TPTP Problem File: SCT182_5.p
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%------------------------------------------------------------------------------
% File : SCT182_5 : TPTP v9.0.0. Released v6.0.0.
% Domain : Social Choice Theory
% Problem : Arrow's Impossibility Theorem line 87
% 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_87 [Bla13]
% Status : Unknown
% Rating : 1.00 v6.4.0
% Syntax : Number of formulae : 122 ( 31 unt; 36 typ; 0 def)
% Number of atoms : 194 ( 76 equ)
% Maximal formula atoms : 10 ( 1 avg)
% Number of connectives : 154 ( 46 ~; 3 |; 10 &)
% ( 21 <=>; 74 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 8 avg)
% Maximal term depth : 10 ( 2 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 40 ( 23 >; 17 *; 0 +; 0 <<)
% Number of predicates : 9 ( 8 usr; 0 prp; 1-5 aty)
% Number of functors : 26 ( 26 usr; 8 con; 0-6 aty)
% Number of variables : 497 ( 451 !; 4 ?; 497 :)
% ( 42 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_UNK_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:14:16
%------------------------------------------------------------------------------
%----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 (32)
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_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_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,C: $tType,B: $tType] : ( ( fun(A,C) * product_prod(A,B) ) > product_prod(C,B) ) ).
tff(sy_c_Product__Type_Oapsnd,type,
product_apsnd:
!>[B: $tType,C: $tType,A: $tType] : ( ( fun(B,C) * product_prod(A,B) ) > product_prod(A,C) ) ).
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_Product__Type_Oscomp,type,
product_scomp:
!>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( fun(A,product_prod(B,C)) * fun(B,fun(C,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_a,type,
a: arrow_411405190le_alt ).
tff(sy_v_b,type,
b: arrow_411405190le_alt ).
tff(sy_v_x,type,
x: arrow_411405190le_alt ).
tff(sy_v_y,type,
y: arrow_411405190le_alt ).
%----Relevant facts (81)
tff(fact_0_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_1_Lin__irrefl,axiom,
! [Ba: arrow_411405190le_alt,Aa: 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),Aa),Ba),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),Ba),Aa),La) ) ) ).
tff(fact_2_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,B3: 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),B3))) ) ).
tff(fact_3_Pair__eq,axiom,
! [A: $tType,B: $tType,B2: B,A3: A,Ba: B,Aa: A] :
( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa),Ba) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2) )
<=> ( ( Aa = A3 )
& ( Ba = B2 ) ) ) ).
tff(fact_4_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_5_in__mkbot,axiom,
! [Z2: 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_276188178_mkbot(La,Z2))
<=> ( ( Ya != Z2 )
& ( ( Xa = Z2 )
=> ( Xa != Ya ) )
& ( ( Xa != Z2 )
=> 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_6_in__mktop,axiom,
! [Z2: 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,Z2))
<=> ( ( Xa != Z2 )
& ( ( Ya = Z2 )
=> ( Xa != Ya ) )
& ( ( Ya != Z2 )
=> 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_7_Pair__inject,axiom,
! [A: $tType,B: $tType,B5: B,A6: A,B4: B,A5: A] :
( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B5) )
=> ~ ( ( A5 = A6 )
=> ( B4 != B5 ) ) ) ).
tff(fact_8_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_9_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,B3: 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),B3))) ) ).
tff(fact_10_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_11_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_12_in__lex__prod,axiom,
! [A: $tType,B: $tType,S: fun(product_prod(B,B),bool),R: fun(product_prod(A,A),bool),B2: B,A3: A,Ba: B,Aa: 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),Aa),Ba)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B2)),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),Aa),A3),R)
| ( ( Aa = A3 )
& member(product_prod(B,B),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Ba),B2),S) ) ) ) ).
tff(fact_13_prod_Orecs,axiom,
! [B: $tType,A: $tType,C: $tType,Ba: C,Aa: 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),Aa),Ba)) = aa(C,A,aa(B,fun(C,A),F1,Aa),Ba) ) ).
tff(fact_14_converse__iff,axiom,
! [A: $tType,B: $tType,R: fun(product_prod(B,A),bool),Ba: B,Aa: A] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa),Ba),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),Ba),Aa),R) ) ).
tff(fact_15_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y2: product_prod(A,product_prod(B,C))] :
~ ! [A2: A,B1: B,C2: C] : ( Y2 != 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)),A2),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B1),C2)) ) ).
tff(fact_16_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,Xa: product_prod(A,product_prod(B,C)),P1: fun(product_prod(A,product_prod(B,C)),bool)] :
( ! [A2: A,B1: B,C2: C] : pp(aa(product_prod(A,product_prod(B,C)),bool,P1,aa(product_prod(B,C),product_prod(A,product_prod(B,C)),aa(A,fun(product_prod(B,C),product_prod(A,product_prod(B,C))),product_Pair(A,product_prod(B,C)),A2),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B1),C2))))
=> pp(aa(product_prod(A,product_prod(B,C)),bool,P1,Xa)) ) ).
tff(fact_17_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y2: product_prod(A,product_prod(B,product_prod(C,D)))] :
~ ! [A2: A,B1: B,C2: C,D1: D] : ( Y2 != 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))),A2),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)),B1),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),C2),D1))) ) ).
tff(fact_18_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,Xa: product_prod(A,product_prod(B,product_prod(C,D))),P1: fun(product_prod(A,product_prod(B,product_prod(C,D))),bool)] :
( ! [A2: A,B1: B,C2: C,D1: D] : pp(aa(product_prod(A,product_prod(B,product_prod(C,D))),bool,P1,aa(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D))),aa(A,fun(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D)))),product_Pair(A,product_prod(B,product_prod(C,D))),A2),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)),B1),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),C2),D1)))))
=> pp(aa(product_prod(A,product_prod(B,product_prod(C,D))),bool,P1,Xa)) ) ).
tff(fact_19_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y2: product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))] :
~ ! [A2: A,B1: B,C2: C,D1: D,E1: E] : ( Y2 != 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)))),A2),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))),B1),aa(product_prod(D,E),product_prod(C,product_prod(D,E)),aa(C,fun(product_prod(D,E),product_prod(C,product_prod(D,E))),product_Pair(C,product_prod(D,E)),C2),aa(E,product_prod(D,E),aa(D,fun(E,product_prod(D,E)),product_Pair(D,E),D1),E1)))) ) ).
tff(fact_20_prod__induct5,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,Xa: 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)] :
( ! [A2: A,B1: B,C2: C,D1: D,E1: E] : pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),bool,P1,aa(product_prod(B,product_prod(C,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,E)))),A2),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))),B1),aa(product_prod(D,E),product_prod(C,product_prod(D,E)),aa(C,fun(product_prod(D,E),product_prod(C,product_prod(D,E))),product_Pair(C,product_prod(D,E)),C2),aa(E,product_prod(D,E),aa(D,fun(E,product_prod(D,E)),product_Pair(D,E),D1),E1))))))
=> pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),bool,P1,Xa)) ) ).
tff(fact_21_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F2: $tType,Y2: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2)))))] :
~ ! [A2: A,B1: B,C2: C,D1: D,E1: E,F3: F2] : ( Y2 != 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))))),A2),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)))),B1),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))),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_22_prod__induct6,axiom,
! [F2: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,Xa: 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)] :
( ! [A2: A,B1: B,C2: C,D1: 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))))),A2),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)))),B1),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))),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(C,product_prod(D,product_prod(E,F2))))),bool,P1,Xa)) ) ).
tff(fact_23_converse__converse,axiom,
! [B: $tType,A: $tType,R: fun(product_prod(A,B),bool)] : ( converse(B,A,converse(A,B,R)) = R ) ).
tff(fact_24_converseI,axiom,
! [B: $tType,A: $tType,R: fun(product_prod(A,B),bool),Ba: B,Aa: A] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa),Ba),R)
=> member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Ba),Aa),converse(A,B,R)) ) ).
tff(fact_25_converseD,axiom,
! [A: $tType,B: $tType,R: fun(product_prod(B,A),bool),Ba: B,Aa: A] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa),Ba),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),Ba),Aa),R) ) ).
tff(fact_26_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_27_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_28_prod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y2: product_prod(A,B)] :
~ ! [A2: A,B1: B] : ( Y2 != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B1) ) ).
tff(fact_29_internal__split__conv,axiom,
! [B: $tType,A: $tType,C: $tType,Ba: C,Aa: 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),Aa),Ba)) = aa(C,A,aa(B,fun(C,A),C1,Aa),Ba) ) ).
tff(fact_30_curry__conv,axiom,
! [A: $tType,B: $tType,C: $tType,Ba: C,Aa: B,F: fun(product_prod(B,C),A)] : ( aa(C,A,aa(B,fun(C,A),product_curry(B,C,A,F),Aa),Ba) = 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),Aa),Ba)) ) ).
tff(fact_31_curryI,axiom,
! [A: $tType,B: $tType,Ba: B,Aa: 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),Aa),Ba)))
=> pp(aa(B,bool,aa(A,fun(B,bool),product_curry(A,B,bool,F),Aa),Ba)) ) ).
tff(fact_32_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_33_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_34_curryE,axiom,
! [A: $tType,B: $tType,Ba: B,Aa: A,F: fun(product_prod(A,B),bool)] :
( pp(aa(B,bool,aa(A,fun(B,bool),product_curry(A,B,bool,F),Aa),Ba))
=> 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),Aa),Ba))) ) ).
tff(fact_35_curryD,axiom,
! [A: $tType,B: $tType,Ba: B,Aa: A,F: fun(product_prod(A,B),bool)] :
( pp(aa(B,bool,aa(A,fun(B,bool),product_curry(A,B,bool,F),Aa),Ba))
=> 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),Aa),Ba))) ) ).
tff(fact_36_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_37_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_38_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_39_antisym__converse,axiom,
! [A: $tType,R: fun(product_prod(A,A),bool)] :
( antisym(A,converse(A,A,R))
<=> antisym(A,R) ) ).
tff(fact_40_converse__Id__on,axiom,
! [A: $tType,A1: fun(A,bool)] : ( converse(A,A,id_on(A,A1)) = id_on(A,A1) ) ).
tff(fact_41_internal__split__def,axiom,
! [C: $tType,B: $tType,A: $tType] : ( produc1605651328_split(A,B,C) = product_prod_case(A,B,C) ) ).
tff(fact_42_splitI,axiom,
! [A: $tType,B: $tType,Ba: B,Aa: A,F: fun(A,fun(B,bool))] :
( pp(aa(B,bool,aa(A,fun(B,bool),F,Aa),Ba))
=> 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),Aa),Ba))) ) ).
tff(fact_43_prod__caseI,axiom,
! [A: $tType,B: $tType,Ba: B,Aa: A,F1: fun(A,fun(B,bool))] :
( pp(aa(B,bool,aa(A,fun(B,bool),F1,Aa),Ba))
=> 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),Aa),Ba))) ) ).
tff(fact_44_mem__splitI,axiom,
! [A: $tType,B: $tType,C: $tType,Ba: C,Aa: B,C1: fun(B,fun(C,fun(A,bool))),Z2: A] :
( member(A,Z2,aa(C,fun(A,bool),aa(B,fun(C,fun(A,bool)),C1,Aa),Ba))
=> member(A,Z2,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),Aa),Ba))) ) ).
tff(fact_45_split__conv,axiom,
! [B: $tType,A: $tType,C: $tType,Ba: C,Aa: 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),Aa),Ba)) = aa(C,A,aa(B,fun(C,A),F,Aa),Ba) ) ).
tff(fact_46_split__weak__cong,axiom,
! [C: $tType,B: $tType,A: $tType,C1: fun(A,fun(B,C)),Q: product_prod(A,B),P: product_prod(A,B)] :
( ( P = Q )
=> ( 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),P) = 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),Q) ) ) ).
tff(fact_47_antisym__Id__on,axiom,
! [A: $tType,A1: fun(A,bool)] : antisym(A,id_on(A,A1)) ).
tff(fact_48_splitD,axiom,
! [A: $tType,B: $tType,Ba: B,Aa: 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),Aa),Ba)))
=> pp(aa(B,bool,aa(A,fun(B,bool),F,Aa),Ba)) ) ).
tff(fact_49_splitD_H,axiom,
! [B: $tType,A: $tType,C: $tType,C1: C,Ba: B,Aa: A,R1: 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)),R1),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa),Ba)),C1))
=> pp(aa(C,bool,aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),R1,Aa),Ba),C1)) ) ).
tff(fact_50_prod_Osimps_I2_J,axiom,
! [B: $tType,A: $tType,C: $tType,Ba: C,Aa: 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),Aa),Ba)) = aa(C,A,aa(B,fun(C,A),F1,Aa),Ba) ) ).
tff(fact_51_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_52_Id__on__eqI,axiom,
! [A: $tType,A1: fun(A,bool),Ba: A,Aa: A] :
( ( Aa = Ba )
=> ( member(A,Aa,A1)
=> member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Aa),Ba),id_on(A,A1)) ) ) ).
tff(fact_53_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_54_antisymD,axiom,
! [A: $tType,Ba: A,Aa: 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),Aa),Ba),R)
=> ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Ba),Aa),R)
=> ( Aa = Ba ) ) ) ) ).
tff(fact_55_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_56_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_57_Id__onE,axiom,
! [A: $tType,A1: fun(A,bool),C1: product_prod(A,A)] :
( member(product_prod(A,A),C1,id_on(A,A1))
=> ~ ! [X: A] :
( member(A,X,A1)
=> ( C1 != aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),X) ) ) ) ).
tff(fact_58_splitE,axiom,
! [A: $tType,B: $tType,P: 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),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),C1,X),Y)) ) ) ).
tff(fact_59_splitI2,axiom,
! [B: $tType,A: $tType,C1: fun(A,fun(B,bool)),P: product_prod(A,B)] :
( ! [A2: A,B1: B] :
( ( P = 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),C1,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),C1),P)) ) ).
tff(fact_60_mem__splitI2,axiom,
! [C: $tType,B: $tType,A: $tType,C1: fun(A,fun(B,fun(C,bool))),Z2: C,P: product_prod(A,B)] :
( ! [A2: A,B1: B] :
( ( P = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B1) )
=> member(C,Z2,aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),C1,A2),B1)) )
=> member(C,Z2,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),P)) ) ).
tff(fact_61_splitI2_H,axiom,
! [A: $tType,B: $tType,C: $tType,Xa: C,C1: fun(A,fun(B,fun(C,bool))),P: product_prod(A,B)] :
( ! [A2: A,B1: B] :
( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B1) = P )
=> pp(aa(C,bool,aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),C1,A2),B1),Xa)) )
=> 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),P),Xa)) ) ).
tff(fact_62_splitE_H,axiom,
! [B: $tType,A: $tType,C: $tType,Z2: C,P: 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),P),Z2))
=> ~ ! [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(C,bool,aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),C1,X),Y),Z2)) ) ) ).
tff(fact_63_mem__splitE,axiom,
! [B: $tType,A: $tType,C: $tType,P: product_prod(B,C),C1: fun(B,fun(C,fun(A,bool))),Z2: A] :
( member(A,Z2,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),P))
=> ~ ! [X: B,Y: C] :
( ( P = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X),Y) )
=> ~ member(A,Z2,aa(C,fun(A,bool),aa(B,fun(C,fun(A,bool)),C1,X),Y)) ) ) ).
tff(fact_64_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_65_splitE2,axiom,
! [B: $tType,A: $tType,C: $tType,Z2: 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),Z2)))
=> ~ ! [X: B,Y: C] :
( ( Z2 = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X),Y) )
=> ~ pp(aa(A,bool,Q1,aa(C,A,aa(B,fun(C,A),P1,X),Y))) ) ) ).
tff(fact_66_split__cong,axiom,
! [C: $tType,B: $tType,A: $tType,P: product_prod(A,B),G: fun(A,fun(B,C)),F: fun(A,fun(B,C)),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,C,aa(A,fun(B,C),F,X),Y) = aa(B,C,aa(A,fun(B,C),G,X),Y) ) )
=> ( ( P = Q )
=> ( aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_prod_case(A,B,C),F),P) = aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_prod_case(A,B,C),G),Q) ) ) ) ).
tff(fact_67_scomp__apply,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,Xa: B,G: fun(C,fun(D,A)),F: fun(B,product_prod(C,D))] : ( aa(B,A,product_scomp(B,C,D,A,F,G),Xa) = aa(product_prod(C,D),A,aa(fun(C,fun(D,A)),fun(product_prod(C,D),A),product_prod_case(C,D,A),G),aa(B,product_prod(C,D),F,Xa)) ) ).
tff(fact_68_Pair__scomp,axiom,
! [A: $tType,B: $tType,C: $tType,F: fun(C,fun(A,B)),Xa: C] : ( product_scomp(A,C,A,B,aa(C,fun(A,product_prod(C,A)),product_Pair(C,A),Xa),F) = aa(C,fun(A,B),F,Xa) ) ).
tff(fact_69_scomp__Pair,axiom,
! [C: $tType,B: $tType,A: $tType,Xa: fun(A,product_prod(B,C))] : ( product_scomp(A,B,C,product_prod(B,C),Xa,product_Pair(B,C)) = Xa ) ).
tff(fact_70_scomp__def,axiom,
! [B: $tType,C: $tType,D: $tType,A: $tType,G: fun(C,fun(D,B)),F: fun(A,product_prod(C,D)),X2: A] : ( aa(A,B,product_scomp(A,C,D,B,F,G),X2) = aa(product_prod(C,D),B,aa(fun(C,fun(D,B)),fun(product_prod(C,D),B),product_prod_case(C,D,B),G),aa(A,product_prod(C,D),F,X2)) ) ).
tff(fact_71_single__valued__Id__on,axiom,
! [A: $tType,A1: fun(A,bool)] : single_valued(A,A,id_on(A,A1)) ).
tff(fact_72_apsnd__conv,axiom,
! [A: $tType,B: $tType,C: $tType,Ya: C,Xa: A,F: fun(C,B)] : ( product_apsnd(C,B,A,F,aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),Xa),Ya)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),aa(C,B,F,Ya)) ) ).
tff(fact_73_apfst__conv,axiom,
! [C: $tType,A: $tType,B: $tType,Ya: B,Xa: C,F: fun(C,A)] : ( product_apfst(C,A,B,F,aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),Xa),Ya)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F,Xa)),Ya) ) ).
tff(fact_74_apsnd__apfst__commute,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,P: product_prod(D,C),G: fun(D,A),F: fun(C,B)] : ( product_apsnd(C,B,A,F,product_apfst(D,A,C,G,P)) = product_apfst(D,A,B,G,product_apsnd(C,B,D,F,P)) ) ).
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__valuedD,axiom,
! [A: $tType,B: $tType,Z2: 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),Z2),R)
=> ( Ya = Z2 ) ) ) ) ).
tff(fact_78_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_79_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) ) ).
tff(fact_80_apfst__convE,axiom,
! [C: $tType,A: $tType,B: $tType,P: product_prod(C,B),F: fun(C,A),Q: product_prod(A,B)] :
( ( Q = product_apfst(C,A,B,F,P) )
=> ~ ! [X: C,Y: B] :
( ( P = aa(B,product_prod(C,B),aa(C,fun(B,product_prod(C,B)),product_Pair(C,B),X),Y) )
=> ( Q != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C,A,F,X)),Y) ) ) ) ).
%----Helper facts (2)
tff(help_pp_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_pp_2_1_U,axiom,
pp(fTrue) ).
%----Conjectures (3)
tff(conj_0,hypothesis,
a != b ).
tff(conj_1,hypothesis,
member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),l,arrow_1985332922le_Lin) ).
tff(conj_2,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_1158827142_above(l,a,b))
<=> ~ ( ( x != y )
& ( ( x = 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),y),l) )
& ( ( x != b )
=> ( ( ( y = b )
=> ( ( x = a )
| 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),a),l) ) )
& ( ( y != 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),x),y),l) ) ) ) ) ) ).
%------------------------------------------------------------------------------