TPTP Problem File: SCT226_5.p
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%------------------------------------------------------------------------------
% File : SCT226_5 : TPTP v9.0.0. Released v6.0.0.
% Domain : Social Choice Theory
% Problem : Arrow's Impossibility Theorem line 221
% 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_221 [Bla13]
% Status : Theorem
% Rating : 0.00 v7.1.0, 0.33 v6.4.0
% Syntax : Number of formulae : 156 ( 49 unt; 45 typ; 0 def)
% Number of atoms : 249 ( 85 equ)
% Maximal formula atoms : 12 ( 1 avg)
% Number of connectives : 197 ( 59 ~; 3 |; 15 &)
% ( 37 <=>; 83 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 6 avg)
% Maximal term depth : 15 ( 2 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 52 ( 27 >; 25 *; 0 +; 0 <<)
% Number of predicates : 10 ( 9 usr; 0 prp; 1-6 aty)
% Number of functors : 33 ( 33 usr; 11 con; 0-6 aty)
% Number of variables : 438 ( 390 !; 11 ?; 438 :)
% ( 37 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:22:53
%------------------------------------------------------------------------------
%----Should-be-implicit typings (6)
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_List_Olist,type,
list: $tType > $tType ).
tff(ty_tc_fun,type,
fun: ( $tType * $tType ) > $tType ).
tff(ty_tc_prod,type,
product_prod: ( $tType * $tType ) > $tType ).
%----Explicit typings (39)
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(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(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(B,C) * 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_FunDef_Oin__rel,type,
in_rel:
!>[A: $tType,B: $tType] : ( ( fun(product_prod(A,B),bool) * A * B ) > $o ) ).
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_List_Odistinct,type,
distinct:
!>[A: $tType] : ( list(A) > $o ) ).
tff(sy_c_List_Oinsert,type,
insert:
!>[A: $tType] : ( ( A * list(A) ) > list(A) ) ).
tff(sy_c_List_Olist_OCons,type,
cons:
!>[A: $tType] : ( ( A * list(A) ) > list(A) ) ).
tff(sy_c_List_Olist_ONil,type,
nil:
!>[A: $tType] : list(A) ).
tff(sy_c_List_Osplice,type,
splice:
!>[A: $tType] : ( ( list(A) * list(A) ) > list(A) ) ).
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] : ( ( 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) * A * B ) > C ) ).
tff(sy_c_Product__Type_Ointernal__split,type,
produc1605651328_split:
!>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,fun(B,C)) * product_prod(A,B) ) > C ) ).
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_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 (100)
tff(fact_0_assms_I3_J,axiom,
arrow_1958449194le_IIA(f) ).
tff(fact_1_u,axiom,
arrow_2069624013nimity(f) ).
tff(fact_2__096a_A_126_061_Ab_096,axiom,
a != b ).
tff(fact_3__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_4__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_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,a),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),p,I3)) ) ).
tff(fact_6__096ALL_Ai_O_A_Ia_A_060_092_060_094bsub_062P_Ai_092_060_094esub_062_Ab_J_A_061_A_Ia_A_060_092_060_094bsub_062below_A_IP_Ai_J_Ac_Ab_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,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,a,c),aa(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),aa(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_319942042_below,aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),p1,I3)),c),b)) ) ).
tff(fact_7__096ALL_Ai_O_A_Ia_A_060_092_060_094bsub_062below_A_IP_Ai_J_Ac_Ab_092_060_094esub_062_Ac_J_A_061_A_Ib_A_060_092_060_094bsub_062below_A_Ibelow_A_IP_Ai_J_Ac_Ab_J_Ab_Aa_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,a,c),aa(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),aa(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_319942042_below,aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),p1,I3)),c),b))
<=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,b,c),aa(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),aa(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_319942042_below,aa(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),aa(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_319942042_below,aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),p1,I3)),c),b)),b),a)) ) ).
tff(fact_8__096ALL_Ai_O_A_Ib_A_060_092_060_094bsub_062below_A_Ibelow_A_IP_Ai_J_Ac_Ab_J_Ab_Aa_092_060_094esub_062_Ac_J_A_061_A_Ib_A_060_092_060_094bsub_062below_A_Ibelow_A_Ibelow_A_IP_Ai_J_Ac_Ab_J_Ab_Aa_J_Aa_Ac_092_060_094esub_062_Aa_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,c),aa(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),aa(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_319942042_below,aa(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),aa(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_319942042_below,aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),p1,I3)),c),b)),b),a))
<=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,b,a),aa(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),aa(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_319942042_below,aa(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),aa(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_319942042_below,aa(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),aa(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_319942042_below,aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),p1,I3)),c),b)),b),a)),a),c)) ) ).
tff(fact_9__096ALL_Ai_O_A_Ib_A_060_092_060_094bsub_062below_A_Ibelow_A_Ibelow_A_IP_Ai_J_Ac_Ab_J_Ab_Aa_J_Aa_Ac_092_060_094esub_062_Aa_J_A_061_A_Ia_A_060_092_060_094bsub_062P_Ai_092_060_094esub_062_Ab_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_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),aa(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_319942042_below,aa(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),aa(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_319942042_below,aa(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),aa(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_319942042_below,aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),p1,I3)),c),b)),b),a)),a),c))
<=> 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)) ) ).
tff(fact_10__096ALL_Ai_O_A_Ib_A_060_092_060_094bsub_062below_A_Ibelow_A_Ibelow_A_IP_Ai_J_Ac_Ab_J_Ab_Aa_J_Aa_Ac_092_060_094esub_062_Aa_J_A_061_A_Ib_A_060_092_060_094bsub_062P_H_Ai_092_060_094esub_062_Aa_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_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),aa(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_319942042_below,aa(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),aa(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_319942042_below,aa(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),aa(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_319942042_below,aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),p1,I3)),c),b)),b),a)),a),c))
<=> 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),p,I3)) ) ).
tff(fact_11_ba,axiom,
( member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,b,a),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,combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combb(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_159774573e_indi,arrow_319942042_below,combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combb(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_159774573e_indi,arrow_319942042_below,combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combb(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_159774573e_indi,arrow_319942042_below,p1),c),b)),b),a)),a),c)))
<=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,b,a),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)) ) ).
tff(fact_12__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)),B6: arrow_411405190le_alt,A3: arrow_411405190le_alt,Ba: arrow_411405190le_alt,Aa: arrow_411405190le_alt] :
( ( Aa != Ba )
=> ( ( A3 != B6 )
=> ( ( Aa != B6 )
=> ( ( Ba != A3 )
=> ( 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)
=> ( ! [I2: 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,I2))
<=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,A3,B6),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),P_a,I2)) )
=> ( 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,A3,B6),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_13__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)),B6: arrow_411405190le_alt,A3: arrow_411405190le_alt,Ba: arrow_411405190le_alt,Aa: arrow_411405190le_alt] :
( ( Aa != Ba )
=> ( ( A3 != B6 )
=> ( ( Aa != B6 )
=> ( ( Ba != A3 )
=> ( 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)
=> ( ! [I2: 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,I2))
<=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,A3,B6),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),P_a,I2)) )
=> ( 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,A3,B6),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_14_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))
<=> ! [A6: A,B4: B] : pp(aa(product_prod(A,B),bool,Pa,product_Pair(A,B,A6,B4))) ) ).
tff(fact_15_Pair__eq,axiom,
! [A: $tType,B: $tType,B6: B,A3: A,Ba: B,Aa: A] :
( ( product_Pair(A,B,Aa,Ba) = product_Pair(A,B,A3,B6) )
<=> ( ( Aa = A3 )
& ( Ba = B6 ) ) ) ).
tff(fact_16_ab,axiom,
( member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,a,b),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))
<=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,a,c),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,combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combb(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_159774573e_indi,arrow_319942042_below,p1),c),b))) ) ).
tff(fact_17_ac,axiom,
( member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,a,c),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,combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combb(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_159774573e_indi,arrow_319942042_below,p1),c),b)))
<=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,b,c),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,combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combb(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_159774573e_indi,arrow_319942042_below,combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combb(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_159774573e_indi,arrow_319942042_below,p1),c),b)),b),a))) ) ).
tff(fact_18_bc,axiom,
( member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,b,c),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,combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combb(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_159774573e_indi,arrow_319942042_below,combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combb(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_159774573e_indi,arrow_319942042_below,p1),c),b)),b),a)))
<=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,b,a),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,combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combb(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_159774573e_indi,arrow_319942042_below,combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combb(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_159774573e_indi,arrow_319942042_below,combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combb(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_159774573e_indi,arrow_319942042_below,p1),c),b)),b),a)),a),c))) ) ).
tff(fact_19_in__mkbot,axiom,
! [Z: arrow_411405190le_alt,L: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Y2: arrow_411405190le_alt,X1: arrow_411405190le_alt] :
( member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,X1,Y2),arrow_276188178_mkbot(L,Z))
<=> ( ( Y2 != Z )
& ( ( X1 = Z )
=> ( X1 != Y2 ) )
& ( ( X1 != Z )
=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,X1,Y2),L) ) ) ) ).
tff(fact_20_in__mktop,axiom,
! [Z: arrow_411405190le_alt,L: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Y2: arrow_411405190le_alt,X1: arrow_411405190le_alt] :
( member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,X1,Y2),arrow_424895264_mktop(L,Z))
<=> ( ( X1 != Z )
& ( ( Y2 = Z )
=> ( X1 != Y2 ) )
& ( ( Y2 != Z )
=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,X1,Y2),L) ) ) ) ).
tff(fact_21_Pair__inject,axiom,
! [A: $tType,B: $tType,B5: B,A2: A,B2: B,A4: A] :
( ( product_Pair(A,B,A4,B2) = product_Pair(A,B,A2,B5) )
=> ~ ( ( A4 = A2 )
=> ( B2 != B5 ) ) ) ).
tff(fact_22_in__rel__def,axiom,
! [B: $tType,A: $tType,Y2: B,X1: A,R2: fun(product_prod(A,B),bool)] :
( in_rel(A,B,R2,X1,Y2)
<=> member(product_prod(A,B),product_Pair(A,B,X1,Y2),R2) ) ).
tff(fact_23__096_I_Fp_O_Abelow_A_IP_Ap_J_Ac_Ab_J_A_058_AProf_096,axiom,
member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combb(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_159774573e_indi,arrow_319942042_below,p1),c),b),arrow_610318064e_Prof) ).
tff(fact_24__096_I_Fp_O_Abelow_A_Ibelow_A_Ibelow_A_IP_Ap_J_Ac_Ab_J_Ab_Aa_J_Aa_Ac_J_A_058_AProf_096,axiom,
member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combb(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_159774573e_indi,arrow_319942042_below,combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combb(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_159774573e_indi,arrow_319942042_below,combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combb(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_159774573e_indi,arrow_319942042_below,p1),c),b)),b),a)),a),c),arrow_610318064e_Prof) ).
tff(fact_25__096_I_Fp_O_Abelow_A_Ibelow_A_IP_Ap_J_Ac_Ab_J_Ab_Aa_J_A_058_AProf_096,axiom,
member(fun(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combb(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_159774573e_indi,arrow_319942042_below,combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),combc(arrow_159774573e_indi,arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),combb(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_159774573e_indi,arrow_319942042_below,p1),c),b)),b),a),arrow_610318064e_Prof) ).
tff(fact_26_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)
=> ! [A6: arrow_411405190le_alt,B4: arrow_411405190le_alt] :
( ! [I1: arrow_159774573e_indi] :
( member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,A6,B4),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),X4,I1))
<=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,A6,B4),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Xa1,I1)) )
=> ( member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,A6,B4),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),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,A6,B4),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_27_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)
=> ! [A6: arrow_411405190le_alt,B4: arrow_411405190le_alt] :
( ! [I1: arrow_159774573e_indi] : member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,A6,B4),aa(arrow_159774573e_indi,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),X4,I1))
=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,A6,B4),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_28_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))
<=> ? [A6: A,B4: B] : pp(aa(product_prod(A,B),bool,Pa,product_Pair(A,B,A6,B4))) ) ).
tff(fact_29_prod_Orecs,axiom,
! [B: $tType,A: $tType,C: $tType,Ba: C,Aa: B,F11: fun(B,fun(C,A))] : ( product_prod_rec(B,C,A,F11,product_Pair(B,C,Aa,Ba)) = aa(C,A,aa(B,fun(C,A),F11,Aa),Ba) ) ).
tff(fact_30_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_31_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_32_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),product_Pair(A,B,X4,Xa1),R2)
<=> member(product_prod(A,B),product_Pair(A,B,X4,Xa1),S) )
<=> ( R2 = S ) ) ).
tff(fact_33_notin__Lin__iff,axiom,
! [Y2: arrow_411405190le_alt,X1: arrow_411405190le_alt,L: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)] :
( member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),L,arrow_1985332922le_Lin)
=> ( ( X1 != Y2 )
=> ( ~ member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,X1,Y2),L)
<=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,Y2,X1),L) ) ) ) ).
tff(fact_34_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),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,Aa,Ba),L)
=> ~ member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,Ba,Aa),L) ) ) ).
tff(fact_35_below__Lin,axiom,
! [L: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Y2: arrow_411405190le_alt,X1: arrow_411405190le_alt] :
( ( X1 != Y2 )
=> ( member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),L,arrow_1985332922le_Lin)
=> member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),aa(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),aa(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_319942042_below,L),X1),Y2),arrow_1985332922le_Lin) ) ) ).
tff(fact_36_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_37_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_38_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_39_in__below,axiom,
! [Y2: 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),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,X1,Y2),aa(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),aa(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)),aa(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),fun(arrow_411405190le_alt,fun(arrow_411405190le_alt,fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool))),arrow_319942042_below,L),Aa),Ba))
<=> ( ( X1 != Y2 )
& ( ( Y2 = Aa )
=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,X1,Ba),L) )
& ( ( Y2 != Aa )
=> ( ( ( X1 = Aa )
=> ( ( Y2 = Ba )
| member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,Ba,Y2),L) ) )
& ( ( X1 != Aa )
=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,X1,Y2),L) ) ) ) ) ) ) ) ).
tff(fact_40_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)))
=> ( ! [X3: 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)),X3,arrow_610318064e_Prof)
=> ! [A1: arrow_411405190le_alt,B1: arrow_411405190le_alt] :
( ( A1 != B1 )
=> ( member(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),X3,I))
=> member(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),Fa,X3)) ) ) )
=> arrow_987702531ctator(Fa,I) ) ) ).
tff(fact_41_PiE,axiom,
! [B: $tType,A: $tType,X1: A,B3: fun(A,fun(B,bool)),A5: fun(A,bool),F: fun(A,B)] :
( member(fun(A,B),F,pi(A,B,A5,B3))
=> ( ~ member(B,aa(A,B,F,X1),aa(A,fun(B,bool),B3,X1))
=> ~ member(A,X1,A5) ) ) ).
tff(fact_42_complete__Lin,axiom,
! [Ba: arrow_411405190le_alt,Aa: arrow_411405190le_alt] :
( ( Aa != Ba )
=> ? [X3: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool)] :
( member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),X3,arrow_1985332922le_Lin)
& member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,Aa,Ba),X3) ) ) ).
tff(fact_43_in__above,axiom,
! [Y2: 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),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,X1,Y2),arrow_1158827142_above(L,Aa,Ba))
<=> ( ( X1 != Y2 )
& ( ( X1 = Ba )
=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,Aa,Y2),L) )
& ( ( X1 != Ba )
=> ( ( ( Y2 = Ba )
=> ( ( X1 = Aa )
| member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,X1,Aa),L) ) )
& ( ( Y2 != Ba )
=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,X1,Y2),L) ) ) ) ) ) ) ) ).
tff(fact_44_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_45_funcset__id,axiom,
! [A: $tType,A5: fun(A,bool)] : member(fun(A,A),combi(A),pi(A,A,A5,combk(fun(A,bool),A,A5))) ).
tff(fact_46_Pi__mem,axiom,
! [B: $tType,A: $tType,X1: A,B3: fun(A,fun(B,bool)),A5: fun(A,bool),F: fun(A,B)] :
( member(fun(A,B),F,pi(A,B,A5,B3))
=> ( member(A,X1,A5)
=> member(B,aa(A,B,F,X1),aa(A,fun(B,bool),B3,X1)) ) ) ).
tff(fact_47_above__Lin,axiom,
! [L: fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),Y2: arrow_411405190le_alt,X1: arrow_411405190le_alt] :
( ( X1 != Y2 )
=> ( member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),L,arrow_1985332922le_Lin)
=> member(fun(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),bool),arrow_1158827142_above(L,X1,Y2),arrow_1985332922le_Lin) ) ) ).
tff(fact_48_funcset__mem,axiom,
! [A: $tType,B: $tType,X1: A,B3: fun(B,bool),A5: fun(A,bool),F: fun(A,B)] :
( member(fun(A,B),F,pi(A,B,A5,combk(fun(B,bool),A,B3)))
=> ( member(A,X1,A5)
=> member(B,aa(A,B,F,X1),B3) ) ) ).
tff(fact_49_Pi__I,axiom,
! [B: $tType,A: $tType,B3: fun(A,fun(B,bool)),F: fun(A,B),A5: fun(A,bool)] :
( ! [X3: A] :
( member(A,X3,A5)
=> member(B,aa(A,B,F,X3),aa(A,fun(B,bool),B3,X3)) )
=> member(fun(A,B),F,pi(A,B,A5,B3)) ) ).
tff(fact_50_funcsetI,axiom,
! [A: $tType,B: $tType,B3: fun(B,bool),F: fun(A,B),A5: fun(A,bool)] :
( ! [X3: A] :
( member(A,X3,A5)
=> member(B,aa(A,B,F,X3),B3) )
=> member(fun(A,B),F,pi(A,B,A5,combk(fun(B,bool),A,B3))) ) ).
tff(fact_51_Pi__I_H,axiom,
! [B: $tType,A: $tType,B3: fun(A,fun(B,bool)),F: fun(A,B),A5: fun(A,bool)] :
( ! [X3: A] :
( member(A,X3,A5)
=> member(B,aa(A,B,F,X3),aa(A,fun(B,bool),B3,X3)) )
=> member(fun(A,B),F,pi(A,B,A5,B3)) ) ).
tff(fact_52_Pi__cong,axiom,
! [B: $tType,A: $tType,B3: fun(A,fun(B,bool)),G: fun(A,B),F: fun(A,B),A5: fun(A,bool)] :
( ! [W: A] :
( member(A,W,A5)
=> ( aa(A,B,F,W) = aa(A,B,G,W) ) )
=> ( member(fun(A,B),F,pi(A,B,A5,B3))
<=> member(fun(A,B),G,pi(A,B,A5,B3)) ) ) ).
tff(fact_53_prod__induct6,axiom,
! [F1: $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,F1))))),Pa: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F1))))),bool)] :
( ! [A1: A,B1: B,C1: C,D1: D,E1: E,F2: F1] : pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F1))))),bool,Pa,product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F1)))),A1,product_Pair(B,product_prod(C,product_prod(D,product_prod(E,F1))),B1,product_Pair(C,product_prod(D,product_prod(E,F1)),C1,product_Pair(D,product_prod(E,F1),D1,product_Pair(E,F1,E1,F2)))))))
=> pp(aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F1))))),bool,Pa,X1)) ) ).
tff(fact_54_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F1: $tType,Y: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F1)))))] :
~ ! [A1: A,B1: B,C1: C,D1: D,E1: E,F2: F1] : ( Y != product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F1)))),A1,product_Pair(B,product_prod(C,product_prod(D,product_prod(E,F1))),B1,product_Pair(C,product_prod(D,product_prod(E,F1)),C1,product_Pair(D,product_prod(E,F1),D1,product_Pair(E,F1,E1,F2))))) ) ).
tff(fact_55_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_56_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))))] :
~ ! [A1: A,B1: B,C1: C,D1: D,E1: E] : ( Y != 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_57_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod(A,product_prod(B,product_prod(C,D)))] :
~ ! [A1: A,B1: B,C1: C,D1: D] : ( Y != 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_58_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_59_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod(A,product_prod(B,C))] :
~ ! [A1: A,B1: B,C1: C] : ( Y != product_Pair(A,product_prod(B,C),A1,product_Pair(B,C,B1,C1)) ) ).
tff(fact_60_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_61_PairE,axiom,
! [A: $tType,B: $tType,P1: product_prod(A,B)] :
~ ! [X3: A,Y3: B] : ( P1 != product_Pair(A,B,X3,Y3) ) ).
tff(fact_62_prod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod(A,B)] :
~ ! [A1: A,B1: B] : ( Y != product_Pair(A,B,A1,B1) ) ).
tff(fact_63_internal__split__conv,axiom,
! [B: $tType,A: $tType,C: $tType,Ba: C,Aa: B,Ca: fun(B,fun(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_64_curry__conv,axiom,
! [A: $tType,B: $tType,C: $tType,Ba: C,Aa: B,F: fun(product_prod(B,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_65_curry__def,axiom,
! [C: $tType,A: $tType,B: $tType,X2: fun(product_prod(A,B),C),Xa: A,Xb: B] : ( product_curry(A,B,C,X2,Xa,Xb) = aa(product_prod(A,B),C,X2,product_Pair(A,B,Xa,Xb)) ) ).
tff(fact_66_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(product_curry(A,B,bool,F,Aa,Ba)) ) ).
tff(fact_67_curryD,axiom,
! [A: $tType,B: $tType,Ba: B,Aa: A,F: fun(product_prod(A,B),bool)] :
( pp(product_curry(A,B,bool,F,Aa,Ba))
=> pp(aa(product_prod(A,B),bool,F,product_Pair(A,B,Aa,Ba))) ) ).
tff(fact_68_curryE,axiom,
! [A: $tType,B: $tType,Ba: B,Aa: A,F: fun(product_prod(A,B),bool)] :
( pp(product_curry(A,B,bool,F,Aa,Ba))
=> pp(aa(product_prod(A,B),bool,F,product_Pair(A,B,Aa,Ba))) ) ).
tff(fact_69_in__inv__imagep,axiom,
! [A: $tType,B: $tType,Y2: B,X1: B,F: fun(B,A),R1: fun(A,fun(A,bool))] :
( inv_imagep(A,B,R1,F,X1,Y2)
<=> pp(aa(A,bool,aa(A,fun(A,bool),R1,aa(B,A,F,X1)),aa(B,A,F,Y2))) ) ).
tff(fact_70_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_71_dist,axiom,
distinct(arrow_411405190le_alt,cons(arrow_411405190le_alt,a,cons(arrow_411405190le_alt,b,cons(arrow_411405190le_alt,c,nil(arrow_411405190le_alt))))) ).
tff(fact_72_top1I,axiom,
! [A: $tType,X1: A] : pp(aa(A,bool,top_top(fun(A,bool)),X1)) ).
tff(fact_73_Pi__UNIV,axiom,
! [B: $tType,A: $tType,A5: fun(A,bool)] : ( pi(A,B,A5,combk(fun(B,bool),A,top_top(fun(B,bool)))) = top_top(fun(fun(A,B),bool)) ) ).
tff(fact_74__096_B_Bthesis_O_A_I_B_Bc_O_Adistinct_A_091a_M_Ab_M_Ac_093_A_061_061_062_Athesis_J_A_061_061_062_Athesis_096,axiom,
~ ! [C1: arrow_411405190le_alt] : ~ distinct(arrow_411405190le_alt,cons(arrow_411405190le_alt,a,cons(arrow_411405190le_alt,b,cons(arrow_411405190le_alt,C1,nil(arrow_411405190le_alt))))) ).
tff(fact_75_ext,axiom,
! [B: $tType,A: $tType,G: fun(A,B),F: fun(A,B)] :
( ! [X3: A] : ( aa(A,B,F,X3) = aa(A,B,G,X3) )
=> ( F = G ) ) ).
tff(fact_76_mem__def,axiom,
! [A: $tType,A5: fun(A,bool),X1: A] :
( member(A,X1,A5)
<=> pp(aa(A,bool,A5,X1)) ) ).
tff(fact_77_Collect__def,axiom,
! [A: $tType,Pa: fun(A,bool)] : ( collect(A,Pa) = Pa ) ).
tff(fact_78_inv__imagep__def,axiom,
! [B: $tType,A: $tType,F: fun(A,B),R1: fun(B,fun(B,bool)),X2: A,Xa: A] :
( inv_imagep(B,A,R1,F,X2,Xa)
<=> pp(aa(B,bool,aa(B,fun(B,bool),R1,aa(A,B,F,X2)),aa(A,B,F,Xa))) ) ).
tff(fact_79_distinct_Osimps_I1_J,axiom,
! [A: $tType] : distinct(A,nil(A)) ).
tff(fact_80_third__alt,axiom,
! [B2: arrow_411405190le_alt,A4: arrow_411405190le_alt] :
( ( A4 != B2 )
=> ? [C1: arrow_411405190le_alt] : distinct(arrow_411405190le_alt,cons(arrow_411405190le_alt,A4,cons(arrow_411405190le_alt,B2,cons(arrow_411405190le_alt,C1,nil(arrow_411405190le_alt))))) ) ).
tff(fact_81_list_Oinject,axiom,
! [A: $tType,List3: list(A),A3: A,List2: list(A),Aa: A] :
( ( cons(A,Aa,List2) = cons(A,A3,List3) )
<=> ( ( Aa = A3 )
& ( List2 = List3 ) ) ) ).
tff(fact_82_not__Cons__self2,axiom,
! [A: $tType,Xs: list(A),X: A] : ( cons(A,X,Xs) != Xs ) ).
tff(fact_83_not__Cons__self,axiom,
! [A: $tType,X: A,Xs: list(A)] : ( Xs != cons(A,X,Xs) ) ).
tff(fact_84_list_Osimps_I3_J,axiom,
! [A: $tType,List1: list(A),A2: A] : ( cons(A,A2,List1) != nil(A) ) ).
tff(fact_85_list_Osimps_I2_J,axiom,
! [A: $tType,List1: list(A),A2: A] : ( nil(A) != cons(A,A2,List1) ) ).
tff(fact_86_distinct__length__2__or__more,axiom,
! [A: $tType,Xs1: list(A),Ba: A,Aa: A] :
( distinct(A,cons(A,Aa,cons(A,Ba,Xs1)))
<=> ( ( Aa != Ba )
& distinct(A,cons(A,Aa,Xs1))
& distinct(A,cons(A,Ba,Xs1)) ) ) ).
tff(fact_87_distinct__singleton,axiom,
! [A: $tType,X: A] : distinct(A,cons(A,X,nil(A))) ).
tff(fact_88_iso__tuple__UNIV__I,axiom,
! [A: $tType,X1: A] : member(A,X1,top_top(fun(A,bool))) ).
tff(fact_89_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_neq__Nil__conv,axiom,
! [A: $tType,Xs1: list(A)] :
( ( Xs1 != nil(A) )
<=> ? [Y1: A,Ys1: list(A)] : ( Xs1 = cons(A,Y1,Ys1) ) ) ).
tff(fact_92_list_Oexhaust,axiom,
! [A: $tType,Y: list(A)] :
( ( Y != nil(A) )
=> ~ ! [A1: A,List: list(A)] : ( Y != cons(A,A1,List) ) ) ).
tff(fact_93_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_94_insert__Nil,axiom,
! [A: $tType,X: A] : ( insert(A,X,nil(A)) = cons(A,X,nil(A)) ) ).
tff(fact_95_distinct__insert,axiom,
! [A: $tType,X: A,Xs: list(A)] :
( distinct(A,Xs)
=> distinct(A,insert(A,X,Xs)) ) ).
tff(fact_96_alt3,axiom,
? [A1: arrow_411405190le_alt,B1: arrow_411405190le_alt,C1: arrow_411405190le_alt] : distinct(arrow_411405190le_alt,cons(arrow_411405190le_alt,A1,cons(arrow_411405190le_alt,B1,cons(arrow_411405190le_alt,C1,nil(arrow_411405190le_alt))))) ).
tff(fact_97_splice_Osimps_I2_J,axiom,
! [A: $tType,Va: list(A),V: A] : ( splice(A,cons(A,V,Va),nil(A)) = cons(A,V,Va) ) ).
tff(fact_98_splice_Osimps_I3_J,axiom,
! [A: $tType,Ys: list(A),Y: A,Xs: list(A),X: A] : ( splice(A,cons(A,X,Xs),cons(A,Y,Ys)) = cons(A,X,cons(A,Y,splice(A,Xs,Ys))) ) ).
tff(fact_99_splice__Nil2,axiom,
! [A: $tType,Xs: list(A)] : ( splice(A,Xs,nil(A)) = Xs ) ).
%----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 (8)
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,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 ) ) ).
%----Conjectures (1)
tff(conj_0,conjecture,
( member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,a,b),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))
<=> member(product_prod(arrow_411405190le_alt,arrow_411405190le_alt),product_Pair(arrow_411405190le_alt,arrow_411405190le_alt,b,a),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)) ) ).
%------------------------------------------------------------------------------