TPTP Problem File: SCT200_5.p
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%------------------------------------------------------------------------------
% File : SCT200_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Social Choice Theory
% Problem : Arrow's Impossibility Theorem line 147
% 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_147 [Bla13]
% Status : Theorem
% Rating : 0.33 v7.4.0, 0.50 v7.1.0, 0.67 v6.4.0
% Syntax : Number of formulae : 131 ( 45 unt; 31 typ; 0 def)
% Number of atoms : 206 ( 112 equ)
% Maximal formula atoms : 12 ( 1 avg)
% Number of connectives : 155 ( 49 ~; 3 |; 15 &)
% ( 15 <=>; 73 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 8 avg)
% Maximal term depth : 10 ( 2 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 38 ( 20 >; 18 *; 0 +; 0 <<)
% Number of predicates : 5 ( 4 usr; 0 prp; 1-5 aty)
% Number of functors : 25 ( 25 usr; 6 con; 0-6 aty)
% Number of variables : 601 ( 555 !; 7 ?; 601 :)
% ( 39 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:18:45
%------------------------------------------------------------------------------
%----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 (27)
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:
!>[A: $tType,B: $tType] : ( ( fun(product_prod(A,B),bool) * A * B ) > $o ) ).
tff(sy_c_Fun_Ocomp,type,
comp:
!>[B: $tType,C1: $tType,A: $tType] : ( ( fun(B,C1) * fun(A,B) ) > fun(A,C1) ) ).
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_Ofst,type,
product_fst:
!>[A: $tType,B: $tType] : ( product_prod(A,B) > A ) ).
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,T2: $tType] : fun(fun(A,fun(B,T2)),fun(product_prod(A,B),T2)) ).
tff(sy_c_Product__Type_Oprod_Oprod__rec,type,
product_prod_rec:
!>[A: $tType,B: $tType,T2: $tType] : ( ( fun(A,fun(B,T2)) * product_prod(A,B) ) > T2 ) ).
tff(sy_c_Product__Type_Oscomp,type,
product_scomp:
!>[A: $tType,B: $tType,C1: $tType,D1: $tType] : ( ( fun(A,product_prod(B,C1)) * fun(B,fun(C1,D1)) ) > fun(A,D1) ) ).
tff(sy_c_Product__Type_Osnd,type,
product_snd:
!>[A: $tType,B: $tType] : ( product_prod(A,B) > B ) ).
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_R____,type,
r: 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 ).
%----Relevant facts (97)
tff(fact_0__096R_A_058_ALin_096,axiom,
member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),r,arrow_1985332922le_Lin) ).
tff(fact_1__096_B_Bthesis_O_A_I_B_BR_O_AR_A_058_ALin_A_061_061_062_Athesis_J_A_061_061_062_Athesis_096,axiom,
~ ! [R1: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)] : ~ member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),R1,arrow_1985332922le_Lin) ).
tff(fact_2_assms,axiom,
a != b ).
tff(fact_3_notin__Lin__iff,axiom,
! [Y1: 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 != Y1 )
=> ( ~ member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),aa(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt),aa(arrow_411405190le_alt,fun(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt),X1),Y1),L)
<=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),aa(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt),aa(arrow_411405190le_alt,fun(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt),Y1),X1),L) ) ) ) ).
tff(fact_4_Lin__irrefl,axiom,
! [Ba: arrow_411405190le_alt,Aa: 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),Aa),Ba),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),Ba),Aa),L) ) ) ).
tff(fact_5_split__paired__All,axiom,
! [A: $tType,B: $tType,P: fun(product_prod(A,B),bool)] :
( ! [X11: product_prod(A,B)] : pp(aa(product_prod(A,B),bool,P,X11))
<=> ! [A5: A,B4: B] : pp(aa(product_prod(A,B),bool,P,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4))) ) ).
tff(fact_6_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_7_in__above,axiom,
! [Y1: arrow_411405190le_alt,X1: arrow_411405190le_alt,L: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Ba: arrow_411405190le_alt,Aa: arrow_411405190le_alt] :
( ( Aa != Ba )
=> ( 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),Y1),arrow_1158827142_above(L,Aa,Ba))
<=> ( ( X1 != Y1 )
& ( ( X1 = Ba )
=> 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),Y1),L) )
& ( ( X1 != Ba )
=> ( ( ( Y1 = Ba )
=> ( ( X1 = Aa )
| 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),Aa),L) ) )
& ( ( Y1 != Ba )
=> 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),Y1),L) ) ) ) ) ) ) ) ).
tff(fact_8_in__below,axiom,
! [Y1: arrow_411405190le_alt,X1: arrow_411405190le_alt,L: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Ba: arrow_411405190le_alt,Aa: arrow_411405190le_alt] :
( ( Aa != Ba )
=> ( 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),Y1),arrow_319942042_below(L,Aa,Ba))
<=> ( ( X1 != Y1 )
& ( ( Y1 = Aa )
=> 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),Ba),L) )
& ( ( Y1 != Aa )
=> ( ( ( X1 = Aa )
=> ( ( Y1 = Ba )
| 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),Y1),L) ) )
& ( ( X1 != Aa )
=> 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),Y1),L) ) ) ) ) ) ) ) ).
tff(fact_9_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_10_above__Lin,axiom,
! [L: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Y1: arrow_411405190le_alt,X1: arrow_411405190le_alt] :
( ( X1 != Y1 )
=> ( member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),L,arrow_1985332922le_Lin)
=> member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),arrow_1158827142_above(L,X1,Y1),arrow_1985332922le_Lin) ) ) ).
tff(fact_11_below__Lin,axiom,
! [L: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Y1: arrow_411405190le_alt,X1: arrow_411405190le_alt] :
( ( X1 != Y1 )
=> ( member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),L,arrow_1985332922le_Lin)
=> member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),arrow_319942042_below(L,X1,Y1),arrow_1985332922le_Lin) ) ) ).
tff(fact_12_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_13_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_14_in__mkbot,axiom,
! [Z: arrow_411405190le_alt,L: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Y1: 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),Y1),arrow_276188178_mkbot(L,Z))
<=> ( ( Y1 != Z )
& ( ( X1 = Z )
=> ( X1 != Y1 ) )
& ( ( 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),Y1),L) ) ) ) ).
tff(fact_15_in__mktop,axiom,
! [Z: arrow_411405190le_alt,L: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Y1: 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),Y1),arrow_424895264_mktop(L,Z))
<=> ( ( X1 != Z )
& ( ( Y1 = Z )
=> ( X1 != Y1 ) )
& ( ( Y1 != Z )
=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),aa(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt),aa(arrow_411405190le_alt,fun(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt),X1),Y1),L) ) ) ) ).
tff(fact_16_Pair__inject,axiom,
! [A: $tType,B: $tType,B5: B,A6: A,B1: B,A1: A] :
( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A1),B1) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B5) )
=> ~ ( ( A1 = A6 )
=> ( B1 != B5 ) ) ) ).
tff(fact_17_split__paired__Ex,axiom,
! [A: $tType,B: $tType,P: fun(product_prod(A,B),bool)] :
( ? [X11: product_prod(A,B)] : pp(aa(product_prod(A,B),bool,P,X11))
<=> ? [A5: A,B4: B] : pp(aa(product_prod(A,B),bool,P,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4))) ) ).
tff(fact_18_prod_Orecs,axiom,
! [B: $tType,A: $tType,C1: $tType,Ba: C1,Aa: B,F11: fun(B,fun(C1,A))] : product_prod_rec(B,C1,A,F11,aa(C1,product_prod(B,C1),aa(B,fun(C1,product_prod(B,C1)),product_Pair(B,C1),Aa),Ba)) = aa(C1,A,aa(B,fun(C1,A),F11,Aa),Ba) ).
tff(fact_19_prod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y2: product_prod(A,B)] :
~ ! [A4: A,B3: B] : Y2 != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3) ).
tff(fact_20_PairE,axiom,
! [A: $tType,B: $tType,P2: product_prod(A,B)] :
~ ! [X3: A,Y: B] : P2 != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y) ).
tff(fact_21_prod__cases3,axiom,
! [A: $tType,B: $tType,C1: $tType,Y2: product_prod(A,product_prod(B,C1))] :
~ ! [A4: A,B3: 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)),A4),aa(C1,product_prod(B,C1),aa(B,fun(C1,product_prod(B,C1)),product_Pair(B,C1),B3),C2)) ).
tff(fact_22_prod__induct3,axiom,
! [C1: $tType,B: $tType,A: $tType,X1: product_prod(A,product_prod(B,C1)),P: fun(product_prod(A,product_prod(B,C1)),bool)] :
( ! [A4: A,B3: B,C2: C1] : pp(aa(product_prod(A,product_prod(B,C1)),bool,P,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)),A4),aa(C1,product_prod(B,C1),aa(B,fun(C1,product_prod(B,C1)),product_Pair(B,C1),B3),C2))))
=> pp(aa(product_prod(A,product_prod(B,C1)),bool,P,X1)) ) ).
tff(fact_23_prod__cases4,axiom,
! [A: $tType,B: $tType,C1: $tType,D1: $tType,Y2: product_prod(A,product_prod(B,product_prod(C1,D1)))] :
~ ! [A4: A,B3: B,C2: C1,D2: D1] : Y2 != aa(product_prod(B,product_prod(C1,D1)),product_prod(A,product_prod(B,product_prod(C1,D1))),aa(A,fun(product_prod(B,product_prod(C1,D1)),product_prod(A,product_prod(B,product_prod(C1,D1)))),product_Pair(A,product_prod(B,product_prod(C1,D1))),A4),aa(product_prod(C1,D1),product_prod(B,product_prod(C1,D1)),aa(B,fun(product_prod(C1,D1),product_prod(B,product_prod(C1,D1))),product_Pair(B,product_prod(C1,D1)),B3),aa(D1,product_prod(C1,D1),aa(C1,fun(D1,product_prod(C1,D1)),product_Pair(C1,D1),C2),D2))) ).
tff(fact_24_prod__induct4,axiom,
! [D1: $tType,C1: $tType,B: $tType,A: $tType,X1: product_prod(A,product_prod(B,product_prod(C1,D1))),P: fun(product_prod(A,product_prod(B,product_prod(C1,D1))),bool)] :
( ! [A4: A,B3: B,C2: C1,D2: D1] : pp(aa(product_prod(A,product_prod(B,product_prod(C1,D1))),bool,P,aa(product_prod(B,product_prod(C1,D1)),product_prod(A,product_prod(B,product_prod(C1,D1))),aa(A,fun(product_prod(B,product_prod(C1,D1)),product_prod(A,product_prod(B,product_prod(C1,D1)))),product_Pair(A,product_prod(B,product_prod(C1,D1))),A4),aa(product_prod(C1,D1),product_prod(B,product_prod(C1,D1)),aa(B,fun(product_prod(C1,D1),product_prod(B,product_prod(C1,D1))),product_Pair(B,product_prod(C1,D1)),B3),aa(D1,product_prod(C1,D1),aa(C1,fun(D1,product_prod(C1,D1)),product_Pair(C1,D1),C2),D2)))))
=> pp(aa(product_prod(A,product_prod(B,product_prod(C1,D1))),bool,P,X1)) ) ).
tff(fact_25_prod__induct6,axiom,
! [F2: $tType,E: $tType,D1: $tType,C1: $tType,B: $tType,A: $tType,X1: product_prod(A,product_prod(B,product_prod(C1,product_prod(D1,product_prod(E,F2))))),P: fun(product_prod(A,product_prod(B,product_prod(C1,product_prod(D1,product_prod(E,F2))))),bool)] :
( ! [A4: A,B3: B,C2: C1,D2: D1,E1: E,F3: F2] : pp(aa(product_prod(A,product_prod(B,product_prod(C1,product_prod(D1,product_prod(E,F2))))),bool,P,aa(product_prod(B,product_prod(C1,product_prod(D1,product_prod(E,F2)))),product_prod(A,product_prod(B,product_prod(C1,product_prod(D1,product_prod(E,F2))))),aa(A,fun(product_prod(B,product_prod(C1,product_prod(D1,product_prod(E,F2)))),product_prod(A,product_prod(B,product_prod(C1,product_prod(D1,product_prod(E,F2)))))),product_Pair(A,product_prod(B,product_prod(C1,product_prod(D1,product_prod(E,F2))))),A4),aa(product_prod(C1,product_prod(D1,product_prod(E,F2))),product_prod(B,product_prod(C1,product_prod(D1,product_prod(E,F2)))),aa(B,fun(product_prod(C1,product_prod(D1,product_prod(E,F2))),product_prod(B,product_prod(C1,product_prod(D1,product_prod(E,F2))))),product_Pair(B,product_prod(C1,product_prod(D1,product_prod(E,F2)))),B3),aa(product_prod(D1,product_prod(E,F2)),product_prod(C1,product_prod(D1,product_prod(E,F2))),aa(C1,fun(product_prod(D1,product_prod(E,F2)),product_prod(C1,product_prod(D1,product_prod(E,F2)))),product_Pair(C1,product_prod(D1,product_prod(E,F2))),C2),aa(product_prod(E,F2),product_prod(D1,product_prod(E,F2)),aa(D1,fun(product_prod(E,F2),product_prod(D1,product_prod(E,F2))),product_Pair(D1,product_prod(E,F2)),D2),aa(F2,product_prod(E,F2),aa(E,fun(F2,product_prod(E,F2)),product_Pair(E,F2),E1),F3)))))))
=> pp(aa(product_prod(A,product_prod(B,product_prod(C1,product_prod(D1,product_prod(E,F2))))),bool,P,X1)) ) ).
tff(fact_26_prod__cases6,axiom,
! [A: $tType,B: $tType,C1: $tType,D1: $tType,E: $tType,F2: $tType,Y2: product_prod(A,product_prod(B,product_prod(C1,product_prod(D1,product_prod(E,F2)))))] :
~ ! [A4: A,B3: B,C2: C1,D2: D1,E1: E,F3: F2] : Y2 != aa(product_prod(B,product_prod(C1,product_prod(D1,product_prod(E,F2)))),product_prod(A,product_prod(B,product_prod(C1,product_prod(D1,product_prod(E,F2))))),aa(A,fun(product_prod(B,product_prod(C1,product_prod(D1,product_prod(E,F2)))),product_prod(A,product_prod(B,product_prod(C1,product_prod(D1,product_prod(E,F2)))))),product_Pair(A,product_prod(B,product_prod(C1,product_prod(D1,product_prod(E,F2))))),A4),aa(product_prod(C1,product_prod(D1,product_prod(E,F2))),product_prod(B,product_prod(C1,product_prod(D1,product_prod(E,F2)))),aa(B,fun(product_prod(C1,product_prod(D1,product_prod(E,F2))),product_prod(B,product_prod(C1,product_prod(D1,product_prod(E,F2))))),product_Pair(B,product_prod(C1,product_prod(D1,product_prod(E,F2)))),B3),aa(product_prod(D1,product_prod(E,F2)),product_prod(C1,product_prod(D1,product_prod(E,F2))),aa(C1,fun(product_prod(D1,product_prod(E,F2)),product_prod(C1,product_prod(D1,product_prod(E,F2)))),product_Pair(C1,product_prod(D1,product_prod(E,F2))),C2),aa(product_prod(E,F2),product_prod(D1,product_prod(E,F2)),aa(D1,fun(product_prod(E,F2),product_prod(D1,product_prod(E,F2))),product_Pair(D1,product_prod(E,F2)),D2),aa(F2,product_prod(E,F2),aa(E,fun(F2,product_prod(E,F2)),product_Pair(E,F2),E1),F3))))) ).
tff(fact_27_prod__induct5,axiom,
! [E: $tType,D1: $tType,C1: $tType,B: $tType,A: $tType,X1: product_prod(A,product_prod(B,product_prod(C1,product_prod(D1,E)))),P: fun(product_prod(A,product_prod(B,product_prod(C1,product_prod(D1,E)))),bool)] :
( ! [A4: A,B3: B,C2: C1,D2: D1,E1: E] : pp(aa(product_prod(A,product_prod(B,product_prod(C1,product_prod(D1,E)))),bool,P,aa(product_prod(B,product_prod(C1,product_prod(D1,E))),product_prod(A,product_prod(B,product_prod(C1,product_prod(D1,E)))),aa(A,fun(product_prod(B,product_prod(C1,product_prod(D1,E))),product_prod(A,product_prod(B,product_prod(C1,product_prod(D1,E))))),product_Pair(A,product_prod(B,product_prod(C1,product_prod(D1,E)))),A4),aa(product_prod(C1,product_prod(D1,E)),product_prod(B,product_prod(C1,product_prod(D1,E))),aa(B,fun(product_prod(C1,product_prod(D1,E)),product_prod(B,product_prod(C1,product_prod(D1,E)))),product_Pair(B,product_prod(C1,product_prod(D1,E))),B3),aa(product_prod(D1,E),product_prod(C1,product_prod(D1,E)),aa(C1,fun(product_prod(D1,E),product_prod(C1,product_prod(D1,E))),product_Pair(C1,product_prod(D1,E)),C2),aa(E,product_prod(D1,E),aa(D1,fun(E,product_prod(D1,E)),product_Pair(D1,E),D2),E1))))))
=> pp(aa(product_prod(A,product_prod(B,product_prod(C1,product_prod(D1,E)))),bool,P,X1)) ) ).
tff(fact_28_prod__cases5,axiom,
! [A: $tType,B: $tType,C1: $tType,D1: $tType,E: $tType,Y2: product_prod(A,product_prod(B,product_prod(C1,product_prod(D1,E))))] :
~ ! [A4: A,B3: B,C2: C1,D2: D1,E1: E] : Y2 != aa(product_prod(B,product_prod(C1,product_prod(D1,E))),product_prod(A,product_prod(B,product_prod(C1,product_prod(D1,E)))),aa(A,fun(product_prod(B,product_prod(C1,product_prod(D1,E))),product_prod(A,product_prod(B,product_prod(C1,product_prod(D1,E))))),product_Pair(A,product_prod(B,product_prod(C1,product_prod(D1,E)))),A4),aa(product_prod(C1,product_prod(D1,E)),product_prod(B,product_prod(C1,product_prod(D1,E))),aa(B,fun(product_prod(C1,product_prod(D1,E)),product_prod(B,product_prod(C1,product_prod(D1,E)))),product_Pair(B,product_prod(C1,product_prod(D1,E))),B3),aa(product_prod(D1,E),product_prod(C1,product_prod(D1,E)),aa(C1,fun(product_prod(D1,E),product_prod(C1,product_prod(D1,E))),product_Pair(C1,product_prod(D1,E)),C2),aa(E,product_prod(D1,E),aa(D1,fun(E,product_prod(D1,E)),product_Pair(D1,E),D2),E1)))) ).
tff(fact_29_internal__split__conv,axiom,
! [B: $tType,A: $tType,C1: $tType,Ba: C1,Aa: 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),Aa),Ba)) = aa(C1,A,aa(B,fun(C1,A),C,Aa),Ba) ).
tff(fact_30_curry__conv,axiom,
! [A: $tType,B: $tType,C1: $tType,Ba: C1,Aa: B,F: fun(product_prod(B,C1),A)] : aa(C1,A,aa(B,fun(C1,A),product_curry(B,C1,A,F),Aa),Ba) = 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),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_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_33_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_34_internal__split__def,axiom,
! [C1: $tType,B: $tType,A: $tType] : produc1605651328_split(A,B,C1) = product_prod_case(A,B,C1) ).
tff(fact_35_in__rel__def,axiom,
! [B: $tType,A: $tType,Y1: B,X1: A,Ra: fun(product_prod(A,B),bool)] :
( in_rel(A,B,Ra,X1,Y1)
<=> 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),Ra) ) ).
tff(fact_36_apsnd__conv,axiom,
! [A: $tType,B: $tType,C1: $tType,Y1: C1,X1: 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),X1),Y1)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X1),aa(C1,B,F,Y1)) ).
tff(fact_37_apfst__conv,axiom,
! [C1: $tType,A: $tType,B: $tType,Y1: B,X1: 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),X1),Y1)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C1,A,F,X1)),Y1) ).
tff(fact_38_surj__pair,axiom,
! [A: $tType,B: $tType,P2: product_prod(A,B)] :
? [X3: A,Y: B] : P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y) ).
tff(fact_39_Nitpick_Orefl_H__def,axiom,
! [A: $tType,R: fun(product_prod(A,A),bool)] :
( refl(A,R)
<=> ! [X5: A] : member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X5),X5),R) ) ).
tff(fact_40_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_41_prod__caseI,axiom,
! [A: $tType,B: $tType,Ba: B,Aa: A,F11: fun(A,fun(B,bool))] :
( pp(aa(B,bool,aa(A,fun(B,bool),F11,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),F11),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa),Ba))) ) ).
tff(fact_42_mem__splitI,axiom,
! [A: $tType,B: $tType,C1: $tType,Ba: C1,Aa: B,C: fun(B,fun(C1,fun(A,bool))),Z: A] :
( member(A,Z,aa(C1,fun(A,bool),aa(B,fun(C1,fun(A,bool)),C,Aa),Ba))
=> member(A,Z,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),Aa),Ba))) ) ).
tff(fact_43_split__conv,axiom,
! [B: $tType,A: $tType,C1: $tType,Ba: C1,Aa: 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),Aa),Ba)) = aa(C1,A,aa(B,fun(C1,A),F,Aa),Ba) ).
tff(fact_44_apsnd__apfst__commute,axiom,
! [A: $tType,B: $tType,C1: $tType,D1: $tType,P1: product_prod(D1,C1),G: fun(D1,A),F: fun(C1,B)] : product_apsnd(C1,B,A,F,product_apfst(D1,A,C1,G,P1)) = product_apfst(D1,A,B,G,product_apsnd(C1,B,D1,F,P1)) ).
tff(fact_45_split__weak__cong,axiom,
! [C1: $tType,B: $tType,A: $tType,C: fun(A,fun(B,C1)),Q1: product_prod(A,B),P1: product_prod(A,B)] :
( ( P1 = Q1 )
=> ( 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),P1) = 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),Q1) ) ) ).
tff(fact_46_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_47_splitD_H,axiom,
! [B: $tType,A: $tType,C1: $tType,C: C1,Ba: B,Aa: A,Ra: 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)),Ra),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa),Ba)),C))
=> pp(aa(C1,bool,aa(B,fun(C1,bool),aa(A,fun(B,fun(C1,bool)),Ra,Aa),Ba),C)) ) ).
tff(fact_48_prod_Osimps_I2_J,axiom,
! [B: $tType,A: $tType,C1: $tType,Ba: C1,Aa: B,F11: 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),F11),aa(C1,product_prod(B,C1),aa(B,fun(C1,product_prod(B,C1)),product_Pair(B,C1),Aa),Ba)) = aa(C1,A,aa(B,fun(C1,A),F11,Aa),Ba) ).
tff(fact_49_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_50_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_51_splitE,axiom,
! [A: $tType,B: $tType,P1: 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),P1))
=> ~ ! [X3: A,Y: B] :
( ( P1 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y) )
=> ~ pp(aa(B,bool,aa(A,fun(B,bool),C,X3),Y)) ) ) ).
tff(fact_52_splitI2,axiom,
! [B: $tType,A: $tType,C: fun(A,fun(B,bool)),P1: product_prod(A,B)] :
( ! [A4: A,B3: B] :
( ( P1 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3) )
=> pp(aa(B,bool,aa(A,fun(B,bool),C,A4),B3)) )
=> pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_prod_case(A,B,bool),C),P1)) ) ).
tff(fact_53_mem__splitI2,axiom,
! [C1: $tType,B: $tType,A: $tType,C: fun(A,fun(B,fun(C1,bool))),Z: C1,P1: product_prod(A,B)] :
( ! [A4: A,B3: B] :
( ( P1 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3) )
=> member(C1,Z,aa(B,fun(C1,bool),aa(A,fun(B,fun(C1,bool)),C,A4),B3)) )
=> member(C1,Z,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),P1)) ) ).
tff(fact_54_splitI2_H,axiom,
! [A: $tType,B: $tType,C1: $tType,X1: C1,C: fun(A,fun(B,fun(C1,bool))),P1: product_prod(A,B)] :
( ! [A4: A,B3: B] :
( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3) = P1 )
=> pp(aa(C1,bool,aa(B,fun(C1,bool),aa(A,fun(B,fun(C1,bool)),C,A4),B3),X1)) )
=> 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),P1),X1)) ) ).
tff(fact_55_mem__splitE,axiom,
! [B: $tType,A: $tType,C1: $tType,P1: product_prod(B,C1),C: fun(B,fun(C1,fun(A,bool))),Z: A] :
( member(A,Z,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),P1))
=> ~ ! [X3: B,Y: C1] :
( ( P1 = aa(C1,product_prod(B,C1),aa(B,fun(C1,product_prod(B,C1)),product_Pair(B,C1),X3),Y) )
=> ~ member(A,Z,aa(C1,fun(A,bool),aa(B,fun(C1,fun(A,bool)),C,X3),Y)) ) ) ).
tff(fact_56_splitE_H,axiom,
! [B: $tType,A: $tType,C1: $tType,Z: C1,P1: 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),P1),Z))
=> ~ ! [X3: A,Y: B] :
( ( P1 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y) )
=> ~ pp(aa(C1,bool,aa(B,fun(C1,bool),aa(A,fun(B,fun(C1,bool)),C,X3),Y),Z)) ) ) ).
tff(fact_57_splitE2,axiom,
! [B: $tType,A: $tType,C1: $tType,Z: product_prod(B,C1),P: fun(B,fun(C1,A)),Q: fun(A,bool)] :
( pp(aa(A,bool,Q,aa(product_prod(B,C1),A,aa(fun(B,fun(C1,A)),fun(product_prod(B,C1),A),product_prod_case(B,C1,A),P),Z)))
=> ~ ! [X3: B,Y: C1] :
( ( Z = aa(C1,product_prod(B,C1),aa(B,fun(C1,product_prod(B,C1)),product_Pair(B,C1),X3),Y) )
=> ~ pp(aa(A,bool,Q,aa(C1,A,aa(B,fun(C1,A),P,X3),Y))) ) ) ).
tff(fact_58_split__cong,axiom,
! [C1: $tType,B: $tType,A: $tType,P1: product_prod(A,B),G: fun(A,fun(B,C1)),F: fun(A,fun(B,C1)),Q1: product_prod(A,B)] :
( ! [X3: A,Y: B] :
( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Y) = Q1 )
=> ( aa(B,C1,aa(A,fun(B,C1),F,X3),Y) = aa(B,C1,aa(A,fun(B,C1),G,X3),Y) ) )
=> ( ( P1 = Q1 )
=> ( 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),P1) = 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),Q1) ) ) ) ).
tff(fact_59_apfst__convE,axiom,
! [C1: $tType,A: $tType,B: $tType,P1: product_prod(C1,B),F: fun(C1,A),Q1: product_prod(A,B)] :
( ( Q1 = product_apfst(C1,A,B,F,P1) )
=> ~ ! [X3: C1,Y: B] :
( ( P1 = aa(B,product_prod(C1,B),aa(C1,fun(B,product_prod(C1,B)),product_Pair(C1,B),X3),Y) )
=> ( Q1 != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C1,A,F,X3)),Y) ) ) ) ).
tff(fact_60_scomp__apply,axiom,
! [A: $tType,C1: $tType,D1: $tType,B: $tType,X1: B,G: fun(C1,fun(D1,A)),F: fun(B,product_prod(C1,D1))] : aa(B,A,product_scomp(B,C1,D1,A,F,G),X1) = aa(product_prod(C1,D1),A,aa(fun(C1,fun(D1,A)),fun(product_prod(C1,D1),A),product_prod_case(C1,D1,A),G),aa(B,product_prod(C1,D1),F,X1)) ).
tff(fact_61_scomp__def,axiom,
! [B: $tType,C1: $tType,D1: $tType,A: $tType,G: fun(C1,fun(D1,B)),F: fun(A,product_prod(C1,D1)),X: A] : aa(A,B,product_scomp(A,C1,D1,B,F,G),X) = aa(product_prod(C1,D1),B,aa(fun(C1,fun(D1,B)),fun(product_prod(C1,D1),B),product_prod_case(C1,D1,B),G),aa(A,product_prod(C1,D1),F,X)) ).
tff(fact_62_scomp__Pair,axiom,
! [C1: $tType,B: $tType,A: $tType,X1: fun(A,product_prod(B,C1))] : product_scomp(A,B,C1,product_prod(B,C1),X1,product_Pair(B,C1)) = X1 ).
tff(fact_63_Pair__scomp,axiom,
! [A: $tType,B: $tType,C1: $tType,F: fun(C1,fun(A,B)),X1: C1] : product_scomp(A,C1,A,B,aa(C1,fun(A,product_prod(C1,A)),product_Pair(C1,A),X1),F) = aa(C1,fun(A,B),F,X1) ).
tff(fact_64_apsnd__apfst,axiom,
! [A: $tType,B: $tType,C1: $tType,D1: $tType,X1: product_prod(D1,C1),G: fun(D1,A),F: fun(C1,B)] : product_apsnd(C1,B,A,F,product_apfst(D1,A,C1,G,X1)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(D1,A,G,product_fst(D1,C1,X1))),aa(C1,B,F,product_snd(D1,C1,X1))) ).
tff(fact_65_apfst__apsnd,axiom,
! [A: $tType,B: $tType,D1: $tType,C1: $tType,X1: product_prod(C1,D1),G: fun(D1,B),F: fun(C1,A)] : product_apfst(C1,A,B,F,product_apsnd(D1,B,C1,G,X1)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(C1,A,F,product_fst(C1,D1,X1))),aa(D1,B,G,product_snd(C1,D1,X1))) ).
tff(fact_66_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_67_apfst__eq__conv,axiom,
! [A: $tType,B: $tType,C1: $tType,G: fun(C1,A),X1: product_prod(C1,B),F: fun(C1,A)] :
( ( product_apfst(C1,A,B,F,X1) = product_apfst(C1,A,B,G,X1) )
<=> ( aa(C1,A,F,product_fst(C1,B,X1)) = aa(C1,A,G,product_fst(C1,B,X1)) ) ) ).
tff(fact_68_fst__apfst,axiom,
! [A: $tType,B: $tType,C1: $tType,X1: product_prod(C1,B),F: fun(C1,A)] : product_fst(A,B,product_apfst(C1,A,B,F,X1)) = aa(C1,A,F,product_fst(C1,B,X1)) ).
tff(fact_69_fst__apsnd,axiom,
! [B: $tType,C1: $tType,A: $tType,X1: product_prod(A,C1),F: fun(C1,B)] : product_fst(A,B,product_apsnd(C1,B,A,F,X1)) = product_fst(A,C1,X1) ).
tff(fact_70_snd__apfst,axiom,
! [B: $tType,A: $tType,C1: $tType,X1: product_prod(C1,A),F: fun(C1,B)] : product_snd(B,A,product_apfst(C1,B,A,F,X1)) = product_snd(C1,A,X1) ).
tff(fact_71_snd__apsnd,axiom,
! [A: $tType,C1: $tType,B: $tType,X1: product_prod(B,C1),F: fun(C1,A)] : product_snd(B,A,product_apsnd(C1,A,B,F,X1)) = aa(C1,A,F,product_snd(B,C1,X1)) ).
tff(fact_72_apsnd__eq__conv,axiom,
! [B: $tType,C1: $tType,A: $tType,G: fun(C1,B),X1: product_prod(A,C1),F: fun(C1,B)] :
( ( product_apsnd(C1,B,A,F,X1) = product_apsnd(C1,B,A,G,X1) )
<=> ( aa(C1,B,F,product_snd(A,C1,X1)) = aa(C1,B,G,product_snd(A,C1,X1)) ) ) ).
tff(fact_73_prod__case__beta,axiom,
! [A: $tType,C1: $tType,B: $tType,P1: product_prod(B,C1),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),P1) = aa(C1,A,aa(B,fun(C1,A),F,product_fst(B,C1,P1)),product_snd(B,C1,P1)) ).
tff(fact_74_snd__eqD,axiom,
! [B: $tType,A: $tType,A1: A,Y2: A,X4: B] :
( ( product_snd(B,A,aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X4),Y2)) = A1 )
=> ( Y2 = A1 ) ) ).
tff(fact_75_ext,axiom,
! [B: $tType,A: $tType,G: fun(A,B),F: fun(A,B)] :
( ! [X3: A] : aa(A,B,F,X3) = aa(A,B,G,X3)
=> ( F = G ) ) ).
tff(fact_76_mem__def,axiom,
! [A: $tType,A2: fun(A,bool),X1: A] :
( member(A,X1,A2)
<=> pp(aa(A,bool,A2,X1)) ) ).
tff(fact_77_snd__conv,axiom,
! [B: $tType,A: $tType,B1: A,A1: B] : product_snd(B,A,aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),A1),B1)) = B1 ).
tff(fact_78_surjective__pairing,axiom,
! [B: $tType,A: $tType,T1: product_prod(A,B)] : T1 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),product_fst(A,B,T1)),product_snd(A,B,T1)) ).
tff(fact_79_pair__collapse,axiom,
! [B: $tType,A: $tType,P2: product_prod(A,B)] : aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),product_fst(A,B,P2)),product_snd(A,B,P2)) = P2 ).
tff(fact_80_prod__eq__iff,axiom,
! [B: $tType,A: $tType,T: product_prod(A,B),S: product_prod(A,B)] :
( ( S = T )
<=> ( ( product_fst(A,B,S) = product_fst(A,B,T) )
& ( product_snd(A,B,S) = product_snd(A,B,T) ) ) ) ).
tff(fact_81_prod__eqI,axiom,
! [B: $tType,A: $tType,Q2: product_prod(A,B),P2: product_prod(A,B)] :
( ( product_fst(A,B,P2) = product_fst(A,B,Q2) )
=> ( ( product_snd(A,B,P2) = product_snd(A,B,Q2) )
=> ( P2 = Q2 ) ) ) ).
tff(fact_82_fst__eqD,axiom,
! [B: $tType,A: $tType,A1: A,Y2: B,X4: A] :
( ( product_fst(A,B,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Y2)) = A1 )
=> ( X4 = A1 ) ) ).
tff(fact_83_fst__conv,axiom,
! [B: $tType,A: $tType,B1: B,A1: A] : product_fst(A,B,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A1),B1)) = A1 ).
tff(fact_84_conjI__realizer,axiom,
! [A: $tType,B: $tType,Q1: B,Q: fun(B,bool),P1: A,P: fun(A,bool)] :
( pp(aa(A,bool,P,P1))
=> ( pp(aa(B,bool,Q,Q1))
=> ( pp(aa(A,bool,P,product_fst(A,B,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),P1),Q1))))
& pp(aa(B,bool,Q,product_snd(A,B,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),P1),Q1)))) ) ) ) ).
tff(fact_85_exI__realizer,axiom,
! [B: $tType,A: $tType,X1: B,Y1: A,P: fun(A,fun(B,bool))] :
( pp(aa(B,bool,aa(A,fun(B,bool),P,Y1),X1))
=> pp(aa(B,bool,aa(A,fun(B,bool),P,product_snd(B,A,aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X1),Y1))),product_fst(B,A,aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X1),Y1)))) ) ).
tff(fact_86_exE__realizer_H,axiom,
! [A: $tType,B: $tType,P1: product_prod(B,A),P: fun(A,fun(B,bool))] :
( pp(aa(B,bool,aa(A,fun(B,bool),P,product_snd(B,A,P1)),product_fst(B,A,P1)))
=> ~ ! [X3: B,Y: A] : ~ pp(aa(B,bool,aa(A,fun(B,bool),P,Y),X3)) ) ).
tff(fact_87_split__comp,axiom,
! [D1: $tType,A: $tType,C1: $tType,B: $tType,X1: product_prod(B,C1),G: fun(B,D1),F: fun(D1,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),comp(D1,fun(C1,A),B,F,G)),X1) = aa(C1,A,aa(D1,fun(C1,A),F,aa(B,D1,G,product_fst(B,C1,X1))),product_snd(B,C1,X1)) ).
tff(fact_88_apsnd__compose,axiom,
! [C1: $tType,B: $tType,D1: $tType,A: $tType,X1: product_prod(A,D1),G: fun(D1,C1),F: fun(C1,B)] : product_apsnd(C1,B,A,F,product_apsnd(D1,C1,A,G,X1)) = product_apsnd(D1,B,A,comp(C1,B,D1,F,G),X1) ).
tff(fact_89_apfst__compose,axiom,
! [C1: $tType,A: $tType,B: $tType,D1: $tType,X1: product_prod(D1,B),G: fun(D1,C1),F: fun(C1,A)] : product_apfst(C1,A,B,F,product_apfst(D1,C1,B,G,X1)) = product_apfst(D1,A,B,comp(C1,A,D1,F,G),X1) ).
tff(fact_90_comp__cong,axiom,
! [C1: $tType,B: $tType,D1: $tType,A: $tType,E: $tType,X2: E,G1: fun(E,D1),F1: fun(D1,A),X1: C1,G: fun(C1,B),F: fun(B,A)] :
( ( aa(B,A,F,aa(C1,B,G,X1)) = aa(D1,A,F1,aa(E,D1,G1,X2)) )
=> ( aa(C1,A,comp(B,A,C1,F,G),X1) = aa(E,A,comp(D1,A,E,F1,G1),X2) ) ) ).
tff(fact_91_o__apply,axiom,
! [A: $tType,B: $tType,C1: $tType,X1: C1,G: fun(C1,B),F: fun(B,A)] : aa(C1,A,comp(B,A,C1,F,G),X1) = aa(B,A,F,aa(C1,B,G,X1)) ).
tff(fact_92_o__eq__elim,axiom,
! [C1: $tType,B: $tType,D1: $tType,A: $tType,D: fun(A,D1),C: fun(D1,B),Ba: fun(A,C1),Aa: fun(C1,B)] :
( ( comp(C1,B,A,Aa,Ba) = comp(D1,B,A,C,D) )
=> ! [V1: A] : aa(C1,B,Aa,aa(A,C1,Ba,V1)) = aa(D1,B,C,aa(A,D1,D,V1)) ) ).
tff(fact_93_o__def,axiom,
! [B: $tType,C1: $tType,A: $tType,G: fun(A,C1),F: fun(C1,B),X: A] : aa(A,B,comp(C1,B,A,F,G),X) = aa(C1,B,F,aa(A,C1,G,X)) ).
tff(fact_94_o__assoc,axiom,
! [C1: $tType,B: $tType,D1: $tType,A: $tType,H: fun(A,D1),G: fun(D1,C1),F: fun(C1,B)] : comp(C1,B,A,F,comp(D1,C1,A,G,H)) = comp(D1,B,A,comp(C1,B,D1,F,G),H) ).
tff(fact_95_o__eq__dest,axiom,
! [C1: $tType,B: $tType,D1: $tType,A: $tType,V: A,D: fun(A,D1),C: fun(D1,B),Ba: fun(A,C1),Aa: fun(C1,B)] :
( ( comp(C1,B,A,Aa,Ba) = comp(D1,B,A,C,D) )
=> ( aa(C1,B,Aa,aa(A,C1,Ba,V)) = aa(D1,B,C,aa(A,D1,D,V)) ) ) ).
tff(fact_96_o__eq__dest__lhs,axiom,
! [C1: $tType,B: $tType,A: $tType,V: A,C: fun(A,B),Ba: fun(A,C1),Aa: fun(C1,B)] :
( ( comp(C1,B,A,Aa,Ba) = C )
=> ( aa(C1,B,Aa,aa(A,C1,Ba,V)) = aa(A,B,C,V) ) ) ).
%----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,
? [X: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)] :
( member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),X,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),a),b),X) ) ).
%------------------------------------------------------------------------------