alt,B4: arrow_411405190le_alt] :
( ! [I1: 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),A4),B4),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),X1,I1))
=> 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),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),Fa,X1)) ) ) ) ).
tff(fact_16_in__mktop,axiom,
! [Z: arrow_411405190le_alt,L: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Y1: arrow_411405190le_alt,X2: 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),X2),Y1),arrow_424895264_mktop(L,Z))
<=> ( ( X2 != Z )
& ( ( Y1 = Z )
=> ( X2 != Y1 ) )
& ( ( Y1 != 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),X2),Y1),L) ) ) ) ).
tff(fact_17_in__mkbot,axiom,
! [Z: arrow_411405190le_alt,L: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Y1: arrow_411405190le_alt,X2: 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),X2),Y1),arrow_276188178_mkbot(L,Z))
<=> ( ( Y1 != Z )
& ( ( X2 = Z )
=> ( X2 != Y1 ) )
& ( ( X2 != 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),X2),Y1),L) ) ) ) ).
tff(fact_18_in__below,axiom,
! [Y1: arrow_411405190le_alt,X2: arrow_411405190le_alt,L: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),B3: arrow_411405190le_alt,A2: arrow_411405190le_alt] :
( ( A2 != B3 )
=> ( 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),X2),Y1),arrow_319942042_below(L,A2,B3))
<=> ( ( X2 != Y1 )
& ( ( Y1 = A2 )
=> 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),X2),B3),L) )
& ( ( Y1 != A2 )
=> ( ( ( X2 = A2 )
=> ( ( Y1 = B3 )
| 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),B3),Y1),L) ) )
& ( ( X2 != A2 )
=> 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),X2),Y1),L) ) ) ) ) ) ) ) ).
tff(fact_19_in__above,axiom,
! [Y1: arrow_411405190le_alt,X2: arrow_411405190le_alt,L: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),B3: arrow_411405190le_alt,A2: arrow_411405190le_alt] :
( ( A2 != B3 )
=> ( 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),X2),Y1),arrow_1158827142_above(L,A2,B3))
<=> ( ( X2 != Y1 )
& ( ( X2 = B3 )
=> 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),A2),Y1),L) )
& ( ( X2 != B3 )
=> ( ( ( Y1 = B3 )
=> ( ( X2 = A2 )
| 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),X2),A2),L) ) )
& ( ( Y1 != B3 )
=> 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),X2),Y1),L) ) ) ) ) ) ) ) ).
tff(fact_20_Lin__irrefl,axiom,
! [B3: arrow_411405190le_alt,A2: 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),A2),B3),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),B3),A2),L) ) ) ).
tff(fact_21_notin__Lin__iff,axiom,
! [Y1: arrow_411405190le_alt,X2: 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)
=> ( ( X2 != Y1 )
=> ( ~ 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),X2),Y1),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),Y1),X2),L) ) ) ) ).
tff(fact_22_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)
<=> ! [X1: 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)),X1,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)
=> ! [A4: arrow_411405190le_alt,B4: arrow_411405190le_alt] :
( ! [I1: 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),A4),B4),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),X1,I1))
<=> 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),B4),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Xa1,I1)) )
=> ( 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),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),Fa,X1))
<=> 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),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),Fa,Xa1)) ) ) ) ) ) ).
tff(fact_23_Pi__I,axiom,
! [B: $tType,A: $tType,B1: fun(A,fun(B,bool)),F: fun(A,B),A1: fun(A,bool)] :
( ! [X3: A] :
( member(A,X3,A1)
=> member(B,aa(A,B,F,X3),aa(A,fun(B,bool),B1,X3)) )
=> member(fun(A,B),F,pi(A,B,A1,B1)) ) ).
tff(fact_24_complete__Lin,axiom,
! [B3: arrow_411405190le_alt,A2: arrow_411405190le_alt] :
( ( A2 != B3 )
=> ? [X3: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)] :
( member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),X3,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),A2),B3),X3) ) ) ).
tff(fact_25_Pair__eq,axiom,
! [A: $tType,B: $tType,B7: B,A7: A,B3: B,A2: A] :
( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B3) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A7),B7) )
<=> ( ( A2 = A7 )
& ( B3 = B7 ) ) ) ).
tff(fact_26_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))
<=> ! [A4: A,B4: 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),A4),B4))) ) ).
tff(fact_27_funcsetI,axiom,
! [A: $tType,B: $tType,B1: fun(B,bool),F: fun(A,B),A1: fun(A,bool)] :
( ! [X3: A] :
( member(A,X3,A1)
=> member(B,aa(A,B,F,X3),B1) )
=> member(fun(A,B),F,pi(A,B,A1,combk(fun(B,bool),A,B1))) ) ).
tff(fact_28_Pi__cong,axiom,
! [B: $tType,A: $tType,B1: fun(A,fun(B,bool)),G: fun(A,B),F: fun(A,B),A1: fun(A,bool)] :
( ! [W: A] :
( member(A,W,A1)
=> ( aa(A,B,F,W) = aa(A,B,G,W) ) )
=> ( member(fun(A,B),F,pi(A,B,A1,B1))
<=> member(fun(A,B),G,pi(A,B,A1,B1)) ) ) ).
tff(fact_29_Pi__I_H,axiom,
! [B: $tType,A: $tType,B1: fun(A,fun(B,bool)),F: fun(A,B),A1: fun(A,bool)] :
( ! [X3: A] :
( member(A,X3,A1)
=> member(B,aa(A,B,F,X3),aa(A,fun(B,bool),B1,X3)) )
=> member(fun(A,B),F,pi(A,B,A1,B1)) ) ).
tff(fact_30_Pair__inject,axiom,
! [A: $tType,B: $tType,B6: B,A6: A,B5: B,A5: A] :
( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B5) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B6) )
=> ~ ( ( A5 = A6 )
=> ( B5 != B6 ) ) ) ).
tff(fact_31_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))
<=> ? [A4: A,B4: 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),A4),B4))) ) ).
tff(fact_32_prod_Orecs,axiom,
! [B: $tType,A: $tType,C: $tType,B3: C,A2: B,F1: fun(B,fun(C,A))] : product_prod_rec(B,C,A,F1,aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A2),B3)) = aa(C,A,aa(B,fun(C,A),F1,A2),B3) ).
tff(fact_33_prod__induct6,axiom,
! [F2: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,X2: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2))))),P1: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2))))),bool)] :
( ! [A3: A,B2: B,C3: C,D2: D,E1: E,F3: F2] : pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2))))),bool,P1,aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2)))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2))))),A3),aa(product_prod(C,product_prod(D,product_prod(E,F2))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2)))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,F2))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,F2)))),B2),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(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2))))),bool,P1,X2)) ) ).
tff(fact_34_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F2: $tType,Y: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2)))))] :
~ ! [A3: A,B2: B,C3: C,D2: D,E1: E,F3: F2] : Y != aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2)))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2))))),A3),aa(product_prod(C,product_prod(D,product_prod(E,F2))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2)))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,F2))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,F2)))),B2),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_35_prod__induct5,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,X2: 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)] :
( ! [A3: A,B2: B,C3: C,D2: 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)))),A3),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))),B2),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(A,product_prod(B,product_prod(C,product_prod(D,E)))),bool,P1,X2)) ) ).
tff(fact_36_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))))] :
~ ! [A3: A,B2: B,C3: C,D2: 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)))),A3),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))),B2),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_37_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod(A,product_prod(B,product_prod(C,D)))] :
~ ! [A3: A,B2: B,C3: C,D2: 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))),A3),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)),B2),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),C3),D2))) ).
tff(fact_38_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,X2: product_prod(A,product_prod(B,product_prod(C,D))),P1: fun(product_prod(A,product_prod(B,product_prod(C,D))),bool)] :
( ! [A3: A,B2: B,C3: C,D2: 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))),A3),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)),B2),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),C3),D2)))))
=> pp(aa(product_prod(A,product_prod(B,product_prod(C,D))),bool,P1,X2)) ) ).
tff(fact_39_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod(A,product_prod(B,C))] :
~ ! [A3: A,B2: B,C3: 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)),A3),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B2),C3)) ).
tff(fact_40_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,X2: product_prod(A,product_prod(B,C)),P1: fun(product_prod(A,product_prod(B,C)),bool)] :
( ! [A3: A,B2: B,C3: 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)),A3),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B2),C3))))
=> pp(aa(product_prod(A,product_prod(B,C)),bool,P1,X2)) ) ).
tff(fact_41_pred__equals__eq2,axiom,
! [B: $tType,A: $tType,S: fun(product_prod(A,B),bool),R2: fun(product_prod(A,B),bool)] :
( ! [X1: A,Xa1: B] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X1),Xa1),R2)
<=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X1),Xa1),S) )
<=> ( R2 = S ) ) ).
tff(fact_42_PairE,axiom,
! [A: $tType,B: $tType,P3: product_prod(A,B)] :
~ ! [X3: A,Y2: B] : P3 != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y2) ).
tff(fact_43_prod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod(A,B)] :
~ ! [A3: A,B2: B] : Y != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2) ).
tff(fact_44_internal__split__conv,axiom,
! [B: $tType,A: $tType,C: $tType,B3: C,A2: B,C1: fun(B,fun(C,A))] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,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),A2),B3)) = aa(C,A,aa(B,fun(C,A),C1,A2),B3) ).
tff(fact_45_curry__conv,axiom,
! [A: $tType,B: $tType,C: $tType,B3: C,A2: B,F: fun(product_prod(B,C),A)] : aa(C,A,aa(B,fun(C,A),product_curry(B,C,A,F),A2),B3) = 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),A2),B3)) ).
tff(fact_46_curry__def,axiom,
! [C: $tType,A: $tType,B: $tType,X4: fun(product_prod(A,B),C),Xa: A,Xb: B] : aa(B,C,aa(A,fun(B,C),product_curry(A,B,C,X4),Xa),Xb) = aa(product_prod(A,B),C,X4,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Xb)) ).
tff(fact_47_curryI,axiom,
! [A: $tType,B: $tType,B3: B,A2: 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),A2),B3)))
=> pp(aa(B,bool,aa(A,fun(B,bool),product_curry(A,B,bool,F),A2),B3)) ) ).
tff(fact_48_curryE,axiom,
! [A: $tType,B: $tType,B3: B,A2: A,F: fun(product_prod(A,B),bool)] :
( pp(aa(B,bool,aa(A,fun(B,bool),product_curry(A,B,bool,F),A2),B3))
=> 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),A2),B3))) ) ).
tff(fact_49_curryD,axiom,
! [A: $tType,B: $tType,B3: B,A2: A,F: fun(product_prod(A,B),bool)] :
( pp(aa(B,bool,aa(A,fun(B,bool),product_curry(A,B,bool,F),A2),B3))
=> 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),A2),B3))) ) ).
tff(fact_50_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_51_in__inv__imagep,axiom,
! [A: $tType,B: $tType,Y1: B,X2: B,F: fun(B,A),R1: fun(A,fun(A,bool))] :
( inv_imagep(A,B,R1,F,X2,Y1)
<=> pp(aa(A,bool,aa(A,fun(A,bool),R1,aa(B,A,F,X2)),aa(B,A,F,Y1))) ) ).
tff(fact_52_internal__split__def,axiom,
! [C: $tType,B: $tType,A: $tType] : produc1605651328_split(A,B,C) = product_prod_case(A,B,C) ).
tff(fact_53_compose__assoc,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,D1: fun(D,bool),H: fun(C,D),C2: fun(C,bool),G: fun(B,C),B1: fun(B,bool),A1: fun(A,bool),F: fun(A,B)] :
( member(fun(A,B),F,pi(A,B,A1,combk(fun(B,bool),A,B1)))
=> ( member(fun(B,C),G,pi(B,C,B1,combk(fun(C,bool),B,C2)))
=> ( member(fun(C,D),H,pi(C,D,C2,combk(fun(D,bool),C,D1)))
=> ( compose(A,C,D,A1,H,compose(A,B,C,A1,G,F)) = compose(A,B,D,A1,compose(B,C,D,B1,H,G),F) ) ) ) ) ).
tff(fact_54_splitI,axiom,
! [A: $tType,B: $tType,B3: B,A2: A,F: fun(A,fun(B,bool))] :
( pp(aa(B,bool,aa(A,fun(B,bool),F,A2),B3))
=> pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_prod_case(A,B,bool),F),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B3))) ) ).
tff(fact_55_prod__caseI,axiom,
! [A: $tType,B: $tType,B3: B,A2: A,F1: fun(A,fun(B,bool))] :
( pp(aa(B,bool,aa(A,fun(B,bool),F1,A2),B3))
=> pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_prod_case(A,B,bool),F1),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B3))) ) ).
tff(fact_56_mem__splitI,axiom,
! [A: $tType,B: $tType,C: $tType,B3: C,A2: B,C1: fun(B,fun(C,fun(A,bool))),Z: A] :
( member(A,Z,aa(C,fun(A,bool),aa(B,fun(C,fun(A,bool)),C1,A2),B3))
=> member(A,Z,aa(product_prod(B,C),fun(A,bool),aa(fun(B,fun(C,fun(A,bool))),fun(product_prod(B,C),fun(A,bool)),product_prod_case(B,C,fun(A,bool)),C1),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A2),B3))) ) ).
tff(fact_57_top1I,axiom,
! [A: $tType,X2: A] : pp(aa(A,bool,top_top(fun(A,bool)),X2)) ).
tff(fact_58_split__conv,axiom,
! [B: $tType,A: $tType,C: $tType,B3: C,A2: B,F: fun(B,fun(C,A))] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_prod_case(B,C,A),F),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A2),B3)) = aa(C,A,aa(B,fun(C,A),F,A2),B3) ).
tff(fact_59_Pi__UNIV,axiom,
! [B: $tType,A: $tType,A1: fun(A,bool)] : pi(A,B,A1,combk(fun(B,bool),A,top_top(fun(B,bool)))) = top_top(fun(fun(A,B),bool)) ).
tff(fact_60_prod_Osimps_I2_J,axiom,
! [B: $tType,A: $tType,C: $tType,B3: C,A2: B,F1: fun(B,fun(C,A))] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_prod_case(B,C,A),F1),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A2),B3)) = aa(C,A,aa(B,fun(C,A),F1,A2),B3) ).
tff(fact_61_splitD,axiom,
! [A: $tType,B: $tType,B3: B,A2: A,F: fun(A,fun(B,bool))] :
( pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_prod_case(A,B,bool),F),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B3)))
=> pp(aa(B,bool,aa(A,fun(B,bool),F,A2),B3)) ) ).
tff(fact_62_compose__eq,axiom,
! [B: $tType,C: $tType,A: $tType,F: fun(A,C),G: fun(C,B),A1: fun(A,bool),X2: A] :
( member(A,X2,A1)
=> ( aa(A,B,compose(A,C,B,A1,G,F),X2) = aa(C,B,G,aa(A,C,F,X2)) ) ) ).
tff(fact_63_split__weak__cong,axiom,
! [C: $tType,B: $tType,A: $tType,C1: fun(A,fun(B,C)),Q2: product_prod(A,B),P2: product_prod(A,B)] :
( ( P2 = Q2 )
=> ( aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_prod_case(A,B,C),C1),P2) = aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_prod_case(A,B,C),C1),Q2) ) ) ).
tff(fact_64_split__twice,axiom,
! [B: $tType,C: $tType,A: $tType,E: $tType,D: $tType,P2: product_prod(D,E),G: fun(D,fun(E,product_prod(B,C))),F: fun(B,fun(C,A))] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_prod_case(B,C,A),F),aa(product_prod(D,E),product_prod(B,C),aa(fun(D,fun(E,product_prod(B,C))),fun(product_prod(D,E),product_prod(B,C)),product_prod_case(D,E,product_prod(B,C)),G),P2)) = aa(product_prod(D,E),A,aa(fun(D,fun(E,A)),fun(product_prod(D,E),A),product_prod_case(D,E,A),aa(fun(D,fun(E,product_prod(B,C))),fun(D,fun(E,A)),aa(fun(fun(E,product_prod(B,C)),fun(E,A)),fun(fun(D,fun(E,product_prod(B,C))),fun(D,fun(E,A))),combb(fun(E,product_prod(B,C)),fun(E,A),D),aa(fun(product_prod(B,C),A),fun(fun(E,product_prod(B,C)),fun(E,A)),combb(product_prod(B,C),A,E),aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_prod_case(B,C,A),F))),G)),P2) ).
tff(fact_65_splitD_H,axiom,
! [B: $tType,A: $tType,C: $tType,C1: C,B3: B,A2: A,R2: fun(A,fun(B,fun(C,bool)))] :
( pp(aa(C,bool,aa(product_prod(A,B),fun(C,bool),aa(fun(A,fun(B,fun(C,bool))),fun(product_prod(A,B),fun(C,bool)),product_prod_case(A,B,fun(C,bool)),R2),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B3)),C1))
=> pp(aa(C,bool,aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),R2,A2),B3),C1)) ) ).
tff(fact_66_split__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(product_prod(A,B),C)] : aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_prod_case(A,B,C),aa(fun(A,fun(B,product_prod(A,B))),fun(A,fun(B,C)),aa(fun(fun(B,product_prod(A,B)),fun(B,C)),fun(fun(A,fun(B,product_prod(A,B))),fun(A,fun(B,C))),combb(fun(B,product_prod(A,B)),fun(B,C),A),aa(fun(product_prod(A,B),C),fun(fun(B,product_prod(A,B)),fun(B,C)),combb(product_prod(A,B),C,B),F)),product_Pair(A,B))) = F ).
tff(fact_67_split__curry,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(product_prod(A,B),C)] : aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_prod_case(A,B,C),product_curry(A,B,C,F)) = F ).
tff(fact_68_curry__split,axiom,
! [C: $tType,B: $tType,A: $tType,F: fun(A,fun(B,C))] : product_curry(A,B,C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_prod_case(A,B,C),F)) = F ).
tff(fact_69_inv__imagep__def,axiom,
! [B: $tType,A: $tType,F: fun(A,B),R1: fun(B,fun(B,bool)),X4: A,Xa: A] :
( inv_imagep(B,A,R1,F,X4,Xa)
<=> pp(aa(B,bool,aa(B,fun(B,bool),R1,aa(A,B,F,X4)),aa(A,B,F,Xa))) ) ).
tff(fact_70_funcset__compose,axiom,
! [B: $tType,A: $tType,C: $tType,C2: fun(C,bool),G: fun(B,C),B1: fun(B,bool),A1: fun(A,bool),F: fun(A,B)] :
( member(fun(A,B),F,pi(A,B,A1,combk(fun(B,bool),A,B1)))
=> ( member(fun(B,C),G,pi(B,C,B1,combk(fun(C,bool),B,C2)))
=> member(fun(A,C),compose(A,B,C,A1,G,F),pi(A,C,A1,combk(fun(C,bool),A,C2))) ) ) ).
tff(fact_71_splitE,axiom,
! [A: $tType,B: $tType,P2: product_prod(A,B),C1: fun(A,fun(B,bool))] :
( pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_prod_case(A,B,bool),C1),P2))
=> ~ ! [X3: A,Y2: B] :
( ( P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y2) )
=> ~ pp(aa(B,bool,aa(A,fun(B,bool),C1,X3),Y2)) ) ) ).
tff(fact_72_splitI2,axiom,
! [B: $tType,A: $tType,C1: fun(A,fun(B,bool)),P2: product_prod(A,B)] :
( ! [A3: A,B2: B] :
( ( P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2) )
=> pp(aa(B,bool,aa(A,fun(B,bool),C1,A3),B2)) )
=> pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_prod_case(A,B,bool),C1),P2)) ) ).
tff(fact_73_mem__splitI2,axiom,
! [C: $tType,B: $tType,A: $tType,C1: fun(A,fun(B,fun(C,bool))),Z: C,P2: product_prod(A,B)] :
( ! [A3: A,B2: B] :
( ( P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2) )
=> member(C,Z,aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),C1,A3),B2)) )
=> member(C,Z,aa(product_prod(A,B),fun(C,bool),aa(fun(A,fun(B,fun(C,bool))),fun(product_prod(A,B),fun(C,bool)),product_prod_case(A,B,fun(C,bool)),C1),P2)) ) ).
tff(fact_74_splitI2_H,axiom,
! [A: $tType,B: $tType,C: $tType,X2: C,C1: fun(A,fun(B,fun(C,bool))),P2: product_prod(A,B)] :
( ! [A3: A,B2: B] :
( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2) = P2 )
=> pp(aa(C,bool,aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),C1,A3),B2),X2)) )
=> pp(aa(C,bool,aa(product_prod(A,B),fun(C,bool),aa(fun(A,fun(B,fun(C,bool))),fun(product_prod(A,B),fun(C,bool)),product_prod_case(A,B,fun(C,bool)),C1),P2),X2)) ) ).
tff(fact_75_ext,axiom,
! [B: $tType,A: $tType,G: fun(A,B),F: fun(A,B)] :
( ! [X3: A] : aa(A,B,F,X3) = aa(A,B,G,X3)
=> ( F = G ) ) ).
tff(fact_76_mem__def,axiom,
! [A: $tType,A1: fun(A,bool),X2: A] :
( member(A,X2,A1)
<=> pp(aa(A,bool,A1,X2)) ) ).
tff(fact_77_Collect__def,axiom,
! [A: $tType,P1: fun(A,bool)] : collect(A,P1) = P1 ).
tff(fact_78_split__part,axiom,
! [B: $tType,A: $tType,Q1: fun(A,fun(B,bool)),P1: bool,X4: product_prod(A,B)] :
( pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_prod_case(A,B,bool),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(fun(B,bool),fun(B,bool)),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),combb(fun(B,bool),fun(B,bool),A),aa(fun(bool,bool),fun(fun(B,bool),fun(B,bool)),combb(bool,bool,B),aa(bool,fun(bool,bool),fconj,P1))),Q1)),X4))
<=> ( pp(P1)
& pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_prod_case(A,B,bool),Q1),X4)) ) ) ).
tff(fact_79_splitE_H,axiom,
! [B: $tType,A: $tType,C: $tType,Z: C,P2: product_prod(A,B),C1: fun(A,fun(B,fun(C,bool)))] :
( pp(aa(C,bool,aa(product_prod(A,B),fun(C,bool),aa(fun(A,fun(B,fun(C,bool))),fun(product_prod(A,B),fun(C,bool)),product_prod_case(A,B,fun(C,bool)),C1),P2),Z))
=> ~ ! [X3: A,Y2: B] :
( ( P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y2) )
=> ~ pp(aa(C,bool,aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),C1,X3),Y2),Z)) ) ) ).
tff(fact_80_mem__splitE,axiom,
! [B: $tType,A: $tType,C: $tType,P2: product_prod(B,C),C1: fun(B,fun(C,fun(A,bool))),Z: A] :
( member(A,Z,aa(product_prod(B,C),fun(A,bool),aa(fun(B,fun(C,fun(A,bool))),fun(product_prod(B,C),fun(A,bool)),product_prod_case(B,C,fun(A,bool)),C1),P2))
=> ~ ! [X3: B,Y2: C] :
( ( P2 = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X3),Y2) )
=> ~ member(A,Z,aa(C,fun(A,bool),aa(B,fun(C,fun(A,bool)),C1,X3),Y2)) ) ) ).
tff(fact_81_cond__split__eta,axiom,
! [C: $tType,B: $tType,A: $tType,G: fun(product_prod(A,B),C),F: fun(A,fun(B,C))] :
( ! [X3: A,Y2: B] : aa(B,C,aa(A,fun(B,C),F,X3),Y2) = aa(product_prod(A,B),C,G,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y2))
=> ( aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_prod_case(A,B,C),F) = G ) ) ).
tff(fact_82_UNIV__I,axiom,
! [A: $tType,X2: A] : member(A,X2,top_top(fun(A,bool))) ).
tff(fact_83_UNIV__def,axiom,
! [A: $tType] : top_top(fun(A,bool)) = collect(A,combk(bool,A,fTrue)) ).
tff(fact_84_iso__tuple__UNIV__I,axiom,
! [A: $tType,X2: A] : member(A,X2,top_top(fun(A,bool))) ).
tff(fact_85_splitE2,axiom,
! [B: $tType,A: $tType,C: $tType,Z: product_prod(B,C),P1: fun(B,fun(C,A)),Q1: fun(A,bool)] :
( pp(aa(A,bool,Q1,aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_prod_case(B,C,A),P1),Z)))
=> ~ ! [X3: B,Y2: C] :
( ( Z = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X3),Y2) )
=> ~ pp(aa(A,bool,Q1,aa(C,A,aa(B,fun(C,A),P1,X3),Y2))) ) ) ).
tff(fact_86_top__apply,axiom,
! [B: $tType,A: $tType] :
( top(A)
=> ! [X2: B] : aa(B,A,top_top(fun(B,A)),X2) = top_top(A) ) ).
tff(fact_87_UNIV__eq__I,axiom,
! [A: $tType,A1: fun(A,bool)] :
( ! [X3: A] : member(A,X3,A1)
=> ( top_top(fun(A,bool)) = A1 ) ) ).
tff(fact_88_UNIV__witness,axiom,
! [A: $tType] :
? [X3: A] : member(A,X3,top_top(fun(A,bool))) ).
tff(fact_89_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_90_The__split__eq,axiom,
! [A: $tType,B: $tType,Y1: B,X2: A] : the(product_prod(A,B),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_prod_case(A,B,bool),combc(A,fun(B,bool),fun(B,bool),aa(fun(A,fun(bool,bool)),fun(A,fun(fun(B,bool),fun(B,bool))),aa(fun(fun(bool,bool),fun(fun(B,bool),fun(B,bool))),fun(fun(A,fun(bool,bool)),fun(A,fun(fun(B,bool),fun(B,bool)))),combb(fun(bool,bool),fun(fun(B,bool),fun(B,bool)),A),combb(bool,bool,B)),aa(fun(A,bool),fun(A,fun(bool,bool)),aa(fun(bool,fun(bool,bool)),fun(fun(A,bool),fun(A,fun(bool,bool))),combb(bool,fun(bool,bool),A),fconj),fequal(A,X2))),fequal(B,Y1)))) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y1) ).
tff(fact_91_Pi__empty,axiom,
! [B: $tType,A: $tType,B1: fun(A,fun(B,bool))] : pi(A,B,bot_bot(fun(A,bool)),B1) = top_top(fun(fun(A,B),bool)) ).
tff(fact_92_all__not__in__conv,axiom,
! [A: $tType,A1: fun(A,bool)] :
( ! [X1: A] : ~ member(A,X1,A1)
<=> ( A1 = bot_bot(fun(A,bool)) ) ) ).
tff(fact_93_empty__Collect__eq,axiom,
! [A: $tType,P1: fun(A,bool)] :
( ( bot_bot(fun(A,bool)) = collect(A,P1) )
<=> ! [X1: A] : ~ pp(aa(A,bool,P1,X1)) ) ).
tff(fact_94_empty__iff,axiom,
! [A: $tType,C1: A] : ~ member(A,C1,bot_bot(fun(A,bool))) ).
tff(fact_95_Collect__empty__eq,axiom,
! [A: $tType,P1: fun(A,bool)] :
( ( collect(A,P1) = bot_bot(fun(A,bool)) )
<=> ! [X1: A] : ~ pp(aa(A,bool,P1,X1)) ) ).
tff(fact_96_emptyE,axiom,
! [A: $tType,A2: A] : ~ member(A,A2,bot_bot(fun(A,bool))) ).
tff(fact_97_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_98_Pi__eq__empty,axiom,
! [A: $tType,B: $tType,B1: fun(A,fun(B,bool)),A1: fun(A,bool)] :
( ( pi(A,B,A1,B1) = bot_bot(fun(fun(A,B),bool)) )
<=> ? [X1: A] :
( member(A,X1,A1)
& ( aa(A,fun(B,bool),B1,X1) = bot_bot(fun(B,bool)) ) ) ) ).
tff(fact_99_UNIV__not__empty,axiom,
! [A: $tType] : top_top(fun(A,bool)) != bot_bot(fun(A,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 (13)
tff(help_pp_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_pp_2_1_U,axiom,
pp(fTrue) ).
tff(help_COMBB_1_1_U,axiom,
! [C: $tType,B: $tType,A: $tType,R: A,Q: fun(A,B),P: fun(B,C)] : aa(A,C,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),combb(B,C,A),P),Q),R) = aa(B,C,P,aa(A,B,Q,R)) ).
tff(help_COMBC_1_1_U,axiom,
! [A: $tType,C: $tType,B: $tType,R: A,Q: B,P: fun(A,fun(B,C))] : aa(A,C,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_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,
! [A: $tType,Y: A,X: A] :
( ~ pp(aa(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,fequal(A,X),Y)) ) ).
%----Conjectures (1)
tff(conj_0,conjecture,
? [I: arrow_159774573e_indi] : arrow_987702531ctator(f,I) ).
%------------------------------------------------------------------------------