TPTP Problem File: SCT202_5.p
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%------------------------------------------------------------------------------
% File : SCT202_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Social Choice Theory
% Problem : Arrow's Impossibility Theorem line 162
% 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_162 [Bla13]
% Status : Theorem
% Rating : 0.67 v7.4.0, 1.00 v7.3.0, 0.75 v7.1.0, 1.00 v6.4.0
% Syntax : Number of formulae : 165 ( 54 unt; 46 typ; 0 def)
% Number of atoms : 250 ( 94 equ)
% Maximal formula atoms : 12 ( 1 avg)
% Number of connectives : 187 ( 56 ~; 10 |; 13 &)
% ( 21 <=>; 87 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 6 avg)
% Maximal term depth : 10 ( 2 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 41 ( 22 >; 19 *; 0 +; 0 <<)
% Number of predicates : 8 ( 7 usr; 0 prp; 1-6 aty)
% Number of functors : 36 ( 36 usr; 13 con; 0-5 aty)
% Number of variables : 530 ( 487 !; 6 ?; 530 :)
% ( 37 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:18:49
%------------------------------------------------------------------------------
%----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 (41)
tff(sy_cl_Orderings_Otop,type,
top:
!>[A: $tType] : $o ).
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_COMBB,type,
combb:
!>[B: $tType,C: $tType,A: $tType] : fun(fun(B,C),fun(fun(A,B),fun(A,C))) ).
tff(sy_c_COMBC,type,
combc:
!>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,fun(B,C)) * B ) > fun(A,C) ) ).
tff(sy_c_COMBI,type,
combi:
!>[A: $tType] : fun(A,A) ).
tff(sy_c_COMBK,type,
combk:
!>[A: $tType,B: $tType] : ( A > fun(B,A) ) ).
tff(sy_c_FuncSet_OPi,type,
pi:
!>[A: $tType,B: $tType] : ( ( fun(A,bool) * fun(A,fun(B,bool)) ) > fun(fun(A,B),bool) ) ).
tff(sy_c_HOL_OThe,type,
the:
!>[A: $tType] : ( fun(A,bool) > A ) ).
tff(sy_c_If,type,
if:
!>[A: $tType] : ( ( bool * A * A ) > A ) ).
tff(sy_c_Order__Relation_Ostrict__linear__order__on,type,
order_215145569der_on:
!>[A: $tType] : ( fun(A,bool) > fun(fun(product_prod(A,A),bool),bool) ) ).
tff(sy_c_Orderings_Otop__class_Otop,type,
top_top:
!>[A: $tType] : A ).
tff(sy_c_Predicate_Oinv__imagep,type,
inv_imagep:
!>[B: $tType,A: $tType] : ( ( fun(B,fun(B,bool)) * fun(A,B) * A * A ) > $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_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_Set_OCollect,type,
collect:
!>[A: $tType] : ( fun(A,bool) > fun(A,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_fconj,type,
fconj: fun(bool,fun(bool,bool)) ).
tff(sy_c_fequal,type,
fequal:
!>[A: $tType] : fun(A,fun(A,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_Q____,type,
q: fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)) ).
tff(sy_v_a_H____,type,
a: arrow_411405190le_alt ).
tff(sy_v_a____,type,
a1: arrow_411405190le_alt ).
tff(sy_v_b_H____,type,
b: arrow_411405190le_alt ).
tff(sy_v_b____,type,
b1: arrow_411405190le_alt ).
%----Relevant facts (100)
tff(fact_0__096P_A_058_AProf_096,axiom,
member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),p1,arrow_610318064e_Prof) ).
tff(fact_1_assms_I3_J,axiom,
arrow_1958449194le_IIA(f) ).
tff(fact_2_u,axiom,
arrow_2069624013nimity(f) ).
tff(fact_3__096P_H_A_058_AProf_096,axiom,
member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),p,arrow_610318064e_Prof) ).
tff(fact_4_dictator__def,axiom,
! [I: 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,I)
<=> ! [X4: 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)),X4,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,X4) = aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),X4,I) ) ) ) ).
tff(fact_5_d1_I2_J,axiom,
a != b ).
tff(fact_6_d2_I1_J,axiom,
a1 != b ).
tff(fact_7_d2_I2_J,axiom,
b1 != a ).
tff(fact_8_d1_I1_J,axiom,
a1 != b1 ).
tff(fact_9_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)
<=> ! [X4: 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)),X4,arrow_610318064e_Prof)
=> ! [A3: arrow_411405190le_alt,B3: arrow_411405190le_alt] :
( ! [I2: arrow_159774573e_indi] : 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),A3),B3),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),X4,I2))
=> 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),A3),B3),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,X4)) ) ) ) ).
tff(fact_10_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)
<=> ! [X4: 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)),X4,arrow_610318064e_Prof)
=> ! [Xa1: 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)),Xa1,arrow_610318064e_Prof)
=> ! [A3: arrow_411405190le_alt,B3: arrow_411405190le_alt] :
( ! [I2: arrow_159774573e_indi] :
( 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),A3),B3),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),X4,I2))
<=> 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),A3),B3),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Xa1,I2)) )
=> ( 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),A3),B3),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,X4))
<=> 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),A3),B3),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,Xa1)) ) ) ) ) ) ).
tff(fact_11_Q__def,axiom,
! [X3: arrow_159774573e_indi] : aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),q,X3) = if(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),aa(arrow_411405190le_alt,bool,aa(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool),fequal(arrow_411405190le_alt),b1),b),if(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),aa(arrow_411405190le_alt,bool,aa(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool),fequal(arrow_411405190le_alt),a1),a),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),p1,X3),arrow_319942042_below(aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),p1,X3),a,a1)),arrow_1158827142_above(if(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),aa(arrow_411405190le_alt,bool,aa(arrow_411405190le_alt,fun(arrow_411405190le_alt,bool),fequal(arrow_411405190le_alt),a1),a),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),p1,X3),arrow_319942042_below(aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),p1,X3),a,a1)),b1,b)) ).
tff(fact_12_const__Lin__Prof,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)
=> member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combk(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),arrow_159774573e_indi,L),arrow_610318064e_Prof) ) ).
tff(fact_13__096a_A_060_092_060_094bsub_062F_AP_092_060_094esub_062_Ab_096,axiom,
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),a1),b1),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,p1)) ).
tff(fact_14__C1_C,axiom,
! [I1: arrow_159774573e_indi] :
( 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),a1),b1),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),p1,I1))
<=> 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,I1)) ) ).
tff(fact_15_assms_I1_J,axiom,
member(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)),f,pi(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_610318064e_Prof,combk(fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool),fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),arrow_1985332922le_Lin))) ).
tff(fact_16_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_17_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_18_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_19_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_20_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_21_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_22_complete__Lin,axiom,
! [Ba: arrow_411405190le_alt,Aa: arrow_411405190le_alt] :
( ( Aa != Ba )
=> ? [X2: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)] :
( member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),X2,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),X2) ) ) ).
tff(fact_23_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,B3: B] : pp(aa(product_prod(A,B),bool,Pa,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3))) ) ).
tff(fact_24_Pair__eq,axiom,
! [A: $tType,B: $tType,B_a: B,A_a: 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),A_a),B_a) )
<=> ( ( Aa = A_a )
& ( Ba = B_a ) ) ) ).
tff(fact_25_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_26_dictatorI,axiom,
! [I: 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))] :
( member(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)),Fa,pi(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_610318064e_Prof,combk(fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool),fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),arrow_1985332922le_Lin)))
=> ( ! [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)
=> ! [A2: arrow_411405190le_alt,B1: arrow_411405190le_alt] :
( ( A2 != B1 )
=> ( member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),aa(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt),aa(arrow_411405190le_alt,fun(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt),A2),B1),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),aa(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt),aa(arrow_411405190le_alt,fun(arrow_411405190le_alt,product_prod(arrow_411405190le_alt,arrow_411405190le_alt)),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt),A2),B1),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)) ) ) )
=> arrow_987702531ctator(Fa,I) ) ) ).
tff(fact_27_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_28_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_29_pred__equals__eq2,axiom,
! [B: $tType,A: $tType,S: fun(product_prod(A,B),bool),R2: fun(product_prod(A,B),bool)] :
( ! [X4: A,Xa1: B] :
( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa1),R2)
<=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X4),Xa1),S) )
<=> ( R2 = S ) ) ).
tff(fact_30_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_31_Pair__inject,axiom,
! [A: $tType,B: $tType,B5: B,A5: A,B4: B,A4: A] :
( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B4) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B5) )
=> ~ ( ( A4 = A5 )
=> ( B4 != B5 ) ) ) ).
tff(fact_32_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_33_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,B3: B] : pp(aa(product_prod(A,B),bool,Pa,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3))) ) ).
tff(fact_34_PiE,axiom,
! [B: $tType,A: $tType,X1: A,B2: fun(A,fun(B,bool)),A1: fun(A,bool),F: fun(A,B)] :
( member(fun(A,B),F,pi(A,B,A1,B2))
=> ( ~ member(B,aa(A,B,F,X1),aa(A,fun(B,bool),B2,X1))
=> ~ member(A,X1,A1) ) ) ).
tff(fact_35_prod_Orecs,axiom,
! [B: $tType,A: $tType,C: $tType,Ba: C,Aa: B,F1: fun(B,fun(C,A))] : product_prod_rec(B,C,A,F1,aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),Aa),Ba)) = aa(C,A,aa(B,fun(C,A),F1,Aa),Ba) ).
tff(fact_36_funcset__id,axiom,
! [A: $tType,A1: fun(A,bool)] : member(fun(A,A),combi(A),pi(A,A,A1,combk(fun(A,bool),A,A1))) ).
tff(fact_37_funcset__mem,axiom,
! [A: $tType,B: $tType,X1: A,B2: fun(B,bool),A1: fun(A,bool),F: fun(A,B)] :
( member(fun(A,B),F,pi(A,B,A1,combk(fun(B,bool),A,B2)))
=> ( member(A,X1,A1)
=> member(B,aa(A,B,F,X1),B2) ) ) ).
tff(fact_38_prod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod(A,B)] :
~ ! [A2: A,B1: B] : Y != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B1) ).
tff(fact_39_Pi__mem,axiom,
! [B: $tType,A: $tType,X1: A,B2: fun(A,fun(B,bool)),A1: fun(A,bool),F: fun(A,B)] :
( member(fun(A,B),F,pi(A,B,A1,B2))
=> ( member(A,X1,A1)
=> member(B,aa(A,B,F,X1),aa(A,fun(B,bool),B2,X1)) ) ) ).
tff(fact_40_Pi__I,axiom,
! [B: $tType,A: $tType,B2: fun(A,fun(B,bool)),F: fun(A,B),A1: fun(A,bool)] :
( ! [X2: A] :
( member(A,X2,A1)
=> member(B,aa(A,B,F,X2),aa(A,fun(B,bool),B2,X2)) )
=> member(fun(A,B),F,pi(A,B,A1,B2)) ) ).
tff(fact_41_funcsetI,axiom,
! [A: $tType,B: $tType,B2: fun(B,bool),F: fun(A,B),A1: fun(A,bool)] :
( ! [X2: A] :
( member(A,X2,A1)
=> member(B,aa(A,B,F,X2),B2) )
=> member(fun(A,B),F,pi(A,B,A1,combk(fun(B,bool),A,B2))) ) ).
tff(fact_42_Pi__I_H,axiom,
! [B: $tType,A: $tType,B2: fun(A,fun(B,bool)),F: fun(A,B),A1: fun(A,bool)] :
( ! [X2: A] :
( member(A,X2,A1)
=> member(B,aa(A,B,F,X2),aa(A,fun(B,bool),B2,X2)) )
=> member(fun(A,B),F,pi(A,B,A1,B2)) ) ).
tff(fact_43_Pi__cong,axiom,
! [B: $tType,A: $tType,B2: fun(A,fun(B,bool)),G: fun(A,B),F: fun(A,B),A1: fun(A,bool)] :
( ! [W: A] :
( member(A,W,A1)
=> ( aa(A,B,F,W) = aa(A,B,G,W) ) )
=> ( member(fun(A,B),F,pi(A,B,A1,B2))
<=> member(fun(A,B),G,pi(A,B,A1,B2)) ) ) ).
tff(fact_44_prod__induct6,axiom,
! [F2: $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,F2))))),Pa: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2))))),bool)] :
( ! [A2: A,B1: B,C2: C,D1: D,E1: E,F3: F2] : pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2))))),bool,Pa,aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2)))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2))))),A2),aa(product_prod(C,product_prod(D,product_prod(E,F2))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2)))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,F2))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,F2)))),B1),aa(product_prod(D,product_prod(E,F2)),product_prod(C,product_prod(D,product_prod(E,F2))),aa(C,fun(product_prod(D,product_prod(E,F2)),product_prod(C,product_prod(D,product_prod(E,F2)))),product_Pair(C,product_prod(D,product_prod(E,F2))),C2),aa(product_prod(E,F2),product_prod(D,product_prod(E,F2)),aa(D,fun(product_prod(E,F2),product_prod(D,product_prod(E,F2))),product_Pair(D,product_prod(E,F2)),D1),aa(F2,product_prod(E,F2),aa(E,fun(F2,product_prod(E,F2)),product_Pair(E,F2),E1),F3)))))))
=> pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2))))),bool,Pa,X1)) ) ).
tff(fact_45_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F2: $tType,Y: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2)))))] :
~ ! [A2: A,B1: B,C2: C,D1: D,E1: E,F3: F2] : Y != aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2)))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2))))),A2),aa(product_prod(C,product_prod(D,product_prod(E,F2))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2)))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,F2))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F2))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,F2)))),B1),aa(product_prod(D,product_prod(E,F2)),product_prod(C,product_prod(D,product_prod(E,F2))),aa(C,fun(product_prod(D,product_prod(E,F2)),product_prod(C,product_prod(D,product_prod(E,F2)))),product_Pair(C,product_prod(D,product_prod(E,F2))),C2),aa(product_prod(E,F2),product_prod(D,product_prod(E,F2)),aa(D,fun(product_prod(E,F2),product_prod(D,product_prod(E,F2))),product_Pair(D,product_prod(E,F2)),D1),aa(F2,product_prod(E,F2),aa(E,fun(F2,product_prod(E,F2)),product_Pair(E,F2),E1),F3))))) ).
tff(fact_46_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))] :
~ ! [A2: A,B1: B,C2: C,D1: D,E1: E] : Y != aa(product_prod(B,product_prod(C,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,E)))),A2),aa(product_prod(C,product_prod(D,E)),product_prod(B,product_prod(C,product_prod(D,E))),aa(B,fun(product_prod(C,product_prod(D,E)),product_prod(B,product_prod(C,product_prod(D,E)))),product_Pair(B,product_prod(C,product_prod(D,E))),B1),aa(product_prod(D,E),product_prod(C,product_prod(D,E)),aa(C,fun(product_prod(D,E),product_prod(C,product_prod(D,E))),product_Pair(C,product_prod(D,E)),C2),aa(E,product_prod(D,E),aa(D,fun(E,product_prod(D,E)),product_Pair(D,E),D1),E1)))) ).
tff(fact_47_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)] :
( ! [A2: A,B1: B,C2: C,D1: D,E1: E] : pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),bool,Pa,aa(product_prod(B,product_prod(C,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,E)))),A2),aa(product_prod(C,product_prod(D,E)),product_prod(B,product_prod(C,product_prod(D,E))),aa(B,fun(product_prod(C,product_prod(D,E)),product_prod(B,product_prod(C,product_prod(D,E)))),product_Pair(B,product_prod(C,product_prod(D,E))),B1),aa(product_prod(D,E),product_prod(C,product_prod(D,E)),aa(C,fun(product_prod(D,E),product_prod(C,product_prod(D,E))),product_Pair(C,product_prod(D,E)),C2),aa(E,product_prod(D,E),aa(D,fun(E,product_prod(D,E)),product_Pair(D,E),D1),E1))))))
=> pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),bool,Pa,X1)) ) ).
tff(fact_48_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod(A,product_prod(B,product_prod(C,D)))] :
~ ! [A2: A,B1: B,C2: C,D1: D] : Y != aa(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D))),aa(A,fun(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D)))),product_Pair(A,product_prod(B,product_prod(C,D))),A2),aa(product_prod(C,D),product_prod(B,product_prod(C,D)),aa(B,fun(product_prod(C,D),product_prod(B,product_prod(C,D))),product_Pair(B,product_prod(C,D)),B1),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),C2),D1))) ).
tff(fact_49_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)] :
( ! [A2: A,B1: B,C2: C,D1: D] : pp(aa(product_prod(A,product_prod(B,product_prod(C,D))),bool,Pa,aa(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D))),aa(A,fun(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D)))),product_Pair(A,product_prod(B,product_prod(C,D))),A2),aa(product_prod(C,D),product_prod(B,product_prod(C,D)),aa(B,fun(product_prod(C,D),product_prod(B,product_prod(C,D))),product_Pair(B,product_prod(C,D)),B1),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),C2),D1)))))
=> pp(aa(product_prod(A,product_prod(B,product_prod(C,D))),bool,Pa,X1)) ) ).
tff(fact_50_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)] :
( ! [A2: A,B1: B,C2: C] : pp(aa(product_prod(A,product_prod(B,C)),bool,Pa,aa(product_prod(B,C),product_prod(A,product_prod(B,C)),aa(A,fun(product_prod(B,C),product_prod(A,product_prod(B,C))),product_Pair(A,product_prod(B,C)),A2),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B1),C2))))
=> pp(aa(product_prod(A,product_prod(B,C)),bool,Pa,X1)) ) ).
tff(fact_51_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod(A,product_prod(B,C))] :
~ ! [A2: A,B1: B,C2: C] : Y != aa(product_prod(B,C),product_prod(A,product_prod(B,C)),aa(A,fun(product_prod(B,C),product_prod(A,product_prod(B,C))),product_Pair(A,product_prod(B,C)),A2),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B1),C2)) ).
tff(fact_52_PairE,axiom,
! [A: $tType,B: $tType,P2: product_prod(A,B)] :
~ ! [X2: A,Y2: B] : P2 != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y2) ).
tff(fact_53_internal__split__conv,axiom,
! [B: $tType,A: $tType,C: $tType,Ba: C,Aa: B,C1: fun(B,fun(C,A))] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),produc1605651328_split(B,C,A),C1),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),Aa),Ba)) = aa(C,A,aa(B,fun(C,A),C1,Aa),Ba) ).
tff(fact_54_curry__conv,axiom,
! [A: $tType,B: $tType,C: $tType,Ba: C,Aa: B,F: fun(product_prod(B,C),A)] : aa(C,A,aa(B,fun(C,A),product_curry(B,C,A,F),Aa),Ba) = aa(product_prod(B,C),A,F,aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),Aa),Ba)) ).
tff(fact_55_curry__def,axiom,
! [C: $tType,A: $tType,B: $tType,X3: fun(product_prod(A,B),C),Xa: A,Xb: B] : aa(B,C,aa(A,fun(B,C),product_curry(A,B,C,X3),Xa),Xb) = aa(product_prod(A,B),C,X3,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Xb)) ).
tff(fact_56_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_57_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_58_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_59_Prof__def,axiom,
arrow_610318064e_Prof = pi(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),top_top(fun(arrow_159774573e_indi,bool)),combk(fun(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),bool),arrow_159774573e_indi,arrow_1985332922le_Lin)) ).
tff(fact_60_in__inv__imagep,axiom,
! [A: $tType,B: $tType,Y1: B,X1: B,F: fun(B,A),R1: fun(A,fun(A,bool))] :
( inv_imagep(A,B,R1,F,X1,Y1)
<=> pp(aa(A,bool,aa(A,fun(A,bool),R1,aa(B,A,F,X1)),aa(B,A,F,Y1))) ) ).
tff(fact_61_internal__split__def,axiom,
! [C: $tType,B: $tType,A: $tType] : produc1605651328_split(A,B,C) = product_prod_case(A,B,C) ).
tff(fact_62_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_63_prod__caseI,axiom,
! [A: $tType,B: $tType,Ba: B,Aa: A,F1: fun(A,fun(B,bool))] :
( pp(aa(B,bool,aa(A,fun(B,bool),F1,Aa),Ba))
=> pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_prod_case(A,B,bool),F1),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa),Ba))) ) ).
tff(fact_64_mem__splitI,axiom,
! [A: $tType,B: $tType,C: $tType,Ba: C,Aa: B,C1: fun(B,fun(C,fun(A,bool))),Z: A] :
( member(A,Z,aa(C,fun(A,bool),aa(B,fun(C,fun(A,bool)),C1,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)),C1),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),Aa),Ba))) ) ).
tff(fact_65_top1I,axiom,
! [A: $tType,X1: A] : pp(aa(A,bool,top_top(fun(A,bool)),X1)) ).
tff(fact_66_split__conv,axiom,
! [B: $tType,A: $tType,C: $tType,Ba: C,Aa: B,F: fun(B,fun(C,A))] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_prod_case(B,C,A),F),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),Aa),Ba)) = aa(C,A,aa(B,fun(C,A),F,Aa),Ba) ).
tff(fact_67_Pi__UNIV,axiom,
! [B: $tType,A: $tType,A1: fun(A,bool)] : pi(A,B,A1,combk(fun(B,bool),A,top_top(fun(B,bool)))) = top_top(fun(fun(A,B),bool)) ).
tff(fact_68_split__weak__cong,axiom,
! [C: $tType,B: $tType,A: $tType,C1: fun(A,fun(B,C)),Q1: product_prod(A,B),P1: product_prod(A,B)] :
( ( P1 = Q1 )
=> ( aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_prod_case(A,B,C),C1),P1) = aa(product_prod(A,B),C,aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_prod_case(A,B,C),C1),Q1) ) ) ).
tff(fact_69_split__twice,axiom,
! [B: $tType,C: $tType,A: $tType,E: $tType,D: $tType,P1: product_prod(D,E),G: fun(D,fun(E,product_prod(B,C))),F: fun(B,fun(C,A))] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_prod_case(B,C,A),F),aa(product_prod(D,E),product_prod(B,C),aa(fun(D,fun(E,product_prod(B,C))),fun(product_prod(D,E),product_prod(B,C)),product_prod_case(D,E,product_prod(B,C)),G),P1)) = aa(product_prod(D,E),A,aa(fun(D,fun(E,A)),fun(product_prod(D,E),A),product_prod_case(D,E,A),aa(fun(D,fun(E,product_prod(B,C))),fun(D,fun(E,A)),aa(fun(fun(E,product_prod(B,C)),fun(E,A)),fun(fun(D,fun(E,product_prod(B,C))),fun(D,fun(E,A))),combb(fun(E,product_prod(B,C)),fun(E,A),D),aa(fun(product_prod(B,C),A),fun(fun(E,product_prod(B,C)),fun(E,A)),combb(product_prod(B,C),A,E),aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_prod_case(B,C,A),F))),G)),P1) ).
tff(fact_70_prod_Osimps_I2_J,axiom,
! [B: $tType,A: $tType,C: $tType,Ba: C,Aa: B,F1: fun(B,fun(C,A))] : aa(product_prod(B,C),A,aa(fun(B,fun(C,A)),fun(product_prod(B,C),A),product_prod_case(B,C,A),F1),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),Aa),Ba)) = aa(C,A,aa(B,fun(C,A),F1,Aa),Ba) ).
tff(fact_71_splitD_H,axiom,
! [B: $tType,A: $tType,C: $tType,C1: C,Ba: B,Aa: A,R2: 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)),R2),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Aa),Ba)),C1))
=> pp(aa(C,bool,aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),R2,Aa),Ba),C1)) ) ).
tff(fact_72_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_73_split__eta,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),aa(fun(A,fun(B,product_prod(A,B))),fun(A,fun(B,C)),aa(fun(fun(B,product_prod(A,B)),fun(B,C)),fun(fun(A,fun(B,product_prod(A,B))),fun(A,fun(B,C))),combb(fun(B,product_prod(A,B)),fun(B,C),A),aa(fun(product_prod(A,B),C),fun(fun(B,product_prod(A,B)),fun(B,C)),combb(product_prod(A,B),C,B),F)),product_Pair(A,B))) = F ).
tff(fact_74_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_75_ext,axiom,
! [B: $tType,A: $tType,G: fun(A,B),F: fun(A,B)] :
( ! [X2: A] : aa(A,B,F,X2) = aa(A,B,G,X2)
=> ( F = G ) ) ).
tff(fact_76_mem__def,axiom,
! [A: $tType,A1: fun(A,bool),X1: A] :
( member(A,X1,A1)
<=> pp(aa(A,bool,A1,X1)) ) ).
tff(fact_77_Collect__def,axiom,
! [A: $tType,Pa: fun(A,bool)] : collect(A,Pa) = Pa ).
tff(fact_78_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_79_inv__imagep__def,axiom,
! [B: $tType,A: $tType,F: fun(A,B),R1: fun(B,fun(B,bool)),X3: A,Xa: A] :
( inv_imagep(B,A,R1,F,X3,Xa)
<=> pp(aa(B,bool,aa(B,fun(B,bool),R1,aa(A,B,F,X3)),aa(A,B,F,Xa))) ) ).
tff(fact_80_splitI2,axiom,
! [B: $tType,A: $tType,C1: fun(A,fun(B,bool)),P1: product_prod(A,B)] :
( ! [A2: A,B1: B] :
( ( P1 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B1) )
=> pp(aa(B,bool,aa(A,fun(B,bool),C1,A2),B1)) )
=> pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_prod_case(A,B,bool),C1),P1)) ) ).
tff(fact_81_splitE,axiom,
! [A: $tType,B: $tType,P1: product_prod(A,B),C1: fun(A,fun(B,bool))] :
( pp(aa(product_prod(A,B),bool,aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_prod_case(A,B,bool),C1),P1))
=> ~ ! [X2: A,Y2: B] :
( ( P1 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y2) )
=> ~ pp(aa(B,bool,aa(A,fun(B,bool),C1,X2),Y2)) ) ) ).
tff(fact_82_mem__splitI2,axiom,
! [C: $tType,B: $tType,A: $tType,C1: fun(A,fun(B,fun(C,bool))),Z: C,P1: product_prod(A,B)] :
( ! [A2: A,B1: B] :
( ( P1 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B1) )
=> member(C,Z,aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),C1,A2),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)),C1),P1)) ) ).
tff(fact_83_split__part,axiom,
! [B: $tType,A: $tType,Qa: fun(A,fun(B,bool)),Pa: bool,X3: product_prod(A,B)] :
( 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),aa(fun(A,fun(B,bool)),fun(A,fun(B,bool)),aa(fun(fun(B,bool),fun(B,bool)),fun(fun(A,fun(B,bool)),fun(A,fun(B,bool))),combb(fun(B,bool),fun(B,bool),A),aa(fun(bool,bool),fun(fun(B,bool),fun(B,bool)),combb(bool,bool,B),aa(bool,fun(bool,bool),fconj,Pa))),Qa)),X3))
<=> ( pp(Pa)
& 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),Qa),X3)) ) ) ).
tff(fact_84_splitE_H,axiom,
! [B: $tType,A: $tType,C: $tType,Z: C,P1: product_prod(A,B),C1: fun(A,fun(B,fun(C,bool)))] :
( pp(aa(C,bool,aa(product_prod(A,B),fun(C,bool),aa(fun(A,fun(B,fun(C,bool))),fun(product_prod(A,B),fun(C,bool)),product_prod_case(A,B,fun(C,bool)),C1),P1),Z))
=> ~ ! [X2: A,Y2: B] :
( ( P1 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y2) )
=> ~ pp(aa(C,bool,aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),C1,X2),Y2),Z)) ) ) ).
tff(fact_85_mem__splitE,axiom,
! [B: $tType,A: $tType,C: $tType,P1: product_prod(B,C),C1: 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)),C1),P1))
=> ~ ! [X2: B,Y2: C] :
( ( P1 = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X2),Y2) )
=> ~ member(A,Z,aa(C,fun(A,bool),aa(B,fun(C,fun(A,bool)),C1,X2),Y2)) ) ) ).
tff(fact_86_splitI2_H,axiom,
! [A: $tType,B: $tType,C: $tType,X1: C,C1: fun(A,fun(B,fun(C,bool))),P1: product_prod(A,B)] :
( ! [A2: A,B1: B] :
( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B1) = P1 )
=> pp(aa(C,bool,aa(B,fun(C,bool),aa(A,fun(B,fun(C,bool)),C1,A2),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)),C1),P1),X1)) ) ).
tff(fact_87_cond__split__eta,axiom,
! [C: $tType,B: $tType,A: $tType,G: fun(product_prod(A,B),C),F: fun(A,fun(B,C))] :
( ! [X2: A,Y2: B] : aa(B,C,aa(A,fun(B,C),F,X2),Y2) = aa(product_prod(A,B),C,G,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y2))
=> ( aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_prod_case(A,B,C),F) = G ) ) ).
tff(fact_88_UNIV__I,axiom,
! [A: $tType,X1: A] : member(A,X1,top_top(fun(A,bool))) ).
tff(fact_89_iso__tuple__UNIV__I,axiom,
! [A: $tType,X1: A] : member(A,X1,top_top(fun(A,bool))) ).
tff(fact_90_UNIV__def,axiom,
! [A: $tType] : top_top(fun(A,bool)) = collect(A,combk(bool,A,fTrue)) ).
tff(fact_91_splitE2,axiom,
! [B: $tType,A: $tType,C: $tType,Z: product_prod(B,C),Pa: fun(B,fun(C,A)),Qa: fun(A,bool)] :
( pp(aa(A,bool,Qa,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)))
=> ~ ! [X2: B,Y2: C] :
( ( Z = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X2),Y2) )
=> ~ pp(aa(A,bool,Qa,aa(C,A,aa(B,fun(C,A),Pa,X2),Y2))) ) ) ).
tff(fact_92_top__apply,axiom,
! [B: $tType,A: $tType] :
( top(A)
=> ! [X1: B] : aa(B,A,top_top(fun(B,A)),X1) = top_top(A) ) ).
tff(fact_93_UNIV__witness,axiom,
! [A: $tType] :
? [X2: A] : member(A,X2,top_top(fun(A,bool))) ).
tff(fact_94_UNIV__eq__I,axiom,
! [A: $tType,A1: fun(A,bool)] :
( ! [X2: A] : member(A,X2,A1)
=> ( top_top(fun(A,bool)) = A1 ) ) ).
tff(fact_95_Lin__def,axiom,
arrow_1985332922le_Lin = collect(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),order_215145569der_on(arrow_411405190le_alt,top_top(fun(arrow_411405190le_alt,bool)))) ).
tff(fact_96_The__split__eq,axiom,
! [A: $tType,B: $tType,Y1: B,X1: A] : the(product_prod(A,B),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_prod_case(A,B,bool),combc(A,fun(B,bool),fun(B,bool),aa(fun(A,fun(bool,bool)),fun(A,fun(fun(B,bool),fun(B,bool))),aa(fun(fun(bool,bool),fun(fun(B,bool),fun(B,bool))),fun(fun(A,fun(bool,bool)),fun(A,fun(fun(B,bool),fun(B,bool)))),combb(fun(bool,bool),fun(fun(B,bool),fun(B,bool)),A),combb(bool,bool,B)),aa(fun(A,bool),fun(A,fun(bool,bool)),aa(fun(bool,fun(bool,bool)),fun(fun(A,bool),fun(A,fun(bool,bool))),combb(bool,fun(bool,bool),A),fconj),aa(A,fun(A,bool),fequal(A),X1))),aa(B,fun(B,bool),fequal(B),Y1)))) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X1),Y1) ).
tff(fact_97_split__paired__The,axiom,
! [B: $tType,A: $tType,Pa: fun(product_prod(A,B),bool)] : the(product_prod(A,B),Pa) = the(product_prod(A,B),aa(fun(A,fun(B,bool)),fun(product_prod(A,B),bool),product_prod_case(A,B,bool),aa(fun(A,fun(B,product_prod(A,B))),fun(A,fun(B,bool)),aa(fun(fun(B,product_prod(A,B)),fun(B,bool)),fun(fun(A,fun(B,product_prod(A,B))),fun(A,fun(B,bool))),combb(fun(B,product_prod(A,B)),fun(B,bool),A),aa(fun(product_prod(A,B),bool),fun(fun(B,product_prod(A,B)),fun(B,bool)),combb(product_prod(A,B),bool,B),Pa)),product_Pair(A,B)))) ).
tff(fact_98_the__sym__eq__trivial,axiom,
! [A: $tType,X1: A] : the(A,aa(A,fun(A,bool),fequal(A),X1)) = X1 ).
tff(fact_99_the__eq__trivial,axiom,
! [A: $tType,Aa: A] : the(A,combc(A,A,bool,fequal(A),Aa)) = Aa ).
%----Arities (2)
tff(arity_fun___Orderings_Otop,axiom,
! [T_1: $tType,T_2: $tType] :
( top(T_2)
=> top(fun(T_1,T_2)) ) ).
tff(arity_HOL_Obool___Orderings_Otop,axiom,
top(bool) ).
%----Helper facts (16)
tff(help_If_1_1_T,axiom,
! [A: $tType,Y: A,X: A] : if(A,fTrue,X,Y) = X ).
tff(help_If_2_1_T,axiom,
! [A: $tType,Y: A,X: A] : if(A,fFalse,X,Y) = Y ).
tff(help_If_3_1_T,axiom,
! [P: bool] :
( ( P = fTrue )
| ( P = fFalse ) ) ).
tff(help_pp_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_pp_2_1_U,axiom,
pp(fTrue) ).
tff(help_COMBB_1_1_U,axiom,
! [C: $tType,B: $tType,A: $tType,R: A,Q: fun(A,B),P: fun(B,C)] : aa(A,C,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),combb(B,C,A),P),Q),R) = aa(B,C,P,aa(A,B,Q,R)) ).
tff(help_COMBC_1_1_U,axiom,
! [A: $tType,C: $tType,B: $tType,R: A,Q: B,P: fun(A,fun(B,C))] : aa(A,C,combc(A,B,C,P,Q),R) = aa(B,C,aa(A,fun(B,C),P,R),Q) ).
tff(help_COMBI_1_1_U,axiom,
! [A: $tType,P: A] : aa(A,A,combi(A),P) = P ).
tff(help_COMBK_1_1_U,axiom,
! [B: $tType,A: $tType,Q: B,P: A] : aa(B,A,combk(A,B,P),Q) = P ).
tff(help_fTrue_1_1_U,axiom,
pp(fTrue) ).
tff(help_fTrue_1_1_T,axiom,
! [P: bool] :
( ( P = fTrue )
| ( P = fFalse ) ) ).
tff(help_fconj_1_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(P)
| ~ pp(Q)
| pp(aa(bool,bool,aa(bool,fun(bool,bool),fconj,P),Q)) ) ).
tff(help_fconj_2_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(aa(bool,bool,aa(bool,fun(bool,bool),fconj,P),Q))
| pp(P) ) ).
tff(help_fconj_3_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(aa(bool,bool,aa(bool,fun(bool,bool),fconj,P),Q))
| pp(Q) ) ).
tff(help_fequal_1_1_T,axiom,
! [A: $tType,Y: A,X: A] :
( ~ pp(aa(A,bool,aa(A,fun(A,bool),fequal(A),X),Y))
| ( X = Y ) ) ).
tff(help_fequal_2_1_T,axiom,
! [A: $tType,Y: A,X: A] :
( ( X != Y )
| pp(aa(A,bool,aa(A,fun(A,bool),fequal(A),X),Y)) ) ).
%----Conjectures (1)
tff(conj_0,conjecture,
member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),q,arrow_610318064e_Prof) ).
%------------------------------------------------------------------------------