TPTP Problem File: SCT229_5.p
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%------------------------------------------------------------------------------
% File : SCT229_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Social Choice Theory
% Problem : Arrow's Impossibility Theorem line 230
% 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_230 [Bla13]
% Status : Theorem
% Rating : 0.00 v6.4.0
% Syntax : Number of formulae : 145 ( 40 unt; 38 typ; 0 def)
% Number of atoms : 262 ( 97 equ)
% Maximal formula atoms : 12 ( 1 avg)
% Number of connectives : 217 ( 62 ~; 3 |; 13 &)
% ( 31 <=>; 108 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 7 avg)
% Maximal term depth : 7 ( 2 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 40 ( 22 >; 18 *; 0 +; 0 <<)
% Number of predicates : 11 ( 9 usr; 1 prp; 0-5 aty)
% Number of functors : 26 ( 26 usr; 10 con; 0-5 aty)
% Number of variables : 517 ( 482 !; 5 ?; 517 :)
% ( 30 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:23:06
%------------------------------------------------------------------------------
%----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 (33)
tff(sy_c_Arrow__Order__Mirabelle__qkbtqzkjxu_OIIA,type,
arrow_1958449194le_IIA: fun(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)) > $o ).
tff(sy_c_Arrow__Order__Mirabelle__qkbtqzkjxu_OLin,type,
arrow_1985332922le_Lin: fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool) ).
tff(sy_c_Arrow__Order__Mirabelle__qkbtqzkjxu_OProf,type,
arrow_610318064e_Prof: fun(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),bool) ).
tff(sy_c_Arrow__Order__Mirabelle__qkbtqzkjxu_Oabove,type,
arrow_1158827142_above: ( fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool) * arrow_411405190le_alt * arrow_411405190le_alt ) > fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool) ).
tff(sy_c_Arrow__Order__Mirabelle__qkbtqzkjxu_Obelow,type,
arrow_319942042_below: ( fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool) * arrow_411405190le_alt * arrow_411405190le_alt ) > fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool) ).
tff(sy_c_Arrow__Order__Mirabelle__qkbtqzkjxu_Odictator,type,
arrow_987702531ctator: ( fun(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)) * arrow_159774573e_indi ) > $o ).
tff(sy_c_Arrow__Order__Mirabelle__qkbtqzkjxu_Omkbot,type,
arrow_276188178_mkbot: ( fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool) * arrow_411405190le_alt ) > fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool) ).
tff(sy_c_Arrow__Order__Mirabelle__qkbtqzkjxu_Omktop,type,
arrow_424895264_mktop: ( fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool) * arrow_411405190le_alt ) > fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool) ).
tff(sy_c_Arrow__Order__Mirabelle__qkbtqzkjxu_Ounanimity,type,
arrow_2069624013nimity: fun(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)) > $o ).
tff(sy_c_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,C: $tType,A: $tType] : ( ( fun(B,C) * fun(A,B) ) > fun(A,C) ) ).
tff(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( ( A * B ) > product_prod(A,B) ) ).
tff(sy_c_Product__Type_Ocurry,type,
product_curry:
!>[A: $tType,B: $tType,C: $tType] : ( fun(product_prod(A,B),C) > fun(A,fun(B,C)) ) ).
tff(sy_c_Product__Type_Ointernal__split,type,
produc1605651328_split:
!>[A: $tType,B: $tType,C: $tType] : fun(fun(A,fun(B,C)),fun(product_prod(A,B),C)) ).
tff(sy_c_Product__Type_Oprod_Oprod__case,type,
product_prod_case:
!>[A: $tType,B: $tType,T: $tType] : fun(fun(A,fun(B,T)),fun(product_prod(A,B),T)) ).
tff(sy_c_Product__Type_Oprod_Oprod__rec,type,
product_prod_rec:
!>[A: $tType,B: $tType,T: $tType] : ( ( fun(A,fun(B,T)) * product_prod(A,B) ) > T ) ).
tff(sy_c_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(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_Ototal__on,type,
total_on:
!>[A: $tType] : ( ( fun(A,bool) * fun(product_prod(A,A),bool) ) > $o ) ).
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_F,type,
f: fun(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)) ).
tff(sy_v_P_H____,type,
p: fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)) ).
tff(sy_v_P____,type,
p1: 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_c____,type,
c: arrow_411405190le_alt ).
%----Relevant facts (97)
tff(fact_0_A_I4_J,axiom,
member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),p1,arrow_610318064e_Prof) ).
tff(fact_1_A_I5_J,axiom,
member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),p,arrow_610318064e_Prof) ).
tff(fact_2_A_I1_J,axiom,
a != b ).
tff(fact_3_A_I2_J,axiom,
b != c ).
tff(fact_4_A_I3_J,axiom,
a != c ).
tff(fact_5_iff,axiom,
! [I3: arrow_159774573e_indi] :
( member(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),p1,I3))
<=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,b,c),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),p,I3)) ) ).
tff(fact_6_cP,axiom,
member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),comp(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),arrow_159774573e_indi,converse(arrow_411405190le_alt,arrow_411405190le_alt),p1),arrow_610318064e_Prof) ).
tff(fact_7_assms_I3_J,axiom,
arrow_1958449194le_IIA(f) ).
tff(fact_8_u,axiom,
arrow_2069624013nimity(f) ).
tff(fact_9__096ALL_Ai_O_A_Ib_A_060_092_060_094bsub_062_Iconverse_Ao_AP_J_Ai_092_060_094esub_062_Aa_J_A_061_A_Ib_A_060_092_060_094bsub_062P_H_Ai_092_060_094esub_062_Ac_J_096,axiom,
! [I3: arrow_159774573e_indi] :
( member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,b,a),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),comp(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),arrow_159774573e_indi,converse(arrow_411405190le_alt,arrow_411405190le_alt),p1),I3))
<=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,b,c),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),p,I3)) ) ).
tff(fact_10_converse__iff,axiom,
! [A: $tType,B: $tType,R: fun(product_prod(B,A),bool),Ba: B,Aa: A] :
( member(product_prod(A,B),product_Pair(A,B,Aa,Ba),aa(fun(product_prod(B,A),bool),fun(product_prod(A,B),bool),converse(B,A),R))
<=> member(product_prod(B,A),product_Pair(B,A,Ba,Aa),R) ) ).
tff(fact_11_o__apply,axiom,
! [A: $tType,B: $tType,C: $tType,X1: C,G: fun(C,B),F: fun(B,A)] : aa(C,A,comp(B,A,C,F,G),X1) = aa(B,A,F,aa(C,B,G,X1)) ).
tff(fact_12_split__paired__All,axiom,
! [A: $tType,B: $tType,Pa: fun(product_prod(A,B),bool)] :
( ! [X11: product_prod(A,B)] : pp(aa(product_prod(A,B),bool,Pa,X11))
<=> ! [A3: A,B2: B] : pp(aa(product_prod(A,B),bool,Pa,product_Pair(A,B,A3,B2))) ) ).
tff(fact_13_Pair__eq,axiom,
! [A: $tType,B: $tType,B5: B,A6: A,Ba: B,Aa: A] :
( ( product_Pair(A,B,Aa,Ba) = product_Pair(A,B,A6,B5) )
<=> ( ( Aa = A6 )
& ( Ba = B5 ) ) ) ).
tff(fact_14_converseI,axiom,
! [B: $tType,A: $tType,R: fun(product_prod(A,B),bool),Ba: B,Aa: A] :
( member(product_prod(A,B),product_Pair(A,B,Aa,Ba),R)
=> member(product_prod(B,A),product_Pair(B,A,Ba,Aa),aa(fun(product_prod(A,B),bool),fun(product_prod(B,A),bool),converse(A,B),R)) ) ).
tff(fact_15_converseD,axiom,
! [A: $tType,B: $tType,R: fun(product_prod(B,A),bool),Ba: B,Aa: A] :
( member(product_prod(A,B),product_Pair(A,B,Aa,Ba),aa(fun(product_prod(B,A),bool),fun(product_prod(A,B),bool),converse(B,A),R))
=> member(product_prod(B,A),product_Pair(B,A,Ba,Aa),R) ) ).
tff(fact_16_dictator__def,axiom,
! [I2: arrow_159774573e_indi,Fa: fun(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))] :
( arrow_987702531ctator(Fa,I2)
<=> ! [X2: fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))] :
( member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),X2,arrow_610318064e_Prof)
=> ( aa(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Fa,X2) = aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),X2,I2) ) ) ) ).
tff(fact_17_converseE,axiom,
! [A: $tType,B: $tType,R: fun(product_prod(B,A),bool),Yx: product_prod(A,B)] :
( member(product_prod(A,B),Yx,aa(fun(product_prod(B,A),bool),fun(product_prod(A,B),bool),converse(B,A),R))
=> ~ ! [X: B,Y: A] :
( ( Yx = product_Pair(A,B,Y,X) )
=> ~ member(product_prod(B,A),product_Pair(B,A,X,Y),R) ) ) ).
tff(fact_18__C2_C,axiom,
! [P_a: fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),Pa: fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),B5: arrow_411405190le_alt,A6: arrow_411405190le_alt,Ba: arrow_411405190le_alt,Aa: arrow_411405190le_alt] :
( ( Aa != Ba )
=> ( ( A6 != B5 )
=> ( ( Aa != B5 )
=> ( ( Ba != A6 )
=> ( member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),Pa,arrow_610318064e_Prof)
=> ( member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),P_a,arrow_610318064e_Prof)
=> ( ! [I1: arrow_159774573e_indi] :
( member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,Aa,Ba),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Pa,I1))
<=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,A6,B5),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),P_a,I1)) )
=> ( member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,Aa,Ba),aa(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),f,Pa))
<=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,A6,B5),aa(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),f,P_a)) ) ) ) ) ) ) ) ) ).
tff(fact_19__C1_C,axiom,
! [P_a: fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),Pa: fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),B5: arrow_411405190le_alt,A6: arrow_411405190le_alt,Ba: arrow_411405190le_alt,Aa: arrow_411405190le_alt] :
( ( Aa != Ba )
=> ( ( A6 != B5 )
=> ( ( Aa != B5 )
=> ( ( Ba != A6 )
=> ( member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),Pa,arrow_610318064e_Prof)
=> ( member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),P_a,arrow_610318064e_Prof)
=> ( ! [I1: arrow_159774573e_indi] :
( member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,Aa,Ba),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Pa,I1))
<=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,A6,B5),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),P_a,I1)) )
=> ( member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,Aa,Ba),aa(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),f,Pa))
=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,A6,B5),aa(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),f,P_a)) ) ) ) ) ) ) ) ) ).
tff(fact_20__C3_C,axiom,
! [P_a: fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),Pa: fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),Ba: arrow_411405190le_alt,Aa: arrow_411405190le_alt] :
( ( Aa != Ba )
=> ( member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),Pa,arrow_610318064e_Prof)
=> ( member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),P_a,arrow_610318064e_Prof)
=> ( ! [I1: arrow_159774573e_indi] :
( member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,Aa,Ba),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Pa,I1))
<=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,Ba,Aa),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),P_a,I1)) )
=> ( member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,Aa,Ba),aa(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),f,Pa))
<=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,Ba,Aa),aa(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),f,P_a)) ) ) ) ) ) ).
tff(fact_21_unanimity__def,axiom,
! [Fa: fun(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))] :
( arrow_2069624013nimity(Fa)
<=> ! [X2: fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))] :
( member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),X2,arrow_610318064e_Prof)
=> ! [A3: arrow_411405190le_alt,B2: arrow_411405190le_alt] :
( ! [I: arrow_159774573e_indi] : member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,A3,B2),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),X2,I))
=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,A3,B2),aa(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Fa,X2)) ) ) ) ).
tff(fact_22_Pair__inject,axiom,
! [A: $tType,B: $tType,B4: B,A5: A,B3: B,A4: A] :
( ( product_Pair(A,B,A4,B3) = product_Pair(A,B,A5,B4) )
=> ~ ( ( A4 = A5 )
=> ( B3 != B4 ) ) ) ).
tff(fact_23_o__eq__elim,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,D2: fun(A,D),Ca: fun(D,B),Ba: fun(A,C),Aa: fun(C,B)] :
( ( comp(C,B,A,Aa,Ba) = comp(D,B,A,Ca,D2) )
=> ! [V1: A] : aa(C,B,Aa,aa(A,C,Ba,V1)) = aa(D,B,Ca,aa(A,D,D2,V1)) ) ).
tff(fact_24_o__eq__dest,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,V: A,D2: fun(A,D),Ca: fun(D,B),Ba: fun(A,C),Aa: fun(C,B)] :
( ( comp(C,B,A,Aa,Ba) = comp(D,B,A,Ca,D2) )
=> ( aa(C,B,Aa,aa(A,C,Ba,V)) = aa(D,B,Ca,aa(A,D,D2,V)) ) ) ).
tff(fact_25_o__assoc,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,H: fun(A,D),G: fun(D,C),F: fun(C,B)] : comp(C,B,A,F,comp(D,C,A,G,H)) = comp(D,B,A,comp(C,B,D,F,G),H) ).
tff(fact_26_o__def,axiom,
! [B: $tType,C: $tType,A: $tType,G: fun(A,C),F: fun(C,B),X4: A] : aa(A,B,comp(C,B,A,F,G),X4) = aa(C,B,F,aa(A,C,G,X4)) ).
tff(fact_27_converse__converse,axiom,
! [B: $tType,A: $tType,R: fun(product_prod(A,B),bool)] : aa(fun(product_prod(B,A),bool),fun(product_prod(A,B),bool),converse(B,A),aa(fun(product_prod(A,B),bool),fun(product_prod(B,A),bool),converse(A,B),R)) = R ).
tff(fact_28_IIA__def,axiom,
! [Fa: fun(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))] :
( arrow_1958449194le_IIA(Fa)
<=> ! [X2: fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))] :
( member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),X2,arrow_610318064e_Prof)
=> ! [Xa: fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))] :
( member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),Xa,arrow_610318064e_Prof)
=> ! [A3: arrow_411405190le_alt,B2: arrow_411405190le_alt] :
( ! [I: arrow_159774573e_indi] :
( member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,A3,B2),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),X2,I))
<=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,A3,B2),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Xa,I)) )
=> ( member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,A3,B2),aa(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Fa,X2))
<=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,A3,B2),aa(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Fa,Xa)) ) ) ) ) ) ).
tff(fact_29_split__paired__Ex,axiom,
! [A: $tType,B: $tType,Pa: fun(product_prod(A,B),bool)] :
( ? [X11: product_prod(A,B)] : pp(aa(product_prod(A,B),bool,Pa,X11))
<=> ? [A3: A,B2: B] : pp(aa(product_prod(A,B),bool,Pa,product_Pair(A,B,A3,B2))) ) ).
tff(fact_30_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,product_Pair(B,C,Aa,Ba)) = aa(C,A,aa(B,fun(C,A),F1,Aa),Ba) ).
tff(fact_31_prod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y3: product_prod(A,B)] :
~ ! [A1: A,B1: B] : Y3 != product_Pair(A,B,A1,B1) ).
tff(fact_32_PairE,axiom,
! [A: $tType,B: $tType,P1: product_prod(A,B)] :
~ ! [X: A,Y: B] : P1 != product_Pair(A,B,X,Y) ).
tff(fact_33_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y3: product_prod(A,product_prod(B,C))] :
~ ! [A1: A,B1: B,C1: C] : Y3 != product_Pair(A,product_prod(B,C),A1,product_Pair(B,C,B1,C1)) ).
tff(fact_34_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,X1: product_prod(A,product_prod(B,C)),Pa: fun(product_prod(A,product_prod(B,C)),bool)] :
( ! [A1: A,B1: B,C1: C] : pp(aa(product_prod(A,product_prod(B,C)),bool,Pa,product_Pair(A,product_prod(B,C),A1,product_Pair(B,C,B1,C1))))
=> pp(aa(product_prod(A,product_prod(B,C)),bool,Pa,X1)) ) ).
tff(fact_35_prod__induct6,axiom,
! [F3: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,X1: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3))))),Pa: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3))))),bool)] :
( ! [A1: A,B1: B,C1: C,D1: D,E1: E,F4: F3] : pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3))))),bool,Pa,product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3)))),A1,product_Pair(B,product_prod(C,product_prod(D,product_prod(E,F3))),B1,product_Pair(C,product_prod(D,product_prod(E,F3)),C1,product_Pair(D,product_prod(E,F3),D1,product_Pair(E,F3,E1,F4)))))))
=> pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3))))),bool,Pa,X1)) ) ).
tff(fact_36_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F3: $tType,Y3: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3)))))] :
~ ! [A1: A,B1: B,C1: C,D1: D,E1: E,F4: F3] : Y3 != product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3)))),A1,product_Pair(B,product_prod(C,product_prod(D,product_prod(E,F3))),B1,product_Pair(C,product_prod(D,product_prod(E,F3)),C1,product_Pair(D,product_prod(E,F3),D1,product_Pair(E,F3,E1,F4))))) ).
tff(fact_37_prod__induct5,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,X1: product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),Pa: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),bool)] :
( ! [A1: A,B1: B,C1: C,D1: D,E1: E] : pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),bool,Pa,product_Pair(A,product_prod(B,product_prod(C,product_prod(D,E))),A1,product_Pair(B,product_prod(C,product_prod(D,E)),B1,product_Pair(C,product_prod(D,E),C1,product_Pair(D,E,D1,E1))))))
=> pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),bool,Pa,X1)) ) ).
tff(fact_38_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y3: product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))] :
~ ! [A1: A,B1: B,C1: C,D1: D,E1: E] : Y3 != product_Pair(A,product_prod(B,product_prod(C,product_prod(D,E))),A1,product_Pair(B,product_prod(C,product_prod(D,E)),B1,product_Pair(C,product_prod(D,E),C1,product_Pair(D,E,D1,E1)))) ).
tff(fact_39_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,X1: product_prod(A,product_prod(B,product_prod(C,D))),Pa: fun(product_prod(A,product_prod(B,product_prod(C,D))),bool)] :
( ! [A1: A,B1: B,C1: C,D1: D] : pp(aa(product_prod(A,product_prod(B,product_prod(C,D))),bool,Pa,product_Pair(A,product_prod(B,product_prod(C,D)),A1,product_Pair(B,product_prod(C,D),B1,product_Pair(C,D,C1,D1)))))
=> pp(aa(product_prod(A,product_prod(B,product_prod(C,D))),bool,Pa,X1)) ) ).
tff(fact_40_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y3: product_prod(A,product_prod(B,product_prod(C,D)))] :
~ ! [A1: A,B1: B,C1: C,D1: D] : Y3 != product_Pair(A,product_prod(B,product_prod(C,D)),A1,product_Pair(B,product_prod(C,D),B1,product_Pair(C,D,C1,D1))) ).
tff(fact_41_internal__split__conv,axiom,
! [B: $tType,A: $tType,C: $tType,Ba: C,Aa: B,Ca: 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),Ca),product_Pair(B,C,Aa,Ba)) = aa(C,A,aa(B,fun(C,A),Ca,Aa),Ba) ).
tff(fact_42_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,product_Pair(B,C,Aa,Ba)) ).
tff(fact_43_in__mktop,axiom,
! [Z: arrow_411405190le_alt,L1: 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),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,X1,Y1),arrow_424895264_mktop(L1,Z))
<=> ( ( X1 != Z )
& ( ( Y1 = Z )
=> ( X1 != Y1 ) )
& ( ( Y1 != Z )
=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,X1,Y1),L1) ) ) ) ).
tff(fact_44_in__mkbot,axiom,
! [Z: arrow_411405190le_alt,L1: 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),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,X1,Y1),arrow_276188178_mkbot(L1,Z))
<=> ( ( Y1 != Z )
& ( ( X1 = Z )
=> ( X1 != Y1 ) )
& ( ( X1 != Z )
=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,X1,Y1),L1) ) ) ) ).
tff(fact_45_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,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_46_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,product_Pair(A,B,Aa,Ba))) ) ).
tff(fact_47_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,product_Pair(A,B,Aa,Ba))) ) ).
tff(fact_48_converse__in__Lin,axiom,
! [L1: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)] :
( member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),converse(arrow_411405190le_alt,arrow_411405190le_alt),L1),arrow_1985332922le_Lin)
<=> member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),L1,arrow_1985332922le_Lin) ) ).
tff(fact_49_comp__cong,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,E: $tType,X3: E,G1: fun(E,D),F2: fun(D,A),X1: C,G: fun(C,B),F: fun(B,A)] :
( ( aa(B,A,F,aa(C,B,G,X1)) = aa(D,A,F2,aa(E,D,G1,X3)) )
=> ( aa(C,A,comp(B,A,C,F,G),X1) = aa(E,A,comp(D,A,E,F2,G1),X3) ) ) ).
tff(fact_50_o__eq__dest__lhs,axiom,
! [C: $tType,B: $tType,A: $tType,V: A,Ca: fun(A,B),Ba: fun(A,C),Aa: fun(C,B)] :
( ( comp(C,B,A,Aa,Ba) = Ca )
=> ( aa(C,B,Aa,aa(A,C,Ba,V)) = aa(A,B,Ca,V) ) ) ).
tff(fact_51_total__on__converse,axiom,
! [A: $tType,R: fun(product_prod(A,A),bool),A2: fun(A,bool)] :
( total_on(A,A2,aa(fun(product_prod(A,A),bool),fun(product_prod(A,A),bool),converse(A,A),R))
<=> total_on(A,A2,R) ) ).
tff(fact_52_Lin__irrefl,axiom,
! [Ba: arrow_411405190le_alt,Aa: arrow_411405190le_alt,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)
=> ( member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,Aa,Ba),L1)
=> ~ member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,Ba,Aa),L1) ) ) ).
tff(fact_53_notin__Lin__iff,axiom,
! [Y1: arrow_411405190le_alt,X1: arrow_411405190le_alt,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)
=> ( ( X1 != Y1 )
=> ( ~ member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,X1,Y1),L1)
<=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,Y1,X1),L1) ) ) ) ).
tff(fact_54_total__on__def,axiom,
! [A: $tType,R: fun(product_prod(A,A),bool),A2: fun(A,bool)] :
( total_on(A,A2,R)
<=> ! [X2: A] :
( member(A,X2,A2)
=> ! [Xa: A] :
( member(A,Xa,A2)
=> ( ( X2 != Xa )
=> ( member(product_prod(A,A),product_Pair(A,A,X2,Xa),R)
| member(product_prod(A,A),product_Pair(A,A,Xa,X2),R) ) ) ) ) ) ).
tff(fact_55_mkbot__Lin,axiom,
! [X1: arrow_411405190le_alt,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)
=> member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),arrow_276188178_mkbot(L1,X1),arrow_1985332922le_Lin) ) ).
tff(fact_56_mktop__Lin,axiom,
! [X1: arrow_411405190le_alt,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)
=> member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),arrow_424895264_mktop(L1,X1),arrow_1985332922le_Lin) ) ).
tff(fact_57_in__rel__def,axiom,
! [B: $tType,A: $tType,Y1: B,X1: A,R1: fun(product_prod(A,B),bool)] :
( in_rel(A,B,R1,X1,Y1)
<=> member(product_prod(A,B),product_Pair(A,B,X1,Y1),R1) ) ).
tff(fact_58_complete__Lin,axiom,
! [Ba: arrow_411405190le_alt,Aa: arrow_411405190le_alt] :
( ( Aa != Ba )
=> ? [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),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,Aa,Ba),X) ) ) ).
tff(fact_59_in__above,axiom,
! [Y1: arrow_411405190le_alt,X1: arrow_411405190le_alt,L1: 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),L1,arrow_1985332922le_Lin)
=> ( member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,X1,Y1),arrow_1158827142_above(L1,Aa,Ba))
<=> ( ( X1 != Y1 )
& ( ( X1 = Ba )
=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,Aa,Y1),L1) )
& ( ( X1 != Ba )
=> ( ( ( Y1 = Ba )
=> ( ( X1 = Aa )
| member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,X1,Aa),L1) ) )
& ( ( Y1 != Ba )
=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,X1,Y1),L1) ) ) ) ) ) ) ) ).
tff(fact_60_in__below,axiom,
! [Y1: arrow_411405190le_alt,X1: arrow_411405190le_alt,L1: 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),L1,arrow_1985332922le_Lin)
=> ( member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,X1,Y1),arrow_319942042_below(L1,Aa,Ba))
<=> ( ( X1 != Y1 )
& ( ( Y1 = Aa )
=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,X1,Ba),L1) )
& ( ( Y1 != Aa )
=> ( ( ( X1 = Aa )
=> ( ( Y1 = Ba )
| member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,Ba,Y1),L1) ) )
& ( ( X1 != Aa )
=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,X1,Y1),L1) ) ) ) ) ) ) ) ).
tff(fact_61_below__Lin,axiom,
! [L1: 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),L1,arrow_1985332922le_Lin)
=> member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),arrow_319942042_below(L1,X1,Y1),arrow_1985332922le_Lin) ) ) ).
tff(fact_62_above__Lin,axiom,
! [L1: 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),L1,arrow_1985332922le_Lin)
=> member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),arrow_1158827142_above(L1,X1,Y1),arrow_1985332922le_Lin) ) ) ).
tff(fact_63_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_64_converse__inv__image,axiom,
! [B: $tType,A: $tType,F: fun(A,B),R1: fun(product_prod(B,B),bool)] : aa(fun(product_prod(A,A),bool),fun(product_prod(A,A),bool),converse(A,A),inv_image(B,A,R1,F)) = inv_image(B,A,aa(fun(product_prod(B,B),bool),fun(product_prod(B,B),bool),converse(B,B),R1),F) ).
tff(fact_65_irrefl__def,axiom,
! [A: $tType,R: fun(product_prod(A,A),bool)] :
( irrefl(A,R)
<=> ! [X2: A] : ~ member(product_prod(A,A),product_Pair(A,A,X2,X2),R) ) ).
tff(fact_66_in__inv__image,axiom,
! [A: $tType,B: $tType,F: fun(A,B),R: fun(product_prod(B,B),bool),Y1: A,X1: A] :
( member(product_prod(A,A),product_Pair(A,A,X1,Y1),inv_image(B,A,R,F))
<=> member(product_prod(B,B),product_Pair(B,B,aa(A,B,F,X1),aa(A,B,F,Y1)),R) ) ).
tff(fact_67_antisym__converse,axiom,
! [A: $tType,R: fun(product_prod(A,A),bool)] :
( antisym(A,aa(fun(product_prod(A,A),bool),fun(product_prod(A,A),bool),converse(A,A),R))
<=> antisym(A,R) ) ).
tff(fact_68_converse__Id__on,axiom,
! [A: $tType,A2: fun(A,bool)] : aa(fun(product_prod(A,A),bool),fun(product_prod(A,A),bool),converse(A,A),id_on(A,A2)) = id_on(A,A2) ).
tff(fact_69_internal__split__def,axiom,
! [C: $tType,B: $tType,A: $tType] : produc1605651328_split(A,B,C) = product_prod_case(A,B,C) ).
tff(fact_70_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),product_Pair(A,B,Aa,Ba))) ) ).
tff(fact_71_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),product_Pair(A,B,Aa,Ba))) ) ).
tff(fact_72_mem__splitI,axiom,
! [A: $tType,B: $tType,C: $tType,Ba: C,Aa: B,Ca: fun(B,fun(C,fun(A,bool))),Z: A] :
( member(A,Z,aa(C,fun(A,bool),aa(B,fun(C,fun(A,bool)),Ca,Aa),Ba))
=> member(A,Z,aa(product_prod(B,C),fun(A,bool),aa(fun(B,fun(C,fun(A,bool))),fun(product_prod(B,C),fun(A,bool)),product_prod_case(B,C,fun(A,bool)),Ca),product_Pair(B,C,Aa,Ba))) ) ).
tff(fact_73_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),product_Pair(B,C,Aa,Ba)) = aa(C,A,aa(B,fun(C,A),F,Aa),Ba) ).
tff(fact_74_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),product_Pair(A,B,Aa,Ba)))
=> pp(aa(B,bool,aa(A,fun(B,bool),F,Aa),Ba)) ) ).
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,A2: fun(A,bool),X1: A] :
( member(A,X1,A2)
<=> pp(aa(A,bool,A2,X1)) ) ).
tff(fact_77_antisym__Id__on,axiom,
! [A: $tType,A2: fun(A,bool)] : antisym(A,id_on(A,A2)) ).
tff(fact_78_split__weak__cong,axiom,
! [C: $tType,B: $tType,A: $tType,Ca: 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),Ca),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),Ca),Q) ) ) ).
tff(fact_79_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),product_Pair(B,C,Aa,Ba)) = aa(C,A,aa(B,fun(C,A),F1,Aa),Ba) ).
tff(fact_80_splitD_H,axiom,
! [B: $tType,A: $tType,C: $tType,Ca: 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),product_Pair(A,B,Aa,Ba)),Ca))
=> pp(aa(C,bool,aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),R1,Aa),Ba),Ca)) ) ).
tff(fact_81_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_82_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_83_antisymD,axiom,
! [A: $tType,Ba: A,Aa: A,R: fun(product_prod(A,A),bool)] :
( antisym(A,R)
=> ( member(product_prod(A,A),product_Pair(A,A,Aa,Ba),R)
=> ( member(product_prod(A,A),product_Pair(A,A,Ba,Aa),R)
=> ( Aa = Ba ) ) ) ) ).
tff(fact_84_antisym__def,axiom,
! [A: $tType,R: fun(product_prod(A,A),bool)] :
( antisym(A,R)
<=> ! [X2: A,Y2: A] :
( member(product_prod(A,A),product_Pair(A,A,X2,Y2),R)
=> ( member(product_prod(A,A),product_Pair(A,A,Y2,X2),R)
=> ( X2 = Y2 ) ) ) ) ).
tff(fact_85_Id__on__eqI,axiom,
! [A: $tType,A2: fun(A,bool),Ba: A,Aa: A] :
( ( Aa = Ba )
=> ( member(A,Aa,A2)
=> member(product_prod(A,A),product_Pair(A,A,Aa,Ba),id_on(A,A2)) ) ) ).
tff(fact_86_Id__on__iff,axiom,
! [A: $tType,A2: fun(A,bool),Y1: A,X1: A] :
( member(product_prod(A,A),product_Pair(A,A,X1,Y1),id_on(A,A2))
<=> ( ( X1 = Y1 )
& member(A,X1,A2) ) ) ).
tff(fact_87_Id__onE,axiom,
! [A: $tType,A2: fun(A,bool),Ca: product_prod(A,A)] :
( member(product_prod(A,A),Ca,id_on(A,A2))
=> ~ ! [X: A] :
( member(A,X,A2)
=> ( Ca != product_Pair(A,A,X,X) ) ) ) ).
tff(fact_88_splitI2,axiom,
! [B: $tType,A: $tType,Ca: fun(A,fun(B,bool)),P: product_prod(A,B)] :
( ! [A1: A,B1: B] :
( ( P = product_Pair(A,B,A1,B1) )
=> pp(aa(B,bool,aa(A,fun(B,bool),Ca,A1),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),Ca),P)) ) ).
tff(fact_89_splitE,axiom,
! [A: $tType,B: $tType,P: product_prod(A,B),Ca: 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),Ca),P))
=> ~ ! [X: A,Y: B] :
( ( P = product_Pair(A,B,X,Y) )
=> ~ pp(aa(B,bool,aa(A,fun(B,bool),Ca,X),Y)) ) ) ).
tff(fact_90_mem__splitI2,axiom,
! [C: $tType,B: $tType,A: $tType,Ca: fun(A,fun(B,fun(C,bool))),Z: C,P: product_prod(A,B)] :
( ! [A1: A,B1: B] :
( ( P = product_Pair(A,B,A1,B1) )
=> member(C,Z,aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),Ca,A1),B1)) )
=> member(C,Z,aa(product_prod(A,B),fun(C,bool),aa(fun(A,fun(B,fun(C,bool))),fun(product_prod(A,B),fun(C,bool)),product_prod_case(A,B,fun(C,bool)),Ca),P)) ) ).
tff(fact_91_splitI2_H,axiom,
! [A: $tType,B: $tType,C: $tType,X1: C,Ca: fun(A,fun(B,fun(C,bool))),P: product_prod(A,B)] :
( ! [A1: A,B1: B] :
( ( product_Pair(A,B,A1,B1) = P )
=> pp(aa(C,bool,aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),Ca,A1),B1),X1)) )
=> 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)),Ca),P),X1)) ) ).
tff(fact_92_splitE_H,axiom,
! [B: $tType,A: $tType,C: $tType,Z: C,P: product_prod(A,B),Ca: 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)),Ca),P),Z))
=> ~ ! [X: A,Y: B] :
( ( P = product_Pair(A,B,X,Y) )
=> ~ pp(aa(C,bool,aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),Ca,X),Y),Z)) ) ) ).
tff(fact_93_mem__splitE,axiom,
! [B: $tType,A: $tType,C: $tType,P: product_prod(B,C),Ca: fun(B,fun(C,fun(A,bool))),Z: A] :
( member(A,Z,aa(product_prod(B,C),fun(A,bool),aa(fun(B,fun(C,fun(A,bool))),fun(product_prod(B,C),fun(A,bool)),product_prod_case(B,C,fun(A,bool)),Ca),P))
=> ~ ! [X: B,Y: C] :
( ( P = product_Pair(B,C,X,Y) )
=> ~ member(A,Z,aa(C,fun(A,bool),aa(B,fun(C,fun(A,bool)),Ca,X),Y)) ) ) ).
tff(fact_94_antisymI,axiom,
! [A: $tType,R: fun(product_prod(A,A),bool)] :
( ! [X: A,Y: A] :
( member(product_prod(A,A),product_Pair(A,A,X,Y),R)
=> ( member(product_prod(A,A),product_Pair(A,A,Y,X),R)
=> ( X = Y ) ) )
=> antisym(A,R) ) ).
tff(fact_95_splitE2,axiom,
! [B: $tType,A: $tType,C: $tType,Z: product_prod(B,C),Pa: 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),Pa),Z)))
=> ~ ! [X: B,Y: C] :
( ( Z = product_Pair(B,C,X,Y) )
=> ~ pp(aa(A,bool,Q1,aa(C,A,aa(B,fun(C,A),Pa,X),Y))) ) ) ).
tff(fact_96_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] :
( ( 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) ) ) ) ).
%----Helper facts (2)
tff(help_pp_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_pp_2_1_U,axiom,
pp(fTrue) ).
%----Conjectures (8)
tff(conj_0,hypothesis,
$true ).
tff(conj_1,hypothesis,
$true ).
tff(conj_2,hypothesis,
$true ).
tff(conj_3,hypothesis,
$true ).
tff(conj_4,hypothesis,
$true ).
tff(conj_5,hypothesis,
$true ).
tff(conj_6,hypothesis,
$true ).
tff(conj_7,conjecture,
b != a ).
%------------------------------------------------------------------------------