TPTP Problem File: SCT195_5.p
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%------------------------------------------------------------------------------
% File : SCT195_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Social Choice Theory
% Problem : Arrow's Impossibility Theorem line 137
% 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_137 [Bla13]
% Status : Theorem
% Rating : 0.00 v7.5.0, 0.33 v7.4.0, 0.50 v7.1.0, 0.67 v6.4.0
% Syntax : Number of formulae : 114 ( 27 unt; 35 typ; 0 def)
% Number of atoms : 198 ( 76 equ)
% Maximal formula atoms : 12 ( 1 avg)
% Number of connectives : 170 ( 51 ~; 6 |; 18 &)
% ( 16 <=>; 79 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 8 avg)
% Maximal term depth : 10 ( 2 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 42 ( 22 >; 20 *; 0 +; 0 <<)
% Number of predicates : 8 ( 7 usr; 0 prp; 1-5 aty)
% Number of functors : 25 ( 25 usr; 7 con; 0-6 aty)
% Number of variables : 467 ( 424 !; 6 ?; 467 :)
% ( 37 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:17:35
%------------------------------------------------------------------------------
%----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 (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_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_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:
!>[A2: $tType,B1: $tType] : ( ( fun(product_prod(A2,B1),bool) * A2 * B1 ) > $o ) ).
tff(sy_c_Order__Relation_Ostrict__linear__order__on,type,
order_215145569der_on:
!>[A2: $tType] : ( ( fun(A2,bool) * fun(product_prod(A2,A2),bool) ) > $o ) ).
tff(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A2: $tType,B1: $tType] : fun(A2,fun(B1,product_prod(A2,B1))) ).
tff(sy_c_Product__Type_Oapfst,type,
product_apfst:
!>[A2: $tType,C: $tType,B1: $tType] : ( ( fun(A2,C) * product_prod(A2,B1) ) > product_prod(C,B1) ) ).
tff(sy_c_Product__Type_Oapsnd,type,
product_apsnd:
!>[B1: $tType,C: $tType,A2: $tType] : ( ( fun(B1,C) * product_prod(A2,B1) ) > product_prod(A2,C) ) ).
tff(sy_c_Product__Type_Ocurry,type,
product_curry:
!>[A2: $tType,B1: $tType,C: $tType] : ( fun(product_prod(A2,B1),C) > fun(A2,fun(B1,C)) ) ).
tff(sy_c_Product__Type_Ointernal__split,type,
produc1605651328_split:
!>[A2: $tType,B1: $tType,C: $tType] : fun(fun(A2,fun(B1,C)),fun(product_prod(A2,B1),C)) ).
tff(sy_c_Product__Type_Oprod_Oprod__case,type,
product_prod_case:
!>[A2: $tType,B1: $tType,T: $tType] : fun(fun(A2,fun(B1,T)),fun(product_prod(A2,B1),T)) ).
tff(sy_c_Product__Type_Oprod_Oprod__rec,type,
product_prod_rec:
!>[A2: $tType,B1: $tType,T: $tType] : ( ( fun(A2,fun(B1,T)) * product_prod(A2,B1) ) > T ) ).
tff(sy_c_Product__Type_Oscomp,type,
product_scomp:
!>[A2: $tType,B1: $tType,C: $tType,D: $tType] : ( ( fun(A2,product_prod(B1,C)) * fun(B1,fun(C,D)) ) > fun(A2,D) ) ).
tff(sy_c_Relation_Oinv__image,type,
inv_image:
!>[B1: $tType,A2: $tType] : ( ( fun(product_prod(B1,B1),bool) * fun(A2,B1) ) > fun(product_prod(A2,A2),bool) ) ).
tff(sy_c_Relation_Oirrefl,type,
irrefl:
!>[A2: $tType] : ( fun(product_prod(A2,A2),bool) > $o ) ).
tff(sy_c_Relation_Ototal__on,type,
total_on:
!>[A2: $tType] : ( ( fun(A2,bool) * fun(product_prod(A2,A2),bool) ) > $o ) ).
tff(sy_c_Relation_Otrans,type,
trans:
!>[A2: $tType] : ( fun(product_prod(A2,A2),bool) > $o ) ).
tff(sy_c_Wellfounded_Olex__prod,type,
lex_prod:
!>[A2: $tType,B1: $tType] : ( ( fun(product_prod(A2,A2),bool) * fun(product_prod(B1,B1),bool) ) > fun(product_prod(product_prod(A2,B1),product_prod(A2,B1)),bool) ) ).
tff(sy_c_aa,type,
aa:
!>[A2: $tType,B1: $tType] : ( ( fun(A2,B1) * A2 ) > B1 ) ).
tff(sy_c_fFalse,type,
fFalse: bool ).
tff(sy_c_fTrue,type,
fTrue: bool ).
tff(sy_c_member,type,
member:
!>[A2: $tType] : ( ( A2 * fun(A2,bool) ) > $o ) ).
tff(sy_c_pp,type,
pp: bool > $o ).
tff(sy_v_F,type,
f: fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)) > fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool) ).
tff(sy_v_P,type,
p: fun(arrow_159774573e_indi,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_i,type,
i: arrow_159774573e_indi ).
%----Relevant facts (72)
tff(fact_0_split__paired__All,axiom,
! [A2: $tType,B1: $tType,P1: fun(product_prod(A2,B1),bool)] :
( ! [X11: product_prod(A2,B1)] : pp(aa(product_prod(A2,B1),bool,P1,X11))
<=> ! [A6: A2,B5: B1] : pp(aa(product_prod(A2,B1),bool,P1,aa(B1,product_prod(A2,B1),aa(A2,fun(B1,product_prod(A2,B1)),product_Pair(A2,B1),A6),B5))) ) ).
tff(fact_1_Pair__eq,axiom,
! [A2: $tType,B1: $tType,B4: B1,A5: A2,B3: B1,A4: A2] :
( ( aa(B1,product_prod(A2,B1),aa(A2,fun(B1,product_prod(A2,B1)),product_Pair(A2,B1),A4),B3) = aa(B1,product_prod(A2,B1),aa(A2,fun(B1,product_prod(A2,B1)),product_Pair(A2,B1),A5),B4) )
<=> ( ( A4 = A5 )
& ( B3 = B4 ) ) ) ).
tff(fact_2_trans__def,axiom,
! [A2: $tType,R: fun(product_prod(A2,A2),bool)] :
( trans(A2,R)
<=> ! [X3: A2,Y4: A2,Z2: A2] :
( member(product_prod(A2,A2),aa(A2,product_prod(A2,A2),aa(A2,fun(A2,product_prod(A2,A2)),product_Pair(A2,A2),X3),Y4),R)
=> ( member(product_prod(A2,A2),aa(A2,product_prod(A2,A2),aa(A2,fun(A2,product_prod(A2,A2)),product_Pair(A2,A2),Y4),Z2),R)
=> member(product_prod(A2,A2),aa(A2,product_prod(A2,A2),aa(A2,fun(A2,product_prod(A2,A2)),product_Pair(A2,A2),X3),Z2),R) ) ) ) ).
tff(fact_3_transD,axiom,
! [A2: $tType,C1: A2,B3: A2,A4: A2,R: fun(product_prod(A2,A2),bool)] :
( trans(A2,R)
=> ( member(product_prod(A2,A2),aa(A2,product_prod(A2,A2),aa(A2,fun(A2,product_prod(A2,A2)),product_Pair(A2,A2),A4),B3),R)
=> ( member(product_prod(A2,A2),aa(A2,product_prod(A2,A2),aa(A2,fun(A2,product_prod(A2,A2)),product_Pair(A2,A2),B3),C1),R)
=> member(product_prod(A2,A2),aa(A2,product_prod(A2,A2),aa(A2,fun(A2,product_prod(A2,A2)),product_Pair(A2,A2),A4),C1),R) ) ) ) ).
tff(fact_4_in__mkbot,axiom,
! [Z: arrow_411405190le_alt,L: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Y2: arrow_411405190le_alt,X1: 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),X1),Y2),arrow_276188178_mkbot(L,Z))
<=> ( ( Y2 != Z )
& ( ( X1 = Z )
=> ( X1 != Y2 ) )
& ( ( X1 != 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),X1),Y2),L) ) ) ) ).
tff(fact_5_in__mktop,axiom,
! [Z: arrow_411405190le_alt,L: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Y2: arrow_411405190le_alt,X1: 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),X1),Y2),arrow_424895264_mktop(L,Z))
<=> ( ( X1 != Z )
& ( ( Y2 = Z )
=> ( X1 != Y2 ) )
& ( ( Y2 != Z )
=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),aa(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt),aa(arrow_411405190le_alt,fun(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt),X1),Y2),L) ) ) ) ).
tff(fact_6_transI,axiom,
! [A2: $tType,R: fun(product_prod(A2,A2),bool)] :
( ! [X2: A2,Y1: A2,Z1: A2] :
( member(product_prod(A2,A2),aa(A2,product_prod(A2,A2),aa(A2,fun(A2,product_prod(A2,A2)),product_Pair(A2,A2),X2),Y1),R)
=> ( member(product_prod(A2,A2),aa(A2,product_prod(A2,A2),aa(A2,fun(A2,product_prod(A2,A2)),product_Pair(A2,A2),Y1),Z1),R)
=> member(product_prod(A2,A2),aa(A2,product_prod(A2,A2),aa(A2,fun(A2,product_prod(A2,A2)),product_Pair(A2,A2),X2),Z1),R) ) )
=> trans(A2,R) ) ).
tff(fact_7_Pair__inject,axiom,
! [A2: $tType,B1: $tType,B7: B1,A8: A2,B6: B1,A7: A2] :
( ( aa(B1,product_prod(A2,B1),aa(A2,fun(B1,product_prod(A2,B1)),product_Pair(A2,B1),A7),B6) = aa(B1,product_prod(A2,B1),aa(A2,fun(B1,product_prod(A2,B1)),product_Pair(A2,B1),A8),B7) )
=> ~ ( ( A7 = A8 )
=> ( B6 != B7 ) ) ) ).
tff(fact_8_trans__lex__prod,axiom,
! [A2: $tType,B1: $tType,R2: fun(product_prod(B1,B1),bool),R11: fun(product_prod(A2,A2),bool)] :
( trans(A2,R11)
=> ( trans(B1,R2)
=> trans(product_prod(A2,B1),lex_prod(A2,B1,R11,R2)) ) ) ).
tff(fact_9_in__rel__def,axiom,
! [B1: $tType,A2: $tType,Y2: B1,X1: A2,R1: fun(product_prod(A2,B1),bool)] :
( in_rel(A2,B1,R1,X1,Y2)
<=> member(product_prod(A2,B1),aa(B1,product_prod(A2,B1),aa(A2,fun(B1,product_prod(A2,B1)),product_Pair(A2,B1),X1),Y2),R1) ) ).
tff(fact_10_split__paired__Ex,axiom,
! [A2: $tType,B1: $tType,P1: fun(product_prod(A2,B1),bool)] :
( ? [X11: product_prod(A2,B1)] : pp(aa(product_prod(A2,B1),bool,P1,X11))
<=> ? [A6: A2,B5: B1] : pp(aa(product_prod(A2,B1),bool,P1,aa(B1,product_prod(A2,B1),aa(A2,fun(B1,product_prod(A2,B1)),product_Pair(A2,B1),A6),B5))) ) ).
tff(fact_11_notin__Lin__iff,axiom,
! [Y2: arrow_411405190le_alt,X1: 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)
=> ( ( X1 != Y2 )
=> ( ~ member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),aa(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt),aa(arrow_411405190le_alt,fun(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt),X1),Y2),L)
<=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),aa(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt),aa(arrow_411405190le_alt,fun(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt),Y2),X1),L) ) ) ) ).
tff(fact_12_in__lex__prod,axiom,
! [A2: $tType,B1: $tType,S: fun(product_prod(B1,B1),bool),R: fun(product_prod(A2,A2),bool),B4: B1,A5: A2,B3: B1,A4: A2] :
( member(product_prod(product_prod(A2,B1),product_prod(A2,B1)),aa(product_prod(A2,B1),product_prod(product_prod(A2,B1),product_prod(A2,B1)),aa(product_prod(A2,B1),fun(product_prod(A2,B1),product_prod(product_prod(A2,B1),product_prod(A2,B1))),product_Pair(product_prod(A2,B1),product_prod(A2,B1)),aa(B1,product_prod(A2,B1),aa(A2,fun(B1,product_prod(A2,B1)),product_Pair(A2,B1),A4),B3)),aa(B1,product_prod(A2,B1),aa(A2,fun(B1,product_prod(A2,B1)),product_Pair(A2,B1),A5),B4)),lex_prod(A2,B1,R,S))
<=> ( member(product_prod(A2,A2),aa(A2,product_prod(A2,A2),aa(A2,fun(A2,product_prod(A2,A2)),product_Pair(A2,A2),A4),A5),R)
| ( ( A4 = A5 )
& member(product_prod(B1,B1),aa(B1,product_prod(B1,B1),aa(B1,fun(B1,product_prod(B1,B1)),product_Pair(B1,B1),B3),B4),S) ) ) ) ).
tff(fact_13_mkbot__Lin,axiom,
! [X1: 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,X1),arrow_1985332922le_Lin) ) ).
tff(fact_14_mktop__Lin,axiom,
! [X1: 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,X1),arrow_1985332922le_Lin) ) ).
tff(fact_15_Lin__irrefl,axiom,
! [B3: arrow_411405190le_alt,A4: 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),A4),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),A4),L) ) ) ).
tff(fact_16_in__below,axiom,
! [Y2: arrow_411405190le_alt,X1: arrow_411405190le_alt,L: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),B3: arrow_411405190le_alt,A4: arrow_411405190le_alt] :
( ( A4 != 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),X1),Y2),arrow_319942042_below(L,A4,B3))
<=> ( ( X1 != Y2 )
& ( ( Y2 = A4 )
=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),aa(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt),aa(arrow_411405190le_alt,fun(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt),X1),B3),L) )
& ( ( Y2 != A4 )
=> ( ( ( X1 = A4 )
=> ( ( Y2 = 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),Y2),L) ) )
& ( ( X1 != A4 )
=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),aa(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt),aa(arrow_411405190le_alt,fun(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt),X1),Y2),L) ) ) ) ) ) ) ) ).
tff(fact_17_in__above,axiom,
! [Y2: arrow_411405190le_alt,X1: arrow_411405190le_alt,L: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),B3: arrow_411405190le_alt,A4: arrow_411405190le_alt] :
( ( A4 != 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),X1),Y2),arrow_1158827142_above(L,A4,B3))
<=> ( ( X1 != Y2 )
& ( ( X1 = 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),A4),Y2),L) )
& ( ( X1 != B3 )
=> ( ( ( Y2 = B3 )
=> ( ( X1 = A4 )
| member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),aa(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt),aa(arrow_411405190le_alt,fun(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt),X1),A4),L) ) )
& ( ( Y2 != 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),X1),Y2),L) ) ) ) ) ) ) ) ).
tff(fact_18_prod_Orecs,axiom,
! [B1: $tType,A2: $tType,C: $tType,B3: C,A4: B1,F1: fun(B1,fun(C,A2))] : product_prod_rec(B1,C,A2,F1,aa(C,product_prod(B1,C),aa(B1,fun(C,product_prod(B1,C)),product_Pair(B1,C),A4),B3)) = aa(C,A2,aa(B1,fun(C,A2),F1,A4),B3) ).
tff(fact_19_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_20_above__Lin,axiom,
! [L: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Y2: arrow_411405190le_alt,X1: arrow_411405190le_alt] :
( ( X1 != Y2 )
=> ( member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),L,arrow_1985332922le_Lin)
=> member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),arrow_1158827142_above(L,X1,Y2),arrow_1985332922le_Lin) ) ) ).
tff(fact_21_below__Lin,axiom,
! [L: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Y2: arrow_411405190le_alt,X1: arrow_411405190le_alt] :
( ( X1 != Y2 )
=> ( member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),L,arrow_1985332922le_Lin)
=> member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),arrow_319942042_below(L,X1,Y2),arrow_1985332922le_Lin) ) ) ).
tff(fact_22_prod_Oexhaust,axiom,
! [A2: $tType,B1: $tType,Y3: product_prod(A2,B1)] :
~ ! [A3: A2,B2: B1] : Y3 != aa(B1,product_prod(A2,B1),aa(A2,fun(B1,product_prod(A2,B1)),product_Pair(A2,B1),A3),B2) ).
tff(fact_23_PairE,axiom,
! [A2: $tType,B1: $tType,P2: product_prod(A2,B1)] :
~ ! [X2: A2,Y1: B1] : P2 != aa(B1,product_prod(A2,B1),aa(A2,fun(B1,product_prod(A2,B1)),product_Pair(A2,B1),X2),Y1) ).
tff(fact_24_prod__cases3,axiom,
! [A2: $tType,B1: $tType,C: $tType,Y3: product_prod(A2,product_prod(B1,C))] :
~ ! [A3: A2,B2: B1,C2: C] : Y3 != aa(product_prod(B1,C),product_prod(A2,product_prod(B1,C)),aa(A2,fun(product_prod(B1,C),product_prod(A2,product_prod(B1,C))),product_Pair(A2,product_prod(B1,C)),A3),aa(C,product_prod(B1,C),aa(B1,fun(C,product_prod(B1,C)),product_Pair(B1,C),B2),C2)) ).
tff(fact_25_prod__induct6,axiom,
! [F2: $tType,E: $tType,D: $tType,C: $tType,B1: $tType,A2: $tType,X1: product_prod(A2,product_prod(B1,product_prod(C,product_prod(D,product_prod(E,F2))))),P1: fun(product_prod(A2,product_prod(B1,product_prod(C,product_prod(D,product_prod(E,F2))))),bool)] :
( ! [A3: A2,B2: B1,C2: C,D1: D,E1: E,F3: F2] : pp(aa(product_prod(A2,product_prod(B1,product_prod(C,product_prod(D,product_prod(E,F2))))),bool,P1,aa(product_prod(B1,product_prod(C,product_prod(D,product_prod(E,F2)))),product_prod(A2,product_prod(B1,product_prod(C,product_prod(D,product_prod(E,F2))))),aa(A2,fun(product_prod(B1,product_prod(C,product_prod(D,product_prod(E,F2)))),product_prod(A2,product_prod(B1,product_prod(C,product_prod(D,product_prod(E,F2)))))),product_Pair(A2,product_prod(B1,product_prod(C,product_prod(D,product_prod(E,F2))))),A3),aa(product_prod(C,product_prod(D,product_prod(E,F2))),product_prod(B1,product_prod(C,product_prod(D,product_prod(E,F2)))),aa(B1,fun(product_prod(C,product_prod(D,product_prod(E,F2))),product_prod(B1,product_prod(C,product_prod(D,product_prod(E,F2))))),product_Pair(B1,product_prod(C,product_prod(D,product_prod(E,F2)))),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))),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(A2,product_prod(B1,product_prod(C,product_prod(D,product_prod(E,F2))))),bool,P1,X1)) ) ).
tff(fact_26_prod__cases6,axiom,
! [A2: $tType,B1: $tType,C: $tType,D: $tType,E: $tType,F2: $tType,Y3: product_prod(A2,product_prod(B1,product_prod(C,product_prod(D,product_prod(E,F2)))))] :
~ ! [A3: A2,B2: B1,C2: C,D1: D,E1: E,F3: F2] : Y3 != aa(product_prod(B1,product_prod(C,product_prod(D,product_prod(E,F2)))),product_prod(A2,product_prod(B1,product_prod(C,product_prod(D,product_prod(E,F2))))),aa(A2,fun(product_prod(B1,product_prod(C,product_prod(D,product_prod(E,F2)))),product_prod(A2,product_prod(B1,product_prod(C,product_prod(D,product_prod(E,F2)))))),product_Pair(A2,product_prod(B1,product_prod(C,product_prod(D,product_prod(E,F2))))),A3),aa(product_prod(C,product_prod(D,product_prod(E,F2))),product_prod(B1,product_prod(C,product_prod(D,product_prod(E,F2)))),aa(B1,fun(product_prod(C,product_prod(D,product_prod(E,F2))),product_prod(B1,product_prod(C,product_prod(D,product_prod(E,F2))))),product_Pair(B1,product_prod(C,product_prod(D,product_prod(E,F2)))),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))),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_27_prod__induct5,axiom,
! [E: $tType,D: $tType,C: $tType,B1: $tType,A2: $tType,X1: product_prod(A2,product_prod(B1,product_prod(C,product_prod(D,E)))),P1: fun(product_prod(A2,product_prod(B1,product_prod(C,product_prod(D,E)))),bool)] :
( ! [A3: A2,B2: B1,C2: C,D1: D,E1: E] : pp(aa(product_prod(A2,product_prod(B1,product_prod(C,product_prod(D,E)))),bool,P1,aa(product_prod(B1,product_prod(C,product_prod(D,E))),product_prod(A2,product_prod(B1,product_prod(C,product_prod(D,E)))),aa(A2,fun(product_prod(B1,product_prod(C,product_prod(D,E))),product_prod(A2,product_prod(B1,product_prod(C,product_prod(D,E))))),product_Pair(A2,product_prod(B1,product_prod(C,product_prod(D,E)))),A3),aa(product_prod(C,product_prod(D,E)),product_prod(B1,product_prod(C,product_prod(D,E))),aa(B1,fun(product_prod(C,product_prod(D,E)),product_prod(B1,product_prod(C,product_prod(D,E)))),product_Pair(B1,product_prod(C,product_prod(D,E))),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)),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(A2,product_prod(B1,product_prod(C,product_prod(D,E)))),bool,P1,X1)) ) ).
tff(fact_28_prod__cases5,axiom,
! [A2: $tType,B1: $tType,C: $tType,D: $tType,E: $tType,Y3: product_prod(A2,product_prod(B1,product_prod(C,product_prod(D,E))))] :
~ ! [A3: A2,B2: B1,C2: C,D1: D,E1: E] : Y3 != aa(product_prod(B1,product_prod(C,product_prod(D,E))),product_prod(A2,product_prod(B1,product_prod(C,product_prod(D,E)))),aa(A2,fun(product_prod(B1,product_prod(C,product_prod(D,E))),product_prod(A2,product_prod(B1,product_prod(C,product_prod(D,E))))),product_Pair(A2,product_prod(B1,product_prod(C,product_prod(D,E)))),A3),aa(product_prod(C,product_prod(D,E)),product_prod(B1,product_prod(C,product_prod(D,E))),aa(B1,fun(product_prod(C,product_prod(D,E)),product_prod(B1,product_prod(C,product_prod(D,E)))),product_Pair(B1,product_prod(C,product_prod(D,E))),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)),C2),aa(E,product_prod(D,E),aa(D,fun(E,product_prod(D,E)),product_Pair(D,E),D1),E1)))) ).
tff(fact_29_prod__induct4,axiom,
! [D: $tType,C: $tType,B1: $tType,A2: $tType,X1: product_prod(A2,product_prod(B1,product_prod(C,D))),P1: fun(product_prod(A2,product_prod(B1,product_prod(C,D))),bool)] :
( ! [A3: A2,B2: B1,C2: C,D1: D] : pp(aa(product_prod(A2,product_prod(B1,product_prod(C,D))),bool,P1,aa(product_prod(B1,product_prod(C,D)),product_prod(A2,product_prod(B1,product_prod(C,D))),aa(A2,fun(product_prod(B1,product_prod(C,D)),product_prod(A2,product_prod(B1,product_prod(C,D)))),product_Pair(A2,product_prod(B1,product_prod(C,D))),A3),aa(product_prod(C,D),product_prod(B1,product_prod(C,D)),aa(B1,fun(product_prod(C,D),product_prod(B1,product_prod(C,D))),product_Pair(B1,product_prod(C,D)),B2),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),C2),D1)))))
=> pp(aa(product_prod(A2,product_prod(B1,product_prod(C,D))),bool,P1,X1)) ) ).
tff(fact_30_prod__cases4,axiom,
! [A2: $tType,B1: $tType,C: $tType,D: $tType,Y3: product_prod(A2,product_prod(B1,product_prod(C,D)))] :
~ ! [A3: A2,B2: B1,C2: C,D1: D] : Y3 != aa(product_prod(B1,product_prod(C,D)),product_prod(A2,product_prod(B1,product_prod(C,D))),aa(A2,fun(product_prod(B1,product_prod(C,D)),product_prod(A2,product_prod(B1,product_prod(C,D)))),product_Pair(A2,product_prod(B1,product_prod(C,D))),A3),aa(product_prod(C,D),product_prod(B1,product_prod(C,D)),aa(B1,fun(product_prod(C,D),product_prod(B1,product_prod(C,D))),product_Pair(B1,product_prod(C,D)),B2),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),C2),D1))) ).
tff(fact_31_prod__induct3,axiom,
! [C: $tType,B1: $tType,A2: $tType,X1: product_prod(A2,product_prod(B1,C)),P1: fun(product_prod(A2,product_prod(B1,C)),bool)] :
( ! [A3: A2,B2: B1,C2: C] : pp(aa(product_prod(A2,product_prod(B1,C)),bool,P1,aa(product_prod(B1,C),product_prod(A2,product_prod(B1,C)),aa(A2,fun(product_prod(B1,C),product_prod(A2,product_prod(B1,C))),product_Pair(A2,product_prod(B1,C)),A3),aa(C,product_prod(B1,C),aa(B1,fun(C,product_prod(B1,C)),product_Pair(B1,C),B2),C2))))
=> pp(aa(product_prod(A2,product_prod(B1,C)),bool,P1,X1)) ) ).
tff(fact_32_internal__split__conv,axiom,
! [B1: $tType,A2: $tType,C: $tType,B3: C,A4: B1,C1: fun(B1,fun(C,A2))] : aa(product_prod(B1,C),A2,aa(fun(B1,fun(C,A2)),fun(product_prod(B1,C),A2),produc1605651328_split(B1,C,A2),C1),aa(C,product_prod(B1,C),aa(B1,fun(C,product_prod(B1,C)),product_Pair(B1,C),A4),B3)) = aa(C,A2,aa(B1,fun(C,A2),C1,A4),B3) ).
tff(fact_33_curry__conv,axiom,
! [A2: $tType,B1: $tType,C: $tType,B3: C,A4: B1,F: fun(product_prod(B1,C),A2)] : aa(C,A2,aa(B1,fun(C,A2),product_curry(B1,C,A2,F),A4),B3) = aa(product_prod(B1,C),A2,F,aa(C,product_prod(B1,C),aa(B1,fun(C,product_prod(B1,C)),product_Pair(B1,C),A4),B3)) ).
tff(fact_34_curryI,axiom,
! [A2: $tType,B1: $tType,B3: B1,A4: A2,F: fun(product_prod(A2,B1),bool)] :
( pp(aa(product_prod(A2,B1),bool,F,aa(B1,product_prod(A2,B1),aa(A2,fun(B1,product_prod(A2,B1)),product_Pair(A2,B1),A4),B3)))
=> pp(aa(B1,bool,aa(A2,fun(B1,bool),product_curry(A2,B1,bool,F),A4),B3)) ) ).
tff(fact_35_irrefl__def,axiom,
! [A2: $tType,R: fun(product_prod(A2,A2),bool)] :
( irrefl(A2,R)
<=> ! [X3: A2] : ~ member(product_prod(A2,A2),aa(A2,product_prod(A2,A2),aa(A2,fun(A2,product_prod(A2,A2)),product_Pair(A2,A2),X3),X3),R) ) ).
tff(fact_36_in__inv__image,axiom,
! [A2: $tType,B1: $tType,F: fun(A2,B1),R: fun(product_prod(B1,B1),bool),Y2: A2,X1: A2] :
( member(product_prod(A2,A2),aa(A2,product_prod(A2,A2),aa(A2,fun(A2,product_prod(A2,A2)),product_Pair(A2,A2),X1),Y2),inv_image(B1,A2,R,F))
<=> member(product_prod(B1,B1),aa(B1,product_prod(B1,B1),aa(B1,fun(B1,product_prod(B1,B1)),product_Pair(B1,B1),aa(A2,B1,F,X1)),aa(A2,B1,F,Y2)),R) ) ).
tff(fact_37_trans__inv__image,axiom,
! [A2: $tType,B1: $tType,F: fun(B1,A2),R: fun(product_prod(A2,A2),bool)] :
( trans(A2,R)
=> trans(B1,inv_image(A2,B1,R,F)) ) ).
tff(fact_38_curryD,axiom,
! [A2: $tType,B1: $tType,B3: B1,A4: A2,F: fun(product_prod(A2,B1),bool)] :
( pp(aa(B1,bool,aa(A2,fun(B1,bool),product_curry(A2,B1,bool,F),A4),B3))
=> pp(aa(product_prod(A2,B1),bool,F,aa(B1,product_prod(A2,B1),aa(A2,fun(B1,product_prod(A2,B1)),product_Pair(A2,B1),A4),B3))) ) ).
tff(fact_39_curryE,axiom,
! [A2: $tType,B1: $tType,B3: B1,A4: A2,F: fun(product_prod(A2,B1),bool)] :
( pp(aa(B1,bool,aa(A2,fun(B1,bool),product_curry(A2,B1,bool,F),A4),B3))
=> pp(aa(product_prod(A2,B1),bool,F,aa(B1,product_prod(A2,B1),aa(A2,fun(B1,product_prod(A2,B1)),product_Pair(A2,B1),A4),B3))) ) ).
tff(fact_40_internal__split__def,axiom,
! [C: $tType,B1: $tType,A2: $tType] : produc1605651328_split(A2,B1,C) = product_prod_case(A2,B1,C) ).
tff(fact_41_apsnd__conv,axiom,
! [A2: $tType,B1: $tType,C: $tType,Y2: C,X1: A2,F: fun(C,B1)] : product_apsnd(C,B1,A2,F,aa(C,product_prod(A2,C),aa(A2,fun(C,product_prod(A2,C)),product_Pair(A2,C),X1),Y2)) = aa(B1,product_prod(A2,B1),aa(A2,fun(B1,product_prod(A2,B1)),product_Pair(A2,B1),X1),aa(C,B1,F,Y2)) ).
tff(fact_42_strict__linear__order__on__def,axiom,
! [A2: $tType,R: fun(product_prod(A2,A2),bool),A1: fun(A2,bool)] :
( order_215145569der_on(A2,A1,R)
<=> ( trans(A2,R)
& irrefl(A2,R)
& total_on(A2,A1,R) ) ) ).
tff(fact_43_apfst__conv,axiom,
! [C: $tType,A2: $tType,B1: $tType,Y2: B1,X1: C,F: fun(C,A2)] : product_apfst(C,A2,B1,F,aa(B1,product_prod(C,B1),aa(C,fun(B1,product_prod(C,B1)),product_Pair(C,B1),X1),Y2)) = aa(B1,product_prod(A2,B1),aa(A2,fun(B1,product_prod(A2,B1)),product_Pair(A2,B1),aa(C,A2,F,X1)),Y2) ).
tff(fact_44_surj__pair,axiom,
! [A2: $tType,B1: $tType,P2: product_prod(A2,B1)] :
? [X2: A2,Y1: B1] : P2 = aa(B1,product_prod(A2,B1),aa(A2,fun(B1,product_prod(A2,B1)),product_Pair(A2,B1),X2),Y1) ).
tff(fact_45_prod__caseI,axiom,
! [A2: $tType,B1: $tType,B3: B1,A4: A2,F1: fun(A2,fun(B1,bool))] :
( pp(aa(B1,bool,aa(A2,fun(B1,bool),F1,A4),B3))
=> pp(aa(product_prod(A2,B1),bool,aa(fun(A2,fun(B1,bool)),fun(product_prod(A2,B1),bool),product_prod_case(A2,B1,bool),F1),aa(B1,product_prod(A2,B1),aa(A2,fun(B1,product_prod(A2,B1)),product_Pair(A2,B1),A4),B3))) ) ).
tff(fact_46_splitI,axiom,
! [A2: $tType,B1: $tType,B3: B1,A4: A2,F: fun(A2,fun(B1,bool))] :
( pp(aa(B1,bool,aa(A2,fun(B1,bool),F,A4),B3))
=> pp(aa(product_prod(A2,B1),bool,aa(fun(A2,fun(B1,bool)),fun(product_prod(A2,B1),bool),product_prod_case(A2,B1,bool),F),aa(B1,product_prod(A2,B1),aa(A2,fun(B1,product_prod(A2,B1)),product_Pair(A2,B1),A4),B3))) ) ).
tff(fact_47_mem__splitI,axiom,
! [A2: $tType,B1: $tType,C: $tType,B3: C,A4: B1,C1: fun(B1,fun(C,fun(A2,bool))),Z: A2] :
( member(A2,Z,aa(C,fun(A2,bool),aa(B1,fun(C,fun(A2,bool)),C1,A4),B3))
=> member(A2,Z,aa(product_prod(B1,C),fun(A2,bool),aa(fun(B1,fun(C,fun(A2,bool))),fun(product_prod(B1,C),fun(A2,bool)),product_prod_case(B1,C,fun(A2,bool)),C1),aa(C,product_prod(B1,C),aa(B1,fun(C,product_prod(B1,C)),product_Pair(B1,C),A4),B3))) ) ).
tff(fact_48_split__conv,axiom,
! [B1: $tType,A2: $tType,C: $tType,B3: C,A4: B1,F: fun(B1,fun(C,A2))] : aa(product_prod(B1,C),A2,aa(fun(B1,fun(C,A2)),fun(product_prod(B1,C),A2),product_prod_case(B1,C,A2),F),aa(C,product_prod(B1,C),aa(B1,fun(C,product_prod(B1,C)),product_Pair(B1,C),A4),B3)) = aa(C,A2,aa(B1,fun(C,A2),F,A4),B3) ).
tff(fact_49_splitD,axiom,
! [A2: $tType,B1: $tType,B3: B1,A4: A2,F: fun(A2,fun(B1,bool))] :
( pp(aa(product_prod(A2,B1),bool,aa(fun(A2,fun(B1,bool)),fun(product_prod(A2,B1),bool),product_prod_case(A2,B1,bool),F),aa(B1,product_prod(A2,B1),aa(A2,fun(B1,product_prod(A2,B1)),product_Pair(A2,B1),A4),B3)))
=> pp(aa(B1,bool,aa(A2,fun(B1,bool),F,A4),B3)) ) ).
tff(fact_50_split__weak__cong,axiom,
! [C: $tType,B1: $tType,A2: $tType,C1: fun(A2,fun(B1,C)),Q: product_prod(A2,B1),P: product_prod(A2,B1)] :
( ( P = Q )
=> ( aa(product_prod(A2,B1),C,aa(fun(A2,fun(B1,C)),fun(product_prod(A2,B1),C),product_prod_case(A2,B1,C),C1),P) = aa(product_prod(A2,B1),C,aa(fun(A2,fun(B1,C)),fun(product_prod(A2,B1),C),product_prod_case(A2,B1,C),C1),Q) ) ) ).
tff(fact_51_apsnd__apfst__commute,axiom,
! [A2: $tType,B1: $tType,C: $tType,D: $tType,P: product_prod(D,C),G: fun(D,A2),F: fun(C,B1)] : product_apsnd(C,B1,A2,F,product_apfst(D,A2,C,G,P)) = product_apfst(D,A2,B1,G,product_apsnd(C,B1,D,F,P)) ).
tff(fact_52_splitD_H,axiom,
! [B1: $tType,A2: $tType,C: $tType,C1: C,B3: B1,A4: A2,R1: fun(A2,fun(B1,fun(C,bool)))] :
( pp(aa(C,bool,aa(product_prod(A2,B1),fun(C,bool),aa(fun(A2,fun(B1,fun(C,bool))),fun(product_prod(A2,B1),fun(C,bool)),product_prod_case(A2,B1,fun(C,bool)),R1),aa(B1,product_prod(A2,B1),aa(A2,fun(B1,product_prod(A2,B1)),product_Pair(A2,B1),A4),B3)),C1))
=> pp(aa(C,bool,aa(B1,fun(C,bool),aa(A2,fun(B1,fun(C,bool)),R1,A4),B3),C1)) ) ).
tff(fact_53_prod_Osimps_I2_J,axiom,
! [B1: $tType,A2: $tType,C: $tType,B3: C,A4: B1,F1: fun(B1,fun(C,A2))] : aa(product_prod(B1,C),A2,aa(fun(B1,fun(C,A2)),fun(product_prod(B1,C),A2),product_prod_case(B1,C,A2),F1),aa(C,product_prod(B1,C),aa(B1,fun(C,product_prod(B1,C)),product_Pair(B1,C),A4),B3)) = aa(C,A2,aa(B1,fun(C,A2),F1,A4),B3) ).
tff(fact_54_split__curry,axiom,
! [C: $tType,B1: $tType,A2: $tType,F: fun(product_prod(A2,B1),C)] : aa(fun(A2,fun(B1,C)),fun(product_prod(A2,B1),C),product_prod_case(A2,B1,C),product_curry(A2,B1,C,F)) = F ).
tff(fact_55_curry__split,axiom,
! [C: $tType,B1: $tType,A2: $tType,F: fun(A2,fun(B1,C))] : product_curry(A2,B1,C,aa(fun(A2,fun(B1,C)),fun(product_prod(A2,B1),C),product_prod_case(A2,B1,C),F)) = F ).
tff(fact_56_total__on__def,axiom,
! [A2: $tType,R: fun(product_prod(A2,A2),bool),A1: fun(A2,bool)] :
( total_on(A2,A1,R)
<=> ! [X3: A2] :
( member(A2,X3,A1)
=> ! [Xa1: A2] :
( member(A2,Xa1,A1)
=> ( ( X3 != Xa1 )
=> ( member(product_prod(A2,A2),aa(A2,product_prod(A2,A2),aa(A2,fun(A2,product_prod(A2,A2)),product_Pair(A2,A2),X3),Xa1),R)
| member(product_prod(A2,A2),aa(A2,product_prod(A2,A2),aa(A2,fun(A2,product_prod(A2,A2)),product_Pair(A2,A2),Xa1),X3),R) ) ) ) ) ) ).
tff(fact_57_mem__splitI2,axiom,
! [C: $tType,B1: $tType,A2: $tType,C1: fun(A2,fun(B1,fun(C,bool))),Z: C,P: product_prod(A2,B1)] :
( ! [A3: A2,B2: B1] :
( ( P = aa(B1,product_prod(A2,B1),aa(A2,fun(B1,product_prod(A2,B1)),product_Pair(A2,B1),A3),B2) )
=> member(C,Z,aa(B1,fun(C,bool),aa(A2,fun(B1,fun(C,bool)),C1,A3),B2)) )
=> member(C,Z,aa(product_prod(A2,B1),fun(C,bool),aa(fun(A2,fun(B1,fun(C,bool))),fun(product_prod(A2,B1),fun(C,bool)),product_prod_case(A2,B1,fun(C,bool)),C1),P)) ) ).
tff(fact_58_splitI2_H,axiom,
! [A2: $tType,B1: $tType,C: $tType,X1: C,C1: fun(A2,fun(B1,fun(C,bool))),P: product_prod(A2,B1)] :
( ! [A3: A2,B2: B1] :
( ( aa(B1,product_prod(A2,B1),aa(A2,fun(B1,product_prod(A2,B1)),product_Pair(A2,B1),A3),B2) = P )
=> pp(aa(C,bool,aa(B1,fun(C,bool),aa(A2,fun(B1,fun(C,bool)),C1,A3),B2),X1)) )
=> pp(aa(C,bool,aa(product_prod(A2,B1),fun(C,bool),aa(fun(A2,fun(B1,fun(C,bool))),fun(product_prod(A2,B1),fun(C,bool)),product_prod_case(A2,B1,fun(C,bool)),C1),P),X1)) ) ).
tff(fact_59_mem__splitE,axiom,
! [B1: $tType,A2: $tType,C: $tType,P: product_prod(B1,C),C1: fun(B1,fun(C,fun(A2,bool))),Z: A2] :
( member(A2,Z,aa(product_prod(B1,C),fun(A2,bool),aa(fun(B1,fun(C,fun(A2,bool))),fun(product_prod(B1,C),fun(A2,bool)),product_prod_case(B1,C,fun(A2,bool)),C1),P))
=> ~ ! [X2: B1,Y1: C] :
( ( P = aa(C,product_prod(B1,C),aa(B1,fun(C,product_prod(B1,C)),product_Pair(B1,C),X2),Y1) )
=> ~ member(A2,Z,aa(C,fun(A2,bool),aa(B1,fun(C,fun(A2,bool)),C1,X2),Y1)) ) ) ).
tff(fact_60_splitE_H,axiom,
! [B1: $tType,A2: $tType,C: $tType,Z: C,P: product_prod(A2,B1),C1: fun(A2,fun(B1,fun(C,bool)))] :
( pp(aa(C,bool,aa(product_prod(A2,B1),fun(C,bool),aa(fun(A2,fun(B1,fun(C,bool))),fun(product_prod(A2,B1),fun(C,bool)),product_prod_case(A2,B1,fun(C,bool)),C1),P),Z))
=> ~ ! [X2: A2,Y1: B1] :
( ( P = aa(B1,product_prod(A2,B1),aa(A2,fun(B1,product_prod(A2,B1)),product_Pair(A2,B1),X2),Y1) )
=> ~ pp(aa(C,bool,aa(B1,fun(C,bool),aa(A2,fun(B1,fun(C,bool)),C1,X2),Y1),Z)) ) ) ).
tff(fact_61_splitE,axiom,
! [A2: $tType,B1: $tType,P: product_prod(A2,B1),C1: fun(A2,fun(B1,bool))] :
( pp(aa(product_prod(A2,B1),bool,aa(fun(A2,fun(B1,bool)),fun(product_prod(A2,B1),bool),product_prod_case(A2,B1,bool),C1),P))
=> ~ ! [X2: A2,Y1: B1] :
( ( P = aa(B1,product_prod(A2,B1),aa(A2,fun(B1,product_prod(A2,B1)),product_Pair(A2,B1),X2),Y1) )
=> ~ pp(aa(B1,bool,aa(A2,fun(B1,bool),C1,X2),Y1)) ) ) ).
tff(fact_62_splitI2,axiom,
! [B1: $tType,A2: $tType,C1: fun(A2,fun(B1,bool)),P: product_prod(A2,B1)] :
( ! [A3: A2,B2: B1] :
( ( P = aa(B1,product_prod(A2,B1),aa(A2,fun(B1,product_prod(A2,B1)),product_Pair(A2,B1),A3),B2) )
=> pp(aa(B1,bool,aa(A2,fun(B1,bool),C1,A3),B2)) )
=> pp(aa(product_prod(A2,B1),bool,aa(fun(A2,fun(B1,bool)),fun(product_prod(A2,B1),bool),product_prod_case(A2,B1,bool),C1),P)) ) ).
tff(fact_63_splitE2,axiom,
! [B1: $tType,A2: $tType,C: $tType,Z: product_prod(B1,C),P1: fun(B1,fun(C,A2)),Q1: fun(A2,bool)] :
( pp(aa(A2,bool,Q1,aa(product_prod(B1,C),A2,aa(fun(B1,fun(C,A2)),fun(product_prod(B1,C),A2),product_prod_case(B1,C,A2),P1),Z)))
=> ~ ! [X2: B1,Y1: C] :
( ( Z = aa(C,product_prod(B1,C),aa(B1,fun(C,product_prod(B1,C)),product_Pair(B1,C),X2),Y1) )
=> ~ pp(aa(A2,bool,Q1,aa(C,A2,aa(B1,fun(C,A2),P1,X2),Y1))) ) ) ).
tff(fact_64_split__cong,axiom,
! [C: $tType,B1: $tType,A2: $tType,P: product_prod(A2,B1),G: fun(A2,fun(B1,C)),F: fun(A2,fun(B1,C)),Q: product_prod(A2,B1)] :
( ! [X2: A2,Y1: B1] :
( ( aa(B1,product_prod(A2,B1),aa(A2,fun(B1,product_prod(A2,B1)),product_Pair(A2,B1),X2),Y1) = Q )
=> ( aa(B1,C,aa(A2,fun(B1,C),F,X2),Y1) = aa(B1,C,aa(A2,fun(B1,C),G,X2),Y1) ) )
=> ( ( P = Q )
=> ( aa(product_prod(A2,B1),C,aa(fun(A2,fun(B1,C)),fun(product_prod(A2,B1),C),product_prod_case(A2,B1,C),F),P) = aa(product_prod(A2,B1),C,aa(fun(A2,fun(B1,C)),fun(product_prod(A2,B1),C),product_prod_case(A2,B1,C),G),Q) ) ) ) ).
tff(fact_65_apfst__convE,axiom,
! [C: $tType,A2: $tType,B1: $tType,P: product_prod(C,B1),F: fun(C,A2),Q: product_prod(A2,B1)] :
( ( Q = product_apfst(C,A2,B1,F,P) )
=> ~ ! [X2: C,Y1: B1] :
( ( P = aa(B1,product_prod(C,B1),aa(C,fun(B1,product_prod(C,B1)),product_Pair(C,B1),X2),Y1) )
=> ( Q != aa(B1,product_prod(A2,B1),aa(A2,fun(B1,product_prod(A2,B1)),product_Pair(A2,B1),aa(C,A2,F,X2)),Y1) ) ) ) ).
tff(fact_66_scomp__apply,axiom,
! [A2: $tType,C: $tType,D: $tType,B1: $tType,X1: B1,G: fun(C,fun(D,A2)),F: fun(B1,product_prod(C,D))] : aa(B1,A2,product_scomp(B1,C,D,A2,F,G),X1) = aa(product_prod(C,D),A2,aa(fun(C,fun(D,A2)),fun(product_prod(C,D),A2),product_prod_case(C,D,A2),G),aa(B1,product_prod(C,D),F,X1)) ).
tff(fact_67_scomp__Pair,axiom,
! [C: $tType,B1: $tType,A2: $tType,X1: fun(A2,product_prod(B1,C))] : product_scomp(A2,B1,C,product_prod(B1,C),X1,product_Pair(B1,C)) = X1 ).
tff(fact_68_Pair__scomp,axiom,
! [A2: $tType,B1: $tType,C: $tType,F: fun(C,fun(A2,B1)),X1: C] : product_scomp(A2,C,A2,B1,aa(C,fun(A2,product_prod(C,A2)),product_Pair(C,A2),X1),F) = aa(C,fun(A2,B1),F,X1) ).
tff(fact_69_scomp__def,axiom,
! [B1: $tType,C: $tType,D: $tType,A2: $tType,G: fun(C,fun(D,B1)),F: fun(A2,product_prod(C,D)),X: A2] : aa(A2,B1,product_scomp(A2,C,D,B1,F,G),X) = aa(product_prod(C,D),B1,aa(fun(C,fun(D,B1)),fun(product_prod(C,D),B1),product_prod_case(C,D,B1),G),aa(A2,product_prod(C,D),F,X)) ).
tff(fact_70_ext,axiom,
! [B1: $tType,A2: $tType,G: fun(A2,B1),F: fun(A2,B1)] :
( ! [X2: A2] : aa(A2,B1,F,X2) = aa(A2,B1,G,X2)
=> ( F = G ) ) ).
tff(fact_71_mem__def,axiom,
! [A2: $tType,A1: fun(A2,bool),X1: A2] :
( member(A2,X1,A1)
<=> pp(aa(A2,bool,A1,X1)) ) ).
%----Helper facts (2)
tff(help_pp_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_pp_2_1_U,axiom,
pp(fTrue) ).
%----Conjectures (5)
tff(conj_0,hypothesis,
! [X: arrow_159774573e_indi] :
( trans(arrow_411405190le_alt,aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),p,X))
& ! [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),Xa),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),p,X))
& ! [Xa: arrow_411405190le_alt,Y: arrow_411405190le_alt] :
( ( Xa != Y )
=> ( 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),Y),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),p,X))
| 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),Y),Xa),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),p,X)) ) ) ) ).
tff(conj_1,hypothesis,
member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),aa(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt),aa(arrow_411405190le_alt,fun(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt),a),b),f(p)) ).
tff(conj_2,hypothesis,
( trans(arrow_411405190le_alt,f(p))
& ! [X: 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),X),X),f(p))
& ! [X: arrow_411405190le_alt,Y: arrow_411405190le_alt] :
( ( X != Y )
=> ( 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),f(p))
| 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),Y),X),f(p)) ) ) ) ).
tff(conj_3,hypothesis,
! [A: arrow_411405190le_alt,B: arrow_411405190le_alt] :
( ( A != B )
=> ( member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),aa(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt),aa(arrow_411405190le_alt,fun(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt),A),B),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),p,i))
=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),aa(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt),aa(arrow_411405190le_alt,fun(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt),A),B),f(p)) ) ) ).
tff(conj_4,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),a),b),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),p,i)) ).
%------------------------------------------------------------------------------