TPTP Problem File: SCT181_5.p
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%------------------------------------------------------------------------------
% File : SCT181_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Social Choice Theory
% Problem : Arrow's Impossibility Theorem line 81
% 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_81 [Bla13]
% Status : Unknown
% Rating : 1.00 v6.4.0
% Syntax : Number of formulae : 118 ( 30 unt; 34 typ; 0 def)
% Number of atoms : 183 ( 70 equ)
% Maximal formula atoms : 6 ( 1 avg)
% Number of connectives : 141 ( 42 ~; 2 |; 7 &)
% ( 20 <=>; 70 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 8 avg)
% Maximal term depth : 10 ( 2 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 37 ( 22 >; 15 *; 0 +; 0 <<)
% Number of predicates : 9 ( 8 usr; 0 prp; 1-5 aty)
% Number of functors : 24 ( 24 usr; 7 con; 0-6 aty)
% Number of variables : 498 ( 450 !; 6 ?; 498 :)
% ( 42 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_UNK_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:14:14
%------------------------------------------------------------------------------
%----Should-be-implicit typings (4)
tff(ty_tc_Arrow__Order__Mirabelle__qkbtqzkjxu_Oalt,type,
arrow_411405190le_alt: $tType ).
tff(ty_tc_HOL_Obool,type,
bool: $tType ).
tff(ty_tc_fun,type,
fun: ( $tType * $tType ) > $tType ).
tff(ty_tc_prod,type,
product_prod: ( $tType * $tType ) > $tType ).
%----Explicit typings (30)
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_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_FunDef_Oin__rel,type,
in_rel:
!>[A: $tType,B: $tType] : ( ( fun(product_prod(A,B),bool) * A * B ) > $o ) ).
tff(sy_c_Nitpick_Orefl_H,type,
refl:
!>[A: $tType] : ( fun(product_prod(A,A),bool) > $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_Oapfst,type,
product_apfst:
!>[A: $tType,C1: $tType,B: $tType] : ( ( fun(A,C1) * product_prod(A,B) ) > product_prod(C1,B) ) ).
tff(sy_c_Product__Type_Oapsnd,type,
product_apsnd:
!>[B: $tType,C1: $tType,A: $tType] : ( ( fun(B,C1) * product_prod(A,B) ) > product_prod(A,C1) ) ).
tff(sy_c_Product__Type_Ocurry,type,
product_curry:
!>[A: $tType,B: $tType,C1: $tType] : ( fun(product_prod(A,B),C1) > fun(A,fun(B,C1)) ) ).
tff(sy_c_Product__Type_Ointernal__split,type,
produc1605651328_split:
!>[A: $tType,B: $tType,C1: $tType] : fun(fun(A,fun(B,C1)),fun(product_prod(A,B),C1)) ).
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_Product__Type_Oscomp,type,
product_scomp:
!>[A: $tType,B: $tType,C1: $tType,D: $tType] : ( ( fun(A,product_prod(B,C1)) * fun(B,fun(C1,D)) ) > fun(A,D) ) ).
tff(sy_c_Relation_OId__on,type,
id_on:
!>[A: $tType] : ( fun(A,bool) > fun(product_prod(A,A),bool) ) ).
tff(sy_c_Relation_Oantisym,type,
antisym:
!>[A: $tType] : ( fun(product_prod(A,A),bool) > $o ) ).
tff(sy_c_Relation_Oconverse,type,
converse:
!>[A: $tType,B: $tType] : ( fun(product_prod(A,B),bool) > fun(product_prod(B,A),bool) ) ).
tff(sy_c_Relation_Oinv__image,type,
inv_image:
!>[B: $tType,A: $tType] : ( ( fun(product_prod(B,B),bool) * fun(A,B) ) > fun(product_prod(A,A),bool) ) ).
tff(sy_c_Relation_Oirrefl,type,
irrefl:
!>[A: $tType] : ( fun(product_prod(A,A),bool) > $o ) ).
tff(sy_c_Relation_Osingle__valued,type,
single_valued:
!>[A: $tType,B: $tType] : ( fun(product_prod(A,B),bool) > $o ) ).
tff(sy_c_Relation_Ototal__on,type,
total_on:
!>[A: $tType] : ( ( fun(A,bool) * fun(product_prod(A,A),bool) ) > $o ) ).
tff(sy_c_Wellfounded_Olex__prod,type,
lex_prod:
!>[A: $tType,B: $tType] : ( ( fun(product_prod(A,A),bool) * fun(product_prod(B,B),bool) ) > fun(product_prod(product_prod(A,B),product_prod(A,B)),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_member,type,
member:
!>[A: $tType] : ( ( A * fun(A,bool) ) > $o ) ).
tff(sy_c_pp,type,
pp: bool > $o ).
tff(sy_v_L,type,
l: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool) ).
tff(sy_v_x,type,
x: arrow_411405190le_alt ).
tff(sy_v_y,type,
y: arrow_411405190le_alt ).
tff(sy_v_z,type,
z: arrow_411405190le_alt ).
%----Relevant facts (81)
tff(fact_0_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))
<=> ! [A5: 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),A5),B4))) ) ).
tff(fact_1_Pair__eq,axiom,
! [A: $tType,B: $tType,B3: B,A4: A,B1: B,A2: A] :
( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B1) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3) )
<=> ( ( A2 = A4 )
& ( B1 = B3 ) ) ) ).
tff(fact_2_in__mktop,axiom,
! [Za: arrow_411405190le_alt,La: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Ya: arrow_411405190le_alt,Xa: 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),Xa),Ya),arrow_424895264_mktop(La,Za))
<=> ( ( Xa != Za )
& ( ( Ya = Za )
=> ( Xa != Ya ) )
& ( ( Ya != Za )
=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),aa(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt),aa(arrow_411405190le_alt,fun(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt),Xa),Ya),La) ) ) ) ).
tff(fact_3_notin__Lin__iff,axiom,
! [Ya: arrow_411405190le_alt,Xa: arrow_411405190le_alt,La: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)] :
( member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),La,arrow_1985332922le_Lin)
=> ( ( Xa != Ya )
=> ( ~ member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),aa(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt),aa(arrow_411405190le_alt,fun(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt),Xa),Ya),La)
<=> 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),Ya),Xa),La) ) ) ) ).
tff(fact_4_Lin__irrefl,axiom,
! [B1: arrow_411405190le_alt,A2: arrow_411405190le_alt,La: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)] :
( member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),La,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),B1),La)
=> ~ 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),B1),A2),La) ) ) ).
tff(fact_5_Pair__inject,axiom,
! [A: $tType,B: $tType,B6: B,A7: A,B5: B,A6: A] :
( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B5) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A7),B6) )
=> ~ ( ( A6 = A7 )
=> ( B5 != B6 ) ) ) ).
tff(fact_6_in__rel__def,axiom,
! [B: $tType,A: $tType,Ya: B,Xa: A,R1: fun(product_prod(A,B),bool)] :
( in_rel(A,B,R1,Xa,Ya)
<=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Ya),R1) ) ).
tff(fact_7_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))
<=> ? [A5: 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),A5),B4))) ) ).
tff(fact_8_Nitpick_Orefl_H__def,axiom,
! [A: $tType,R: fun(product_prod(A,A),bool)] :
( refl(A,R)
<=> ! [X1: A] : member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X1),X1),R) ) ).
tff(fact_9_in__lex__prod,axiom,
! [A: $tType,B: $tType,S: fun(product_prod(B,B),bool),R: fun(product_prod(A,A),bool),B3: B,A4: A,B1: B,A2: A] :
( member(product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B)),aa(product_prod(A,B),fun(product_prod(A,B),product_prod(product_prod(A,B),product_prod(A,B))),product_Pair(product_prod(A,B),product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B1)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3)),lex_prod(A,B,R,S))
<=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),A4),R)
| ( ( A2 = A4 )
& member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),B1),B3),S) ) ) ) ).
tff(fact_10_prod_Orecs,axiom,
! [B: $tType,A: $tType,C1: $tType,B1: C1,A2: B,F1: fun(B,fun(C1,A))] : product_prod_rec(B,C1,A,F1,aa(C1,product_prod(B,C1),aa(B,fun(C1,product_prod(B,C1)),product_Pair(B,C1),A2),B1)) = aa(C1,A,aa(B,fun(C1,A),F1,A2),B1) ).
tff(fact_11_prod__induct6,axiom,
! [F2: $tType,E: $tType,D: $tType,C1: $tType,B: $tType,A: $tType,Xa: product_prod(A,product_prod(B,product_prod(C1,product_prod(D,product_prod(E,F2))))),P1: fun(product_prod(A,product_prod(B,product_prod(C1,product_prod(D,product_prod(E,F2))))),bool)] :
( ! [A3: A,B2: B,C2: C1,D1: D,E1: E,F3: F2] : pp(aa(product_prod(A,product_prod(B,product_prod(C1,product_prod(D,product_prod(E,F2))))),bool,P1,aa(product_prod(B,product_prod(C1,product_prod(D,product_prod(E,F2)))),product_prod(A,product_prod(B,product_prod(C1,product_prod(D,product_prod(E,F2))))),aa(A,fun(product_prod(B,product_prod(C1,product_prod(D,product_prod(E,F2)))),product_prod(A,product_prod(B,product_prod(C1,product_prod(D,product_prod(E,F2)))))),product_Pair(A,product_prod(B,product_prod(C1,product_prod(D,product_prod(E,F2))))),A3),aa(product_prod(C1,product_prod(D,product_prod(E,F2))),product_prod(B,product_prod(C1,product_prod(D,product_prod(E,F2)))),aa(B,fun(product_prod(C1,product_prod(D,product_prod(E,F2))),product_prod(B,product_prod(C1,product_prod(D,product_prod(E,F2))))),product_Pair(B,product_prod(C1,product_prod(D,product_prod(E,F2)))),B2),aa(product_prod(D,product_prod(E,F2)),product_prod(C1,product_prod(D,product_prod(E,F2))),aa(C1,fun(product_prod(D,product_prod(E,F2)),product_prod(C1,product_prod(D,product_prod(E,F2)))),product_Pair(C1,product_prod(D,product_prod(E,F2))),C2),aa(product_prod(E,F2),product_prod(D,product_prod(E,F2)),aa(D,fun(product_prod(E,F2),product_prod(D,product_prod(E,F2))),product_Pair(D,product_prod(E,F2)),D1),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(C1,product_prod(D,product_prod(E,F2))))),bool,P1,Xa)) ) ).
tff(fact_12_prod__cases6,axiom,
! [A: $tType,B: $tType,C1: $tType,D: $tType,E: $tType,F2: $tType,Y2: product_prod(A,product_prod(B,product_prod(C1,product_prod(D,product_prod(E,F2)))))] :
~ ! [A3: A,B2: B,C2: C1,D1: D,E1: E,F3: F2] : Y2 != aa(product_prod(B,product_prod(C1,product_prod(D,product_prod(E,F2)))),product_prod(A,product_prod(B,product_prod(C1,product_prod(D,product_prod(E,F2))))),aa(A,fun(product_prod(B,product_prod(C1,product_prod(D,product_prod(E,F2)))),product_prod(A,product_prod(B,product_prod(C1,product_prod(D,product_prod(E,F2)))))),product_Pair(A,product_prod(B,product_prod(C1,product_prod(D,product_prod(E,F2))))),A3),aa(product_prod(C1,product_prod(D,product_prod(E,F2))),product_prod(B,product_prod(C1,product_prod(D,product_prod(E,F2)))),aa(B,fun(product_prod(C1,product_prod(D,product_prod(E,F2))),product_prod(B,product_prod(C1,product_prod(D,product_prod(E,F2))))),product_Pair(B,product_prod(C1,product_prod(D,product_prod(E,F2)))),B2),aa(product_prod(D,product_prod(E,F2)),product_prod(C1,product_prod(D,product_prod(E,F2))),aa(C1,fun(product_prod(D,product_prod(E,F2)),product_prod(C1,product_prod(D,product_prod(E,F2)))),product_Pair(C1,product_prod(D,product_prod(E,F2))),C2),aa(product_prod(E,F2),product_prod(D,product_prod(E,F2)),aa(D,fun(product_prod(E,F2),product_prod(D,product_prod(E,F2))),product_Pair(D,product_prod(E,F2)),D1),aa(F2,product_prod(E,F2),aa(E,fun(F2,product_prod(E,F2)),product_Pair(E,F2),E1),F3))))) ).
tff(fact_13_prod__cases5,axiom,
! [A: $tType,B: $tType,C1: $tType,D: $tType,E: $tType,Y2: product_prod(A,product_prod(B,product_prod(C1,product_prod(D,E))))] :
~ ! [A3: A,B2: B,C2: C1,D1: D,E1: E] : Y2 != aa(product_prod(B,product_prod(C1,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C1,product_prod(D,E)))),aa(A,fun(product_prod(B,product_prod(C1,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C1,product_prod(D,E))))),product_Pair(A,product_prod(B,product_prod(C1,product_prod(D,E)))),A3),aa(product_prod(C1,product_prod(D,E)),product_prod(B,product_prod(C1,product_prod(D,E))),aa(B,fun(product_prod(C1,product_prod(D,E)),product_prod(B,product_prod(C1,product_prod(D,E)))),product_Pair(B,product_prod(C1,product_prod(D,E))),B2),aa(product_prod(D,E),product_prod(C1,product_prod(D,E)),aa(C1,fun(product_prod(D,E),product_prod(C1,product_prod(D,E))),product_Pair(C1,product_prod(D,E)),C2),aa(E,product_prod(D,E),aa(D,fun(E,product_prod(D,E)),product_Pair(D,E),D1),E1)))) ).
tff(fact_14_prod__induct5,axiom,
! [E: $tType,D: $tType,C1: $tType,B: $tType,A: $tType,Xa: product_prod(A,product_prod(B,product_prod(C1,product_prod(D,E)))),P1: fun(product_prod(A,product_prod(B,product_prod(C1,product_prod(D,E)))),bool)] :
( ! [A3: A,B2: B,C2: C1,D1: D,E1: E] : pp(aa(product_prod(A,product_prod(B,product_prod(C1,product_prod(D,E)))),bool,P1,aa(product_prod(B,product_prod(C1,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C1,product_prod(D,E)))),aa(A,fun(product_prod(B,product_prod(C1,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C1,product_prod(D,E))))),product_Pair(A,product_prod(B,product_prod(C1,product_prod(D,E)))),A3),aa(product_prod(C1,product_prod(D,E)),product_prod(B,product_prod(C1,product_prod(D,E))),aa(B,fun(product_prod(C1,product_prod(D,E)),product_prod(B,product_prod(C1,product_prod(D,E)))),product_Pair(B,product_prod(C1,product_prod(D,E))),B2),aa(product_prod(D,E),product_prod(C1,product_prod(D,E)),aa(C1,fun(product_prod(D,E),product_prod(C1,product_prod(D,E))),product_Pair(C1,product_prod(D,E)),C2),aa(E,product_prod(D,E),aa(D,fun(E,product_prod(D,E)),product_Pair(D,E),D1),E1))))))
=> pp(aa(product_prod(A,product_prod(B,product_prod(C1,product_prod(D,E)))),bool,P1,Xa)) ) ).
tff(fact_15_prod__cases4,axiom,
! [A: $tType,B: $tType,C1: $tType,D: $tType,Y2: product_prod(A,product_prod(B,product_prod(C1,D)))] :
~ ! [A3: A,B2: B,C2: C1,D1: D] : Y2 != aa(product_prod(B,product_prod(C1,D)),product_prod(A,product_prod(B,product_prod(C1,D))),aa(A,fun(product_prod(B,product_prod(C1,D)),product_prod(A,product_prod(B,product_prod(C1,D)))),product_Pair(A,product_prod(B,product_prod(C1,D))),A3),aa(product_prod(C1,D),product_prod(B,product_prod(C1,D)),aa(B,fun(product_prod(C1,D),product_prod(B,product_prod(C1,D))),product_Pair(B,product_prod(C1,D)),B2),aa(D,product_prod(C1,D),aa(C1,fun(D,product_prod(C1,D)),product_Pair(C1,D),C2),D1))) ).
tff(fact_16_prod__induct4,axiom,
! [D: $tType,C1: $tType,B: $tType,A: $tType,Xa: product_prod(A,product_prod(B,product_prod(C1,D))),P1: fun(product_prod(A,product_prod(B,product_prod(C1,D))),bool)] :
( ! [A3: A,B2: B,C2: C1,D1: D] : pp(aa(product_prod(A,product_prod(B,product_prod(C1,D))),bool,P1,aa(product_prod(B,product_prod(C1,D)),product_prod(A,product_prod(B,product_prod(C1,D))),aa(A,fun(product_prod(B,product_prod(C1,D)),product_prod(A,product_prod(B,product_prod(C1,D)))),product_Pair(A,product_prod(B,product_prod(C1,D))),A3),aa(product_prod(C1,D),product_prod(B,product_prod(C1,D)),aa(B,fun(product_prod(C1,D),product_prod(B,product_prod(C1,D))),product_Pair(B,product_prod(C1,D)),B2),aa(D,product_prod(C1,D),aa(C1,fun(D,product_prod(C1,D)),product_Pair(C1,D),C2),D1)))))
=> pp(aa(product_prod(A,product_prod(B,product_prod(C1,D))),bool,P1,Xa)) ) ).
tff(fact_17_prod__cases3,axiom,
! [A: $tType,B: $tType,C1: $tType,Y2: product_prod(A,product_prod(B,C1))] :
~ ! [A3: A,B2: B,C2: C1] : Y2 != aa(product_prod(B,C1),product_prod(A,product_prod(B,C1)),aa(A,fun(product_prod(B,C1),product_prod(A,product_prod(B,C1))),product_Pair(A,product_prod(B,C1)),A3),aa(C1,product_prod(B,C1),aa(B,fun(C1,product_prod(B,C1)),product_Pair(B,C1),B2),C2)) ).
tff(fact_18_prod__induct3,axiom,
! [C1: $tType,B: $tType,A: $tType,Xa: product_prod(A,product_prod(B,C1)),P1: fun(product_prod(A,product_prod(B,C1)),bool)] :
( ! [A3: A,B2: B,C2: C1] : pp(aa(product_prod(A,product_prod(B,C1)),bool,P1,aa(product_prod(B,C1),product_prod(A,product_prod(B,C1)),aa(A,fun(product_prod(B,C1),product_prod(A,product_prod(B,C1))),product_Pair(A,product_prod(B,C1)),A3),aa(C1,product_prod(B,C1),aa(B,fun(C1,product_prod(B,C1)),product_Pair(B,C1),B2),C2))))
=> pp(aa(product_prod(A,product_prod(B,C1)),bool,P1,Xa)) ) ).
tff(fact_19_linear__alt,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) ).
tff(fact_20_prod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y2: product_prod(A,B)] :
~ ! [A3: A,B2: B] : Y2 != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2) ).
tff(fact_21_PairE,axiom,
! [A: $tType,B: $tType,P2: product_prod(A,B)] :
~ ! [X: A,Y: B] : P2 != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y) ).
tff(fact_22_internal__split__conv,axiom,
! [B: $tType,A: $tType,C1: $tType,B1: C1,A2: B,C: fun(B,fun(C1,A))] : aa(product_prod(B,C1),A,aa(fun(B,fun(C1,A)),fun(product_prod(B,C1),A),produc1605651328_split(B,C1,A),C),aa(C1,product_prod(B,C1),aa(B,fun(C1,product_prod(B,C1)),product_Pair(B,C1),A2),B1)) = aa(C1,A,aa(B,fun(C1,A),C,A2),B1) ).
tff(fact_23_curry__conv,axiom,
! [A: $tType,B: $tType,C1: $tType,B1: C1,A2: B,F: fun(product_prod(B,C1),A)] : aa(C1,A,aa(B,fun(C1,A),product_curry(B,C1,A,F),A2),B1) = aa(product_prod(B,C1),A,F,aa(C1,product_prod(B,C1),aa(B,fun(C1,product_prod(B,C1)),product_Pair(B,C1),A2),B1)) ).
tff(fact_24_curryI,axiom,
! [A: $tType,B: $tType,B1: 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),B1)))
=> pp(aa(B,bool,aa(A,fun(B,bool),product_curry(A,B,bool,F),A2),B1)) ) ).
tff(fact_25_curryD,axiom,
! [A: $tType,B: $tType,B1: 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),B1))
=> 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),B1))) ) ).
tff(fact_26_curryE,axiom,
! [A: $tType,B: $tType,B1: 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),B1))
=> 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),B1))) ) ).
tff(fact_27_apsnd__conv,axiom,
! [A: $tType,B: $tType,C1: $tType,Ya: C1,Xa: A,F: fun(C1,B)] : product_apsnd(C1,B,A,F,aa(C1,product_prod(A,C1),aa(A,fun(C1,product_prod(A,C1)),product_Pair(A,C1),Xa),Ya)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),aa(C1,B,F,Ya)) ).
tff(fact_28_internal__split__def,axiom,
! [C1: $tType,B: $tType,A: $tType] : produc1605651328_split(A,B,C1) = product_prod_case(A,B,C1) ).
tff(fact_29_apfst__conv,axiom,
! [C1: $tType,A: $tType,B: $tType,Ya: B,Xa: C1,F: fun(C1,A)] : product_apfst(C1,A,B,F,aa(B,product_prod(C1,B),aa(C1,fun(B,product_prod(C1,B)),product_Pair(C1,B),Xa),Ya)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C1,A,F,Xa)),Ya) ).
tff(fact_30_surj__pair,axiom,
! [A: $tType,B: $tType,P2: product_prod(A,B)] :
? [X: A,Y: B] : P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y) ).
tff(fact_31_apsnd__apfst__commute,axiom,
! [A: $tType,B: $tType,C1: $tType,D: $tType,P: product_prod(D,C1),G: fun(D,A),F: fun(C1,B)] : product_apsnd(C1,B,A,F,product_apfst(D,A,C1,G,P)) = product_apfst(D,A,B,G,product_apsnd(C1,B,D,F,P)) ).
tff(fact_32_converse__in__Lin,axiom,
! [La: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)] :
( member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),converse(arrow_411405190le_alt,arrow_411405190le_alt,La),arrow_1985332922le_Lin)
<=> member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),La,arrow_1985332922le_Lin) ) ).
tff(fact_33_Pair__scomp,axiom,
! [A: $tType,B: $tType,C1: $tType,F: fun(C1,fun(A,B)),Xa: C1] : product_scomp(A,C1,A,B,aa(C1,fun(A,product_prod(C1,A)),product_Pair(C1,A),Xa),F) = aa(C1,fun(A,B),F,Xa) ).
tff(fact_34_splitI,axiom,
! [A: $tType,B: $tType,B1: B,A2: A,F: fun(A,fun(B,bool))] :
( pp(aa(B,bool,aa(A,fun(B,bool),F,A2),B1))
=> 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),B1))) ) ).
tff(fact_35_prod__caseI,axiom,
! [A: $tType,B: $tType,B1: B,A2: A,F1: fun(A,fun(B,bool))] :
( pp(aa(B,bool,aa(A,fun(B,bool),F1,A2),B1))
=> 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),B1))) ) ).
tff(fact_36_mem__splitI,axiom,
! [A: $tType,B: $tType,C1: $tType,B1: C1,A2: B,C: fun(B,fun(C1,fun(A,bool))),Za: A] :
( member(A,Za,aa(C1,fun(A,bool),aa(B,fun(C1,fun(A,bool)),C,A2),B1))
=> member(A,Za,aa(product_prod(B,C1),fun(A,bool),aa(fun(B,fun(C1,fun(A,bool))),fun(product_prod(B,C1),fun(A,bool)),product_prod_case(B,C1,fun(A,bool)),C),aa(C1,product_prod(B,C1),aa(B,fun(C1,product_prod(B,C1)),product_Pair(B,C1),A2),B1))) ) ).
tff(fact_37_split__conv,axiom,
! [B: $tType,A: $tType,C1: $tType,B1: C1,A2: B,F: fun(B,fun(C1,A))] : aa(product_prod(B,C1),A,aa(fun(B,fun(C1,A)),fun(product_prod(B,C1),A),product_prod_case(B,C1,A),F),aa(C1,product_prod(B,C1),aa(B,fun(C1,product_prod(B,C1)),product_Pair(B,C1),A2),B1)) = aa(C1,A,aa(B,fun(C1,A),F,A2),B1) ).
tff(fact_38_scomp__apply,axiom,
! [A: $tType,C1: $tType,D: $tType,B: $tType,Xa: B,G: fun(C1,fun(D,A)),F: fun(B,product_prod(C1,D))] : aa(B,A,product_scomp(B,C1,D,A,F,G),Xa) = aa(product_prod(C1,D),A,aa(fun(C1,fun(D,A)),fun(product_prod(C1,D),A),product_prod_case(C1,D,A),G),aa(B,product_prod(C1,D),F,Xa)) ).
tff(fact_39_scomp__def,axiom,
! [B: $tType,C1: $tType,D: $tType,A: $tType,G: fun(C1,fun(D,B)),F: fun(A,product_prod(C1,D)),X2: A] : aa(A,B,product_scomp(A,C1,D,B,F,G),X2) = aa(product_prod(C1,D),B,aa(fun(C1,fun(D,B)),fun(product_prod(C1,D),B),product_prod_case(C1,D,B),G),aa(A,product_prod(C1,D),F,X2)) ).
tff(fact_40_split__weak__cong,axiom,
! [C1: $tType,B: $tType,A: $tType,C: fun(A,fun(B,C1)),Q: product_prod(A,B),P: product_prod(A,B)] :
( ( P = Q )
=> ( aa(product_prod(A,B),C1,aa(fun(A,fun(B,C1)),fun(product_prod(A,B),C1),product_prod_case(A,B,C1),C),P) = aa(product_prod(A,B),C1,aa(fun(A,fun(B,C1)),fun(product_prod(A,B),C1),product_prod_case(A,B,C1),C),Q) ) ) ).
tff(fact_41_splitD,axiom,
! [A: $tType,B: $tType,B1: 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),B1)))
=> pp(aa(B,bool,aa(A,fun(B,bool),F,A2),B1)) ) ).
tff(fact_42_splitD_H,axiom,
! [B: $tType,A: $tType,C1: $tType,C: C1,B1: B,A2: A,R1: fun(A,fun(B,fun(C1,bool)))] :
( pp(aa(C1,bool,aa(product_prod(A,B),fun(C1,bool),aa(fun(A,fun(B,fun(C1,bool))),fun(product_prod(A,B),fun(C1,bool)),product_prod_case(A,B,fun(C1,bool)),R1),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B1)),C))
=> pp(aa(C1,bool,aa(B,fun(C1,bool),aa(A,fun(B,fun(C1,bool)),R1,A2),B1),C)) ) ).
tff(fact_43_prod_Osimps_I2_J,axiom,
! [B: $tType,A: $tType,C1: $tType,B1: C1,A2: B,F1: fun(B,fun(C1,A))] : aa(product_prod(B,C1),A,aa(fun(B,fun(C1,A)),fun(product_prod(B,C1),A),product_prod_case(B,C1,A),F1),aa(C1,product_prod(B,C1),aa(B,fun(C1,product_prod(B,C1)),product_Pair(B,C1),A2),B1)) = aa(C1,A,aa(B,fun(C1,A),F1,A2),B1) ).
tff(fact_44_scomp__Pair,axiom,
! [C1: $tType,B: $tType,A: $tType,Xa: fun(A,product_prod(B,C1))] : product_scomp(A,B,C1,product_prod(B,C1),Xa,product_Pair(B,C1)) = Xa ).
tff(fact_45_curry__split,axiom,
! [C1: $tType,B: $tType,A: $tType,F: fun(A,fun(B,C1))] : product_curry(A,B,C1,aa(fun(A,fun(B,C1)),fun(product_prod(A,B),C1),product_prod_case(A,B,C1),F)) = F ).
tff(fact_46_split__curry,axiom,
! [C1: $tType,B: $tType,A: $tType,F: fun(product_prod(A,B),C1)] : aa(fun(A,fun(B,C1)),fun(product_prod(A,B),C1),product_prod_case(A,B,C1),product_curry(A,B,C1,F)) = F ).
tff(fact_47_converse__iff,axiom,
! [A: $tType,B: $tType,R: fun(product_prod(B,A),bool),B1: B,A2: A] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B1),converse(B,A,R))
<=> member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),B1),A2),R) ) ).
tff(fact_48_splitI2,axiom,
! [B: $tType,A: $tType,C: fun(A,fun(B,bool)),P: product_prod(A,B)] :
( ! [A3: A,B2: B] :
( ( P = 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),C,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),C),P)) ) ).
tff(fact_49_splitE,axiom,
! [A: $tType,B: $tType,P: product_prod(A,B),C: 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),C),P))
=> ~ ! [X: A,Y: B] :
( ( P = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y) )
=> ~ pp(aa(B,bool,aa(A,fun(B,bool),C,X),Y)) ) ) ).
tff(fact_50_mem__splitI2,axiom,
! [C1: $tType,B: $tType,A: $tType,C: fun(A,fun(B,fun(C1,bool))),Za: C1,P: product_prod(A,B)] :
( ! [A3: A,B2: B] :
( ( P = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2) )
=> member(C1,Za,aa(B,fun(C1,bool),aa(A,fun(B,fun(C1,bool)),C,A3),B2)) )
=> member(C1,Za,aa(product_prod(A,B),fun(C1,bool),aa(fun(A,fun(B,fun(C1,bool))),fun(product_prod(A,B),fun(C1,bool)),product_prod_case(A,B,fun(C1,bool)),C),P)) ) ).
tff(fact_51_splitI2_H,axiom,
! [A: $tType,B: $tType,C1: $tType,Xa: C1,C: fun(A,fun(B,fun(C1,bool))),P: 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) = P )
=> pp(aa(C1,bool,aa(B,fun(C1,bool),aa(A,fun(B,fun(C1,bool)),C,A3),B2),Xa)) )
=> pp(aa(C1,bool,aa(product_prod(A,B),fun(C1,bool),aa(fun(A,fun(B,fun(C1,bool))),fun(product_prod(A,B),fun(C1,bool)),product_prod_case(A,B,fun(C1,bool)),C),P),Xa)) ) ).
tff(fact_52_mem__splitE,axiom,
! [B: $tType,A: $tType,C1: $tType,P: product_prod(B,C1),C: fun(B,fun(C1,fun(A,bool))),Za: A] :
( member(A,Za,aa(product_prod(B,C1),fun(A,bool),aa(fun(B,fun(C1,fun(A,bool))),fun(product_prod(B,C1),fun(A,bool)),product_prod_case(B,C1,fun(A,bool)),C),P))
=> ~ ! [X: B,Y: C1] :
( ( P = aa(C1,product_prod(B,C1),aa(B,fun(C1,product_prod(B,C1)),product_Pair(B,C1),X),Y) )
=> ~ member(A,Za,aa(C1,fun(A,bool),aa(B,fun(C1,fun(A,bool)),C,X),Y)) ) ) ).
tff(fact_53_converse__converse,axiom,
! [B: $tType,A: $tType,R: fun(product_prod(A,B),bool)] : converse(B,A,converse(A,B,R)) = R ).
tff(fact_54_converseI,axiom,
! [B: $tType,A: $tType,R: fun(product_prod(A,B),bool),B1: B,A2: A] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B1),R)
=> member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),B1),A2),converse(A,B,R)) ) ).
tff(fact_55_converseD,axiom,
! [A: $tType,B: $tType,R: fun(product_prod(B,A),bool),B1: B,A2: A] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B1),converse(B,A,R))
=> member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),B1),A2),R) ) ).
tff(fact_56_splitE_H,axiom,
! [B: $tType,A: $tType,C1: $tType,Za: C1,P: product_prod(A,B),C: fun(A,fun(B,fun(C1,bool)))] :
( pp(aa(C1,bool,aa(product_prod(A,B),fun(C1,bool),aa(fun(A,fun(B,fun(C1,bool))),fun(product_prod(A,B),fun(C1,bool)),product_prod_case(A,B,fun(C1,bool)),C),P),Za))
=> ~ ! [X: A,Y: B] :
( ( P = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y) )
=> ~ pp(aa(C1,bool,aa(B,fun(C1,bool),aa(A,fun(B,fun(C1,bool)),C,X),Y),Za)) ) ) ).
tff(fact_57_converseE,axiom,
! [A: $tType,B: $tType,R: fun(product_prod(B,A),bool),Yx: product_prod(A,B)] :
( member(product_prod(A,B),Yx,converse(B,A,R))
=> ~ ! [X: B,Y: A] :
( ( Yx = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y),X) )
=> ~ member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X),Y),R) ) ) ).
tff(fact_58_splitE2,axiom,
! [B: $tType,A: $tType,C1: $tType,Za: product_prod(B,C1),P1: fun(B,fun(C1,A)),Q1: fun(A,bool)] :
( pp(aa(A,bool,Q1,aa(product_prod(B,C1),A,aa(fun(B,fun(C1,A)),fun(product_prod(B,C1),A),product_prod_case(B,C1,A),P1),Za)))
=> ~ ! [X: B,Y: C1] :
( ( Za = aa(C1,product_prod(B,C1),aa(B,fun(C1,product_prod(B,C1)),product_Pair(B,C1),X),Y) )
=> ~ pp(aa(A,bool,Q1,aa(C1,A,aa(B,fun(C1,A),P1,X),Y))) ) ) ).
tff(fact_59_split__cong,axiom,
! [C1: $tType,B: $tType,A: $tType,P: product_prod(A,B),G: fun(A,fun(B,C1)),F: fun(A,fun(B,C1)),Q: product_prod(A,B)] :
( ! [X: A,Y: B] :
( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y) = Q )
=> ( aa(B,C1,aa(A,fun(B,C1),F,X),Y) = aa(B,C1,aa(A,fun(B,C1),G,X),Y) ) )
=> ( ( P = Q )
=> ( aa(product_prod(A,B),C1,aa(fun(A,fun(B,C1)),fun(product_prod(A,B),C1),product_prod_case(A,B,C1),F),P) = aa(product_prod(A,B),C1,aa(fun(A,fun(B,C1)),fun(product_prod(A,B),C1),product_prod_case(A,B,C1),G),Q) ) ) ) ).
tff(fact_60_apfst__convE,axiom,
! [C1: $tType,A: $tType,B: $tType,P: product_prod(C1,B),F: fun(C1,A),Q: product_prod(A,B)] :
( ( Q = product_apfst(C1,A,B,F,P) )
=> ~ ! [X: C1,Y: B] :
( ( P = aa(B,product_prod(C1,B),aa(C1,fun(B,product_prod(C1,B)),product_Pair(C1,B),X),Y) )
=> ( Q != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C1,A,F,X)),Y) ) ) ) ).
tff(fact_61_total__on__converse,axiom,
! [A: $tType,R: fun(product_prod(A,A),bool),A1: fun(A,bool)] :
( total_on(A,A1,converse(A,A,R))
<=> total_on(A,A1,R) ) ).
tff(fact_62_converse__inv__image,axiom,
! [B: $tType,A: $tType,F: fun(A,B),R1: fun(product_prod(B,B),bool)] : converse(A,A,inv_image(B,A,R1,F)) = inv_image(B,A,converse(B,B,R1),F) ).
tff(fact_63_in__inv__image,axiom,
! [A: $tType,B: $tType,F: fun(A,B),R: fun(product_prod(B,B),bool),Ya: A,Xa: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Ya),inv_image(B,A,R,F))
<=> member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),aa(A,B,F,Xa)),aa(A,B,F,Ya)),R) ) ).
tff(fact_64_total__on__def,axiom,
! [A: $tType,R: fun(product_prod(A,A),bool),A1: fun(A,bool)] :
( total_on(A,A1,R)
<=> ! [X1: A] :
( member(A,X1,A1)
=> ! [Xa1: A] :
( member(A,Xa1,A1)
=> ( ( X1 != Xa1 )
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X1),Xa1),R)
| member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa1),X1),R) ) ) ) ) ) ).
tff(fact_65_irrefl__def,axiom,
! [A: $tType,R: fun(product_prod(A,A),bool)] :
( irrefl(A,R)
<=> ! [X1: A] : ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X1),X1),R) ) ).
tff(fact_66_antisym__converse,axiom,
! [A: $tType,R: fun(product_prod(A,A),bool)] :
( antisym(A,converse(A,A,R))
<=> antisym(A,R) ) ).
tff(fact_67_converse__Id__on,axiom,
! [A: $tType,A1: fun(A,bool)] : converse(A,A,id_on(A,A1)) = id_on(A,A1) ).
tff(fact_68_antisym__Id__on,axiom,
! [A: $tType,A1: fun(A,bool)] : antisym(A,id_on(A,A1)) ).
tff(fact_69_antisym__def,axiom,
! [A: $tType,R: fun(product_prod(A,A),bool)] :
( antisym(A,R)
<=> ! [X1: A,Y1: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X1),Y1),R)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y1),X1),R)
=> ( X1 = Y1 ) ) ) ) ).
tff(fact_70_antisymD,axiom,
! [A: $tType,B1: A,A2: A,R: fun(product_prod(A,A),bool)] :
( antisym(A,R)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B1),R)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),B1),A2),R)
=> ( A2 = B1 ) ) ) ) ).
tff(fact_71_Id__on__iff,axiom,
! [A: $tType,A1: fun(A,bool),Ya: A,Xa: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Ya),id_on(A,A1))
<=> ( ( Xa = Ya )
& member(A,Xa,A1) ) ) ).
tff(fact_72_Id__on__eqI,axiom,
! [A: $tType,A1: fun(A,bool),B1: A,A2: A] :
( ( A2 = B1 )
=> ( member(A,A2,A1)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B1),id_on(A,A1)) ) ) ).
tff(fact_73_Id__onE,axiom,
! [A: $tType,A1: fun(A,bool),C: product_prod(A,A)] :
( member(product_prod(A,A),C,id_on(A,A1))
=> ~ ! [X: A] :
( member(A,X,A1)
=> ( C != aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X) ) ) ) ).
tff(fact_74_antisymI,axiom,
! [A: $tType,R: fun(product_prod(A,A),bool)] :
( ! [X: A,Y: A] :
( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y),R)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Y),X),R)
=> ( X = Y ) ) )
=> antisym(A,R) ) ).
tff(fact_75_ext,axiom,
! [B: $tType,A: $tType,G: fun(A,B),F: fun(A,B)] :
( ! [X: A] : aa(A,B,F,X) = aa(A,B,G,X)
=> ( F = G ) ) ).
tff(fact_76_mem__def,axiom,
! [A: $tType,A1: fun(A,bool),Xa: A] :
( member(A,Xa,A1)
<=> pp(aa(A,bool,A1,Xa)) ) ).
tff(fact_77_single__valued__Id__on,axiom,
! [A: $tType,A1: fun(A,bool)] : single_valued(A,A,id_on(A,A1)) ).
tff(fact_78_single__valuedD,axiom,
! [A: $tType,B: $tType,Za: B,Ya: B,Xa: A,R: fun(product_prod(A,B),bool)] :
( single_valued(A,B,R)
=> ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Ya),R)
=> ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Za),R)
=> ( Ya = Za ) ) ) ) ).
tff(fact_79_single__valued__def,axiom,
! [A: $tType,B: $tType,R: fun(product_prod(A,B),bool)] :
( single_valued(A,B,R)
<=> ! [X1: A,Y1: 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),Y1),R)
=> ! [Z1: 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),Z1),R)
=> ( Y1 = Z1 ) ) ) ) ).
tff(fact_80_single__valuedI,axiom,
! [B: $tType,A: $tType,R: fun(product_prod(A,B),bool)] :
( ! [X: A,Y: B] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y),R)
=> ! [Z: B] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Z),R)
=> ( Y = Z ) ) )
=> single_valued(A,B,R) ) ).
%----Helper facts (2)
tff(help_pp_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_pp_2_1_U,axiom,
pp(fTrue) ).
%----Conjectures (1)
tff(conj_0,conjecture,
~ ( 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),x),y),arrow_276188178_mkbot(l,z))
<=> ~ ( ( y != z )
& ( ( x = z )
=> ( x != y ) )
& ( ( x != 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),x),y),l) ) ) ) ).
%------------------------------------------------------------------------------